LMFDB - Embedded newform 1792.2.f.a.1791.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(1791,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("1792.1791");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.f (of order $2$, degree $1$, 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:	$14.3091920422$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 896)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1791.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1792.1791
Dual form			1792.2.f.a.1791.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -0.732051i q^{5} +(-2.00000 - 1.73205i) q^{7} +4.46410 q^{9} +O(q^{10})$	$q-2.73205 q^{3} -0.732051i q^{5} +(-2.00000 - 1.73205i) q^{7} +4.46410 q^{9} +5.46410i q^{11} -4.73205i q^{13} +2.00000i q^{15} -4.00000i q^{17} -1.26795 q^{19} +(5.46410 + 4.73205i) q^{21} -5.46410i q^{23} +4.46410 q^{25} -4.00000 q^{27} -6.92820 q^{29} +6.92820 q^{31} -14.9282i q^{33} +(-1.26795 + 1.46410i) q^{35} -4.00000 q^{37} +12.9282i q^{39} -2.92820i q^{41} -2.53590i q^{43} -3.26795i q^{45} -6.92820 q^{47} +(1.00000 + 6.92820i) q^{49} +10.9282i q^{51} +6.92820 q^{53} +4.00000 q^{55} +3.46410 q^{57} -3.80385 q^{59} +11.6603i q^{61} +(-8.92820 - 7.73205i) q^{63} -3.46410 q^{65} -2.53590i q^{67} +14.9282i q^{69} -4.53590i q^{71} +6.92820i q^{73} -12.1962 q^{75} +(9.46410 - 10.9282i) q^{77} -3.46410i q^{79} -2.46410 q^{81} -10.7321 q^{83} -2.92820 q^{85} +18.9282 q^{87} +14.9282i q^{89} +(-8.19615 + 9.46410i) q^{91} -18.9282 q^{93} +0.928203i q^{95} +12.0000i q^{97} +24.3923i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9} - 12 q^{19} + 8 q^{21} + 4 q^{25} - 16 q^{27} - 12 q^{35} - 16 q^{37} + 4 q^{49} + 16 q^{55} - 36 q^{59} - 8 q^{63} - 28 q^{75} + 24 q^{77} + 4 q^{81} - 36 q^{83} + 16 q^{85} + 48 q^{87} - 12 q^{91} - 48 q^{93}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9 - 12 * q^19 + 8 * q^21 + 4 * q^25 - 16 * q^27 - 12 * q^35 - 16 * q^37 + 4 * q^49 + 16 * q^55 - 36 * q^59 - 8 * q^63 - 28 * q^75 + 24 * q^77 + 4 * q^81 - 36 * q^83 + 16 * q^85 + 48 * q^87 - 12 * q^91 - 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−2.73205			−1.57735			−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205			−1.57735			−0.788675	+	0.614810i	$0.789233\pi$
$4$		0			0
$5$		−	0.732051i		−	0.327383i	−0.986512	−	0.163692i	$-0.947660\pi$
$5$		−	0.732051i		−	0.327383i	0.986512	−	0.163692i	$-0.0523402\pi$
$6$		0			0
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$8$		0			0
$9$	4.46410			1.48803
$10$		0			0
$11$			5.46410i			1.64749i	0.566961	+	0.823744i	$0.308119\pi$
$11$			5.46410i			1.64749i	−0.566961	+	0.823744i	$0.691881\pi$
$12$		0			0
$13$		−	4.73205i		−	1.31243i	−0.754572	−	0.656217i	$-0.772155\pi$
$13$		−	4.73205i		−	1.31243i	0.754572	−	0.656217i	$-0.227845\pi$
$14$		0			0
$15$			2.00000i			0.516398i
$16$		0			0
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		0			0
$19$	−1.26795			−0.290887			−0.145444	−	0.989367i	$-0.546461\pi$
$19$	−1.26795			−0.290887			−0.145444	+	0.989367i	$0.546461\pi$
$20$		0			0
$21$	5.46410	+	4.73205i	1.19236	+	1.03262i
$22$		0			0
$23$		−	5.46410i		−	1.13934i	−0.821872	−	0.569672i	$-0.807070\pi$
$23$		−	5.46410i		−	1.13934i	0.821872	−	0.569672i	$-0.192930\pi$
$24$		0			0
$25$	4.46410			0.892820
$26$		0			0
$27$	−4.00000			−0.769800
$28$		0			0
$29$	−6.92820			−1.28654			−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820			−1.28654			−0.643268	+	0.765641i	$0.722422\pi$
$30$		0			0
$31$	6.92820			1.24434			0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820			1.24434			0.622171	+	0.782881i	$0.286251\pi$
$32$		0			0
$33$		−	14.9282i		−	2.59867i
$34$		0			0
$35$	−1.26795	+	1.46410i	−0.214323	+	0.247478i
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$			12.9282i			2.07017i
$40$		0			0
$41$		−	2.92820i		−	0.457309i	−0.973508	−	0.228654i	$-0.926567\pi$
$41$		−	2.92820i		−	0.457309i	0.973508	−	0.228654i	$-0.0734325\pi$
$42$		0			0
$43$		−	2.53590i		−	0.386721i	−0.981128	−	0.193360i	$-0.938061\pi$
$43$		−	2.53590i		−	0.386721i	0.981128	−	0.193360i	$-0.0619387\pi$
$44$		0			0
$45$		−	3.26795i		−	0.487157i
$46$		0			0
$47$	−6.92820			−1.01058			−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820			−1.01058			−0.505291	+	0.862949i	$0.668615\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$			10.9282i			1.53025i
$52$		0			0
$53$	6.92820			0.951662			0.475831	−	0.879537i	$-0.342147\pi$
$53$	6.92820			0.951662			0.475831	+	0.879537i	$0.342147\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$	3.46410			0.458831
$58$		0			0
$59$	−3.80385			−0.495219			−0.247609	−	0.968860i	$-0.579645\pi$
$59$	−3.80385			−0.495219			−0.247609	+	0.968860i	$0.579645\pi$
$60$		0			0
$61$			11.6603i			1.49294i	0.665418	+	0.746471i	$0.268253\pi$
$61$			11.6603i			1.49294i	−0.665418	+	0.746471i	$0.731747\pi$
$62$		0			0
$63$	−8.92820	−	7.73205i	−1.12485	−	0.974147i
$64$		0			0
$65$	−3.46410			−0.429669
$66$		0			0
$67$		−	2.53590i		−	0.309809i	−0.987929	−	0.154905i	$-0.950493\pi$
$67$		−	2.53590i		−	0.309809i	0.987929	−	0.154905i	$-0.0495070\pi$
$68$		0			0
$69$			14.9282i			1.79714i
$70$		0			0
$71$		−	4.53590i		−	0.538312i	−0.963097	−	0.269156i	$-0.913255\pi$
$71$		−	4.53590i		−	0.538312i	0.963097	−	0.269156i	$-0.0867447\pi$
$72$		0			0
$73$			6.92820i			0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$			6.92820i			0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$		0			0
$75$	−12.1962			−1.40829
$76$		0			0
$77$	9.46410	−	10.9282i	1.07853	−	1.24538i
$78$		0			0
$79$		−	3.46410i		−	0.389742i	−0.980829	−	0.194871i	$-0.937571\pi$
$79$		−	3.46410i		−	0.389742i	0.980829	−	0.194871i	$-0.0624288\pi$
$80$		0			0
$81$	−2.46410			−0.273789
$82$		0			0
$83$	−10.7321			−1.17800			−0.588998	−	0.808135i	$-0.700477\pi$
$83$	−10.7321			−1.17800			−0.588998	+	0.808135i	$0.700477\pi$
$84$		0			0
$85$	−2.92820			−0.317608
$86$		0			0
$87$	18.9282			2.02932
$88$		0			0
$89$			14.9282i			1.58239i	0.611566	+	0.791193i	$0.290540\pi$
$89$			14.9282i			1.58239i	−0.611566	+	0.791193i	$0.709460\pi$
$90$		0			0
$91$	−8.19615	+	9.46410i	−0.859190	+	0.992107i
$92$		0			0
$93$	−18.9282			−1.96276
$94$		0			0
$95$			0.928203i			0.0952316i
$96$		0			0
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		0			0
$99$			24.3923i			2.45152i
$100$		0			0
$101$		−	6.19615i		−	0.616540i	−0.951299	−	0.308270i	$-0.900250\pi$
$101$		−	6.19615i		−	0.616540i	0.951299	−	0.308270i	$-0.0997501\pi$
$102$		0			0
$103$	−18.9282			−1.86505			−0.932526	−	0.361104i	$-0.882400\pi$
$103$	−18.9282			−1.86505			−0.932526	+	0.361104i	$0.882400\pi$
$104$		0			0
$105$	3.46410	−	4.00000i	0.338062	−	0.390360i
$106$		0			0
$107$			8.39230i			0.811315i	0.914025	+	0.405657i	$0.132957\pi$
$107$			8.39230i			0.811315i	−0.914025	+	0.405657i	$0.867043\pi$
$108$		0			0
$109$	6.92820			0.663602			0.331801	−	0.943349i	$-0.392344\pi$
$109$	6.92820			0.663602			0.331801	+	0.943349i	$0.392344\pi$
$110$		0			0
$111$	10.9282			1.03726
$112$		0			0
$113$	−9.46410			−0.890308			−0.445154	−	0.895454i	$-0.646851\pi$
$113$	−9.46410			−0.890308			−0.445154	+	0.895454i	$0.646851\pi$
$114$		0			0
$115$	−4.00000			−0.373002
$116$		0			0
$117$		−	21.1244i		−	1.95295i
$118$		0			0
$119$	−6.92820	+	8.00000i	−0.635107	+	0.733359i
$120$		0			0
$121$	−18.8564			−1.71422
$122$		0			0
$123$			8.00000i			0.721336i
$124$		0			0
$125$		−	6.92820i		−	0.619677i
$126$		0			0
$127$			21.4641i			1.90463i	0.305115	+	0.952316i	$0.401305\pi$
$127$			21.4641i			1.90463i	−0.305115	+	0.952316i	$0.598695\pi$
$128$		0			0
$129$			6.92820i			0.609994i
$130$		0			0
$131$	−10.7321			−0.937664			−0.468832	−	0.883287i	$-0.655325\pi$
$131$	−10.7321			−0.937664			−0.468832	+	0.883287i	$0.655325\pi$
$132$		0			0
$133$	2.53590	+	2.19615i	0.219890	+	0.190431i
$134$		0			0
$135$			2.92820i			0.252020i
$136$		0			0
$137$	−19.8564			−1.69645			−0.848224	−	0.529638i	$-0.822328\pi$
$137$	−19.8564			−1.69645			−0.848224	+	0.529638i	$0.822328\pi$
$138$		0			0
$139$	4.19615			0.355913			0.177957	−	0.984038i	$-0.443051\pi$
$139$	4.19615			0.355913			0.177957	+	0.984038i	$0.443051\pi$
$140$		0			0
$141$	18.9282			1.59404
$142$		0			0
$143$	25.8564			2.16222
$144$		0			0
$145$			5.07180i			0.421190i
$146$		0			0
$147$	−2.73205	−	18.9282i	−0.225336	−	1.56117i
$148$		0			0
$149$	−12.0000			−0.983078			−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000			−0.983078			−0.491539	+	0.870855i	$0.663566\pi$
$150$		0			0
$151$		−	11.3205i		−	0.921250i	−0.887595	−	0.460625i	$-0.847625\pi$
$151$		−	11.3205i		−	0.921250i	0.887595	−	0.460625i	$-0.152375\pi$
$152$		0			0
$153$		−	17.8564i		−	1.44360i
$154$		0			0
$155$		−	5.07180i		−	0.407377i
$156$		0			0
$157$			19.2679i			1.53775i	0.639399	+	0.768875i	$0.279183\pi$
$157$			19.2679i			1.53775i	−0.639399	+	0.768875i	$0.720817\pi$
$158$		0			0
$159$	−18.9282			−1.50110
$160$		0			0
$161$	−9.46410	+	10.9282i	−0.745876	+	0.861263i
$162$		0			0
$163$		−	16.3923i		−	1.28394i	−0.766728	−	0.641972i	$-0.778116\pi$
$163$		−	16.3923i		−	1.28394i	0.766728	−	0.641972i	$-0.221884\pi$
$164$		0			0
$165$	−10.9282			−0.850759
$166$		0			0
$167$	−5.07180			−0.392467			−0.196234	−	0.980557i	$-0.562871\pi$
$167$	−5.07180			−0.392467			−0.196234	+	0.980557i	$0.562871\pi$
$168$		0			0
$169$	−9.39230			−0.722485
$170$		0			0
$171$	−5.66025			−0.432850
$172$		0			0
$173$			3.26795i			0.248458i	0.992254	+	0.124229i	$0.0396457\pi$
$173$			3.26795i			0.248458i	−0.992254	+	0.124229i	$0.960354\pi$
$174$		0			0
$175$	−8.92820	−	7.73205i	−0.674909	−	0.584488i
$176$		0			0
$177$	10.3923			0.781133
$178$		0			0
$179$			0.392305i			0.0293222i	0.999893	+	0.0146611i	$0.00466695\pi$
$179$			0.392305i			0.0293222i	−0.999893	+	0.0146611i	$0.995333\pi$
$180$		0			0
$181$		−	16.7321i		−	1.24368i	−0.783143	−	0.621842i	$-0.786385\pi$
$181$		−	16.7321i		−	1.24368i	0.783143	−	0.621842i	$-0.213615\pi$
$182$		0			0
$183$		−	31.8564i		−	2.35489i
$184$		0			0
$185$			2.92820i			0.215286i
$186$		0			0
$187$	21.8564			1.59830
$188$		0			0
$189$	8.00000	+	6.92820i	0.581914	+	0.503953i
$190$		0			0
$191$			11.4641i			0.829513i	0.909932	+	0.414757i	$0.136133\pi$
$191$			11.4641i			0.829513i	−0.909932	+	0.414757i	$0.863867\pi$
$192$		0			0
$193$	4.39230			0.316165			0.158083	−	0.987426i	$-0.449469\pi$
$193$	4.39230			0.316165			0.158083	+	0.987426i	$0.449469\pi$
$194$		0			0
$195$	9.46410			0.677738
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	16.7846			1.18983			0.594915	−	0.803789i	$-0.297186\pi$
$199$	16.7846			1.18983			0.594915	+	0.803789i	$0.297186\pi$
$200$		0			0
$201$			6.92820i			0.488678i
$202$		0			0
$203$	13.8564	+	12.0000i	0.972529	+	0.842235i
$204$		0			0
$205$	−2.14359			−0.149715
$206$		0			0
$207$		−	24.3923i		−	1.69538i
$208$		0			0
$209$		−	6.92820i		−	0.479234i
$210$		0			0
$211$			7.60770i			0.523735i	0.965104	+	0.261868i	$0.0843384\pi$
$211$			7.60770i			0.523735i	−0.965104	+	0.261868i	$0.915662\pi$
$212$		0			0
$213$			12.3923i			0.849107i
$214$		0			0
$215$	−1.85641			−0.126606
$216$		0			0
$217$	−13.8564	−	12.0000i	−0.940634	−	0.814613i
$218$		0			0
$219$		−	18.9282i		−	1.27905i
$220$		0			0
$221$	−18.9282			−1.27325
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$	19.9282			1.32855
$226$		0			0
$227$	−1.26795			−0.0841567			−0.0420784	−	0.999114i	$-0.513398\pi$
$227$	−1.26795			−0.0841567			−0.0420784	+	0.999114i	$0.513398\pi$
$228$		0			0
$229$		−	0.339746i		−	0.0224510i	−0.999937	−	0.0112255i	$-0.996427\pi$
$229$		−	0.339746i		−	0.0224510i	0.999937	−	0.0112255i	$-0.00357327\pi$
$230$		0			0
$231$	−25.8564	+	29.8564i	−1.70123	+	1.96441i
$232$		0			0
$233$	19.8564			1.30084			0.650418	−	0.759576i	$-0.274594\pi$
$233$	19.8564			1.30084			0.650418	+	0.759576i	$0.274594\pi$
$234$		0			0
$235$			5.07180i			0.330848i
$236$		0			0
$237$			9.46410i			0.614759i
$238$		0			0
$239$		−	0.392305i		−	0.0253761i	−0.999920	−	0.0126880i	$-0.995961\pi$
$239$		−	0.392305i		−	0.0253761i	0.999920	−	0.0126880i	$-0.00403884\pi$
$240$		0			0
$241$			12.0000i			0.772988i	0.922292	+	0.386494i	$0.126314\pi$
$241$			12.0000i			0.772988i	−0.922292	+	0.386494i	$0.873686\pi$
$242$		0			0
$243$	18.7321			1.20166
$244$		0			0
$245$	5.07180	−	0.732051i	0.324025	−	0.0467690i
$246$		0			0
$247$			6.00000i			0.381771i
$248$		0			0
$249$	29.3205			1.85811
$250$		0			0
$251$	−22.0526			−1.39195			−0.695973	−	0.718068i	$-0.745027\pi$
$251$	−22.0526			−1.39195			−0.695973	+	0.718068i	$0.745027\pi$
$252$		0			0
$253$	29.8564			1.87706
$254$		0			0
$255$	8.00000			0.500979
$256$		0			0
$257$			21.8564i			1.36337i	0.731648	+	0.681683i	$0.238751\pi$
$257$			21.8564i			1.36337i	−0.731648	+	0.681683i	$0.761249\pi$
$258$		0			0
$259$	8.00000	+	6.92820i	0.497096	+	0.430498i
$260$		0			0
$261$	−30.9282			−1.91441
$262$		0			0
$263$			11.4641i			0.706907i	0.935452	+	0.353453i	$0.114993\pi$
$263$			11.4641i			0.706907i	−0.935452	+	0.353453i	$0.885007\pi$
$264$		0			0
$265$		−	5.07180i		−	0.311558i
$266$		0			0
$267$		−	40.7846i		−	2.49598i
$268$		0			0
$269$			1.12436i			0.0685532i	0.999412	+	0.0342766i	$0.0109127\pi$
$269$			1.12436i			0.0685532i	−0.999412	+	0.0342766i	$0.989087\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$	22.3923	−	25.8564i	1.35524	−	1.56490i
$274$		0			0
$275$			24.3923i			1.47091i
$276$		0			0
$277$	−28.7846			−1.72950			−0.864750	−	0.502203i	$-0.832523\pi$
$277$	−28.7846			−1.72950			−0.864750	+	0.502203i	$0.832523\pi$
$278$		0			0
$279$	30.9282			1.85162
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	15.8038			0.939441			0.469721	−	0.882815i	$-0.344355\pi$
$283$	15.8038			0.939441			0.469721	+	0.882815i	$0.344355\pi$
$284$		0			0
$285$		−	2.53590i		−	0.150214i
$286$		0			0
$287$	−5.07180	+	5.85641i	−0.299379	+	0.345693i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		−	32.7846i		−	1.92187i
$292$		0			0
$293$			1.80385i			0.105382i	0.998611	+	0.0526910i	$0.0167798\pi$
$293$			1.80385i			0.105382i	−0.998611	+	0.0526910i	$0.983220\pi$
$294$		0			0
$295$			2.78461i			0.162126i
$296$		0			0
$297$		−	21.8564i		−	1.26824i
$298$		0			0
$299$	−25.8564			−1.49531
$300$		0			0
$301$	−4.39230	+	5.07180i	−0.253168	+	0.292334i
$302$		0			0
$303$			16.9282i			0.972500i
$304$		0			0
$305$	8.53590			0.488764
$306$		0			0
$307$	13.2679			0.757242			0.378621	−	0.925552i	$-0.376398\pi$
$307$	13.2679			0.757242			0.378621	+	0.925552i	$0.376398\pi$
$308$		0			0
$309$	51.7128			2.94184
$310$		0			0
$311$	25.8564			1.46618			0.733091	−	0.680130i	$-0.238077\pi$
$311$	25.8564			1.46618			0.733091	+	0.680130i	$0.238077\pi$
$312$		0			0
$313$		−	32.7846i		−	1.85310i	−0.376177	−	0.926548i	$-0.622762\pi$
$313$		−	32.7846i		−	1.85310i	0.376177	−	0.926548i	$-0.377238\pi$
$314$		0			0
$315$	−5.66025	+	6.53590i	−0.318919	+	0.368256i
$316$		0			0
$317$	1.85641			0.104266			0.0521331	−	0.998640i	$-0.483398\pi$
$317$	1.85641			0.104266			0.0521331	+	0.998640i	$0.483398\pi$
$318$		0			0
$319$		−	37.8564i		−	2.11955i
$320$		0			0
$321$		−	22.9282i		−	1.27973i
$322$		0			0
$323$			5.07180i			0.282202i
$324$		0			0
$325$		−	21.1244i		−	1.17177i
$326$		0			0
$327$	−18.9282			−1.04673
$328$		0			0
$329$	13.8564	+	12.0000i	0.763928	+	0.661581i
$330$		0			0
$331$		−	11.3205i		−	0.622231i	−0.950372	−	0.311116i	$-0.899297\pi$
$331$		−	11.3205i		−	0.622231i	0.950372	−	0.311116i	$-0.100703\pi$
$332$		0			0
$333$	−17.8564			−0.978525
$334$		0			0
$335$	−1.85641			−0.101426
$336$		0			0
$337$	−3.60770			−0.196524			−0.0982618	−	0.995161i	$-0.531328\pi$
$337$	−3.60770			−0.196524			−0.0982618	+	0.995161i	$0.531328\pi$
$338$		0			0
$339$	25.8564			1.40433
$340$		0			0
$341$			37.8564i			2.05004i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$	10.9282			0.588355
$346$		0			0
$347$			13.4641i			0.722791i	0.932413	+	0.361395i	$0.117700\pi$
$347$			13.4641i			0.722791i	−0.932413	+	0.361395i	$0.882300\pi$
$348$		0			0
$349$			11.6603i			0.624159i	0.950056	+	0.312080i	$0.101026\pi$
$349$			11.6603i			0.624159i	−0.950056	+	0.312080i	$0.898974\pi$
$350$		0			0
$351$			18.9282i			1.01031i
$352$		0			0
$353$		−	29.8564i		−	1.58910i	−0.607201	−	0.794548i	$-0.707708\pi$
$353$		−	29.8564i		−	1.58910i	0.607201	−	0.794548i	$-0.292292\pi$
$354$		0			0
$355$	−3.32051			−0.176234
$356$		0			0
$357$	18.9282	−	21.8564i	1.00179	−	1.15676i
$358$		0			0
$359$			0.392305i			0.0207051i	0.999946	+	0.0103525i	$0.00329537\pi$
$359$			0.392305i			0.0207051i	−0.999946	+	0.0103525i	$0.996705\pi$
$360$		0			0
$361$	−17.3923			−0.915384
$362$		0			0
$363$	51.5167			2.70392
$364$		0			0
$365$	5.07180			0.265470
$366$		0			0
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$		0			0
$369$		−	13.0718i		−	0.680491i
$370$		0			0
$371$	−13.8564	−	12.0000i	−0.719389	−	0.623009i
$372$		0			0
$373$	−17.0718			−0.883944			−0.441972	−	0.897029i	$-0.645721\pi$
$373$	−17.0718			−0.883944			−0.441972	+	0.897029i	$0.645721\pi$
$374$		0			0
$375$			18.9282i			0.977448i
$376$		0			0
$377$			32.7846i			1.68849i
$378$		0			0
$379$		−	21.4641i		−	1.10254i	−0.834328	−	0.551268i	$-0.814144\pi$
$379$		−	21.4641i		−	1.10254i	0.834328	−	0.551268i	$-0.185856\pi$
$380$		0			0
$381$		−	58.6410i		−	3.00427i
$382$		0			0
$383$	−6.92820			−0.354015			−0.177007	−	0.984210i	$-0.556642\pi$
$383$	−6.92820			−0.354015			−0.177007	+	0.984210i	$0.556642\pi$
$384$		0			0
$385$	−8.00000	−	6.92820i	−0.407718	−	0.353094i
$386$		0			0
$387$		−	11.3205i		−	0.575454i
$388$		0			0
$389$	1.85641			0.0941235			0.0470618	−	0.998892i	$-0.485014\pi$
$389$	1.85641			0.0941235			0.0470618	+	0.998892i	$0.485014\pi$
$390$		0			0
$391$	−21.8564			−1.10533
$392$		0			0
$393$	29.3205			1.47902
$394$		0			0
$395$	−2.53590			−0.127595
$396$		0			0
$397$		−	29.9090i		−	1.50109i	−0.660821	−	0.750544i	$-0.729792\pi$
$397$		−	29.9090i		−	1.50109i	0.660821	−	0.750544i	$-0.270208\pi$
$398$		0			0
$399$	−6.92820	−	6.00000i	−0.346844	−	0.300376i
$400$		0			0
$401$	−23.3205			−1.16457			−0.582285	−	0.812985i	$-0.697841\pi$
$401$	−23.3205			−1.16457			−0.582285	+	0.812985i	$0.697841\pi$
$402$		0			0
$403$		−	32.7846i		−	1.63312i
$404$		0			0
$405$			1.80385i			0.0896339i
$406$		0			0
$407$		−	21.8564i		−	1.08338i
$408$		0			0
$409$			5.07180i			0.250784i	0.992107	+	0.125392i	$0.0400189\pi$
$409$			5.07180i			0.250784i	−0.992107	+	0.125392i	$0.959981\pi$
$410$		0			0
$411$	54.2487			2.67589
$412$		0			0
$413$	7.60770	+	6.58846i	0.374350	+	0.324197i
$414$		0			0
$415$			7.85641i			0.385656i
$416$		0			0
$417$	−11.4641			−0.561399
$418$		0			0
$419$	3.12436			0.152635			0.0763174	−	0.997084i	$-0.475684\pi$
$419$	3.12436			0.152635			0.0763174	+	0.997084i	$0.475684\pi$
$420$		0			0
$421$	20.0000			0.974740			0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000			0.974740			0.487370	+	0.873195i	$0.337956\pi$
$422$		0			0
$423$	−30.9282			−1.50378
$424$		0			0
$425$		−	17.8564i		−	0.866163i
$426$		0			0
$427$	20.1962	−	23.3205i	0.977360	−	1.12856i
$428$		0			0
$429$	−70.6410			−3.41058
$430$		0			0
$431$			5.46410i			0.263197i	0.991303	+	0.131598i	$0.0420109\pi$
$431$			5.46410i			0.263197i	−0.991303	+	0.131598i	$0.957989\pi$
$432$		0			0
$433$			1.85641i			0.0892132i	0.999005	+	0.0446066i	$0.0142034\pi$
$433$			1.85641i			0.0892132i	−0.999005	+	0.0446066i	$0.985797\pi$
$434$		0			0
$435$		−	13.8564i		−	0.664364i
$436$		0			0
$437$			6.92820i			0.331421i
$438$		0			0
$439$	4.00000			0.190910			0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000			0.190910			0.0954548	+	0.995434i	$0.469569\pi$
$440$		0			0
$441$	4.46410	+	30.9282i	0.212576	+	1.47277i
$442$		0			0
$443$		−	5.46410i		−	0.259607i	−0.991540	−	0.129804i	$-0.958565\pi$
$443$		−	5.46410i		−	0.259607i	0.991540	−	0.129804i	$-0.0414347\pi$
$444$		0			0
$445$	10.9282			0.518047
$446$		0			0
$447$	32.7846			1.55066
$448$		0			0
$449$	4.14359			0.195548			0.0977741	−	0.995209i	$-0.468828\pi$
$449$	4.14359			0.195548			0.0977741	+	0.995209i	$0.468828\pi$
$450$		0			0
$451$	16.0000			0.753411
$452$		0			0
$453$			30.9282i			1.45313i
$454$		0			0
$455$	6.92820	+	6.00000i	0.324799	+	0.281284i
$456$		0			0
$457$	−12.3923			−0.579688			−0.289844	−	0.957074i	$-0.593603\pi$
$457$	−12.3923			−0.579688			−0.289844	+	0.957074i	$0.593603\pi$
$458$		0			0
$459$			16.0000i			0.746816i
$460$		0			0
$461$			17.5167i			0.815832i	0.913019	+	0.407916i	$0.133744\pi$
$461$			17.5167i			0.815832i	−0.913019	+	0.407916i	$0.866256\pi$
$462$		0			0
$463$			27.4641i			1.27637i	0.769885	+	0.638183i	$0.220313\pi$
$463$			27.4641i			1.27637i	−0.769885	+	0.638183i	$0.779687\pi$
$464$		0			0
$465$			13.8564i			0.642575i
$466$		0			0
$467$	12.5885			0.582524			0.291262	−	0.956643i	$-0.405925\pi$
$467$	12.5885			0.582524			0.291262	+	0.956643i	$0.405925\pi$
$468$		0			0
$469$	−4.39230	+	5.07180i	−0.202818	+	0.234194i
$470$		0			0
$471$		−	52.6410i		−	2.42557i
$472$		0			0
$473$	13.8564			0.637118
$474$		0			0
$475$	−5.66025			−0.259710
$476$		0			0
$477$	30.9282			1.41611
$478$		0			0
$479$	−17.0718			−0.780030			−0.390015	−	0.920808i	$-0.627530\pi$
$479$	−17.0718			−0.780030			−0.390015	+	0.920808i	$0.627530\pi$
$480$		0			0
$481$			18.9282i			0.863052i
$482$		0			0
$483$	25.8564	−	29.8564i	1.17651	−	1.35851i
$484$		0			0
$485$	8.78461			0.398889
$486$		0			0
$487$			16.3923i			0.742806i	0.928472	+	0.371403i	$0.121123\pi$
$487$			16.3923i			0.742806i	−0.928472	+	0.371403i	$0.878877\pi$
$488$		0			0
$489$			44.7846i			2.02523i
$490$		0			0
$491$			19.3205i			0.871922i	0.899965	+	0.435961i	$0.143591\pi$
$491$			19.3205i			0.871922i	−0.899965	+	0.435961i	$0.856409\pi$
$492$		0			0
$493$			27.7128i			1.24812i
$494$		0			0
$495$	17.8564			0.802586
$496$		0			0
$497$	−7.85641	+	9.07180i	−0.352408	+	0.406926i
$498$		0			0
$499$		−	16.3923i		−	0.733820i	−0.930256	−	0.366910i	$-0.880416\pi$
$499$		−	16.3923i		−	0.733820i	0.930256	−	0.366910i	$-0.119584\pi$
$500$		0			0
$501$	13.8564			0.619059
$502$		0			0
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$	−4.53590			−0.201845
$506$		0			0
$507$	25.6603			1.13961
$508$		0			0
$509$		−	42.1962i		−	1.87031i	−0.354237	−	0.935156i	$-0.615259\pi$
$509$		−	42.1962i		−	1.87031i	0.354237	−	0.935156i	$-0.384741\pi$
$510$		0			0
$511$	12.0000	−	13.8564i	0.530849	−	0.612971i
$512$		0			0
$513$	5.07180			0.223925
$514$		0			0
$515$			13.8564i			0.610586i
$516$		0			0
$517$		−	37.8564i		−	1.66492i
$518$		0			0
$519$		−	8.92820i		−	0.391905i
$520$		0			0
$521$			13.0718i			0.572686i	0.958127	+	0.286343i	$0.0924397\pi$
$521$			13.0718i			0.572686i	−0.958127	+	0.286343i	$0.907560\pi$
$522$		0			0
$523$	−29.3731			−1.28439			−0.642197	−	0.766539i	$-0.721977\pi$
$523$	−29.3731			−1.28439			−0.642197	+	0.766539i	$0.721977\pi$
$524$		0			0
$525$	24.3923	+	21.1244i	1.06457	+	0.921942i
$526$		0			0
$527$		−	27.7128i		−	1.20719i
$528$		0			0
$529$	−6.85641			−0.298105
$530$		0			0
$531$	−16.9808			−0.736902
$532$		0			0
$533$	−13.8564			−0.600188
$534$		0			0
$535$	6.14359			0.265611
$536$		0			0
$537$		−	1.07180i		−	0.0462514i
$538$		0			0
$539$	−37.8564	+	5.46410i	−1.63059	+	0.235356i
$540$		0			0
$541$	−28.0000			−1.20381			−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000			−1.20381			−0.601907	+	0.798566i	$0.705592\pi$
$542$		0			0
$543$			45.7128i			1.96172i
$544$		0			0
$545$		−	5.07180i		−	0.217252i
$546$		0			0
$547$			21.4641i			0.917739i	0.888504	+	0.458869i	$0.151745\pi$
$547$			21.4641i			0.917739i	−0.888504	+	0.458869i	$0.848255\pi$
$548$		0			0
$549$			52.0526i			2.22155i
$550$		0			0
$551$	8.78461			0.374237
$552$		0			0
$553$	−6.00000	+	6.92820i	−0.255146	+	0.294617i
$554$		0			0
$555$		−	8.00000i		−	0.339581i
$556$		0			0
$557$	−30.9282			−1.31047			−0.655235	−	0.755425i	$-0.727430\pi$
$557$	−30.9282			−1.31047			−0.655235	+	0.755425i	$0.727430\pi$
$558$		0			0
$559$	−12.0000			−0.507546
$560$		0			0
$561$	−59.7128			−2.52108
$562$		0			0
$563$	−28.9808			−1.22139			−0.610697	−	0.791865i	$-0.709111\pi$
$563$	−28.9808			−1.22139			−0.610697	+	0.791865i	$0.709111\pi$
$564$		0			0
$565$			6.92820i			0.291472i
$566$		0			0
$567$	4.92820	+	4.26795i	0.206965	+	0.179237i
$568$		0			0
$569$	−28.3923			−1.19027			−0.595134	−	0.803627i	$-0.702901\pi$
$569$	−28.3923			−1.19027			−0.595134	+	0.803627i	$0.702901\pi$
$570$		0			0
$571$		−	30.2487i		−	1.26587i	−0.774205	−	0.632935i	$-0.781850\pi$
$571$		−	30.2487i		−	1.26587i	0.774205	−	0.632935i	$-0.218150\pi$
$572$		0			0
$573$		−	31.3205i		−	1.30843i
$574$		0			0
$575$		−	24.3923i		−	1.01723i
$576$		0			0
$577$			10.1436i			0.422283i	0.977455	+	0.211142i	$0.0677181\pi$
$577$			10.1436i			0.422283i	−0.977455	+	0.211142i	$0.932282\pi$
$578$		0			0
$579$	−12.0000			−0.498703
$580$		0			0
$581$	21.4641	+	18.5885i	0.890481	+	0.771179i
$582$		0			0
$583$			37.8564i			1.56785i
$584$		0			0
$585$	−15.4641			−0.639362
$586$		0			0
$587$	−3.80385			−0.157002			−0.0785008	−	0.996914i	$-0.525013\pi$
$587$	−3.80385			−0.157002			−0.0785008	+	0.996914i	$0.525013\pi$
$588$		0			0
$589$	−8.78461			−0.361964
$590$		0			0
$591$	−32.7846			−1.34858
$592$		0			0
$593$			19.7128i			0.809508i	0.914426	+	0.404754i	$0.132643\pi$
$593$			19.7128i			0.809508i	−0.914426	+	0.404754i	$0.867357\pi$
$594$		0			0
$595$	5.85641	+	5.07180i	0.240089	+	0.207923i
$596$		0			0
$597$	−45.8564			−1.87678
$598$		0			0
$599$			12.5359i			0.512203i	0.966650	+	0.256101i	$0.0824381\pi$
$599$			12.5359i			0.512203i	−0.966650	+	0.256101i	$0.917562\pi$
$600$		0			0
$601$			34.6410i			1.41304i	0.707695	+	0.706518i	$0.249735\pi$
$601$			34.6410i			1.41304i	−0.707695	+	0.706518i	$0.750265\pi$
$602$		0			0
$603$		−	11.3205i		−	0.461007i
$604$		0			0
$605$			13.8038i			0.561206i
$606$		0			0
$607$	−8.00000			−0.324710			−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000			−0.324710			−0.162355	+	0.986732i	$0.551909\pi$
$608$		0			0
$609$	−37.8564	−	32.7846i	−1.53402	−	1.32850i
$610$		0			0
$611$			32.7846i			1.32632i
$612$		0			0
$613$	25.8564			1.04433			0.522165	−	0.852844i	$-0.325124\pi$
$613$	25.8564			1.04433			0.522165	+	0.852844i	$0.325124\pi$
$614$		0			0
$615$	5.85641			0.236153
$616$		0			0
$617$	−4.39230			−0.176828			−0.0884138	−	0.996084i	$-0.528180\pi$
$617$	−4.39230			−0.176828			−0.0884138	+	0.996084i	$0.528180\pi$
$618$		0			0
$619$	−24.1962			−0.972525			−0.486263	−	0.873813i	$-0.661640\pi$
$619$	−24.1962			−0.972525			−0.486263	+	0.873813i	$0.661640\pi$
$620$		0			0
$621$			21.8564i			0.877067i
$622$		0			0
$623$	25.8564	−	29.8564i	1.03592	−	1.19617i
$624$		0			0
$625$	17.2487			0.689948
$626$		0			0
$627$			18.9282i			0.755920i
$628$		0			0
$629$			16.0000i			0.637962i
$630$		0			0
$631$			24.2487i			0.965326i	0.875806	+	0.482663i	$0.160330\pi$
$631$			24.2487i			0.965326i	−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$		−	20.7846i		−	0.826114i
$634$		0			0
$635$	15.7128			0.623544
$636$		0			0
$637$	32.7846	−	4.73205i	1.29897	−	0.187491i
$638$		0			0
$639$		−	20.2487i		−	0.801027i
$640$		0			0
$641$	−33.4641			−1.32175			−0.660876	−	0.750495i	$-0.729815\pi$
$641$	−33.4641			−1.32175			−0.660876	+	0.750495i	$0.729815\pi$
$642$		0			0
$643$	22.7321			0.896465			0.448232	−	0.893917i	$-0.352054\pi$
$643$	22.7321			0.896465			0.448232	+	0.893917i	$0.352054\pi$
$644$		0			0
$645$	5.07180			0.199702
$646$		0			0
$647$	−22.6410			−0.890110			−0.445055	−	0.895503i	$-0.646816\pi$
$647$	−22.6410			−0.890110			−0.445055	+	0.895503i	$0.646816\pi$
$648$		0			0
$649$		−	20.7846i		−	0.815867i
$650$		0			0
$651$	37.8564	+	32.7846i	1.48371	+	1.28493i
$652$		0			0
$653$	−12.0000			−0.469596			−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000			−0.469596			−0.234798	+	0.972044i	$0.575443\pi$
$654$		0			0
$655$			7.85641i			0.306975i
$656$		0			0
$657$			30.9282i			1.20662i
$658$		0			0
$659$			36.1051i			1.40646i	0.710965	+	0.703228i	$0.248259\pi$
$659$			36.1051i			1.40646i	−0.710965	+	0.703228i	$0.751741\pi$
$660$		0			0
$661$		−	14.1962i		−	0.552166i	−0.961134	−	0.276083i	$-0.910963\pi$
$661$		−	14.1962i		−	0.552166i	0.961134	−	0.276083i	$-0.0890365\pi$
$662$		0			0
$663$	51.7128			2.00836
$664$		0			0
$665$	1.60770	−	1.85641i	0.0623437	−	0.0719884i
$666$		0			0
$667$			37.8564i			1.46581i
$668$		0			0
$669$	21.8564			0.845017
$670$		0			0
$671$	−63.7128			−2.45961
$672$		0			0
$673$	30.0000			1.15642			0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000			1.15642			0.578208	+	0.815890i	$0.303752\pi$
$674$		0			0
$675$	−17.8564			−0.687293
$676$		0			0
$677$		−	18.8756i		−	0.725450i	−0.931896	−	0.362725i	$-0.881846\pi$
$677$		−	18.8756i		−	0.725450i	0.931896	−	0.362725i	$-0.118154\pi$
$678$		0			0
$679$	20.7846	−	24.0000i	0.797640	−	0.921035i
$680$		0			0
$681$	3.46410			0.132745
$682$		0			0
$683$		−	29.4641i		−	1.12741i	−0.825975	−	0.563706i	$-0.809375\pi$
$683$		−	29.4641i		−	1.12741i	0.825975	−	0.563706i	$-0.190625\pi$
$684$		0			0
$685$			14.5359i			0.555388i
$686$		0			0
$687$			0.928203i			0.0354132i
$688$		0			0
$689$		−	32.7846i		−	1.24899i
$690$		0			0
$691$	32.9808			1.25465			0.627324	−	0.778759i	$-0.284150\pi$
$691$	32.9808			1.25465			0.627324	+	0.778759i	$0.284150\pi$
$692$		0			0
$693$	42.2487	−	48.7846i	1.60490	−	1.85317i
$694$		0			0
$695$		−	3.07180i		−	0.116520i
$696$		0			0
$697$	−11.7128			−0.443654
$698$		0			0
$699$	−54.2487			−2.05187
$700$		0			0
$701$	−1.85641			−0.0701155			−0.0350578	−	0.999385i	$-0.511162\pi$
$701$	−1.85641			−0.0701155			−0.0350578	+	0.999385i	$0.511162\pi$
$702$		0			0
$703$	5.07180			0.191286
$704$		0			0
$705$		−	13.8564i		−	0.521862i
$706$		0			0
$707$	−10.7321	+	12.3923i	−0.403620	+	0.466061i
$708$		0			0
$709$	−30.9282			−1.16153			−0.580767	−	0.814070i	$-0.697247\pi$
$709$	−30.9282			−1.16153			−0.580767	+	0.814070i	$0.697247\pi$
$710$		0			0
$711$		−	15.4641i		−	0.579949i
$712$		0			0
$713$		−	37.8564i		−	1.41773i
$714$		0			0
$715$		−	18.9282i		−	0.707875i
$716$		0			0
$717$			1.07180i			0.0400270i
$718$		0			0
$719$	17.0718			0.636671			0.318335	−	0.947978i	$-0.396876\pi$
$719$	17.0718			0.636671			0.318335	+	0.947978i	$0.396876\pi$
$720$		0			0
$721$	37.8564	+	32.7846i	1.40985	+	1.22096i
$722$		0			0
$723$		−	32.7846i		−	1.21927i
$724$		0			0
$725$	−30.9282			−1.14864
$726$		0			0
$727$	−8.78461			−0.325803			−0.162902	−	0.986642i	$-0.552085\pi$
$727$	−8.78461			−0.325803			−0.162902	+	0.986642i	$0.552085\pi$
$728$		0			0
$729$	−43.7846			−1.62165
$730$		0			0
$731$	−10.1436			−0.375174
$732$		0			0
$733$		−	2.19615i		−	0.0811167i	−0.999177	−	0.0405584i	$-0.987086\pi$
$733$		−	2.19615i		−	0.0811167i	0.999177	−	0.0405584i	$-0.0129137\pi$
$734$		0			0
$735$	−13.8564	+	2.00000i	−0.511101	+	0.0737711i
$736$		0			0
$737$	13.8564			0.510407
$738$		0			0
$739$			11.3205i			0.416432i	0.978083	+	0.208216i	$0.0667656\pi$
$739$			11.3205i			0.416432i	−0.978083	+	0.208216i	$0.933234\pi$
$740$		0			0
$741$		−	16.3923i		−	0.602186i
$742$		0			0
$743$			10.5359i			0.386525i	0.981147	+	0.193262i	$0.0619068\pi$
$743$			10.5359i			0.386525i	−0.981147	+	0.193262i	$0.938093\pi$
$744$		0			0
$745$			8.78461i			0.321843i
$746$		0			0
$747$	−47.9090			−1.75290
$748$		0			0
$749$	14.5359	−	16.7846i	0.531130	−	0.613296i
$750$		0			0
$751$		−	26.5359i		−	0.968309i	−0.874983	−	0.484154i	$-0.839127\pi$
$751$		−	26.5359i		−	0.968309i	0.874983	−	0.484154i	$-0.160873\pi$
$752$		0			0
$753$	60.2487			2.19559
$754$		0			0
$755$	−8.28719			−0.301602
$756$		0			0
$757$	−29.5692			−1.07471			−0.537356	−	0.843356i	$-0.680577\pi$
$757$	−29.5692			−1.07471			−0.537356	+	0.843356i	$0.680577\pi$
$758$		0			0
$759$	−81.5692			−2.96078
$760$		0			0
$761$			16.7846i			0.608442i	0.952602	+	0.304221i	$0.0983961\pi$
$761$			16.7846i			0.608442i	−0.952602	+	0.304221i	$0.901604\pi$
$762$		0			0
$763$	−13.8564	−	12.0000i	−0.501636	−	0.434429i
$764$		0			0
$765$	−13.0718			−0.472612
$766$		0			0
$767$			18.0000i			0.649942i
$768$		0			0
$769$			1.85641i			0.0669437i	0.999440	+	0.0334719i	$0.0106564\pi$
$769$			1.85641i			0.0669437i	−0.999440	+	0.0334719i	$0.989344\pi$
$770$		0			0
$771$		−	59.7128i		−	2.15050i
$772$		0			0
$773$		−	36.4449i		−	1.31083i	−0.755269	−	0.655415i	$-0.772494\pi$
$773$		−	36.4449i		−	1.31083i	0.755269	−	0.655415i	$-0.227506\pi$
$774$		0			0
$775$	30.9282			1.11097
$776$		0			0
$777$	−21.8564	−	18.9282i	−0.784094	−	0.679046i
$778$		0			0
$779$			3.71281i			0.133025i
$780$		0			0
$781$	24.7846			0.886863
$782$		0			0
$783$	27.7128			0.990375
$784$		0			0
$785$	14.1051			0.503433
$786$		0			0
$787$	20.5885			0.733899			0.366950	−	0.930241i	$-0.380402\pi$
$787$	20.5885			0.733899			0.366950	+	0.930241i	$0.380402\pi$
$788$		0			0
$789$		−	31.3205i		−	1.11504i
$790$		0			0
$791$	18.9282	+	16.3923i	0.673009	+	0.582843i
$792$		0			0
$793$	55.1769			1.95939
$794$		0			0
$795$			13.8564i			0.491436i
$796$		0			0
$797$			41.1244i			1.45670i	0.685206	+	0.728350i	$0.259712\pi$
$797$			41.1244i			1.45670i	−0.685206	+	0.728350i	$0.740288\pi$
$798$		0			0
$799$			27.7128i			0.980409i
$800$		0			0
$801$			66.6410i			2.35464i
$802$		0			0
$803$	−37.8564			−1.33592
$804$		0			0
$805$	8.00000	+	6.92820i	0.281963	+	0.244187i
$806$		0			0
$807$		−	3.07180i		−	0.108132i
$808$		0			0
$809$	37.1769			1.30707			0.653535	−	0.756896i	$-0.273285\pi$
$809$	37.1769			1.30707			0.653535	+	0.756896i	$0.273285\pi$
$810$		0			0
$811$	49.3731			1.73372			0.866861	−	0.498549i	$-0.166134\pi$
$811$	49.3731			1.73372			0.866861	+	0.498549i	$0.166134\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−12.0000			−0.420342
$816$		0			0
$817$			3.21539i			0.112492i
$818$		0			0
$819$	−36.5885	+	42.2487i	−1.27850	+	1.47629i
$820$		0			0
$821$	−6.92820			−0.241796			−0.120898	−	0.992665i	$-0.538577\pi$
$821$	−6.92820			−0.241796			−0.120898	+	0.992665i	$0.538577\pi$
$822$		0			0
$823$		−	20.5359i		−	0.715836i	−0.933753	−	0.357918i	$-0.883487\pi$
$823$		−	20.5359i		−	0.715836i	0.933753	−	0.357918i	$-0.116513\pi$
$824$		0			0
$825$		−	66.6410i		−	2.32014i
$826$		0			0
$827$			43.3205i			1.50640i	0.657791	+	0.753201i	$0.271491\pi$
$827$			43.3205i			1.50640i	−0.657791	+	0.753201i	$0.728509\pi$
$828$		0			0
$829$		−	2.19615i		−	0.0762755i	−0.999272	−	0.0381378i	$-0.987857\pi$
$829$		−	2.19615i		−	0.0762755i	0.999272	−	0.0381378i	$-0.0121426\pi$
$830$		0			0
$831$	78.6410			2.72803
$832$		0			0
$833$	27.7128	−	4.00000i	0.960192	−	0.138592i
$834$		0			0
$835$			3.71281i			0.128487i
$836$		0			0
$837$	−27.7128			−0.957895
$838$		0			0
$839$	5.07180			0.175098			0.0875489	−	0.996160i	$-0.472097\pi$
$839$	5.07180			0.175098			0.0875489	+	0.996160i	$0.472097\pi$
$840$		0			0
$841$	19.0000			0.655172
$842$		0			0
$843$	16.3923			0.564581
$844$		0			0
$845$			6.87564i			0.236529i
$846$		0			0
$847$	37.7128	+	32.6603i	1.29583	+	1.12222i
$848$		0			0
$849$	−43.1769			−1.48183
$850$		0			0
$851$			21.8564i			0.749228i
$852$		0			0
$853$			51.3731i			1.75898i	0.475917	+	0.879490i	$0.342116\pi$
$853$			51.3731i			1.75898i	−0.475917	+	0.879490i	$0.657884\pi$
$854$		0			0
$855$			4.14359i			0.141708i
$856$		0			0
$857$		−	21.0718i		−	0.719799i	−0.932991	−	0.359899i	$-0.882811\pi$
$857$		−	21.0718i		−	0.719799i	0.932991	−	0.359899i	$-0.117189\pi$
$858$		0			0
$859$	19.5167			0.665900			0.332950	−	0.942945i	$-0.391956\pi$
$859$	19.5167			0.665900			0.332950	+	0.942945i	$0.391956\pi$
$860$		0			0
$861$	13.8564	−	16.0000i	0.472225	−	0.545279i
$862$		0			0
$863$			39.1769i			1.33360i	0.745238	+	0.666799i	$0.232336\pi$
$863$			39.1769i			1.33360i	−0.745238	+	0.666799i	$0.767664\pi$
$864$		0			0
$865$	2.39230			0.0813408
$866$		0			0
$867$	−2.73205			−0.0927853
$868$		0			0
$869$	18.9282			0.642095
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$			53.5692i			1.81304i
$874$		0			0
$875$	−12.0000	+	13.8564i	−0.405674	+	0.468432i
$876$		0			0
$877$	29.5692			0.998482			0.499241	−	0.866463i	$-0.333612\pi$
$877$	29.5692			0.998482			0.499241	+	0.866463i	$0.333612\pi$
$878$		0			0
$879$		−	4.92820i		−	0.166224i
$880$		0			0
$881$		−	35.7128i		−	1.20320i	−0.798799	−	0.601598i	$-0.794531\pi$
$881$		−	35.7128i		−	1.20320i	0.798799	−	0.601598i	$-0.205469\pi$
$882$		0			0
$883$			2.53590i			0.0853398i	0.999089	+	0.0426699i	$0.0135864\pi$
$883$			2.53590i			0.0853398i	−0.999089	+	0.0426699i	$0.986414\pi$
$884$		0			0
$885$		−	7.60770i		−	0.255730i
$886$		0			0
$887$	−29.0718			−0.976135			−0.488068	−	0.872806i	$-0.662298\pi$
$887$	−29.0718			−0.976135			−0.488068	+	0.872806i	$0.662298\pi$
$888$		0			0
$889$	37.1769	−	42.9282i	1.24687	−	1.43977i
$890$		0			0
$891$		−	13.4641i		−	0.451064i
$892$		0			0
$893$	8.78461			0.293966
$894$		0			0
$895$	0.287187			0.00959961
$896$		0			0
$897$	70.6410			2.35863
$898$		0			0
$899$	−48.0000			−1.60089
$900$		0			0
$901$		−	27.7128i		−	0.923248i
$902$		0			0
$903$	12.0000	−	13.8564i	0.399335	−	0.461112i
$904$		0			0
$905$	−12.2487			−0.407161
$906$		0			0
$907$			44.1051i			1.46449i	0.681043	+	0.732243i	$0.261527\pi$
$907$			44.1051i			1.46449i	−0.681043	+	0.732243i	$0.738473\pi$
$908$		0			0
$909$		−	27.6603i		−	0.917433i
$910$		0			0
$911$		−	32.3923i		−	1.07320i	−0.843835	−	0.536602i	$-0.819707\pi$
$911$		−	32.3923i		−	1.07320i	0.843835	−	0.536602i	$-0.180293\pi$
$912$		0			0
$913$		−	58.6410i		−	1.94073i
$914$		0			0
$915$	−23.3205			−0.770952
$916$		0			0
$917$	21.4641	+	18.5885i	0.708807	+	0.613845i
$918$		0			0
$919$			27.4641i			0.905957i	0.891521	+	0.452979i	$0.149639\pi$
$919$			27.4641i			0.905957i	−0.891521	+	0.452979i	$0.850361\pi$
$920$		0			0
$921$	−36.2487			−1.19444
$922$		0			0
$923$	−21.4641			−0.706500
$924$		0			0
$925$	−17.8564			−0.587115
$926$		0			0
$927$	−84.4974			−2.77526
$928$		0			0
$929$			4.00000i			0.131236i	0.997845	+	0.0656179i	$0.0209018\pi$
$929$			4.00000i			0.131236i	−0.997845	+	0.0656179i	$0.979098\pi$
$930$		0			0
$931$	−1.26795	−	8.78461i	−0.0415554	−	0.287904i
$932$		0			0
$933$	−70.6410			−2.31268
$934$		0			0
$935$		−	16.0000i		−	0.523256i
$936$		0			0
$937$		−	30.9282i		−	1.01038i	−0.863008	−	0.505190i	$-0.831422\pi$
$937$		−	30.9282i		−	1.01038i	0.863008	−	0.505190i	$-0.168578\pi$
$938$		0			0
$939$			89.5692i			2.92298i
$940$		0			0
$941$			17.1244i			0.558238i	0.960257	+	0.279119i	$0.0900424\pi$
$941$			17.1244i			0.558238i	−0.960257	+	0.279119i	$0.909958\pi$
$942$		0			0
$943$	−16.0000			−0.521032
$944$		0			0
$945$	5.07180	−	5.85641i	0.164986	−	0.190509i
$946$		0			0
$947$		−	13.4641i		−	0.437525i	−0.975778	−	0.218762i	$-0.929798\pi$
$947$		−	13.4641i		−	0.437525i	0.975778	−	0.218762i	$-0.0702019\pi$
$948$		0			0
$949$	32.7846			1.06423
$950$		0			0
$951$	−5.07180			−0.164464
$952$		0			0
$953$	33.7128			1.09207			0.546033	−	0.837764i	$-0.316137\pi$
$953$	33.7128			1.09207			0.546033	+	0.837764i	$0.316137\pi$
$954$		0			0
$955$	8.39230			0.271569
$956$		0			0
$957$			103.426i			3.34328i
$958$		0			0
$959$	39.7128	+	34.3923i	1.28239	+	1.11059i
$960$		0			0
$961$	17.0000			0.548387
$962$		0			0
$963$			37.4641i			1.20726i
$964$		0			0
$965$		−	3.21539i		−	0.103507i
$966$		0			0
$967$		−	11.3205i		−	0.364043i	−0.983295	−	0.182021i	$-0.941736\pi$
$967$		−	11.3205i		−	0.364043i	0.983295	−	0.182021i	$-0.0582640\pi$
$968$		0			0
$969$		−	13.8564i		−	0.445132i
$970$		0			0
$971$	−55.5167			−1.78161			−0.890807	−	0.454381i	$-0.849860\pi$
$971$	−55.5167			−1.78161			−0.890807	+	0.454381i	$0.849860\pi$
$972$		0			0
$973$	−8.39230	−	7.26795i	−0.269045	−	0.233000i
$974$		0			0
$975$			57.7128i			1.84829i
$976$		0			0
$977$	31.8564			1.01918			0.509588	−	0.860418i	$-0.329798\pi$
$977$	31.8564			1.01918			0.509588	+	0.860418i	$0.329798\pi$
$978$		0			0
$979$	−81.5692			−2.60696
$980$		0			0
$981$	30.9282			0.987462
$982$		0			0
$983$	−5.07180			−0.161765			−0.0808826	−	0.996724i	$-0.525774\pi$
$983$	−5.07180			−0.161765			−0.0808826	+	0.996724i	$0.525774\pi$
$984$		0			0
$985$		−	8.78461i		−	0.279901i
$986$		0			0
$987$	−37.8564	−	32.7846i	−1.20498	−	1.04355i
$988$		0			0
$989$	−13.8564			−0.440608
$990$		0			0
$991$		−	31.1769i		−	0.990367i	−0.868788	−	0.495184i	$-0.835101\pi$
$991$		−	31.1769i		−	0.990367i	0.868788	−	0.495184i	$-0.164899\pi$
$992$		0			0
$993$			30.9282i			0.981477i
$994$		0			0
$995$		−	12.2872i		−	0.389530i
$996$		0			0
$997$		−	2.87564i		−	0.0910726i	−0.998963	−	0.0455363i	$-0.985500\pi$
$997$		−	2.87564i		−	0.0910726i	0.998963	−	0.0455363i	$-0.0144997\pi$
$998$		0			0
$999$	16.0000			0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.f.a.1791.1		4
4.3	odd	2		1792.2.f.i.1791.3		4
7.6	odd	2		1792.2.f.i.1791.4		4
8.3	odd	2		1792.2.f.b.1791.2		4
8.5	even	2		1792.2.f.h.1791.4		4
16.3	odd	4		896.2.e.e.447.4	yes	4
16.5	even	4		896.2.e.b.447.4	yes	4
16.11	odd	4		896.2.e.a.447.1	✓	4
16.13	even	4		896.2.e.f.447.1	yes	4
28.27	even	2	inner	1792.2.f.a.1791.2		4
56.13	odd	2		1792.2.f.b.1791.1		4
56.27	even	2		1792.2.f.h.1791.3		4
112.13	odd	4		896.2.e.a.447.4	yes	4
112.27	even	4		896.2.e.f.447.4	yes	4
112.69	odd	4		896.2.e.e.447.1	yes	4
112.83	even	4		896.2.e.b.447.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.e.a.447.1	✓	4	16.11	odd	4
896.2.e.a.447.4	yes	4	112.13	odd	4
896.2.e.b.447.1	yes	4	112.83	even	4
896.2.e.b.447.4	yes	4	16.5	even	4
896.2.e.e.447.1	yes	4	112.69	odd	4
896.2.e.e.447.4	yes	4	16.3	odd	4
896.2.e.f.447.1	yes	4	16.13	even	4
896.2.e.f.447.4	yes	4	112.27	even	4
1792.2.f.a.1791.1		4	1.1	even	1	trivial
1792.2.f.a.1791.2		4	28.27	even	2	inner
1792.2.f.b.1791.1		4	56.13	odd	2
1792.2.f.b.1791.2		4	8.3	odd	2
1792.2.f.h.1791.3		4	56.27	even	2
1792.2.f.h.1791.4		4	8.5	even	2
1792.2.f.i.1791.3		4	4.3	odd	2
1792.2.f.i.1791.4		4	7.6	odd	2

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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -0.732051i q^{5} +(-2.00000 - 1.73205i) q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q-2.73205 q^{3} -0.732051i q^{5} +(-2.00000 - 1.73205i) q^{7} +4.46410 q^{9} +5.46410i q^{11} -4.73205i q^{13} +2.00000i q^{15} -4.00000i q^{17} -1.26795 q^{19} +(5.46410 + 4.73205i) q^{21} -5.46410i q^{23} +4.46410 q^{25} -4.00000 q^{27} -6.92820 q^{29} +6.92820 q^{31} -14.9282i q^{33} +(-1.26795 + 1.46410i) q^{35} -4.00000 q^{37} +12.9282i q^{39} -2.92820i q^{41} -2.53590i q^{43} -3.26795i q^{45} -6.92820 q^{47} +(1.00000 + 6.92820i) q^{49} +10.9282i q^{51} +6.92820 q^{53} +4.00000 q^{55} +3.46410 q^{57} -3.80385 q^{59} +11.6603i q^{61} +(-8.92820 - 7.73205i) q^{63} -3.46410 q^{65} -2.53590i q^{67} +14.9282i q^{69} -4.53590i q^{71} +6.92820i q^{73} -12.1962 q^{75} +(9.46410 - 10.9282i) q^{77} -3.46410i q^{79} -2.46410 q^{81} -10.7321 q^{83} -2.92820 q^{85} +18.9282 q^{87} +14.9282i q^{89} +(-8.19615 + 9.46410i) q^{91} -18.9282 q^{93} +0.928203i q^{95} +12.0000i q^{97} +24.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9} - 12 q^{19} + 8 q^{21} + 4 q^{25} - 16 q^{27} - 12 q^{35} - 16 q^{37} + 4 q^{49} + 16 q^{55} - 36 q^{59} - 8 q^{63} - 28 q^{75} + 24 q^{77} + 4 q^{81} - 36 q^{83} + 16 q^{85} + 48 q^{87} - 12 q^{91} - 48 q^{93}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9 - 12 * q^19 + 8 * q^21 + 4 * q^25 - 16 * q^27 - 12 * q^35 - 16 * q^37 + 4 * q^49 + 16 * q^55 - 36 * q^59 - 8 * q^63 - 28 * q^75 + 24 * q^77 + 4 * q^81 - 36 * q^83 + 16 * q^85 + 48 * q^87 - 12 * q^91 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

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Database

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