LMFDB - Embedded newform 684.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,4,Mod(1,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, 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("684.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	684.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:	$40.3573064439$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 76)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	684.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.8614 q^{5} -26.4891 q^{7} +O(q^{10})$	$q-11.8614 q^{5} -26.4891 q^{7} +49.8614 q^{11} -49.0951 q^{13} -17.2337 q^{17} -19.0000 q^{19} -166.965 q^{23} +15.6930 q^{25} +109.198 q^{29} -273.783 q^{31} +314.198 q^{35} +167.022 q^{37} -15.1684 q^{41} +413.557 q^{43} +161.438 q^{47} +358.674 q^{49} +490.791 q^{53} -591.426 q^{55} +335.970 q^{59} +725.149 q^{61} +582.337 q^{65} +497.709 q^{67} +798.554 q^{71} -311.174 q^{73} -1320.79 q^{77} -665.723 q^{79} +372.440 q^{83} +204.416 q^{85} +673.783 q^{89} +1300.49 q^{91} +225.367 q^{95} -960.505 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{5} - 30 q^{7}+O(q^{10}) $ 2 * q + 5 * q^5 - 30 * q^7	$ 2 q + 5 q^{5} - 30 q^{7} + 71 q^{11} - 35 q^{13} - 38 q^{19} + 5 q^{23} + 175 q^{25} - 155 q^{29} - 88 q^{31} + 255 q^{35} + 380 q^{37} + 142 q^{41} + 155 q^{43} + 455 q^{47} + 28 q^{49} + 275 q^{53} - 235 q^{55} + 873 q^{59} + 445 q^{61} + 820 q^{65} + 645 q^{67} + 1712 q^{71} - 990 q^{73} - 1395 q^{77} - 1274 q^{79} + 90 q^{83} + 495 q^{85} + 888 q^{89} + 1251 q^{91} - 95 q^{95} + 710 q^{97}+O(q^{100}) $ 2 * q + 5 * q^5 - 30 * q^7 + 71 * q^11 - 35 * q^13 - 38 * q^19 + 5 * q^23 + 175 * q^25 - 155 * q^29 - 88 * q^31 + 255 * q^35 + 380 * q^37 + 142 * q^41 + 155 * q^43 + 455 * q^47 + 28 * q^49 + 275 * q^53 - 235 * q^55 + 873 * q^59 + 445 * q^61 + 820 * q^65 + 645 * q^67 + 1712 * q^71 - 990 * q^73 - 1395 * q^77 - 1274 * q^79 + 90 * q^83 + 495 * q^85 + 888 * q^89 + 1251 * q^91 - 95 * q^95 + 710 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−11.8614	−1.06092	−0.530458	−	0.847711i	$-0.677980\pi$
$5$	−11.8614	−1.06092	−0.530458	+	0.847711i	$0.677980\pi$
$6$	0	0
$7$	−26.4891	−1.43028	−0.715139	−	0.698982i	$-0.753637\pi$
$7$	−26.4891	−1.43028	−0.715139	+	0.698982i	$0.753637\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	49.8614	1.36671	0.683354	−	0.730088i	$-0.260521\pi$
$11$	49.8614	1.36671	0.683354	+	0.730088i	$0.260521\pi$
$12$	0	0
$13$	−49.0951	−1.04743	−0.523713	−	0.851895i	$-0.675453\pi$
$13$	−49.0951	−1.04743	−0.523713	+	0.851895i	$0.675453\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−17.2337	−0.245870	−0.122935	−	0.992415i	$-0.539231\pi$
$17$	−17.2337	−0.245870	−0.122935	+	0.992415i	$0.539231\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−166.965	−1.51368	−0.756838	−	0.653603i	$-0.773257\pi$
$23$	−166.965	−1.51368	−0.756838	+	0.653603i	$0.773257\pi$
$24$	0	0
$25$	15.6930	0.125544
$26$	0	0
$27$	0	0
$28$	0	0
$29$	109.198	0.699228	0.349614	−	0.936894i	$-0.386313\pi$
$29$	109.198	0.699228	0.349614	+	0.936894i	$0.386313\pi$
$30$	0	0
$31$	−273.783	−1.58622	−0.793110	−	0.609079i	$-0.791539\pi$
$31$	−273.783	−1.58622	−0.793110	+	0.609079i	$0.791539\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	314.198	1.51741
$36$	0	0
$37$	167.022	0.742114	0.371057	−	0.928610i	$-0.378995\pi$
$37$	167.022	0.742114	0.371057	+	0.928610i	$0.378995\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−15.1684	−0.0577783	−0.0288892	−	0.999583i	$-0.509197\pi$
$41$	−15.1684	−0.0577783	−0.0288892	+	0.999583i	$0.509197\pi$
$42$	0	0
$43$	413.557	1.46667	0.733335	−	0.679867i	$-0.237963\pi$
$43$	413.557	1.46667	0.733335	+	0.679867i	$0.237963\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	161.438	0.501023	0.250512	−	0.968114i	$-0.419401\pi$
$47$	161.438	0.501023	0.250512	+	0.968114i	$0.419401\pi$
$48$	0	0
$49$	358.674	1.04570
$50$	0	0
$51$	0	0
$52$	0	0
$53$	490.791	1.27199	0.635993	−	0.771695i	$-0.280591\pi$
$53$	490.791	1.27199	0.635993	+	0.771695i	$0.280591\pi$
$54$	0	0
$55$	−591.426	−1.44996
$56$	0	0
$57$	0	0
$58$	0	0
$59$	335.970	0.741349	0.370674	−	0.928763i	$-0.379127\pi$
$59$	335.970	0.741349	0.370674	+	0.928763i	$0.379127\pi$
$60$	0	0
$61$	725.149	1.52206	0.761032	−	0.648715i	$-0.224693\pi$
$61$	725.149	1.52206	0.761032	+	0.648715i	$0.224693\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	582.337	1.11123
$66$	0	0
$67$	497.709	0.907535	0.453768	−	0.891120i	$-0.350080\pi$
$67$	497.709	0.907535	0.453768	+	0.891120i	$0.350080\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	798.554	1.33480	0.667401	−	0.744698i	$-0.267407\pi$
$71$	798.554	1.33480	0.667401	+	0.744698i	$0.267407\pi$
$72$	0	0
$73$	−311.174	−0.498906	−0.249453	−	0.968387i	$-0.580251\pi$
$73$	−311.174	−0.498906	−0.249453	+	0.968387i	$0.580251\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1320.79	−1.95477
$78$	0	0
$79$	−665.723	−0.948097	−0.474049	−	0.880499i	$-0.657208\pi$
$79$	−665.723	−0.948097	−0.474049	+	0.880499i	$0.657208\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	372.440	0.492537	0.246269	−	0.969202i	$-0.420795\pi$
$83$	372.440	0.492537	0.246269	+	0.969202i	$0.420795\pi$
$84$	0	0
$85$	204.416	0.260847
$86$	0	0
$87$	0	0
$88$	0	0
$89$	673.783	0.802481	0.401240	−	0.915973i	$-0.368579\pi$
$89$	673.783	0.802481	0.401240	+	0.915973i	$0.368579\pi$
$90$	0	0
$91$	1300.49	1.49811
$92$	0	0
$93$	0	0
$94$	0	0
$95$	225.367	0.243391
$96$	0	0
$97$	−960.505	−1.00541	−0.502704	−	0.864459i	$-0.667661\pi$
$97$	−960.505	−1.00541	−0.502704	+	0.864459i	$0.667661\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1889.68	−1.86169	−0.930845	−	0.365415i	$-0.880927\pi$
$101$	−1889.68	−1.86169	−0.930845	+	0.365415i	$0.880927\pi$
$102$	0	0
$103$	760.217	0.727247	0.363624	−	0.931546i	$-0.381539\pi$
$103$	760.217	0.727247	0.363624	+	0.931546i	$0.381539\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−681.730	−0.615938	−0.307969	−	0.951396i	$-0.599649\pi$
$107$	−681.730	−0.615938	−0.307969	+	0.951396i	$0.599649\pi$
$108$	0	0
$109$	−1310.76	−1.15182	−0.575910	−	0.817513i	$-0.695352\pi$
$109$	−1310.76	−1.15182	−0.575910	+	0.817513i	$0.695352\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	713.413	0.593914	0.296957	−	0.954891i	$-0.404028\pi$
$113$	713.413	0.593914	0.296957	+	0.954891i	$0.404028\pi$
$114$	0	0
$115$	1980.43	1.60588
$116$	0	0
$117$	0	0
$118$	0	0
$119$	456.505	0.351662
$120$	0	0
$121$	1155.16	0.867889
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1296.54	0.927725
$126$	0	0
$127$	−107.847	−0.0753535	−0.0376768	−	0.999290i	$-0.511996\pi$
$127$	−107.847	−0.0753535	−0.0376768	+	0.999290i	$0.511996\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−373.247	−0.248937	−0.124469	−	0.992224i	$-0.539723\pi$
$131$	−373.247	−0.248937	−0.124469	+	0.992224i	$0.539723\pi$
$132$	0	0
$133$	503.293	0.328128
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3118.25	1.94460	0.972301	−	0.233732i	$-0.0750940\pi$
$137$	3118.25	1.94460	0.972301	+	0.233732i	$0.0750940\pi$
$138$	0	0
$139$	−56.3774	−0.0344019	−0.0172010	−	0.999852i	$-0.505476\pi$
$139$	−56.3774	−0.0344019	−0.0172010	+	0.999852i	$0.505476\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2447.95	−1.43152
$144$	0	0
$145$	−1295.25	−0.741823
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1810.51	0.995457	0.497728	−	0.867333i	$-0.334168\pi$
$149$	1810.51	0.995457	0.497728	+	0.867333i	$0.334168\pi$
$150$	0	0
$151$	2894.72	1.56006	0.780029	−	0.625744i	$-0.215204\pi$
$151$	2894.72	1.56006	0.780029	+	0.625744i	$0.215204\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3247.45	1.68285
$156$	0	0
$157$	1381.71	0.702371	0.351185	−	0.936306i	$-0.385779\pi$
$157$	1381.71	0.702371	0.351185	+	0.936306i	$0.385779\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4422.75	2.16498
$162$	0	0
$163$	−3740.83	−1.79757	−0.898785	−	0.438389i	$-0.855549\pi$
$163$	−3740.83	−1.79757	−0.898785	+	0.438389i	$0.855549\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3085.17	−1.42957	−0.714783	−	0.699347i	$-0.753474\pi$
$167$	−3085.17	−1.42957	−0.714783	+	0.699347i	$0.753474\pi$
$168$	0	0
$169$	213.328	0.0970998
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−293.000	−0.128765	−0.0643827	−	0.997925i	$-0.520508\pi$
$173$	−293.000	−0.128765	−0.0643827	+	0.997925i	$0.520508\pi$
$174$	0	0
$175$	−415.693	−0.179562
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2996.32	1.25115	0.625574	−	0.780165i	$-0.284865\pi$
$179$	2996.32	1.25115	0.625574	+	0.780165i	$0.284865\pi$
$180$	0	0
$181$	−3265.41	−1.34097	−0.670487	−	0.741922i	$-0.733915\pi$
$181$	−3265.41	−1.34097	−0.670487	+	0.741922i	$0.733915\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1981.11	−0.787321
$186$	0	0
$187$	−859.296	−0.336032
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1502.27	0.569113	0.284556	−	0.958659i	$-0.408154\pi$
$191$	1502.27	0.569113	0.284556	+	0.958659i	$0.408154\pi$
$192$	0	0
$193$	−4103.30	−1.53037	−0.765186	−	0.643809i	$-0.777353\pi$
$193$	−4103.30	−1.53037	−0.765186	+	0.643809i	$0.777353\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5420.95	−1.96054	−0.980271	−	0.197658i	$-0.936666\pi$
$197$	−5420.95	−1.96054	−0.980271	+	0.197658i	$0.936666\pi$
$198$	0	0
$199$	1666.05	0.593484	0.296742	−	0.954958i	$-0.404100\pi$
$199$	1666.05	0.593484	0.296742	+	0.954958i	$0.404100\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2892.57	−1.00009
$204$	0	0
$205$	179.919	0.0612980
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−947.367	−0.313544
$210$	0	0
$211$	5865.90	1.91386	0.956931	−	0.290314i	$-0.0937598\pi$
$211$	5865.90	1.91386	0.956931	+	0.290314i	$0.0937598\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4905.37	−1.55602
$216$	0	0
$217$	7252.26	2.26873
$218$	0	0
$219$	0	0
$220$	0	0
$221$	846.090	0.257530
$222$	0	0
$223$	−5402.94	−1.62246	−0.811229	−	0.584729i	$-0.801201\pi$
$223$	−5402.94	−1.62246	−0.811229	+	0.584729i	$0.801201\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2571.34	0.751833	0.375916	−	0.926654i	$-0.377328\pi$
$227$	2571.34	0.751833	0.375916	+	0.926654i	$0.377328\pi$
$228$	0	0
$229$	3256.08	0.939597	0.469799	−	0.882774i	$-0.344327\pi$
$229$	3256.08	0.939597	0.469799	+	0.882774i	$0.344327\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1205.58	−0.338972	−0.169486	−	0.985533i	$-0.554211\pi$
$233$	−1205.58	−0.338972	−0.169486	+	0.985533i	$0.554211\pi$
$234$	0	0
$235$	−1914.88	−0.531544
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3404.52	0.921424	0.460712	−	0.887550i	$-0.347594\pi$
$239$	3404.52	0.921424	0.460712	+	0.887550i	$0.347594\pi$
$240$	0	0
$241$	3301.51	0.882443	0.441221	−	0.897398i	$-0.354545\pi$
$241$	3301.51	0.882443	0.441221	+	0.897398i	$0.354545\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4254.38	−1.10940
$246$	0	0
$247$	932.807	0.240296
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5330.81	1.34055	0.670275	−	0.742113i	$-0.266176\pi$
$251$	5330.81	1.34055	0.670275	+	0.742113i	$0.266176\pi$
$252$	0	0
$253$	−8325.09	−2.06875
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3812.14	−0.925272	−0.462636	−	0.886548i	$-0.653096\pi$
$257$	−3812.14	−0.925272	−0.462636	+	0.886548i	$0.653096\pi$
$258$	0	0
$259$	−4424.26	−1.06143
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−899.159	−0.210816	−0.105408	−	0.994429i	$-0.533615\pi$
$263$	−899.159	−0.210816	−0.105408	+	0.994429i	$0.533615\pi$
$264$	0	0
$265$	−5821.47	−1.34947
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2645.77	−0.599685	−0.299842	−	0.953989i	$-0.596934\pi$
$269$	−2645.77	−0.599685	−0.299842	+	0.953989i	$0.596934\pi$
$270$	0	0
$271$	516.529	0.115782	0.0578909	−	0.998323i	$-0.481562\pi$
$271$	516.529	0.115782	0.0578909	+	0.998323i	$0.481562\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	782.473	0.171582
$276$	0	0
$277$	2365.14	0.513023	0.256512	−	0.966541i	$-0.417427\pi$
$277$	2365.14	0.513023	0.256512	+	0.966541i	$0.417427\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1526.54	0.324077	0.162038	−	0.986784i	$-0.448193\pi$
$281$	1526.54	0.324077	0.162038	+	0.986784i	$0.448193\pi$
$282$	0	0
$283$	4764.01	1.00067	0.500337	−	0.865831i	$-0.333209\pi$
$283$	4764.01	1.00067	0.500337	+	0.865831i	$0.333209\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	401.799	0.0826391
$288$	0	0
$289$	−4616.00	−0.939548
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3189.07	−0.635862	−0.317931	−	0.948114i	$-0.602988\pi$
$293$	−3189.07	−0.635862	−0.317931	+	0.948114i	$0.602988\pi$
$294$	0	0
$295$	−3985.08	−0.786509
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8197.14	1.58546
$300$	0	0
$301$	−10954.8	−2.09775
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8601.29	−1.61478
$306$	0	0
$307$	5998.56	1.11517	0.557583	−	0.830121i	$-0.311729\pi$
$307$	5998.56	1.11517	0.557583	+	0.830121i	$0.311729\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5114.96	−0.932614	−0.466307	−	0.884623i	$-0.654416\pi$
$311$	−5114.96	−0.932614	−0.466307	+	0.884623i	$0.654416\pi$
$312$	0	0
$313$	2131.63	0.384942	0.192471	−	0.981303i	$-0.438350\pi$
$313$	2131.63	0.384942	0.192471	+	0.981303i	$0.438350\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4457.03	−0.789691	−0.394845	−	0.918748i	$-0.629202\pi$
$317$	−4457.03	−0.789691	−0.394845	+	0.918748i	$0.629202\pi$
$318$	0	0
$319$	5444.78	0.955640
$320$	0	0
$321$	0	0
$322$	0	0
$323$	327.440	0.0564064
$324$	0	0
$325$	−770.448	−0.131498
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4276.34	−0.716602
$330$	0	0
$331$	−4143.68	−0.688088	−0.344044	−	0.938954i	$-0.611797\pi$
$331$	−4143.68	−0.688088	−0.344044	+	0.938954i	$0.611797\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5903.53	−0.962819
$336$	0	0
$337$	−5148.60	−0.832231	−0.416115	−	0.909312i	$-0.636609\pi$
$337$	−5148.60	−0.832231	−0.416115	+	0.909312i	$0.636609\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−13651.2	−2.16790
$342$	0	0
$343$	−415.184	−0.0653581
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5873.56	0.908672	0.454336	−	0.890830i	$-0.349877\pi$
$347$	5873.56	0.908672	0.454336	+	0.890830i	$0.349877\pi$
$348$	0	0
$349$	5167.58	0.792590	0.396295	−	0.918123i	$-0.370296\pi$
$349$	5167.58	0.792590	0.396295	+	0.918123i	$0.370296\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7191.17	−1.08427	−0.542135	−	0.840291i	$-0.682384\pi$
$353$	−7191.17	−1.08427	−0.542135	+	0.840291i	$0.682384\pi$
$354$	0	0
$355$	−9471.98	−1.41611
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7049.62	1.03639	0.518196	−	0.855262i	$-0.326604\pi$
$359$	7049.62	1.03639	0.518196	+	0.855262i	$0.326604\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3690.96	0.529298
$366$	0	0
$367$	6935.53	0.986463	0.493231	−	0.869898i	$-0.335816\pi$
$367$	6935.53	0.986463	0.493231	+	0.869898i	$0.335816\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−13000.6	−1.81929
$372$	0	0
$373$	−6346.92	−0.881048	−0.440524	−	0.897741i	$-0.645207\pi$
$373$	−6346.92	−0.881048	−0.440524	+	0.897741i	$0.645207\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5361.10	−0.732389
$378$	0	0
$379$	11363.0	1.54004	0.770022	−	0.638017i	$-0.220245\pi$
$379$	11363.0	1.54004	0.770022	+	0.638017i	$0.220245\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3848.09	0.513390	0.256695	−	0.966493i	$-0.417366\pi$
$383$	3848.09	0.513390	0.256695	+	0.966493i	$0.417366\pi$
$384$	0	0
$385$	15666.4	2.07385
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1936.75	−0.252435	−0.126217	−	0.992003i	$-0.540284\pi$
$389$	−1936.75	−0.252435	−0.126217	+	0.992003i	$0.540284\pi$
$390$	0	0
$391$	2877.42	0.372167
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7896.41	1.00585
$396$	0	0
$397$	3851.40	0.486892	0.243446	−	0.969914i	$-0.421722\pi$
$397$	3851.40	0.486892	0.243446	+	0.969914i	$0.421722\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1381.47	0.172038	0.0860189	−	0.996294i	$-0.472585\pi$
$401$	1381.47	0.172038	0.0860189	+	0.996294i	$0.472585\pi$
$402$	0	0
$403$	13441.4	1.66145
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8327.94	1.01425
$408$	0	0
$409$	368.878	0.0445962	0.0222981	−	0.999751i	$-0.492902\pi$
$409$	368.878	0.0445962	0.0222981	+	0.999751i	$0.492902\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8899.56	−1.06034
$414$	0	0
$415$	−4417.66	−0.522541
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1814.85	0.211602	0.105801	−	0.994387i	$-0.466259\pi$
$419$	1814.85	0.211602	0.105801	+	0.994387i	$0.466259\pi$
$420$	0	0
$421$	4614.01	0.534141	0.267070	−	0.963677i	$-0.413944\pi$
$421$	4614.01	0.534141	0.267070	+	0.963677i	$0.413944\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−270.448	−0.0308674
$426$	0	0
$427$	−19208.6	−2.17697
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2739.14	0.306125	0.153062	−	0.988217i	$-0.451086\pi$
$431$	2739.14	0.306125	0.153062	+	0.988217i	$0.451086\pi$
$432$	0	0
$433$	−10827.9	−1.20174	−0.600872	−	0.799345i	$-0.705180\pi$
$433$	−10827.9	−1.20174	−0.600872	+	0.799345i	$0.705180\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3172.33	0.347261
$438$	0	0
$439$	−10359.5	−1.12627	−0.563137	−	0.826364i	$-0.690406\pi$
$439$	−10359.5	−1.12627	−0.563137	+	0.826364i	$0.690406\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8040.36	−0.862322	−0.431161	−	0.902275i	$-0.641896\pi$
$443$	−8040.36	−0.862322	−0.431161	+	0.902275i	$0.641896\pi$
$444$	0	0
$445$	−7992.01	−0.851365
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14071.8	1.47904	0.739522	−	0.673132i	$-0.235052\pi$
$449$	14071.8	1.47904	0.739522	+	0.673132i	$0.235052\pi$
$450$	0	0
$451$	−756.320	−0.0789661
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15425.6	−1.58937
$456$	0	0
$457$	−4372.74	−0.447589	−0.223795	−	0.974636i	$-0.571845\pi$
$457$	−4372.74	−0.447589	−0.223795	+	0.974636i	$0.571845\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17819.4	1.80029	0.900146	−	0.435589i	$-0.143460\pi$
$461$	17819.4	1.80029	0.900146	+	0.435589i	$0.143460\pi$
$462$	0	0
$463$	7114.51	0.714124	0.357062	−	0.934081i	$-0.383778\pi$
$463$	7114.51	0.714124	0.357062	+	0.934081i	$0.383778\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8208.41	0.813361	0.406681	−	0.913570i	$-0.366686\pi$
$467$	8208.41	0.813361	0.406681	+	0.913570i	$0.366686\pi$
$468$	0	0
$469$	−13183.9	−1.29803
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20620.5	2.00451
$474$	0	0
$475$	−298.166	−0.0288017
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11889.5	1.13413	0.567064	−	0.823674i	$-0.308079\pi$
$479$	11889.5	1.13413	0.567064	+	0.823674i	$0.308079\pi$
$480$	0	0
$481$	−8199.95	−0.777309
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11392.9	1.06665
$486$	0	0
$487$	13922.4	1.29545	0.647725	−	0.761875i	$-0.275721\pi$
$487$	13922.4	1.29545	0.647725	+	0.761875i	$0.275721\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2718.22	−0.249840	−0.124920	−	0.992167i	$-0.539867\pi$
$491$	−2718.22	−0.249840	−0.124920	+	0.992167i	$0.539867\pi$
$492$	0	0
$493$	−1881.89	−0.171919
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−21153.0	−1.90914
$498$	0	0
$499$	−7830.51	−0.702489	−0.351244	−	0.936284i	$-0.614241\pi$
$499$	−7830.51	−0.702489	−0.351244	+	0.936284i	$0.614241\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6554.35	0.581002	0.290501	−	0.956875i	$-0.406178\pi$
$503$	6554.35	0.581002	0.290501	+	0.956875i	$0.406178\pi$
$504$	0	0
$505$	22414.3	1.97510
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12611.4	−1.09821	−0.549105	−	0.835753i	$-0.685031\pi$
$509$	−12611.4	−1.09821	−0.549105	+	0.835753i	$0.685031\pi$
$510$	0	0
$511$	8242.73	0.713575
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9017.25	−0.771548
$516$	0	0
$517$	8049.50	0.684752
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1695.01	−0.142533	−0.0712666	−	0.997457i	$-0.522704\pi$
$521$	−1695.01	−0.142533	−0.0712666	+	0.997457i	$0.522704\pi$
$522$	0	0
$523$	−21001.4	−1.75589	−0.877944	−	0.478764i	$-0.841085\pi$
$523$	−21001.4	−1.75589	−0.877944	+	0.478764i	$0.841085\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4718.28	0.390003
$528$	0	0
$529$	15710.2	1.29121
$530$	0	0
$531$	0	0
$532$	0	0
$533$	744.696	0.0605185
$534$	0	0
$535$	8086.28	0.653459
$536$	0	0
$537$	0	0
$538$	0	0
$539$	17884.0	1.42916
$540$	0	0
$541$	−555.994	−0.0441849	−0.0220925	−	0.999756i	$-0.507033\pi$
$541$	−555.994	−0.0441849	−0.0220925	+	0.999756i	$0.507033\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	15547.5	1.22198
$546$	0	0
$547$	−2987.93	−0.233555	−0.116778	−	0.993158i	$-0.537256\pi$
$547$	−2987.93	−0.233555	−0.116778	+	0.993158i	$0.537256\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2074.77	−0.160414
$552$	0	0
$553$	17634.4	1.35604
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4357.38	0.331469	0.165735	−	0.986170i	$-0.447001\pi$
$557$	4357.38	0.331469	0.165735	+	0.986170i	$0.447001\pi$
$558$	0	0
$559$	−20303.6	−1.53623
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1669.67	−0.124988	−0.0624941	−	0.998045i	$-0.519905\pi$
$563$	−1669.67	−0.124988	−0.0624941	+	0.998045i	$0.519905\pi$
$564$	0	0
$565$	−8462.08	−0.630093
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20440.1	1.50596	0.752982	−	0.658041i	$-0.228615\pi$
$569$	20440.1	1.50596	0.752982	+	0.658041i	$0.228615\pi$
$570$	0	0
$571$	−6920.98	−0.507240	−0.253620	−	0.967304i	$-0.581621\pi$
$571$	−6920.98	−0.507240	−0.253620	+	0.967304i	$0.581621\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2620.17	−0.190032
$576$	0	0
$577$	−5021.59	−0.362307	−0.181154	−	0.983455i	$-0.557983\pi$
$577$	−5021.59	−0.362307	−0.181154	+	0.983455i	$0.557983\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9865.61	−0.704466
$582$	0	0
$583$	24471.5	1.73843
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6784.80	0.477068	0.238534	−	0.971134i	$-0.423333\pi$
$587$	6784.80	0.477068	0.238534	+	0.971134i	$0.423333\pi$
$588$	0	0
$589$	5201.87	0.363904
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22058.5	−1.52754	−0.763771	−	0.645487i	$-0.776654\pi$
$593$	−22058.5	−1.52754	−0.763771	+	0.645487i	$0.776654\pi$
$594$	0	0
$595$	−5414.80	−0.373084
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15616.7	1.06524	0.532620	−	0.846354i	$-0.321207\pi$
$599$	15616.7	1.06524	0.532620	+	0.846354i	$0.321207\pi$
$600$	0	0
$601$	26769.6	1.81690	0.908448	−	0.417998i	$-0.137268\pi$
$601$	26769.6	1.81690	0.908448	+	0.417998i	$0.137268\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−13701.8	−0.920757
$606$	0	0
$607$	5164.26	0.345323	0.172661	−	0.984981i	$-0.444763\pi$
$607$	5164.26	0.345323	0.172661	+	0.984981i	$0.444763\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7925.79	−0.524784
$612$	0	0
$613$	4226.03	0.278447	0.139223	−	0.990261i	$-0.455539\pi$
$613$	4226.03	0.278447	0.139223	+	0.990261i	$0.455539\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14233.5	−0.928717	−0.464358	−	0.885647i	$-0.653715\pi$
$617$	−14233.5	−0.928717	−0.464358	+	0.885647i	$0.653715\pi$
$618$	0	0
$619$	−3737.82	−0.242707	−0.121353	−	0.992609i	$-0.538723\pi$
$619$	−3737.82	−0.242707	−0.121353	+	0.992609i	$0.538723\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17847.9	−1.14777
$624$	0	0
$625$	−17340.4	−1.10978
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2878.40	−0.182463
$630$	0	0
$631$	−2893.31	−0.182537	−0.0912684	−	0.995826i	$-0.529092\pi$
$631$	−2893.31	−0.182537	−0.0912684	+	0.995826i	$0.529092\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1279.22	0.0799438
$636$	0	0
$637$	−17609.1	−1.09529
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6867.26	−0.423152	−0.211576	−	0.977362i	$-0.567860\pi$
$641$	−6867.26	−0.423152	−0.211576	+	0.977362i	$0.567860\pi$
$642$	0	0
$643$	4975.32	0.305144	0.152572	−	0.988292i	$-0.451244\pi$
$643$	4975.32	0.305144	0.152572	+	0.988292i	$0.451244\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24033.2	1.46035	0.730174	−	0.683262i	$-0.239439\pi$
$647$	24033.2	1.46035	0.730174	+	0.683262i	$0.239439\pi$
$648$	0	0
$649$	16751.9	1.01321
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25718.4	1.54125	0.770626	−	0.637288i	$-0.219944\pi$
$653$	25718.4	1.54125	0.770626	+	0.637288i	$0.219944\pi$
$654$	0	0
$655$	4427.24	0.264102
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7970.84	0.471168	0.235584	−	0.971854i	$-0.424300\pi$
$659$	7970.84	0.471168	0.235584	+	0.971854i	$0.424300\pi$
$660$	0	0
$661$	14207.8	0.836038	0.418019	−	0.908438i	$-0.362725\pi$
$661$	14207.8	0.836038	0.418019	+	0.908438i	$0.362725\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5969.77	−0.348117
$666$	0	0
$667$	−18232.2	−1.05840
$668$	0	0
$669$	0	0
$670$	0	0
$671$	36157.0	2.08021
$672$	0	0
$673$	17080.0	0.978287	0.489143	−	0.872203i	$-0.337309\pi$
$673$	17080.0	0.978287	0.489143	+	0.872203i	$0.337309\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16410.9	0.931641	0.465821	−	0.884879i	$-0.345759\pi$
$677$	16410.9	0.931641	0.465821	+	0.884879i	$0.345759\pi$
$678$	0	0
$679$	25442.9	1.43801
$680$	0	0
$681$	0	0
$682$	0	0
$683$	183.169	0.0102617	0.00513086	−	0.999987i	$-0.498367\pi$
$683$	183.169	0.0102617	0.00513086	+	0.999987i	$0.498367\pi$
$684$	0	0
$685$	−36986.9	−2.06306
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−24095.4	−1.33231
$690$	0	0
$691$	14974.0	0.824366	0.412183	−	0.911101i	$-0.364766\pi$
$691$	14974.0	0.824366	0.412183	+	0.911101i	$0.364766\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	668.715	0.0364976
$696$	0	0
$697$	261.408	0.0142059
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−4341.08	−0.233895	−0.116947	−	0.993138i	$-0.537311\pi$
$701$	−4341.08	−0.233895	−0.116947	+	0.993138i	$0.537311\pi$
$702$	0	0
$703$	−3173.41	−0.170253
$704$	0	0
$705$	0	0
$706$	0	0
$707$	50056.1	2.66273
$708$	0	0
$709$	23931.9	1.26767	0.633837	−	0.773467i	$-0.281479\pi$
$709$	23931.9	1.26767	0.633837	+	0.773467i	$0.281479\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	45712.0	2.40102
$714$	0	0
$715$	29036.1	1.51873
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16258.6	−0.843314	−0.421657	−	0.906756i	$-0.638551\pi$
$719$	−16258.6	−0.843314	−0.421657	+	0.906756i	$0.638551\pi$
$720$	0	0
$721$	−20137.5	−1.04017
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1713.65	0.0877837
$726$	0	0
$727$	−2608.95	−0.133096	−0.0665479	−	0.997783i	$-0.521199\pi$
$727$	−2608.95	−0.133096	−0.0665479	+	0.997783i	$0.521199\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7127.11	−0.360610
$732$	0	0
$733$	9843.63	0.496020	0.248010	−	0.968757i	$-0.420223\pi$
$733$	9843.63	0.496020	0.248010	+	0.968757i	$0.420223\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24816.5	1.24033
$738$	0	0
$739$	2419.46	0.120435	0.0602174	−	0.998185i	$-0.480821\pi$
$739$	2419.46	0.120435	0.0602174	+	0.998185i	$0.480821\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10442.9	−0.515631	−0.257816	−	0.966194i	$-0.583003\pi$
$743$	−10442.9	−0.515631	−0.257816	+	0.966194i	$0.583003\pi$
$744$	0	0
$745$	−21475.2	−1.05610
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18058.4	0.880963
$750$	0	0
$751$	29434.6	1.43021	0.715103	−	0.699019i	$-0.246380\pi$
$751$	29434.6	1.43021	0.715103	+	0.699019i	$0.246380\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−34335.4	−1.65509
$756$	0	0
$757$	−20638.3	−0.990902	−0.495451	−	0.868636i	$-0.664997\pi$
$757$	−20638.3	−0.990902	−0.495451	+	0.868636i	$0.664997\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4103.09	−0.195449	−0.0977246	−	0.995213i	$-0.531156\pi$
$761$	−4103.09	−0.195449	−0.0977246	+	0.995213i	$0.531156\pi$
$762$	0	0
$763$	34721.0	1.64742
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16494.5	−0.776508
$768$	0	0
$769$	−14020.0	−0.657444	−0.328722	−	0.944427i	$-0.606618\pi$
$769$	−14020.0	−0.657444	−0.328722	+	0.944427i	$0.606618\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−32823.8	−1.52728	−0.763642	−	0.645640i	$-0.776591\pi$
$773$	−32823.8	−1.52728	−0.763642	+	0.645640i	$0.776591\pi$
$774$	0	0
$775$	−4296.46	−0.199140
$776$	0	0
$777$	0	0
$778$	0	0
$779$	288.200	0.0132553
$780$	0	0
$781$	39817.0	1.82428
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16389.0	−0.745157
$786$	0	0
$787$	33903.0	1.53559	0.767796	−	0.640694i	$-0.221353\pi$
$787$	33903.0	1.53559	0.767796	+	0.640694i	$0.221353\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−18897.7	−0.849462
$792$	0	0
$793$	−35601.3	−1.59425
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−20830.2	−0.925778	−0.462889	−	0.886416i	$-0.653187\pi$
$797$	−20830.2	−0.925778	−0.462889	+	0.886416i	$0.653187\pi$
$798$	0	0
$799$	−2782.16	−0.123186
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−15515.6	−0.681859
$804$	0	0
$805$	−52460.0	−2.29686
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26505.7	1.15190	0.575952	−	0.817484i	$-0.304632\pi$
$809$	26505.7	1.15190	0.575952	+	0.817484i	$0.304632\pi$
$810$	0	0
$811$	−10829.0	−0.468876	−0.234438	−	0.972131i	$-0.575325\pi$
$811$	−10829.0	−0.468876	−0.234438	+	0.972131i	$0.575325\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	44371.5	1.90707
$816$	0	0
$817$	−7857.58	−0.336477
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42046.8	−1.78738	−0.893692	−	0.448681i	$-0.851894\pi$
$821$	−42046.8	−1.78738	−0.893692	+	0.448681i	$0.851894\pi$
$822$	0	0
$823$	−12949.0	−0.548448	−0.274224	−	0.961666i	$-0.588421\pi$
$823$	−12949.0	−0.548448	−0.274224	+	0.961666i	$0.588421\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13874.0	0.583368	0.291684	−	0.956515i	$-0.405784\pi$
$827$	13874.0	0.583368	0.291684	+	0.956515i	$0.405784\pi$
$828$	0	0
$829$	−2387.20	−0.100013	−0.0500065	−	0.998749i	$-0.515924\pi$
$829$	−2387.20	−0.100013	−0.0500065	+	0.998749i	$0.515924\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6181.27	−0.257105
$834$	0	0
$835$	36594.4	1.51665
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−47308.8	−1.94670	−0.973350	−	0.229324i	$-0.926349\pi$
$839$	−47308.8	−1.94670	−0.973350	+	0.229324i	$0.926349\pi$
$840$	0	0
$841$	−12464.7	−0.511080
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2530.37	−0.103015
$846$	0	0
$847$	−30599.2	−1.24132
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−27886.7	−1.12332
$852$	0	0
$853$	−18211.3	−0.730998	−0.365499	−	0.930812i	$-0.619102\pi$
$853$	−18211.3	−0.730998	−0.365499	+	0.930812i	$0.619102\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14824.5	0.590893	0.295447	−	0.955359i	$-0.404532\pi$
$857$	14824.5	0.590893	0.295447	+	0.955359i	$0.404532\pi$
$858$	0	0
$859$	−44191.1	−1.75527	−0.877637	−	0.479326i	$-0.840881\pi$
$859$	−44191.1	−1.75527	−0.877637	+	0.479326i	$0.840881\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34927.0	1.37767	0.688835	−	0.724918i	$-0.258122\pi$
$863$	34927.0	1.37767	0.688835	+	0.724918i	$0.258122\pi$
$864$	0	0
$865$	3475.40	0.136609
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−33193.9	−1.29577
$870$	0	0
$871$	−24435.1	−0.950575
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−34344.1	−1.32691
$876$	0	0
$877$	−16914.8	−0.651278	−0.325639	−	0.945494i	$-0.605579\pi$
$877$	−16914.8	−0.651278	−0.325639	+	0.945494i	$0.605579\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12634.8	−0.483177	−0.241588	−	0.970379i	$-0.577668\pi$
$881$	−12634.8	−0.483177	−0.241588	+	0.970379i	$0.577668\pi$
$882$	0	0
$883$	−47260.3	−1.80117	−0.900586	−	0.434678i	$-0.856862\pi$
$883$	−47260.3	−1.80117	−0.900586	+	0.434678i	$0.856862\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4758.84	0.180142	0.0900711	−	0.995935i	$-0.471291\pi$
$887$	4758.84	0.180142	0.0900711	+	0.995935i	$0.471291\pi$
$888$	0	0
$889$	2856.78	0.107777
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3067.31	−0.114943
$894$	0	0
$895$	−35540.6	−1.32736
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−29896.6	−1.10913
$900$	0	0
$901$	−8458.13	−0.312743
$902$	0	0
$903$	0	0
$904$	0	0
$905$	38732.4	1.42266
$906$	0	0
$907$	−4111.61	−0.150522	−0.0752612	−	0.997164i	$-0.523979\pi$
$907$	−4111.61	−0.150522	−0.0752612	+	0.997164i	$0.523979\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	35113.1	1.27700	0.638501	−	0.769621i	$-0.279555\pi$
$911$	35113.1	1.27700	0.638501	+	0.769621i	$0.279555\pi$
$912$	0	0
$913$	18570.4	0.673155
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9887.00	0.356049
$918$	0	0
$919$	37509.1	1.34637	0.673184	−	0.739475i	$-0.264926\pi$
$919$	37509.1	1.34637	0.673184	+	0.739475i	$0.264926\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−39205.1	−1.39811
$924$	0	0
$925$	2621.07	0.0931677
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−18393.8	−0.649604	−0.324802	−	0.945782i	$-0.605298\pi$
$929$	−18393.8	−0.649604	−0.324802	+	0.945782i	$0.605298\pi$
$930$	0	0
$931$	−6814.80	−0.239899
$932$	0	0
$933$	0	0
$934$	0	0
$935$	10192.5	0.356502
$936$	0	0
$937$	−17106.7	−0.596427	−0.298214	−	0.954499i	$-0.596391\pi$
$937$	−17106.7	−0.596427	−0.298214	+	0.954499i	$0.596391\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	38100.7	1.31992	0.659961	−	0.751300i	$-0.270573\pi$
$941$	38100.7	1.31992	0.659961	+	0.751300i	$0.270573\pi$
$942$	0	0
$943$	2532.59	0.0874576
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52712.7	1.80880	0.904400	−	0.426686i	$-0.140319\pi$
$947$	52712.7	1.80880	0.904400	+	0.426686i	$0.140319\pi$
$948$	0	0
$949$	15277.1	0.522567
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6744.38	0.229246	0.114623	−	0.993409i	$-0.463434\pi$
$953$	6744.38	0.229246	0.114623	+	0.993409i	$0.463434\pi$
$954$	0	0
$955$	−17819.0	−0.603781
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−82599.8	−2.78132
$960$	0	0
$961$	45165.9	1.51609
$962$	0	0
$963$	0	0
$964$	0	0
$965$	48670.9	1.62360
$966$	0	0
$967$	−28449.8	−0.946104	−0.473052	−	0.881034i	$-0.656848\pi$
$967$	−28449.8	−0.946104	−0.473052	+	0.881034i	$0.656848\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13018.7	0.430266	0.215133	−	0.976585i	$-0.430981\pi$
$971$	13018.7	0.430266	0.215133	+	0.976585i	$0.430981\pi$
$972$	0	0
$973$	1493.39	0.0492043
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20043.8	0.656355	0.328178	−	0.944616i	$-0.393566\pi$
$977$	20043.8	0.656355	0.328178	+	0.944616i	$0.393566\pi$
$978$	0	0
$979$	33595.7	1.09676
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1823.72	0.0591735	0.0295868	−	0.999562i	$-0.490581\pi$
$983$	1823.72	0.0591735	0.0295868	+	0.999562i	$0.490581\pi$
$984$	0	0
$985$	64300.1	2.07997
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−69049.4	−2.22006
$990$	0	0
$991$	−26635.4	−0.853786	−0.426893	−	0.904302i	$-0.640392\pi$
$991$	−26635.4	−0.853786	−0.426893	+	0.904302i	$0.640392\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−19761.7	−0.629637
$996$	0	0
$997$	50874.6	1.61606	0.808031	−	0.589139i	$-0.200533\pi$
$997$	50874.6	1.61606	0.808031	+	0.589139i	$0.200533\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.4.a.g.1.1		2
3.2	odd	2		76.4.a.a.1.1	✓	2
12.11	even	2		304.4.a.f.1.2		2
15.2	even	4		1900.4.c.b.1749.4		4
15.8	even	4		1900.4.c.b.1749.1		4
15.14	odd	2		1900.4.a.b.1.2		2
24.5	odd	2		1216.4.a.o.1.2		2
24.11	even	2		1216.4.a.h.1.1		2
57.56	even	2		1444.4.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.a.a.1.1	✓	2	3.2	odd	2
304.4.a.f.1.2		2	12.11	even	2
684.4.a.g.1.1		2	1.1	even	1	trivial
1216.4.a.h.1.1		2	24.11	even	2
1216.4.a.o.1.2		2	24.5	odd	2
1444.4.a.d.1.2		2	57.56	even	2
1900.4.a.b.1.2		2	15.14	odd	2
1900.4.c.b.1749.1		4	15.8	even	4
1900.4.c.b.1749.4		4	15.2	even	4

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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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	684.a (trivial)

\(f(q)\)	\(=\)	\(q-11.8614 q^{5} -26.4891 q^{7} +O(q^{10})\)	\(q-11.8614 q^{5} -26.4891 q^{7} +49.8614 q^{11} -49.0951 q^{13} -17.2337 q^{17} -19.0000 q^{19} -166.965 q^{23} +15.6930 q^{25} +109.198 q^{29} -273.783 q^{31} +314.198 q^{35} +167.022 q^{37} -15.1684 q^{41} +413.557 q^{43} +161.438 q^{47} +358.674 q^{49} +490.791 q^{53} -591.426 q^{55} +335.970 q^{59} +725.149 q^{61} +582.337 q^{65} +497.709 q^{67} +798.554 q^{71} -311.174 q^{73} -1320.79 q^{77} -665.723 q^{79} +372.440 q^{83} +204.416 q^{85} +673.783 q^{89} +1300.49 q^{91} +225.367 q^{95} -960.505 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{5} - 30 q^{7}+O(q^{10}) \) 2 * q + 5 * q^5 - 30 * q^7	\( 2 q + 5 q^{5} - 30 q^{7} + 71 q^{11} - 35 q^{13} - 38 q^{19} + 5 q^{23} + 175 q^{25} - 155 q^{29} - 88 q^{31} + 255 q^{35} + 380 q^{37} + 142 q^{41} + 155 q^{43} + 455 q^{47} + 28 q^{49} + 275 q^{53} - 235 q^{55} + 873 q^{59} + 445 q^{61} + 820 q^{65} + 645 q^{67} + 1712 q^{71} - 990 q^{73} - 1395 q^{77} - 1274 q^{79} + 90 q^{83} + 495 q^{85} + 888 q^{89} + 1251 q^{91} - 95 q^{95} + 710 q^{97}+O(q^{100}) \) 2 * q + 5 * q^5 - 30 * q^7 + 71 * q^11 - 35 * q^13 - 38 * q^19 + 5 * q^23 + 175 * q^25 - 155 * q^29 - 88 * q^31 + 255 * q^35 + 380 * q^37 + 142 * q^41 + 155 * q^43 + 455 * q^47 + 28 * q^49 + 275 * q^53 - 235 * q^55 + 873 * q^59 + 445 * q^61 + 820 * q^65 + 645 * q^67 + 1712 * q^71 - 990 * q^73 - 1395 * q^77 - 1274 * q^79 + 90 * q^83 + 495 * q^85 + 888 * q^89 + 1251 * q^91 - 95 * q^95 + 710 * q^97

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