LMFDB - Embedded newform 324.2.h.a.215.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,2,Mod(107,324)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("324.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	324.h (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.58715302549$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			215.2
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	324.215
Dual form			324.2.h.a.107.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +(-2.44949 - 1.41421i) q^{5} +(1.50000 - 0.866025i) q^{7} -2.82843i q^{8} +O(q^{10})$	\(q+1.41421i q^{2} -2.00000 q^{4} +(-2.44949 - 1.41421i) q^{5} +(1.50000 - 0.866025i) q^{7} -2.82843i q^{8} +(2.00000 - 3.46410i) q^{10} +(-2.44949 - 4.24264i) q^{11} +(0.500000 - 0.866025i) q^{13} +(1.22474 + 2.12132i) q^{14} +4.00000 q^{16} +2.82843i q^{17} -5.19615i q^{19} +(4.89898 + 2.82843i) q^{20} +(6.00000 - 3.46410i) q^{22} +(2.44949 - 4.24264i) q^{23} +(1.50000 + 2.59808i) q^{25} +(1.22474 + 0.707107i) q^{26} +(-3.00000 + 1.73205i) q^{28} +(-4.89898 + 2.82843i) q^{29} +(-3.00000 - 1.73205i) q^{31} +5.65685i q^{32} -4.00000 q^{34} -4.89898 q^{35} -1.00000 q^{37} +7.34847 q^{38} +(-4.00000 + 6.92820i) q^{40} +(4.89898 + 2.82843i) q^{41} +(-3.00000 + 1.73205i) q^{43} +(4.89898 + 8.48528i) q^{44} +(6.00000 + 3.46410i) q^{46} +(-2.44949 - 4.24264i) q^{47} +(-2.00000 + 3.46410i) q^{49} +(-3.67423 + 2.12132i) q^{50} +(-1.00000 + 1.73205i) q^{52} -5.65685i q^{53} +13.8564i q^{55} +(-2.44949 - 4.24264i) q^{56} +(-4.00000 - 6.92820i) q^{58} +(2.44949 - 4.24264i) q^{59} +(-5.50000 - 9.52628i) q^{61} +(2.44949 - 4.24264i) q^{62} -8.00000 q^{64} +(-2.44949 + 1.41421i) q^{65} +(10.5000 + 6.06218i) q^{67} -5.65685i q^{68} -6.92820i q^{70} -1.00000 q^{73} -1.41421i q^{74} +10.3923i q^{76} +(-7.34847 - 4.24264i) q^{77} +(1.50000 - 0.866025i) q^{79} +(-9.79796 - 5.65685i) q^{80} +(-4.00000 + 6.92820i) q^{82} +(4.89898 + 8.48528i) q^{83} +(4.00000 - 6.92820i) q^{85} +(-2.44949 - 4.24264i) q^{86} +(-12.0000 + 6.92820i) q^{88} +2.82843i q^{89} -1.73205i q^{91} +(-4.89898 + 8.48528i) q^{92} +(6.00000 - 3.46410i) q^{94} +(-7.34847 + 12.7279i) q^{95} +(6.50000 + 11.2583i) q^{97} +(-4.89898 - 2.82843i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 6 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 + 6 * q^7	$ 4 q - 8 q^{4} + 6 q^{7} + 8 q^{10} + 2 q^{13} + 16 q^{16} + 24 q^{22} + 6 q^{25} - 12 q^{28} - 12 q^{31} - 16 q^{34} - 4 q^{37} - 16 q^{40} - 12 q^{43} + 24 q^{46} - 8 q^{49} - 4 q^{52} - 16 q^{58} - 22 q^{61} - 32 q^{64} + 42 q^{67} - 4 q^{73} + 6 q^{79} - 16 q^{82} + 16 q^{85} - 48 q^{88} + 24 q^{94} + 26 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 6 * q^7 + 8 * q^10 + 2 * q^13 + 16 * q^16 + 24 * q^22 + 6 * q^25 - 12 * q^28 - 12 * q^31 - 16 * q^34 - 4 * q^37 - 16 * q^40 - 12 * q^43 + 24 * q^46 - 8 * q^49 - 4 * q^52 - 16 * q^58 - 22 * q^61 - 32 * q^64 + 42 * q^67 - 4 * q^73 + 6 * q^79 - 16 * q^82 + 16 * q^85 - 48 * q^88 + 24 * q^94 + 26 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.41421i			1.00000i
$2$			1.41421i			1.00000i
$3$		0			0
$3$		0			0
$4$	−2.00000			−1.00000
$5$	−2.44949	−	1.41421i	−1.09545	−	0.632456i	−0.160424	−	0.987048i	$-0.551286\pi$
$5$	−2.44949	−	1.41421i	−1.09545	−	0.632456i	−0.935021	+	0.354593i	$0.884620\pi$
$6$		0			0
$7$	1.50000	−	0.866025i	0.566947	−	0.327327i	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	1.50000	−	0.866025i	0.566947	−	0.327327i	0.755929	+	0.654654i	$0.227186\pi$
$8$		−	2.82843i		−	1.00000i
$9$		0			0
$10$	2.00000	−	3.46410i	0.632456	−	1.09545i
$11$	−2.44949	−	4.24264i	−0.738549	−	1.27920i	−0.953149	−	0.302502i	$-0.902178\pi$
$11$	−2.44949	−	4.24264i	−0.738549	−	1.27920i	0.214600	−	0.976702i	$-0.431155\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	0.926995	+	0.375073i	$0.122382\pi$
$14$	1.22474	+	2.12132i	0.327327	+	0.566947i
$15$		0			0
$16$	4.00000			1.00000
$17$			2.82843i			0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$			2.82843i			0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$		0			0
$19$		−	5.19615i		−	1.19208i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$		−	5.19615i		−	1.19208i	0.802955	−	0.596040i	$-0.203260\pi$
$20$	4.89898	+	2.82843i	1.09545	+	0.632456i
$21$		0			0
$22$	6.00000	−	3.46410i	1.27920	−	0.738549i
$23$	2.44949	−	4.24264i	0.510754	−	0.884652i	−0.489168	−	0.872189i	$-0.662700\pi$
$23$	2.44949	−	4.24264i	0.510754	−	0.884652i	0.999922	−	0.0124624i	$-0.00396701\pi$
$24$		0			0
$25$	1.50000	+	2.59808i	0.300000	+	0.519615i
$26$	1.22474	+	0.707107i	0.240192	+	0.138675i
$27$		0			0
$28$	−3.00000	+	1.73205i	−0.566947	+	0.327327i
$29$	−4.89898	+	2.82843i	−0.909718	+	0.525226i	−0.880340	−	0.474343i	$-0.842686\pi$
$29$	−4.89898	+	2.82843i	−0.909718	+	0.525226i	−0.0293774	+	0.999568i	$0.509352\pi$
$30$		0			0
$31$	−3.00000	−	1.73205i	−0.538816	−	0.311086i	0.205783	−	0.978598i	$-0.434026\pi$
$31$	−3.00000	−	1.73205i	−0.538816	−	0.311086i	−0.744599	+	0.667512i	$0.767359\pi$
$32$			5.65685i			1.00000i
$33$		0			0
$34$	−4.00000			−0.685994
$35$	−4.89898			−0.828079
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$	7.34847			1.19208
$39$		0			0
$40$	−4.00000	+	6.92820i	−0.632456	+	1.09545i
$41$	4.89898	+	2.82843i	0.765092	+	0.441726i	0.831121	−	0.556092i	$-0.187700\pi$
$41$	4.89898	+	2.82843i	0.765092	+	0.441726i	−0.0660290	+	0.997818i	$0.521033\pi$
$42$		0			0
$43$	−3.00000	+	1.73205i	−0.457496	+	0.264135i	−0.710991	−	0.703201i	$-0.751753\pi$
$43$	−3.00000	+	1.73205i	−0.457496	+	0.264135i	0.253495	+	0.967337i	$0.418420\pi$
$44$	4.89898	+	8.48528i	0.738549	+	1.27920i
$45$		0			0
$46$	6.00000	+	3.46410i	0.884652	+	0.510754i
$47$	−2.44949	−	4.24264i	−0.357295	−	0.618853i	0.630213	−	0.776422i	$-0.282968\pi$
$47$	−2.44949	−	4.24264i	−0.357295	−	0.618853i	−0.987508	+	0.157569i	$0.949634\pi$
$48$		0			0
$49$	−2.00000	+	3.46410i	−0.285714	+	0.494872i
$50$	−3.67423	+	2.12132i	−0.519615	+	0.300000i
$51$		0			0
$52$	−1.00000	+	1.73205i	−0.138675	+	0.240192i
$53$		−	5.65685i		−	0.777029i	−0.921443	−	0.388514i	$-0.872988\pi$
$53$		−	5.65685i		−	0.777029i	0.921443	−	0.388514i	$-0.127012\pi$
$54$		0			0
$55$			13.8564i			1.86840i
$56$	−2.44949	−	4.24264i	−0.327327	−	0.566947i
$57$		0			0
$58$	−4.00000	−	6.92820i	−0.525226	−	0.909718i
$59$	2.44949	−	4.24264i	0.318896	−	0.552345i	−0.661362	−	0.750067i	$-0.730021\pi$
$59$	2.44949	−	4.24264i	0.318896	−	0.552345i	0.980258	+	0.197722i	$0.0633545\pi$
$60$		0			0
$61$	−5.50000	−	9.52628i	−0.704203	−	1.21972i	−0.966978	−	0.254858i	$-0.917971\pi$
$61$	−5.50000	−	9.52628i	−0.704203	−	1.21972i	0.262776	−	0.964857i	$-0.415362\pi$
$62$	2.44949	−	4.24264i	0.311086	−	0.538816i
$63$		0			0
$64$	−8.00000			−1.00000
$65$	−2.44949	+	1.41421i	−0.303822	+	0.175412i
$66$		0			0
$67$	10.5000	+	6.06218i	1.28278	+	0.740613i	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	10.5000	+	6.06218i	1.28278	+	0.740613i	0.305424	+	0.952217i	$0.401202\pi$
$68$		−	5.65685i		−	0.685994i
$69$		0			0
$70$		−	6.92820i		−	0.828079i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−1.00000			−0.117041			−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000			−0.117041			−0.0585206	+	0.998286i	$0.518638\pi$
$74$		−	1.41421i		−	0.164399i
$75$		0			0
$76$			10.3923i			1.19208i
$77$	−7.34847	−	4.24264i	−0.837436	−	0.483494i
$78$		0			0
$79$	1.50000	−	0.866025i	0.168763	−	0.0974355i	−0.413239	−	0.910622i	$-0.635603\pi$
$79$	1.50000	−	0.866025i	0.168763	−	0.0974355i	0.582003	+	0.813187i	$0.302269\pi$
$80$	−9.79796	−	5.65685i	−1.09545	−	0.632456i
$81$		0			0
$82$	−4.00000	+	6.92820i	−0.441726	+	0.765092i
$83$	4.89898	+	8.48528i	0.537733	+	0.931381i	0.999026	+	0.0441327i	$0.0140524\pi$
$83$	4.89898	+	8.48528i	0.537733	+	0.931381i	−0.461293	+	0.887248i	$0.652614\pi$
$84$		0			0
$85$	4.00000	−	6.92820i	0.433861	−	0.751469i
$86$	−2.44949	−	4.24264i	−0.264135	−	0.457496i
$87$		0			0
$88$	−12.0000	+	6.92820i	−1.27920	+	0.738549i
$89$			2.82843i			0.299813i	0.988700	+	0.149906i	$0.0478972\pi$
$89$			2.82843i			0.299813i	−0.988700	+	0.149906i	$0.952103\pi$
$90$		0			0
$91$		−	1.73205i		−	0.181568i
$92$	−4.89898	+	8.48528i	−0.510754	+	0.884652i
$93$		0			0
$94$	6.00000	−	3.46410i	0.618853	−	0.357295i
$95$	−7.34847	+	12.7279i	−0.753937	+	1.30586i
$96$		0			0
$97$	6.50000	+	11.2583i	0.659975	+	1.14311i	0.980622	+	0.195911i	$0.0627665\pi$
$97$	6.50000	+	11.2583i	0.659975	+	1.14311i	−0.320647	+	0.947199i	$0.603900\pi$
$98$	−4.89898	−	2.82843i	−0.494872	−	0.285714i
$99$		0			0
$100$	−3.00000	−	5.19615i	−0.300000	−	0.519615i
$101$	9.79796	−	5.65685i	0.974933	−	0.562878i	0.0741967	−	0.997244i	$-0.476361\pi$
$101$	9.79796	−	5.65685i	0.974933	−	0.562878i	0.900737	+	0.434366i	$0.143027\pi$
$102$		0			0
$103$	−7.50000	−	4.33013i	−0.738997	−	0.426660i	0.0827075	−	0.996574i	$-0.473643\pi$
$103$	−7.50000	−	4.33013i	−0.738997	−	0.426660i	−0.821705	+	0.569914i	$0.806977\pi$
$104$	−2.44949	−	1.41421i	−0.240192	−	0.138675i
$105$		0			0
$106$	8.00000			0.777029
$107$	14.6969			1.42081			0.710403	−	0.703795i	$-0.248513\pi$
$107$	14.6969			1.42081			0.710403	+	0.703795i	$0.248513\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$	−19.5959			−1.86840
$111$		0			0
$112$	6.00000	−	3.46410i	0.566947	−	0.327327i
$113$	−2.44949	−	1.41421i	−0.230429	−	0.133038i	0.380341	−	0.924846i	$-0.375807\pi$
$113$	−2.44949	−	1.41421i	−0.230429	−	0.133038i	−0.610770	+	0.791808i	$0.709140\pi$
$114$		0			0
$115$	−12.0000	+	6.92820i	−1.11901	+	0.646058i
$116$	9.79796	−	5.65685i	0.909718	−	0.525226i
$117$		0			0
$118$	6.00000	+	3.46410i	0.552345	+	0.318896i
$119$	2.44949	+	4.24264i	0.224544	+	0.388922i
$120$		0			0
$121$	−6.50000	+	11.2583i	−0.590909	+	1.02348i
$122$	13.4722	−	7.77817i	1.21972	−	0.704203i
$123$		0			0
$124$	6.00000	+	3.46410i	0.538816	+	0.311086i
$125$			5.65685i			0.505964i
$126$		0			0
$127$		−	10.3923i		−	0.922168i	−0.887357	−	0.461084i	$-0.847461\pi$
$127$		−	10.3923i		−	0.922168i	0.887357	−	0.461084i	$-0.152539\pi$
$128$		−	11.3137i		−	1.00000i
$129$		0			0
$130$	−2.00000	−	3.46410i	−0.175412	−	0.303822i
$131$	−4.89898	+	8.48528i	−0.428026	+	0.741362i	−0.996698	−	0.0812020i	$-0.974124\pi$
$131$	−4.89898	+	8.48528i	−0.428026	+	0.741362i	0.568672	+	0.822564i	$0.307457\pi$
$132$		0			0
$133$	−4.50000	−	7.79423i	−0.390199	−	0.675845i
$134$	−8.57321	+	14.8492i	−0.740613	+	1.28278i
$135$		0			0
$136$	8.00000			0.685994
$137$	17.1464	−	9.89949i	1.46492	−	0.845771i	0.465686	−	0.884950i	$-0.345808\pi$
$137$	17.1464	−	9.89949i	1.46492	−	0.845771i	0.999232	+	0.0391791i	$0.0124743\pi$
$138$		0			0
$139$	−7.50000	−	4.33013i	−0.636142	−	0.367277i	0.146985	−	0.989139i	$-0.453043\pi$
$139$	−7.50000	−	4.33013i	−0.636142	−	0.367277i	−0.783127	+	0.621862i	$0.786376\pi$
$140$	9.79796			0.828079
$141$		0			0
$142$		0			0
$143$	−4.89898			−0.409673
$144$		0			0
$145$	16.0000			1.32873
$146$		−	1.41421i		−	0.117041i
$147$		0			0
$148$	2.00000			0.164399
$149$	19.5959	+	11.3137i	1.60536	+	0.926855i	0.990390	+	0.138305i	$0.0441654\pi$
$149$	19.5959	+	11.3137i	1.60536	+	0.926855i	0.614970	+	0.788550i	$0.289168\pi$
$150$		0			0
$151$	19.5000	−	11.2583i	1.58689	−	0.916190i	0.593072	−	0.805150i	$-0.297915\pi$
$151$	19.5000	−	11.2583i	1.58689	−	0.916190i	0.993816	−	0.111040i	$-0.0354182\pi$
$152$	−14.6969			−1.19208
$153$		0			0
$154$	6.00000	−	10.3923i	0.483494	−	0.837436i
$155$	4.89898	+	8.48528i	0.393496	+	0.681554i
$156$		0			0
$157$	5.00000	−	8.66025i	0.399043	−	0.691164i	−0.594565	−	0.804048i	$-0.702676\pi$
$157$	5.00000	−	8.66025i	0.399043	−	0.691164i	0.993608	+	0.112884i	$0.0360089\pi$
$158$	1.22474	+	2.12132i	0.0974355	+	0.168763i
$159$		0			0
$160$	8.00000	−	13.8564i	0.632456	−	1.09545i
$161$		−	8.48528i		−	0.668734i
$162$		0			0
$163$		−	5.19615i		−	0.406994i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$		−	5.19615i		−	0.406994i	0.979076	−	0.203497i	$-0.0652307\pi$
$164$	−9.79796	−	5.65685i	−0.765092	−	0.441726i
$165$		0			0
$166$	−12.0000	+	6.92820i	−0.931381	+	0.537733i
$167$	2.44949	−	4.24264i	0.189547	−	0.328305i	−0.755552	−	0.655089i	$-0.772631\pi$
$167$	2.44949	−	4.24264i	0.189547	−	0.328305i	0.945099	+	0.326783i	$0.105965\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$	9.79796	+	5.65685i	0.751469	+	0.433861i
$171$		0			0
$172$	6.00000	−	3.46410i	0.457496	−	0.264135i
$173$	−4.89898	+	2.82843i	−0.372463	+	0.215041i	−0.674534	−	0.738244i	$-0.735655\pi$
$173$	−4.89898	+	2.82843i	−0.372463	+	0.215041i	0.302071	+	0.953285i	$0.402322\pi$
$174$		0			0
$175$	4.50000	+	2.59808i	0.340168	+	0.196396i
$176$	−9.79796	−	16.9706i	−0.738549	−	1.27920i
$177$		0			0
$178$	−4.00000			−0.299813
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	11.0000			0.817624			0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000			0.817624			0.408812	+	0.912619i	$0.365943\pi$
$182$	2.44949			0.181568
$183$		0			0
$184$	−12.0000	−	6.92820i	−0.884652	−	0.510754i
$185$	2.44949	+	1.41421i	0.180090	+	0.103975i
$186$		0			0
$187$	12.0000	−	6.92820i	0.877527	−	0.506640i
$188$	4.89898	+	8.48528i	0.357295	+	0.618853i
$189$		0			0
$190$	−18.0000	−	10.3923i	−1.30586	−	0.753937i
$191$	12.2474	+	21.2132i	0.886194	+	1.53493i	0.844339	+	0.535810i	$0.179994\pi$
$191$	12.2474	+	21.2132i	0.886194	+	1.53493i	0.0418556	+	0.999124i	$0.486673\pi$
$192$		0			0
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	−0.847469	−	0.530845i	$-0.821875\pi$
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	0.883460	+	0.468507i	$0.155208\pi$
$194$	−15.9217	+	9.19239i	−1.14311	+	0.659975i
$195$		0			0
$196$	4.00000	−	6.92820i	0.285714	−	0.494872i
$197$			2.82843i			0.201517i	0.994911	+	0.100759i	$0.0321270\pi$
$197$			2.82843i			0.201517i	−0.994911	+	0.100759i	$0.967873\pi$
$198$		0			0
$199$		−	5.19615i		−	0.368345i	−0.982894	−	0.184173i	$-0.941039\pi$
$199$		−	5.19615i		−	0.368345i	0.982894	−	0.184173i	$-0.0589606\pi$
$200$	7.34847	−	4.24264i	0.519615	−	0.300000i
$201$		0			0
$202$	8.00000	+	13.8564i	0.562878	+	0.974933i
$203$	−4.89898	+	8.48528i	−0.343841	+	0.595550i
$204$		0			0
$205$	−8.00000	−	13.8564i	−0.558744	−	0.967773i
$206$	6.12372	−	10.6066i	0.426660	−	0.738997i
$207$		0			0
$208$	2.00000	−	3.46410i	0.138675	−	0.240192i
$209$	−22.0454	+	12.7279i	−1.52491	+	0.880409i
$210$		0			0
$211$	10.5000	+	6.06218i	0.722850	+	0.417338i	0.815801	−	0.578333i	$-0.196297\pi$
$211$	10.5000	+	6.06218i	0.722850	+	0.417338i	−0.0929509	+	0.995671i	$0.529630\pi$
$212$			11.3137i			0.777029i
$213$		0			0
$214$			20.7846i			1.42081i
$215$	9.79796			0.668215
$216$		0			0
$217$	−6.00000			−0.407307
$218$		−	14.1421i		−	0.957826i
$219$		0			0
$220$		−	27.7128i		−	1.86840i
$221$	2.44949	+	1.41421i	0.164771	+	0.0951303i
$222$		0			0
$223$	−21.0000	+	12.1244i	−1.40626	+	0.811907i	−0.995025	−	0.0996209i	$-0.968237\pi$
$223$	−21.0000	+	12.1244i	−1.40626	+	0.811907i	−0.411239	+	0.911528i	$0.634904\pi$
$224$	4.89898	+	8.48528i	0.327327	+	0.566947i
$225$		0			0
$226$	2.00000	−	3.46410i	0.133038	−	0.230429i
$227$	−9.79796	−	16.9706i	−0.650313	−	1.12638i	−0.983047	−	0.183355i	$-0.941304\pi$
$227$	−9.79796	−	16.9706i	−0.650313	−	1.12638i	0.332733	−	0.943021i	$-0.392029\pi$
$228$		0			0
$229$	5.00000	−	8.66025i	0.330409	−	0.572286i	−0.652183	−	0.758062i	$-0.726147\pi$
$229$	5.00000	−	8.66025i	0.330409	−	0.572286i	0.982592	+	0.185776i	$0.0594799\pi$
$230$	−9.79796	−	16.9706i	−0.646058	−	1.11901i
$231$		0			0
$232$	8.00000	+	13.8564i	0.525226	+	0.909718i
$233$		−	22.6274i		−	1.48237i	−0.671300	−	0.741186i	$-0.734264\pi$
$233$		−	22.6274i		−	1.48237i	0.671300	−	0.741186i	$-0.265736\pi$
$234$		0			0
$235$			13.8564i			0.903892i
$236$	−4.89898	+	8.48528i	−0.318896	+	0.552345i
$237$		0			0
$238$	−6.00000	+	3.46410i	−0.388922	+	0.224544i
$239$	9.79796	−	16.9706i	0.633777	−	1.09773i	−0.352996	−	0.935625i	$-0.614837\pi$
$239$	9.79796	−	16.9706i	0.633777	−	1.09773i	0.986773	−	0.162109i	$-0.0518298\pi$
$240$		0			0
$241$	12.5000	+	21.6506i	0.805196	+	1.39464i	0.916159	+	0.400815i	$0.131273\pi$
$241$	12.5000	+	21.6506i	0.805196	+	1.39464i	−0.110963	+	0.993825i	$0.535394\pi$
$242$	−15.9217	−	9.19239i	−1.02348	−	0.590909i
$243$		0			0
$244$	11.0000	+	19.0526i	0.704203	+	1.21972i
$245$	9.79796	−	5.65685i	0.625969	−	0.361403i
$246$		0			0
$247$	−4.50000	−	2.59808i	−0.286328	−	0.165312i
$248$	−4.89898	+	8.48528i	−0.311086	+	0.538816i
$249$		0			0
$250$	−8.00000			−0.505964
$251$	−29.3939			−1.85533			−0.927663	−	0.373420i	$-0.878185\pi$
$251$	−29.3939			−1.85533			−0.927663	+	0.373420i	$0.878185\pi$
$252$		0			0
$253$	−24.0000			−1.50887
$254$	14.6969			0.922168
$255$		0			0
$256$	16.0000			1.00000
$257$	−24.4949	−	14.1421i	−1.52795	−	0.882162i	−0.999448	−	0.0332301i	$-0.989421\pi$
$257$	−24.4949	−	14.1421i	−1.52795	−	0.882162i	−0.528502	−	0.848932i	$-0.677246\pi$
$258$		0			0
$259$	−1.50000	+	0.866025i	−0.0932055	+	0.0538122i
$260$	4.89898	−	2.82843i	0.303822	−	0.175412i
$261$		0			0
$262$	−12.0000	−	6.92820i	−0.741362	−	0.428026i
$263$	−9.79796	−	16.9706i	−0.604168	−	1.04645i	−0.992182	−	0.124796i	$-0.960172\pi$
$263$	−9.79796	−	16.9706i	−0.604168	−	1.04645i	0.388014	−	0.921653i	$-0.373161\pi$
$264$		0			0
$265$	−8.00000	+	13.8564i	−0.491436	+	0.851192i
$266$	11.0227	−	6.36396i	0.675845	−	0.390199i
$267$		0			0
$268$	−21.0000	−	12.1244i	−1.28278	−	0.740613i
$269$			2.82843i			0.172452i	0.996276	+	0.0862261i	$0.0274808\pi$
$269$			2.82843i			0.172452i	−0.996276	+	0.0862261i	$0.972519\pi$
$270$		0			0
$271$		−	5.19615i		−	0.315644i	−0.987468	−	0.157822i	$-0.949553\pi$
$271$		−	5.19615i		−	0.315644i	0.987468	−	0.157822i	$-0.0504472\pi$
$272$			11.3137i			0.685994i
$273$		0			0
$274$	14.0000	+	24.2487i	0.845771	+	1.46492i
$275$	7.34847	−	12.7279i	0.443129	−	0.767523i
$276$		0			0
$277$	−7.00000	−	12.1244i	−0.420589	−	0.728482i	0.575408	−	0.817867i	$-0.304843\pi$
$277$	−7.00000	−	12.1244i	−0.420589	−	0.728482i	−0.995997	+	0.0893846i	$0.971510\pi$
$278$	6.12372	−	10.6066i	0.367277	−	0.636142i
$279$		0			0
$280$			13.8564i			0.828079i
$281$	−4.89898	+	2.82843i	−0.292249	+	0.168730i	−0.638955	−	0.769244i	$-0.720633\pi$
$281$	−4.89898	+	2.82843i	−0.292249	+	0.168730i	0.346707	+	0.937974i	$0.387300\pi$
$282$		0			0
$283$	−21.0000	−	12.1244i	−1.24832	−	0.720718i	−0.277546	−	0.960712i	$-0.589521\pi$
$283$	−21.0000	−	12.1244i	−1.24832	−	0.720718i	−0.970774	+	0.239994i	$0.922854\pi$
$284$		0			0
$285$		0			0
$286$		−	6.92820i		−	0.409673i
$287$	9.79796			0.578355
$288$		0			0
$289$	9.00000			0.529412
$290$			22.6274i			1.32873i
$291$		0			0
$292$	2.00000			0.117041
$293$	−2.44949	−	1.41421i	−0.143101	−	0.0826192i	0.426740	−	0.904374i	$-0.359662\pi$
$293$	−2.44949	−	1.41421i	−0.143101	−	0.0826192i	−0.569841	+	0.821755i	$0.692995\pi$
$294$		0			0
$295$	−12.0000	+	6.92820i	−0.698667	+	0.403376i
$296$			2.82843i			0.164399i
$297$		0			0
$298$	−16.0000	+	27.7128i	−0.926855	+	1.60536i
$299$	−2.44949	−	4.24264i	−0.141658	−	0.245358i
$300$		0			0
$301$	−3.00000	+	5.19615i	−0.172917	+	0.299501i
$302$	15.9217	+	27.5772i	0.916190	+	1.58689i
$303$		0			0
$304$		−	20.7846i		−	1.19208i
$305$			31.1127i			1.78151i
$306$		0			0
$307$			10.3923i			0.593120i	0.955014	+	0.296560i	$0.0958395\pi$
$307$			10.3923i			0.593120i	−0.955014	+	0.296560i	$0.904160\pi$
$308$	14.6969	+	8.48528i	0.837436	+	0.483494i
$309$		0			0
$310$	−12.0000	+	6.92820i	−0.681554	+	0.393496i
$311$	−12.2474	+	21.2132i	−0.694489	+	1.20289i	0.275864	+	0.961197i	$0.411036\pi$
$311$	−12.2474	+	21.2132i	−0.694489	+	1.20289i	−0.970353	+	0.241694i	$0.922297\pi$
$312$		0			0
$313$	−5.50000	−	9.52628i	−0.310878	−	0.538457i	0.667674	−	0.744453i	$-0.267290\pi$
$313$	−5.50000	−	9.52628i	−0.310878	−	0.538457i	−0.978553	+	0.205996i	$0.933957\pi$
$314$	12.2474	+	7.07107i	0.691164	+	0.399043i
$315$		0			0
$316$	−3.00000	+	1.73205i	−0.168763	+	0.0974355i
$317$	−4.89898	+	2.82843i	−0.275154	+	0.158860i	−0.631228	−	0.775598i	$-0.717449\pi$
$317$	−4.89898	+	2.82843i	−0.275154	+	0.158860i	0.356073	+	0.934458i	$0.384115\pi$
$318$		0			0
$319$	24.0000	+	13.8564i	1.34374	+	0.775810i
$320$	19.5959	+	11.3137i	1.09545	+	0.632456i
$321$		0			0
$322$	12.0000			0.668734
$323$	14.6969			0.817760
$324$		0			0
$325$	3.00000			0.166410
$326$	7.34847			0.406994
$327$		0			0
$328$	8.00000	−	13.8564i	0.441726	−	0.765092i
$329$	−7.34847	−	4.24264i	−0.405134	−	0.233904i
$330$		0			0
$331$	19.5000	−	11.2583i	1.07182	−	0.618814i	0.143140	−	0.989703i	$-0.454280\pi$
$331$	19.5000	−	11.2583i	1.07182	−	0.618814i	0.928677	+	0.370889i	$0.120947\pi$
$332$	−9.79796	−	16.9706i	−0.537733	−	0.931381i
$333$		0			0
$334$	6.00000	+	3.46410i	0.328305	+	0.189547i
$335$	−17.1464	−	29.6985i	−0.936809	−	1.62260i
$336$		0			0
$337$	−5.50000	+	9.52628i	−0.299604	+	0.518930i	−0.976045	−	0.217567i	$-0.930188\pi$
$337$	−5.50000	+	9.52628i	−0.299604	+	0.518930i	0.676441	+	0.736497i	$0.263521\pi$
$338$	−14.6969	+	8.48528i	−0.799408	+	0.461538i
$339$		0			0
$340$	−8.00000	+	13.8564i	−0.433861	+	0.751469i
$341$			16.9706i			0.919007i
$342$		0			0
$343$			19.0526i			1.02874i
$344$	4.89898	+	8.48528i	0.264135	+	0.457496i
$345$		0			0
$346$	−4.00000	−	6.92820i	−0.215041	−	0.372463i
$347$	9.79796	−	16.9706i	0.525982	−	0.911028i	−0.473560	−	0.880762i	$-0.657031\pi$
$347$	9.79796	−	16.9706i	0.525982	−	0.911028i	0.999542	−	0.0302659i	$-0.00963541\pi$
$348$		0			0
$349$	−11.5000	−	19.9186i	−0.615581	−	1.06622i	−0.990282	−	0.139072i	$-0.955588\pi$
$349$	−11.5000	−	19.9186i	−0.615581	−	1.06622i	0.374701	−	0.927146i	$-0.377745\pi$
$350$	−3.67423	+	6.36396i	−0.196396	+	0.340168i
$351$		0			0
$352$	24.0000	−	13.8564i	1.27920	−	0.738549i
$353$	9.79796	−	5.65685i	0.521493	−	0.301084i	−0.216052	−	0.976382i	$-0.569318\pi$
$353$	9.79796	−	5.65685i	0.521493	−	0.301084i	0.737545	+	0.675298i	$0.235985\pi$
$354$		0			0
$355$		0			0
$356$		−	5.65685i		−	0.299813i
$357$		0			0
$358$		0			0
$359$	−14.6969			−0.775675			−0.387837	−	0.921728i	$-0.626778\pi$
$359$	−14.6969			−0.775675			−0.387837	+	0.921728i	$0.626778\pi$
$360$		0			0
$361$	−8.00000			−0.421053
$362$			15.5563i			0.817624i
$363$		0			0
$364$			3.46410i			0.181568i
$365$	2.44949	+	1.41421i	0.128212	+	0.0740233i
$366$		0			0
$367$	−16.5000	+	9.52628i	−0.861293	+	0.497268i	−0.864445	−	0.502727i	$-0.832330\pi$
$367$	−16.5000	+	9.52628i	−0.861293	+	0.497268i	0.00315207	+	0.999995i	$0.498997\pi$
$368$	9.79796	−	16.9706i	0.510754	−	0.884652i
$369$		0			0
$370$	−2.00000	+	3.46410i	−0.103975	+	0.180090i
$371$	−4.89898	−	8.48528i	−0.254342	−	0.440534i
$372$		0			0
$373$	12.5000	−	21.6506i	0.647225	−	1.12103i	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	12.5000	−	21.6506i	0.647225	−	1.12103i	0.983783	−	0.179364i	$-0.0574041\pi$
$374$	9.79796	+	16.9706i	0.506640	+	0.877527i
$375$		0			0
$376$	−12.0000	+	6.92820i	−0.618853	+	0.357295i
$377$			5.65685i			0.291343i
$378$		0			0
$379$			15.5885i			0.800725i	0.916357	+	0.400363i	$0.131116\pi$
$379$			15.5885i			0.800725i	−0.916357	+	0.400363i	$0.868884\pi$
$380$	14.6969	−	25.4558i	0.753937	−	1.30586i
$381$		0			0
$382$	−30.0000	+	17.3205i	−1.53493	+	0.886194i
$383$	9.79796	−	16.9706i	0.500652	−	0.867155i	−0.499347	−	0.866402i	$-0.666427\pi$
$383$	9.79796	−	16.9706i	0.500652	−	0.867155i	1.00000		0.000753393i	$-0.000239813\pi$
$384$		0			0
$385$	12.0000	+	20.7846i	0.611577	+	1.05928i
$386$	1.22474	+	0.707107i	0.0623379	+	0.0359908i
$387$		0			0
$388$	−13.0000	−	22.5167i	−0.659975	−	1.14311i
$389$	−12.2474	+	7.07107i	−0.620970	+	0.358517i	−0.777247	−	0.629196i	$-0.783384\pi$
$389$	−12.2474	+	7.07107i	−0.620970	+	0.358517i	0.156276	+	0.987713i	$0.450051\pi$
$390$		0			0
$391$	12.0000	+	6.92820i	0.606866	+	0.350374i
$392$	9.79796	+	5.65685i	0.494872	+	0.285714i
$393$		0			0
$394$	−4.00000			−0.201517
$395$	−4.89898			−0.246494
$396$		0			0
$397$	−10.0000			−0.501886			−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000			−0.501886			−0.250943	+	0.968002i	$0.580741\pi$
$398$	7.34847			0.368345
$399$		0			0
$400$	6.00000	+	10.3923i	0.300000	+	0.519615i
$401$	19.5959	+	11.3137i	0.978573	+	0.564980i	0.901839	−	0.432072i	$-0.142217\pi$
$401$	19.5959	+	11.3137i	0.978573	+	0.564980i	0.0767343	+	0.997052i	$0.475551\pi$
$402$		0			0
$403$	−3.00000	+	1.73205i	−0.149441	+	0.0862796i
$404$	−19.5959	+	11.3137i	−0.974933	+	0.562878i
$405$		0			0
$406$	−12.0000	−	6.92820i	−0.595550	−	0.343841i
$407$	2.44949	+	4.24264i	0.121417	+	0.210300i
$408$		0			0
$409$	0.500000	−	0.866025i	0.0247234	−	0.0428222i	−0.853399	−	0.521258i	$-0.825463\pi$
$409$	0.500000	−	0.866025i	0.0247234	−	0.0428222i	0.878122	+	0.478436i	$0.158796\pi$
$410$	19.5959	−	11.3137i	0.967773	−	0.558744i
$411$		0			0
$412$	15.0000	+	8.66025i	0.738997	+	0.426660i
$413$		−	8.48528i		−	0.417533i
$414$		0			0
$415$		−	27.7128i		−	1.36037i
$416$	4.89898	+	2.82843i	0.240192	+	0.138675i
$417$		0			0
$418$	−18.0000	−	31.1769i	−0.880409	−	1.52491i
$419$	2.44949	−	4.24264i	0.119665	−	0.207267i	−0.799970	−	0.600040i	$-0.795151\pi$
$419$	2.44949	−	4.24264i	0.119665	−	0.207267i	0.919635	+	0.392774i	$0.128484\pi$
$420$		0			0
$421$	12.5000	+	21.6506i	0.609213	+	1.05519i	0.991370	+	0.131090i	$0.0418478\pi$
$421$	12.5000	+	21.6506i	0.609213	+	1.05519i	−0.382158	+	0.924097i	$0.624819\pi$
$422$	−8.57321	+	14.8492i	−0.417338	+	0.722850i
$423$		0			0
$424$	−16.0000			−0.777029
$425$	−7.34847	+	4.24264i	−0.356453	+	0.205798i
$426$		0			0
$427$	−16.5000	−	9.52628i	−0.798491	−	0.461009i
$428$	−29.3939			−1.42081
$429$		0			0
$430$			13.8564i			0.668215i
$431$	−14.6969			−0.707927			−0.353963	−	0.935259i	$-0.615166\pi$
$431$	−14.6969			−0.707927			−0.353963	+	0.935259i	$0.615166\pi$
$432$		0			0
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$		−	8.48528i		−	0.407307i
$435$		0			0
$436$	20.0000			0.957826
$437$	−22.0454	−	12.7279i	−1.05457	−	0.608859i
$438$		0			0
$439$	15.0000	−	8.66025i	0.715911	−	0.413331i	−0.0973349	−	0.995252i	$-0.531032\pi$
$439$	15.0000	−	8.66025i	0.715911	−	0.413331i	0.813246	+	0.581920i	$0.197698\pi$
$440$	39.1918			1.86840
$441$		0			0
$442$	−2.00000	+	3.46410i	−0.0951303	+	0.164771i
$443$	4.89898	+	8.48528i	0.232758	+	0.403148i	0.958619	−	0.284693i	$-0.0918918\pi$
$443$	4.89898	+	8.48528i	0.232758	+	0.403148i	−0.725861	+	0.687841i	$0.758558\pi$
$444$		0			0
$445$	4.00000	−	6.92820i	0.189618	−	0.328428i
$446$	−17.1464	−	29.6985i	−0.811907	−	1.40626i
$447$		0			0
$448$	−12.0000	+	6.92820i	−0.566947	+	0.327327i
$449$		−	31.1127i		−	1.46830i	−0.678988	−	0.734150i	$-0.737581\pi$
$449$		−	31.1127i		−	1.46830i	0.678988	−	0.734150i	$-0.262419\pi$
$450$		0			0
$451$		−	27.7128i		−	1.30495i
$452$	4.89898	+	2.82843i	0.230429	+	0.133038i
$453$		0			0
$454$	24.0000	−	13.8564i	1.12638	−	0.650313i
$455$	−2.44949	+	4.24264i	−0.114834	+	0.198898i
$456$		0			0
$457$	−1.00000	−	1.73205i	−0.0467780	−	0.0810219i	0.841688	−	0.539964i	$-0.181562\pi$
$457$	−1.00000	−	1.73205i	−0.0467780	−	0.0810219i	−0.888466	+	0.458942i	$0.848229\pi$
$458$	12.2474	+	7.07107i	0.572286	+	0.330409i
$459$		0			0
$460$	24.0000	−	13.8564i	1.11901	−	0.646058i
$461$	−12.2474	+	7.07107i	−0.570421	+	0.329332i	−0.757317	−	0.653047i	$-0.773490\pi$
$461$	−12.2474	+	7.07107i	−0.570421	+	0.329332i	0.186897	+	0.982380i	$0.440157\pi$
$462$		0			0
$463$	10.5000	+	6.06218i	0.487976	+	0.281733i	0.723735	−	0.690078i	$-0.242424\pi$
$463$	10.5000	+	6.06218i	0.487976	+	0.281733i	−0.235758	+	0.971812i	$0.575757\pi$
$464$	−19.5959	+	11.3137i	−0.909718	+	0.525226i
$465$		0			0
$466$	32.0000			1.48237
$467$	14.6969			0.680093			0.340047	−	0.940409i	$-0.389557\pi$
$467$	14.6969			0.680093			0.340047	+	0.940409i	$0.389557\pi$
$468$		0			0
$469$	21.0000			0.969690
$470$	−19.5959			−0.903892
$471$		0			0
$472$	−12.0000	−	6.92820i	−0.552345	−	0.318896i
$473$	14.6969	+	8.48528i	0.675766	+	0.390154i
$474$		0			0
$475$	13.5000	−	7.79423i	0.619422	−	0.357624i
$476$	−4.89898	−	8.48528i	−0.224544	−	0.388922i
$477$		0			0
$478$	24.0000	+	13.8564i	1.09773	+	0.633777i
$479$	4.89898	+	8.48528i	0.223840	+	0.387702i	0.955971	−	0.293462i	$-0.0948073\pi$
$479$	4.89898	+	8.48528i	0.223840	+	0.387702i	−0.732131	+	0.681164i	$0.761474\pi$
$480$		0			0
$481$	−0.500000	+	0.866025i	−0.0227980	+	0.0394874i
$482$	−30.6186	+	17.6777i	−1.39464	+	0.805196i
$483$		0			0
$484$	13.0000	−	22.5167i	0.590909	−	1.02348i
$485$		−	36.7696i		−	1.66962i
$486$		0			0
$487$		−	25.9808i		−	1.17730i	−0.808388	−	0.588650i	$-0.799659\pi$
$487$		−	25.9808i		−	1.17730i	0.808388	−	0.588650i	$-0.200341\pi$
$488$	−26.9444	+	15.5563i	−1.21972	+	0.704203i
$489$		0			0
$490$	8.00000	+	13.8564i	0.361403	+	0.625969i
$491$	−12.2474	+	21.2132i	−0.552720	+	0.957338i	0.445357	+	0.895353i	$0.353077\pi$
$491$	−12.2474	+	21.2132i	−0.552720	+	0.957338i	−0.998077	+	0.0619856i	$0.980257\pi$
$492$		0			0
$493$	−8.00000	−	13.8564i	−0.360302	−	0.624061i
$494$	3.67423	−	6.36396i	0.165312	−	0.286328i
$495$		0			0
$496$	−12.0000	−	6.92820i	−0.538816	−	0.311086i
$497$		0			0
$498$		0			0
$499$	−21.0000	−	12.1244i	−0.940089	−	0.542761i	−0.0501009	−	0.998744i	$-0.515954\pi$
$499$	−21.0000	−	12.1244i	−0.940089	−	0.542761i	−0.889988	+	0.455983i	$0.849288\pi$
$500$		−	11.3137i		−	0.505964i
$501$		0			0
$502$		−	41.5692i		−	1.85533i
$503$	−14.6969			−0.655304			−0.327652	−	0.944798i	$-0.606257\pi$
$503$	−14.6969			−0.655304			−0.327652	+	0.944798i	$0.606257\pi$
$504$		0			0
$505$	−32.0000			−1.42398
$506$		−	33.9411i		−	1.50887i
$507$		0			0
$508$			20.7846i			0.922168i
$509$	−17.1464	−	9.89949i	−0.760002	−	0.438787i	0.0692944	−	0.997596i	$-0.477925\pi$
$509$	−17.1464	−	9.89949i	−0.760002	−	0.438787i	−0.829296	+	0.558809i	$0.811259\pi$
$510$		0			0
$511$	−1.50000	+	0.866025i	−0.0663561	+	0.0383107i
$512$			22.6274i			1.00000i
$513$		0			0
$514$	20.0000	−	34.6410i	0.882162	−	1.52795i
$515$	12.2474	+	21.2132i	0.539687	+	0.934765i
$516$		0			0
$517$	−12.0000	+	20.7846i	−0.527759	+	0.914106i
$518$	−1.22474	−	2.12132i	−0.0538122	−	0.0932055i
$519$		0			0
$520$	4.00000	+	6.92820i	0.175412	+	0.303822i
$521$			19.7990i			0.867409i	0.901055	+	0.433705i	$0.142794\pi$
$521$			19.7990i			0.867409i	−0.901055	+	0.433705i	$0.857206\pi$
$522$		0			0
$523$			15.5885i			0.681636i	0.940129	+	0.340818i	$0.110704\pi$
$523$			15.5885i			0.681636i	−0.940129	+	0.340818i	$0.889296\pi$
$524$	9.79796	−	16.9706i	0.428026	−	0.741362i
$525$		0			0
$526$	24.0000	−	13.8564i	1.04645	−	0.604168i
$527$	4.89898	−	8.48528i	0.213403	−	0.369625i
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.0217391	−	0.0376533i
$530$	−19.5959	−	11.3137i	−0.851192	−	0.491436i
$531$		0			0
$532$	9.00000	+	15.5885i	0.390199	+	0.675845i
$533$	4.89898	−	2.82843i	0.212198	−	0.122513i
$534$		0			0
$535$	−36.0000	−	20.7846i	−1.55642	−	0.898597i
$536$	17.1464	−	29.6985i	0.740613	−	1.28278i
$537$		0			0
$538$	−4.00000			−0.172452
$539$	19.5959			0.844056
$540$		0			0
$541$	−1.00000			−0.0429934			−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000			−0.0429934			−0.0214967	+	0.999769i	$0.506843\pi$
$542$	7.34847			0.315644
$543$		0			0
$544$	−16.0000			−0.685994
$545$	24.4949	+	14.1421i	1.04925	+	0.605783i
$546$		0			0
$547$	19.5000	−	11.2583i	0.833760	−	0.481371i	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	19.5000	−	11.2583i	0.833760	−	0.481371i	0.855138	+	0.518400i	$0.173472\pi$
$548$	−34.2929	+	19.7990i	−1.46492	+	0.845771i
$549$		0			0
$550$	18.0000	+	10.3923i	0.767523	+	0.443129i
$551$	14.6969	+	25.4558i	0.626111	+	1.08446i
$552$		0			0
$553$	1.50000	−	2.59808i	0.0637865	−	0.110481i
$554$	17.1464	−	9.89949i	0.728482	−	0.420589i
$555$		0			0
$556$	15.0000	+	8.66025i	0.636142	+	0.367277i
$557$			2.82843i			0.119844i	0.998203	+	0.0599222i	$0.0190852\pi$
$557$			2.82843i			0.119844i	−0.998203	+	0.0599222i	$0.980915\pi$
$558$		0			0
$559$			3.46410i			0.146516i
$560$	−19.5959			−0.828079
$561$		0			0
$562$	−4.00000	−	6.92820i	−0.168730	−	0.292249i
$563$	−4.89898	+	8.48528i	−0.206467	+	0.357612i	−0.950599	−	0.310421i	$-0.899530\pi$
$563$	−4.89898	+	8.48528i	−0.206467	+	0.357612i	0.744132	+	0.668033i	$0.232863\pi$
$564$		0			0
$565$	4.00000	+	6.92820i	0.168281	+	0.291472i
$566$	17.1464	−	29.6985i	0.720718	−	1.24832i
$567$		0			0
$568$		0			0
$569$	17.1464	−	9.89949i	0.718816	−	0.415008i	−0.0955010	−	0.995429i	$-0.530445\pi$
$569$	17.1464	−	9.89949i	0.718816	−	0.415008i	0.814317	+	0.580421i	$0.197112\pi$
$570$		0			0
$571$	28.5000	+	16.4545i	1.19269	+	0.688599i	0.958915	−	0.283693i	$-0.0915595\pi$
$571$	28.5000	+	16.4545i	1.19269	+	0.688599i	0.233773	+	0.972291i	$0.424893\pi$
$572$	9.79796			0.409673
$573$		0			0
$574$			13.8564i			0.578355i
$575$	14.6969			0.612905
$576$		0			0
$577$	35.0000			1.45707			0.728535	−	0.685009i	$-0.240202\pi$
$577$	35.0000			1.45707			0.728535	+	0.685009i	$0.240202\pi$
$578$			12.7279i			0.529412i
$579$		0			0
$580$	−32.0000			−1.32873
$581$	14.6969	+	8.48528i	0.609732	+	0.352029i
$582$		0			0
$583$	−24.0000	+	13.8564i	−0.993978	+	0.573874i
$584$			2.82843i			0.117041i
$585$		0			0
$586$	2.00000	−	3.46410i	0.0826192	−	0.143101i
$587$	−2.44949	−	4.24264i	−0.101101	−	0.175113i	0.811037	−	0.584994i	$-0.198903\pi$
$587$	−2.44949	−	4.24264i	−0.101101	−	0.175113i	−0.912139	+	0.409882i	$0.865570\pi$
$588$		0			0
$589$	−9.00000	+	15.5885i	−0.370839	+	0.642311i
$590$	−9.79796	−	16.9706i	−0.403376	−	0.698667i
$591$		0			0
$592$	−4.00000			−0.164399
$593$			28.2843i			1.16150i	0.814083	+	0.580748i	$0.197240\pi$
$593$			28.2843i			1.16150i	−0.814083	+	0.580748i	$0.802760\pi$
$594$		0			0
$595$		−	13.8564i		−	0.568057i
$596$	−39.1918	−	22.6274i	−1.60536	−	0.926855i
$597$		0			0
$598$	6.00000	−	3.46410i	0.245358	−	0.141658i
$599$	−4.89898	+	8.48528i	−0.200167	+	0.346699i	−0.948582	−	0.316531i	$-0.897482\pi$
$599$	−4.89898	+	8.48528i	−0.200167	+	0.346699i	0.748415	+	0.663231i	$0.230815\pi$
$600$		0			0
$601$	−1.00000	−	1.73205i	−0.0407909	−	0.0706518i	0.844909	−	0.534910i	$-0.179654\pi$
$601$	−1.00000	−	1.73205i	−0.0407909	−	0.0706518i	−0.885700	+	0.464258i	$0.846321\pi$
$602$	−7.34847	−	4.24264i	−0.299501	−	0.172917i
$603$		0			0
$604$	−39.0000	+	22.5167i	−1.58689	+	0.916190i
$605$	31.8434	−	18.3848i	1.29462	−	0.747447i
$606$		0			0
$607$	10.5000	+	6.06218i	0.426182	+	0.246056i	0.697719	−	0.716372i	$-0.254199\pi$
$607$	10.5000	+	6.06218i	0.426182	+	0.246056i	−0.271537	+	0.962428i	$0.587532\pi$
$608$	29.3939			1.19208
$609$		0			0
$610$	−44.0000			−1.78151
$611$	−4.89898			−0.198191
$612$		0			0
$613$	11.0000			0.444286			0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000			0.444286			0.222143	+	0.975014i	$0.428695\pi$
$614$	−14.6969			−0.593120
$615$		0			0
$616$	−12.0000	+	20.7846i	−0.483494	+	0.837436i
$617$	−2.44949	−	1.41421i	−0.0986127	−	0.0569341i	0.449883	−	0.893088i	$-0.351466\pi$
$617$	−2.44949	−	1.41421i	−0.0986127	−	0.0569341i	−0.548495	+	0.836154i	$0.684799\pi$
$618$		0			0
$619$	−16.5000	+	9.52628i	−0.663191	+	0.382893i	−0.793492	−	0.608581i	$-0.791739\pi$
$619$	−16.5000	+	9.52628i	−0.663191	+	0.382893i	0.130301	+	0.991475i	$0.458406\pi$
$620$	−9.79796	−	16.9706i	−0.393496	−	0.681554i
$621$		0			0
$622$	−30.0000	−	17.3205i	−1.20289	−	0.694489i
$623$	2.44949	+	4.24264i	0.0981367	+	0.169978i
$624$		0			0
$625$	15.5000	−	26.8468i	0.620000	−	1.07387i
$626$	13.4722	−	7.77817i	0.538457	−	0.310878i
$627$		0			0
$628$	−10.0000	+	17.3205i	−0.399043	+	0.691164i
$629$		−	2.82843i		−	0.112777i
$630$		0			0
$631$			36.3731i			1.44799i	0.689806	+	0.723994i	$0.257696\pi$
$631$			36.3731i			1.44799i	−0.689806	+	0.723994i	$0.742304\pi$
$632$	−2.44949	−	4.24264i	−0.0974355	−	0.168763i
$633$		0			0
$634$	−4.00000	−	6.92820i	−0.158860	−	0.275154i
$635$	−14.6969	+	25.4558i	−0.583230	+	1.01018i
$636$		0			0
$637$	2.00000	+	3.46410i	0.0792429	+	0.137253i
$638$	−19.5959	+	33.9411i	−0.775810	+	1.34374i
$639$		0			0
$640$	−16.0000	+	27.7128i	−0.632456	+	1.09545i
$641$	39.1918	−	22.6274i	1.54798	−	0.893729i	0.549689	−	0.835369i	$-0.314746\pi$
$641$	39.1918	−	22.6274i	1.54798	−	0.893729i	0.998296	−	0.0583597i	$-0.0185870\pi$
$642$		0			0
$643$	15.0000	+	8.66025i	0.591542	+	0.341527i	0.765707	−	0.643189i	$-0.222389\pi$
$643$	15.0000	+	8.66025i	0.591542	+	0.341527i	−0.174165	+	0.984717i	$0.555723\pi$
$644$			16.9706i			0.668734i
$645$		0			0
$646$			20.7846i			0.817760i
$647$	29.3939			1.15559			0.577796	−	0.816181i	$-0.303913\pi$
$647$	29.3939			1.15559			0.577796	+	0.816181i	$0.303913\pi$
$648$		0			0
$649$	−24.0000			−0.942082
$650$			4.24264i			0.166410i
$651$		0			0
$652$			10.3923i			0.406994i
$653$	4.89898	+	2.82843i	0.191712	+	0.110685i	0.592784	−	0.805362i	$-0.298029\pi$
$653$	4.89898	+	2.82843i	0.191712	+	0.110685i	−0.401072	+	0.916047i	$0.631362\pi$
$654$		0			0
$655$	24.0000	−	13.8564i	0.937758	−	0.541415i
$656$	19.5959	+	11.3137i	0.765092	+	0.441726i
$657$		0			0
$658$	6.00000	−	10.3923i	0.233904	−	0.405134i
$659$	4.89898	+	8.48528i	0.190837	+	0.330540i	0.945528	−	0.325541i	$-0.105546\pi$
$659$	4.89898	+	8.48528i	0.190837	+	0.330540i	−0.754691	+	0.656081i	$0.772213\pi$
$660$		0			0
$661$	18.5000	−	32.0429i	0.719567	−	1.24633i	−0.241605	−	0.970375i	$-0.577674\pi$
$661$	18.5000	−	32.0429i	0.719567	−	1.24633i	0.961172	−	0.275951i	$-0.0889928\pi$
$662$	15.9217	+	27.5772i	0.618814	+	1.07182i
$663$		0			0
$664$	24.0000	−	13.8564i	0.931381	−	0.537733i
$665$			25.4558i			0.987135i
$666$		0			0
$667$			27.7128i			1.07304i
$668$	−4.89898	+	8.48528i	−0.189547	+	0.328305i
$669$		0			0
$670$	42.0000	−	24.2487i	1.62260	−	0.936809i
$671$	−26.9444	+	46.6690i	−1.04018	+	1.80164i
$672$		0			0
$673$	−5.50000	−	9.52628i	−0.212009	−	0.367211i	0.740334	−	0.672239i	$-0.234667\pi$
$673$	−5.50000	−	9.52628i	−0.212009	−	0.367211i	−0.952343	+	0.305028i	$0.901334\pi$
$674$	−13.4722	−	7.77817i	−0.518930	−	0.299604i
$675$		0			0
$676$	−12.0000	−	20.7846i	−0.461538	−	0.799408i
$677$	−12.2474	+	7.07107i	−0.470708	+	0.271763i	−0.716536	−	0.697550i	$-0.754273\pi$
$677$	−12.2474	+	7.07107i	−0.470708	+	0.271763i	0.245828	+	0.969313i	$0.420940\pi$
$678$		0			0
$679$	19.5000	+	11.2583i	0.748341	+	0.432055i
$680$	−19.5959	−	11.3137i	−0.751469	−	0.433861i
$681$		0			0
$682$	−24.0000			−0.919007
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	−56.0000			−2.13965
$686$	−26.9444			−1.02874
$687$		0			0
$688$	−12.0000	+	6.92820i	−0.457496	+	0.264135i
$689$	−4.89898	−	2.82843i	−0.186636	−	0.107754i
$690$		0			0
$691$	−39.0000	+	22.5167i	−1.48363	+	0.856574i	−0.999827	−	0.0186028i	$-0.994078\pi$
$691$	−39.0000	+	22.5167i	−1.48363	+	0.856574i	−0.483803	+	0.875177i	$0.660745\pi$
$692$	9.79796	−	5.65685i	0.372463	−	0.215041i
$693$		0			0
$694$	24.0000	+	13.8564i	0.911028	+	0.525982i
$695$	12.2474	+	21.2132i	0.464572	+	0.804663i
$696$		0			0
$697$	−8.00000	+	13.8564i	−0.303022	+	0.524849i
$698$	28.1691	−	16.2635i	1.06622	−	0.615581i
$699$		0			0
$700$	−9.00000	−	5.19615i	−0.340168	−	0.196396i
$701$			2.82843i			0.106828i	0.998572	+	0.0534141i	$0.0170103\pi$
$701$			2.82843i			0.106828i	−0.998572	+	0.0534141i	$0.982990\pi$
$702$		0			0
$703$			5.19615i			0.195977i
$704$	19.5959	+	33.9411i	0.738549	+	1.27920i
$705$		0			0
$706$	8.00000	+	13.8564i	0.301084	+	0.521493i
$707$	9.79796	−	16.9706i	0.368490	−	0.638244i
$708$		0			0
$709$	12.5000	+	21.6506i	0.469447	+	0.813107i	0.999390	−	0.0349269i	$-0.0111198\pi$
$709$	12.5000	+	21.6506i	0.469447	+	0.813107i	−0.529943	+	0.848034i	$0.677787\pi$
$710$		0			0
$711$		0			0
$712$	8.00000			0.299813
$713$	−14.6969	+	8.48528i	−0.550405	+	0.317776i
$714$		0			0
$715$	12.0000	+	6.92820i	0.448775	+	0.259100i
$716$		0			0
$717$		0			0
$718$		−	20.7846i		−	0.775675i
$719$	29.3939			1.09621			0.548103	−	0.836411i	$-0.315350\pi$
$719$	29.3939			1.09621			0.548103	+	0.836411i	$0.315350\pi$
$720$		0			0
$721$	−15.0000			−0.558629
$722$		−	11.3137i		−	0.421053i
$723$		0			0
$724$	−22.0000			−0.817624
$725$	−14.6969	−	8.48528i	−0.545831	−	0.315135i
$726$		0			0
$727$	15.0000	−	8.66025i	0.556319	−	0.321191i	−0.195348	−	0.980734i	$-0.562584\pi$
$727$	15.0000	−	8.66025i	0.556319	−	0.321191i	0.751667	+	0.659543i	$0.229250\pi$
$728$	−4.89898			−0.181568
$729$		0			0
$730$	−2.00000	+	3.46410i	−0.0740233	+	0.128212i
$731$	−4.89898	−	8.48528i	−0.181195	−	0.313839i
$732$		0			0
$733$	−19.0000	+	32.9090i	−0.701781	+	1.21552i	0.266060	+	0.963957i	$0.414278\pi$
$733$	−19.0000	+	32.9090i	−0.701781	+	1.21552i	−0.967841	+	0.251564i	$0.919055\pi$
$734$	−13.4722	−	23.3345i	−0.497268	−	0.861293i
$735$		0			0
$736$	24.0000	+	13.8564i	0.884652	+	0.510754i
$737$		−	59.3970i		−	2.18792i
$738$		0			0
$739$			10.3923i			0.382287i	0.981562	+	0.191144i	$0.0612196\pi$
$739$			10.3923i			0.382287i	−0.981562	+	0.191144i	$0.938780\pi$
$740$	−4.89898	−	2.82843i	−0.180090	−	0.103975i
$741$		0			0
$742$	12.0000	−	6.92820i	0.440534	−	0.254342i
$743$	17.1464	−	29.6985i	0.629041	−	1.08953i	−0.358703	−	0.933452i	$-0.616781\pi$
$743$	17.1464	−	29.6985i	0.629041	−	1.08953i	0.987744	−	0.156080i	$-0.0498858\pi$
$744$		0			0
$745$	−32.0000	−	55.4256i	−1.17239	−	2.03064i
$746$	30.6186	+	17.6777i	1.12103	+	0.647225i
$747$		0			0
$748$	−24.0000	+	13.8564i	−0.877527	+	0.506640i
$749$	22.0454	−	12.7279i	0.805522	−	0.465068i
$750$		0			0
$751$	−25.5000	−	14.7224i	−0.930508	−	0.537229i	−0.0435359	−	0.999052i	$-0.513862\pi$
$751$	−25.5000	−	14.7224i	−0.930508	−	0.537229i	−0.886972	+	0.461823i	$0.847196\pi$
$752$	−9.79796	−	16.9706i	−0.357295	−	0.618853i
$753$		0			0
$754$	−8.00000			−0.291343
$755$	−63.6867			−2.31780
$756$		0			0
$757$	47.0000			1.70824			0.854122	−	0.520073i	$-0.174095\pi$
$757$	47.0000			1.70824			0.854122	+	0.520073i	$0.174095\pi$
$758$	−22.0454			−0.800725
$759$		0			0
$760$	36.0000	+	20.7846i	1.30586	+	0.753937i
$761$	−2.44949	−	1.41421i	−0.0887939	−	0.0512652i	0.454946	−	0.890519i	$-0.349659\pi$
$761$	−2.44949	−	1.41421i	−0.0887939	−	0.0512652i	−0.543740	+	0.839254i	$0.682992\pi$
$762$		0			0
$763$	−15.0000	+	8.66025i	−0.543036	+	0.313522i
$764$	−24.4949	−	42.4264i	−0.886194	−	1.53493i
$765$		0			0
$766$	24.0000	+	13.8564i	0.867155	+	0.500652i
$767$	−2.44949	−	4.24264i	−0.0884459	−	0.153193i
$768$		0			0
$769$	0.500000	−	0.866025i	0.0180305	−	0.0312297i	−0.856869	−	0.515534i	$-0.827594\pi$
$769$	0.500000	−	0.866025i	0.0180305	−	0.0312297i	0.874900	+	0.484304i	$0.160927\pi$
$770$	−29.3939	+	16.9706i	−1.05928	+	0.611577i
$771$		0			0
$772$	−1.00000	+	1.73205i	−0.0359908	+	0.0623379i
$773$			28.2843i			1.01731i	0.860969	+	0.508657i	$0.169858\pi$
$773$			28.2843i			1.01731i	−0.860969	+	0.508657i	$0.830142\pi$
$774$		0			0
$775$		−	10.3923i		−	0.373303i
$776$	31.8434	−	18.3848i	1.14311	−	0.659975i
$777$		0			0
$778$	−10.0000	−	17.3205i	−0.358517	−	0.620970i
$779$	14.6969	−	25.4558i	0.526572	−	0.912050i
$780$		0			0
$781$		0			0
$782$	−9.79796	+	16.9706i	−0.350374	+	0.606866i
$783$		0			0
$784$	−8.00000	+	13.8564i	−0.285714	+	0.494872i
$785$	−24.4949	+	14.1421i	−0.874260	+	0.504754i
$786$		0			0
$787$	−7.50000	−	4.33013i	−0.267346	−	0.154352i	0.360335	−	0.932823i	$-0.382662\pi$
$787$	−7.50000	−	4.33013i	−0.267346	−	0.154352i	−0.627681	+	0.778471i	$0.715996\pi$
$788$		−	5.65685i		−	0.201517i
$789$		0			0
$790$		−	6.92820i		−	0.246494i
$791$	−4.89898			−0.174188
$792$		0			0
$793$	−11.0000			−0.390621
$794$		−	14.1421i		−	0.501886i
$795$		0			0
$796$			10.3923i			0.368345i
$797$	19.5959	+	11.3137i	0.694123	+	0.400752i	0.805155	−	0.593065i	$-0.202082\pi$
$797$	19.5959	+	11.3137i	0.694123	+	0.400752i	−0.111032	+	0.993817i	$0.535416\pi$
$798$		0			0
$799$	12.0000	−	6.92820i	0.424529	−	0.245102i
$800$	−14.6969	+	8.48528i	−0.519615	+	0.300000i
$801$		0			0
$802$	−16.0000	+	27.7128i	−0.564980	+	0.978573i
$803$	2.44949	+	4.24264i	0.0864406	+	0.149720i
$804$		0			0
$805$	−12.0000	+	20.7846i	−0.422944	+	0.732561i
$806$	−2.44949	−	4.24264i	−0.0862796	−	0.149441i
$807$		0			0
$808$	−16.0000	−	27.7128i	−0.562878	−	0.974933i
$809$		−	22.6274i		−	0.795538i	−0.917486	−	0.397769i	$-0.869785\pi$
$809$		−	22.6274i		−	0.795538i	0.917486	−	0.397769i	$-0.130215\pi$
$810$		0			0
$811$		−	31.1769i		−	1.09477i	−0.836881	−	0.547385i	$-0.815623\pi$
$811$		−	31.1769i		−	1.09477i	0.836881	−	0.547385i	$-0.184377\pi$
$812$	9.79796	−	16.9706i	0.343841	−	0.595550i
$813$		0			0
$814$	−6.00000	+	3.46410i	−0.210300	+	0.121417i
$815$	−7.34847	+	12.7279i	−0.257406	+	0.445840i
$816$		0			0
$817$	9.00000	+	15.5885i	0.314870	+	0.545371i
$818$	1.22474	+	0.707107i	0.0428222	+	0.0247234i
$819$		0			0
$820$	16.0000	+	27.7128i	0.558744	+	0.967773i
$821$	−26.9444	+	15.5563i	−0.940366	+	0.542920i	−0.890075	−	0.455814i	$-0.849348\pi$
$821$	−26.9444	+	15.5563i	−0.940366	+	0.542920i	−0.0502907	+	0.998735i	$0.516015\pi$
$822$		0			0
$823$	−43.5000	−	25.1147i	−1.51631	−	0.875445i	−0.999816	−	0.0191564i	$-0.993902\pi$
$823$	−43.5000	−	25.1147i	−1.51631	−	0.875445i	−0.516498	−	0.856288i	$-0.672765\pi$
$824$	−12.2474	+	21.2132i	−0.426660	+	0.738997i
$825$		0			0
$826$	12.0000			0.417533
$827$	14.6969			0.511063			0.255531	−	0.966801i	$-0.417750\pi$
$827$	14.6969			0.511063			0.255531	+	0.966801i	$0.417750\pi$
$828$		0			0
$829$	11.0000			0.382046			0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000			0.382046			0.191023	+	0.981586i	$0.438820\pi$
$830$	39.1918			1.36037
$831$		0			0
$832$	−4.00000	+	6.92820i	−0.138675	+	0.240192i
$833$	−9.79796	−	5.65685i	−0.339479	−	0.195998i
$834$		0			0
$835$	−12.0000	+	6.92820i	−0.415277	+	0.239760i
$836$	44.0908	−	25.4558i	1.52491	−	0.880409i
$837$		0			0
$838$	6.00000	+	3.46410i	0.207267	+	0.119665i
$839$	−24.4949	−	42.4264i	−0.845658	−	1.46472i	−0.885049	−	0.465498i	$-0.845875\pi$
$839$	−24.4949	−	42.4264i	−0.845658	−	1.46472i	0.0393910	−	0.999224i	$-0.487458\pi$
$840$		0			0
$841$	1.50000	−	2.59808i	0.0517241	−	0.0895888i
$842$	−30.6186	+	17.6777i	−1.05519	+	0.609213i
$843$		0			0
$844$	−21.0000	−	12.1244i	−0.722850	−	0.417338i
$845$		−	33.9411i		−	1.16761i
$846$		0			0
$847$			22.5167i			0.773682i
$848$		−	22.6274i		−	0.777029i
$849$		0			0
$850$	−6.00000	−	10.3923i	−0.205798	−	0.356453i
$851$	−2.44949	+	4.24264i	−0.0839674	+	0.145436i
$852$		0			0
$853$	24.5000	+	42.4352i	0.838864	+	1.45296i	0.890845	+	0.454307i	$0.150113\pi$
$853$	24.5000	+	42.4352i	0.838864	+	1.45296i	−0.0519811	+	0.998648i	$0.516554\pi$
$854$	13.4722	−	23.3345i	0.461009	−	0.798491i
$855$		0			0
$856$		−	41.5692i		−	1.42081i
$857$	−48.9898	+	28.2843i	−1.67346	+	0.966172i	−0.707781	+	0.706432i	$0.750304\pi$
$857$	−48.9898	+	28.2843i	−1.67346	+	0.966172i	−0.965678	+	0.259740i	$0.916363\pi$
$858$		0			0
$859$	10.5000	+	6.06218i	0.358255	+	0.206839i	0.668315	−	0.743878i	$-0.267016\pi$
$859$	10.5000	+	6.06218i	0.358255	+	0.206839i	−0.310060	+	0.950717i	$0.600349\pi$
$860$	−19.5959			−0.668215
$861$		0			0
$862$		−	20.7846i		−	0.707927i
$863$	−14.6969			−0.500290			−0.250145	−	0.968208i	$-0.580478\pi$
$863$	−14.6969			−0.500290			−0.250145	+	0.968208i	$0.580478\pi$
$864$		0			0
$865$	16.0000			0.544016
$866$			36.7696i			1.24948i
$867$		0			0
$868$	12.0000			0.407307
$869$	−7.34847	−	4.24264i	−0.249280	−	0.143922i
$870$		0			0
$871$	10.5000	−	6.06218i	0.355779	−	0.205409i
$872$			28.2843i			0.957826i
$873$		0			0
$874$	18.0000	−	31.1769i	0.608859	−	1.05457i
$875$	4.89898	+	8.48528i	0.165616	+	0.286855i
$876$		0			0
$877$	0.500000	−	0.866025i	0.0168838	−	0.0292436i	−0.857460	−	0.514551i	$-0.827959\pi$
$877$	0.500000	−	0.866025i	0.0168838	−	0.0292436i	0.874344	+	0.485307i	$0.161292\pi$
$878$	12.2474	+	21.2132i	0.413331	+	0.715911i
$879$		0			0
$880$			55.4256i			1.86840i
$881$			19.7990i			0.667045i	0.942742	+	0.333522i	$0.108237\pi$
$881$			19.7990i			0.667045i	−0.942742	+	0.333522i	$0.891763\pi$
$882$		0			0
$883$		−	5.19615i		−	0.174864i	−0.996170	−	0.0874322i	$-0.972134\pi$
$883$		−	5.19615i		−	0.174864i	0.996170	−	0.0874322i	$-0.0278661\pi$
$884$	−4.89898	−	2.82843i	−0.164771	−	0.0951303i
$885$		0			0
$886$	−12.0000	+	6.92820i	−0.403148	+	0.232758i
$887$	24.4949	−	42.4264i	0.822458	−	1.42454i	−0.0813883	−	0.996682i	$-0.525935\pi$
$887$	24.4949	−	42.4264i	0.822458	−	1.42454i	0.903847	−	0.427857i	$-0.140731\pi$
$888$		0			0
$889$	−9.00000	−	15.5885i	−0.301850	−	0.522820i
$890$	9.79796	+	5.65685i	0.328428	+	0.189618i
$891$		0			0
$892$	42.0000	−	24.2487i	1.40626	−	0.811907i
$893$	−22.0454	+	12.7279i	−0.737721	+	0.425924i
$894$		0			0
$895$		0			0
$896$	−9.79796	−	16.9706i	−0.327327	−	0.566947i
$897$		0			0
$898$	44.0000			1.46830
$899$	19.5959			0.653560
$900$		0			0
$901$	16.0000			0.533037
$902$	39.1918			1.30495
$903$		0			0
$904$	−4.00000	+	6.92820i	−0.133038	+	0.230429i
$905$	−26.9444	−	15.5563i	−0.895662	−	0.517111i
$906$		0			0
$907$	1.50000	−	0.866025i	0.0498067	−	0.0287559i	−0.474890	−	0.880045i	$-0.657512\pi$
$907$	1.50000	−	0.866025i	0.0498067	−	0.0287559i	0.524697	+	0.851289i	$0.324179\pi$
$908$	19.5959	+	33.9411i	0.650313	+	1.12638i
$909$		0			0
$910$	−6.00000	−	3.46410i	−0.198898	−	0.114834i
$911$	4.89898	+	8.48528i	0.162310	+	0.281130i	0.935697	−	0.352805i	$-0.114772\pi$
$911$	4.89898	+	8.48528i	0.162310	+	0.281130i	−0.773386	+	0.633935i	$0.781439\pi$
$912$		0			0
$913$	24.0000	−	41.5692i	0.794284	−	1.37574i
$914$	2.44949	−	1.41421i	0.0810219	−	0.0467780i
$915$		0			0
$916$	−10.0000	+	17.3205i	−0.330409	+	0.572286i
$917$			16.9706i			0.560417i
$918$		0			0
$919$		−	10.3923i		−	0.342811i	−0.985201	−	0.171405i	$-0.945169\pi$
$919$		−	10.3923i		−	0.342811i	0.985201	−	0.171405i	$-0.0548307\pi$
$920$	19.5959	+	33.9411i	0.646058	+	1.11901i
$921$		0			0
$922$	−10.0000	−	17.3205i	−0.329332	−	0.570421i
$923$		0			0
$924$		0			0
$925$	−1.50000	−	2.59808i	−0.0493197	−	0.0854242i
$926$	−8.57321	+	14.8492i	−0.281733	+	0.487976i
$927$		0			0
$928$	−16.0000	−	27.7128i	−0.525226	−	0.909718i
$929$	39.1918	−	22.6274i	1.28584	−	0.742381i	0.307932	−	0.951408i	$-0.400363\pi$
$929$	39.1918	−	22.6274i	1.28584	−	0.742381i	0.977910	+	0.209027i	$0.0670296\pi$
$930$		0			0
$931$	18.0000	+	10.3923i	0.589926	+	0.340594i
$932$			45.2548i			1.48237i
$933$		0			0
$934$			20.7846i			0.680093i
$935$	−39.1918			−1.28171
$936$		0			0
$937$	−37.0000			−1.20874			−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000			−1.20874			−0.604369	+	0.796705i	$0.706575\pi$
$938$			29.6985i			0.969690i
$939$		0			0
$940$		−	27.7128i		−	0.903892i
$941$	41.6413	+	24.0416i	1.35747	+	0.783735i	0.989282	−	0.146017i	$-0.0466455\pi$
$941$	41.6413	+	24.0416i	1.35747	+	0.783735i	0.368186	+	0.929752i	$0.379979\pi$
$942$		0			0
$943$	24.0000	−	13.8564i	0.781548	−	0.451227i
$944$	9.79796	−	16.9706i	0.318896	−	0.552345i
$945$		0			0
$946$	−12.0000	+	20.7846i	−0.390154	+	0.675766i
$947$	−17.1464	−	29.6985i	−0.557184	−	0.965071i	−0.997730	−	0.0673410i	$-0.978548\pi$
$947$	−17.1464	−	29.6985i	−0.557184	−	0.965071i	0.440546	−	0.897730i	$-0.354785\pi$
$948$		0			0
$949$	−0.500000	+	0.866025i	−0.0162307	+	0.0281124i
$950$	11.0227	+	19.0919i	0.357624	+	0.619422i
$951$		0			0
$952$	12.0000	−	6.92820i	0.388922	−	0.224544i
$953$			2.82843i			0.0916217i	0.998950	+	0.0458109i	$0.0145872\pi$
$953$			2.82843i			0.0916217i	−0.998950	+	0.0458109i	$0.985413\pi$
$954$		0			0
$955$		−	69.2820i		−	2.24191i
$956$	−19.5959	+	33.9411i	−0.633777	+	1.09773i
$957$		0			0
$958$	−12.0000	+	6.92820i	−0.387702	+	0.223840i
$959$	17.1464	−	29.6985i	0.553687	−	0.959014i
$960$		0			0
$961$	−9.50000	−	16.4545i	−0.306452	−	0.530790i
$962$	−1.22474	−	0.707107i	−0.0394874	−	0.0227980i
$963$		0			0
$964$	−25.0000	−	43.3013i	−0.805196	−	1.39464i
$965$	−2.44949	+	1.41421i	−0.0788519	+	0.0455251i
$966$		0			0
$967$	46.5000	+	26.8468i	1.49534	+	0.863334i	0.999986	−	0.00535699i	$-0.00170519\pi$
$967$	46.5000	+	26.8468i	1.49534	+	0.863334i	0.495354	+	0.868691i	$0.335039\pi$
$968$	31.8434	+	18.3848i	1.02348	+	0.590909i
$969$		0			0
$970$	52.0000			1.66962
$971$	−14.6969			−0.471647			−0.235824	−	0.971796i	$-0.575779\pi$
$971$	−14.6969			−0.471647			−0.235824	+	0.971796i	$0.575779\pi$
$972$		0			0
$973$	−15.0000			−0.480878
$974$	36.7423			1.17730
$975$		0			0
$976$	−22.0000	−	38.1051i	−0.704203	−	1.21972i
$977$	4.89898	+	2.82843i	0.156732	+	0.0904894i	0.576315	−	0.817228i	$-0.304490\pi$
$977$	4.89898	+	2.82843i	0.156732	+	0.0904894i	−0.419583	+	0.907717i	$0.637824\pi$
$978$		0			0
$979$	12.0000	−	6.92820i	0.383522	−	0.221426i
$980$	−19.5959	+	11.3137i	−0.625969	+	0.361403i
$981$		0			0
$982$	−30.0000	−	17.3205i	−0.957338	−	0.552720i
$983$	−17.1464	−	29.6985i	−0.546886	−	0.947235i	−0.998486	−	0.0550138i	$-0.982480\pi$
$983$	−17.1464	−	29.6985i	−0.546886	−	0.947235i	0.451599	−	0.892221i	$-0.350854\pi$
$984$		0			0
$985$	4.00000	−	6.92820i	0.127451	−	0.220751i
$986$	19.5959	−	11.3137i	0.624061	−	0.360302i
$987$		0			0
$988$	9.00000	+	5.19615i	0.286328	+	0.165312i
$989$			16.9706i			0.539633i
$990$		0			0
$991$			36.3731i			1.15543i	0.816239	+	0.577714i	$0.196055\pi$
$991$			36.3731i			1.15543i	−0.816239	+	0.577714i	$0.803945\pi$
$992$	9.79796	−	16.9706i	0.311086	−	0.538816i
$993$		0			0
$994$		0			0
$995$	−7.34847	+	12.7279i	−0.232962	+	0.403502i
$996$		0			0
$997$	17.0000	+	29.4449i	0.538395	+	0.932528i	0.998991	+	0.0449179i	$0.0143026\pi$
$997$	17.0000	+	29.4449i	0.538395	+	0.932528i	−0.460595	+	0.887610i	$0.652364\pi$
$998$	17.1464	−	29.6985i	0.542761	−	0.940089i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.2.h.a.215.2		4
3.2	odd	2	inner	324.2.h.a.215.1		4
4.3	odd	2		324.2.h.b.215.1		4
9.2	odd	6		324.2.h.b.107.1		4
9.4	even	3		108.2.b.b.107.1	✓	4
9.5	odd	6		108.2.b.b.107.4	yes	4
9.7	even	3		324.2.h.b.107.2		4
12.11	even	2		324.2.h.b.215.2		4
36.7	odd	6	inner	324.2.h.a.107.2		4
36.11	even	6	inner	324.2.h.a.107.1		4
36.23	even	6		108.2.b.b.107.2	yes	4
36.31	odd	6		108.2.b.b.107.3	yes	4
72.5	odd	6		1728.2.c.d.1727.4		4
72.13	even	6		1728.2.c.d.1727.2		4
72.59	even	6		1728.2.c.d.1727.3		4
72.67	odd	6		1728.2.c.d.1727.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.2.b.b.107.1	✓	4	9.4	even	3
108.2.b.b.107.2	yes	4	36.23	even	6
108.2.b.b.107.3	yes	4	36.31	odd	6
108.2.b.b.107.4	yes	4	9.5	odd	6
324.2.h.a.107.1		4	36.11	even	6	inner
324.2.h.a.107.2		4	36.7	odd	6	inner
324.2.h.a.215.1		4	3.2	odd	2	inner
324.2.h.a.215.2		4	1.1	even	1	trivial
324.2.h.b.107.1		4	9.2	odd	6
324.2.h.b.107.2		4	9.7	even	3
324.2.h.b.215.1		4	4.3	odd	2
324.2.h.b.215.2		4	12.11	even	2
1728.2.c.d.1727.1		4	72.67	odd	6
1728.2.c.d.1727.2		4	72.13	even	6
1728.2.c.d.1727.3		4	72.59	even	6
1728.2.c.d.1727.4		4	72.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.h (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +(-2.44949 - 1.41421i) q^{5} +(1.50000 - 0.866025i) q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} +(-2.44949 - 1.41421i) q^{5} +(1.50000 - 0.866025i) q^{7} -2.82843i q^{8} +(2.00000 - 3.46410i) q^{10} +(-2.44949 - 4.24264i) q^{11} +(0.500000 - 0.866025i) q^{13} +(1.22474 + 2.12132i) q^{14} +4.00000 q^{16} +2.82843i q^{17} -5.19615i q^{19} +(4.89898 + 2.82843i) q^{20} +(6.00000 - 3.46410i) q^{22} +(2.44949 - 4.24264i) q^{23} +(1.50000 + 2.59808i) q^{25} +(1.22474 + 0.707107i) q^{26} +(-3.00000 + 1.73205i) q^{28} +(-4.89898 + 2.82843i) q^{29} +(-3.00000 - 1.73205i) q^{31} +5.65685i q^{32} -4.00000 q^{34} -4.89898 q^{35} -1.00000 q^{37} +7.34847 q^{38} +(-4.00000 + 6.92820i) q^{40} +(4.89898 + 2.82843i) q^{41} +(-3.00000 + 1.73205i) q^{43} +(4.89898 + 8.48528i) q^{44} +(6.00000 + 3.46410i) q^{46} +(-2.44949 - 4.24264i) q^{47} +(-2.00000 + 3.46410i) q^{49} +(-3.67423 + 2.12132i) q^{50} +(-1.00000 + 1.73205i) q^{52} -5.65685i q^{53} +13.8564i q^{55} +(-2.44949 - 4.24264i) q^{56} +(-4.00000 - 6.92820i) q^{58} +(2.44949 - 4.24264i) q^{59} +(-5.50000 - 9.52628i) q^{61} +(2.44949 - 4.24264i) q^{62} -8.00000 q^{64} +(-2.44949 + 1.41421i) q^{65} +(10.5000 + 6.06218i) q^{67} -5.65685i q^{68} -6.92820i q^{70} -1.00000 q^{73} -1.41421i q^{74} +10.3923i q^{76} +(-7.34847 - 4.24264i) q^{77} +(1.50000 - 0.866025i) q^{79} +(-9.79796 - 5.65685i) q^{80} +(-4.00000 + 6.92820i) q^{82} +(4.89898 + 8.48528i) q^{83} +(4.00000 - 6.92820i) q^{85} +(-2.44949 - 4.24264i) q^{86} +(-12.0000 + 6.92820i) q^{88} +2.82843i q^{89} -1.73205i q^{91} +(-4.89898 + 8.48528i) q^{92} +(6.00000 - 3.46410i) q^{94} +(-7.34847 + 12.7279i) q^{95} +(6.50000 + 11.2583i) q^{97} +(-4.89898 - 2.82843i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 6 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 + 6 * q^7	\( 4 q - 8 q^{4} + 6 q^{7} + 8 q^{10} + 2 q^{13} + 16 q^{16} + 24 q^{22} + 6 q^{25} - 12 q^{28} - 12 q^{31} - 16 q^{34} - 4 q^{37} - 16 q^{40} - 12 q^{43} + 24 q^{46} - 8 q^{49} - 4 q^{52} - 16 q^{58} - 22 q^{61} - 32 q^{64} + 42 q^{67} - 4 q^{73} + 6 q^{79} - 16 q^{82} + 16 q^{85} - 48 q^{88} + 24 q^{94} + 26 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 6 * q^7 + 8 * q^10 + 2 * q^13 + 16 * q^16 + 24 * q^22 + 6 * q^25 - 12 * q^28 - 12 * q^31 - 16 * q^34 - 4 * q^37 - 16 * q^40 - 12 * q^43 + 24 * q^46 - 8 * q^49 - 4 * q^52 - 16 * q^58 - 22 * q^61 - 32 * q^64 + 42 * q^67 - 4 * q^73 + 6 * q^79 - 16 * q^82 + 16 * q^85 - 48 * q^88 + 24 * q^94 + 26 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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