LMFDB - Embedded newform 476.4.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [476,4,Mod(1,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("476.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 476 = 2^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	476.a (trivial)

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:	yes
Analytic conductor:	$28.0849091627$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 87x^{4} + 184x^{3} + 2031x^{2} - 4232x - 7516 $ x^6 - 2x^5 - 87x^4 + 184x^3 + 2031x^2 - 4232*x - 7516
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.20043$ of defining polynomial
Character	$\chi$	$=$	476.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.20043 q^{3} +6.13806 q^{5} -7.00000 q^{7} -25.5590 q^{9} +O(q^{10})$	$q-1.20043 q^{3} +6.13806 q^{5} -7.00000 q^{7} -25.5590 q^{9} +36.3784 q^{11} -0.689604 q^{13} -7.36832 q^{15} -17.0000 q^{17} -10.3187 q^{19} +8.40303 q^{21} -13.7415 q^{23} -87.3243 q^{25} +63.0935 q^{27} -178.679 q^{29} -161.389 q^{31} -43.6698 q^{33} -42.9664 q^{35} +376.875 q^{37} +0.827823 q^{39} -34.7408 q^{41} -35.0321 q^{43} -156.882 q^{45} -618.382 q^{47} +49.0000 q^{49} +20.4074 q^{51} -237.243 q^{53} +223.292 q^{55} +12.3869 q^{57} +280.414 q^{59} +453.692 q^{61} +178.913 q^{63} -4.23283 q^{65} -970.778 q^{67} +16.4957 q^{69} -912.977 q^{71} -1039.04 q^{73} +104.827 q^{75} -254.648 q^{77} -799.796 q^{79} +614.352 q^{81} -774.476 q^{83} -104.347 q^{85} +214.492 q^{87} -1312.08 q^{89} +4.82723 q^{91} +193.737 q^{93} -63.3365 q^{95} +644.564 q^{97} -929.793 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9	$ 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9} - 20 q^{11} + 42 q^{13} - 136 q^{15} - 102 q^{17} - 104 q^{19} - 14 q^{21} - 230 q^{23} + 108 q^{25} - 130 q^{27} - 52 q^{29} - 564 q^{31} - 346 q^{33} - 70 q^{35} - 564 q^{37} - 626 q^{39} - 548 q^{41} - 648 q^{43} - 174 q^{45} - 366 q^{47} + 294 q^{49} - 34 q^{51} - 74 q^{53} - 1460 q^{55} - 316 q^{57} - 558 q^{59} - 620 q^{61} - 112 q^{63} - 1378 q^{65} - 164 q^{67} - 540 q^{69} - 822 q^{71} + 940 q^{73} - 2698 q^{75} + 140 q^{77} - 1838 q^{79} - 3094 q^{81} - 1118 q^{83} - 170 q^{85} - 1354 q^{87} - 1634 q^{89} - 294 q^{91} + 268 q^{93} - 2642 q^{95} + 182 q^{97} + 12 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9 - 20 * q^11 + 42 * q^13 - 136 * q^15 - 102 * q^17 - 104 * q^19 - 14 * q^21 - 230 * q^23 + 108 * q^25 - 130 * q^27 - 52 * q^29 - 564 * q^31 - 346 * q^33 - 70 * q^35 - 564 * q^37 - 626 * q^39 - 548 * q^41 - 648 * q^43 - 174 * q^45 - 366 * q^47 + 294 * q^49 - 34 * q^51 - 74 * q^53 - 1460 * q^55 - 316 * q^57 - 558 * q^59 - 620 * q^61 - 112 * q^63 - 1378 * q^65 - 164 * q^67 - 540 * q^69 - 822 * q^71 + 940 * q^73 - 2698 * q^75 + 140 * q^77 - 1838 * q^79 - 3094 * q^81 - 1118 * q^83 - 170 * q^85 - 1354 * q^87 - 1634 * q^89 - 294 * q^91 + 268 * q^93 - 2642 * q^95 + 182 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.20043	−0.231023	−0.115512	−	0.993306i	$-0.536851\pi$
$3$	−1.20043	−0.231023	−0.115512	+	0.993306i	$0.536851\pi$
$4$	0	0
$5$	6.13806	0.549004	0.274502	−	0.961586i	$-0.411487\pi$
$5$	6.13806	0.549004	0.274502	+	0.961586i	$0.411487\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−25.5590	−0.946628
$10$	0	0
$11$	36.3784	0.997135	0.498568	−	0.866851i	$-0.333860\pi$
$11$	36.3784	0.997135	0.498568	+	0.866851i	$0.333860\pi$
$12$	0	0
$13$	−0.689604	−0.0147124	−0.00735622	−	0.999973i	$-0.502342\pi$
$13$	−0.689604	−0.0147124	−0.00735622	+	0.999973i	$0.502342\pi$
$14$	0	0
$15$	−7.36832	−0.126833
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−10.3187	−0.124593	−0.0622964	−	0.998058i	$-0.519842\pi$
$19$	−10.3187	−0.124593	−0.0622964	+	0.998058i	$0.519842\pi$
$20$	0	0
$21$	8.40303	0.0873186
$22$	0	0
$23$	−13.7415	−0.124578	−0.0622891	−	0.998058i	$-0.519840\pi$
$23$	−13.7415	−0.124578	−0.0622891	+	0.998058i	$0.519840\pi$
$24$	0	0
$25$	−87.3243	−0.698594
$26$	0	0
$27$	63.0935	0.449717
$28$	0	0
$29$	−178.679	−1.14413	−0.572067	−	0.820207i	$-0.693858\pi$
$29$	−178.679	−1.14413	−0.572067	+	0.820207i	$0.693858\pi$
$30$	0	0
$31$	−161.389	−0.935044	−0.467522	−	0.883982i	$-0.654853\pi$
$31$	−161.389	−0.935044	−0.467522	+	0.883982i	$0.654853\pi$
$32$	0	0
$33$	−43.6698	−0.230362
$34$	0	0
$35$	−42.9664	−0.207504
$36$	0	0
$37$	376.875	1.67454	0.837268	−	0.546793i	$-0.184151\pi$
$37$	376.875	1.67454	0.837268	+	0.546793i	$0.184151\pi$
$38$	0	0
$39$	0.827823	0.00339892
$40$	0	0
$41$	−34.7408	−0.132332	−0.0661659	−	0.997809i	$-0.521077\pi$
$41$	−34.7408	−0.132332	−0.0661659	+	0.997809i	$0.521077\pi$
$42$	0	0
$43$	−35.0321	−0.124240	−0.0621202	−	0.998069i	$-0.519786\pi$
$43$	−35.0321	−0.124240	−0.0621202	+	0.998069i	$0.519786\pi$
$44$	0	0
$45$	−156.882	−0.519703
$46$	0	0
$47$	−618.382	−1.91915	−0.959577	−	0.281446i	$-0.909186\pi$
$47$	−618.382	−1.91915	−0.959577	+	0.281446i	$0.909186\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	20.4074	0.0560314
$52$	0	0
$53$	−237.243	−0.614866	−0.307433	−	0.951570i	$-0.599470\pi$
$53$	−237.243	−0.614866	−0.307433	+	0.951570i	$0.599470\pi$
$54$	0	0
$55$	223.292	0.547432
$56$	0	0
$57$	12.3869	0.0287839
$58$	0	0
$59$	280.414	0.618758	0.309379	−	0.950939i	$-0.399879\pi$
$59$	280.414	0.618758	0.309379	+	0.950939i	$0.399879\pi$
$60$	0	0
$61$	453.692	0.952285	0.476142	−	0.879368i	$-0.342035\pi$
$61$	453.692	0.952285	0.476142	+	0.879368i	$0.342035\pi$
$62$	0	0
$63$	178.913	0.357792
$64$	0	0
$65$	−4.23283	−0.00807720
$66$	0	0
$67$	−970.778	−1.77014	−0.885070	−	0.465458i	$-0.845890\pi$
$67$	−970.778	−1.77014	−0.885070	+	0.465458i	$0.845890\pi$
$68$	0	0
$69$	16.4957	0.0287805
$70$	0	0
$71$	−912.977	−1.52606	−0.763031	−	0.646362i	$-0.776290\pi$
$71$	−912.977	−1.52606	−0.763031	+	0.646362i	$0.776290\pi$
$72$	0	0
$73$	−1039.04	−1.66590	−0.832949	−	0.553350i	$-0.813349\pi$
$73$	−1039.04	−1.66590	−0.832949	+	0.553350i	$0.813349\pi$
$74$	0	0
$75$	104.827	0.161392
$76$	0	0
$77$	−254.648	−0.376882
$78$	0	0
$79$	−799.796	−1.13904	−0.569520	−	0.821978i	$-0.692871\pi$
$79$	−799.796	−1.13904	−0.569520	+	0.821978i	$0.692871\pi$
$80$	0	0
$81$	614.352	0.842733
$82$	0	0
$83$	−774.476	−1.02421	−0.512107	−	0.858921i	$-0.671135\pi$
$83$	−774.476	−1.02421	−0.512107	+	0.858921i	$0.671135\pi$
$84$	0	0
$85$	−104.347	−0.133153
$86$	0	0
$87$	214.492	0.264322
$88$	0	0
$89$	−1312.08	−1.56270	−0.781350	−	0.624094i	$-0.785468\pi$
$89$	−1312.08	−1.56270	−0.781350	+	0.624094i	$0.785468\pi$
$90$	0	0
$91$	4.82723	0.00556078
$92$	0	0
$93$	193.737	0.216017
$94$	0	0
$95$	−63.3365	−0.0684020
$96$	0	0
$97$	644.564	0.674697	0.337348	−	0.941380i	$-0.390470\pi$
$97$	644.564	0.674697	0.337348	+	0.941380i	$0.390470\pi$
$98$	0	0
$99$	−929.793	−0.943916
$100$	0	0
$101$	978.471	0.963975	0.481987	−	0.876178i	$-0.339915\pi$
$101$	978.471	0.963975	0.481987	+	0.876178i	$0.339915\pi$
$102$	0	0
$103$	−1946.22	−1.86181	−0.930906	−	0.365258i	$-0.880981\pi$
$103$	−1946.22	−1.86181	−0.930906	+	0.365258i	$0.880981\pi$
$104$	0	0
$105$	51.5783	0.0479383
$106$	0	0
$107$	81.6320	0.0737539	0.0368769	−	0.999320i	$-0.488259\pi$
$107$	81.6320	0.0737539	0.0368769	+	0.999320i	$0.488259\pi$
$108$	0	0
$109$	1163.58	1.02248	0.511242	−	0.859437i	$-0.329186\pi$
$109$	1163.58	1.02248	0.511242	+	0.859437i	$0.329186\pi$
$110$	0	0
$111$	−452.413	−0.386857
$112$	0	0
$113$	956.048	0.795906	0.397953	−	0.917406i	$-0.369721\pi$
$113$	956.048	0.795906	0.397953	+	0.917406i	$0.369721\pi$
$114$	0	0
$115$	−84.3461	−0.0683940
$116$	0	0
$117$	17.6256	0.0139272
$118$	0	0
$119$	119.000	0.0916698
$120$	0	0
$121$	−7.61517	−0.00572139
$122$	0	0
$123$	41.7040	0.0305717
$124$	0	0
$125$	−1303.26	−0.932536
$126$	0	0
$127$	2857.29	1.99640	0.998202	−	0.0599358i	$-0.0190896\pi$
$127$	2857.29	1.99640	0.998202	+	0.0599358i	$0.0190896\pi$
$128$	0	0
$129$	42.0536	0.0287025
$130$	0	0
$131$	1349.96	0.900355	0.450178	−	0.892939i	$-0.351361\pi$
$131$	1349.96	0.900355	0.450178	+	0.892939i	$0.351361\pi$
$132$	0	0
$133$	72.2306	0.0470916
$134$	0	0
$135$	387.271	0.246896
$136$	0	0
$137$	67.5343	0.0421157	0.0210578	−	0.999778i	$-0.493297\pi$
$137$	67.5343	0.0421157	0.0210578	+	0.999778i	$0.493297\pi$
$138$	0	0
$139$	1998.04	1.21922	0.609609	−	0.792702i	$-0.291327\pi$
$139$	1998.04	1.21922	0.609609	+	0.792702i	$0.291327\pi$
$140$	0	0
$141$	742.326	0.443370
$142$	0	0
$143$	−25.0867	−0.0146703
$144$	0	0
$145$	−1096.74	−0.628134
$146$	0	0
$147$	−58.8212	−0.0330033
$148$	0	0
$149$	−2040.46	−1.12189	−0.560943	−	0.827854i	$-0.689561\pi$
$149$	−2040.46	−1.12189	−0.560943	+	0.827854i	$0.689561\pi$
$150$	0	0
$151$	−3265.03	−1.75963	−0.879816	−	0.475314i	$-0.842334\pi$
$151$	−3265.03	−1.75963	−0.879816	+	0.475314i	$0.842334\pi$
$152$	0	0
$153$	434.502	0.229591
$154$	0	0
$155$	−990.616	−0.513343
$156$	0	0
$157$	3713.72	1.88782	0.943908	−	0.330209i	$-0.107119\pi$
$157$	3713.72	1.88782	0.943908	+	0.330209i	$0.107119\pi$
$158$	0	0
$159$	284.795	0.142048
$160$	0	0
$161$	96.1905	0.0470862
$162$	0	0
$163$	−0.303639	−0.000145907 0	−7.29536e−5	−	1.00000i	$-0.500023\pi$
$163$	−0.303639	−0.000145907 0	−7.29536e−5		1.00000i	$0.500023\pi$
$164$	0	0
$165$	−268.048	−0.126470
$166$	0	0
$167$	1883.41	0.872710	0.436355	−	0.899775i	$-0.356269\pi$
$167$	1883.41	0.872710	0.436355	+	0.899775i	$0.356269\pi$
$168$	0	0
$169$	−2196.52	−0.999784
$170$	0	0
$171$	263.734	0.117943
$172$	0	0
$173$	971.466	0.426932	0.213466	−	0.976951i	$-0.431525\pi$
$173$	971.466	0.426932	0.213466	+	0.976951i	$0.431525\pi$
$174$	0	0
$175$	611.270	0.264044
$176$	0	0
$177$	−336.618	−0.142948
$178$	0	0
$179$	3257.86	1.36036	0.680179	−	0.733046i	$-0.261902\pi$
$179$	3257.86	1.36036	0.680179	+	0.733046i	$0.261902\pi$
$180$	0	0
$181$	−705.061	−0.289540	−0.144770	−	0.989465i	$-0.546244\pi$
$181$	−705.061	−0.289540	−0.144770	+	0.989465i	$0.546244\pi$
$182$	0	0
$183$	−544.627	−0.220000
$184$	0	0
$185$	2313.28	0.919328
$186$	0	0
$187$	−618.432	−0.241841
$188$	0	0
$189$	−441.655	−0.169977
$190$	0	0
$191$	−2457.96	−0.931160	−0.465580	−	0.885006i	$-0.654154\pi$
$191$	−2457.96	−0.931160	−0.465580	+	0.885006i	$0.654154\pi$
$192$	0	0
$193$	−293.233	−0.109365	−0.0546823	−	0.998504i	$-0.517415\pi$
$193$	−293.233	−0.109365	−0.0546823	+	0.998504i	$0.517415\pi$
$194$	0	0
$195$	5.08123	0.00186602
$196$	0	0
$197$	5232.06	1.89223	0.946114	−	0.323835i	$-0.104972\pi$
$197$	5232.06	1.89223	0.946114	+	0.323835i	$0.104972\pi$
$198$	0	0
$199$	678.290	0.241622	0.120811	−	0.992676i	$-0.461451\pi$
$199$	678.290	0.241622	0.120811	+	0.992676i	$0.461451\pi$
$200$	0	0
$201$	1165.35	0.408944
$202$	0	0
$203$	1250.75	0.432442
$204$	0	0
$205$	−213.241	−0.0726507
$206$	0	0
$207$	351.218	0.117929
$208$	0	0
$209$	−375.376	−0.124236
$210$	0	0
$211$	−3818.82	−1.24596	−0.622982	−	0.782236i	$-0.714079\pi$
$211$	−3818.82	−1.24596	−0.622982	+	0.782236i	$0.714079\pi$
$212$	0	0
$213$	1095.97	0.352556
$214$	0	0
$215$	−215.029	−0.0682086
$216$	0	0
$217$	1129.72	0.353413
$218$	0	0
$219$	1247.30	0.384861
$220$	0	0
$221$	11.7233	0.00356829
$222$	0	0
$223$	3917.92	1.17652	0.588258	−	0.808673i	$-0.299814\pi$
$223$	3917.92	1.17652	0.588258	+	0.808673i	$0.299814\pi$
$224$	0	0
$225$	2231.92	0.661309
$226$	0	0
$227$	−1026.52	−0.300143	−0.150072	−	0.988675i	$-0.547950\pi$
$227$	−1026.52	−0.300143	−0.150072	+	0.988675i	$0.547950\pi$
$228$	0	0
$229$	2241.92	0.646944	0.323472	−	0.946238i	$-0.395150\pi$
$229$	2241.92	0.646944	0.323472	+	0.946238i	$0.395150\pi$
$230$	0	0
$231$	305.688	0.0870685
$232$	0	0
$233$	1051.52	0.295653	0.147827	−	0.989013i	$-0.452772\pi$
$233$	1051.52	0.295653	0.147827	+	0.989013i	$0.452772\pi$
$234$	0	0
$235$	−3795.66	−1.05362
$236$	0	0
$237$	960.102	0.263145
$238$	0	0
$239$	−1235.73	−0.334446	−0.167223	−	0.985919i	$-0.553480\pi$
$239$	−1235.73	−0.334446	−0.167223	+	0.985919i	$0.553480\pi$
$240$	0	0
$241$	1460.79	0.390448	0.195224	−	0.980759i	$-0.437457\pi$
$241$	1460.79	0.390448	0.195224	+	0.980759i	$0.437457\pi$
$242$	0	0
$243$	−2441.01	−0.644408
$244$	0	0
$245$	300.765	0.0784292
$246$	0	0
$247$	7.11579	0.00183306
$248$	0	0
$249$	929.707	0.236618
$250$	0	0
$251$	6639.31	1.66960	0.834800	−	0.550553i	$-0.185583\pi$
$251$	6639.31	1.66960	0.834800	+	0.550553i	$0.185583\pi$
$252$	0	0
$253$	−499.893	−0.124221
$254$	0	0
$255$	125.261	0.0307615
$256$	0	0
$257$	2563.12	0.622113	0.311056	−	0.950391i	$-0.399317\pi$
$257$	2563.12	0.622113	0.311056	+	0.950391i	$0.399317\pi$
$258$	0	0
$259$	−2638.12	−0.632915
$260$	0	0
$261$	4566.85	1.08307
$262$	0	0
$263$	−6569.42	−1.54026	−0.770129	−	0.637888i	$-0.779808\pi$
$263$	−6569.42	−1.54026	−0.770129	+	0.637888i	$0.779808\pi$
$264$	0	0
$265$	−1456.21	−0.337564
$266$	0	0
$267$	1575.06	0.361020
$268$	0	0
$269$	−5302.00	−1.20174	−0.600871	−	0.799346i	$-0.705180\pi$
$269$	−5302.00	−1.20174	−0.600871	+	0.799346i	$0.705180\pi$
$270$	0	0
$271$	137.042	0.0307185	0.0153593	−	0.999882i	$-0.495111\pi$
$271$	137.042	0.0307185	0.0153593	+	0.999882i	$0.495111\pi$
$272$	0	0
$273$	−5.79476	−0.00128467
$274$	0	0
$275$	−3176.71	−0.696593
$276$	0	0
$277$	−1584.91	−0.343783	−0.171891	−	0.985116i	$-0.554988\pi$
$277$	−1584.91	−0.343783	−0.171891	+	0.985116i	$0.554988\pi$
$278$	0	0
$279$	4124.94	0.885139
$280$	0	0
$281$	4264.67	0.905369	0.452685	−	0.891671i	$-0.350466\pi$
$281$	4264.67	0.905369	0.452685	+	0.891671i	$0.350466\pi$
$282$	0	0
$283$	−3093.22	−0.649727	−0.324864	−	0.945761i	$-0.605318\pi$
$283$	−3093.22	−0.649727	−0.324864	+	0.945761i	$0.605318\pi$
$284$	0	0
$285$	76.0312	0.0158025
$286$	0	0
$287$	243.186	0.0500167
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−773.756	−0.155871
$292$	0	0
$293$	7457.61	1.48696	0.743478	−	0.668760i	$-0.233175\pi$
$293$	7457.61	1.48696	0.743478	+	0.668760i	$0.233175\pi$
$294$	0	0
$295$	1721.19	0.339701
$296$	0	0
$297$	2295.24	0.448428
$298$	0	0
$299$	9.47619	0.00183285
$300$	0	0
$301$	245.225	0.0469585
$302$	0	0
$303$	−1174.59	−0.222701
$304$	0	0
$305$	2784.79	0.522808
$306$	0	0
$307$	−5390.94	−1.00221	−0.501103	−	0.865388i	$-0.667072\pi$
$307$	−5390.94	−1.00221	−0.501103	+	0.865388i	$0.667072\pi$
$308$	0	0
$309$	2336.31	0.430122
$310$	0	0
$311$	9362.62	1.70709	0.853546	−	0.521018i	$-0.174447\pi$
$311$	9362.62	1.70709	0.853546	+	0.521018i	$0.174447\pi$
$312$	0	0
$313$	−666.288	−0.120322	−0.0601610	−	0.998189i	$-0.519161\pi$
$313$	−666.288	−0.120322	−0.0601610	+	0.998189i	$0.519161\pi$
$314$	0	0
$315$	1098.18	0.196429
$316$	0	0
$317$	663.470	0.117553	0.0587763	−	0.998271i	$-0.481280\pi$
$317$	663.470	0.117553	0.0587763	+	0.998271i	$0.481280\pi$
$318$	0	0
$319$	−6500.05	−1.14086
$320$	0	0
$321$	−97.9937	−0.0170389
$322$	0	0
$323$	175.417	0.0302182
$324$	0	0
$325$	60.2192	0.0102780
$326$	0	0
$327$	−1396.80	−0.236218
$328$	0	0
$329$	4328.67	0.725372
$330$	0	0
$331$	4725.46	0.784697	0.392348	−	0.919817i	$-0.371663\pi$
$331$	4725.46	0.784697	0.392348	+	0.919817i	$0.371663\pi$
$332$	0	0
$333$	−9632.53	−1.58516
$334$	0	0
$335$	−5958.69	−0.971815
$336$	0	0
$337$	4847.68	0.783591	0.391795	−	0.920052i	$-0.371854\pi$
$337$	4847.68	0.783591	0.391795	+	0.920052i	$0.371854\pi$
$338$	0	0
$339$	−1147.67	−0.183873
$340$	0	0
$341$	−5871.07	−0.932365
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	101.252	0.0158006
$346$	0	0
$347$	−11806.2	−1.82649	−0.913244	−	0.407414i	$-0.866431\pi$
$347$	−11806.2	−1.82649	−0.913244	+	0.407414i	$0.866431\pi$
$348$	0	0
$349$	10034.3	1.53903	0.769515	−	0.638629i	$-0.220498\pi$
$349$	10034.3	1.53903	0.769515	+	0.638629i	$0.220498\pi$
$350$	0	0
$351$	−43.5095	−0.00661643
$352$	0	0
$353$	7890.60	1.18973	0.594864	−	0.803826i	$-0.297206\pi$
$353$	7890.60	1.18973	0.594864	+	0.803826i	$0.297206\pi$
$354$	0	0
$355$	−5603.90	−0.837815
$356$	0	0
$357$	−142.852	−0.0211779
$358$	0	0
$359$	−5048.88	−0.742256	−0.371128	−	0.928582i	$-0.621029\pi$
$359$	−5048.88	−0.742256	−0.371128	+	0.928582i	$0.621029\pi$
$360$	0	0
$361$	−6752.53	−0.984477
$362$	0	0
$363$	9.14150	0.00132178
$364$	0	0
$365$	−6377.69	−0.914585
$366$	0	0
$367$	−6344.35	−0.902377	−0.451189	−	0.892429i	$-0.649000\pi$
$367$	−6344.35	−0.902377	−0.451189	+	0.892429i	$0.649000\pi$
$368$	0	0
$369$	887.939	0.125269
$370$	0	0
$371$	1660.70	0.232397
$372$	0	0
$373$	−3040.24	−0.422031	−0.211016	−	0.977483i	$-0.567677\pi$
$373$	−3040.24	−0.422031	−0.211016	+	0.977483i	$0.567677\pi$
$374$	0	0
$375$	1564.47	0.215438
$376$	0	0
$377$	123.218	0.0168330
$378$	0	0
$379$	−7019.69	−0.951391	−0.475695	−	0.879610i	$-0.657804\pi$
$379$	−7019.69	−0.951391	−0.475695	+	0.879610i	$0.657804\pi$
$380$	0	0
$381$	−3429.98	−0.461216
$382$	0	0
$383$	2077.93	0.277225	0.138612	−	0.990347i	$-0.455736\pi$
$383$	2077.93	0.277225	0.138612	+	0.990347i	$0.455736\pi$
$384$	0	0
$385$	−1563.05	−0.206910
$386$	0	0
$387$	895.383	0.117610
$388$	0	0
$389$	−3400.38	−0.443203	−0.221602	−	0.975137i	$-0.571128\pi$
$389$	−3400.38	−0.443203	−0.221602	+	0.975137i	$0.571128\pi$
$390$	0	0
$391$	233.605	0.0302147
$392$	0	0
$393$	−1620.54	−0.208003
$394$	0	0
$395$	−4909.19	−0.625338
$396$	0	0
$397$	−9356.02	−1.18278	−0.591392	−	0.806385i	$-0.701421\pi$
$397$	−9356.02	−1.18278	−0.591392	+	0.806385i	$0.701421\pi$
$398$	0	0
$399$	−86.7080	−0.0108793
$400$	0	0
$401$	−12529.2	−1.56029	−0.780145	−	0.625598i	$-0.784855\pi$
$401$	−12529.2	−1.56029	−0.780145	+	0.625598i	$0.784855\pi$
$402$	0	0
$403$	111.295	0.0137568
$404$	0	0
$405$	3770.93	0.462664
$406$	0	0
$407$	13710.1	1.66974
$408$	0	0
$409$	−4209.88	−0.508961	−0.254481	−	0.967078i	$-0.581904\pi$
$409$	−4209.88	−0.508961	−0.254481	+	0.967078i	$0.581904\pi$
$410$	0	0
$411$	−81.0704	−0.00972970
$412$	0	0
$413$	−1962.90	−0.233869
$414$	0	0
$415$	−4753.78	−0.562298
$416$	0	0
$417$	−2398.51	−0.281668
$418$	0	0
$419$	2024.50	0.236046	0.118023	−	0.993011i	$-0.462344\pi$
$419$	2024.50	0.236046	0.118023	+	0.993011i	$0.462344\pi$
$420$	0	0
$421$	3703.54	0.428740	0.214370	−	0.976753i	$-0.431230\pi$
$421$	3703.54	0.428740	0.214370	+	0.976753i	$0.431230\pi$
$422$	0	0
$423$	15805.2	1.81673
$424$	0	0
$425$	1484.51	0.169434
$426$	0	0
$427$	−3175.85	−0.359930
$428$	0	0
$429$	30.1149	0.00338918
$430$	0	0
$431$	−1158.73	−0.129499	−0.0647497	−	0.997902i	$-0.520625\pi$
$431$	−1158.73	−0.129499	−0.0647497	+	0.997902i	$0.520625\pi$
$432$	0	0
$433$	−5409.03	−0.600327	−0.300164	−	0.953888i	$-0.597041\pi$
$433$	−5409.03	−0.600327	−0.300164	+	0.953888i	$0.597041\pi$
$434$	0	0
$435$	1316.57	0.145114
$436$	0	0
$437$	141.794	0.0155216
$438$	0	0
$439$	−14134.4	−1.53667	−0.768337	−	0.640046i	$-0.778915\pi$
$439$	−14134.4	−1.53667	−0.768337	+	0.640046i	$0.778915\pi$
$440$	0	0
$441$	−1252.39	−0.135233
$442$	0	0
$443$	−2605.66	−0.279455	−0.139728	−	0.990190i	$-0.544623\pi$
$443$	−2605.66	−0.279455	−0.139728	+	0.990190i	$0.544623\pi$
$444$	0	0
$445$	−8053.62	−0.857929
$446$	0	0
$447$	2449.44	0.259182
$448$	0	0
$449$	12089.7	1.27071	0.635354	−	0.772221i	$-0.280854\pi$
$449$	12089.7	1.27071	0.635354	+	0.772221i	$0.280854\pi$
$450$	0	0
$451$	−1263.81	−0.131953
$452$	0	0
$453$	3919.45	0.406516
$454$	0	0
$455$	29.6298	0.00305289
$456$	0	0
$457$	3619.53	0.370491	0.185245	−	0.982692i	$-0.440692\pi$
$457$	3619.53	0.370491	0.185245	+	0.982692i	$0.440692\pi$
$458$	0	0
$459$	−1072.59	−0.109072
$460$	0	0
$461$	−1428.57	−0.144328	−0.0721638	−	0.997393i	$-0.522990\pi$
$461$	−1428.57	−0.144328	−0.0721638	+	0.997393i	$0.522990\pi$
$462$	0	0
$463$	−10604.9	−1.06448	−0.532239	−	0.846594i	$-0.678649\pi$
$463$	−10604.9	−1.06448	−0.532239	+	0.846594i	$0.678649\pi$
$464$	0	0
$465$	1189.17	0.118594
$466$	0	0
$467$	2355.58	0.233411	0.116706	−	0.993167i	$-0.462767\pi$
$467$	2355.58	0.233411	0.116706	+	0.993167i	$0.462767\pi$
$468$	0	0
$469$	6795.44	0.669050
$470$	0	0
$471$	−4458.07	−0.436130
$472$	0	0
$473$	−1274.41	−0.123885
$474$	0	0
$475$	901.069	0.0870398
$476$	0	0
$477$	6063.70	0.582049
$478$	0	0
$479$	4942.22	0.471432	0.235716	−	0.971822i	$-0.424257\pi$
$479$	4942.22	0.471432	0.235716	+	0.971822i	$0.424257\pi$
$480$	0	0
$481$	−259.894	−0.0246365
$482$	0	0
$483$	−115.470	−0.0108780
$484$	0	0
$485$	3956.37	0.370411
$486$	0	0
$487$	−3143.32	−0.292479	−0.146240	−	0.989249i	$-0.546717\pi$
$487$	−3143.32	−0.292479	−0.146240	+	0.989249i	$0.546717\pi$
$488$	0	0
$489$	0.364499	3.37080e−5 0
$490$	0	0
$491$	−9083.42	−0.834886	−0.417443	−	0.908703i	$-0.637074\pi$
$491$	−9083.42	−0.834886	−0.417443	+	0.908703i	$0.637074\pi$
$492$	0	0
$493$	3037.54	0.277493
$494$	0	0
$495$	−5707.12	−0.518214
$496$	0	0
$497$	6390.84	0.576797
$498$	0	0
$499$	2535.57	0.227470	0.113735	−	0.993511i	$-0.463718\pi$
$499$	2535.57	0.227470	0.113735	+	0.993511i	$0.463718\pi$
$500$	0	0
$501$	−2260.91	−0.201616
$502$	0	0
$503$	5647.73	0.500635	0.250318	−	0.968164i	$-0.419465\pi$
$503$	5647.73	0.500635	0.250318	+	0.968164i	$0.419465\pi$
$504$	0	0
$505$	6005.91	0.529226
$506$	0	0
$507$	2636.78	0.230973
$508$	0	0
$509$	−17122.7	−1.49106	−0.745530	−	0.666472i	$-0.767804\pi$
$509$	−17122.7	−1.49106	−0.745530	+	0.666472i	$0.767804\pi$
$510$	0	0
$511$	7273.28	0.629650
$512$	0	0
$513$	−651.040	−0.0560315
$514$	0	0
$515$	−11946.0	−1.02214
$516$	0	0
$517$	−22495.7	−1.91366
$518$	0	0
$519$	−1166.18	−0.0986312
$520$	0	0
$521$	13900.7	1.16891	0.584455	−	0.811426i	$-0.301308\pi$
$521$	13900.7	1.16891	0.584455	+	0.811426i	$0.301308\pi$
$522$	0	0
$523$	−7298.14	−0.610182	−0.305091	−	0.952323i	$-0.598687\pi$
$523$	−7298.14	−0.610182	−0.305091	+	0.952323i	$0.598687\pi$
$524$	0	0
$525$	−733.788	−0.0610003
$526$	0	0
$527$	2743.62	0.226781
$528$	0	0
$529$	−11978.2	−0.984480
$530$	0	0
$531$	−7167.08	−0.585734
$532$	0	0
$533$	23.9574	0.00194692
$534$	0	0
$535$	501.062	0.0404912
$536$	0	0
$537$	−3910.84	−0.314274
$538$	0	0
$539$	1782.54	0.142448
$540$	0	0
$541$	−24507.1	−1.94758	−0.973791	−	0.227444i	$-0.926963\pi$
$541$	−24507.1	−1.94758	−0.973791	+	0.227444i	$0.926963\pi$
$542$	0	0
$543$	846.379	0.0668906
$544$	0	0
$545$	7142.11	0.561348
$546$	0	0
$547$	3435.13	0.268511	0.134255	−	0.990947i	$-0.457136\pi$
$547$	3435.13	0.268511	0.134255	+	0.990947i	$0.457136\pi$
$548$	0	0
$549$	−11595.9	−0.901459
$550$	0	0
$551$	1843.73	0.142551
$552$	0	0
$553$	5598.57	0.430516
$554$	0	0
$555$	−2776.94	−0.212386
$556$	0	0
$557$	14481.3	1.10160	0.550801	−	0.834636i	$-0.314322\pi$
$557$	14481.3	1.10160	0.550801	+	0.834636i	$0.314322\pi$
$558$	0	0
$559$	24.1583	0.00182788
$560$	0	0
$561$	742.386	0.0558709
$562$	0	0
$563$	10727.5	0.803034	0.401517	−	0.915851i	$-0.368483\pi$
$563$	10727.5	0.803034	0.401517	+	0.915851i	$0.368483\pi$
$564$	0	0
$565$	5868.28	0.436956
$566$	0	0
$567$	−4300.47	−0.318523
$568$	0	0
$569$	6075.02	0.447589	0.223794	−	0.974636i	$-0.428156\pi$
$569$	6075.02	0.447589	0.223794	+	0.974636i	$0.428156\pi$
$570$	0	0
$571$	−638.333	−0.0467835	−0.0233918	−	0.999726i	$-0.507447\pi$
$571$	−638.333	−0.0467835	−0.0233918	+	0.999726i	$0.507447\pi$
$572$	0	0
$573$	2950.61	0.215120
$574$	0	0
$575$	1199.97	0.0870297
$576$	0	0
$577$	27455.7	1.98093	0.990465	−	0.137761i	$-0.0439906\pi$
$577$	27455.7	1.98093	0.990465	+	0.137761i	$0.0439906\pi$
$578$	0	0
$579$	352.006	0.0252658
$580$	0	0
$581$	5421.34	0.387117
$582$	0	0
$583$	−8630.53	−0.613104
$584$	0	0
$585$	108.187	0.00764610
$586$	0	0
$587$	19411.6	1.36491	0.682455	−	0.730927i	$-0.260912\pi$
$587$	19411.6	1.36491	0.682455	+	0.730927i	$0.260912\pi$
$588$	0	0
$589$	1665.32	0.116500
$590$	0	0
$591$	−6280.73	−0.437149
$592$	0	0
$593$	19824.0	1.37281	0.686405	−	0.727220i	$-0.259188\pi$
$593$	19824.0	1.37281	0.686405	+	0.727220i	$0.259188\pi$
$594$	0	0
$595$	730.429	0.0503272
$596$	0	0
$597$	−814.242	−0.0558203
$598$	0	0
$599$	−12085.4	−0.824366	−0.412183	−	0.911101i	$-0.635234\pi$
$599$	−12085.4	−0.824366	−0.412183	+	0.911101i	$0.635234\pi$
$600$	0	0
$601$	16882.9	1.14587	0.572934	−	0.819602i	$-0.305805\pi$
$601$	16882.9	1.14587	0.572934	+	0.819602i	$0.305805\pi$
$602$	0	0
$603$	24812.1	1.67566
$604$	0	0
$605$	−46.7424	−0.00314107
$606$	0	0
$607$	−13132.5	−0.878139	−0.439069	−	0.898453i	$-0.644692\pi$
$607$	−13132.5	−0.878139	−0.439069	+	0.898453i	$0.644692\pi$
$608$	0	0
$609$	−1501.45	−0.0999042
$610$	0	0
$611$	426.439	0.0282354
$612$	0	0
$613$	−24352.6	−1.60455	−0.802276	−	0.596953i	$-0.796378\pi$
$613$	−24352.6	−1.60455	−0.802276	+	0.596953i	$0.796378\pi$
$614$	0	0
$615$	255.982	0.0167840
$616$	0	0
$617$	−23590.9	−1.53927	−0.769637	−	0.638481i	$-0.779563\pi$
$617$	−23590.9	−1.53927	−0.769637	+	0.638481i	$0.779563\pi$
$618$	0	0
$619$	20180.2	1.31036	0.655179	−	0.755474i	$-0.272593\pi$
$619$	20180.2	1.31036	0.655179	+	0.755474i	$0.272593\pi$
$620$	0	0
$621$	−866.999	−0.0560249
$622$	0	0
$623$	9184.56	0.590645
$624$	0	0
$625$	2916.06	0.186628
$626$	0	0
$627$	450.614	0.0287014
$628$	0	0
$629$	−6406.87	−0.406135
$630$	0	0
$631$	−1639.67	−0.103445	−0.0517227	−	0.998661i	$-0.516471\pi$
$631$	−1639.67	−0.103445	−0.0517227	+	0.998661i	$0.516471\pi$
$632$	0	0
$633$	4584.24	0.287847
$634$	0	0
$635$	17538.2	1.09603
$636$	0	0
$637$	−33.7906	−0.00210178
$638$	0	0
$639$	23334.7	1.44461
$640$	0	0
$641$	−13881.4	−0.855358	−0.427679	−	0.903931i	$-0.640669\pi$
$641$	−13881.4	−0.855358	−0.427679	+	0.903931i	$0.640669\pi$
$642$	0	0
$643$	8348.10	0.512002	0.256001	−	0.966677i	$-0.417595\pi$
$643$	8348.10	0.512002	0.256001	+	0.966677i	$0.417595\pi$
$644$	0	0
$645$	258.128	0.0157578
$646$	0	0
$647$	−14384.3	−0.874041	−0.437020	−	0.899452i	$-0.643966\pi$
$647$	−14384.3	−0.874041	−0.437020	+	0.899452i	$0.643966\pi$
$648$	0	0
$649$	10201.0	0.616986
$650$	0	0
$651$	−1356.16	−0.0816467
$652$	0	0
$653$	32379.6	1.94045	0.970224	−	0.242211i	$-0.0778726\pi$
$653$	32379.6	1.94045	0.970224	+	0.242211i	$0.0778726\pi$
$654$	0	0
$655$	8286.13	0.494299
$656$	0	0
$657$	26556.8	1.57699
$658$	0	0
$659$	10207.3	0.603368	0.301684	−	0.953408i	$-0.402451\pi$
$659$	10207.3	0.603368	0.301684	+	0.953408i	$0.402451\pi$
$660$	0	0
$661$	−141.054	−0.00830009	−0.00415004	−	0.999991i	$-0.501321\pi$
$661$	−141.054	−0.00830009	−0.00415004	+	0.999991i	$0.501321\pi$
$662$	0	0
$663$	−14.0730	−0.000824359 0
$664$	0	0
$665$	443.356	0.0258535
$666$	0	0
$667$	2455.32	0.142534
$668$	0	0
$669$	−4703.20	−0.271803
$670$	0	0
$671$	16504.6	0.949557
$672$	0	0
$673$	−17637.5	−1.01022	−0.505109	−	0.863056i	$-0.668548\pi$
$673$	−17637.5	−1.01022	−0.505109	+	0.863056i	$0.668548\pi$
$674$	0	0
$675$	−5509.59	−0.314169
$676$	0	0
$677$	−13675.7	−0.776364	−0.388182	−	0.921583i	$-0.626897\pi$
$677$	−13675.7	−0.776364	−0.388182	+	0.921583i	$0.626897\pi$
$678$	0	0
$679$	−4511.95	−0.255011
$680$	0	0
$681$	1232.27	0.0693401
$682$	0	0
$683$	5950.15	0.333347	0.166674	−	0.986012i	$-0.446697\pi$
$683$	5950.15	0.333347	0.166674	+	0.986012i	$0.446697\pi$
$684$	0	0
$685$	414.529	0.0231217
$686$	0	0
$687$	−2691.27	−0.149459
$688$	0	0
$689$	163.604	0.00904618
$690$	0	0
$691$	−13729.5	−0.755853	−0.377926	−	0.925836i	$-0.623363\pi$
$691$	−13729.5	−0.755853	−0.377926	+	0.925836i	$0.623363\pi$
$692$	0	0
$693$	6508.55	0.356767
$694$	0	0
$695$	12264.1	0.669356
$696$	0	0
$697$	590.594	0.0320952
$698$	0	0
$699$	−1262.28	−0.0683028
$700$	0	0
$701$	112.998	0.00608829	0.00304414	−	0.999995i	$-0.499031\pi$
$701$	112.998	0.00608829	0.00304414	+	0.999995i	$0.499031\pi$
$702$	0	0
$703$	−3888.84	−0.208635
$704$	0	0
$705$	4556.44	0.243412
$706$	0	0
$707$	−6849.29	−0.364348
$708$	0	0
$709$	−14666.1	−0.776864	−0.388432	−	0.921477i	$-0.626983\pi$
$709$	−14666.1	−0.776864	−0.388432	+	0.921477i	$0.626983\pi$
$710$	0	0
$711$	20442.0	1.07825
$712$	0	0
$713$	2217.73	0.116486
$714$	0	0
$715$	−153.983	−0.00805406
$716$	0	0
$717$	1483.41	0.0772649
$718$	0	0
$719$	−19663.1	−1.01991	−0.509953	−	0.860203i	$-0.670337\pi$
$719$	−19663.1	−1.01991	−0.509953	+	0.860203i	$0.670337\pi$
$720$	0	0
$721$	13623.5	0.703699
$722$	0	0
$723$	−1753.59	−0.0902027
$724$	0	0
$725$	15603.0	0.799285
$726$	0	0
$727$	28643.9	1.46127	0.730636	−	0.682767i	$-0.239224\pi$
$727$	28643.9	1.46127	0.730636	+	0.682767i	$0.239224\pi$
$728$	0	0
$729$	−13657.2	−0.693860
$730$	0	0
$731$	595.545	0.0301327
$732$	0	0
$733$	−22193.8	−1.11835	−0.559173	−	0.829051i	$-0.688881\pi$
$733$	−22193.8	−1.11835	−0.559173	+	0.829051i	$0.688881\pi$
$734$	0	0
$735$	−361.048	−0.0181190
$736$	0	0
$737$	−35315.3	−1.76507
$738$	0	0
$739$	−669.140	−0.0333082	−0.0166541	−	0.999861i	$-0.505301\pi$
$739$	−669.140	−0.0333082	−0.0166541	+	0.999861i	$0.505301\pi$
$740$	0	0
$741$	−8.54203	−0.000423481 0
$742$	0	0
$743$	30441.1	1.50306	0.751531	−	0.659698i	$-0.229316\pi$
$743$	30441.1	1.50306	0.751531	+	0.659698i	$0.229316\pi$
$744$	0	0
$745$	−12524.5	−0.615921
$746$	0	0
$747$	19794.8	0.969551
$748$	0	0
$749$	−571.424	−0.0278763
$750$	0	0
$751$	32417.4	1.57514	0.787569	−	0.616227i	$-0.211340\pi$
$751$	32417.4	1.57514	0.787569	+	0.616227i	$0.211340\pi$
$752$	0	0
$753$	−7970.05	−0.385717
$754$	0	0
$755$	−20040.9	−0.966046
$756$	0	0
$757$	12569.0	0.603472	0.301736	−	0.953392i	$-0.402434\pi$
$757$	12569.0	0.603472	0.301736	+	0.953392i	$0.402434\pi$
$758$	0	0
$759$	600.088	0.0286981
$760$	0	0
$761$	11222.7	0.534587	0.267294	−	0.963615i	$-0.413871\pi$
$761$	11222.7	0.534587	0.267294	+	0.963615i	$0.413871\pi$
$762$	0	0
$763$	−8145.05	−0.386462
$764$	0	0
$765$	2667.00	0.126047
$766$	0	0
$767$	−193.374	−0.00910345
$768$	0	0
$769$	−30494.0	−1.42996	−0.714981	−	0.699144i	$-0.753565\pi$
$769$	−30494.0	−1.42996	−0.714981	+	0.699144i	$0.753565\pi$
$770$	0	0
$771$	−3076.85	−0.143723
$772$	0	0
$773$	28760.0	1.33820	0.669098	−	0.743174i	$-0.266681\pi$
$773$	28760.0	1.33820	0.669098	+	0.743174i	$0.266681\pi$
$774$	0	0
$775$	14093.2	0.653216
$776$	0	0
$777$	3166.89	0.146218
$778$	0	0
$779$	358.479	0.0164876
$780$	0	0
$781$	−33212.6	−1.52169
$782$	0	0
$783$	−11273.5	−0.514536
$784$	0	0
$785$	22795.0	1.03642
$786$	0	0
$787$	10035.2	0.454531	0.227266	−	0.973833i	$-0.427021\pi$
$787$	10035.2	0.454531	0.227266	+	0.973833i	$0.427021\pi$
$788$	0	0
$789$	7886.15	0.355836
$790$	0	0
$791$	−6692.34	−0.300824
$792$	0	0
$793$	−312.868	−0.0140104
$794$	0	0
$795$	1748.09	0.0779852
$796$	0	0
$797$	23580.1	1.04799	0.523997	−	0.851720i	$-0.324440\pi$
$797$	23580.1	1.04799	0.523997	+	0.851720i	$0.324440\pi$
$798$	0	0
$799$	10512.5	0.465463
$800$	0	0
$801$	33535.4	1.47930
$802$	0	0
$803$	−37798.6	−1.66112
$804$	0	0
$805$	590.423	0.0258505
$806$	0	0
$807$	6364.70	0.277631
$808$	0	0
$809$	14215.9	0.617806	0.308903	−	0.951093i	$-0.400038\pi$
$809$	14215.9	0.617806	0.308903	+	0.951093i	$0.400038\pi$
$810$	0	0
$811$	12948.6	0.560649	0.280324	−	0.959905i	$-0.409558\pi$
$811$	12948.6	0.560649	0.280324	+	0.959905i	$0.409558\pi$
$812$	0	0
$813$	−164.510	−0.00709670
$814$	0	0
$815$	−1.86376	−8.01037e−5 0
$816$	0	0
$817$	361.484	0.0154795
$818$	0	0
$819$	−123.379	−0.00526399
$820$	0	0
$821$	−36221.5	−1.53976	−0.769878	−	0.638191i	$-0.779683\pi$
$821$	−36221.5	−1.53976	−0.769878	+	0.638191i	$0.779683\pi$
$822$	0	0
$823$	−34555.9	−1.46360	−0.731801	−	0.681518i	$-0.761320\pi$
$823$	−34555.9	−1.46360	−0.731801	+	0.681518i	$0.761320\pi$
$824$	0	0
$825$	3813.43	0.160929
$826$	0	0
$827$	30236.4	1.27137	0.635684	−	0.771949i	$-0.280718\pi$
$827$	30236.4	1.27137	0.635684	+	0.771949i	$0.280718\pi$
$828$	0	0
$829$	−31282.3	−1.31059	−0.655295	−	0.755373i	$-0.727456\pi$
$829$	−31282.3	−1.31059	−0.655295	+	0.755373i	$0.727456\pi$
$830$	0	0
$831$	1902.57	0.0794218
$832$	0	0
$833$	−833.000	−0.0346479
$834$	0	0
$835$	11560.5	0.479121
$836$	0	0
$837$	−10182.6	−0.420505
$838$	0	0
$839$	−15441.4	−0.635395	−0.317698	−	0.948192i	$-0.602910\pi$
$839$	−15441.4	−0.635395	−0.317698	+	0.948192i	$0.602910\pi$
$840$	0	0
$841$	7537.22	0.309042
$842$	0	0
$843$	−5119.45	−0.209162
$844$	0	0
$845$	−13482.4	−0.548886
$846$	0	0
$847$	53.3062	0.00216248
$848$	0	0
$849$	3713.20	0.150102
$850$	0	0
$851$	−5178.82	−0.208611
$852$	0	0
$853$	10617.4	0.426181	0.213090	−	0.977032i	$-0.431647\pi$
$853$	10617.4	0.426181	0.213090	+	0.977032i	$0.431647\pi$
$854$	0	0
$855$	1618.82	0.0647513
$856$	0	0
$857$	−30583.6	−1.21904	−0.609518	−	0.792772i	$-0.708637\pi$
$857$	−30583.6	−1.21904	−0.609518	+	0.792772i	$0.708637\pi$
$858$	0	0
$859$	18157.9	0.721233	0.360616	−	0.932714i	$-0.382566\pi$
$859$	18157.9	0.721233	0.360616	+	0.932714i	$0.382566\pi$
$860$	0	0
$861$	−291.928	−0.0115550
$862$	0	0
$863$	−11553.2	−0.455709	−0.227854	−	0.973695i	$-0.573171\pi$
$863$	−11553.2	−0.455709	−0.227854	+	0.973695i	$0.573171\pi$
$864$	0	0
$865$	5962.91	0.234387
$866$	0	0
$867$	−346.925	−0.0135896
$868$	0	0
$869$	−29095.3	−1.13578
$870$	0	0
$871$	669.452	0.0260431
$872$	0	0
$873$	−16474.4	−0.638687
$874$	0	0
$875$	9122.81	0.352465
$876$	0	0
$877$	−4660.41	−0.179442	−0.0897211	−	0.995967i	$-0.528598\pi$
$877$	−4660.41	−0.179442	−0.0897211	+	0.995967i	$0.528598\pi$
$878$	0	0
$879$	−8952.36	−0.343522
$880$	0	0
$881$	4283.04	0.163790	0.0818951	−	0.996641i	$-0.473903\pi$
$881$	4283.04	0.163790	0.0818951	+	0.996641i	$0.473903\pi$
$882$	0	0
$883$	−18630.2	−0.710030	−0.355015	−	0.934861i	$-0.615524\pi$
$883$	−18630.2	−0.710030	−0.355015	+	0.934861i	$0.615524\pi$
$884$	0	0
$885$	−2066.18	−0.0784789
$886$	0	0
$887$	−5053.80	−0.191308	−0.0956539	−	0.995415i	$-0.530494\pi$
$887$	−5053.80	−0.191308	−0.0956539	+	0.995415i	$0.530494\pi$
$888$	0	0
$889$	−20001.0	−0.754570
$890$	0	0
$891$	22349.1	0.840319
$892$	0	0
$893$	6380.87	0.239113
$894$	0	0
$895$	19996.9	0.746842
$896$	0	0
$897$	−11.3755	−0.000423431 0
$898$	0	0
$899$	28836.9	1.06981
$900$	0	0
$901$	4033.14	0.149127
$902$	0	0
$903$	−294.376	−0.0108485
$904$	0	0
$905$	−4327.71	−0.158959
$906$	0	0
$907$	−13600.6	−0.497906	−0.248953	−	0.968516i	$-0.580086\pi$
$907$	−13600.6	−0.497906	−0.248953	+	0.968516i	$0.580086\pi$
$908$	0	0
$909$	−25008.7	−0.912526
$910$	0	0
$911$	9210.17	0.334958	0.167479	−	0.985876i	$-0.446437\pi$
$911$	9210.17	0.334958	0.167479	+	0.985876i	$0.446437\pi$
$912$	0	0
$913$	−28174.2	−1.02128
$914$	0	0
$915$	−3342.95	−0.120781
$916$	0	0
$917$	−9449.72	−0.340302
$918$	0	0
$919$	−38434.7	−1.37959	−0.689796	−	0.724004i	$-0.742300\pi$
$919$	−38434.7	−1.37959	−0.689796	+	0.724004i	$0.742300\pi$
$920$	0	0
$921$	6471.46	0.231533
$922$	0	0
$923$	629.593	0.0224521
$924$	0	0
$925$	−32910.3	−1.16982
$926$	0	0
$927$	49743.4	1.76244
$928$	0	0
$929$	−20810.7	−0.734958	−0.367479	−	0.930032i	$-0.619779\pi$
$929$	−20810.7	−0.734958	−0.367479	+	0.930032i	$0.619779\pi$
$930$	0	0
$931$	−505.614	−0.0177990
$932$	0	0
$933$	−11239.2	−0.394378
$934$	0	0
$935$	−3795.97	−0.132772
$936$	0	0
$937$	29319.0	1.02221	0.511105	−	0.859518i	$-0.329236\pi$
$937$	29319.0	1.02221	0.511105	+	0.859518i	$0.329236\pi$
$938$	0	0
$939$	799.833	0.0277972
$940$	0	0
$941$	−42231.3	−1.46302	−0.731510	−	0.681831i	$-0.761184\pi$
$941$	−42231.3	−1.46302	−0.731510	+	0.681831i	$0.761184\pi$
$942$	0	0
$943$	477.391	0.0164857
$944$	0	0
$945$	−2710.90	−0.0933181
$946$	0	0
$947$	4544.19	0.155931	0.0779654	−	0.996956i	$-0.475158\pi$
$947$	4544.19	0.155931	0.0779654	+	0.996956i	$0.475158\pi$
$948$	0	0
$949$	716.526	0.0245094
$950$	0	0
$951$	−796.451	−0.0271574
$952$	0	0
$953$	46006.7	1.56380	0.781900	−	0.623404i	$-0.214251\pi$
$953$	46006.7	1.56380	0.781900	+	0.623404i	$0.214251\pi$
$954$	0	0
$955$	−15087.1	−0.511211
$956$	0	0
$957$	7802.88	0.263564
$958$	0	0
$959$	−472.740	−0.0159182
$960$	0	0
$961$	−3744.54	−0.125694
$962$	0	0
$963$	−2086.43	−0.0698175
$964$	0	0
$965$	−1799.88	−0.0600416
$966$	0	0
$967$	−11560.1	−0.384434	−0.192217	−	0.981352i	$-0.561568\pi$
$967$	−11560.1	−0.384434	−0.192217	+	0.981352i	$0.561568\pi$
$968$	0	0
$969$	−210.577	−0.00698111
$970$	0	0
$971$	2778.50	0.0918295	0.0459147	−	0.998945i	$-0.485380\pi$
$971$	2778.50	0.0918295	0.0459147	+	0.998945i	$0.485380\pi$
$972$	0	0
$973$	−13986.3	−0.460821
$974$	0	0
$975$	−72.2891	−0.00237446
$976$	0	0
$977$	44345.9	1.45215	0.726076	−	0.687615i	$-0.241342\pi$
$977$	44345.9	1.45215	0.726076	+	0.687615i	$0.241342\pi$
$978$	0	0
$979$	−47731.3	−1.55822
$980$	0	0
$981$	−29739.9	−0.967911
$982$	0	0
$983$	−35261.2	−1.14411	−0.572054	−	0.820216i	$-0.693853\pi$
$983$	−35261.2	−1.14411	−0.572054	+	0.820216i	$0.693853\pi$
$984$	0	0
$985$	32114.7	1.03884
$986$	0	0
$987$	−5196.28	−0.167578
$988$	0	0
$989$	481.393	0.0154777
$990$	0	0
$991$	−37985.8	−1.21762	−0.608809	−	0.793317i	$-0.708352\pi$
$991$	−37985.8	−1.21762	−0.608809	+	0.793317i	$0.708352\pi$
$992$	0	0
$993$	−5672.60	−0.181283
$994$	0	0
$995$	4163.38	0.132651
$996$	0	0
$997$	35915.9	1.14089	0.570445	−	0.821336i	$-0.306771\pi$
$997$	35915.9	1.14089	0.570445	+	0.821336i	$0.306771\pi$
$998$	0	0
$999$	23778.3	0.753067

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	476.4.a.c.1.3	✓	6
4.3	odd	2		1904.4.a.n.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.4.a.c.1.3	✓	6	1.1	even	1	trivial
1904.4.a.n.1.4		6	4.3	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:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	476.a (trivial)

\(f(q)\)	\(=\)	\(q-1.20043 q^{3} +6.13806 q^{5} -7.00000 q^{7} -25.5590 q^{9} +O(q^{10})\)	\(q-1.20043 q^{3} +6.13806 q^{5} -7.00000 q^{7} -25.5590 q^{9} +36.3784 q^{11} -0.689604 q^{13} -7.36832 q^{15} -17.0000 q^{17} -10.3187 q^{19} +8.40303 q^{21} -13.7415 q^{23} -87.3243 q^{25} +63.0935 q^{27} -178.679 q^{29} -161.389 q^{31} -43.6698 q^{33} -42.9664 q^{35} +376.875 q^{37} +0.827823 q^{39} -34.7408 q^{41} -35.0321 q^{43} -156.882 q^{45} -618.382 q^{47} +49.0000 q^{49} +20.4074 q^{51} -237.243 q^{53} +223.292 q^{55} +12.3869 q^{57} +280.414 q^{59} +453.692 q^{61} +178.913 q^{63} -4.23283 q^{65} -970.778 q^{67} +16.4957 q^{69} -912.977 q^{71} -1039.04 q^{73} +104.827 q^{75} -254.648 q^{77} -799.796 q^{79} +614.352 q^{81} -774.476 q^{83} -104.347 q^{85} +214.492 q^{87} -1312.08 q^{89} +4.82723 q^{91} +193.737 q^{93} -63.3365 q^{95} +644.564 q^{97} -929.793 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9	\( 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9} - 20 q^{11} + 42 q^{13} - 136 q^{15} - 102 q^{17} - 104 q^{19} - 14 q^{21} - 230 q^{23} + 108 q^{25} - 130 q^{27} - 52 q^{29} - 564 q^{31} - 346 q^{33} - 70 q^{35} - 564 q^{37} - 626 q^{39} - 548 q^{41} - 648 q^{43} - 174 q^{45} - 366 q^{47} + 294 q^{49} - 34 q^{51} - 74 q^{53} - 1460 q^{55} - 316 q^{57} - 558 q^{59} - 620 q^{61} - 112 q^{63} - 1378 q^{65} - 164 q^{67} - 540 q^{69} - 822 q^{71} + 940 q^{73} - 2698 q^{75} + 140 q^{77} - 1838 q^{79} - 3094 q^{81} - 1118 q^{83} - 170 q^{85} - 1354 q^{87} - 1634 q^{89} - 294 q^{91} + 268 q^{93} - 2642 q^{95} + 182 q^{97} + 12 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9 - 20 * q^11 + 42 * q^13 - 136 * q^15 - 102 * q^17 - 104 * q^19 - 14 * q^21 - 230 * q^23 + 108 * q^25 - 130 * q^27 - 52 * q^29 - 564 * q^31 - 346 * q^33 - 70 * q^35 - 564 * q^37 - 626 * q^39 - 548 * q^41 - 648 * q^43 - 174 * q^45 - 366 * q^47 + 294 * q^49 - 34 * q^51 - 74 * q^53 - 1460 * q^55 - 316 * q^57 - 558 * q^59 - 620 * q^61 - 112 * q^63 - 1378 * q^65 - 164 * q^67 - 540 * q^69 - 822 * q^71 + 940 * q^73 - 2698 * q^75 + 140 * q^77 - 1838 * q^79 - 3094 * q^81 - 1118 * q^83 - 170 * q^85 - 1354 * q^87 - 1634 * q^89 - 294 * q^91 + 268 * q^93 - 2642 * q^95 + 182 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more