LMFDB - Embedded newform 912.3.m.c.799.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,3,Mod(799,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("912.799");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	912.m (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:	$24.8502001097$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 61x^{10} + 1243x^{8} + 9566x^{6} + 25219x^{4} + 13245x^{2} + 841 $ x^12 + 61x^10 + 1243x^8 + 9566x^6 + 25219x^4 + 13245*x^2 + 841
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.5
Root			$-0.769253i$ of defining polynomial
Character	$\chi$	$=$	912.799
Dual form			912.3.m.c.799.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +5.32253 q^{5} -1.51700i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +5.32253 q^{5} -1.51700i q^{7} -3.00000 q^{9} -10.6649i q^{11} -20.0811 q^{13} -9.21889i q^{15} -25.9327 q^{17} -4.35890i q^{19} -2.62751 q^{21} +39.6351i q^{23} +3.32932 q^{25} +5.19615i q^{27} -44.4200 q^{29} -31.8897i q^{31} -18.4722 q^{33} -8.07425i q^{35} +1.58503 q^{37} +34.7814i q^{39} +27.5248 q^{41} -46.4321i q^{43} -15.9676 q^{45} -46.6506i q^{47} +46.6987 q^{49} +44.9168i q^{51} +46.4525 q^{53} -56.7643i q^{55} -7.54983 q^{57} -37.8483i q^{59} -90.5803 q^{61} +4.55099i q^{63} -106.882 q^{65} +100.284i q^{67} +68.6500 q^{69} +82.4118i q^{71} -98.7982 q^{73} -5.76656i q^{75} -16.1786 q^{77} +103.210i q^{79} +9.00000 q^{81} -164.542i q^{83} -138.028 q^{85} +76.9376i q^{87} +57.8355 q^{89} +30.4629i q^{91} -55.2346 q^{93} -23.2004i q^{95} +1.55097 q^{97} +31.9947i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{5} - 36 q^{9}+O(q^{10}) $ 12 * q - 4 * q^5 - 36 * q^9	$ 12 q - 4 q^{5} - 36 q^{9} + 20 q^{17} + 88 q^{25} - 24 q^{29} - 72 q^{33} - 40 q^{37} - 32 q^{41} + 12 q^{45} + 128 q^{49} + 184 q^{53} - 276 q^{61} - 232 q^{65} + 120 q^{69} - 92 q^{73} + 308 q^{77} + 108 q^{81} - 244 q^{85} + 72 q^{89} + 144 q^{93} - 280 q^{97}+O(q^{100}) $ 12 * q - 4 * q^5 - 36 * q^9 + 20 * q^17 + 88 * q^25 - 24 * q^29 - 72 * q^33 - 40 * q^37 - 32 * q^41 + 12 * q^45 + 128 * q^49 + 184 * q^53 - 276 * q^61 - 232 * q^65 + 120 * q^69 - 92 * q^73 + 308 * q^77 + 108 * q^81 - 244 * q^85 + 72 * q^89 + 144 * q^93 - 280 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	5.32253	1.06451	0.532253	−	0.846585i	$-0.321346\pi$
$5$	5.32253	1.06451	0.532253	+	0.846585i	$0.321346\pi$
$6$	0	0
$7$	− 1.51700i	− 0.216714i	−0.994112	−	0.108357i	$-0.965441\pi$
$7$	− 1.51700i	− 0.216714i	0.994112	−	0.108357i	$-0.0345589\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 10.6649i	− 0.969537i	−0.874643	−	0.484768i	$-0.838904\pi$
$11$	− 10.6649i	− 0.969537i	0.874643	−	0.484768i	$-0.161096\pi$
$12$	0	0
$13$	−20.0811	−1.54470	−0.772349	−	0.635199i	$-0.780918\pi$
$13$	−20.0811	−1.54470	−0.772349	+	0.635199i	$0.780918\pi$
$14$	0	0
$15$	− 9.21889i	− 0.614593i
$16$	0	0
$17$	−25.9327	−1.52545	−0.762727	−	0.646721i	$-0.776140\pi$
$17$	−25.9327	−1.52545	−0.762727	+	0.646721i	$0.776140\pi$
$18$	0	0
$19$	− 4.35890i	− 0.229416i
$19$	− 4.35890i	− 0.229416i
$20$	0	0
$21$	−2.62751	−0.125120
$22$	0	0
$23$	39.6351i	1.72327i	0.507532	+	0.861633i	$0.330558\pi$
$23$	39.6351i	1.72327i	−0.507532	+	0.861633i	$0.669442\pi$
$24$	0	0
$25$	3.32932	0.133173
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−44.4200	−1.53172	−0.765861	−	0.643006i	$-0.777687\pi$
$29$	−44.4200	−1.53172	−0.765861	+	0.643006i	$0.777687\pi$
$30$	0	0
$31$	− 31.8897i	− 1.02870i	−0.857580	−	0.514350i	$-0.828033\pi$
$31$	− 31.8897i	− 1.02870i	0.857580	−	0.514350i	$-0.171967\pi$
$32$	0	0
$33$	−18.4722	−0.559762
$34$	0	0
$35$	− 8.07425i	− 0.230693i
$36$	0	0
$37$	1.58503	0.0428387	0.0214194	−	0.999771i	$-0.493181\pi$
$37$	1.58503	0.0428387	0.0214194	+	0.999771i	$0.493181\pi$
$38$	0	0
$39$	34.7814i	0.891831i
$40$	0	0
$41$	27.5248	0.671336	0.335668	−	0.941980i	$-0.391038\pi$
$41$	27.5248	0.671336	0.335668	+	0.941980i	$0.391038\pi$
$42$	0	0
$43$	− 46.4321i	− 1.07982i	−0.841724	−	0.539908i	$-0.818459\pi$
$43$	− 46.4321i	− 1.07982i	0.841724	−	0.539908i	$-0.181541\pi$
$44$	0	0
$45$	−15.9676	−0.354835
$46$	0	0
$47$	− 46.6506i	− 0.992566i	−0.868161	−	0.496283i	$-0.834698\pi$
$47$	− 46.6506i	− 0.992566i	0.868161	−	0.496283i	$-0.165302\pi$
$48$	0	0
$49$	46.6987	0.953035
$50$	0	0
$51$	44.9168i	0.880721i
$52$	0	0
$53$	46.4525	0.876462	0.438231	−	0.898862i	$-0.355605\pi$
$53$	46.4525	0.876462	0.438231	+	0.898862i	$0.355605\pi$
$54$	0	0
$55$	− 56.7643i	− 1.03208i
$56$	0	0
$57$	−7.54983	−0.132453
$58$	0	0
$59$	− 37.8483i	− 0.641496i	−0.947165	−	0.320748i	$-0.896066\pi$
$59$	− 37.8483i	− 0.641496i	0.947165	−	0.320748i	$-0.103934\pi$
$60$	0	0
$61$	−90.5803	−1.48492	−0.742462	−	0.669888i	$-0.766342\pi$
$61$	−90.5803	−1.48492	−0.742462	+	0.669888i	$0.766342\pi$
$62$	0	0
$63$	4.55099i	0.0722379i
$64$	0	0
$65$	−106.882	−1.64434
$66$	0	0
$67$	100.284i	1.49678i	0.663258	+	0.748391i	$0.269173\pi$
$67$	100.284i	1.49678i	−0.663258	+	0.748391i	$0.730827\pi$
$68$	0	0
$69$	68.6500	0.994928
$70$	0	0
$71$	82.4118i	1.16073i	0.814357	+	0.580365i	$0.197090\pi$
$71$	82.4118i	1.16073i	−0.814357	+	0.580365i	$0.802910\pi$
$72$	0	0
$73$	−98.7982	−1.35340	−0.676700	−	0.736259i	$-0.736591\pi$
$73$	−98.7982	−1.35340	−0.676700	+	0.736259i	$0.736591\pi$
$74$	0	0
$75$	− 5.76656i	− 0.0768874i
$76$	0	0
$77$	−16.1786	−0.210112
$78$	0	0
$79$	103.210i	1.30645i	0.757164	+	0.653225i	$0.226585\pi$
$79$	103.210i	1.30645i	−0.757164	+	0.653225i	$0.773415\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 164.542i	− 1.98243i	−0.132250	−	0.991216i	$-0.542220\pi$
$83$	− 164.542i	− 1.98243i	0.132250	−	0.991216i	$-0.457780\pi$
$84$	0	0
$85$	−138.028	−1.62385
$86$	0	0
$87$	76.9376i	0.884340i
$88$	0	0
$89$	57.8355	0.649837	0.324918	−	0.945742i	$-0.394663\pi$
$89$	57.8355	0.649837	0.324918	+	0.945742i	$0.394663\pi$
$90$	0	0
$91$	30.4629i	0.334757i
$92$	0	0
$93$	−55.2346	−0.593921
$94$	0	0
$95$	− 23.2004i	− 0.244214i
$96$	0	0
$97$	1.55097	0.0159893	0.00799467	−	0.999968i	$-0.497455\pi$
$97$	1.55097	0.0159893	0.00799467	+	0.999968i	$0.497455\pi$
$98$	0	0
$99$	31.9947i	0.323179i
$100$	0	0
$101$	−58.9060	−0.583227	−0.291614	−	0.956536i	$-0.594192\pi$
$101$	−58.9060	−0.583227	−0.291614	+	0.956536i	$0.594192\pi$
$102$	0	0
$103$	− 154.851i	− 1.50341i	−0.659499	−	0.751705i	$-0.729232\pi$
$103$	− 154.851i	− 1.50341i	0.659499	−	0.751705i	$-0.270768\pi$
$104$	0	0
$105$	−13.9850	−0.133191
$106$	0	0
$107$	6.64113i	0.0620666i	0.999518	+	0.0310333i	$0.00987980\pi$
$107$	6.64113i	0.0620666i	−0.999518	+	0.0310333i	$0.990120\pi$
$108$	0	0
$109$	15.5942	0.143066	0.0715331	−	0.997438i	$-0.477211\pi$
$109$	15.5942	0.143066	0.0715331	+	0.997438i	$0.477211\pi$
$110$	0	0
$111$	− 2.74536i	− 0.0247329i
$112$	0	0
$113$	−101.858	−0.901395	−0.450698	−	0.892677i	$-0.648825\pi$
$113$	−101.858	−0.901395	−0.450698	+	0.892677i	$0.648825\pi$
$114$	0	0
$115$	210.959i	1.83443i
$116$	0	0
$117$	60.2432	0.514899
$118$	0	0
$119$	39.3398i	0.330587i
$120$	0	0
$121$	7.25981	0.0599985
$122$	0	0
$123$	− 47.6743i	− 0.387596i
$124$	0	0
$125$	−115.343	−0.922743
$126$	0	0
$127$	20.6143i	0.162317i	0.996701	+	0.0811587i	$0.0258621\pi$
$127$	20.6143i	0.162317i	−0.996701	+	0.0811587i	$0.974138\pi$
$128$	0	0
$129$	−80.4228	−0.623432
$130$	0	0
$131$	− 174.075i	− 1.32881i	−0.747371	−	0.664407i	$-0.768684\pi$
$131$	− 174.075i	− 1.32881i	0.747371	−	0.664407i	$-0.231316\pi$
$132$	0	0
$133$	−6.61243	−0.0497175
$134$	0	0
$135$	27.6567i	0.204864i
$136$	0	0
$137$	−196.739	−1.43605	−0.718025	−	0.696017i	$-0.754954\pi$
$137$	−196.739	−1.43605	−0.718025	+	0.696017i	$0.754954\pi$
$138$	0	0
$139$	193.372i	1.39117i	0.718446	+	0.695583i	$0.244854\pi$
$139$	193.372i	1.39117i	−0.718446	+	0.695583i	$0.755146\pi$
$140$	0	0
$141$	−80.8012	−0.573058
$142$	0	0
$143$	214.163i	1.49764i
$144$	0	0
$145$	−236.427	−1.63053
$146$	0	0
$147$	− 80.8846i	− 0.550235i
$148$	0	0
$149$	89.5244	0.600835	0.300418	−	0.953808i	$-0.402874\pi$
$149$	89.5244	0.600835	0.300418	+	0.953808i	$0.402874\pi$
$150$	0	0
$151$	111.282i	0.736967i	0.929634	+	0.368483i	$0.120123\pi$
$151$	111.282i	0.736967i	−0.929634	+	0.368483i	$0.879877\pi$
$152$	0	0
$153$	77.7981	0.508485
$154$	0	0
$155$	− 169.734i	− 1.09506i
$156$	0	0
$157$	206.598	1.31591	0.657955	−	0.753057i	$-0.271422\pi$
$157$	206.598	1.31591	0.657955	+	0.753057i	$0.271422\pi$
$158$	0	0
$159$	− 80.4581i	− 0.506026i
$160$	0	0
$161$	60.1263	0.373455
$162$	0	0
$163$	− 270.012i	− 1.65651i	−0.560349	−	0.828256i	$-0.689333\pi$
$163$	− 270.012i	− 1.65651i	0.560349	−	0.828256i	$-0.310667\pi$
$164$	0	0
$165$	−98.3186	−0.595870
$166$	0	0
$167$	− 312.111i	− 1.86893i	−0.356059	−	0.934463i	$-0.615880\pi$
$167$	− 312.111i	− 1.86893i	0.356059	−	0.934463i	$-0.384120\pi$
$168$	0	0
$169$	234.249	1.38609
$170$	0	0
$171$	13.0767i	0.0764719i
$172$	0	0
$173$	−3.57648	−0.0206733	−0.0103367	−	0.999947i	$-0.503290\pi$
$173$	−3.57648	−0.0206733	−0.0103367	+	0.999947i	$0.503290\pi$
$174$	0	0
$175$	− 5.05057i	− 0.0288604i
$176$	0	0
$177$	−65.5552	−0.370368
$178$	0	0
$179$	− 75.6210i	− 0.422464i	−0.977436	−	0.211232i	$-0.932252\pi$
$179$	− 75.6210i	− 0.422464i	0.977436	−	0.211232i	$-0.0677475\pi$
$180$	0	0
$181$	274.805	1.51826	0.759129	−	0.650940i	$-0.225625\pi$
$181$	274.805	1.51826	0.759129	+	0.650940i	$0.225625\pi$
$182$	0	0
$183$	156.890i	0.857321i
$184$	0	0
$185$	8.43638	0.0456021
$186$	0	0
$187$	276.570i	1.47898i
$188$	0	0
$189$	7.88254	0.0417065
$190$	0	0
$191$	− 113.605i	− 0.594790i	−0.954754	−	0.297395i	$-0.903882\pi$
$191$	− 113.605i	− 0.594790i	0.954754	−	0.297395i	$-0.0961179\pi$
$192$	0	0
$193$	−366.062	−1.89670	−0.948348	−	0.317231i	$-0.897247\pi$
$193$	−366.062	−1.89670	−0.948348	+	0.317231i	$0.897247\pi$
$194$	0	0
$195$	185.125i	0.949360i
$196$	0	0
$197$	133.490	0.677613	0.338807	−	0.940856i	$-0.389977\pi$
$197$	133.490	0.677613	0.338807	+	0.940856i	$0.389977\pi$
$198$	0	0
$199$	63.1501i	0.317337i	0.987332	+	0.158669i	$0.0507201\pi$
$199$	63.1501i	0.317337i	−0.987332	+	0.158669i	$0.949280\pi$
$200$	0	0
$201$	173.698	0.864167
$202$	0	0
$203$	67.3849i	0.331945i
$204$	0	0
$205$	146.501	0.714641
$206$	0	0
$207$	− 118.905i	− 0.574422i
$208$	0	0
$209$	−46.4872	−0.222427
$210$	0	0
$211$	− 43.9449i	− 0.208270i	−0.994563	−	0.104135i	$-0.966793\pi$
$211$	− 43.9449i	− 0.208270i	0.994563	−	0.104135i	$-0.0332073\pi$
$212$	0	0
$213$	142.741	0.670147
$214$	0	0
$215$	− 247.136i	− 1.14947i
$216$	0	0
$217$	−48.3765	−0.222933
$218$	0	0
$219$	171.123i	0.781386i
$220$	0	0
$221$	520.756	2.35636
$222$	0	0
$223$	304.568i	1.36578i	0.730523	+	0.682889i	$0.239277\pi$
$223$	304.568i	1.36578i	−0.730523	+	0.682889i	$0.760723\pi$
$224$	0	0
$225$	−9.98797	−0.0443910
$226$	0	0
$227$	− 243.845i	− 1.07421i	−0.843516	−	0.537104i	$-0.819518\pi$
$227$	− 243.845i	− 1.07421i	0.843516	−	0.537104i	$-0.180482\pi$
$228$	0	0
$229$	269.966	1.17889	0.589445	−	0.807809i	$-0.299347\pi$
$229$	269.966	1.17889	0.589445	+	0.807809i	$0.299347\pi$
$230$	0	0
$231$	28.0222i	0.121308i
$232$	0	0
$233$	−60.1384	−0.258105	−0.129052	−	0.991638i	$-0.541194\pi$
$233$	−60.1384	−0.258105	−0.129052	+	0.991638i	$0.541194\pi$
$234$	0	0
$235$	− 248.299i	− 1.05659i
$236$	0	0
$237$	178.764	0.754280
$238$	0	0
$239$	− 364.980i	− 1.52711i	−0.645742	−	0.763556i	$-0.723452\pi$
$239$	− 364.980i	− 1.52711i	0.645742	−	0.763556i	$-0.276548\pi$
$240$	0	0
$241$	−231.191	−0.959298	−0.479649	−	0.877460i	$-0.659236\pi$
$241$	−231.191	−0.959298	−0.479649	+	0.877460i	$0.659236\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	248.555	1.01451
$246$	0	0
$247$	87.5313i	0.354378i
$248$	0	0
$249$	−284.995	−1.14456
$250$	0	0
$251$	− 19.5801i	− 0.0780085i	−0.999239	−	0.0390042i	$-0.987581\pi$
$251$	− 19.5801i	− 0.0780085i	0.999239	−	0.0390042i	$-0.0124186\pi$
$252$	0	0
$253$	422.705	1.67077
$254$	0	0
$255$	239.071i	0.937533i
$256$	0	0
$257$	−17.1006	−0.0665391	−0.0332696	−	0.999446i	$-0.510592\pi$
$257$	−17.1006	−0.0665391	−0.0332696	+	0.999446i	$0.510592\pi$
$258$	0	0
$259$	− 2.40449i	− 0.00928373i
$260$	0	0
$261$	133.260	0.510574
$262$	0	0
$263$	− 57.3106i	− 0.217911i	−0.994047	−	0.108956i	$-0.965249\pi$
$263$	− 57.3106i	− 0.217911i	0.994047	−	0.108956i	$-0.0347506\pi$
$264$	0	0
$265$	247.245	0.932999
$266$	0	0
$267$	− 100.174i	− 0.375183i
$268$	0	0
$269$	−28.7230	−0.106777	−0.0533885	−	0.998574i	$-0.517002\pi$
$269$	−28.7230	−0.106777	−0.0533885	+	0.998574i	$0.517002\pi$
$270$	0	0
$271$	114.957i	0.424197i	0.977248	+	0.212099i	$0.0680298\pi$
$271$	114.957i	0.424197i	−0.977248	+	0.212099i	$0.931970\pi$
$272$	0	0
$273$	52.7632	0.193272
$274$	0	0
$275$	− 35.5069i	− 0.129116i
$276$	0	0
$277$	−464.495	−1.67688	−0.838439	−	0.544995i	$-0.816531\pi$
$277$	−464.495	−1.67688	−0.838439	+	0.544995i	$0.816531\pi$
$278$	0	0
$279$	95.6692i	0.342900i
$280$	0	0
$281$	131.163	0.466772	0.233386	−	0.972384i	$-0.425019\pi$
$281$	131.163	0.466772	0.233386	+	0.972384i	$0.425019\pi$
$282$	0	0
$283$	144.862i	0.511881i	0.966693	+	0.255940i	$0.0823851\pi$
$283$	144.862i	0.511881i	−0.966693	+	0.255940i	$0.917615\pi$
$284$	0	0
$285$	−40.1842	−0.140997
$286$	0	0
$287$	− 41.7549i	− 0.145488i
$288$	0	0
$289$	383.506	1.32701
$290$	0	0
$291$	− 2.68635i	− 0.00923145i
$292$	0	0
$293$	430.757	1.47016	0.735080	−	0.677981i	$-0.237145\pi$
$293$	430.757	1.47016	0.735080	+	0.677981i	$0.237145\pi$
$294$	0	0
$295$	− 201.449i	− 0.682877i
$296$	0	0
$297$	55.4165	0.186587
$298$	0	0
$299$	− 795.915i	− 2.66192i
$300$	0	0
$301$	−70.4373	−0.234011
$302$	0	0
$303$	102.028i	0.336726i
$304$	0	0
$305$	−482.117	−1.58071
$306$	0	0
$307$	578.208i	1.88341i	0.336436	+	0.941706i	$0.390778\pi$
$307$	578.208i	1.88341i	−0.336436	+	0.941706i	$0.609222\pi$
$308$	0	0
$309$	−268.210	−0.867994
$310$	0	0
$311$	− 99.1339i	− 0.318759i	−0.987217	−	0.159379i	$-0.949051\pi$
$311$	− 99.1339i	− 0.318759i	0.987217	−	0.159379i	$-0.0509493\pi$
$312$	0	0
$313$	−294.871	−0.942080	−0.471040	−	0.882112i	$-0.656121\pi$
$313$	−294.871	−0.942080	−0.471040	+	0.882112i	$0.656121\pi$
$314$	0	0
$315$	24.2228i	0.0768976i
$316$	0	0
$317$	440.217	1.38870	0.694348	−	0.719639i	$-0.255693\pi$
$317$	440.217	1.38870	0.694348	+	0.719639i	$0.255693\pi$
$318$	0	0
$319$	473.735i	1.48506i
$320$	0	0
$321$	11.5028	0.0358342
$322$	0	0
$323$	113.038i	0.349963i
$324$	0	0
$325$	−66.8564	−0.205712
$326$	0	0
$327$	− 27.0100i	− 0.0825993i
$328$	0	0
$329$	−70.7688	−0.215103
$330$	0	0
$331$	− 325.854i	− 0.984454i	−0.870467	−	0.492227i	$-0.836183\pi$
$331$	− 325.854i	− 0.984454i	0.870467	−	0.492227i	$-0.163817\pi$
$332$	0	0
$333$	−4.75510	−0.0142796
$334$	0	0
$335$	533.767i	1.59333i
$336$	0	0
$337$	63.9035	0.189625	0.0948123	−	0.995495i	$-0.469775\pi$
$337$	63.9035	0.189625	0.0948123	+	0.995495i	$0.469775\pi$
$338$	0	0
$339$	176.423i	0.520421i
$340$	0	0
$341$	−340.101	−0.997363
$342$	0	0
$343$	− 145.174i	− 0.423249i
$344$	0	0
$345$	365.392	1.05911
$346$	0	0
$347$	439.669i	1.26706i	0.773720	+	0.633528i	$0.218394\pi$
$347$	439.669i	1.26706i	−0.773720	+	0.633528i	$0.781606\pi$
$348$	0	0
$349$	459.775	1.31741	0.658703	−	0.752403i	$-0.271105\pi$
$349$	459.775	1.31741	0.658703	+	0.752403i	$0.271105\pi$
$350$	0	0
$351$	− 104.344i	− 0.297277i
$352$	0	0
$353$	116.450	0.329887	0.164943	−	0.986303i	$-0.447256\pi$
$353$	116.450	0.329887	0.164943	+	0.986303i	$0.447256\pi$
$354$	0	0
$355$	438.639i	1.23560i
$356$	0	0
$357$	68.1385	0.190864
$358$	0	0
$359$	60.8754i	0.169569i	0.996399	+	0.0847847i	$0.0270202\pi$
$359$	60.8754i	0.169569i	−0.996399	+	0.0847847i	$0.972980\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	− 12.5744i	− 0.0346401i
$364$	0	0
$365$	−525.856	−1.44070
$366$	0	0
$367$	− 653.185i	− 1.77980i	−0.456159	−	0.889898i	$-0.650775\pi$
$367$	− 653.185i	− 1.77980i	0.456159	−	0.889898i	$-0.349225\pi$
$368$	0	0
$369$	−82.5743	−0.223779
$370$	0	0
$371$	− 70.4682i	− 0.189941i
$372$	0	0
$373$	173.776	0.465887	0.232944	−	0.972490i	$-0.425164\pi$
$373$	173.776	0.465887	0.232944	+	0.972490i	$0.425164\pi$
$374$	0	0
$375$	199.780i	0.532746i
$376$	0	0
$377$	892.000	2.36605
$378$	0	0
$379$	− 120.525i	− 0.318007i	−0.987278	−	0.159003i	$-0.949172\pi$
$379$	− 120.525i	− 0.318007i	0.987278	−	0.159003i	$-0.0508281\pi$
$380$	0	0
$381$	35.7050	0.0937140
$382$	0	0
$383$	212.557i	0.554980i	0.960729	+	0.277490i	$0.0895025\pi$
$383$	212.557i	0.554980i	−0.960729	+	0.277490i	$0.910497\pi$
$384$	0	0
$385$	−86.1111	−0.223665
$386$	0	0
$387$	139.296i	0.359939i
$388$	0	0
$389$	203.069	0.522028	0.261014	−	0.965335i	$-0.415943\pi$
$389$	203.069	0.522028	0.261014	+	0.965335i	$0.415943\pi$
$390$	0	0
$391$	− 1027.85i	− 2.62876i
$392$	0	0
$393$	−301.506	−0.767191
$394$	0	0
$395$	549.336i	1.39072i
$396$	0	0
$397$	−254.424	−0.640867	−0.320433	−	0.947271i	$-0.603828\pi$
$397$	−254.424	−0.640867	−0.320433	+	0.947271i	$0.603828\pi$
$398$	0	0
$399$	11.4531i	0.0287044i
$400$	0	0
$401$	−1.25127	−0.00312037	−0.00156019	−	0.999999i	$-0.500497\pi$
$401$	−1.25127	−0.00312037	−0.00156019	+	0.999999i	$0.500497\pi$
$402$	0	0
$403$	640.379i	1.58903i
$404$	0	0
$405$	47.9028	0.118278
$406$	0	0
$407$	− 16.9042i	− 0.0415337i
$408$	0	0
$409$	333.713	0.815923	0.407962	−	0.912999i	$-0.366240\pi$
$409$	333.713	0.815923	0.407962	+	0.912999i	$0.366240\pi$
$410$	0	0
$411$	340.762i	0.829104i
$412$	0	0
$413$	−57.4157	−0.139021
$414$	0	0
$415$	− 875.779i	− 2.11031i
$416$	0	0
$417$	334.930	0.803190
$418$	0	0
$419$	87.5217i	0.208882i	0.994531	+	0.104441i	$0.0333054\pi$
$419$	87.5217i	0.208882i	−0.994531	+	0.104441i	$0.966695\pi$
$420$	0	0
$421$	390.964	0.928656	0.464328	−	0.885663i	$-0.346296\pi$
$421$	390.964	0.928656	0.464328	+	0.885663i	$0.346296\pi$
$422$	0	0
$423$	139.952i	0.330855i
$424$	0	0
$425$	−86.3384	−0.203149
$426$	0	0
$427$	137.410i	0.321803i
$428$	0	0
$429$	370.941	0.864663
$430$	0	0
$431$	− 128.733i	− 0.298685i	−0.988786	−	0.149342i	$-0.952284\pi$
$431$	− 128.733i	− 0.298685i	0.988786	−	0.149342i	$-0.0477157\pi$
$432$	0	0
$433$	−718.320	−1.65894	−0.829468	−	0.558554i	$-0.811357\pi$
$433$	−718.320	−1.65894	−0.829468	+	0.558554i	$0.811357\pi$
$434$	0	0
$435$	409.503i	0.941386i
$436$	0	0
$437$	172.766	0.395344
$438$	0	0
$439$	− 31.3496i	− 0.0714113i	−0.999362	−	0.0357057i	$-0.988632\pi$
$439$	− 31.3496i	− 0.0714113i	0.999362	−	0.0357057i	$-0.0113679\pi$
$440$	0	0
$441$	−140.096	−0.317678
$442$	0	0
$443$	− 162.714i	− 0.367300i	−0.982992	−	0.183650i	$-0.941209\pi$
$443$	− 162.714i	− 0.367300i	0.982992	−	0.183650i	$-0.0587913\pi$
$444$	0	0
$445$	307.831	0.691755
$446$	0	0
$447$	− 155.061i	− 0.346892i
$448$	0	0
$449$	37.7009	0.0839664	0.0419832	−	0.999118i	$-0.486632\pi$
$449$	37.7009	0.0839664	0.0419832	+	0.999118i	$0.486632\pi$
$450$	0	0
$451$	− 293.549i	− 0.650885i
$452$	0	0
$453$	192.746	0.425488
$454$	0	0
$455$	162.140i	0.356351i
$456$	0	0
$457$	224.412	0.491054	0.245527	−	0.969390i	$-0.421039\pi$
$457$	224.412	0.491054	0.245527	+	0.969390i	$0.421039\pi$
$458$	0	0
$459$	− 134.750i	− 0.293574i
$460$	0	0
$461$	−457.231	−0.991825	−0.495912	−	0.868373i	$-0.665166\pi$
$461$	−457.231	−0.991825	−0.495912	+	0.868373i	$0.665166\pi$
$462$	0	0
$463$	576.614i	1.24539i	0.782466	+	0.622693i	$0.213962\pi$
$463$	576.614i	1.24539i	−0.782466	+	0.622693i	$0.786038\pi$
$464$	0	0
$465$	−293.988	−0.632232
$466$	0	0
$467$	− 481.334i	− 1.03069i	−0.856982	−	0.515347i	$-0.827663\pi$
$467$	− 481.334i	− 1.03069i	0.856982	−	0.515347i	$-0.172337\pi$
$468$	0	0
$469$	152.131	0.324373
$470$	0	0
$471$	− 357.838i	− 0.759741i
$472$	0	0
$473$	−495.194	−1.04692
$474$	0	0
$475$	− 14.5122i	− 0.0305520i
$476$	0	0
$477$	−139.357	−0.292154
$478$	0	0
$479$	− 319.707i	− 0.667446i	−0.942671	−	0.333723i	$-0.891695\pi$
$479$	− 319.707i	− 0.667446i	0.942671	−	0.333723i	$-0.108305\pi$
$480$	0	0
$481$	−31.8291	−0.0661728
$482$	0	0
$483$	− 104.142i	− 0.215614i
$484$	0	0
$485$	8.25506	0.0170207
$486$	0	0
$487$	209.819i	0.430840i	0.976521	+	0.215420i	$0.0691121\pi$
$487$	209.819i	0.430840i	−0.976521	+	0.215420i	$0.930888\pi$
$488$	0	0
$489$	−467.674	−0.956388
$490$	0	0
$491$	− 494.297i	− 1.00671i	−0.864078	−	0.503357i	$-0.832098\pi$
$491$	− 494.297i	− 1.00671i	0.864078	−	0.503357i	$-0.167902\pi$
$492$	0	0
$493$	1151.93	2.33657
$494$	0	0
$495$	170.293i	0.344026i
$496$	0	0
$497$	125.018	0.251546
$498$	0	0
$499$	− 23.0217i	− 0.0461357i	−0.999734	−	0.0230678i	$-0.992657\pi$
$499$	− 23.0217i	− 0.0461357i	0.999734	−	0.0230678i	$-0.00734337\pi$
$500$	0	0
$501$	−540.592	−1.07903
$502$	0	0
$503$	− 164.552i	− 0.327142i	−0.986532	−	0.163571i	$-0.947699\pi$
$503$	− 164.552i	− 0.327142i	0.986532	−	0.163571i	$-0.0523012\pi$
$504$	0	0
$505$	−313.529	−0.620849
$506$	0	0
$507$	− 405.731i	− 0.800259i
$508$	0	0
$509$	318.792	0.626310	0.313155	−	0.949702i	$-0.398614\pi$
$509$	318.792	0.626310	0.313155	+	0.949702i	$0.398614\pi$
$510$	0	0
$511$	149.876i	0.293300i
$512$	0	0
$513$	22.6495	0.0441511
$514$	0	0
$515$	− 824.200i	− 1.60039i
$516$	0	0
$517$	−497.524	−0.962330
$518$	0	0
$519$	6.19465i	0.0119357i
$520$	0	0
$521$	−678.757	−1.30280	−0.651399	−	0.758736i	$-0.725817\pi$
$521$	−678.757	−1.30280	−0.651399	+	0.758736i	$0.725817\pi$
$522$	0	0
$523$	199.492i	0.381437i	0.981645	+	0.190719i	$0.0610818\pi$
$523$	199.492i	0.381437i	−0.981645	+	0.190719i	$0.938918\pi$
$524$	0	0
$525$	−8.74784	−0.0166626
$526$	0	0
$527$	826.987i	1.56924i
$528$	0	0
$529$	−1041.94	−1.96965
$530$	0	0
$531$	113.545i	0.213832i
$532$	0	0
$533$	−552.726	−1.03701
$534$	0	0
$535$	35.3476i	0.0660703i
$536$	0	0
$537$	−130.979	−0.243910
$538$	0	0
$539$	− 498.037i	− 0.924003i
$540$	0	0
$541$	−468.348	−0.865708	−0.432854	−	0.901464i	$-0.642493\pi$
$541$	−468.348	−0.865708	−0.432854	+	0.901464i	$0.642493\pi$
$542$	0	0
$543$	− 475.976i	− 0.876567i
$544$	0	0
$545$	83.0006	0.152295
$546$	0	0
$547$	93.4661i	0.170870i	0.996344	+	0.0854352i	$0.0272281\pi$
$547$	93.4661i	0.170870i	−0.996344	+	0.0854352i	$0.972772\pi$
$548$	0	0
$549$	271.741	0.494975
$550$	0	0
$551$	193.622i	0.351401i
$552$	0	0
$553$	156.568	0.283126
$554$	0	0
$555$	− 14.6122i	− 0.0263284i
$556$	0	0
$557$	296.185	0.531751	0.265876	−	0.964007i	$-0.414339\pi$
$557$	296.185	0.531751	0.265876	+	0.964007i	$0.414339\pi$
$558$	0	0
$559$	932.406i	1.66799i
$560$	0	0
$561$	479.033	0.853892
$562$	0	0
$563$	292.251i	0.519097i	0.965730	+	0.259548i	$0.0835737\pi$
$563$	292.251i	0.519097i	−0.965730	+	0.259548i	$0.916426\pi$
$564$	0	0
$565$	−542.141	−0.959541
$566$	0	0
$567$	− 13.6530i	− 0.0240793i
$568$	0	0
$569$	527.208	0.926551	0.463276	−	0.886214i	$-0.346674\pi$
$569$	527.208	0.926551	0.463276	+	0.886214i	$0.346674\pi$
$570$	0	0
$571$	75.0971i	0.131519i	0.997836	+	0.0657593i	$0.0209470\pi$
$571$	75.0971i	0.131519i	−0.997836	+	0.0657593i	$0.979053\pi$
$572$	0	0
$573$	−196.770	−0.343402
$574$	0	0
$575$	131.958i	0.229492i
$576$	0	0
$577$	870.821	1.50922	0.754611	−	0.656172i	$-0.227826\pi$
$577$	870.821	1.50922	0.754611	+	0.656172i	$0.227826\pi$
$578$	0	0
$579$	634.039i	1.09506i
$580$	0	0
$581$	−249.609	−0.429620
$582$	0	0
$583$	− 495.411i	− 0.849762i
$584$	0	0
$585$	320.646	0.548113
$586$	0	0
$587$	− 872.310i	− 1.48605i	−0.669265	−	0.743024i	$-0.733391\pi$
$587$	− 872.310i	− 1.48605i	0.669265	−	0.743024i	$-0.266609\pi$
$588$	0	0
$589$	−139.004	−0.236000
$590$	0	0
$591$	− 231.211i	− 0.391220i
$592$	0	0
$593$	−467.746	−0.788779	−0.394390	−	0.918943i	$-0.629044\pi$
$593$	−467.746	−0.788779	−0.394390	+	0.918943i	$0.629044\pi$
$594$	0	0
$595$	209.387i	0.351911i
$596$	0	0
$597$	109.379	0.183215
$598$	0	0
$599$	154.268i	0.257542i	0.991674	+	0.128771i	$0.0411032\pi$
$599$	154.268i	0.257542i	−0.991674	+	0.128771i	$0.958897\pi$
$600$	0	0
$601$	−77.3914	−0.128771	−0.0643855	−	0.997925i	$-0.520509\pi$
$601$	−77.3914	−0.128771	−0.0643855	+	0.997925i	$0.520509\pi$
$602$	0	0
$603$	− 300.853i	− 0.498927i
$604$	0	0
$605$	38.6406	0.0638687
$606$	0	0
$607$	91.0406i	0.149985i	0.997184	+	0.0749923i	$0.0238932\pi$
$607$	91.0406i	0.149985i	−0.997184	+	0.0749923i	$0.976107\pi$
$608$	0	0
$609$	116.714	0.191649
$610$	0	0
$611$	936.794i	1.53321i
$612$	0	0
$613$	−327.588	−0.534402	−0.267201	−	0.963641i	$-0.586099\pi$
$613$	−327.588	−0.534402	−0.267201	+	0.963641i	$0.586099\pi$
$614$	0	0
$615$	− 253.748i	− 0.412598i
$616$	0	0
$617$	−902.097	−1.46207	−0.731034	−	0.682341i	$-0.760962\pi$
$617$	−902.097	−1.46207	−0.731034	+	0.682341i	$0.760962\pi$
$618$	0	0
$619$	356.416i	0.575793i	0.957662	+	0.287896i	$0.0929559\pi$
$619$	356.416i	0.575793i	−0.957662	+	0.287896i	$0.907044\pi$
$620$	0	0
$621$	−205.950	−0.331643
$622$	0	0
$623$	− 87.7361i	− 0.140828i
$624$	0	0
$625$	−697.149	−1.11544
$626$	0	0
$627$	80.5183i	0.128418i
$628$	0	0
$629$	−41.1042	−0.0653485
$630$	0	0
$631$	759.264i	1.20327i	0.798771	+	0.601635i	$0.205484\pi$
$631$	759.264i	1.20327i	−0.798771	+	0.601635i	$0.794516\pi$
$632$	0	0
$633$	−76.1147	−0.120244
$634$	0	0
$635$	109.720i	0.172788i
$636$	0	0
$637$	−937.760	−1.47215
$638$	0	0
$639$	− 247.235i	− 0.386910i
$640$	0	0
$641$	−912.838	−1.42408	−0.712042	−	0.702136i	$-0.752230\pi$
$641$	−912.838	−1.42408	−0.712042	+	0.702136i	$0.752230\pi$
$642$	0	0
$643$	− 69.3051i	− 0.107784i	−0.998547	−	0.0538920i	$-0.982837\pi$
$643$	− 69.3051i	− 0.107784i	0.998547	−	0.0538920i	$-0.0171627\pi$
$644$	0	0
$645$	−428.053	−0.663647
$646$	0	0
$647$	382.031i	0.590465i	0.955425	+	0.295232i	$0.0953971\pi$
$647$	382.031i	0.590465i	−0.955425	+	0.295232i	$0.904603\pi$
$648$	0	0
$649$	−403.648	−0.621954
$650$	0	0
$651$	83.7906i	0.128711i
$652$	0	0
$653$	0.546434	0.000836806 0	0.000418403	−	1.00000i	$-0.499867\pi$
$653$	0.546434	0.000836806 0	0.000418403		1.00000i	$0.499867\pi$
$654$	0	0
$655$	− 926.517i	− 1.41453i
$656$	0	0
$657$	296.395	0.451133
$658$	0	0
$659$	− 1196.64i	− 1.81584i	−0.419143	−	0.907920i	$-0.637670\pi$
$659$	− 1196.64i	− 1.81584i	0.419143	−	0.907920i	$-0.362330\pi$
$660$	0	0
$661$	−665.059	−1.00614	−0.503070	−	0.864245i	$-0.667796\pi$
$661$	−665.059	−1.00614	−0.503070	+	0.864245i	$0.667796\pi$
$662$	0	0
$663$	− 901.977i	− 1.36045i
$664$	0	0
$665$	−35.1948	−0.0529246
$666$	0	0
$667$	− 1760.59i	− 2.63957i
$668$	0	0
$669$	527.528	0.788532
$670$	0	0
$671$	966.031i	1.43969i
$672$	0	0
$673$	178.037	0.264543	0.132271	−	0.991214i	$-0.457773\pi$
$673$	178.037	0.264543	0.132271	+	0.991214i	$0.457773\pi$
$674$	0	0
$675$	17.2997i	0.0256291i
$676$	0	0
$677$	−668.313	−0.987169	−0.493585	−	0.869698i	$-0.664314\pi$
$677$	−668.313	−0.987169	−0.493585	+	0.869698i	$0.664314\pi$
$678$	0	0
$679$	− 2.35281i	− 0.00346511i
$680$	0	0
$681$	−422.353	−0.620195
$682$	0	0
$683$	253.264i	0.370811i	0.982662	+	0.185405i	$0.0593598\pi$
$683$	253.264i	0.370811i	−0.982662	+	0.185405i	$0.940640\pi$
$684$	0	0
$685$	−1047.15	−1.52868
$686$	0	0
$687$	− 467.594i	− 0.680632i
$688$	0	0
$689$	−932.815	−1.35387
$690$	0	0
$691$	− 830.873i	− 1.20242i	−0.799091	−	0.601211i	$-0.794685\pi$
$691$	− 830.873i	− 1.20242i	0.799091	−	0.601211i	$-0.205315\pi$
$692$	0	0
$693$	48.5358	0.0700373
$694$	0	0
$695$	1029.23i	1.48090i
$696$	0	0
$697$	−713.792	−1.02409
$698$	0	0
$699$	104.163i	0.149017i
$700$	0	0
$701$	426.515	0.608438	0.304219	−	0.952602i	$-0.401604\pi$
$701$	426.515	0.608438	0.304219	+	0.952602i	$0.401604\pi$
$702$	0	0
$703$	− 6.90900i	− 0.00982787i
$704$	0	0
$705$	−430.067	−0.610024
$706$	0	0
$707$	89.3600i	0.126393i
$708$	0	0
$709$	−143.702	−0.202682	−0.101341	−	0.994852i	$-0.532313\pi$
$709$	−143.702	−0.202682	−0.101341	+	0.994852i	$0.532313\pi$
$710$	0	0
$711$	− 309.629i	− 0.435484i
$712$	0	0
$713$	1263.95	1.77273
$714$	0	0
$715$	1139.89i	1.59425i
$716$	0	0
$717$	−632.163	−0.881678
$718$	0	0
$719$	− 276.110i	− 0.384020i	−0.981393	−	0.192010i	$-0.938499\pi$
$719$	− 276.110i	− 0.384020i	0.981393	−	0.192010i	$-0.0615005\pi$
$720$	0	0
$721$	−234.909	−0.325809
$722$	0	0
$723$	400.434i	0.553851i
$724$	0	0
$725$	−147.888	−0.203984
$726$	0	0
$727$	− 85.2768i	− 0.117300i	−0.998279	−	0.0586498i	$-0.981320\pi$
$727$	− 85.2768i	− 0.117300i	0.998279	−	0.0586498i	$-0.0186795\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	1204.11i	1.64721i
$732$	0	0
$733$	−585.716	−0.799067	−0.399533	−	0.916719i	$-0.630828\pi$
$733$	−585.716	−0.799067	−0.399533	+	0.916719i	$0.630828\pi$
$734$	0	0
$735$	− 430.511i	− 0.585729i
$736$	0	0
$737$	1069.52	1.45119
$738$	0	0
$739$	− 1297.15i	− 1.75527i	−0.479328	−	0.877636i	$-0.659120\pi$
$739$	− 1297.15i	− 1.75527i	0.479328	−	0.877636i	$-0.340880\pi$
$740$	0	0
$741$	151.609	0.204600
$742$	0	0
$743$	1384.78i	1.86376i	0.362764	+	0.931881i	$0.381833\pi$
$743$	1384.78i	1.86376i	−0.362764	+	0.931881i	$0.618167\pi$
$744$	0	0
$745$	476.496	0.639592
$746$	0	0
$747$	493.626i	0.660811i
$748$	0	0
$749$	10.0746	0.0134507
$750$	0	0
$751$	199.884i	0.266157i	0.991105	+	0.133079i	$0.0424863\pi$
$751$	199.884i	0.266157i	−0.991105	+	0.133079i	$0.957514\pi$
$752$	0	0
$753$	−33.9138	−0.0450382
$754$	0	0
$755$	592.302i	0.784505i
$756$	0	0
$757$	599.426	0.791844	0.395922	−	0.918284i	$-0.370425\pi$
$757$	599.426	0.791844	0.395922	+	0.918284i	$0.370425\pi$
$758$	0	0
$759$	− 732.146i	− 0.964620i
$760$	0	0
$761$	688.083	0.904182	0.452091	−	0.891972i	$-0.350678\pi$
$761$	688.083	0.904182	0.452091	+	0.891972i	$0.350678\pi$
$762$	0	0
$763$	− 23.6563i	− 0.0310044i
$764$	0	0
$765$	414.083	0.541285
$766$	0	0
$767$	760.034i	0.990918i
$768$	0	0
$769$	1002.76	1.30398	0.651992	−	0.758226i	$-0.273934\pi$
$769$	1002.76	1.30398	0.651992	+	0.758226i	$0.273934\pi$
$770$	0	0
$771$	29.6190i	0.0384164i
$772$	0	0
$773$	137.272	0.177583	0.0887915	−	0.996050i	$-0.471700\pi$
$773$	137.272	0.177583	0.0887915	+	0.996050i	$0.471700\pi$
$774$	0	0
$775$	− 106.171i	− 0.136995i
$776$	0	0
$777$	−4.16469	−0.00535996
$778$	0	0
$779$	− 119.978i	− 0.154015i
$780$	0	0
$781$	878.914	1.12537
$782$	0	0
$783$	− 230.813i	− 0.294780i
$784$	0	0
$785$	1099.62	1.40079
$786$	0	0
$787$	969.224i	1.23154i	0.787925	+	0.615771i	$0.211155\pi$
$787$	969.224i	1.23154i	−0.787925	+	0.615771i	$0.788845\pi$
$788$	0	0
$789$	−99.2650	−0.125811
$790$	0	0
$791$	154.518i	0.195345i
$792$	0	0
$793$	1818.95	2.29376
$794$	0	0
$795$	− 428.240i	− 0.538667i
$796$	0	0
$797$	−1458.98	−1.83058	−0.915292	−	0.402790i	$-0.868040\pi$
$797$	−1458.98	−1.83058	−0.915292	+	0.402790i	$0.868040\pi$
$798$	0	0
$799$	1209.78i	1.51411i
$800$	0	0
$801$	−173.506	−0.216612
$802$	0	0
$803$	1053.67i	1.31217i
$804$	0	0
$805$	320.024	0.397545
$806$	0	0
$807$	49.7497i	0.0616477i
$808$	0	0
$809$	−449.947	−0.556177	−0.278089	−	0.960555i	$-0.589701\pi$
$809$	−449.947	−0.556177	−0.278089	+	0.960555i	$0.589701\pi$
$810$	0	0
$811$	1086.35i	1.33952i	0.742579	+	0.669759i	$0.233602\pi$
$811$	1086.35i	1.33952i	−0.742579	+	0.669759i	$0.766398\pi$
$812$	0	0
$813$	199.112	0.244910
$814$	0	0
$815$	− 1437.14i	− 1.76337i
$816$	0	0
$817$	−202.393	−0.247727
$818$	0	0
$819$	− 91.3886i	− 0.111586i
$820$	0	0
$821$	−499.306	−0.608168	−0.304084	−	0.952645i	$-0.598350\pi$
$821$	−499.306	−0.608168	−0.304084	+	0.952645i	$0.598350\pi$
$822$	0	0
$823$	1139.14i	1.38413i	0.721837	+	0.692063i	$0.243298\pi$
$823$	1139.14i	1.38413i	−0.721837	+	0.692063i	$0.756702\pi$
$824$	0	0
$825$	−61.4998	−0.0745452
$826$	0	0
$827$	8.60576i	0.0104060i	0.999986	+	0.00520300i	$0.00165617\pi$
$827$	8.60576i	0.0104060i	−0.999986	+	0.00520300i	$0.998344\pi$
$828$	0	0
$829$	1326.72	1.60039	0.800194	−	0.599741i	$-0.204730\pi$
$829$	1326.72	1.60039	0.800194	+	0.599741i	$0.204730\pi$
$830$	0	0
$831$	804.529i	0.968146i
$832$	0	0
$833$	−1211.02	−1.45381
$834$	0	0
$835$	− 1661.22i	− 1.98948i
$836$	0	0
$837$	165.704	0.197974
$838$	0	0
$839$	− 674.341i	− 0.803743i	−0.915696	−	0.401872i	$-0.868360\pi$
$839$	− 674.341i	− 0.803743i	0.915696	−	0.401872i	$-0.131640\pi$
$840$	0	0
$841$	1132.13	1.34617
$842$	0	0
$843$	− 227.181i	− 0.269491i
$844$	0	0
$845$	1246.80	1.47550
$846$	0	0
$847$	− 11.0131i	− 0.0130025i
$848$	0	0
$849$	250.909	0.295535
$850$	0	0
$851$	62.8230i	0.0738225i
$852$	0	0
$853$	217.616	0.255118	0.127559	−	0.991831i	$-0.459286\pi$
$853$	217.616	0.255118	0.127559	+	0.991831i	$0.459286\pi$
$854$	0	0
$855$	69.6011i	0.0814048i
$856$	0	0
$857$	−881.427	−1.02850	−0.514251	−	0.857640i	$-0.671930\pi$
$857$	−881.427	−1.02850	−0.514251	+	0.857640i	$0.671930\pi$
$858$	0	0
$859$	− 938.186i	− 1.09218i	−0.837725	−	0.546092i	$-0.816115\pi$
$859$	− 938.186i	− 1.09218i	0.837725	−	0.546092i	$-0.183885\pi$
$860$	0	0
$861$	−72.3216	−0.0839973
$862$	0	0
$863$	729.700i	0.845539i	0.906237	+	0.422770i	$0.138942\pi$
$863$	729.700i	0.845539i	−0.906237	+	0.422770i	$0.861058\pi$
$864$	0	0
$865$	−19.0359	−0.0220069
$866$	0	0
$867$	− 664.251i	− 0.766149i
$868$	0	0
$869$	1100.72	1.26665
$870$	0	0
$871$	− 2013.82i	− 2.31207i
$872$	0	0
$873$	−4.65290	−0.00532978
$874$	0	0
$875$	174.974i	0.199971i
$876$	0	0
$877$	−775.530	−0.884299	−0.442149	−	0.896941i	$-0.645784\pi$
$877$	−775.530	−0.884299	−0.442149	+	0.896941i	$0.645784\pi$
$878$	0	0
$879$	− 746.093i	− 0.848797i
$880$	0	0
$881$	−897.202	−1.01839	−0.509195	−	0.860651i	$-0.670057\pi$
$881$	−897.202	−1.01839	−0.509195	+	0.860651i	$0.670057\pi$
$882$	0	0
$883$	− 757.581i	− 0.857962i	−0.903313	−	0.428981i	$-0.858873\pi$
$883$	− 757.581i	− 0.857962i	0.903313	−	0.428981i	$-0.141127\pi$
$884$	0	0
$885$	−348.919	−0.394259
$886$	0	0
$887$	601.943i	0.678628i	0.940673	+	0.339314i	$0.110195\pi$
$887$	601.943i	0.678628i	−0.940673	+	0.339314i	$0.889805\pi$
$888$	0	0
$889$	31.2718	0.0351764
$890$	0	0
$891$	− 95.9841i	− 0.107726i
$892$	0	0
$893$	−203.345	−0.227710
$894$	0	0
$895$	− 402.495i	− 0.449715i
$896$	0	0
$897$	−1378.57	−1.53686
$898$	0	0
$899$	1416.54i	1.57568i
$900$	0	0
$901$	−1204.64	−1.33700
$902$	0	0
$903$	122.001i	0.135106i
$904$	0	0
$905$	1462.66	1.61620
$906$	0	0
$907$	1426.37i	1.57263i	0.617826	+	0.786315i	$0.288013\pi$
$907$	1426.37i	1.57263i	−0.617826	+	0.786315i	$0.711987\pi$
$908$	0	0
$909$	176.718	0.194409
$910$	0	0
$911$	− 502.815i	− 0.551937i	−0.961167	−	0.275969i	$-0.911001\pi$
$911$	− 502.815i	− 0.551937i	0.961167	−	0.275969i	$-0.0889986\pi$
$912$	0	0
$913$	−1754.82	−1.92204
$914$	0	0
$915$	835.050i	0.912623i
$916$	0	0
$917$	−264.070	−0.287972
$918$	0	0
$919$	− 438.460i	− 0.477105i	−0.971130	−	0.238553i	$-0.923327\pi$
$919$	− 438.460i	− 0.477105i	0.971130	−	0.238553i	$-0.0766729\pi$
$920$	0	0
$921$	1001.49	1.08739
$922$	0	0
$923$	− 1654.92i	− 1.79297i
$924$	0	0
$925$	5.27709	0.00570496
$926$	0	0
$927$	464.554i	0.501137i
$928$	0	0
$929$	228.035	0.245462	0.122731	−	0.992440i	$-0.460835\pi$
$929$	228.035	0.245462	0.122731	+	0.992440i	$0.460835\pi$
$930$	0	0
$931$	− 203.555i	− 0.218641i
$932$	0	0
$933$	−171.705	−0.184035
$934$	0	0
$935$	1472.05i	1.57439i
$936$	0	0
$937$	−750.124	−0.800559	−0.400280	−	0.916393i	$-0.631087\pi$
$937$	−750.124	−0.800559	−0.400280	+	0.916393i	$0.631087\pi$
$938$	0	0
$939$	510.732i	0.543910i
$940$	0	0
$941$	1191.03	1.26571	0.632853	−	0.774272i	$-0.281884\pi$
$941$	1191.03	1.26571	0.632853	+	0.774272i	$0.281884\pi$
$942$	0	0
$943$	1090.95i	1.15689i
$944$	0	0
$945$	41.9550	0.0443969
$946$	0	0
$947$	− 420.827i	− 0.444379i	−0.975004	−	0.222189i	$-0.928680\pi$
$947$	− 420.827i	− 0.444379i	0.975004	−	0.222189i	$-0.0713203\pi$
$948$	0	0
$949$	1983.97	2.09059
$950$	0	0
$951$	− 762.478i	− 0.801764i
$952$	0	0
$953$	−494.580	−0.518971	−0.259486	−	0.965747i	$-0.583553\pi$
$953$	−494.580	−0.518971	−0.259486	+	0.965747i	$0.583553\pi$
$954$	0	0
$955$	− 604.666i	− 0.633158i
$956$	0	0
$957$	820.532	0.857401
$958$	0	0
$959$	298.452i	0.311212i
$960$	0	0
$961$	−55.9542	−0.0582250
$962$	0	0
$963$	− 19.9234i	− 0.0206889i
$964$	0	0
$965$	−1948.38	−2.01905
$966$	0	0
$967$	− 994.616i	− 1.02856i	−0.857623	−	0.514279i	$-0.828060\pi$
$967$	− 994.616i	− 1.02856i	0.857623	−	0.514279i	$-0.171940\pi$
$968$	0	0
$969$	195.788	0.202051
$970$	0	0
$971$	− 1512.76i	− 1.55794i	−0.627062	−	0.778969i	$-0.715743\pi$
$971$	− 1512.76i	− 1.55794i	0.627062	−	0.778969i	$-0.284257\pi$
$972$	0	0
$973$	293.345	0.301485
$974$	0	0
$975$	115.799i	0.118768i
$976$	0	0
$977$	454.879	0.465588	0.232794	−	0.972526i	$-0.425213\pi$
$977$	454.879	0.465588	0.232794	+	0.972526i	$0.425213\pi$
$978$	0	0
$979$	− 616.810i	− 0.630041i
$980$	0	0
$981$	−46.7826	−0.0476887
$982$	0	0
$983$	1799.80i	1.83093i	0.402401	+	0.915464i	$0.368176\pi$
$983$	1799.80i	1.83093i	−0.402401	+	0.915464i	$0.631824\pi$
$984$	0	0
$985$	710.504	0.721324
$986$	0	0
$987$	122.575i	0.124190i
$988$	0	0
$989$	1840.34	1.86081
$990$	0	0
$991$	− 1579.90i	− 1.59425i	−0.603817	−	0.797123i	$-0.706354\pi$
$991$	− 1579.90i	− 1.59425i	0.603817	−	0.797123i	$-0.293646\pi$
$992$	0	0
$993$	−564.396	−0.568375
$994$	0	0
$995$	336.118i	0.337807i
$996$	0	0
$997$	−1577.10	−1.58184	−0.790922	−	0.611917i	$-0.790399\pi$
$997$	−1577.10	−1.58184	−0.790922	+	0.611917i	$0.790399\pi$
$998$	0	0
$999$	8.23607i	0.00824431i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.3.m.c.799.5	✓	12
3.2	odd	2		2736.3.m.d.1711.3		12
4.3	odd	2	inner	912.3.m.c.799.11	yes	12
12.11	even	2		2736.3.m.d.1711.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.3.m.c.799.5	✓	12	1.1	even	1	trivial
912.3.m.c.799.11	yes	12	4.3	odd	2	inner
2736.3.m.d.1711.3		12	3.2	odd	2
2736.3.m.d.1711.4		12	12.11	even	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	912.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +5.32253 q^{5} -1.51700i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +5.32253 q^{5} -1.51700i q^{7} -3.00000 q^{9} -10.6649i q^{11} -20.0811 q^{13} -9.21889i q^{15} -25.9327 q^{17} -4.35890i q^{19} -2.62751 q^{21} +39.6351i q^{23} +3.32932 q^{25} +5.19615i q^{27} -44.4200 q^{29} -31.8897i q^{31} -18.4722 q^{33} -8.07425i q^{35} +1.58503 q^{37} +34.7814i q^{39} +27.5248 q^{41} -46.4321i q^{43} -15.9676 q^{45} -46.6506i q^{47} +46.6987 q^{49} +44.9168i q^{51} +46.4525 q^{53} -56.7643i q^{55} -7.54983 q^{57} -37.8483i q^{59} -90.5803 q^{61} +4.55099i q^{63} -106.882 q^{65} +100.284i q^{67} +68.6500 q^{69} +82.4118i q^{71} -98.7982 q^{73} -5.76656i q^{75} -16.1786 q^{77} +103.210i q^{79} +9.00000 q^{81} -164.542i q^{83} -138.028 q^{85} +76.9376i q^{87} +57.8355 q^{89} +30.4629i q^{91} -55.2346 q^{93} -23.2004i q^{95} +1.55097 q^{97} +31.9947i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{5} - 36 q^{9}+O(q^{10}) \) 12 * q - 4 * q^5 - 36 * q^9	\( 12 q - 4 q^{5} - 36 q^{9} + 20 q^{17} + 88 q^{25} - 24 q^{29} - 72 q^{33} - 40 q^{37} - 32 q^{41} + 12 q^{45} + 128 q^{49} + 184 q^{53} - 276 q^{61} - 232 q^{65} + 120 q^{69} - 92 q^{73} + 308 q^{77} + 108 q^{81} - 244 q^{85} + 72 q^{89} + 144 q^{93} - 280 q^{97}+O(q^{100}) \) 12 * q - 4 * q^5 - 36 * q^9 + 20 * q^17 + 88 * q^25 - 24 * q^29 - 72 * q^33 - 40 * q^37 - 32 * q^41 + 12 * q^45 + 128 * q^49 + 184 * q^53 - 276 * q^61 - 232 * q^65 + 120 * q^69 - 92 * q^73 + 308 * q^77 + 108 * q^81 - 244 * q^85 + 72 * q^89 + 144 * q^93 - 280 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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