LMFDB - Embedded newform 2912.2.k.i.1793.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(1793,2912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2912.1793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.k (of order $2$, degree $1$, 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:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 37x^{12} + 544x^{10} + 4060x^{8} + 16288x^{6} + 34160x^{4} + 33216x^{2} + 10816 $ x^14 + 37x^12 + 544x^10 + 4060x^8 + 16288x^6 + 34160x^4 + 33216x^2 + 10816
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.1
Root			$-3.28037i$ of defining polynomial
Character	$\chi$	$=$	2912.1793
Dual form			2912.2.k.i.1793.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.28037 q^{3} -0.760790i q^{5} +1.00000i q^{7} +7.76086 q^{9} +1.60001i q^{11} +(-1.07939 - 3.44019i) q^{13} +2.49568i q^{15} -4.36086 q^{17} +7.34163i q^{19} -3.28037i q^{21} +6.38477 q^{23} +4.42120 q^{25} -15.6174 q^{27} -4.95692 q^{29} -0.261137i q^{31} -5.24862i q^{33} +0.760790 q^{35} -4.74360i q^{37} +(3.54082 + 11.2851i) q^{39} -0.895672i q^{41} -4.18466 q^{43} -5.90439i q^{45} -5.63856i q^{47} -1.00000 q^{49} +14.3053 q^{51} +8.81277 q^{53} +1.21727 q^{55} -24.0833i q^{57} -3.49774i q^{59} +5.77605 q^{61} +7.76086i q^{63} +(-2.61726 + 0.821193i) q^{65} -4.82282i q^{67} -20.9444 q^{69} -7.45664i q^{71} +3.84128i q^{73} -14.5032 q^{75} -1.60001 q^{77} -9.48055 q^{79} +27.9483 q^{81} -1.08431i q^{83} +3.31770i q^{85} +16.2606 q^{87} +16.6986i q^{89} +(3.44019 - 1.07939i) q^{91} +0.856628i q^{93} +5.58544 q^{95} +6.31970i q^{97} +12.4174i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{3} + 32 q^{9} - 6 q^{13} + 36 q^{17} - 18 q^{23} - 18 q^{25} - 14 q^{27} - 16 q^{29} + 8 q^{35} + 34 q^{39} + 60 q^{43} - 14 q^{49} + 12 q^{51} + 4 q^{53} - 60 q^{55} + 6 q^{61} + 20 q^{65}+ \cdots + 68 q^{95}+O(q^{100}) $ 14 * q - 2 * q^3 + 32 * q^9 - 6 * q^13 + 36 * q^17 - 18 * q^23 - 18 * q^25 - 14 * q^27 - 16 * q^29 + 8 * q^35 + 34 * q^39 + 60 * q^43 - 14 * q^49 + 12 * q^51 + 4 * q^53 - 60 * q^55 + 6 * q^61 + 20 * q^65 - 2 * q^69 - 114 * q^75 - 2 * q^77 - 26 * q^79 + 22 * q^81 - 8 * q^87 + 16 * q^91 + 68 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	−3.28037			−1.89393			−0.946963	−	0.321344i	$-0.895866\pi$
$3$	−3.28037			−1.89393			−0.946963	+	0.321344i	$0.895866\pi$
$4$		0			0
$5$		−	0.760790i		−	0.340236i	−0.985424	−	0.170118i	$-0.945585\pi$
$5$		−	0.760790i		−	0.340236i	0.985424	−	0.170118i	$-0.0544148\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	7.76086			2.58695
$10$		0			0
$11$			1.60001i			0.482420i	0.970473	+	0.241210i	$0.0775442\pi$
$11$			1.60001i			0.482420i	−0.970473	+	0.241210i	$0.922456\pi$
$12$		0			0
$13$	−1.07939	−	3.44019i	−0.299370	−	0.954137i
$13$	−1.07939	−	3.44019i	−0.299370	−	0.954137i
$14$		0			0
$15$			2.49568i			0.644381i
$16$		0			0
$17$	−4.36086			−1.05766			−0.528832	−	0.848726i	$-0.677370\pi$
$17$	−4.36086			−1.05766			−0.528832	+	0.848726i	$0.677370\pi$
$18$		0			0
$19$			7.34163i			1.68428i	0.539255	+	0.842142i	$0.318706\pi$
$19$			7.34163i			1.68428i	−0.539255	+	0.842142i	$0.681294\pi$
$20$		0			0
$21$		−	3.28037i		−	0.715836i
$22$		0			0
$23$	6.38477			1.33132			0.665658	−	0.746257i	$-0.268151\pi$
$23$	6.38477			1.33132			0.665658	+	0.746257i	$0.268151\pi$
$24$		0			0
$25$	4.42120			0.884240
$26$		0			0
$27$	−15.6174			−3.00557
$28$		0			0
$29$	−4.95692			−0.920478			−0.460239	−	0.887795i	$-0.652236\pi$
$29$	−4.95692			−0.920478			−0.460239	+	0.887795i	$0.652236\pi$
$30$		0			0
$31$		−	0.261137i		−	0.0469016i	−0.999725	−	0.0234508i	$-0.992535\pi$
$31$		−	0.261137i		−	0.0469016i	0.999725	−	0.0234508i	$-0.00746531\pi$
$32$		0			0
$33$		−	5.24862i		−	0.913667i
$34$		0			0
$35$	0.760790			0.128597
$36$		0			0
$37$		−	4.74360i		−	0.779843i	−0.920848	−	0.389922i	$-0.872502\pi$
$37$		−	4.74360i		−	0.779843i	0.920848	−	0.389922i	$-0.127498\pi$
$38$		0			0
$39$	3.54082	+	11.2851i	0.566985	+	1.80706i
$40$		0			0
$41$		−	0.895672i		−	0.139880i	−0.997551	−	0.0699402i	$-0.977719\pi$
$41$		−	0.895672i		−	0.139880i	0.997551	−	0.0699402i	$-0.0222809\pi$
$42$		0			0
$43$	−4.18466			−0.638155			−0.319077	−	0.947729i	$-0.603373\pi$
$43$	−4.18466			−0.638155			−0.319077	+	0.947729i	$0.603373\pi$
$44$		0			0
$45$		−	5.90439i		−	0.880174i
$46$		0			0
$47$		−	5.63856i		−	0.822469i	−0.911530	−	0.411235i	$-0.865098\pi$
$47$		−	5.63856i		−	0.822469i	0.911530	−	0.411235i	$-0.134902\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	14.3053			2.00314
$52$		0			0
$53$	8.81277			1.21053			0.605264	−	0.796025i	$-0.293068\pi$
$53$	8.81277			1.21053			0.605264	+	0.796025i	$0.293068\pi$
$54$		0			0
$55$	1.21727			0.164136
$56$		0			0
$57$		−	24.0833i		−	3.18991i
$58$		0			0
$59$		−	3.49774i		−	0.455367i	−0.973735	−	0.227684i	$-0.926885\pi$
$59$		−	3.49774i		−	0.455367i	0.973735	−	0.227684i	$-0.0731152\pi$
$60$		0			0
$61$	5.77605			0.739548			0.369774	−	0.929122i	$-0.379435\pi$
$61$	5.77605			0.739548			0.369774	+	0.929122i	$0.379435\pi$
$62$		0			0
$63$			7.76086i			0.977776i
$64$		0			0
$65$	−2.61726	+	0.821193i	−0.324632	+	0.101856i
$66$		0			0
$67$		−	4.82282i		−	0.589201i	−0.955621	−	0.294600i	$-0.904813\pi$
$67$		−	4.82282i		−	0.589201i	0.955621	−	0.294600i	$-0.0951866\pi$
$68$		0			0
$69$	−20.9444			−2.52141
$70$		0			0
$71$		−	7.45664i		−	0.884941i	−0.896783	−	0.442470i	$-0.854102\pi$
$71$		−	7.45664i		−	0.884941i	0.896783	−	0.442470i	$-0.145898\pi$
$72$		0			0
$73$			3.84128i			0.449588i	0.974406	+	0.224794i	$0.0721709\pi$
$73$			3.84128i			0.449588i	−0.974406	+	0.224794i	$0.927829\pi$
$74$		0			0
$75$	−14.5032			−1.67468
$76$		0			0
$77$	−1.60001			−0.182338
$78$		0			0
$79$	−9.48055			−1.06665			−0.533323	−	0.845912i	$-0.679057\pi$
$79$	−9.48055			−1.06665			−0.533323	+	0.845912i	$0.679057\pi$
$80$		0			0
$81$	27.9483			3.10537
$82$		0			0
$83$		−	1.08431i		−	0.119018i	−0.998228	−	0.0595092i	$-0.981046\pi$
$83$		−	1.08431i		−	0.119018i	0.998228	−	0.0595092i	$-0.0189536\pi$
$84$		0			0
$85$			3.31770i			0.359855i
$86$		0			0
$87$	16.2606			1.74332
$88$		0			0
$89$			16.6986i			1.77005i	0.465544	+	0.885025i	$0.345859\pi$
$89$			16.6986i			1.77005i	−0.465544	+	0.885025i	$0.654141\pi$
$90$		0			0
$91$	3.44019	−	1.07939i	0.360630	−	0.113151i
$92$		0			0
$93$			0.856628i			0.0888282i
$94$		0			0
$95$	5.58544			0.573054
$96$		0			0
$97$			6.31970i			0.641668i	0.947135	+	0.320834i	$0.103963\pi$
$97$			6.31970i			0.641668i	−0.947135	+	0.320834i	$0.896037\pi$
$98$		0			0
$99$			12.4174i			1.24800i
$100$		0			0
$101$	−15.5455			−1.54683			−0.773416	−	0.633899i	$-0.781454\pi$
$101$	−15.5455			−1.54683			−0.773416	+	0.633899i	$0.781454\pi$
$102$		0			0
$103$	−4.00535			−0.394659			−0.197329	−	0.980337i	$-0.563227\pi$
$103$	−4.00535			−0.394659			−0.197329	+	0.980337i	$0.563227\pi$
$104$		0			0
$105$	−2.49568			−0.243553
$106$		0			0
$107$	−6.33427			−0.612357			−0.306178	−	0.951974i	$-0.599050\pi$
$107$	−6.33427			−0.612357			−0.306178	+	0.951974i	$0.599050\pi$
$108$		0			0
$109$			11.7216i			1.12273i	0.827570	+	0.561363i	$0.189723\pi$
$109$			11.7216i			1.12273i	−0.827570	+	0.561363i	$0.810277\pi$
$110$		0			0
$111$			15.5608i			1.47696i
$112$		0			0
$113$	12.8870			1.21230			0.606152	−	0.795349i	$-0.292712\pi$
$113$	12.8870			1.21230			0.606152	+	0.795349i	$0.292712\pi$
$114$		0			0
$115$		−	4.85747i		−	0.452962i
$116$		0			0
$117$	−8.37703	−	26.6988i	−0.774456	−	2.46831i
$118$		0			0
$119$		−	4.36086i		−	0.399760i
$120$		0			0
$121$	8.43998			0.767271
$122$		0			0
$123$			2.93814i			0.264923i
$124$		0			0
$125$		−	7.16756i		−	0.641086i
$126$		0			0
$127$	−10.6000			−0.940599			−0.470300	−	0.882507i	$-0.655854\pi$
$127$	−10.6000			−0.940599			−0.470300	+	0.882507i	$0.655854\pi$
$128$		0			0
$129$	13.7273			1.20862
$130$		0			0
$131$	−16.2889			−1.42317			−0.711585	−	0.702600i	$-0.752023\pi$
$131$	−16.2889			−1.42317			−0.711585	+	0.702600i	$0.752023\pi$
$132$		0			0
$133$	−7.34163			−0.636600
$134$		0			0
$135$			11.8816i			1.02260i
$136$		0			0
$137$			5.67900i			0.485190i	0.970128	+	0.242595i	$0.0779986\pi$
$137$			5.67900i			0.485190i	−0.970128	+	0.242595i	$0.922001\pi$
$138$		0			0
$139$	16.0196			1.35876			0.679382	−	0.733785i	$-0.262248\pi$
$139$	16.0196			1.35876			0.679382	+	0.733785i	$0.262248\pi$
$140$		0			0
$141$			18.4966i			1.55770i
$142$		0			0
$143$	5.50432	−	1.72704i	0.460295	−	0.144422i
$144$		0			0
$145$			3.77118i			0.313179i
$146$		0			0
$147$	3.28037			0.270561
$148$		0			0
$149$		−	1.77742i		−	0.145612i	−0.997346	−	0.0728060i	$-0.976805\pi$
$149$		−	1.77742i		−	0.145612i	0.997346	−	0.0728060i	$-0.0231954\pi$
$150$		0			0
$151$			21.0220i			1.71075i	0.518009	+	0.855375i	$0.326673\pi$
$151$			21.0220i			1.71075i	−0.518009	+	0.855375i	$0.673327\pi$
$152$		0			0
$153$	−33.8440			−2.73613
$154$		0			0
$155$	−0.198671			−0.0159576
$156$		0			0
$157$	−6.65780			−0.531350			−0.265675	−	0.964063i	$-0.585595\pi$
$157$	−6.65780			−0.531350			−0.265675	+	0.964063i	$0.585595\pi$
$158$		0			0
$159$	−28.9092			−2.29265
$160$		0			0
$161$			6.38477i			0.503190i
$162$		0			0
$163$			23.2347i			1.81988i	0.414741	+	0.909940i	$0.363872\pi$
$163$			23.2347i			1.81988i	−0.414741	+	0.909940i	$0.636128\pi$
$164$		0			0
$165$	−3.99310			−0.310862
$166$		0			0
$167$			18.4614i			1.42858i	0.699847	+	0.714292i	$0.253251\pi$
$167$			18.4614i			1.42858i	−0.699847	+	0.714292i	$0.746749\pi$
$168$		0			0
$169$	−10.6698	+	7.42664i	−0.820755	+	0.571280i
$170$		0			0
$171$			56.9773i			4.35716i
$172$		0			0
$173$	18.5291			1.40874			0.704372	−	0.709831i	$-0.251229\pi$
$173$	18.5291			1.40874			0.704372	+	0.709831i	$0.251229\pi$
$174$		0			0
$175$			4.42120i			0.334211i
$176$		0			0
$177$			11.4739i			0.862431i
$178$		0			0
$179$	7.89780			0.590310			0.295155	−	0.955449i	$-0.404629\pi$
$179$	7.89780			0.590310			0.295155	+	0.955449i	$0.404629\pi$
$180$		0			0
$181$	0.576860			0.0428777			0.0214388	−	0.999770i	$-0.493175\pi$
$181$	0.576860			0.0428777			0.0214388	+	0.999770i	$0.493175\pi$
$182$		0			0
$183$	−18.9476			−1.40065
$184$		0			0
$185$	−3.60889			−0.265331
$186$		0			0
$187$		−	6.97741i		−	0.510238i
$188$		0			0
$189$		−	15.6174i		−	1.13600i
$190$		0			0
$191$	−2.33031			−0.168615			−0.0843075	−	0.996440i	$-0.526868\pi$
$191$	−2.33031			−0.168615			−0.0843075	+	0.996440i	$0.526868\pi$
$192$		0			0
$193$		−	7.11386i		−	0.512067i	−0.966668	−	0.256033i	$-0.917584\pi$
$193$		−	7.11386i		−	0.512067i	0.966668	−	0.256033i	$-0.0824157\pi$
$194$		0			0
$195$	8.58560	−	2.69382i	0.614828	−	0.192908i
$196$		0			0
$197$			20.6314i			1.46993i	0.678107	+	0.734963i	$0.262800\pi$
$197$			20.6314i			1.46993i	−0.678107	+	0.734963i	$0.737200\pi$
$198$		0			0
$199$	−1.93506			−0.137173			−0.0685865	−	0.997645i	$-0.521849\pi$
$199$	−1.93506			−0.137173			−0.0685865	+	0.997645i	$0.521849\pi$
$200$		0			0
$201$			15.8206i			1.11590i
$202$		0			0
$203$		−	4.95692i		−	0.347908i
$204$		0			0
$205$	−0.681418			−0.0475923
$206$		0			0
$207$	49.5513			3.44405
$208$		0			0
$209$	−11.7466			−0.812532
$210$		0			0
$211$	−20.9989			−1.44563			−0.722813	−	0.691044i	$-0.757151\pi$
$211$	−20.9989			−1.44563			−0.722813	+	0.691044i	$0.757151\pi$
$212$		0			0
$213$			24.4606i			1.67601i
$214$		0			0
$215$			3.18365i			0.217123i
$216$		0			0
$217$	0.261137			0.0177272
$218$		0			0
$219$		−	12.6008i		−	0.851486i
$220$		0			0
$221$	4.70709	+	15.0022i	0.316633	+	1.00916i
$222$		0			0
$223$		−	23.4561i		−	1.57073i	−0.619030	−	0.785367i	$-0.712474\pi$
$223$		−	23.4561i		−	1.57073i	0.619030	−	0.785367i	$-0.287526\pi$
$224$		0			0
$225$	34.3123			2.28749
$226$		0			0
$227$			15.3245i			1.01712i	0.861027	+	0.508560i	$0.169822\pi$
$227$			15.3245i			1.01712i	−0.861027	+	0.508560i	$0.830178\pi$
$228$		0			0
$229$			23.2295i			1.53505i	0.641020	+	0.767524i	$0.278512\pi$
$229$			23.2295i			1.53505i	−0.641020	+	0.767524i	$0.721488\pi$
$230$		0			0
$231$	5.24862			0.345334
$232$		0			0
$233$	−26.1915			−1.71586			−0.857932	−	0.513763i	$-0.828251\pi$
$233$	−26.1915			−1.71586			−0.857932	+	0.513763i	$0.828251\pi$
$234$		0			0
$235$	−4.28977			−0.279833
$236$		0			0
$237$	31.0998			2.02015
$238$		0			0
$239$			16.7670i			1.08457i	0.840196	+	0.542283i	$0.182440\pi$
$239$			16.7670i			1.08457i	−0.840196	+	0.542283i	$0.817560\pi$
$240$		0			0
$241$			29.4420i			1.89653i	0.317487	+	0.948263i	$0.397161\pi$
$241$			29.4420i			1.89653i	−0.317487	+	0.948263i	$0.602839\pi$
$242$		0			0
$243$	−44.8289			−2.87577
$244$		0			0
$245$			0.760790i			0.0486051i
$246$		0			0
$247$	25.2566	−	7.92451i	1.60704	−	0.504225i
$248$		0			0
$249$			3.55694i			0.225412i
$250$		0			0
$251$	−14.6733			−0.926173			−0.463086	−	0.886313i	$-0.653258\pi$
$251$	−14.6733			−0.926173			−0.463086	+	0.886313i	$0.653258\pi$
$252$		0			0
$253$			10.2157i			0.642254i
$254$		0			0
$255$		−	10.8833i		−	0.681539i
$256$		0			0
$257$	−4.36837			−0.272491			−0.136246	−	0.990675i	$-0.543504\pi$
$257$	−4.36837			−0.272491			−0.136246	+	0.990675i	$0.543504\pi$
$258$		0			0
$259$	4.74360			0.294753
$260$		0			0
$261$	−38.4700			−2.38123
$262$		0			0
$263$	−12.3748			−0.763063			−0.381532	−	0.924356i	$-0.624603\pi$
$263$	−12.3748			−0.763063			−0.381532	+	0.924356i	$0.624603\pi$
$264$		0			0
$265$		−	6.70467i		−	0.411865i
$266$		0			0
$267$		−	54.7777i		−	3.35234i
$268$		0			0
$269$	22.3515			1.36279			0.681397	−	0.731914i	$-0.261373\pi$
$269$	22.3515			1.36279			0.681397	+	0.731914i	$0.261373\pi$
$270$		0			0
$271$			25.1788i			1.52951i	0.644324	+	0.764753i	$0.277139\pi$
$271$			25.1788i			1.52951i	−0.644324	+	0.764753i	$0.722861\pi$
$272$		0			0
$273$	−11.2851	+	3.54082i	−0.683006	+	0.214300i
$274$		0			0
$275$			7.07394i			0.426575i
$276$		0			0
$277$	9.85789			0.592303			0.296152	−	0.955141i	$-0.404297\pi$
$277$	9.85789			0.592303			0.296152	+	0.955141i	$0.404297\pi$
$278$		0			0
$279$		−	2.02665i		−	0.121332i
$280$		0			0
$281$		−	27.3164i		−	1.62956i	−0.579769	−	0.814781i	$-0.696857\pi$
$281$		−	27.3164i		−	1.62956i	0.579769	−	0.814781i	$-0.303143\pi$
$282$		0			0
$283$	14.6654			0.871769			0.435884	−	0.900003i	$-0.356436\pi$
$283$	14.6654			0.871769			0.435884	+	0.900003i	$0.356436\pi$
$284$		0			0
$285$	−18.3223			−1.08532
$286$		0			0
$287$	0.895672			0.0528698
$288$		0			0
$289$	2.01713			0.118655
$290$		0			0
$291$		−	20.7310i		−	1.21527i
$292$		0			0
$293$		−	3.30881i		−	0.193303i	−0.995318	−	0.0966514i	$-0.969187\pi$
$293$		−	3.30881i		−	0.193303i	0.995318	−	0.0966514i	$-0.0308132\pi$
$294$		0			0
$295$	−2.66105			−0.154932
$296$		0			0
$297$		−	24.9879i		−	1.44995i
$298$		0			0
$299$	−6.89169	−	21.9648i	−0.398556	−	1.27026i
$300$		0			0
$301$		−	4.18466i		−	0.241200i
$302$		0			0
$303$	50.9950			2.92958
$304$		0			0
$305$		−	4.39436i		−	0.251621i
$306$		0			0
$307$		−	31.1628i		−	1.77856i	−0.457367	−	0.889278i	$-0.651207\pi$
$307$		−	31.1628i		−	1.77856i	0.457367	−	0.889278i	$-0.348793\pi$
$308$		0			0
$309$	13.1390			0.747454
$310$		0			0
$311$	20.1037			1.13998			0.569988	−	0.821653i	$-0.306948\pi$
$311$	20.1037			1.13998			0.569988	+	0.821653i	$0.306948\pi$
$312$		0			0
$313$	−32.0485			−1.81149			−0.905743	−	0.423827i	$-0.860686\pi$
$313$	−32.0485			−1.81149			−0.905743	+	0.423827i	$0.860686\pi$
$314$		0			0
$315$	5.90439			0.332674
$316$		0			0
$317$			3.06402i			0.172092i	0.996291	+	0.0860462i	$0.0274233\pi$
$317$			3.06402i			0.172092i	−0.996291	+	0.0860462i	$0.972577\pi$
$318$		0			0
$319$		−	7.93110i		−	0.444057i
$320$		0			0
$321$	20.7788			1.15976
$322$		0			0
$323$		−	32.0158i		−	1.78141i
$324$		0			0
$325$	−4.77222	−	15.2098i	−0.264715	−	0.843686i
$326$		0			0
$327$		−	38.4512i		−	2.12636i
$328$		0			0
$329$	5.63856			0.310864
$330$		0			0
$331$			29.2691i			1.60878i	0.594104	+	0.804388i	$0.297507\pi$
$331$			29.2691i			1.60878i	−0.594104	+	0.804388i	$0.702493\pi$
$332$		0			0
$333$		−	36.8144i		−	2.01742i
$334$		0			0
$335$	−3.66915			−0.200467
$336$		0			0
$337$	−9.23557			−0.503093			−0.251547	−	0.967845i	$-0.580939\pi$
$337$	−9.23557			−0.503093			−0.251547	+	0.967845i	$0.580939\pi$
$338$		0			0
$339$	−42.2741			−2.29601
$340$		0			0
$341$	0.417821			0.0226263
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			15.9343i			0.857875i
$346$		0			0
$347$	−11.3517			−0.609390			−0.304695	−	0.952450i	$-0.598555\pi$
$347$	−11.3517			−0.609390			−0.304695	+	0.952450i	$0.598555\pi$
$348$		0			0
$349$		−	3.01015i		−	0.161130i	−0.996749	−	0.0805650i	$-0.974328\pi$
$349$		−	3.01015i		−	0.161130i	0.996749	−	0.0805650i	$-0.0256724\pi$
$350$		0			0
$351$	16.8573	+	53.7268i	0.899778	+	2.86773i
$352$		0			0
$353$			4.70892i			0.250630i	0.992117	+	0.125315i	$0.0399942\pi$
$353$			4.70892i			0.250630i	−0.992117	+	0.125315i	$0.960006\pi$
$354$		0			0
$355$	−5.67294			−0.301089
$356$		0			0
$357$			14.3053i			0.757115i
$358$		0			0
$359$		−	23.8698i		−	1.25980i	−0.776676	−	0.629901i	$-0.783096\pi$
$359$		−	23.8698i		−	1.25980i	0.776676	−	0.629901i	$-0.216904\pi$
$360$		0			0
$361$	−34.8995			−1.83681
$362$		0			0
$363$	−27.6863			−1.45315
$364$		0			0
$365$	2.92241			0.152966
$366$		0			0
$367$	18.6643			0.974268			0.487134	−	0.873327i	$-0.338042\pi$
$367$	18.6643			0.974268			0.487134	+	0.873327i	$0.338042\pi$
$368$		0			0
$369$		−	6.95118i		−	0.361864i
$370$		0			0
$371$			8.81277i			0.457536i
$372$		0			0
$373$	21.8345			1.13055			0.565273	−	0.824904i	$-0.308771\pi$
$373$	21.8345			1.13055			0.565273	+	0.824904i	$0.308771\pi$
$374$		0			0
$375$			23.5123i			1.21417i
$376$		0			0
$377$	5.35048	+	17.0528i	0.275563	+	0.878262i
$378$		0			0
$379$			17.8739i			0.918123i	0.888404	+	0.459062i	$0.151814\pi$
$379$			17.8739i			0.918123i	−0.888404	+	0.459062i	$0.848186\pi$
$380$		0			0
$381$	34.7720			1.78142
$382$		0			0
$383$		−	2.10363i		−	0.107491i	−0.998555	−	0.0537453i	$-0.982884\pi$
$383$		−	2.10363i		−	0.107491i	0.998555	−	0.0537453i	$-0.0171159\pi$
$384$		0			0
$385$			1.21727i			0.0620378i
$386$		0			0
$387$	−32.4766			−1.65088
$388$		0			0
$389$	−20.1523			−1.02176			−0.510882	−	0.859651i	$-0.670681\pi$
$389$	−20.1523			−1.02176			−0.510882	+	0.859651i	$0.670681\pi$
$390$		0			0
$391$	−27.8431			−1.40809
$392$		0			0
$393$	53.4338			2.69538
$394$		0			0
$395$			7.21271i			0.362911i
$396$		0			0
$397$			9.26937i			0.465216i	0.972571	+	0.232608i	$0.0747259\pi$
$397$			9.26937i			0.465216i	−0.972571	+	0.232608i	$0.925274\pi$
$398$		0			0
$399$	24.0833			1.20567
$400$		0			0
$401$		−	3.30880i		−	0.165233i	−0.996581	−	0.0826167i	$-0.973672\pi$
$401$		−	3.30880i		−	0.165233i	0.996581	−	0.0826167i	$-0.0263277\pi$
$402$		0			0
$403$	−0.898362	+	0.281870i	−0.0447506	+	0.0140409i
$404$		0			0
$405$		−	21.2628i		−	1.05656i
$406$		0			0
$407$	7.58979			0.376212
$408$		0			0
$409$		−	36.6226i		−	1.81087i	−0.424482	−	0.905437i	$-0.639544\pi$
$409$		−	36.6226i		−	1.81087i	0.424482	−	0.905437i	$-0.360456\pi$
$410$		0			0
$411$		−	18.6292i		−	0.918913i
$412$		0			0
$413$	3.49774			0.172113
$414$		0			0
$415$	−0.824932			−0.0404943
$416$		0			0
$417$	−52.5502			−2.57340
$418$		0			0
$419$	−10.5762			−0.516679			−0.258339	−	0.966054i	$-0.583175\pi$
$419$	−10.5762			−0.516679			−0.258339	+	0.966054i	$0.583175\pi$
$420$		0			0
$421$			14.6278i			0.712917i	0.934311	+	0.356458i	$0.116016\pi$
$421$			14.6278i			0.712917i	−0.934311	+	0.356458i	$0.883984\pi$
$422$		0			0
$423$		−	43.7601i		−	2.12769i
$424$		0			0
$425$	−19.2802			−0.935229
$426$		0			0
$427$			5.77605i			0.279523i
$428$		0			0
$429$	−18.0562	+	5.66533i	−0.871764	+	0.273525i
$430$		0			0
$431$			2.30428i			0.110993i	0.998459	+	0.0554966i	$0.0176742\pi$
$431$			2.30428i			0.110993i	−0.998459	+	0.0554966i	$0.982326\pi$
$432$		0			0
$433$	3.63305			0.174593			0.0872965	−	0.996182i	$-0.472177\pi$
$433$	3.63305			0.174593			0.0872965	+	0.996182i	$0.472177\pi$
$434$		0			0
$435$		−	12.3709i		−	0.593138i
$436$		0			0
$437$			46.8746i			2.24232i
$438$		0			0
$439$	−18.2832			−0.872608			−0.436304	−	0.899799i	$-0.643713\pi$
$439$	−18.2832			−0.872608			−0.436304	+	0.899799i	$0.643713\pi$
$440$		0			0
$441$	−7.76086			−0.369565
$442$		0			0
$443$	40.3161			1.91547			0.957737	−	0.287646i	$-0.0928725\pi$
$443$	40.3161			1.91547			0.957737	+	0.287646i	$0.0928725\pi$
$444$		0			0
$445$	12.7041			0.602234
$446$		0			0
$447$			5.83061i			0.275778i
$448$		0			0
$449$		−	7.99236i		−	0.377183i	−0.982056	−	0.188591i	$-0.939608\pi$
$449$		−	7.99236i		−	0.377183i	0.982056	−	0.188591i	$-0.0603922\pi$
$450$		0			0
$451$	1.43308			0.0674811
$452$		0			0
$453$		−	68.9602i		−	3.24003i
$454$		0			0
$455$	−0.821193	−	2.61726i	−0.0384981	−	0.122699i
$456$		0			0
$457$			14.6746i			0.686450i	0.939253	+	0.343225i	$0.111519\pi$
$457$			14.6746i			0.686450i	−0.939253	+	0.343225i	$0.888481\pi$
$458$		0			0
$459$	68.1053			3.17889
$460$		0			0
$461$			27.0796i			1.26122i	0.776100	+	0.630610i	$0.217195\pi$
$461$			27.0796i			1.26122i	−0.776100	+	0.630610i	$0.782805\pi$
$462$		0			0
$463$			28.5491i			1.32679i	0.748270	+	0.663395i	$0.230885\pi$
$463$			28.5491i			1.32679i	−0.748270	+	0.663395i	$0.769115\pi$
$464$		0			0
$465$	0.651714			0.0302225
$466$		0			0
$467$	−15.1313			−0.700195			−0.350097	−	0.936713i	$-0.613851\pi$
$467$	−15.1313			−0.700195			−0.350097	+	0.936713i	$0.613851\pi$
$468$		0			0
$469$	4.82282			0.222697
$470$		0			0
$471$	21.8401			1.00634
$472$		0			0
$473$		−	6.69548i		−	0.307859i
$474$		0			0
$475$			32.4588i			1.48931i
$476$		0			0
$477$	68.3947			3.13158
$478$		0			0
$479$		−	12.5192i		−	0.572019i	−0.958227	−	0.286009i	$-0.907671\pi$
$479$		−	12.5192i		−	0.572019i	0.958227	−	0.286009i	$-0.0923289\pi$
$480$		0			0
$481$	−16.3189	+	5.12022i	−0.744077	+	0.233462i
$482$		0			0
$483$		−	20.9444i		−	0.953005i
$484$		0			0
$485$	4.80797			0.218318
$486$		0			0
$487$			1.81778i			0.0823714i	0.999152	+	0.0411857i	$0.0131135\pi$
$487$			1.81778i			0.0823714i	−0.999152	+	0.0411857i	$0.986886\pi$
$488$		0			0
$489$		−	76.2184i		−	3.44672i
$490$		0			0
$491$	−11.9704			−0.540217			−0.270108	−	0.962830i	$-0.587059\pi$
$491$	−11.9704			−0.540217			−0.270108	+	0.962830i	$0.587059\pi$
$492$		0			0
$493$	21.6165			0.973557
$494$		0			0
$495$	9.44705			0.424613
$496$		0			0
$497$	7.45664			0.334476
$498$		0			0
$499$			33.9002i			1.51758i	0.651336	+	0.758790i	$0.274209\pi$
$499$			33.9002i			1.51758i	−0.651336	+	0.758790i	$0.725791\pi$
$500$		0			0
$501$		−	60.5603i		−	2.70563i
$502$		0			0
$503$	−22.3180			−0.995109			−0.497554	−	0.867433i	$-0.665769\pi$
$503$	−22.3180			−0.995109			−0.497554	+	0.867433i	$0.665769\pi$
$504$		0			0
$505$			11.8268i			0.526288i
$506$		0			0
$507$	35.0010	−	24.3622i	1.55445	−	1.08196i
$508$		0			0
$509$		−	26.5145i		−	1.17523i	−0.809139	−	0.587617i	$-0.800066\pi$
$509$		−	26.5145i		−	1.17523i	0.809139	−	0.587617i	$-0.199934\pi$
$510$		0			0
$511$	−3.84128			−0.169928
$512$		0			0
$513$		−	114.657i		−	5.06223i
$514$		0			0
$515$			3.04723i			0.134277i
$516$		0			0
$517$	9.02173			0.396775
$518$		0			0
$519$	−60.7825			−2.66805
$520$		0			0
$521$	12.2357			0.536054			0.268027	−	0.963411i	$-0.413628\pi$
$521$	12.2357			0.536054			0.268027	+	0.963411i	$0.413628\pi$
$522$		0			0
$523$	19.9448			0.872125			0.436062	−	0.899916i	$-0.356373\pi$
$523$	19.9448			0.872125			0.436062	+	0.899916i	$0.356373\pi$
$524$		0			0
$525$		−	14.5032i		−	0.632971i
$526$		0			0
$527$			1.13878i			0.0496062i
$528$		0			0
$529$	17.7653			0.772404
$530$		0			0
$531$		−	27.1455i		−	1.17801i
$532$		0			0
$533$	−3.08128	+	0.966783i	−0.133465	+	0.0418760i
$534$		0			0
$535$			4.81905i			0.208346i
$536$		0			0
$537$	−25.9078			−1.11800
$538$		0			0
$539$		−	1.60001i		−	0.0689171i
$540$		0			0
$541$			27.8440i			1.19711i	0.801083	+	0.598553i	$0.204257\pi$
$541$			27.8440i			1.19711i	−0.801083	+	0.598553i	$0.795743\pi$
$542$		0			0
$543$	−1.89232			−0.0812071
$544$		0			0
$545$	8.91768			0.381991
$546$		0			0
$547$	−1.81588			−0.0776414			−0.0388207	−	0.999246i	$-0.512360\pi$
$547$	−1.81588			−0.0776414			−0.0388207	+	0.999246i	$0.512360\pi$
$548$		0			0
$549$	44.8271			1.91317
$550$		0			0
$551$		−	36.3919i		−	1.55035i
$552$		0			0
$553$		−	9.48055i		−	0.403154i
$554$		0			0
$555$	11.8385			0.502516
$556$		0			0
$557$			29.2770i			1.24051i	0.784401	+	0.620253i	$0.212970\pi$
$557$			29.2770i			1.24051i	−0.784401	+	0.620253i	$0.787030\pi$
$558$		0			0
$559$	4.51690	+	14.3960i	0.191045	+	0.608887i
$560$		0			0
$561$			22.8885i			0.966353i
$562$		0			0
$563$	27.5425			1.16078			0.580389	−	0.814340i	$-0.302901\pi$
$563$	27.5425			1.16078			0.580389	+	0.814340i	$0.302901\pi$
$564$		0			0
$565$		−	9.80427i		−	0.412469i
$566$		0			0
$567$			27.9483i			1.17372i
$568$		0			0
$569$	−34.5835			−1.44981			−0.724907	−	0.688847i	$-0.758117\pi$
$569$	−34.5835			−1.44981			−0.724907	+	0.688847i	$0.758117\pi$
$570$		0			0
$571$	13.4640			0.563452			0.281726	−	0.959495i	$-0.409093\pi$
$571$	13.4640			0.563452			0.281726	+	0.959495i	$0.409093\pi$
$572$		0			0
$573$	7.64428			0.319344
$574$		0			0
$575$	28.2283			1.17720
$576$		0			0
$577$			33.8343i			1.40854i	0.709931	+	0.704271i	$0.248726\pi$
$577$			33.8343i			1.40854i	−0.709931	+	0.704271i	$0.751274\pi$
$578$		0			0
$579$			23.3361i			0.969816i
$580$		0			0
$581$	1.08431			0.0449847
$582$		0			0
$583$			14.1005i			0.583982i
$584$		0			0
$585$	−20.3122	+	6.37316i	−0.839806	+	0.263498i
$586$		0			0
$587$		−	33.4515i		−	1.38069i	−0.723480	−	0.690345i	$-0.757459\pi$
$587$		−	33.4515i		−	1.38069i	0.723480	−	0.690345i	$-0.242541\pi$
$588$		0			0
$589$	1.91717			0.0789957
$590$		0			0
$591$		−	67.6787i		−	2.78393i
$592$		0			0
$593$		−	28.2128i		−	1.15856i	−0.815129	−	0.579280i	$-0.803334\pi$
$593$		−	28.2128i		−	1.15856i	0.815129	−	0.579280i	$-0.196666\pi$
$594$		0			0
$595$	−3.31770			−0.136013
$596$		0			0
$597$	6.34773			0.259795
$598$		0			0
$599$	24.7302			1.01045			0.505225	−	0.862988i	$-0.331410\pi$
$599$	24.7302			1.01045			0.505225	+	0.862988i	$0.331410\pi$
$600$		0			0
$601$	−28.9306			−1.18010			−0.590051	−	0.807366i	$-0.700892\pi$
$601$	−28.9306			−1.18010			−0.590051	+	0.807366i	$0.700892\pi$
$602$		0			0
$603$		−	37.4292i		−	1.52423i
$604$		0			0
$605$		−	6.42106i		−	0.261053i
$606$		0			0
$607$	−11.1920			−0.454268			−0.227134	−	0.973864i	$-0.572935\pi$
$607$	−11.1920			−0.454268			−0.227134	+	0.973864i	$0.572935\pi$
$608$		0			0
$609$			16.2606i			0.658911i
$610$		0			0
$611$	−19.3977	+	6.08624i	−0.784748	+	0.246223i
$612$		0			0
$613$		−	35.7076i		−	1.44222i	−0.692822	−	0.721109i	$-0.743633\pi$
$613$		−	35.7076i		−	1.44222i	0.692822	−	0.721109i	$-0.256367\pi$
$614$		0			0
$615$	2.23531			0.0901363
$616$		0			0
$617$			44.7674i			1.80227i	0.433541	+	0.901134i	$0.357264\pi$
$617$			44.7674i			1.80227i	−0.433541	+	0.901134i	$0.642736\pi$
$618$		0			0
$619$			30.5204i			1.22672i	0.789805	+	0.613358i	$0.210182\pi$
$619$			30.5204i			1.22672i	−0.789805	+	0.613358i	$0.789818\pi$
$620$		0			0
$621$	−99.7135			−4.00137
$622$		0			0
$623$	−16.6986			−0.669016
$624$		0			0
$625$	16.6530			0.666119
$626$		0			0
$627$	38.5334			1.53888
$628$		0			0
$629$			20.6862i			0.824813i
$630$		0			0
$631$		−	20.2780i		−	0.807254i	−0.914924	−	0.403627i	$-0.867749\pi$
$631$		−	20.2780i		−	0.807254i	0.914924	−	0.403627i	$-0.132251\pi$
$632$		0			0
$633$	68.8844			2.73791
$634$		0			0
$635$			8.06439i			0.320026i
$636$		0			0
$637$	1.07939	+	3.44019i	0.0427672	+	0.136305i
$638$		0			0
$639$		−	57.8700i		−	2.28930i
$640$		0			0
$641$	10.0626			0.397449			0.198725	−	0.980055i	$-0.436320\pi$
$641$	10.0626			0.397449			0.198725	+	0.980055i	$0.436320\pi$
$642$		0			0
$643$			13.6462i			0.538152i	0.963119	+	0.269076i	$0.0867183\pi$
$643$			13.6462i			0.538152i	−0.963119	+	0.269076i	$0.913282\pi$
$644$		0			0
$645$		−	10.4436i		−	0.411215i
$646$		0			0
$647$	3.41634			0.134310			0.0671550	−	0.997743i	$-0.478608\pi$
$647$	3.41634			0.134310			0.0671550	+	0.997743i	$0.478608\pi$
$648$		0			0
$649$	5.59641			0.219678
$650$		0			0
$651$	−0.856628			−0.0335739
$652$		0			0
$653$	−19.3094			−0.755634			−0.377817	−	0.925880i	$-0.623325\pi$
$653$	−19.3094			−0.755634			−0.377817	+	0.925880i	$0.623325\pi$
$654$		0			0
$655$			12.3925i			0.484214i
$656$		0			0
$657$			29.8116i			1.16306i
$658$		0			0
$659$	−13.0159			−0.507027			−0.253514	−	0.967332i	$-0.581586\pi$
$659$	−13.0159			−0.507027			−0.253514	+	0.967332i	$0.581586\pi$
$660$		0			0
$661$		−	19.1782i		−	0.745947i	−0.927842	−	0.372973i	$-0.878338\pi$
$661$		−	19.1782i		−	0.745947i	0.927842	−	0.372973i	$-0.121662\pi$
$662$		0			0
$663$	−15.4410	−	49.2128i	−0.599680	−	1.91127i
$664$		0			0
$665$			5.58544i			0.216594i
$666$		0			0
$667$	−31.6488			−1.22545
$668$		0			0
$669$			76.9447i			2.97485i
$670$		0			0
$671$			9.24171i			0.356772i
$672$		0			0
$673$	−5.37316			−0.207120			−0.103560	−	0.994623i	$-0.533023\pi$
$673$	−5.37316			−0.207120			−0.103560	+	0.994623i	$0.533023\pi$
$674$		0			0
$675$	−69.0476			−2.65764
$676$		0			0
$677$	−37.6640			−1.44755			−0.723773	−	0.690038i	$-0.757594\pi$
$677$	−37.6640			−1.44755			−0.723773	+	0.690038i	$0.757594\pi$
$678$		0			0
$679$	−6.31970			−0.242528
$680$		0			0
$681$		−	50.2699i		−	1.92635i
$682$		0			0
$683$		−	32.9170i		−	1.25954i	−0.776784	−	0.629768i	$-0.783150\pi$
$683$		−	32.9170i		−	1.25954i	0.776784	−	0.629768i	$-0.216850\pi$
$684$		0			0
$685$	4.32053			0.165079
$686$		0			0
$687$		−	76.2015i		−	2.90727i
$688$		0			0
$689$	−9.51245	−	30.3176i	−0.362396	−	1.15501i
$690$		0			0
$691$			29.1383i			1.10847i	0.832359	+	0.554237i	$0.186990\pi$
$691$			29.1383i			1.10847i	−0.832359	+	0.554237i	$0.813010\pi$
$692$		0			0
$693$	−12.4174			−0.471699
$694$		0			0
$695$		−	12.1875i		−	0.462300i
$696$		0			0
$697$			3.90590i			0.147947i
$698$		0			0
$699$	85.9180			3.24972
$700$		0			0
$701$	44.8363			1.69344			0.846722	−	0.532035i	$-0.178572\pi$
$701$	44.8363			1.69344			0.846722	+	0.532035i	$0.178572\pi$
$702$		0			0
$703$	34.8257			1.31348
$704$		0			0
$705$	14.0720			0.529984
$706$		0			0
$707$		−	15.5455i		−	0.584648i
$708$		0			0
$709$			22.9875i			0.863313i	0.902038	+	0.431657i	$0.142071\pi$
$709$			22.9875i			0.863313i	−0.902038	+	0.431657i	$0.857929\pi$
$710$		0			0
$711$	−73.5772			−2.75936
$712$		0			0
$713$		−	1.66730i		−	0.0624409i
$714$		0			0
$715$	−1.31391	−	4.18764i	−0.0491376	−	0.156609i
$716$		0			0
$717$		−	55.0020i		−	2.05409i
$718$		0			0
$719$	13.7872			0.514177			0.257088	−	0.966388i	$-0.417237\pi$
$719$	13.7872			0.514177			0.257088	+	0.966388i	$0.417237\pi$
$720$		0			0
$721$		−	4.00535i		−	0.149167i
$722$		0			0
$723$		−	96.5808i		−	3.59188i
$724$		0			0
$725$	−21.9155			−0.813923
$726$		0			0
$727$	4.70998			0.174683			0.0873417	−	0.996178i	$-0.472163\pi$
$727$	4.70998			0.174683			0.0873417	+	0.996178i	$0.472163\pi$
$728$		0			0
$729$	63.2104			2.34113
$730$		0			0
$731$	18.2487			0.674954
$732$		0			0
$733$			35.8427i			1.32388i	0.749557	+	0.661940i	$0.230266\pi$
$733$			35.8427i			1.32388i	−0.749557	+	0.661940i	$0.769734\pi$
$734$		0			0
$735$		−	2.49568i		−	0.0920544i
$736$		0			0
$737$	7.71653			0.284242
$738$		0			0
$739$		−	37.2790i		−	1.37133i	−0.727918	−	0.685664i	$-0.759512\pi$
$739$		−	37.2790i		−	1.37133i	0.727918	−	0.685664i	$-0.240488\pi$
$740$		0			0
$741$	−82.8511	+	25.9954i	−3.04361	+	0.954964i
$742$		0			0
$743$			5.68269i			0.208478i	0.994552	+	0.104239i	$0.0332407\pi$
$743$			5.68269i			0.208478i	−0.994552	+	0.104239i	$0.966759\pi$
$744$		0			0
$745$	−1.35225			−0.0495424
$746$		0			0
$747$		−	8.41517i		−	0.307895i
$748$		0			0
$749$		−	6.33427i		−	0.231449i
$750$		0			0
$751$	38.4665			1.40366			0.701832	−	0.712343i	$-0.252366\pi$
$751$	38.4665			1.40366			0.701832	+	0.712343i	$0.252366\pi$
$752$		0			0
$753$	48.1340			1.75410
$754$		0			0
$755$	15.9934			0.582058
$756$		0			0
$757$	29.6453			1.07748			0.538739	−	0.842473i	$-0.318901\pi$
$757$	29.6453			1.07748			0.538739	+	0.842473i	$0.318901\pi$
$758$		0			0
$759$		−	33.5112i		−	1.21638i
$760$		0			0
$761$			20.7622i			0.752628i	0.926492	+	0.376314i	$0.122809\pi$
$761$			20.7622i			0.752628i	−0.926492	+	0.376314i	$0.877191\pi$
$762$		0			0
$763$	−11.7216			−0.424350
$764$		0			0
$765$			25.7482i			0.930929i
$766$		0			0
$767$	−12.0329	+	3.77544i	−0.434483	+	0.136323i
$768$		0			0
$769$			39.8384i			1.43661i	0.695729	+	0.718305i	$0.255082\pi$
$769$			39.8384i			1.43661i	−0.695729	+	0.718305i	$0.744918\pi$
$770$		0			0
$771$	14.3299			0.516078
$772$		0			0
$773$		−	34.9153i		−	1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$773$		−	34.9153i		−	1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$774$		0			0
$775$		−	1.15454i		−	0.0414723i
$776$		0			0
$777$	−15.5608			−0.558240
$778$		0			0
$779$	6.57569			0.235598
$780$		0			0
$781$	11.9307			0.426913
$782$		0			0
$783$	77.4142			2.76656
$784$		0			0
$785$			5.06519i			0.180784i
$786$		0			0
$787$		−	0.524948i		−	0.0187124i	−0.999956	−	0.00935619i	$-0.997022\pi$
$787$		−	0.524948i		−	0.0187124i	0.999956	−	0.00935619i	$-0.00297821\pi$
$788$		0			0
$789$	40.5940			1.44518
$790$		0			0
$791$			12.8870i			0.458208i
$792$		0			0
$793$	−6.23464	−	19.8707i	−0.221398	−	0.705630i
$794$		0			0
$795$			21.9938i			0.780041i
$796$		0			0
$797$	16.3824			0.580293			0.290146	−	0.956982i	$-0.406296\pi$
$797$	16.3824			0.580293			0.290146	+	0.956982i	$0.406296\pi$
$798$		0			0
$799$			24.5890i			0.869897i
$800$		0			0
$801$			129.596i			4.57903i
$802$		0			0
$803$	−6.14607			−0.216890
$804$		0			0
$805$	4.85747			0.171203
$806$		0			0
$807$	−73.3213			−2.58103
$808$		0			0
$809$	−23.2403			−0.817084			−0.408542	−	0.912740i	$-0.633963\pi$
$809$	−23.2403			−0.817084			−0.408542	+	0.912740i	$0.633963\pi$
$810$		0			0
$811$		−	15.0032i		−	0.526834i	−0.964682	−	0.263417i	$-0.915150\pi$
$811$		−	15.0032i		−	0.526834i	0.964682	−	0.263417i	$-0.0848495\pi$
$812$		0			0
$813$		−	82.5960i		−	2.89677i
$814$		0			0
$815$	17.6767			0.619188
$816$		0			0
$817$		−	30.7222i		−	1.07483i
$818$		0			0
$819$	26.6988	−	8.37703i	0.932933	−	0.292717i
$820$		0			0
$821$			32.6183i			1.13839i	0.822204	+	0.569193i	$0.192744\pi$
$821$			32.6183i			1.13839i	−0.822204	+	0.569193i	$0.807256\pi$
$822$		0			0
$823$	−3.16682			−0.110389			−0.0551943	−	0.998476i	$-0.517578\pi$
$823$	−3.16682			−0.110389			−0.0551943	+	0.998476i	$0.517578\pi$
$824$		0			0
$825$		−	23.2052i		−	0.807901i
$826$		0			0
$827$			6.27434i			0.218180i	0.994032	+	0.109090i	$0.0347937\pi$
$827$			6.27434i			0.218180i	−0.994032	+	0.109090i	$0.965206\pi$
$828$		0			0
$829$	11.5935			0.402660			0.201330	−	0.979523i	$-0.435474\pi$
$829$	11.5935			0.402660			0.201330	+	0.979523i	$0.435474\pi$
$830$		0			0
$831$	−32.3376			−1.12178
$832$		0			0
$833$	4.36086			0.151095
$834$		0			0
$835$	14.0452			0.486056
$836$		0			0
$837$			4.07828i			0.140966i
$838$		0			0
$839$		−	17.7903i		−	0.614188i	−0.951679	−	0.307094i	$-0.900643\pi$
$839$		−	17.7903i		−	0.614188i	0.951679	−	0.307094i	$-0.0993566\pi$
$840$		0			0
$841$	−4.42891			−0.152721
$842$		0			0
$843$			89.6081i			3.08627i
$844$		0			0
$845$	5.65012	+	8.11749i	0.194370	+	0.279250i
$846$		0			0
$847$			8.43998i			0.290001i
$848$		0			0
$849$	−48.1081			−1.65106
$850$		0			0
$851$		−	30.2868i		−	1.03822i
$852$		0			0
$853$			4.84692i			0.165955i	0.996551	+	0.0829776i	$0.0264430\pi$
$853$			4.84692i			0.165955i	−0.996551	+	0.0829776i	$0.973557\pi$
$854$		0			0
$855$	43.3478			1.48246
$856$		0			0
$857$	−37.4954			−1.28082			−0.640409	−	0.768034i	$-0.721235\pi$
$857$	−37.4954			−1.28082			−0.640409	+	0.768034i	$0.721235\pi$
$858$		0			0
$859$	−7.77239			−0.265191			−0.132595	−	0.991170i	$-0.542331\pi$
$859$	−7.77239			−0.265191			−0.132595	+	0.991170i	$0.542331\pi$
$860$		0			0
$861$	−2.93814			−0.100132
$862$		0			0
$863$			7.76065i			0.264175i	0.991238	+	0.132088i	$0.0421681\pi$
$863$			7.76065i			0.264175i	−0.991238	+	0.132088i	$0.957832\pi$
$864$		0			0
$865$		−	14.0968i		−	0.479305i
$866$		0			0
$867$	−6.61695			−0.224723
$868$		0			0
$869$		−	15.1689i		−	0.514571i
$870$		0			0
$871$	−16.5914	+	5.20572i	−0.562178	+	0.176389i
$872$		0			0
$873$			49.0463i			1.65997i
$874$		0			0
$875$	7.16756			0.242308
$876$		0			0
$877$			6.85760i			0.231565i	0.993275	+	0.115782i	$0.0369375\pi$
$877$			6.85760i			0.231565i	−0.993275	+	0.115782i	$0.963063\pi$
$878$		0			0
$879$			10.8541i			0.366101i
$880$		0			0
$881$	−12.9270			−0.435521			−0.217761	−	0.976002i	$-0.569875\pi$
$881$	−12.9270			−0.435521			−0.217761	+	0.976002i	$0.569875\pi$
$882$		0			0
$883$	−31.5595			−1.06206			−0.531031	−	0.847353i	$-0.678195\pi$
$883$	−31.5595			−1.06206			−0.531031	+	0.847353i	$0.678195\pi$
$884$		0			0
$885$	8.72923			0.293430
$886$		0			0
$887$	53.3613			1.79170			0.895848	−	0.444361i	$-0.146569\pi$
$887$	53.3613			1.79170			0.895848	+	0.444361i	$0.146569\pi$
$888$		0			0
$889$		−	10.6000i		−	0.355513i
$890$		0			0
$891$			44.7175i			1.49809i
$892$		0			0
$893$	41.3962			1.38527
$894$		0			0
$895$		−	6.00857i		−	0.200844i
$896$		0			0
$897$	22.6073	+	72.0529i	0.754836	+	2.40577i
$898$		0			0
$899$			1.29444i			0.0431719i
$900$		0			0
$901$	−38.4313			−1.28033
$902$		0			0
$903$			13.7273i			0.456815i
$904$		0			0
$905$		−	0.438870i		−	0.0145885i
$906$		0			0
$907$	−42.1777			−1.40049			−0.700244	−	0.713904i	$-0.746925\pi$
$907$	−42.1777			−1.40049			−0.700244	+	0.713904i	$0.746925\pi$
$908$		0			0
$909$	−120.646			−4.00158
$910$		0			0
$911$	−17.0673			−0.565465			−0.282732	−	0.959199i	$-0.591241\pi$
$911$	−17.0673			−0.565465			−0.282732	+	0.959199i	$0.591241\pi$
$912$		0			0
$913$	1.73490			0.0574169
$914$		0			0
$915$			14.4152i			0.476551i
$916$		0			0
$917$		−	16.2889i		−	0.537908i
$918$		0			0
$919$	−53.7739			−1.77384			−0.886919	−	0.461924i	$-0.847159\pi$
$919$	−53.7739			−1.77384			−0.886919	+	0.461924i	$0.847159\pi$
$920$		0			0
$921$			102.226i			3.36845i
$922$		0			0
$923$	−25.6523	+	8.04866i	−0.844355	+	0.264925i
$924$		0			0
$925$		−	20.9724i		−	0.689568i
$926$		0			0
$927$	−31.0849			−1.02096
$928$		0			0
$929$		−	33.7464i		−	1.10718i	−0.832788	−	0.553592i	$-0.813257\pi$
$929$		−	33.7464i		−	1.10718i	0.832788	−	0.553592i	$-0.186743\pi$
$930$		0			0
$931$		−	7.34163i		−	0.240612i
$932$		0			0
$933$	−65.9476			−2.15903
$934$		0			0
$935$	−5.30834			−0.173601
$936$		0			0
$937$	−52.3784			−1.71113			−0.855563	−	0.517698i	$-0.826789\pi$
$937$	−52.3784			−1.71113			−0.855563	+	0.517698i	$0.826789\pi$
$938$		0			0
$939$	105.131			3.43082
$940$		0			0
$941$			30.6587i			0.999446i	0.866185	+	0.499723i	$0.166565\pi$
$941$			30.6587i			0.999446i	−0.866185	+	0.499723i	$0.833435\pi$
$942$		0			0
$943$		−	5.71866i		−	0.186225i
$944$		0			0
$945$	−11.8816			−0.386507
$946$		0			0
$947$		−	31.4545i		−	1.02213i	−0.859541	−	0.511067i	$-0.829250\pi$
$947$		−	31.4545i		−	1.02213i	0.859541	−	0.511067i	$-0.170750\pi$
$948$		0			0
$949$	13.2147	−	4.14626i	0.428968	−	0.134593i
$950$		0			0
$951$		−	10.0511i		−	0.325930i
$952$		0			0
$953$	−13.2590			−0.429503			−0.214751	−	0.976669i	$-0.568894\pi$
$953$	−13.2590			−0.429503			−0.214751	+	0.976669i	$0.568894\pi$
$954$		0			0
$955$			1.77287i			0.0573689i
$956$		0			0
$957$			26.0170i			0.841010i
$958$		0			0
$959$	−5.67900			−0.183384
$960$		0			0
$961$	30.9318			0.997800
$962$		0			0
$963$	−49.1593			−1.58414
$964$		0			0
$965$	−5.41215			−0.174223
$966$		0			0
$967$			13.4181i			0.431498i	0.976449	+	0.215749i	$0.0692194\pi$
$967$			13.4181i			0.431498i	−0.976449	+	0.215749i	$0.930781\pi$
$968$		0			0
$969$			105.024i			3.37385i
$970$		0			0
$971$	−9.65428			−0.309821			−0.154910	−	0.987929i	$-0.549509\pi$
$971$	−9.65428			−0.309821			−0.154910	+	0.987929i	$0.549509\pi$
$972$		0			0
$973$			16.0196i			0.513564i
$974$		0			0
$975$	15.6547	+	49.8937i	0.501350	+	1.59788i
$976$		0			0
$977$			12.4119i			0.397092i	0.980092	+	0.198546i	$0.0636220\pi$
$977$			12.4119i			0.397092i	−0.980092	+	0.198546i	$0.936378\pi$
$978$		0			0
$979$	−26.7179			−0.853907
$980$		0			0
$981$			90.9697i			2.90444i
$982$		0			0
$983$			13.2171i			0.421560i	0.977534	+	0.210780i	$0.0676004\pi$
$983$			13.2171i			0.421560i	−0.977534	+	0.210780i	$0.932400\pi$
$984$		0			0
$985$	15.6962			0.500121
$986$		0			0
$987$	−18.4966			−0.588753
$988$		0			0
$989$	−26.7181			−0.849586
$990$		0			0
$991$	−5.18039			−0.164561			−0.0822803	−	0.996609i	$-0.526220\pi$
$991$	−5.18039			−0.164561			−0.0822803	+	0.996609i	$0.526220\pi$
$992$		0			0
$993$		−	96.0137i		−	3.04690i
$994$		0			0
$995$			1.47218i			0.0466712i
$996$		0			0
$997$	−22.6948			−0.718752			−0.359376	−	0.933193i	$-0.617010\pi$
$997$	−22.6948			−0.718752			−0.359376	+	0.933193i	$0.617010\pi$
$998$		0			0
$999$			74.0827i			2.34387i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.k.i.1793.1	✓	14
4.3	odd	2		2912.2.k.j.1793.13	yes	14
13.12	even	2	inner	2912.2.k.i.1793.2	yes	14
52.51	odd	2		2912.2.k.j.1793.14	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2912.2.k.i.1793.1	✓	14	1.1	even	1	trivial
2912.2.k.i.1793.2	yes	14	13.12	even	2	inner
2912.2.k.j.1793.13	yes	14	4.3	odd	2
2912.2.k.j.1793.14	yes	14	52.51	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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.28037 q^{3} -0.760790i q^{5} +1.00000i q^{7} +7.76086 q^{9} +1.60001i q^{11} +(-1.07939 - 3.44019i) q^{13} +2.49568i q^{15} -4.36086 q^{17} +7.34163i q^{19} -3.28037i q^{21} +6.38477 q^{23} +4.42120 q^{25} -15.6174 q^{27} -4.95692 q^{29} -0.261137i q^{31} -5.24862i q^{33} +0.760790 q^{35} -4.74360i q^{37} +(3.54082 + 11.2851i) q^{39} -0.895672i q^{41} -4.18466 q^{43} -5.90439i q^{45} -5.63856i q^{47} -1.00000 q^{49} +14.3053 q^{51} +8.81277 q^{53} +1.21727 q^{55} -24.0833i q^{57} -3.49774i q^{59} +5.77605 q^{61} +7.76086i q^{63} +(-2.61726 + 0.821193i) q^{65} -4.82282i q^{67} -20.9444 q^{69} -7.45664i q^{71} +3.84128i q^{73} -14.5032 q^{75} -1.60001 q^{77} -9.48055 q^{79} +27.9483 q^{81} -1.08431i q^{83} +3.31770i q^{85} +16.2606 q^{87} +16.6986i q^{89} +(3.44019 - 1.07939i) q^{91} +0.856628i q^{93} +5.58544 q^{95} +6.31970i q^{97} +12.4174i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} + 32 q^{9} - 6 q^{13} + 36 q^{17} - 18 q^{23} - 18 q^{25} - 14 q^{27} - 16 q^{29} + 8 q^{35} + 34 q^{39} + 60 q^{43} - 14 q^{49} + 12 q^{51} + 4 q^{53} - 60 q^{55} + 6 q^{61} + 20 q^{65}+ \cdots + 68 q^{95}+O(q^{100}) \) 14 * q - 2 * q^3 + 32 * q^9 - 6 * q^13 + 36 * q^17 - 18 * q^23 - 18 * q^25 - 14 * q^27 - 16 * q^29 + 8 * q^35 + 34 * q^39 + 60 * q^43 - 14 * q^49 + 12 * q^51 + 4 * q^53 - 60 * q^55 + 6 * q^61 + 20 * q^65 - 2 * q^69 - 114 * q^75 - 2 * q^77 - 26 * q^79 + 22 * q^81 - 8 * q^87 + 16 * q^91 + 68 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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