LMFDB - Embedded newform 324.2.h.b.215.1

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			215.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	324.215
Dual form			324.2.h.b.107.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-2.44949 - 1.41421i) q^{5} +(-1.50000 + 0.866025i) q^{7} -2.82843i q^{8} +O(q^{10})$	\(q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-2.44949 - 1.41421i) q^{5} +(-1.50000 + 0.866025i) q^{7} -2.82843i q^{8} +(2.00000 + 3.46410i) q^{10} +(2.44949 + 4.24264i) q^{11} +(0.500000 - 0.866025i) q^{13} +2.44949 q^{14} +(-2.00000 + 3.46410i) q^{16} +2.82843i q^{17} +5.19615i q^{19} -5.65685i q^{20} -6.92820i q^{22} +(-2.44949 + 4.24264i) q^{23} +(1.50000 + 2.59808i) q^{25} +(-1.22474 + 0.707107i) q^{26} +(-3.00000 - 1.73205i) q^{28} +(-4.89898 + 2.82843i) q^{29} +(3.00000 + 1.73205i) q^{31} +(4.89898 - 2.82843i) q^{32} +(2.00000 - 3.46410i) q^{34} +4.89898 q^{35} -1.00000 q^{37} +(3.67423 - 6.36396i) q^{38} +(-4.00000 + 6.92820i) q^{40} +(4.89898 + 2.82843i) q^{41} +(3.00000 - 1.73205i) q^{43} +(-4.89898 + 8.48528i) q^{44} +(6.00000 - 3.46410i) q^{46} +(2.44949 + 4.24264i) q^{47} +(-2.00000 + 3.46410i) q^{49} -4.24264i q^{50} +2.00000 q^{52} -5.65685i q^{53} -13.8564i q^{55} +(2.44949 + 4.24264i) q^{56} +8.00000 q^{58} +(-2.44949 + 4.24264i) q^{59} +(-5.50000 - 9.52628i) q^{61} +(-2.44949 - 4.24264i) q^{62} -8.00000 q^{64} +(-2.44949 + 1.41421i) q^{65} +(-10.5000 - 6.06218i) q^{67} +(-4.89898 + 2.82843i) q^{68} +(-6.00000 - 3.46410i) q^{70} -1.00000 q^{73} +(1.22474 + 0.707107i) q^{74} +(-9.00000 + 5.19615i) q^{76} +(-7.34847 - 4.24264i) q^{77} +(-1.50000 + 0.866025i) q^{79} +(9.79796 - 5.65685i) q^{80} +(-4.00000 - 6.92820i) q^{82} +(-4.89898 - 8.48528i) q^{83} +(4.00000 - 6.92820i) q^{85} -4.89898 q^{86} +(12.0000 - 6.92820i) q^{88} +2.82843i q^{89} +1.73205i q^{91} -9.79796 q^{92} -6.92820i q^{94} +(7.34847 - 12.7279i) q^{95} +(6.50000 + 11.2583i) q^{97} +(4.89898 - 2.82843i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 6 q^{7}+O(q^{10}) $ 4 * q + 4 * q^4 - 6 * q^7	$ 4 q + 4 q^{4} - 6 q^{7} + 8 q^{10} + 2 q^{13} - 8 q^{16} + 6 q^{25} - 12 q^{28} + 12 q^{31} + 8 q^{34} - 4 q^{37} - 16 q^{40} + 12 q^{43} + 24 q^{46} - 8 q^{49} + 8 q^{52} + 32 q^{58} - 22 q^{61} - 32 q^{64} - 42 q^{67} - 24 q^{70} - 4 q^{73} - 36 q^{76} - 6 q^{79} - 16 q^{82} + 16 q^{85} + 48 q^{88} + 26 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 6 * q^7 + 8 * q^10 + 2 * q^13 - 8 * q^16 + 6 * q^25 - 12 * q^28 + 12 * q^31 + 8 * q^34 - 4 * q^37 - 16 * q^40 + 12 * q^43 + 24 * q^46 - 8 * q^49 + 8 * q^52 + 32 * q^58 - 22 * q^61 - 32 * q^64 - 42 * q^67 - 24 * q^70 - 4 * q^73 - 36 * q^76 - 6 * q^79 - 16 * q^82 + 16 * q^85 + 48 * q^88 + 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 $

$2$	−1.22474	−	0.707107i	−0.866025	−	0.500000i
$2$	−1.22474	−	0.707107i	−0.866025	−	0.500000i
$3$		0			0
$3$		0			0
$4$	1.00000	+	1.73205i	0.500000	+	0.866025i
$5$	−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	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−1.50000	+	0.866025i	−0.566947	+	0.327327i	0.188982	+	0.981981i	$0.439481\pi$
$8$		−	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	$0.0978220\pi$
$11$	2.44949	+	4.24264i	0.738549	+	1.27920i	−0.214600	+	0.976702i	$0.568845\pi$
$12$		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	$0.122382\pi$
$14$	2.44949			0.654654
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$			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	$0.203260\pi$
$19$			5.19615i			1.19208i	−0.802955	+	0.596040i	$0.796740\pi$
$20$		−	5.65685i		−	1.26491i
$21$		0			0
$22$		−	6.92820i		−	1.47710i
$23$	−2.44949	+	4.24264i	−0.510754	+	0.884652i	0.489168	+	0.872189i	$0.337300\pi$
$23$	−2.44949	+	4.24264i	−0.510754	+	0.884652i	−0.999922	+	0.0124624i	$0.996033\pi$
$24$		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	0.744599	−	0.667512i	$-0.232641\pi$
$31$	3.00000	+	1.73205i	0.538816	+	0.311086i	−0.205783	+	0.978598i	$0.565974\pi$
$32$	4.89898	−	2.82843i	0.866025	−	0.500000i
$33$		0			0
$34$	2.00000	−	3.46410i	0.342997	−	0.594089i
$35$	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	$0.526194\pi$
$38$	3.67423	−	6.36396i	0.596040	−	1.03237i
$39$		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	−0.253495	−	0.967337i	$-0.581580\pi$
$43$	3.00000	−	1.73205i	0.457496	−	0.264135i	0.710991	+	0.703201i	$0.248247\pi$
$44$	−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	0.987508	−	0.157569i	$-0.0503658\pi$
$47$	2.44949	+	4.24264i	0.357295	+	0.618853i	−0.630213	+	0.776422i	$0.717032\pi$
$48$		0			0
$49$	−2.00000	+	3.46410i	−0.285714	+	0.494872i
$50$		−	4.24264i		−	0.600000i
$51$		0			0
$52$	2.00000			0.277350
$53$		−	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
$58$	8.00000			1.05045
$59$	−2.44949	+	4.24264i	−0.318896	+	0.552345i	−0.980258	−	0.197722i	$-0.936646\pi$
$59$	−2.44949	+	4.24264i	−0.318896	+	0.552345i	0.661362	+	0.750067i	$0.269979\pi$
$60$		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	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−10.5000	−	6.06218i	−1.28278	−	0.740613i	−0.977356	+	0.211604i	$0.932131\pi$
$68$	−4.89898	+	2.82843i	−0.594089	+	0.342997i
$69$		0			0
$70$	−6.00000	−	3.46410i	−0.717137	−	0.414039i
$71$		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	$0.518638\pi$
$74$	1.22474	+	0.707107i	0.142374	+	0.0821995i
$75$		0			0
$76$	−9.00000	+	5.19615i	−1.03237	+	0.596040i
$77$	−7.34847	−	4.24264i	−0.837436	−	0.483494i
$78$		0			0
$79$	−1.50000	+	0.866025i	−0.168763	+	0.0974355i	−0.582003	−	0.813187i	$-0.697731\pi$
$79$	−1.50000	+	0.866025i	−0.168763	+	0.0974355i	0.413239	+	0.910622i	$0.364397\pi$
$80$	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	$-0.985948\pi$
$83$	−4.89898	−	8.48528i	−0.537733	−	0.931381i	0.461293	−	0.887248i	$-0.347386\pi$
$84$		0			0
$85$	4.00000	−	6.92820i	0.433861	−	0.751469i
$86$	−4.89898			−0.528271
$87$		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
$92$	−9.79796			−1.02151
$93$		0			0
$94$		−	6.92820i		−	0.714590i
$95$	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	0.821705	−	0.569914i	$-0.193023\pi$
$103$	7.50000	+	4.33013i	0.738997	+	0.426660i	−0.0827075	+	0.996574i	$0.526357\pi$
$104$	−2.44949	−	1.41421i	−0.240192	−	0.138675i
$105$		0			0
$106$	−4.00000	+	6.92820i	−0.388514	+	0.672927i
$107$	−14.6969			−1.42081			−0.710403	−	0.703795i	$-0.751487\pi$
$107$	−14.6969			−1.42081			−0.710403	+	0.703795i	$0.751487\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$	−9.79796	+	16.9706i	−0.934199	+	1.61808i
$111$		0			0
$112$		−	6.92820i		−	0.654654i
$113$	−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
$122$			15.5563i			1.40841i
$123$		0			0
$124$			6.92820i			0.622171i
$125$			5.65685i			0.505964i
$126$		0			0
$127$			10.3923i			0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$			10.3923i			0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$	9.79796	+	5.65685i	0.866025	+	0.500000i
$129$		0			0
$130$	4.00000			0.350823
$131$	4.89898	−	8.48528i	0.428026	−	0.741362i	−0.568672	−	0.822564i	$-0.692543\pi$
$131$	4.89898	−	8.48528i	0.428026	−	0.741362i	0.996698	+	0.0812020i	$0.0258759\pi$
$132$		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	0.783127	−	0.621862i	$-0.213624\pi$
$139$	7.50000	+	4.33013i	0.636142	+	0.367277i	−0.146985	+	0.989139i	$0.546957\pi$
$140$	4.89898	+	8.48528i	0.414039	+	0.717137i
$141$		0			0
$142$		0			0
$143$	4.89898			0.409673
$144$		0			0
$145$	16.0000			1.32873
$146$	1.22474	+	0.707107i	0.101361	+	0.0585206i
$147$		0			0
$148$	−1.00000	−	1.73205i	−0.0821995	−	0.142374i
$149$	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	$0.702085\pi$
$151$	−19.5000	+	11.2583i	−1.58689	+	0.916190i	−0.993816	+	0.111040i	$0.964582\pi$
$152$	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	$0.0360089\pi$
$158$	2.44949			0.194871
$159$		0			0
$160$	−16.0000			−1.26491
$161$		−	8.48528i		−	0.668734i
$162$		0			0
$163$			5.19615i			0.406994i	0.979076	+	0.203497i	$0.0652307\pi$
$163$			5.19615i			0.406994i	−0.979076	+	0.203497i	$0.934769\pi$
$164$			11.3137i			0.883452i
$165$		0			0
$166$			13.8564i			1.07547i
$167$	−2.44949	+	4.24264i	−0.189547	+	0.328305i	−0.945099	−	0.326783i	$-0.894035\pi$
$167$	−2.44949	+	4.24264i	−0.189547	+	0.328305i	0.755552	+	0.655089i	$0.227369\pi$
$168$		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
$176$	−19.5959			−1.47710
$177$		0			0
$178$	2.00000	−	3.46410i	0.149906	−	0.259645i
$179$		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	$0.365943\pi$
$182$	1.22474	−	2.12132i	0.0907841	−	0.157243i
$183$		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	$-0.820006\pi$
$191$	−12.2474	−	21.2132i	−0.886194	−	1.53493i	−0.0418556	−	0.999124i	$-0.513327\pi$
$192$		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	$0.155208\pi$
$194$		−	18.3848i		−	1.31995i
$195$		0			0
$196$	−8.00000			−0.571429
$197$			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	$0.0589606\pi$
$199$			5.19615i			0.368345i	−0.982894	+	0.184173i	$0.941039\pi$
$200$	7.34847	−	4.24264i	0.519615	−	0.300000i
$201$		0			0
$202$	−16.0000			−1.12576
$203$	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	0.0929509	−	0.995671i	$-0.470370\pi$
$211$	−10.5000	−	6.06218i	−0.722850	−	0.417338i	−0.815801	+	0.578333i	$0.803703\pi$
$212$	9.79796	−	5.65685i	0.672927	−	0.388514i
$213$		0			0
$214$	18.0000	+	10.3923i	1.23045	+	0.710403i
$215$	−9.79796			−0.668215
$216$		0			0
$217$	−6.00000			−0.407307
$218$	12.2474	+	7.07107i	0.829502	+	0.478913i
$219$		0			0
$220$	24.0000	−	13.8564i	1.61808	−	0.934199i
$221$	2.44949	+	1.41421i	0.164771	+	0.0951303i
$222$		0			0
$223$	21.0000	−	12.1244i	1.40626	−	0.811907i	0.411239	−	0.911528i	$-0.365096\pi$
$223$	21.0000	−	12.1244i	1.40626	−	0.811907i	0.995025	+	0.0996209i	$0.0317630\pi$
$224$	−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	$0.0586957\pi$
$227$	9.79796	+	16.9706i	0.650313	+	1.12638i	−0.332733	+	0.943021i	$0.607971\pi$
$228$		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	$0.0594799\pi$
$230$	−19.5959			−1.29212
$231$		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
$236$	−9.79796			−0.637793
$237$		0			0
$238$			6.92820i			0.449089i
$239$	−9.79796	+	16.9706i	−0.633777	+	1.09773i	0.352996	+	0.935625i	$0.385163\pi$
$239$	−9.79796	+	16.9706i	−0.633777	+	1.09773i	−0.986773	+	0.162109i	$0.948170\pi$
$240$		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
$250$	4.00000	−	6.92820i	0.252982	−	0.438178i
$251$	29.3939			1.85533			0.927663	−	0.373420i	$-0.121815\pi$
$251$	29.3939			1.85533			0.927663	+	0.373420i	$0.121815\pi$
$252$		0			0
$253$	−24.0000			−1.50887
$254$	7.34847	−	12.7279i	0.461084	−	0.798621i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	−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	$0.0398278\pi$
$263$	9.79796	+	16.9706i	0.604168	+	1.04645i	−0.388014	+	0.921653i	$0.626839\pi$
$264$		0			0
$265$	−8.00000	+	13.8564i	−0.491436	+	0.851192i
$266$			12.7279i			0.780399i
$267$		0			0
$268$		−	24.2487i		−	1.48123i
$269$			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	$0.0504472\pi$
$271$			5.19615i			0.315644i	−0.987468	+	0.157822i	$0.949553\pi$
$272$	−9.79796	−	5.65685i	−0.594089	−	0.342997i
$273$		0			0
$274$	−28.0000			−1.69154
$275$	−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	0.970774	−	0.239994i	$-0.0771455\pi$
$283$	21.0000	+	12.1244i	1.24832	+	0.720718i	0.277546	+	0.960712i	$0.410479\pi$
$284$		0			0
$285$		0			0
$286$	−6.00000	−	3.46410i	−0.354787	−	0.204837i
$287$	−9.79796			−0.578355
$288$		0			0
$289$	9.00000			0.529412
$290$	−19.5959	−	11.3137i	−1.15071	−	0.664364i
$291$		0			0
$292$	−1.00000	−	1.73205i	−0.0585206	−	0.101361i
$293$	−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
$302$	31.8434			1.83238
$303$		0			0
$304$	−18.0000	−	10.3923i	−1.03237	−	0.596040i
$305$			31.1127i			1.78151i
$306$		0			0
$307$		−	10.3923i		−	0.593120i	−0.955014	−	0.296560i	$-0.904160\pi$
$307$		−	10.3923i		−	0.593120i	0.955014	−	0.296560i	$-0.0958395\pi$
$308$		−	16.9706i		−	0.966988i
$309$		0			0
$310$			13.8564i			0.786991i
$311$	12.2474	−	21.2132i	0.694489	−	1.20289i	−0.275864	−	0.961197i	$-0.588964\pi$
$311$	12.2474	−	21.2132i	0.694489	−	1.20289i	0.970353	−	0.241694i	$-0.0777029\pi$
$312$		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
$322$	−6.00000	+	10.3923i	−0.334367	+	0.579141i
$323$	−14.6969			−0.817760
$324$		0			0
$325$	3.00000			0.166410
$326$	3.67423	−	6.36396i	0.203497	−	0.352467i
$327$		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	−0.928677	−	0.370889i	$-0.879053\pi$
$331$	−19.5000	+	11.2583i	−1.07182	+	0.618814i	−0.143140	+	0.989703i	$0.545720\pi$
$332$	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	$0.263521\pi$
$338$		−	16.9706i		−	0.923077i
$339$		0			0
$340$	16.0000			0.867722
$341$			16.9706i			0.919007i
$342$		0			0
$343$		−	19.0526i		−	1.02874i
$344$	−4.89898	−	8.48528i	−0.264135	−	0.457496i
$345$		0			0
$346$	8.00000			0.430083
$347$	−9.79796	+	16.9706i	−0.525982	+	0.911028i	0.473560	+	0.880762i	$0.342969\pi$
$347$	−9.79796	+	16.9706i	−0.525982	+	0.911028i	−0.999542	+	0.0302659i	$0.990365\pi$
$348$		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
$356$	−4.89898	+	2.82843i	−0.259645	+	0.149906i
$357$		0			0
$358$		0			0
$359$	14.6969			0.775675			0.387837	−	0.921728i	$-0.373222\pi$
$359$	14.6969			0.775675			0.387837	+	0.921728i	$0.373222\pi$
$360$		0			0
$361$	−8.00000			−0.421053
$362$	−13.4722	−	7.77817i	−0.708083	−	0.408812i
$363$		0			0
$364$	−3.00000	+	1.73205i	−0.157243	+	0.0907841i
$365$	2.44949	+	1.41421i	0.128212	+	0.0740233i
$366$		0			0
$367$	16.5000	−	9.52628i	0.861293	−	0.497268i	−0.00315207	−	0.999995i	$-0.501003\pi$
$367$	16.5000	−	9.52628i	0.861293	−	0.497268i	0.864445	+	0.502727i	$0.167670\pi$
$368$	−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	$-0.0574041\pi$
$374$	19.5959			1.01328
$375$		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	$-0.868884\pi$
$379$		−	15.5885i		−	0.800725i	0.916357	−	0.400363i	$-0.131116\pi$
$380$	29.3939			1.50787
$381$		0			0
$382$			34.6410i			1.77239i
$383$	−9.79796	+	16.9706i	−0.500652	+	0.867155i	0.499347	+	0.866402i	$0.333573\pi$
$383$	−9.79796	+	16.9706i	−0.500652	+	0.867155i	−1.00000		0.000753393i	$0.999760\pi$
$384$		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
$394$	2.00000	−	3.46410i	0.100759	−	0.174519i
$395$	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	$0.580741\pi$
$398$	3.67423	−	6.36396i	0.184173	−	0.318997i
$399$		0			0
$400$	−12.0000			−0.600000
$401$	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	$0.158796\pi$
$410$			22.6274i			1.11749i
$411$		0			0
$412$			17.3205i			0.853320i
$413$		−	8.48528i		−	0.417533i
$414$		0			0
$415$			27.7128i			1.36037i
$416$		−	5.65685i		−	0.277350i
$417$		0			0
$418$	36.0000			1.76082
$419$	−2.44949	+	4.24264i	−0.119665	+	0.207267i	−0.919635	−	0.392774i	$-0.871516\pi$
$419$	−2.44949	+	4.24264i	−0.119665	+	0.207267i	0.799970	+	0.600040i	$0.204849\pi$
$420$		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
$428$	−14.6969	−	25.4558i	−0.710403	−	1.23045i
$429$		0			0
$430$	12.0000	+	6.92820i	0.578691	+	0.334108i
$431$	14.6969			0.707927			0.353963	−	0.935259i	$-0.384834\pi$
$431$	14.6969			0.707927			0.353963	+	0.935259i	$0.384834\pi$
$432$		0			0
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$	7.34847	+	4.24264i	0.352738	+	0.203653i
$435$		0			0
$436$	−10.0000	−	17.3205i	−0.478913	−	0.829502i
$437$	−22.0454	−	12.7279i	−1.05457	−	0.608859i
$438$		0			0
$439$	−15.0000	+	8.66025i	−0.715911	+	0.413331i	−0.813246	−	0.581920i	$-0.802302\pi$
$439$	−15.0000	+	8.66025i	−0.715911	+	0.413331i	0.0973349	+	0.995252i	$0.468968\pi$
$440$	−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	0.725861	−	0.687841i	$-0.241442\pi$
$443$	−4.89898	−	8.48528i	−0.232758	−	0.403148i	−0.958619	+	0.284693i	$0.908108\pi$
$444$		0			0
$445$	4.00000	−	6.92820i	0.189618	−	0.328428i
$446$	−34.2929			−1.62381
$447$		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
$452$		−	5.65685i		−	0.266076i
$453$		0			0
$454$		−	27.7128i		−	1.30063i
$455$	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	0.235758	−	0.971812i	$-0.424243\pi$
$463$	−10.5000	−	6.06218i	−0.487976	−	0.281733i	−0.723735	+	0.690078i	$0.757576\pi$
$464$		−	22.6274i		−	1.05045i
$465$		0			0
$466$	−16.0000	+	27.7128i	−0.741186	+	1.28377i
$467$	−14.6969			−0.680093			−0.340047	−	0.940409i	$-0.610443\pi$
$467$	−14.6969			−0.680093			−0.340047	+	0.940409i	$0.610443\pi$
$468$		0			0
$469$	21.0000			0.969690
$470$	−9.79796	+	16.9706i	−0.451946	+	0.782794i
$471$		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	0.732131	−	0.681164i	$-0.238526\pi$
$479$	−4.89898	−	8.48528i	−0.223840	−	0.387702i	−0.955971	+	0.293462i	$0.905193\pi$
$480$		0			0
$481$	−0.500000	+	0.866025i	−0.0227980	+	0.0394874i
$482$		−	35.3553i		−	1.61039i
$483$		0			0
$484$	−26.0000			−1.18182
$485$		−	36.7696i		−	1.66962i
$486$		0			0
$487$			25.9808i			1.17730i	0.808388	+	0.588650i	$0.200341\pi$
$487$			25.9808i			1.17730i	−0.808388	+	0.588650i	$0.799659\pi$
$488$	−26.9444	+	15.5563i	−1.21972	+	0.704203i
$489$		0			0
$490$	−16.0000			−0.722806
$491$	12.2474	−	21.2132i	0.552720	−	0.957338i	−0.445357	−	0.895353i	$-0.646923\pi$
$491$	12.2474	−	21.2132i	0.552720	−	0.957338i	0.998077	−	0.0619856i	$-0.0197433\pi$
$492$		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	0.889988	−	0.455983i	$-0.150712\pi$
$499$	21.0000	+	12.1244i	0.940089	+	0.542761i	0.0501009	+	0.998744i	$0.484046\pi$
$500$	−9.79796	+	5.65685i	−0.438178	+	0.252982i
$501$		0			0
$502$	−36.0000	−	20.7846i	−1.60676	−	0.927663i
$503$	14.6969			0.655304			0.327652	−	0.944798i	$-0.393743\pi$
$503$	14.6969			0.655304			0.327652	+	0.944798i	$0.393743\pi$
$504$		0			0
$505$	−32.0000			−1.42398
$506$	29.3939	+	16.9706i	1.30672	+	0.754434i
$507$		0			0
$508$	−18.0000	+	10.3923i	−0.798621	+	0.461084i
$509$	−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
$518$	−2.44949			−0.107624
$519$		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	$-0.889296\pi$
$523$		−	15.5885i		−	0.681636i	0.940129	−	0.340818i	$-0.110704\pi$
$524$	19.5959			0.856052
$525$		0			0
$526$		−	27.7128i		−	1.20834i
$527$	−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
$538$	2.00000	−	3.46410i	0.0862261	−	0.149348i
$539$	−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	$0.506843\pi$
$542$	3.67423	−	6.36396i	0.157822	−	0.273356i
$543$		0			0
$544$	8.00000	+	13.8564i	0.342997	+	0.594089i
$545$	24.4949	+	14.1421i	1.04925	+	0.605783i
$546$		0			0
$547$	−19.5000	+	11.2583i	−0.833760	+	0.481371i	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−19.5000	+	11.2583i	−0.833760	+	0.481371i	0.0213785	+	0.999771i	$0.493195\pi$
$548$	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
$554$			19.7990i			0.841178i
$555$		0			0
$556$			17.3205i			0.734553i
$557$			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
$560$	−9.79796	+	16.9706i	−0.414039	+	0.717137i
$561$		0			0
$562$	8.00000			0.337460
$563$	4.89898	−	8.48528i	0.206467	−	0.357612i	−0.744132	−	0.668033i	$-0.767137\pi$
$563$	4.89898	−	8.48528i	0.206467	−	0.357612i	0.950599	+	0.310421i	$0.100470\pi$
$564$		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	−0.233773	−	0.972291i	$-0.575107\pi$
$571$	−28.5000	−	16.4545i	−1.19269	−	0.688599i	−0.958915	+	0.283693i	$0.908440\pi$
$572$	4.89898	+	8.48528i	0.204837	+	0.354787i
$573$		0			0
$574$	12.0000	+	6.92820i	0.500870	+	0.289178i
$575$	−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	$0.240202\pi$
$578$	−11.0227	−	6.36396i	−0.458484	−	0.264706i
$579$		0			0
$580$	16.0000	+	27.7128i	0.664364	+	1.15071i
$581$	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	0.912139	−	0.409882i	$-0.134430\pi$
$587$	2.44949	+	4.24264i	0.101101	+	0.175113i	−0.811037	+	0.584994i	$0.801097\pi$
$588$		0			0
$589$	−9.00000	+	15.5885i	−0.370839	+	0.642311i
$590$	−19.5959			−0.806751
$591$		0			0
$592$	2.00000	−	3.46410i	0.0821995	−	0.142374i
$593$			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
$596$			45.2548i			1.85371i
$597$		0			0
$598$		−	6.92820i		−	0.283315i
$599$	4.89898	−	8.48528i	0.200167	−	0.346699i	−0.748415	−	0.663231i	$-0.769185\pi$
$599$	4.89898	−	8.48528i	0.200167	−	0.346699i	0.948582	+	0.316531i	$0.102518\pi$
$600$		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	0.271537	−	0.962428i	$-0.412468\pi$
$607$	−10.5000	−	6.06218i	−0.426182	−	0.246056i	−0.697719	+	0.716372i	$0.745801\pi$
$608$	14.6969	+	25.4558i	0.596040	+	1.03237i
$609$		0			0
$610$	22.0000	−	38.1051i	0.890754	−	1.54283i
$611$	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	$0.428695\pi$
$614$	−7.34847	+	12.7279i	−0.296560	+	0.513657i
$615$		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	−0.130301	−	0.991475i	$-0.541594\pi$
$619$	16.5000	−	9.52628i	0.663191	−	0.382893i	0.793492	+	0.608581i	$0.208261\pi$
$620$	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
$626$			15.5563i			0.621757i
$627$		0			0
$628$	20.0000			0.798087
$629$		−	2.82843i		−	0.112777i
$630$		0			0
$631$		−	36.3731i		−	1.44799i	−0.689806	−	0.723994i	$-0.742304\pi$
$631$		−	36.3731i		−	1.44799i	0.689806	−	0.723994i	$-0.257696\pi$
$632$	2.44949	+	4.24264i	0.0974355	+	0.168763i
$633$		0			0
$634$	8.00000			0.317721
$635$	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	0.174165	−	0.984717i	$-0.444277\pi$
$643$	−15.0000	−	8.66025i	−0.591542	−	0.341527i	−0.765707	+	0.643189i	$0.777611\pi$
$644$	14.6969	−	8.48528i	0.579141	−	0.334367i
$645$		0			0
$646$	18.0000	+	10.3923i	0.708201	+	0.408880i
$647$	−29.3939			−1.15559			−0.577796	−	0.816181i	$-0.696087\pi$
$647$	−29.3939			−1.15559			−0.577796	+	0.816181i	$0.696087\pi$
$648$		0			0
$649$	−24.0000			−0.942082
$650$	−3.67423	−	2.12132i	−0.144115	−	0.0832050i
$651$		0			0
$652$	−9.00000	+	5.19615i	−0.352467	+	0.203497i
$653$	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	0.754691	−	0.656081i	$-0.227787\pi$
$659$	−4.89898	−	8.48528i	−0.190837	−	0.330540i	−0.945528	+	0.325541i	$0.894454\pi$
$660$		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	$-0.0889928\pi$
$662$	31.8434			1.23763
$663$		0			0
$664$	−24.0000	+	13.8564i	−0.931381	+	0.537733i
$665$			25.4558i			0.987135i
$666$		0			0
$667$		−	27.7128i		−	1.07304i
$668$	−9.79796			−0.379094
$669$		0			0
$670$		−	48.4974i		−	1.87362i
$671$	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
$682$	12.0000	−	20.7846i	0.459504	−	0.795884i
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	−56.0000			−2.13965
$686$	−13.4722	+	23.3345i	−0.514371	+	0.890916i
$687$		0			0
$688$			13.8564i			0.528271i
$689$	−4.89898	−	2.82843i	−0.186636	−	0.107754i
$690$		0			0
$691$	39.0000	−	22.5167i	1.48363	−	0.856574i	0.483803	−	0.875177i	$-0.339255\pi$
$691$	39.0000	−	22.5167i	1.48363	−	0.856574i	0.999827	+	0.0186028i	$0.00592180\pi$
$692$	−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
$698$			32.5269i			1.23116i
$699$		0			0
$700$		−	10.3923i		−	0.392792i
$701$			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
$706$	−16.0000			−0.602168
$707$	−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
$718$	−18.0000	−	10.3923i	−0.671754	−	0.387837i
$719$	−29.3939			−1.09621			−0.548103	−	0.836411i	$-0.684650\pi$
$719$	−29.3939			−1.09621			−0.548103	+	0.836411i	$0.684650\pi$
$720$		0			0
$721$	−15.0000			−0.558629
$722$	9.79796	+	5.65685i	0.364642	+	0.210526i
$723$		0			0
$724$	11.0000	+	19.0526i	0.408812	+	0.708083i
$725$	−14.6969	−	8.48528i	−0.545831	−	0.315135i
$726$		0			0
$727$	−15.0000	+	8.66025i	−0.556319	+	0.321191i	−0.751667	−	0.659543i	$-0.770750\pi$
$727$	−15.0000	+	8.66025i	−0.556319	+	0.321191i	0.195348	+	0.980734i	$0.437416\pi$
$728$	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	$0.919055\pi$
$734$	−26.9444			−0.994535
$735$		0			0
$736$			27.7128i			1.02151i
$737$		−	59.3970i		−	2.18792i
$738$		0			0
$739$		−	10.3923i		−	0.382287i	−0.981562	−	0.191144i	$-0.938780\pi$
$739$		−	10.3923i		−	0.382287i	0.981562	−	0.191144i	$-0.0612196\pi$
$740$			5.65685i			0.207950i
$741$		0			0
$742$		−	13.8564i		−	0.508685i
$743$	−17.1464	+	29.6985i	−0.629041	+	1.08953i	0.358703	+	0.933452i	$0.383219\pi$
$743$	−17.1464	+	29.6985i	−0.629041	+	1.08953i	−0.987744	+	0.156080i	$0.950114\pi$
$744$		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	0.886972	−	0.461823i	$-0.152804\pi$
$751$	25.5000	+	14.7224i	0.930508	+	0.537229i	0.0435359	+	0.999052i	$0.486138\pi$
$752$	−19.5959			−0.714590
$753$		0			0
$754$	4.00000	−	6.92820i	0.145671	−	0.252310i
$755$	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	$0.174095\pi$
$758$	−11.0227	+	19.0919i	−0.400363	+	0.693448i
$759$		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	$0.160927\pi$
$770$		−	33.9411i		−	1.22315i
$771$		0			0
$772$	2.00000			0.0719816
$773$			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
$778$	20.0000			0.717035
$779$	−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	0.627681	−	0.778471i	$-0.284004\pi$
$787$	7.50000	+	4.33013i	0.267346	+	0.154352i	−0.360335	+	0.932823i	$0.617338\pi$
$788$	−4.89898	+	2.82843i	−0.174519	+	0.100759i
$789$		0			0
$790$	−6.00000	−	3.46410i	−0.213470	−	0.123247i
$791$	4.89898			0.174188
$792$		0			0
$793$	−11.0000			−0.390621
$794$	12.2474	+	7.07107i	0.434646	+	0.250943i
$795$		0			0
$796$	−9.00000	+	5.19615i	−0.318997	+	0.184173i
$797$	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
$806$	−4.89898			−0.172559
$807$		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	$0.184377\pi$
$811$			31.1769i			1.09477i	−0.836881	+	0.547385i	$0.815623\pi$
$812$	19.5959			0.687682
$813$		0			0
$814$			6.92820i			0.242833i
$815$	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	$0.00609804\pi$
$823$	43.5000	+	25.1147i	1.51631	+	0.875445i	0.516498	+	0.856288i	$0.327235\pi$
$824$	12.2474	−	21.2132i	0.426660	−	0.738997i
$825$		0			0
$826$	−6.00000	+	10.3923i	−0.208767	+	0.361595i
$827$	−14.6969			−0.511063			−0.255531	−	0.966801i	$-0.582250\pi$
$827$	−14.6969			−0.511063			−0.255531	+	0.966801i	$0.582250\pi$
$828$		0			0
$829$	11.0000			0.382046			0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000			0.382046			0.191023	+	0.981586i	$0.438820\pi$
$830$	19.5959	−	33.9411i	0.680184	−	1.17811i
$831$		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	$0.154125\pi$
$839$	24.4949	+	42.4264i	0.845658	+	1.46472i	−0.0393910	+	0.999224i	$0.512542\pi$
$840$		0			0
$841$	1.50000	−	2.59808i	0.0517241	−	0.0895888i
$842$		−	35.3553i		−	1.21843i
$843$		0			0
$844$		−	24.2487i		−	0.834675i
$845$		−	33.9411i		−	1.16761i
$846$		0			0
$847$		−	22.5167i		−	0.773682i
$848$	19.5959	+	11.3137i	0.672927	+	0.388514i
$849$		0			0
$850$	12.0000			0.411597
$851$	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	0.310060	−	0.950717i	$-0.399651\pi$
$859$	−10.5000	−	6.06218i	−0.358255	−	0.206839i	−0.668315	+	0.743878i	$0.732984\pi$
$860$	−9.79796	−	16.9706i	−0.334108	−	0.578691i
$861$		0			0
$862$	−18.0000	−	10.3923i	−0.613082	−	0.353963i
$863$	14.6969			0.500290			0.250145	−	0.968208i	$-0.419522\pi$
$863$	14.6969			0.500290			0.250145	+	0.968208i	$0.419522\pi$
$864$		0			0
$865$	16.0000			0.544016
$866$	−31.8434	−	18.3848i	−1.08208	−	0.624740i
$867$		0			0
$868$	−6.00000	−	10.3923i	−0.203653	−	0.352738i
$869$	−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	$0.161292\pi$
$878$	24.4949			0.826663
$879$		0			0
$880$	48.0000	+	27.7128i	1.61808	+	0.934199i
$881$			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	$0.0278661\pi$
$883$			5.19615i			0.174864i	−0.996170	+	0.0874322i	$0.972134\pi$
$884$			5.65685i			0.190261i
$885$		0			0
$886$			13.8564i			0.465515i
$887$	−24.4949	+	42.4264i	−0.822458	+	1.42454i	0.0813883	+	0.996682i	$0.474065\pi$
$887$	−24.4949	+	42.4264i	−0.822458	+	1.42454i	−0.903847	+	0.427857i	$0.859269\pi$
$888$		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
$896$	−19.5959			−0.654654
$897$		0			0
$898$	−22.0000	+	38.1051i	−0.734150	+	1.27158i
$899$	−19.5959			−0.653560
$900$		0			0
$901$	16.0000			0.533037
$902$	19.5959	−	33.9411i	0.652473	−	1.13012i
$903$		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	−0.524697	−	0.851289i	$-0.675821\pi$
$907$	−1.50000	+	0.866025i	−0.0498067	+	0.0287559i	0.474890	+	0.880045i	$0.342488\pi$
$908$	−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	0.773386	−	0.633935i	$-0.218561\pi$
$911$	−4.89898	−	8.48528i	−0.162310	−	0.281130i	−0.935697	+	0.352805i	$0.885228\pi$
$912$		0			0
$913$	24.0000	−	41.5692i	0.794284	−	1.37574i
$914$			2.82843i			0.0935561i
$915$		0			0
$916$	20.0000			0.660819
$917$			16.9706i			0.560417i
$918$		0			0
$919$			10.3923i			0.342811i	0.985201	+	0.171405i	$0.0548307\pi$
$919$			10.3923i			0.342811i	−0.985201	+	0.171405i	$0.945169\pi$
$920$	−19.5959	−	33.9411i	−0.646058	−	1.11901i
$921$		0			0
$922$	20.0000			0.658665
$923$		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
$932$	39.1918	−	22.6274i	1.28377	−	0.741186i
$933$		0			0
$934$	18.0000	+	10.3923i	0.588978	+	0.340047i
$935$	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	$0.706575\pi$
$938$	−25.7196	−	14.8492i	−0.839776	−	0.484845i
$939$		0			0
$940$	24.0000	−	13.8564i	0.782794	−	0.451946i
$941$	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	$0.0214515\pi$
$947$	17.1464	+	29.6985i	0.557184	+	0.965071i	−0.440546	+	0.897730i	$0.645215\pi$
$948$		0			0
$949$	−0.500000	+	0.866025i	−0.0162307	+	0.0281124i
$950$	22.0454			0.715247
$951$		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
$956$	−39.1918			−1.26755
$957$		0			0
$958$			13.8564i			0.447680i
$959$	−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	−0.495354	−	0.868691i	$-0.664961\pi$
$967$	−46.5000	−	26.8468i	−1.49534	−	0.863334i	−0.999986	+	0.00535699i	$0.998295\pi$
$968$	31.8434	+	18.3848i	1.02348	+	0.590909i
$969$		0			0
$970$	−26.0000	+	45.0333i	−0.834810	+	1.44593i
$971$	14.6969			0.471647			0.235824	−	0.971796i	$-0.424221\pi$
$971$	14.6969			0.471647			0.235824	+	0.971796i	$0.424221\pi$
$972$		0			0
$973$	−15.0000			−0.480878
$974$	18.3712	−	31.8198i	0.588650	−	1.01957i
$975$		0			0
$976$	44.0000			1.40841
$977$	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	$0.0175203\pi$
$983$	17.1464	+	29.6985i	0.546886	+	0.947235i	−0.451599	+	0.892221i	$0.649146\pi$
$984$		0			0
$985$	4.00000	−	6.92820i	0.127451	−	0.220751i
$986$			22.6274i			0.720604i
$987$		0			0
$988$			10.3923i			0.330623i
$989$			16.9706i			0.539633i
$990$		0			0
$991$		−	36.3731i		−	1.15543i	−0.816239	−	0.577714i	$-0.803945\pi$
$991$		−	36.3731i		−	1.15543i	0.816239	−	0.577714i	$-0.196055\pi$
$992$	19.5959			0.622171
$993$		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.b.215.1		4
3.2	odd	2	inner	324.2.h.b.215.2		4
4.3	odd	2		324.2.h.a.215.2		4
9.2	odd	6		324.2.h.a.107.1		4
9.4	even	3		108.2.b.b.107.3	yes	4
9.5	odd	6		108.2.b.b.107.2	yes	4
9.7	even	3		324.2.h.a.107.2		4
12.11	even	2		324.2.h.a.215.1		4
36.7	odd	6	inner	324.2.h.b.107.2		4
36.11	even	6	inner	324.2.h.b.107.1		4
36.23	even	6		108.2.b.b.107.4	yes	4
36.31	odd	6		108.2.b.b.107.1	✓	4
72.5	odd	6		1728.2.c.d.1727.3		4
72.13	even	6		1728.2.c.d.1727.1		4
72.59	even	6		1728.2.c.d.1727.4		4
72.67	odd	6		1728.2.c.d.1727.2		4

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

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-2.44949 - 1.41421i) q^{5} +(-1.50000 + 0.866025i) q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-2.44949 - 1.41421i) q^{5} +(-1.50000 + 0.866025i) q^{7} -2.82843i q^{8} +(2.00000 + 3.46410i) q^{10} +(2.44949 + 4.24264i) q^{11} +(0.500000 - 0.866025i) q^{13} +2.44949 q^{14} +(-2.00000 + 3.46410i) q^{16} +2.82843i q^{17} +5.19615i q^{19} -5.65685i q^{20} -6.92820i q^{22} +(-2.44949 + 4.24264i) q^{23} +(1.50000 + 2.59808i) q^{25} +(-1.22474 + 0.707107i) q^{26} +(-3.00000 - 1.73205i) q^{28} +(-4.89898 + 2.82843i) q^{29} +(3.00000 + 1.73205i) q^{31} +(4.89898 - 2.82843i) q^{32} +(2.00000 - 3.46410i) q^{34} +4.89898 q^{35} -1.00000 q^{37} +(3.67423 - 6.36396i) q^{38} +(-4.00000 + 6.92820i) q^{40} +(4.89898 + 2.82843i) q^{41} +(3.00000 - 1.73205i) q^{43} +(-4.89898 + 8.48528i) q^{44} +(6.00000 - 3.46410i) q^{46} +(2.44949 + 4.24264i) q^{47} +(-2.00000 + 3.46410i) q^{49} -4.24264i q^{50} +2.00000 q^{52} -5.65685i q^{53} -13.8564i q^{55} +(2.44949 + 4.24264i) q^{56} +8.00000 q^{58} +(-2.44949 + 4.24264i) q^{59} +(-5.50000 - 9.52628i) q^{61} +(-2.44949 - 4.24264i) q^{62} -8.00000 q^{64} +(-2.44949 + 1.41421i) q^{65} +(-10.5000 - 6.06218i) q^{67} +(-4.89898 + 2.82843i) q^{68} +(-6.00000 - 3.46410i) q^{70} -1.00000 q^{73} +(1.22474 + 0.707107i) q^{74} +(-9.00000 + 5.19615i) q^{76} +(-7.34847 - 4.24264i) q^{77} +(-1.50000 + 0.866025i) q^{79} +(9.79796 - 5.65685i) q^{80} +(-4.00000 - 6.92820i) q^{82} +(-4.89898 - 8.48528i) q^{83} +(4.00000 - 6.92820i) q^{85} -4.89898 q^{86} +(12.0000 - 6.92820i) q^{88} +2.82843i q^{89} +1.73205i q^{91} -9.79796 q^{92} -6.92820i q^{94} +(7.34847 - 12.7279i) q^{95} +(6.50000 + 11.2583i) q^{97} +(4.89898 - 2.82843i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 6 q^{7}+O(q^{10}) \) 4 * q + 4 * q^4 - 6 * q^7	\( 4 q + 4 q^{4} - 6 q^{7} + 8 q^{10} + 2 q^{13} - 8 q^{16} + 6 q^{25} - 12 q^{28} + 12 q^{31} + 8 q^{34} - 4 q^{37} - 16 q^{40} + 12 q^{43} + 24 q^{46} - 8 q^{49} + 8 q^{52} + 32 q^{58} - 22 q^{61} - 32 q^{64} - 42 q^{67} - 24 q^{70} - 4 q^{73} - 36 q^{76} - 6 q^{79} - 16 q^{82} + 16 q^{85} + 48 q^{88} + 26 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 6 * q^7 + 8 * q^10 + 2 * q^13 - 8 * q^16 + 6 * q^25 - 12 * q^28 + 12 * q^31 + 8 * q^34 - 4 * q^37 - 16 * q^40 + 12 * q^43 + 24 * q^46 - 8 * q^49 + 8 * q^52 + 32 * q^58 - 22 * q^61 - 32 * q^64 - 42 * q^67 - 24 * q^70 - 4 * q^73 - 36 * q^76 - 6 * q^79 - 16 * q^82 + 16 * q^85 + 48 * q^88 + 26 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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