LMFDB - Embedded newform 3654.2.g.b.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,-4,0,-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:	yes
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.b.2899.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} +1.00000i q^{8} +2.00000i q^{10} -4.00000i q^{11} +2.00000 q^{13} +1.00000i q^{14} +1.00000 q^{16} +2.00000i q^{17} -4.00000i q^{19} +2.00000 q^{20} -4.00000 q^{22} +6.00000 q^{23} -1.00000 q^{25} -2.00000i q^{26} +1.00000 q^{28} +(2.00000 - 5.00000i) q^{29} +4.00000i q^{31} -1.00000i q^{32} +2.00000 q^{34} +2.00000 q^{35} +4.00000i q^{37} -4.00000 q^{38} -2.00000i q^{40} -2.00000i q^{41} +6.00000i q^{43} +4.00000i q^{44} -6.00000i q^{46} -12.0000i q^{47} +1.00000 q^{49} +1.00000i q^{50} -2.00000 q^{52} +8.00000 q^{53} +8.00000i q^{55} -1.00000i q^{56} +(-5.00000 - 2.00000i) q^{58} -8.00000 q^{59} -2.00000i q^{61} +4.00000 q^{62} -1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} -2.00000i q^{68} -2.00000i q^{70} -2.00000 q^{71} -10.0000i q^{73} +4.00000 q^{74} +4.00000i q^{76} +4.00000i q^{77} +10.0000i q^{79} -2.00000 q^{80} -2.00000 q^{82} -16.0000 q^{83} -4.00000i q^{85} +6.00000 q^{86} +4.00000 q^{88} -2.00000i q^{89} -2.00000 q^{91} -6.00000 q^{92} -12.0000 q^{94} +8.00000i q^{95} -10.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{5} - 2 q^{7} + 4 q^{13} + 2 q^{16} + 4 q^{20} - 8 q^{22} + 12 q^{23} - 2 q^{25} + 2 q^{28} + 4 q^{29} + 4 q^{34} + 4 q^{35} - 8 q^{38} + 2 q^{49} - 4 q^{52} + 16 q^{53} - 10 q^{58}+ \cdots - 24 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^5 - 2 * q^7 + 4 * q^13 + 2 * q^16 + 4 * q^20 - 8 * q^22 + 12 * q^23 - 2 * q^25 + 2 * q^28 + 4 * q^29 + 4 * q^34 + 4 * q^35 - 8 * q^38 + 2 * q^49 - 4 * q^52 + 16 * q^53 - 10 * q^58 - 16 * q^59 + 8 * q^62 - 2 * q^64 - 8 * q^65 - 8 * q^67 - 4 * q^71 + 8 * q^74 - 4 * q^80 - 4 * q^82 - 32 * q^83 + 12 * q^86 + 8 * q^88 - 4 * q^91 - 12 * q^92 - 24 * 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
$5$	−2.00000			−0.894427			−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000			−0.894427			−0.447214	+	0.894427i	$0.647584\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$			1.00000i			0.353553i
$9$		0			0
$10$			2.00000i			0.632456i
$11$		−	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	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$			1.00000i			0.267261i
$15$		0			0
$16$	1.00000			0.250000
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	2.00000			0.447214
$21$		0			0
$22$	−4.00000			−0.852803
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		−	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
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	2.00000			0.342997
$35$	2.00000			0.338062
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	−4.00000			−0.648886
$39$		0			0
$40$		−	2.00000i		−	0.316228i
$41$		−	2.00000i		−	0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$		−	2.00000i		−	0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$			4.00000i			0.603023i
$45$		0			0
$46$		−	6.00000i		−	0.884652i
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$			1.00000i			0.141421i
$51$		0			0
$52$	−2.00000			−0.277350
$53$	8.00000			1.09888			0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000			1.09888			0.549442	+	0.835532i	$0.314840\pi$
$54$		0			0
$55$			8.00000i			1.07872i
$56$		−	1.00000i		−	0.133631i
$57$		0			0
$58$	−5.00000	−	2.00000i	−0.656532	−	0.262613i
$59$	−8.00000			−1.04151			−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000			−1.04151			−0.520756	+	0.853706i	$0.674350\pi$
$60$		0			0
$61$		−	2.00000i		−	0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$61$		−	2.00000i		−	0.256074i	0.991769	−	0.128037i	$-0.0408676\pi$
$62$	4.00000			0.508001
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−4.00000			−0.496139
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$		−	2.00000i		−	0.242536i
$69$		0			0
$70$		−	2.00000i		−	0.239046i
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$		−	10.0000i		−	1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$		−	10.0000i		−	1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	4.00000			0.464991
$75$		0			0
$76$			4.00000i			0.458831i
$77$			4.00000i			0.455842i
$78$		0			0
$79$			10.0000i			1.12509i	0.826767	+	0.562544i	$0.190177\pi$
$79$			10.0000i			1.12509i	−0.826767	+	0.562544i	$0.809823\pi$
$80$	−2.00000			−0.223607
$81$		0			0
$82$	−2.00000			−0.220863
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$		−	4.00000i		−	0.433861i
$86$	6.00000			0.646997
$87$		0			0
$88$	4.00000			0.426401
$89$		−	2.00000i		−	0.212000i	−0.994366	−	0.106000i	$-0.966196\pi$
$89$		−	2.00000i		−	0.212000i	0.994366	−	0.106000i	$-0.0338043\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$	−6.00000			−0.625543
$93$		0			0
$94$	−12.0000			−1.23771
$95$			8.00000i			0.820783i
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$		−	1.00000i		−	0.101015i
$99$		0			0
$100$	1.00000			0.100000
$101$		−	6.00000i		−	0.597022i	−0.954406	−	0.298511i	$-0.903510\pi$
$101$		−	6.00000i		−	0.597022i	0.954406	−	0.298511i	$-0.0964900\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$			2.00000i			0.196116i
$105$		0			0
$106$		−	8.00000i		−	0.777029i
$107$	−14.0000			−1.35343			−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000			−1.35343			−0.676716	+	0.736245i	$0.736597\pi$
$108$		0			0
$109$	−18.0000			−1.72409			−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000			−1.72409			−0.862044	+	0.506834i	$0.830816\pi$
$110$	8.00000			0.762770
$111$		0			0
$112$	−1.00000			−0.0944911
$113$		−	2.00000i		−	0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$		−	2.00000i		−	0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$		0			0
$115$	−12.0000			−1.11901
$116$	−2.00000	+	5.00000i	−0.185695	+	0.464238i
$117$		0			0
$118$			8.00000i			0.736460i
$119$		−	2.00000i		−	0.183340i
$120$		0			0
$121$	−5.00000			−0.454545
$122$	−2.00000			−0.181071
$123$		0			0
$124$		−	4.00000i		−	0.359211i
$125$	12.0000			1.07331
$126$		0			0
$127$			18.0000i			1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$			18.0000i			1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$			4.00000i			0.350823i
$131$		−	12.0000i		−	1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$		−	12.0000i		−	1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$		0			0
$133$			4.00000i			0.346844i
$134$			4.00000i			0.345547i
$135$		0			0
$136$	−2.00000			−0.171499
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$	−2.00000			−0.169031
$141$		0			0
$142$			2.00000i			0.167836i
$143$		−	8.00000i		−	0.668994i
$144$		0			0
$145$	−4.00000	+	10.0000i	−0.332182	+	0.830455i
$146$	−10.0000			−0.827606
$147$		0			0
$148$		−	4.00000i		−	0.328798i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000			0.324443
$153$		0			0
$154$	4.00000			0.322329
$155$		−	8.00000i		−	0.642575i
$156$		0			0
$157$			18.0000i			1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$			18.0000i			1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	10.0000			0.795557
$159$		0			0
$160$			2.00000i			0.158114i
$161$	−6.00000			−0.472866
$162$		0			0
$163$		−	10.0000i		−	0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$		−	10.0000i		−	0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$			2.00000i			0.156174i
$165$		0			0
$166$			16.0000i			1.24184i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$	−4.00000			−0.306786
$171$		0			0
$172$		−	6.00000i		−	0.457496i
$173$	−18.0000			−1.36851			−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000			−1.36851			−0.684257	+	0.729241i	$0.739873\pi$
$174$		0			0
$175$	1.00000			0.0755929
$176$		−	4.00000i		−	0.301511i
$177$		0			0
$178$	−2.00000			−0.149906
$179$	18.0000			1.34538			0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000			1.34538			0.672692	+	0.739923i	$0.265138\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$			2.00000i			0.148250i
$183$		0			0
$184$			6.00000i			0.442326i
$185$		−	8.00000i		−	0.588172i
$186$		0			0
$187$	8.00000			0.585018
$188$			12.0000i			0.875190i
$189$		0			0
$190$	8.00000			0.580381
$191$		−	24.0000i		−	1.73658i	−0.496058	−	0.868290i	$-0.665220\pi$
$191$		−	24.0000i		−	1.73658i	0.496058	−	0.868290i	$-0.334780\pi$
$192$		0			0
$193$			8.00000i			0.575853i	0.957653	+	0.287926i	$0.0929658\pi$
$193$			8.00000i			0.575853i	−0.957653	+	0.287926i	$0.907034\pi$
$194$	−10.0000			−0.717958
$195$		0			0
$196$	−1.00000			−0.0714286
$197$	−8.00000			−0.569976			−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000			−0.569976			−0.284988	+	0.958531i	$0.591990\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		−	1.00000i		−	0.0707107i
$201$		0			0
$202$	−6.00000			−0.422159
$203$	−2.00000	+	5.00000i	−0.140372	+	0.350931i
$204$		0			0
$205$			4.00000i			0.279372i
$206$		−	4.00000i		−	0.278693i
$207$		0			0
$208$	2.00000			0.138675
$209$	−16.0000			−1.10674
$210$		0			0
$211$			10.0000i			0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$			10.0000i			0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$	−8.00000			−0.549442
$213$		0			0
$214$			14.0000i			0.957020i
$215$		−	12.0000i		−	0.818393i
$216$		0			0
$217$		−	4.00000i		−	0.271538i
$218$			18.0000i			1.21911i
$219$		0			0
$220$		−	8.00000i		−	0.539360i
$221$			4.00000i			0.269069i
$222$		0			0
$223$	−12.0000			−0.803579			−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000			−0.803579			−0.401790	+	0.915732i	$0.631612\pi$
$224$			1.00000i			0.0668153i
$225$		0			0
$226$	−2.00000			−0.133038
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		−	18.0000i		−	1.18947i	−0.803921	−	0.594737i	$-0.797256\pi$
$229$		−	18.0000i		−	1.18947i	0.803921	−	0.594737i	$-0.202744\pi$
$230$			12.0000i			0.791257i
$231$		0			0
$232$	5.00000	+	2.00000i	0.328266	+	0.131306i
$233$	20.0000			1.31024			0.655122	−	0.755523i	$-0.272617\pi$
$233$	20.0000			1.31024			0.655122	+	0.755523i	$0.272617\pi$
$234$		0			0
$235$			24.0000i			1.56559i
$236$	8.00000			0.520756
$237$		0			0
$238$	−2.00000			−0.129641
$239$	−30.0000			−1.94054			−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000			−1.94054			−0.970269	+	0.242028i	$0.922188\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$			5.00000i			0.321412i
$243$		0			0
$244$			2.00000i			0.128037i
$245$	−2.00000			−0.127775
$246$		0			0
$247$		−	8.00000i		−	0.509028i
$248$	−4.00000			−0.254000
$249$		0			0
$250$		−	12.0000i		−	0.758947i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		−	24.0000i		−	1.50887i
$254$	18.0000			1.12942
$255$		0			0
$256$	1.00000			0.0625000
$257$	2.00000			0.124757			0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000			0.124757			0.0623783	+	0.998053i	$0.480131\pi$
$258$		0			0
$259$		−	4.00000i		−	0.248548i
$260$	4.00000			0.248069
$261$		0			0
$262$	−12.0000			−0.741362
$263$		−	8.00000i		−	0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$		−	8.00000i		−	0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$		0			0
$265$	−16.0000			−0.982872
$266$	4.00000			0.245256
$267$		0			0
$268$	4.00000			0.244339
$269$			14.0000i			0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$			14.0000i			0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$		0			0
$271$		−	16.0000i		−	0.971931i	−0.873978	−	0.485965i	$-0.838468\pi$
$271$		−	16.0000i		−	0.971931i	0.873978	−	0.485965i	$-0.161532\pi$
$272$			2.00000i			0.121268i
$273$		0			0
$274$	6.00000			0.362473
$275$			4.00000i			0.241209i
$276$		0			0
$277$	−6.00000			−0.360505			−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000			−0.360505			−0.180253	+	0.983620i	$0.557691\pi$
$278$			20.0000i			1.19952i
$279$		0			0
$280$			2.00000i			0.119523i
$281$	20.0000			1.19310			0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000			1.19310			0.596550	+	0.802576i	$0.296538\pi$
$282$		0			0
$283$	−12.0000			−0.713326			−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000			−0.713326			−0.356663	+	0.934233i	$0.616086\pi$
$284$	2.00000			0.118678
$285$		0			0
$286$	−8.00000			−0.473050
$287$			2.00000i			0.118056i
$288$		0			0
$289$	13.0000			0.764706
$290$	10.0000	+	4.00000i	0.587220	+	0.234888i
$291$		0			0
$292$			10.0000i			0.585206i
$293$			22.0000i			1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$			22.0000i			1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$		0			0
$295$	16.0000			0.931556
$296$	−4.00000			−0.232495
$297$		0			0
$298$		0			0
$299$	12.0000			0.693978
$300$		0			0
$301$		−	6.00000i		−	0.345834i
$302$			16.0000i			0.920697i
$303$		0			0
$304$		−	4.00000i		−	0.229416i
$305$			4.00000i			0.229039i
$306$		0			0
$307$			24.0000i			1.36975i	0.728659	+	0.684876i	$0.240144\pi$
$307$			24.0000i			1.36975i	−0.728659	+	0.684876i	$0.759856\pi$
$308$		−	4.00000i		−	0.227921i
$309$		0			0
$310$	−8.00000			−0.454369
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	14.0000			0.791327			0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000			0.791327			0.395663	+	0.918396i	$0.370515\pi$
$314$	18.0000			1.01580
$315$		0			0
$316$		−	10.0000i		−	0.562544i
$317$		−	14.0000i		−	0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$		−	14.0000i		−	0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$		0			0
$319$	−20.0000	−	8.00000i	−1.11979	−	0.447914i
$320$	2.00000			0.111803
$321$		0			0
$322$			6.00000i			0.334367i
$323$	8.00000			0.445132
$324$		0			0
$325$	−2.00000			−0.110940
$326$	−10.0000			−0.553849
$327$		0			0
$328$	2.00000			0.110432
$329$			12.0000i			0.661581i
$330$		0			0
$331$		−	10.0000i		−	0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$		−	10.0000i		−	0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$	16.0000			0.878114
$333$		0			0
$334$		0			0
$335$	8.00000			0.437087
$336$		0			0
$337$			28.0000i			1.52526i	0.646837	+	0.762629i	$0.276092\pi$
$337$			28.0000i			1.52526i	−0.646837	+	0.762629i	$0.723908\pi$
$338$			9.00000i			0.489535i
$339$		0			0
$340$			4.00000i			0.216930i
$341$	16.0000			0.866449
$342$		0			0
$343$	−1.00000			−0.0539949
$344$	−6.00000			−0.323498
$345$		0			0
$346$			18.0000i			0.967686i
$347$	6.00000			0.322097			0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000			0.322097			0.161048	+	0.986947i	$0.448512\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		−	1.00000i		−	0.0534522i
$351$		0			0
$352$	−4.00000			−0.213201
$353$	−18.0000			−0.958043			−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000			−0.958043			−0.479022	+	0.877803i	$0.659008\pi$
$354$		0			0
$355$	4.00000			0.212298
$356$			2.00000i			0.106000i
$357$		0			0
$358$		−	18.0000i		−	0.951330i
$359$		−	16.0000i		−	0.844448i	−0.906492	−	0.422224i	$-0.861250\pi$
$359$		−	16.0000i		−	0.844448i	0.906492	−	0.422224i	$-0.138750\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$			10.0000i			0.525588i
$363$		0			0
$364$	2.00000			0.104828
$365$			20.0000i			1.04685i
$366$		0			0
$367$		−	16.0000i		−	0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$		−	16.0000i		−	0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	6.00000			0.312772
$369$		0			0
$370$	−8.00000			−0.415900
$371$	−8.00000			−0.415339
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		−	8.00000i		−	0.413670i
$375$		0			0
$376$	12.0000			0.618853
$377$	4.00000	−	10.0000i	0.206010	−	0.515026i
$378$		0			0
$379$		−	34.0000i		−	1.74646i	−0.487306	−	0.873231i	$-0.662020\pi$
$379$		−	34.0000i		−	1.74646i	0.487306	−	0.873231i	$-0.337980\pi$
$380$		−	8.00000i		−	0.410391i
$381$		0			0
$382$	−24.0000			−1.22795
$383$	12.0000			0.613171			0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000			0.613171			0.306586	+	0.951843i	$0.400813\pi$
$384$		0			0
$385$		−	8.00000i		−	0.407718i
$386$	8.00000			0.407189
$387$		0			0
$388$			10.0000i			0.507673i
$389$		−	6.00000i		−	0.304212i	−0.988364	−	0.152106i	$-0.951394\pi$
$389$		−	6.00000i		−	0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$390$		0			0
$391$			12.0000i			0.606866i
$392$			1.00000i			0.0505076i
$393$		0			0
$394$			8.00000i			0.403034i
$395$		−	20.0000i		−	1.00631i
$396$		0			0
$397$	−30.0000			−1.50566			−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000			−1.50566			−0.752828	+	0.658217i	$0.771311\pi$
$398$			8.00000i			0.401004i
$399$		0			0
$400$	−1.00000			−0.0500000
$401$	−20.0000			−0.998752			−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000			−0.998752			−0.499376	+	0.866385i	$0.666437\pi$
$402$		0			0
$403$			8.00000i			0.398508i
$404$			6.00000i			0.298511i
$405$		0			0
$406$	5.00000	+	2.00000i	0.248146	+	0.0992583i
$407$	16.0000			0.793091
$408$		0			0
$409$			2.00000i			0.0988936i	0.998777	+	0.0494468i	$0.0157458\pi$
$409$			2.00000i			0.0988936i	−0.998777	+	0.0494468i	$0.984254\pi$
$410$	4.00000			0.197546
$411$		0			0
$412$	−4.00000			−0.197066
$413$	8.00000			0.393654
$414$		0			0
$415$	32.0000			1.57082
$416$		−	2.00000i		−	0.0980581i
$417$		0			0
$418$			16.0000i			0.782586i
$419$	−36.0000			−1.75872			−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000			−1.75872			−0.879358	+	0.476162i	$0.842028\pi$
$420$		0			0
$421$		−	8.00000i		−	0.389896i	−0.980814	−	0.194948i	$-0.937546\pi$
$421$		−	8.00000i		−	0.389896i	0.980814	−	0.194948i	$-0.0624538\pi$
$422$	10.0000			0.486792
$423$		0			0
$424$			8.00000i			0.388514i
$425$		−	2.00000i		−	0.0970143i
$426$		0			0
$427$			2.00000i			0.0967868i
$428$	14.0000			0.676716
$429$		0			0
$430$	−12.0000			−0.578691
$431$	22.0000			1.05970			0.529851	−	0.848091i	$-0.322248\pi$
$431$	22.0000			1.05970			0.529851	+	0.848091i	$0.322248\pi$
$432$		0			0
$433$		−	34.0000i		−	1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$		−	34.0000i		−	1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	−4.00000			−0.192006
$435$		0			0
$436$	18.0000			0.862044
$437$		−	24.0000i		−	1.14808i
$438$		0			0
$439$	36.0000			1.71819			0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000			1.71819			0.859093	+	0.511819i	$0.171028\pi$
$440$	−8.00000			−0.381385
$441$		0			0
$442$	4.00000			0.190261
$443$			4.00000i			0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$			4.00000i			0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$		0			0
$445$			4.00000i			0.189618i
$446$			12.0000i			0.568216i
$447$		0			0
$448$	1.00000			0.0472456
$449$			6.00000i			0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$			6.00000i			0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$		0			0
$451$	−8.00000			−0.376705
$452$			2.00000i			0.0940721i
$453$		0			0
$454$		0			0
$455$	4.00000			0.187523
$456$		0			0
$457$	−18.0000			−0.842004			−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000			−0.842004			−0.421002	+	0.907060i	$0.638322\pi$
$458$	−18.0000			−0.841085
$459$		0			0
$460$	12.0000			0.559503
$461$		−	14.0000i		−	0.652045i	−0.945362	−	0.326023i	$-0.894291\pi$
$461$		−	14.0000i		−	0.652045i	0.945362	−	0.326023i	$-0.105709\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$	2.00000	−	5.00000i	0.0928477	−	0.232119i
$465$		0			0
$466$		−	20.0000i		−	0.926482i
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$	24.0000			1.10704
$471$		0			0
$472$		−	8.00000i		−	0.368230i
$473$	24.0000			1.10352
$474$		0			0
$475$			4.00000i			0.183533i
$476$			2.00000i			0.0916698i
$477$		0			0
$478$			30.0000i			1.37217i
$479$		−	4.00000i		−	0.182765i	−0.995816	−	0.0913823i	$-0.970871\pi$
$479$		−	4.00000i		−	0.182765i	0.995816	−	0.0913823i	$-0.0291285\pi$
$480$		0			0
$481$			8.00000i			0.364769i
$482$		−	22.0000i		−	1.00207i
$483$		0			0
$484$	5.00000			0.227273
$485$			20.0000i			0.908153i
$486$		0			0
$487$	32.0000			1.45006			0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000			1.45006			0.725029	+	0.688718i	$0.241826\pi$
$488$	2.00000			0.0905357
$489$		0			0
$490$			2.00000i			0.0903508i
$491$			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
$493$	10.0000	+	4.00000i	0.450377	+	0.180151i
$494$	−8.00000			−0.359937
$495$		0			0
$496$			4.00000i			0.179605i
$497$	2.00000			0.0897123
$498$		0			0
$499$	−12.0000			−0.537194			−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000			−0.537194			−0.268597	+	0.963253i	$0.586560\pi$
$500$	−12.0000			−0.536656
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$			12.0000i			0.533993i
$506$	−24.0000			−1.06693
$507$		0			0
$508$		−	18.0000i		−	0.798621i
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$			10.0000i			0.442374i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	2.00000i		−	0.0882162i
$515$	−8.00000			−0.352522
$516$		0			0
$517$	−48.0000			−2.11104
$518$	−4.00000			−0.175750
$519$		0			0
$520$		−	4.00000i		−	0.175412i
$521$	−34.0000			−1.48957			−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000			−1.48957			−0.744784	+	0.667306i	$0.767447\pi$
$522$		0			0
$523$	16.0000			0.699631			0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000			0.699631			0.349816	+	0.936819i	$0.386244\pi$
$524$			12.0000i			0.524222i
$525$		0			0
$526$	−8.00000			−0.348817
$527$	−8.00000			−0.348485
$528$		0			0
$529$	13.0000			0.565217
$530$			16.0000i			0.694996i
$531$		0			0
$532$		−	4.00000i		−	0.173422i
$533$		−	4.00000i		−	0.173259i
$534$		0			0
$535$	28.0000			1.21055
$536$		−	4.00000i		−	0.172774i
$537$		0			0
$538$	14.0000			0.603583
$539$		−	4.00000i		−	0.172292i
$540$		0			0
$541$		−	32.0000i		−	1.37579i	−0.725811	−	0.687894i	$-0.758536\pi$
$541$		−	32.0000i		−	1.37579i	0.725811	−	0.687894i	$-0.241464\pi$
$542$	−16.0000			−0.687259
$543$		0			0
$544$	2.00000			0.0857493
$545$	36.0000			1.54207
$546$		0			0
$547$	20.0000			0.855138			0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000			0.855138			0.427569	+	0.903983i	$0.359370\pi$
$548$		−	6.00000i		−	0.256307i
$549$		0			0
$550$	4.00000			0.170561
$551$	−20.0000	−	8.00000i	−0.852029	−	0.340811i
$552$		0			0
$553$		−	10.0000i		−	0.425243i
$554$			6.00000i			0.254916i
$555$		0			0
$556$	20.0000			0.848189
$557$	−24.0000			−1.01691			−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000			−1.01691			−0.508456	+	0.861088i	$0.669784\pi$
$558$		0			0
$559$			12.0000i			0.507546i
$560$	2.00000			0.0845154
$561$		0			0
$562$		−	20.0000i		−	0.843649i
$563$		−	20.0000i		−	0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$		−	20.0000i		−	0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$		0			0
$565$			4.00000i			0.168281i
$566$			12.0000i			0.504398i
$567$		0			0
$568$		−	2.00000i		−	0.0839181i
$569$			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
$571$	−44.0000			−1.84134			−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000			−1.84134			−0.920671	+	0.390339i	$0.872358\pi$
$572$			8.00000i			0.334497i
$573$		0			0
$574$	2.00000			0.0834784
$575$	−6.00000			−0.250217
$576$		0			0
$577$			2.00000i			0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$			2.00000i			0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$		−	13.0000i		−	0.540729i
$579$		0			0
$580$	4.00000	−	10.0000i	0.166091	−	0.415227i
$581$	16.0000			0.663792
$582$		0			0
$583$		−	32.0000i		−	1.32530i
$584$	10.0000			0.413803
$585$		0			0
$586$	22.0000			0.908812
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		−	16.0000i		−	0.658710i
$591$		0			0
$592$			4.00000i			0.164399i
$593$	14.0000			0.574911			0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000			0.574911			0.287456	+	0.957794i	$0.407191\pi$
$594$		0			0
$595$			4.00000i			0.163984i
$596$		0			0
$597$		0			0
$598$		−	12.0000i		−	0.490716i
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			30.0000i			1.22373i	0.790964	+	0.611863i	$0.209580\pi$
$601$			30.0000i			1.22373i	−0.790964	+	0.611863i	$0.790420\pi$
$602$	−6.00000			−0.244542
$603$		0			0
$604$	16.0000			0.651031
$605$	10.0000			0.406558
$606$		0			0
$607$		−	28.0000i		−	1.13648i	−0.822861	−	0.568242i	$-0.807624\pi$
$607$		−	28.0000i		−	1.13648i	0.822861	−	0.568242i	$-0.192376\pi$
$608$	−4.00000			−0.162221
$609$		0			0
$610$	4.00000			0.161955
$611$		−	24.0000i		−	0.970936i
$612$		0			0
$613$	14.0000			0.565455			0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000			0.565455			0.282727	+	0.959200i	$0.408761\pi$
$614$	24.0000			0.968561
$615$		0			0
$616$	−4.00000			−0.161165
$617$			26.0000i			1.04672i	0.852111	+	0.523360i	$0.175322\pi$
$617$			26.0000i			1.04672i	−0.852111	+	0.523360i	$0.824678\pi$
$618$		0			0
$619$			44.0000i			1.76851i	0.467005	+	0.884255i	$0.345333\pi$
$619$			44.0000i			1.76851i	−0.467005	+	0.884255i	$0.654667\pi$
$620$			8.00000i			0.321288i
$621$		0			0
$622$		0			0
$623$			2.00000i			0.0801283i
$624$		0			0
$625$	−19.0000			−0.760000
$626$		−	14.0000i		−	0.559553i
$627$		0			0
$628$		−	18.0000i		−	0.718278i
$629$	−8.00000			−0.318981
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$	−10.0000			−0.397779
$633$		0			0
$634$	−14.0000			−0.556011
$635$		−	36.0000i		−	1.42862i
$636$		0			0
$637$	2.00000			0.0792429
$638$	−8.00000	+	20.0000i	−0.316723	+	0.791808i
$639$		0			0
$640$		−	2.00000i		−	0.0790569i
$641$			30.0000i			1.18493i	0.805597	+	0.592464i	$0.201845\pi$
$641$			30.0000i			1.18493i	−0.805597	+	0.592464i	$0.798155\pi$
$642$		0			0
$643$	−4.00000			−0.157745			−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000			−0.157745			−0.0788723	+	0.996885i	$0.525132\pi$
$644$	6.00000			0.236433
$645$		0			0
$646$		−	8.00000i		−	0.314756i
$647$	−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
$649$			32.0000i			1.25611i
$650$			2.00000i			0.0784465i
$651$		0			0
$652$			10.0000i			0.391630i
$653$			6.00000i			0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$			6.00000i			0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$		0			0
$655$			24.0000i			0.937758i
$656$		−	2.00000i		−	0.0780869i
$657$		0			0
$658$	12.0000			0.467809
$659$			28.0000i			1.09073i	0.838200	+	0.545363i	$0.183608\pi$
$659$			28.0000i			1.09073i	−0.838200	+	0.545363i	$0.816392\pi$
$660$		0			0
$661$	−26.0000			−1.01128			−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000			−1.01128			−0.505641	+	0.862744i	$0.668744\pi$
$662$	−10.0000			−0.388661
$663$		0			0
$664$		−	16.0000i		−	0.620920i
$665$		−	8.00000i		−	0.310227i
$666$		0			0
$667$	12.0000	−	30.0000i	0.464642	−	1.16160i
$668$		0			0
$669$		0			0
$670$		−	8.00000i		−	0.309067i
$671$	−8.00000			−0.308837
$672$		0			0
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$	28.0000			1.07852
$675$		0			0
$676$	9.00000			0.346154
$677$			22.0000i			0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$			22.0000i			0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$		0			0
$679$			10.0000i			0.383765i
$680$	4.00000			0.153393
$681$		0			0
$682$		−	16.0000i		−	0.612672i
$683$	−6.00000			−0.229584			−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000			−0.229584			−0.114792	+	0.993390i	$0.536620\pi$
$684$		0			0
$685$		−	12.0000i		−	0.458496i
$686$			1.00000i			0.0381802i
$687$		0			0
$688$			6.00000i			0.228748i
$689$	16.0000			0.609551
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$	18.0000			0.684257
$693$		0			0
$694$		−	6.00000i		−	0.227757i
$695$	40.0000			1.51729
$696$		0			0
$697$	4.00000			0.151511
$698$			10.0000i			0.378506i
$699$		0			0
$700$	−1.00000			−0.0377964
$701$	−36.0000			−1.35970			−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000			−1.35970			−0.679851	+	0.733351i	$0.737955\pi$
$702$		0			0
$703$	16.0000			0.603451
$704$			4.00000i			0.150756i
$705$		0			0
$706$			18.0000i			0.677439i
$707$			6.00000i			0.225653i
$708$		0			0
$709$	46.0000			1.72757			0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000			1.72757			0.863783	+	0.503864i	$0.168089\pi$
$710$		−	4.00000i		−	0.150117i
$711$		0			0
$712$	2.00000			0.0749532
$713$			24.0000i			0.898807i
$714$		0			0
$715$			16.0000i			0.598366i
$716$	−18.0000			−0.672692
$717$		0			0
$718$	−16.0000			−0.597115
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$	−4.00000			−0.148968
$722$		−	3.00000i		−	0.111648i
$723$		0			0
$724$	10.0000			0.371647
$725$	−2.00000	+	5.00000i	−0.0742781	+	0.185695i
$726$		0			0
$727$		−	24.0000i		−	0.890111i	−0.895503	−	0.445055i	$-0.853184\pi$
$727$		−	24.0000i		−	0.890111i	0.895503	−	0.445055i	$-0.146816\pi$
$728$		−	2.00000i		−	0.0741249i
$729$		0			0
$730$	20.0000			0.740233
$731$	−12.0000			−0.443836
$732$		0			0
$733$		−	42.0000i		−	1.55131i	−0.631160	−	0.775653i	$-0.717421\pi$
$733$		−	42.0000i		−	1.55131i	0.631160	−	0.775653i	$-0.282579\pi$
$734$	−16.0000			−0.590571
$735$		0			0
$736$		−	6.00000i		−	0.221163i
$737$			16.0000i			0.589368i
$738$		0			0
$739$		−	30.0000i		−	1.10357i	−0.833987	−	0.551784i	$-0.813947\pi$
$739$		−	30.0000i		−	1.10357i	0.833987	−	0.551784i	$-0.186053\pi$
$740$			8.00000i			0.294086i
$741$		0			0
$742$			8.00000i			0.293689i
$743$			8.00000i			0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$			8.00000i			0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$		0			0
$745$		0			0
$746$		−	10.0000i		−	0.366126i
$747$		0			0
$748$	−8.00000			−0.292509
$749$	14.0000			0.511549
$750$		0			0
$751$			34.0000i			1.24068i	0.784334	+	0.620339i	$0.213005\pi$
$751$			34.0000i			1.24068i	−0.784334	+	0.620339i	$0.786995\pi$
$752$		−	12.0000i		−	0.437595i
$753$		0			0
$754$	−10.0000	−	4.00000i	−0.364179	−	0.145671i
$755$	32.0000			1.16460
$756$		0			0
$757$		−	4.00000i		−	0.145382i	−0.997354	−	0.0726912i	$-0.976841\pi$
$757$		−	4.00000i		−	0.145382i	0.997354	−	0.0726912i	$-0.0231588\pi$
$758$	−34.0000			−1.23494
$759$		0			0
$760$	−8.00000			−0.290191
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$	18.0000			0.651644
$764$			24.0000i			0.868290i
$765$		0			0
$766$		−	12.0000i		−	0.433578i
$767$	−16.0000			−0.577727
$768$		0			0
$769$			26.0000i			0.937584i	0.883309	+	0.468792i	$0.155311\pi$
$769$			26.0000i			0.937584i	−0.883309	+	0.468792i	$0.844689\pi$
$770$	−8.00000			−0.288300
$771$		0			0
$772$		−	8.00000i		−	0.287926i
$773$		−	34.0000i		−	1.22290i	−0.791285	−	0.611448i	$-0.790588\pi$
$773$		−	34.0000i		−	1.22290i	0.791285	−	0.611448i	$-0.209412\pi$
$774$		0			0
$775$		−	4.00000i		−	0.143684i
$776$	10.0000			0.358979
$777$		0			0
$778$	−6.00000			−0.215110
$779$	−8.00000			−0.286630
$780$		0			0
$781$			8.00000i			0.286263i
$782$	12.0000			0.429119
$783$		0			0
$784$	1.00000			0.0357143
$785$		−	36.0000i		−	1.28490i
$786$		0			0
$787$	−32.0000			−1.14068			−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000			−1.14068			−0.570338	+	0.821410i	$0.693188\pi$
$788$	8.00000			0.284988
$789$		0			0
$790$	−20.0000			−0.711568
$791$			2.00000i			0.0711118i
$792$		0			0
$793$		−	4.00000i		−	0.142044i
$794$			30.0000i			1.06466i
$795$		0			0
$796$	8.00000			0.283552
$797$		−	22.0000i		−	0.779280i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$		−	22.0000i		−	0.779280i	0.920967	−	0.389640i	$-0.127401\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$			1.00000i			0.0353553i
$801$		0			0
$802$			20.0000i			0.706225i
$803$	−40.0000			−1.41157
$804$		0			0
$805$	12.0000			0.422944
$806$	8.00000			0.281788
$807$		0			0
$808$	6.00000			0.211079
$809$		−	10.0000i		−	0.351581i	−0.984428	−	0.175791i	$-0.943752\pi$
$809$		−	10.0000i		−	0.351581i	0.984428	−	0.175791i	$-0.0562482\pi$
$810$		0			0
$811$	24.0000			0.842754			0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000			0.842754			0.421377	+	0.906886i	$0.361547\pi$
$812$	2.00000	−	5.00000i	0.0701862	−	0.175466i
$813$		0			0
$814$		−	16.0000i		−	0.560800i
$815$			20.0000i			0.700569i
$816$		0			0
$817$	24.0000			0.839654
$818$	2.00000			0.0699284
$819$		0			0
$820$		−	4.00000i		−	0.139686i
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		−	2.00000i		−	0.0697156i	−0.999392	−	0.0348578i	$-0.988902\pi$
$823$		−	2.00000i		−	0.0697156i	0.999392	−	0.0348578i	$-0.0110978\pi$
$824$			4.00000i			0.139347i
$825$		0			0
$826$		−	8.00000i		−	0.278356i
$827$		−	12.0000i		−	0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$		−	12.0000i		−	0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$		0			0
$829$		−	22.0000i		−	0.764092i	−0.924143	−	0.382046i	$-0.875220\pi$
$829$		−	22.0000i		−	0.764092i	0.924143	−	0.382046i	$-0.124780\pi$
$830$		−	32.0000i		−	1.11074i
$831$		0			0
$832$	−2.00000			−0.0693375
$833$			2.00000i			0.0692959i
$834$		0			0
$835$		0			0
$836$	16.0000			0.553372
$837$		0			0
$838$			36.0000i			1.24360i
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−21.0000	−	20.0000i	−0.724138	−	0.689655i
$842$	−8.00000			−0.275698
$843$		0			0
$844$		−	10.0000i		−	0.344214i
$845$	18.0000			0.619219
$846$		0			0
$847$	5.00000			0.171802
$848$	8.00000			0.274721
$849$		0			0
$850$	−2.00000			−0.0685994
$851$			24.0000i			0.822709i
$852$		0			0
$853$			22.0000i			0.753266i	0.926363	+	0.376633i	$0.122918\pi$
$853$			22.0000i			0.753266i	−0.926363	+	0.376633i	$0.877082\pi$
$854$	2.00000			0.0684386
$855$		0			0
$856$		−	14.0000i		−	0.478510i
$857$	10.0000			0.341593			0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000			0.341593			0.170797	+	0.985306i	$0.445366\pi$
$858$		0			0
$859$		−	40.0000i		−	1.36478i	−0.730987	−	0.682391i	$-0.760940\pi$
$859$		−	40.0000i		−	1.36478i	0.730987	−	0.682391i	$-0.239060\pi$
$860$			12.0000i			0.409197i
$861$		0			0
$862$		−	22.0000i		−	0.749323i
$863$	−54.0000			−1.83818			−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000			−1.83818			−0.919091	+	0.394046i	$0.871075\pi$
$864$		0			0
$865$	36.0000			1.22404
$866$	−34.0000			−1.15537
$867$		0			0
$868$			4.00000i			0.135769i
$869$	40.0000			1.35691
$870$		0			0
$871$	−8.00000			−0.271070
$872$		−	18.0000i		−	0.609557i
$873$		0			0
$874$	−24.0000			−0.811812
$875$	−12.0000			−0.405674
$876$		0			0
$877$	46.0000			1.55331			0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000			1.55331			0.776655	+	0.629926i	$0.216915\pi$
$878$		−	36.0000i		−	1.21494i
$879$		0			0
$880$			8.00000i			0.269680i
$881$			22.0000i			0.741199i	0.928793	+	0.370599i	$0.120848\pi$
$881$			22.0000i			0.741199i	−0.928793	+	0.370599i	$0.879152\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		−	4.00000i		−	0.134535i
$885$		0			0
$886$	4.00000			0.134383
$887$		−	48.0000i		−	1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$		−	48.0000i		−	1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$		0			0
$889$		−	18.0000i		−	0.603701i
$890$	4.00000			0.134080
$891$		0			0
$892$	12.0000			0.401790
$893$	−48.0000			−1.60626
$894$		0			0
$895$	−36.0000			−1.20335
$896$		−	1.00000i		−	0.0334077i
$897$		0			0
$898$	6.00000			0.200223
$899$	20.0000	+	8.00000i	0.667037	+	0.266815i
$900$		0			0
$901$			16.0000i			0.533037i
$902$			8.00000i			0.266371i
$903$		0			0
$904$	2.00000			0.0665190
$905$	20.0000			0.664822
$906$		0			0
$907$		−	10.0000i		−	0.332045i	−0.986122	−	0.166022i	$-0.946908\pi$
$907$		−	10.0000i		−	0.332045i	0.986122	−	0.166022i	$-0.0530924\pi$
$908$		0			0
$909$		0			0
$910$		−	4.00000i		−	0.132599i
$911$		−	48.0000i		−	1.59031i	−0.606406	−	0.795155i	$-0.707389\pi$
$911$		−	48.0000i		−	1.59031i	0.606406	−	0.795155i	$-0.292611\pi$
$912$		0			0
$913$			64.0000i			2.11809i
$914$			18.0000i			0.595387i
$915$		0			0
$916$			18.0000i			0.594737i
$917$			12.0000i			0.396275i
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$		−	12.0000i		−	0.395628i
$921$		0			0
$922$	−14.0000			−0.461065
$923$	−4.00000			−0.131662
$924$		0			0
$925$		−	4.00000i		−	0.131519i
$926$		−	32.0000i		−	1.05159i
$927$		0			0
$928$	−5.00000	−	2.00000i	−0.164133	−	0.0656532i
$929$	−14.0000			−0.459325			−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000			−0.459325			−0.229663	+	0.973270i	$0.573762\pi$
$930$		0			0
$931$		−	4.00000i		−	0.131095i
$932$	−20.0000			−0.655122
$933$		0			0
$934$	12.0000			0.392652
$935$	−16.0000			−0.523256
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		−	4.00000i		−	0.130605i
$939$		0			0
$940$		−	24.0000i		−	0.782794i
$941$	6.00000			0.195594			0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000			0.195594			0.0977972	+	0.995206i	$0.468820\pi$
$942$		0			0
$943$		−	12.0000i		−	0.390774i
$944$	−8.00000			−0.260378
$945$		0			0
$946$		−	24.0000i		−	0.780307i
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$		−	20.0000i		−	0.649227i
$950$	4.00000			0.129777
$951$		0			0
$952$	2.00000			0.0648204
$953$	16.0000			0.518291			0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000			0.518291			0.259145	+	0.965838i	$0.416559\pi$
$954$		0			0
$955$			48.0000i			1.55324i
$956$	30.0000			0.970269
$957$		0			0
$958$	−4.00000			−0.129234
$959$		−	6.00000i		−	0.193750i
$960$		0			0
$961$	15.0000			0.483871
$962$	8.00000			0.257930
$963$		0			0
$964$	−22.0000			−0.708572
$965$		−	16.0000i		−	0.515058i
$966$		0			0
$967$		−	34.0000i		−	1.09337i	−0.837340	−	0.546683i	$-0.815890\pi$
$967$		−	34.0000i		−	1.09337i	0.837340	−	0.546683i	$-0.184110\pi$
$968$		−	5.00000i		−	0.160706i
$969$		0			0
$970$	20.0000			0.642161
$971$		−	40.0000i		−	1.28366i	−0.766846	−	0.641831i	$-0.778175\pi$
$971$		−	40.0000i		−	1.28366i	0.766846	−	0.641831i	$-0.221825\pi$
$972$		0			0
$973$	20.0000			0.641171
$974$		−	32.0000i		−	1.02535i
$975$		0			0
$976$		−	2.00000i		−	0.0640184i
$977$	−12.0000			−0.383914			−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000			−0.383914			−0.191957	+	0.981403i	$0.561483\pi$
$978$		0			0
$979$	−8.00000			−0.255681
$980$	2.00000			0.0638877
$981$		0			0
$982$	12.0000			0.382935
$983$		−	52.0000i		−	1.65854i	−0.558846	−	0.829271i	$-0.688756\pi$
$983$		−	52.0000i		−	1.65854i	0.558846	−	0.829271i	$-0.311244\pi$
$984$		0			0
$985$	16.0000			0.509802
$986$	4.00000	−	10.0000i	0.127386	−	0.318465i
$987$		0			0
$988$			8.00000i			0.254514i
$989$			36.0000i			1.14473i
$990$		0			0
$991$	−56.0000			−1.77890			−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000			−1.77890			−0.889449	+	0.457034i	$0.848912\pi$
$992$	4.00000			0.127000
$993$		0			0
$994$		−	2.00000i		−	0.0634361i
$995$	16.0000			0.507234
$996$		0			0
$997$		−	10.0000i		−	0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$		−	10.0000i		−	0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$			12.0000i			0.379853i
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3654.2.g.b.2899.1	✓	2	1.1	even	1	trivial
3654.2.g.b.2899.2	yes	2	29.28	even	2	inner
3654.2.g.g.2899.1	yes	2	87.86	odd	2
3654.2.g.g.2899.2	yes	2	3.2	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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} +1.00000i q^{8} +2.00000i q^{10} -4.00000i q^{11} +2.00000 q^{13} +1.00000i q^{14} +1.00000 q^{16} +2.00000i q^{17} -4.00000i q^{19} +2.00000 q^{20} -4.00000 q^{22} +6.00000 q^{23} -1.00000 q^{25} -2.00000i q^{26} +1.00000 q^{28} +(2.00000 - 5.00000i) q^{29} +4.00000i q^{31} -1.00000i q^{32} +2.00000 q^{34} +2.00000 q^{35} +4.00000i q^{37} -4.00000 q^{38} -2.00000i q^{40} -2.00000i q^{41} +6.00000i q^{43} +4.00000i q^{44} -6.00000i q^{46} -12.0000i q^{47} +1.00000 q^{49} +1.00000i q^{50} -2.00000 q^{52} +8.00000 q^{53} +8.00000i q^{55} -1.00000i q^{56} +(-5.00000 - 2.00000i) q^{58} -8.00000 q^{59} -2.00000i q^{61} +4.00000 q^{62} -1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} -2.00000i q^{68} -2.00000i q^{70} -2.00000 q^{71} -10.0000i q^{73} +4.00000 q^{74} +4.00000i q^{76} +4.00000i q^{77} +10.0000i q^{79} -2.00000 q^{80} -2.00000 q^{82} -16.0000 q^{83} -4.00000i q^{85} +6.00000 q^{86} +4.00000 q^{88} -2.00000i q^{89} -2.00000 q^{91} -6.00000 q^{92} -12.0000 q^{94} +8.00000i q^{95} -10.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{5} - 2 q^{7} + 4 q^{13} + 2 q^{16} + 4 q^{20} - 8 q^{22} + 12 q^{23} - 2 q^{25} + 2 q^{28} + 4 q^{29} + 4 q^{34} + 4 q^{35} - 8 q^{38} + 2 q^{49} - 4 q^{52} + 16 q^{53} - 10 q^{58}+ \cdots - 24 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^5 - 2 * q^7 + 4 * q^13 + 2 * q^16 + 4 * q^20 - 8 * q^22 + 12 * q^23 - 2 * q^25 + 2 * q^28 + 4 * q^29 + 4 * q^34 + 4 * q^35 - 8 * q^38 + 2 * q^49 - 4 * q^52 + 16 * q^53 - 10 * q^58 - 16 * q^59 + 8 * q^62 - 2 * q^64 - 8 * q^65 - 8 * q^67 - 4 * q^71 + 8 * q^74 - 4 * q^80 - 4 * q^82 - 32 * q^83 + 12 * q^86 + 8 * q^88 - 4 * q^91 - 12 * q^92 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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