LMFDB - Embedded newform 3654.2.g.e.2899.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3654,2,Mod(2899,3654)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3654, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3654.2899"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3654.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,-2,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$29.1773368986$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1218)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3654.2899
Dual form			3654.2.g.e.2899.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{7} +1.00000i q^{8} -4.00000i q^{11} -2.00000 q^{13} +1.00000i q^{14} +1.00000 q^{16} +6.00000i q^{17} +4.00000i q^{19} -4.00000 q^{22} +4.00000 q^{23} -5.00000 q^{25} +2.00000i q^{26} +1.00000 q^{28} +(2.00000 - 5.00000i) q^{29} +2.00000i q^{31} -1.00000i q^{32} +6.00000 q^{34} -10.0000i q^{37} +4.00000 q^{38} +10.0000i q^{41} +4.00000i q^{44} -4.00000i q^{46} +4.00000i q^{47} +1.00000 q^{49} +5.00000i q^{50} +2.00000 q^{52} +12.0000 q^{53} -1.00000i q^{56} +(-5.00000 - 2.00000i) q^{58} +10.0000 q^{59} +14.0000i q^{61} +2.00000 q^{62} -1.00000 q^{64} -6.00000i q^{68} +4.00000i q^{73} -10.0000 q^{74} -4.00000i q^{76} +4.00000i q^{77} -10.0000i q^{79} +10.0000 q^{82} -6.00000 q^{83} +4.00000 q^{88} +2.00000i q^{89} +2.00000 q^{91} -4.00000 q^{92} +4.00000 q^{94} -8.00000i q^{97} -1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{7} - 4 q^{13} + 2 q^{16} - 8 q^{22} + 8 q^{23} - 10 q^{25} + 2 q^{28} + 4 q^{29} + 12 q^{34} + 8 q^{38} + 2 q^{49} + 4 q^{52} + 24 q^{53} - 10 q^{58} + 20 q^{59} + 4 q^{62} - 2 q^{64}+ \cdots + 8 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^7 - 4 * q^13 + 2 * q^16 - 8 * q^22 + 8 * q^23 - 10 * q^25 + 2 * q^28 + 4 * q^29 + 12 * q^34 + 8 * q^38 + 2 * q^49 + 4 * q^52 + 24 * q^53 - 10 * q^58 + 20 * q^59 + 4 * q^62 - 2 * q^64 - 20 * q^74 + 20 * q^82 - 12 * q^83 + 8 * q^88 + 4 * q^91 - 8 * q^92 + 8 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3654\mathbb{Z}\right)^\times$.

$n$	$379$	$407$	$2089$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$			1.00000i			0.353553i
$9$		0			0
$10$		0			0
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$			1.00000i			0.267261i
$15$		0			0
$16$	1.00000			0.250000
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		0			0
$22$	−4.00000			−0.852803
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$			2.00000i			0.392232i
$27$		0			0
$28$	1.00000			0.188982
$29$	2.00000	−	5.00000i	0.371391	−	0.928477i
$29$	2.00000	−	5.00000i	0.371391	−	0.928477i
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	6.00000			1.02899
$35$		0			0
$36$		0			0
$37$		−	10.0000i		−	1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$		−	10.0000i		−	1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	4.00000			0.648886
$39$		0			0
$40$		0			0
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$			4.00000i			0.603023i
$45$		0			0
$46$		−	4.00000i		−	0.589768i
$47$			4.00000i			0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$			4.00000i			0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$			5.00000i			0.707107i
$51$		0			0
$52$	2.00000			0.277350
$53$	12.0000			1.64833			0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000			1.64833			0.824163	+	0.566352i	$0.191646\pi$
$54$		0			0
$55$		0			0
$56$		−	1.00000i		−	0.133631i
$57$		0			0
$58$	−5.00000	−	2.00000i	−0.656532	−	0.262613i
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$		0			0
$61$			14.0000i			1.79252i	0.443533	+	0.896258i	$0.353725\pi$
$61$			14.0000i			1.79252i	−0.443533	+	0.896258i	$0.646275\pi$
$62$	2.00000			0.254000
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		−	6.00000i		−	0.727607i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−10.0000			−1.16248
$75$		0			0
$76$		−	4.00000i		−	0.458831i
$77$			4.00000i			0.455842i
$78$		0			0
$79$		−	10.0000i		−	1.12509i	−0.826767	−	0.562544i	$-0.809823\pi$
$79$		−	10.0000i		−	1.12509i	0.826767	−	0.562544i	$-0.190177\pi$
$80$		0			0
$81$		0			0
$82$	10.0000			1.10432
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	4.00000			0.426401
$89$			2.00000i			0.212000i	0.994366	+	0.106000i	$0.0338043\pi$
$89$			2.00000i			0.212000i	−0.994366	+	0.106000i	$0.966196\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$	−4.00000			−0.417029
$93$		0			0
$94$	4.00000			0.412568
$95$		0			0
$96$		0			0
$97$		−	8.00000i		−	0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$		−	8.00000i		−	0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$		−	1.00000i		−	0.101015i
$99$		0			0
$100$	5.00000			0.500000
$101$		−	18.0000i		−	1.79107i	−0.444994	−	0.895533i	$-0.646794\pi$
$101$		−	18.0000i		−	1.79107i	0.444994	−	0.895533i	$-0.353206\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		−	2.00000i		−	0.196116i
$105$		0			0
$106$		−	12.0000i		−	1.16554i
$107$	6.00000			0.580042			0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000			0.580042			0.290021	+	0.957020i	$0.406338\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$		0			0
$112$	−1.00000			−0.0944911
$113$			18.0000i			1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$			18.0000i			1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$		0			0
$115$		0			0
$116$	−2.00000	+	5.00000i	−0.185695	+	0.464238i
$117$		0			0
$118$		−	10.0000i		−	0.920575i
$119$		−	6.00000i		−	0.550019i
$120$		0			0
$121$	−5.00000			−0.454545
$122$	14.0000			1.26750
$123$		0			0
$124$		−	2.00000i		−	0.179605i
$125$		0			0
$126$		0			0
$127$			14.0000i			1.24230i	0.783692	+	0.621150i	$0.213334\pi$
$127$			14.0000i			1.24230i	−0.783692	+	0.621150i	$0.786666\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$		0			0
$131$		−	4.00000i		−	0.349482i	−0.984614	−	0.174741i	$-0.944091\pi$
$131$		−	4.00000i		−	0.349482i	0.984614	−	0.174741i	$-0.0559088\pi$
$132$		0			0
$133$		−	4.00000i		−	0.346844i
$134$		0			0
$135$		0			0
$136$	−6.00000			−0.514496
$137$		−	14.0000i		−	1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$137$		−	14.0000i		−	1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			8.00000i			0.668994i
$144$		0			0
$145$		0			0
$146$	4.00000			0.331042
$147$		0			0
$148$			10.0000i			0.821995i
$149$	16.0000			1.31077			0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000			1.31077			0.655386	+	0.755295i	$0.272506\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$	−4.00000			−0.324443
$153$		0			0
$154$	4.00000			0.322329
$155$		0			0
$156$		0			0
$157$			22.0000i			1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$			22.0000i			1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	−10.0000			−0.795557
$159$		0			0
$160$		0			0
$161$	−4.00000			−0.315244
$162$		0			0
$163$			8.00000i			0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$			8.00000i			0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$		−	10.0000i		−	0.780869i
$165$		0			0
$166$			6.00000i			0.465690i
$167$	20.0000			1.54765			0.773823	−	0.633402i	$-0.218342\pi$
$167$	20.0000			1.54765			0.773823	+	0.633402i	$0.218342\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.00000			0.608229			0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000			0.608229			0.304114	+	0.952636i	$0.401639\pi$
$174$		0			0
$175$	5.00000			0.377964
$176$		−	4.00000i		−	0.301511i
$177$		0			0
$178$	2.00000			0.149906
$179$	22.0000			1.64436			0.822179	−	0.569230i	$-0.192758\pi$
$179$	22.0000			1.64436			0.822179	+	0.569230i	$0.192758\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		−	2.00000i		−	0.148250i
$183$		0			0
$184$			4.00000i			0.294884i
$185$		0			0
$186$		0			0
$187$	24.0000			1.75505
$188$		−	4.00000i		−	0.291730i
$189$		0			0
$190$		0			0
$191$			12.0000i			0.868290i	0.900843	+	0.434145i	$0.142949\pi$
$191$			12.0000i			0.868290i	−0.900843	+	0.434145i	$0.857051\pi$
$192$		0			0
$193$			12.0000i			0.863779i	0.901927	+	0.431889i	$0.142153\pi$
$193$			12.0000i			0.863779i	−0.901927	+	0.431889i	$0.857847\pi$
$194$	−8.00000			−0.574367
$195$		0			0
$196$	−1.00000			−0.0714286
$197$	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
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		−	5.00000i		−	0.353553i
$201$		0			0
$202$	−18.0000			−1.26648
$203$	−2.00000	+	5.00000i	−0.140372	+	0.350931i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−2.00000			−0.138675
$209$	16.0000			1.10674
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−12.0000			−0.824163
$213$		0			0
$214$		−	6.00000i		−	0.410152i
$215$		0			0
$216$		0			0
$217$		−	2.00000i		−	0.135769i
$218$		−	10.0000i		−	0.677285i
$219$		0			0
$220$		0			0
$221$		−	12.0000i		−	0.807207i
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$			1.00000i			0.0668153i
$225$		0			0
$226$	18.0000			1.19734
$227$	−2.00000			−0.132745			−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000			−0.132745			−0.0663723	+	0.997795i	$0.521143\pi$
$228$		0			0
$229$			6.00000i			0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$			6.00000i			0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$		0			0
$231$		0			0
$232$	5.00000	+	2.00000i	0.328266	+	0.131306i
$233$	−26.0000			−1.70332			−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000			−1.70332			−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$		0			0
$236$	−10.0000			−0.650945
$237$		0			0
$238$	−6.00000			−0.388922
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−22.0000			−1.41714			−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000			−1.41714			−0.708572	+	0.705638i	$0.750660\pi$
$242$			5.00000i			0.321412i
$243$		0			0
$244$		−	14.0000i		−	0.896258i
$245$		0			0
$246$		0			0
$247$		−	8.00000i		−	0.509028i
$248$	−2.00000			−0.127000
$249$		0			0
$250$		0			0
$251$		−	12.0000i		−	0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$		−	12.0000i		−	0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$		0			0
$253$		−	16.0000i		−	1.00591i
$254$	14.0000			0.878438
$255$		0			0
$256$	1.00000			0.0625000
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$		0			0
$259$			10.0000i			0.621370i
$260$		0			0
$261$		0			0
$262$	−4.00000			−0.247121
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$		0			0
$266$	−4.00000			−0.245256
$267$		0			0
$268$		0			0
$269$		−	18.0000i		−	1.09748i	−0.835993	−	0.548740i	$-0.815108\pi$
$269$		−	18.0000i		−	1.09748i	0.835993	−	0.548740i	$-0.184892\pi$
$270$		0			0
$271$			26.0000i			1.57939i	0.613501	+	0.789694i	$0.289761\pi$
$271$			26.0000i			1.57939i	−0.613501	+	0.789694i	$0.710239\pi$
$272$			6.00000i			0.363803i
$273$		0			0
$274$	−14.0000			−0.845771
$275$			20.0000i			1.20605i
$276$		0			0
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$		−	8.00000i		−	0.479808i
$279$		0			0
$280$		0			0
$281$	30.0000			1.78965			0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000			1.78965			0.894825	+	0.446417i	$0.147300\pi$
$282$		0			0
$283$	24.0000			1.42665			0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000			1.42665			0.713326	+	0.700832i	$0.247188\pi$
$284$		0			0
$285$		0			0
$286$	8.00000			0.473050
$287$		−	10.0000i		−	0.590281i
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$		0			0
$292$		−	4.00000i		−	0.234082i
$293$		−	26.0000i		−	1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$		−	26.0000i		−	1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$		0			0
$295$		0			0
$296$	10.0000			0.581238
$297$		0			0
$298$		−	16.0000i		−	0.926855i
$299$	−8.00000			−0.462652
$300$		0			0
$301$		0			0
$302$		−	16.0000i		−	0.920697i
$303$		0			0
$304$			4.00000i			0.229416i
$305$		0			0
$306$		0			0
$307$		−	16.0000i		−	0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$		−	16.0000i		−	0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$		−	4.00000i		−	0.227921i
$309$		0			0
$310$		0			0
$311$		−	16.0000i		−	0.907277i	−0.891186	−	0.453638i	$-0.850126\pi$
$311$		−	16.0000i		−	0.907277i	0.891186	−	0.453638i	$-0.149874\pi$
$312$		0			0
$313$	−6.00000			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$314$	22.0000			1.24153
$315$		0			0
$316$			10.0000i			0.562544i
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$	−20.0000	−	8.00000i	−1.11979	−	0.447914i
$320$		0			0
$321$		0			0
$322$			4.00000i			0.222911i
$323$	−24.0000			−1.33540
$324$		0			0
$325$	10.0000			0.554700
$326$	8.00000			0.443079
$327$		0			0
$328$	−10.0000			−0.552158
$329$		−	4.00000i		−	0.220527i
$330$		0			0
$331$			20.0000i			1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$			20.0000i			1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	6.00000			0.329293
$333$		0			0
$334$		−	20.0000i		−	1.09435i
$335$		0			0
$336$		0			0
$337$		−	36.0000i		−	1.96104i	−0.196407	−	0.980522i	$-0.562927\pi$
$337$		−	36.0000i		−	1.96104i	0.196407	−	0.980522i	$-0.437073\pi$
$338$			9.00000i			0.489535i
$339$		0			0
$340$		0			0
$341$	8.00000			0.433224
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		−	8.00000i		−	0.430083i
$347$	22.0000			1.18102			0.590511	−	0.807030i	$-0.298926\pi$
$347$	22.0000			1.18102			0.590511	+	0.807030i	$0.298926\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		−	5.00000i		−	0.267261i
$351$		0			0
$352$	−4.00000			−0.213201
$353$	−14.0000			−0.745145			−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000			−0.745145			−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$		0			0
$356$		−	2.00000i		−	0.106000i
$357$		0			0
$358$		−	22.0000i		−	1.16274i
$359$			24.0000i			1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$			24.0000i			1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		−	2.00000i		−	0.105118i
$363$		0			0
$364$	−2.00000			−0.104828
$365$		0			0
$366$		0			0
$367$			34.0000i			1.77479i	0.461014	+	0.887393i	$0.347486\pi$
$367$			34.0000i			1.77479i	−0.461014	+	0.887393i	$0.652514\pi$
$368$	4.00000			0.208514
$369$		0			0
$370$		0			0
$371$	−12.0000			−0.623009
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		−	24.0000i		−	1.24101i
$375$		0			0
$376$	−4.00000			−0.206284
$377$	−4.00000	+	10.0000i	−0.206010	+	0.515026i
$378$		0			0
$379$			20.0000i			1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$			20.0000i			1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$		0			0
$381$		0			0
$382$	12.0000			0.613973
$383$	−32.0000			−1.63512			−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000			−1.63512			−0.817562	+	0.575841i	$0.804675\pi$
$384$		0			0
$385$		0			0
$386$	12.0000			0.610784
$387$		0			0
$388$			8.00000i			0.406138i
$389$			10.0000i			0.507020i	0.967333	+	0.253510i	$0.0815851\pi$
$389$			10.0000i			0.507020i	−0.967333	+	0.253510i	$0.918415\pi$
$390$		0			0
$391$			24.0000i			1.21373i
$392$			1.00000i			0.0505076i
$393$		0			0
$394$		−	12.0000i		−	0.604551i
$395$		0			0
$396$		0			0
$397$	−26.0000			−1.30490			−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000			−1.30490			−0.652451	+	0.757831i	$0.726259\pi$
$398$		0			0
$399$		0			0
$400$	−5.00000			−0.250000
$401$	10.0000			0.499376			0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000			0.499376			0.249688	+	0.968326i	$0.419672\pi$
$402$		0			0
$403$		−	4.00000i		−	0.199254i
$404$			18.0000i			0.895533i
$405$		0			0
$406$	5.00000	+	2.00000i	0.248146	+	0.0992583i
$407$	−40.0000			−1.98273
$408$		0			0
$409$		−	8.00000i		−	0.395575i	−0.980245	−	0.197787i	$-0.936624\pi$
$409$		−	8.00000i		−	0.395575i	0.980245	−	0.197787i	$-0.0633755\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−10.0000			−0.492068
$414$		0			0
$415$		0			0
$416$			2.00000i			0.0980581i
$417$		0			0
$418$		−	16.0000i		−	0.782586i
$419$	−30.0000			−1.46560			−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000			−1.46560			−0.732798	+	0.680446i	$0.761786\pi$
$420$		0			0
$421$		−	10.0000i		−	0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$		−	10.0000i		−	0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$		0			0
$423$		0			0
$424$			12.0000i			0.582772i
$425$		−	30.0000i		−	1.45521i
$426$		0			0
$427$		−	14.0000i		−	0.677507i
$428$	−6.00000			−0.290021
$429$		0			0
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$		−	12.0000i		−	0.576683i	−0.957528	−	0.288342i	$-0.906896\pi$
$433$		−	12.0000i		−	0.576683i	0.957528	−	0.288342i	$-0.0931039\pi$
$434$	−2.00000			−0.0960031
$435$		0			0
$436$	−10.0000			−0.478913
$437$			16.0000i			0.765384i
$438$		0			0
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$		0			0
$442$	−12.0000			−0.570782
$443$			20.0000i			0.950229i	0.879924	+	0.475114i	$0.157593\pi$
$443$			20.0000i			0.950229i	−0.879924	+	0.475114i	$0.842407\pi$
$444$		0			0
$445$		0			0
$446$		−	16.0000i		−	0.757622i
$447$		0			0
$448$	1.00000			0.0472456
$449$		−	10.0000i		−	0.471929i	−0.971762	−	0.235965i	$-0.924175\pi$
$449$		−	10.0000i		−	0.471929i	0.971762	−	0.235965i	$-0.0758249\pi$
$450$		0			0
$451$	40.0000			1.88353
$452$		−	18.0000i		−	0.846649i
$453$		0			0
$454$			2.00000i			0.0938647i
$455$		0			0
$456$		0			0
$457$	30.0000			1.40334			0.701670	−	0.712502i	$-0.252438\pi$
$457$	30.0000			1.40334			0.701670	+	0.712502i	$0.252438\pi$
$458$	6.00000			0.280362
$459$		0			0
$460$		0			0
$461$			18.0000i			0.838344i	0.907907	+	0.419172i	$0.137680\pi$
$461$			18.0000i			0.838344i	−0.907907	+	0.419172i	$0.862320\pi$
$462$		0			0
$463$	−32.0000			−1.48717			−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000			−1.48717			−0.743583	+	0.668644i	$0.766875\pi$
$464$	2.00000	−	5.00000i	0.0928477	−	0.232119i
$465$		0			0
$466$			26.0000i			1.20443i
$467$			36.0000i			1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$			36.0000i			1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$			10.0000i			0.460287i
$473$		0			0
$474$		0			0
$475$		−	20.0000i		−	0.917663i
$476$			6.00000i			0.275010i
$477$		0			0
$478$		0			0
$479$			36.0000i			1.64488i	0.568850	+	0.822441i	$0.307388\pi$
$479$			36.0000i			1.64488i	−0.568850	+	0.822441i	$0.692612\pi$
$480$		0			0
$481$			20.0000i			0.911922i
$482$			22.0000i			1.00207i
$483$		0			0
$484$	5.00000			0.227273
$485$		0			0
$486$		0			0
$487$	32.0000			1.45006			0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000			1.45006			0.725029	+	0.688718i	$0.241826\pi$
$488$	−14.0000			−0.633750
$489$		0			0
$490$		0			0
$491$			12.0000i			0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$			12.0000i			0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$		0			0
$493$	30.0000	+	12.0000i	1.35113	+	0.540453i
$494$	−8.00000			−0.359937
$495$		0			0
$496$			2.00000i			0.0898027i
$497$		0			0
$498$		0			0
$499$	−32.0000			−1.43252			−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000			−1.43252			−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$	−12.0000			−0.535586
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$	−16.0000			−0.711287
$507$		0			0
$508$		−	14.0000i		−	0.621150i
$509$	−20.0000			−0.886484			−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000			−0.886484			−0.443242	+	0.896402i	$0.646172\pi$
$510$		0			0
$511$		−	4.00000i		−	0.176950i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			6.00000i			0.264649i
$515$		0			0
$516$		0			0
$517$	16.0000			0.703679
$518$	10.0000			0.439375
$519$		0			0
$520$		0			0
$521$	14.0000			0.613351			0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000			0.613351			0.306676	+	0.951814i	$0.400783\pi$
$522$		0			0
$523$	−12.0000			−0.524723			−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000			−0.524723			−0.262362	+	0.964970i	$0.584501\pi$
$524$			4.00000i			0.174741i
$525$		0			0
$526$	−24.0000			−1.04645
$527$	−12.0000			−0.522728
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		0			0
$532$			4.00000i			0.173422i
$533$		−	20.0000i		−	0.866296i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−18.0000			−0.776035
$539$		−	4.00000i		−	0.172292i
$540$		0			0
$541$			6.00000i			0.257960i	0.991647	+	0.128980i	$0.0411703\pi$
$541$			6.00000i			0.257960i	−0.991647	+	0.128980i	$0.958830\pi$
$542$	26.0000			1.11680
$543$		0			0
$544$	6.00000			0.257248
$545$		0			0
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$			14.0000i			0.598050i
$549$		0			0
$550$	20.0000			0.852803
$551$	20.0000	+	8.00000i	0.852029	+	0.340811i
$552$		0			0
$553$			10.0000i			0.425243i
$554$			10.0000i			0.424859i
$555$		0			0
$556$	−8.00000			−0.339276
$557$	16.0000			0.677942			0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000			0.677942			0.338971	+	0.940797i	$0.389921\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		−	30.0000i		−	1.26547i
$563$		−	12.0000i		−	0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$		−	12.0000i		−	0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$		0			0
$565$		0			0
$566$		−	24.0000i		−	1.00880i
$567$		0			0
$568$		0			0
$569$			6.00000i			0.251533i	0.992060	+	0.125767i	$0.0401390\pi$
$569$			6.00000i			0.251533i	−0.992060	+	0.125767i	$0.959861\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		−	8.00000i		−	0.334497i
$573$		0			0
$574$	−10.0000			−0.417392
$575$	−20.0000			−0.834058
$576$		0			0
$577$			24.0000i			0.999133i	0.866276	+	0.499567i	$0.166507\pi$
$577$			24.0000i			0.999133i	−0.866276	+	0.499567i	$0.833493\pi$
$578$			19.0000i			0.790296i
$579$		0			0
$580$		0			0
$581$	6.00000			0.248922
$582$		0			0
$583$		−	48.0000i		−	1.98796i
$584$	−4.00000			−0.165521
$585$		0			0
$586$	−26.0000			−1.07405
$587$	−14.0000			−0.577842			−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000			−0.577842			−0.288921	+	0.957353i	$0.593296\pi$
$588$		0			0
$589$	−8.00000			−0.329634
$590$		0			0
$591$		0			0
$592$		−	10.0000i		−	0.410997i
$593$	−30.0000			−1.23195			−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000			−1.23195			−0.615976	+	0.787765i	$0.711238\pi$
$594$		0			0
$595$		0			0
$596$	−16.0000			−0.655386
$597$		0			0
$598$			8.00000i			0.327144i
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		−	40.0000i		−	1.63163i	−0.578310	−	0.815817i	$-0.696288\pi$
$601$		−	40.0000i		−	1.63163i	0.578310	−	0.815817i	$-0.303712\pi$
$602$		0			0
$603$		0			0
$604$	−16.0000			−0.651031
$605$		0			0
$606$		0			0
$607$			6.00000i			0.243532i	0.992559	+	0.121766i	$0.0388558\pi$
$607$			6.00000i			0.243532i	−0.992559	+	0.121766i	$0.961144\pi$
$608$	4.00000			0.162221
$609$		0			0
$610$		0			0
$611$		−	8.00000i		−	0.323645i
$612$		0			0
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$	−16.0000			−0.645707
$615$		0			0
$616$	−4.00000			−0.161165
$617$		−	6.00000i		−	0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$		−	6.00000i		−	0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$	−16.0000			−0.641542
$623$		−	2.00000i		−	0.0801283i
$624$		0			0
$625$	25.0000			1.00000
$626$			6.00000i			0.239808i
$627$		0			0
$628$		−	22.0000i		−	0.877896i
$629$	60.0000			2.39236
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$	10.0000			0.397779
$633$		0			0
$634$	6.00000			0.238290
$635$		0			0
$636$		0			0
$637$	−2.00000			−0.0792429
$638$	−8.00000	+	20.0000i	−0.316723	+	0.791808i
$639$		0			0
$640$		0			0
$641$		−	30.0000i		−	1.18493i	−0.805597	−	0.592464i	$-0.798155\pi$
$641$		−	30.0000i		−	1.18493i	0.805597	−	0.592464i	$-0.201845\pi$
$642$		0			0
$643$	−28.0000			−1.10421			−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000			−1.10421			−0.552106	+	0.833774i	$0.686176\pi$
$644$	4.00000			0.157622
$645$		0			0
$646$			24.0000i			0.944267i
$647$	−28.0000			−1.10079			−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000			−1.10079			−0.550397	+	0.834903i	$0.685524\pi$
$648$		0			0
$649$		−	40.0000i		−	1.57014i
$650$		−	10.0000i		−	0.392232i
$651$		0			0
$652$		−	8.00000i		−	0.313304i
$653$			34.0000i			1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$			34.0000i			1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$		0			0
$655$		0			0
$656$			10.0000i			0.390434i
$657$		0			0
$658$	−4.00000			−0.155936
$659$			44.0000i			1.71400i	0.515319	+	0.856998i	$0.327673\pi$
$659$			44.0000i			1.71400i	−0.515319	+	0.856998i	$0.672327\pi$
$660$		0			0
$661$	−14.0000			−0.544537			−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000			−0.544537			−0.272268	+	0.962221i	$0.587774\pi$
$662$	20.0000			0.777322
$663$		0			0
$664$		−	6.00000i		−	0.232845i
$665$		0			0
$666$		0			0
$667$	8.00000	−	20.0000i	0.309761	−	0.774403i
$668$	−20.0000			−0.773823
$669$		0			0
$670$		0			0
$671$	56.0000			2.16186
$672$		0			0
$673$	−2.00000			−0.0770943			−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000			−0.0770943			−0.0385472	+	0.999257i	$0.512273\pi$
$674$	−36.0000			−1.38667
$675$		0			0
$676$	9.00000			0.346154
$677$		−	6.00000i		−	0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$		−	6.00000i		−	0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$		0			0
$679$			8.00000i			0.307012i
$680$		0			0
$681$		0			0
$682$		−	8.00000i		−	0.306336i
$683$	−30.0000			−1.14792			−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000			−1.14792			−0.573959	+	0.818884i	$0.694593\pi$
$684$		0			0
$685$		0			0
$686$			1.00000i			0.0381802i
$687$		0			0
$688$		0			0
$689$	−24.0000			−0.914327
$690$		0			0
$691$	36.0000			1.36950			0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000			1.36950			0.684752	+	0.728776i	$0.259910\pi$
$692$	−8.00000			−0.304114
$693$		0			0
$694$		−	22.0000i		−	0.835109i
$695$		0			0
$696$		0			0
$697$	−60.0000			−2.27266
$698$		−	26.0000i		−	0.984115i
$699$		0			0
$700$	−5.00000			−0.188982
$701$	−12.0000			−0.453234			−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000			−0.453234			−0.226617	+	0.973984i	$0.572767\pi$
$702$		0			0
$703$	40.0000			1.50863
$704$			4.00000i			0.150756i
$705$		0			0
$706$			14.0000i			0.526897i
$707$			18.0000i			0.676960i
$708$		0			0
$709$	−18.0000			−0.676004			−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000			−0.676004			−0.338002	+	0.941145i	$0.609751\pi$
$710$		0			0
$711$		0			0
$712$	−2.00000			−0.0749532
$713$			8.00000i			0.299602i
$714$		0			0
$715$		0			0
$716$	−22.0000			−0.822179
$717$		0			0
$718$	24.0000			0.895672
$719$	−36.0000			−1.34257			−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000			−1.34257			−0.671287	+	0.741198i	$0.734258\pi$
$720$		0			0
$721$		0			0
$722$		−	3.00000i		−	0.111648i
$723$		0			0
$724$	−2.00000			−0.0743294
$725$	−10.0000	+	25.0000i	−0.371391	+	0.928477i
$726$		0			0
$727$			2.00000i			0.0741759i	0.999312	+	0.0370879i	$0.0118082\pi$
$727$			2.00000i			0.0741759i	−0.999312	+	0.0370879i	$0.988192\pi$
$728$			2.00000i			0.0741249i
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$			2.00000i			0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$			2.00000i			0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	34.0000			1.25496
$735$		0			0
$736$		−	4.00000i		−	0.147442i
$737$		0			0
$738$		0			0
$739$		−	40.0000i		−	1.47142i	−0.677295	−	0.735712i	$-0.736848\pi$
$739$		−	40.0000i		−	1.47142i	0.677295	−	0.735712i	$-0.263152\pi$
$740$		0			0
$741$		0			0
$742$			12.0000i			0.440534i
$743$			36.0000i			1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$			36.0000i			1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$		0			0
$745$		0			0
$746$			10.0000i			0.366126i
$747$		0			0
$748$	−24.0000			−0.877527
$749$	−6.00000			−0.219235
$750$		0			0
$751$			14.0000i			0.510867i	0.966827	+	0.255434i	$0.0822182\pi$
$751$			14.0000i			0.510867i	−0.966827	+	0.255434i	$0.917782\pi$
$752$			4.00000i			0.145865i
$753$		0			0
$754$	10.0000	+	4.00000i	0.364179	+	0.145671i
$755$		0			0
$756$		0			0
$757$			10.0000i			0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$			10.0000i			0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	20.0000			0.726433
$759$		0			0
$760$		0			0
$761$	38.0000			1.37750			0.688749	−	0.724999i	$-0.258160\pi$
$761$	38.0000			1.37750			0.688749	+	0.724999i	$0.258160\pi$
$762$		0			0
$763$	−10.0000			−0.362024
$764$		−	12.0000i		−	0.434145i
$765$		0			0
$766$			32.0000i			1.15621i
$767$	−20.0000			−0.722158
$768$		0			0
$769$			4.00000i			0.144244i	0.997396	+	0.0721218i	$0.0229770\pi$
$769$			4.00000i			0.144244i	−0.997396	+	0.0721218i	$0.977023\pi$
$770$		0			0
$771$		0			0
$772$		−	12.0000i		−	0.431889i
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$		0			0
$775$		−	10.0000i		−	0.359211i
$776$	8.00000			0.287183
$777$		0			0
$778$	10.0000			0.358517
$779$	−40.0000			−1.43315
$780$		0			0
$781$		0			0
$782$	24.0000			0.858238
$783$		0			0
$784$	1.00000			0.0357143
$785$		0			0
$786$		0			0
$787$	8.00000			0.285169			0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000			0.285169			0.142585	+	0.989783i	$0.454459\pi$
$788$	−12.0000			−0.427482
$789$		0			0
$790$		0			0
$791$		−	18.0000i		−	0.640006i
$792$		0			0
$793$		−	28.0000i		−	0.994309i
$794$			26.0000i			0.922705i
$795$		0			0
$796$		0			0
$797$			30.0000i			1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$			30.0000i			1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$			5.00000i			0.176777i
$801$		0			0
$802$		−	10.0000i		−	0.353112i
$803$	16.0000			0.564628
$804$		0			0
$805$		0			0
$806$	−4.00000			−0.140894
$807$		0			0
$808$	18.0000			0.633238
$809$			18.0000i			0.632846i	0.948618	+	0.316423i	$0.102482\pi$
$809$			18.0000i			0.632846i	−0.948618	+	0.316423i	$0.897518\pi$
$810$		0			0
$811$	−4.00000			−0.140459			−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000			−0.140459			−0.0702295	+	0.997531i	$0.522373\pi$
$812$	2.00000	−	5.00000i	0.0701862	−	0.175466i
$813$		0			0
$814$			40.0000i			1.40200i
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−8.00000			−0.279713
$819$		0			0
$820$		0			0
$821$	−20.0000			−0.698005			−0.349002	−	0.937122i	$-0.613479\pi$
$821$	−20.0000			−0.698005			−0.349002	+	0.937122i	$0.613479\pi$
$822$		0			0
$823$			54.0000i			1.88232i	0.337959	+	0.941161i	$0.390263\pi$
$823$			54.0000i			1.88232i	−0.337959	+	0.941161i	$0.609737\pi$
$824$		0			0
$825$		0			0
$826$			10.0000i			0.347945i
$827$			20.0000i			0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$			20.0000i			0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$		0			0
$829$		−	34.0000i		−	1.18087i	−0.807086	−	0.590434i	$-0.798956\pi$
$829$		−	34.0000i		−	1.18087i	0.807086	−	0.590434i	$-0.201044\pi$
$830$		0			0
$831$		0			0
$832$	2.00000			0.0693375
$833$			6.00000i			0.207888i
$834$		0			0
$835$		0			0
$836$	−16.0000			−0.553372
$837$		0			0
$838$			30.0000i			1.03633i
$839$		−	8.00000i		−	0.276191i	−0.990419	−	0.138095i	$-0.955902\pi$
$839$		−	8.00000i		−	0.276191i	0.990419	−	0.138095i	$-0.0440980\pi$
$840$		0			0
$841$	−21.0000	−	20.0000i	−0.724138	−	0.689655i
$842$	−10.0000			−0.344623
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	5.00000			0.171802
$848$	12.0000			0.412082
$849$		0			0
$850$	−30.0000			−1.02899
$851$		−	40.0000i		−	1.37118i
$852$		0			0
$853$			26.0000i			0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$			26.0000i			0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	−14.0000			−0.479070
$855$		0			0
$856$			6.00000i			0.205076i
$857$	30.0000			1.02478			0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000			1.02478			0.512390	+	0.858753i	$0.328760\pi$
$858$		0			0
$859$			40.0000i			1.36478i	0.730987	+	0.682391i	$0.239060\pi$
$859$			40.0000i			1.36478i	−0.730987	+	0.682391i	$0.760940\pi$
$860$		0			0
$861$		0			0
$862$		−	12.0000i		−	0.408722i
$863$	36.0000			1.22545			0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000			1.22545			0.612727	+	0.790295i	$0.290072\pi$
$864$		0			0
$865$		0			0
$866$	−12.0000			−0.407777
$867$		0			0
$868$			2.00000i			0.0678844i
$869$	−40.0000			−1.35691
$870$		0			0
$871$		0			0
$872$			10.0000i			0.338643i
$873$		0			0
$874$	16.0000			0.541208
$875$		0			0
$876$		0			0
$877$	−38.0000			−1.28317			−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000			−1.28317			−0.641584	+	0.767052i	$0.721723\pi$
$878$			8.00000i			0.269987i
$879$		0			0
$880$		0			0
$881$		−	54.0000i		−	1.81931i	−0.415369	−	0.909653i	$-0.636347\pi$
$881$		−	54.0000i		−	1.81931i	0.415369	−	0.909653i	$-0.363653\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$			12.0000i			0.403604i
$885$		0			0
$886$	20.0000			0.671913
$887$			20.0000i			0.671534i	0.941945	+	0.335767i	$0.108996\pi$
$887$			20.0000i			0.671534i	−0.941945	+	0.335767i	$0.891004\pi$
$888$		0			0
$889$		−	14.0000i		−	0.469545i
$890$		0			0
$891$		0			0
$892$	−16.0000			−0.535720
$893$	−16.0000			−0.535420
$894$		0			0
$895$		0			0
$896$		−	1.00000i		−	0.0334077i
$897$		0			0
$898$	−10.0000			−0.333704
$899$	10.0000	+	4.00000i	0.333519	+	0.133407i
$900$		0			0
$901$			72.0000i			2.39867i
$902$		−	40.0000i		−	1.33185i
$903$		0			0
$904$	−18.0000			−0.598671
$905$		0			0
$906$		0			0
$907$		−	20.0000i		−	0.664089i	−0.943264	−	0.332045i	$-0.892262\pi$
$907$		−	20.0000i		−	0.664089i	0.943264	−	0.332045i	$-0.107738\pi$
$908$	2.00000			0.0663723
$909$		0			0
$910$		0			0
$911$		−	20.0000i		−	0.662630i	−0.943520	−	0.331315i	$-0.892508\pi$
$911$		−	20.0000i		−	0.662630i	0.943520	−	0.331315i	$-0.107492\pi$
$912$		0			0
$913$			24.0000i			0.794284i
$914$		−	30.0000i		−	0.992312i
$915$		0			0
$916$		−	6.00000i		−	0.198246i
$917$			4.00000i			0.132092i
$918$		0			0
$919$	−56.0000			−1.84727			−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000			−1.84727			−0.923635	+	0.383274i	$0.874797\pi$
$920$		0			0
$921$		0			0
$922$	18.0000			0.592798
$923$		0			0
$924$		0			0
$925$			50.0000i			1.64399i
$926$			32.0000i			1.05159i
$927$		0			0
$928$	−5.00000	−	2.00000i	−0.164133	−	0.0656532i
$929$	6.00000			0.196854			0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000			0.196854			0.0984268	+	0.995144i	$0.468619\pi$
$930$		0			0
$931$			4.00000i			0.131095i
$932$	26.0000			0.851658
$933$		0			0
$934$	36.0000			1.17796
$935$		0			0
$936$		0			0
$937$	50.0000			1.63343			0.816714	−	0.577042i	$-0.195793\pi$
$937$	50.0000			1.63343			0.816714	+	0.577042i	$0.195793\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−40.0000			−1.30396			−0.651981	−	0.758235i	$-0.726062\pi$
$941$	−40.0000			−1.30396			−0.651981	+	0.758235i	$0.726062\pi$
$942$		0			0
$943$			40.0000i			1.30258i
$944$	10.0000			0.325472
$945$		0			0
$946$		0			0
$947$		−	36.0000i		−	1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$		−	36.0000i		−	1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$		0			0
$949$		−	8.00000i		−	0.259691i
$950$	−20.0000			−0.648886
$951$		0			0
$952$	6.00000			0.194461
$953$	−46.0000			−1.49009			−0.745043	−	0.667016i	$-0.767571\pi$
$953$	−46.0000			−1.49009			−0.745043	+	0.667016i	$0.767571\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$	36.0000			1.16311
$959$			14.0000i			0.452084i
$960$		0			0
$961$	27.0000			0.870968
$962$	20.0000			0.644826
$963$		0			0
$964$	22.0000			0.708572
$965$		0			0
$966$		0			0
$967$		−	6.00000i		−	0.192947i	−0.995336	−	0.0964735i	$-0.969244\pi$
$967$		−	6.00000i		−	0.192947i	0.995336	−	0.0964735i	$-0.0307563\pi$
$968$		−	5.00000i		−	0.160706i
$969$		0			0
$970$		0			0
$971$			20.0000i			0.641831i	0.947108	+	0.320915i	$0.103990\pi$
$971$			20.0000i			0.641831i	−0.947108	+	0.320915i	$0.896010\pi$
$972$		0			0
$973$	−8.00000			−0.256468
$974$		−	32.0000i		−	1.02535i
$975$		0			0
$976$			14.0000i			0.448129i
$977$	14.0000			0.447900			0.223950	−	0.974601i	$-0.428105\pi$
$977$	14.0000			0.447900			0.223950	+	0.974601i	$0.428105\pi$
$978$		0			0
$979$	8.00000			0.255681
$980$		0			0
$981$		0			0
$982$	12.0000			0.382935
$983$			48.0000i			1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$			48.0000i			1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$		0			0
$985$		0			0
$986$	12.0000	−	30.0000i	0.382158	−	0.955395i
$987$		0			0
$988$			8.00000i			0.254514i
$989$		0			0
$990$		0			0
$991$	56.0000			1.77890			0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000			1.77890			0.889449	+	0.457034i	$0.151088\pi$
$992$	2.00000			0.0635001
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	26.0000i		−	0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$		−	26.0000i		−	0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$			32.0000i			1.01294i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3654.2.g.e.2899.1		2
3.2	odd	2		1218.2.g.b.463.2	yes	2
29.28	even	2	inner	3654.2.g.e.2899.2		2
87.86	odd	2		1218.2.g.b.463.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1218.2.g.b.463.1	✓	2	87.86	odd	2
1218.2.g.b.463.2	yes	2	3.2	odd	2
3654.2.g.e.2899.1		2	1.1	even	1	trivial
3654.2.g.e.2899.2		2	29.28	even	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3654.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{7} +1.00000i q^{8} -4.00000i q^{11} -2.00000 q^{13} +1.00000i q^{14} +1.00000 q^{16} +6.00000i q^{17} +4.00000i q^{19} -4.00000 q^{22} +4.00000 q^{23} -5.00000 q^{25} +2.00000i q^{26} +1.00000 q^{28} +(2.00000 - 5.00000i) q^{29} +2.00000i q^{31} -1.00000i q^{32} +6.00000 q^{34} -10.0000i q^{37} +4.00000 q^{38} +10.0000i q^{41} +4.00000i q^{44} -4.00000i q^{46} +4.00000i q^{47} +1.00000 q^{49} +5.00000i q^{50} +2.00000 q^{52} +12.0000 q^{53} -1.00000i q^{56} +(-5.00000 - 2.00000i) q^{58} +10.0000 q^{59} +14.0000i q^{61} +2.00000 q^{62} -1.00000 q^{64} -6.00000i q^{68} +4.00000i q^{73} -10.0000 q^{74} -4.00000i q^{76} +4.00000i q^{77} -10.0000i q^{79} +10.0000 q^{82} -6.00000 q^{83} +4.00000 q^{88} +2.00000i q^{89} +2.00000 q^{91} -4.00000 q^{92} +4.00000 q^{94} -8.00000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{7} - 4 q^{13} + 2 q^{16} - 8 q^{22} + 8 q^{23} - 10 q^{25} + 2 q^{28} + 4 q^{29} + 12 q^{34} + 8 q^{38} + 2 q^{49} + 4 q^{52} + 24 q^{53} - 10 q^{58} + 20 q^{59} + 4 q^{62} - 2 q^{64}+ \cdots + 8 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^7 - 4 * q^13 + 2 * q^16 - 8 * q^22 + 8 * q^23 - 10 * q^25 + 2 * q^28 + 4 * q^29 + 12 * q^34 + 8 * q^38 + 2 * q^49 + 4 * q^52 + 24 * q^53 - 10 * q^58 + 20 * q^59 + 4 * q^62 - 2 * q^64 - 20 * q^74 + 20 * q^82 - 12 * q^83 + 8 * q^88 + 4 * q^91 - 8 * q^92 + 8 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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