LMFDB - Embedded newform 2100.4.k.q.1849.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,4,Mod(1849,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1849");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2100.k (of order $2$, degree $1$, not 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:	$123.904011012$
Analytic rank:	$0$
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} + 411x^{4} + 42405x^{2} + 36100 $ x^6 + 411x^4 + 42405x^2 + 36100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.2
Root			$0.926522i$ of defining polynomial
Character	$\chi$	$=$	2100.1849
Dual form			2100.4.k.q.1849.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} +19.9969 q^{11} -36.5561i q^{13} -10.5561i q^{17} +80.6590 q^{19} +21.0000 q^{21} +148.330i q^{23} +27.0000i q^{27} -162.777 q^{29} -46.1029 q^{31} -59.9908i q^{33} +383.533i q^{37} -109.668 q^{39} +345.421 q^{41} -138.771i q^{43} -56.6713i q^{47} -49.0000 q^{49} -31.6682 q^{51} -168.327i q^{53} -241.977i q^{57} -604.292 q^{59} +362.570 q^{61} -63.0000i q^{63} -44.8556i q^{67} +444.991 q^{69} -933.941 q^{71} -271.593i q^{73} +139.979i q^{77} +357.547 q^{79} +81.0000 q^{81} +490.564i q^{83} +488.332i q^{87} +859.974 q^{89} +255.892 q^{91} +138.309i q^{93} -182.402i q^{97} -179.972 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 54 q^{9} + 16 q^{11} - 204 q^{19} + 126 q^{21} - 246 q^{29} + 286 q^{31} - 282 q^{39} + 134 q^{41} - 294 q^{49} + 186 q^{51} - 82 q^{59} + 1234 q^{61} - 210 q^{69} + 1508 q^{71} - 860 q^{79} + 486 q^{81}+ \cdots - 144 q^{99}+O(q^{100}) $ 6 * q - 54 * q^9 + 16 * q^11 - 204 * q^19 + 126 * q^21 - 246 * q^29 + 286 * q^31 - 282 * q^39 + 134 * q^41 - 294 * q^49 + 186 * q^51 - 82 * q^59 + 1234 * q^61 - 210 * q^69 + 1508 * q^71 - 860 * q^79 + 486 * q^81 + 424 * q^89 + 658 * q^91 - 144 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	19.9969	0.548118	0.274059	−	0.961713i	$-0.411634\pi$
$11$	19.9969	0.548118	0.274059	+	0.961713i	$0.411634\pi$
$12$	0	0
$13$	− 36.5561i	− 0.779910i	−0.920834	−	0.389955i	$-0.872491\pi$
$13$	− 36.5561i	− 0.779910i	0.920834	−	0.389955i	$-0.127509\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 10.5561i	− 0.150601i	−0.997161	−	0.0753006i	$-0.976008\pi$
$17$	− 10.5561i	− 0.150601i	0.997161	−	0.0753006i	$-0.0239916\pi$
$18$	0	0
$19$	80.6590	0.973918	0.486959	−	0.873425i	$-0.338106\pi$
$19$	80.6590	0.973918	0.486959	+	0.873425i	$0.338106\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	148.330i	1.34474i	0.740216	+	0.672369i	$0.234723\pi$
$23$	148.330i	1.34474i	−0.740216	+	0.672369i	$0.765277\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−162.777	−1.04231	−0.521155	−	0.853462i	$-0.674499\pi$
$29$	−162.777	−1.04231	−0.521155	+	0.853462i	$0.674499\pi$
$30$	0	0
$31$	−46.1029	−0.267107	−0.133554	−	0.991042i	$-0.542639\pi$
$31$	−46.1029	−0.267107	−0.133554	+	0.991042i	$0.542639\pi$
$32$	0	0
$33$	− 59.9908i	− 0.316456i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	383.533i	1.70412i	0.523444	+	0.852060i	$0.324647\pi$
$37$	383.533i	1.70412i	−0.523444	+	0.852060i	$0.675353\pi$
$38$	0	0
$39$	−109.668	−0.450281
$40$	0	0
$41$	345.421	1.31575	0.657874	−	0.753128i	$-0.271456\pi$
$41$	345.421	1.31575	0.657874	+	0.753128i	$0.271456\pi$
$42$	0	0
$43$	− 138.771i	− 0.492149i	−0.969251	−	0.246074i	$-0.920859\pi$
$43$	− 138.771i	− 0.492149i	0.969251	−	0.246074i	$-0.0791407\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 56.6713i	− 0.175880i	−0.996126	−	0.0879399i	$-0.971972\pi$
$47$	− 56.6713i	− 0.175880i	0.996126	−	0.0879399i	$-0.0280283\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−31.6682	−0.0869497
$52$	0	0
$53$	− 168.327i	− 0.436255i	−0.975920	−	0.218127i	$-0.930005\pi$
$53$	− 168.327i	− 0.436255i	0.975920	−	0.218127i	$-0.0699949\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 241.977i	− 0.562292i
$58$	0	0
$59$	−604.292	−1.33343	−0.666713	−	0.745315i	$-0.732299\pi$
$59$	−604.292	−1.33343	−0.666713	+	0.745315i	$0.732299\pi$
$60$	0	0
$61$	362.570	0.761021	0.380511	−	0.924777i	$-0.375748\pi$
$61$	362.570	0.761021	0.380511	+	0.924777i	$0.375748\pi$
$62$	0	0
$63$	− 63.0000i	− 0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 44.8556i	− 0.0817908i	−0.999163	−	0.0408954i	$-0.986979\pi$
$67$	− 44.8556i	− 0.0817908i	0.999163	−	0.0408954i	$-0.0130210\pi$
$68$	0	0
$69$	444.991	0.776385
$70$	0	0
$71$	−933.941	−1.56111	−0.780553	−	0.625090i	$-0.785062\pi$
$71$	−933.941	−1.56111	−0.780553	+	0.625090i	$0.785062\pi$
$72$	0	0
$73$	− 271.593i	− 0.435446i	−0.976011	−	0.217723i	$-0.930137\pi$
$73$	− 271.593i	− 0.435446i	0.976011	−	0.217723i	$-0.0698629\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	139.979i	0.207169i
$78$	0	0
$79$	357.547	0.509204	0.254602	−	0.967046i	$-0.418055\pi$
$79$	357.547	0.509204	0.254602	+	0.967046i	$0.418055\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	490.564i	0.648751i	0.945928	+	0.324376i	$0.105154\pi$
$83$	490.564i	0.648751i	−0.945928	+	0.324376i	$0.894846\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	488.332i	0.601778i
$88$	0	0
$89$	859.974	1.02424	0.512118	−	0.858915i	$-0.328861\pi$
$89$	859.974	1.02424	0.512118	+	0.858915i	$0.328861\pi$
$90$	0	0
$91$	255.892	0.294778
$92$	0	0
$93$	138.309i	0.154215i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 182.402i	− 0.190929i	−0.995433	−	0.0954647i	$-0.969566\pi$
$97$	− 182.402i	− 0.190929i	0.995433	−	0.0954647i	$-0.0304337\pi$
$98$	0	0
$99$	−179.972	−0.182706
$100$	0	0
$101$	1205.91	1.18805	0.594024	−	0.804448i	$-0.297539\pi$
$101$	1205.91	1.18805	0.594024	+	0.804448i	$0.297539\pi$
$102$	0	0
$103$	1372.99i	1.31344i	0.754134	+	0.656720i	$0.228057\pi$
$103$	1372.99i	1.31344i	−0.754134	+	0.656720i	$0.771943\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	183.843i	0.166101i	0.996545	+	0.0830505i	$0.0264663\pi$
$107$	183.843i	0.166101i	−0.996545	+	0.0830505i	$0.973534\pi$
$108$	0	0
$109$	−510.934	−0.448978	−0.224489	−	0.974477i	$-0.572071\pi$
$109$	−510.934	−0.448978	−0.224489	+	0.974477i	$0.572071\pi$
$110$	0	0
$111$	1150.60	0.983874
$112$	0	0
$113$	633.525i	0.527408i	0.964604	+	0.263704i	$0.0849441\pi$
$113$	633.525i	0.527408i	−0.964604	+	0.263704i	$0.915056\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	329.005i	0.259970i
$118$	0	0
$119$	73.8924	0.0569219
$120$	0	0
$121$	−931.123	−0.699566
$122$	0	0
$123$	− 1036.26i	− 0.759648i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2732.95i	1.90953i	0.297363	+	0.954764i	$0.403893\pi$
$127$	2732.95i	1.90953i	−0.297363	+	0.954764i	$0.596107\pi$
$128$	0	0
$129$	−416.313	−0.284142
$130$	0	0
$131$	1494.90	0.997026	0.498513	−	0.866882i	$-0.333880\pi$
$131$	1494.90	0.997026	0.498513	+	0.866882i	$0.333880\pi$
$132$	0	0
$133$	564.613i	0.368106i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1265.40i	0.789126i	0.918869	+	0.394563i	$0.129104\pi$
$137$	1265.40i	0.789126i	−0.918869	+	0.394563i	$0.870896\pi$
$138$	0	0
$139$	−2196.54	−1.34035	−0.670173	−	0.742205i	$-0.733780\pi$
$139$	−2196.54	−1.34035	−0.670173	+	0.742205i	$0.733780\pi$
$140$	0	0
$141$	−170.014	−0.101544
$142$	0	0
$143$	− 731.009i	− 0.427483i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	645.613	0.354971	0.177486	−	0.984123i	$-0.443204\pi$
$149$	645.613	0.354971	0.177486	+	0.984123i	$0.443204\pi$
$150$	0	0
$151$	−77.9861	−0.0420293	−0.0210146	−	0.999779i	$-0.506690\pi$
$151$	−77.9861	−0.0420293	−0.0210146	+	0.999779i	$0.506690\pi$
$152$	0	0
$153$	95.0045i	0.0502004i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1387.31i	0.705218i	0.935771	+	0.352609i	$0.114706\pi$
$157$	1387.31i	0.705218i	−0.935771	+	0.352609i	$0.885294\pi$
$158$	0	0
$159$	−504.981	−0.251872
$160$	0	0
$161$	−1038.31	−0.508263
$162$	0	0
$163$	− 722.085i	− 0.346982i	−0.984835	−	0.173491i	$-0.944495\pi$
$163$	− 722.085i	− 0.346982i	0.984835	−	0.173491i	$-0.0555047\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1211.47i	0.561356i	0.959802	+	0.280678i	$0.0905592\pi$
$167$	1211.47i	0.561356i	−0.959802	+	0.280678i	$0.909441\pi$
$168$	0	0
$169$	860.654	0.391741
$170$	0	0
$171$	−725.931	−0.324639
$172$	0	0
$173$	883.783i	0.388398i	0.980962	+	0.194199i	$0.0622107\pi$
$173$	883.783i	0.388398i	−0.980962	+	0.194199i	$0.937789\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1812.88i	0.769853i
$178$	0	0
$179$	2933.72	1.22501	0.612504	−	0.790468i	$-0.290162\pi$
$179$	2933.72	1.22501	0.612504	+	0.790468i	$0.290162\pi$
$180$	0	0
$181$	1729.22	0.710122	0.355061	−	0.934843i	$-0.384460\pi$
$181$	1729.22	0.710122	0.355061	+	0.934843i	$0.384460\pi$
$182$	0	0
$183$	− 1087.71i	− 0.439376i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 211.089i	− 0.0825473i
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	2069.63	0.784048	0.392024	−	0.919955i	$-0.371775\pi$
$191$	2069.63	0.784048	0.392024	+	0.919955i	$0.371775\pi$
$192$	0	0
$193$	− 4587.33i	− 1.71090i	−0.517887	−	0.855449i	$-0.673281\pi$
$193$	− 4587.33i	− 1.71090i	0.517887	−	0.855449i	$-0.326719\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3100.30i	1.12126i	0.828068	+	0.560628i	$0.189440\pi$
$197$	3100.30i	1.12126i	−0.828068	+	0.560628i	$0.810560\pi$
$198$	0	0
$199$	4885.27	1.74024	0.870120	−	0.492840i	$-0.164041\pi$
$199$	4885.27	1.74024	0.870120	+	0.492840i	$0.164041\pi$
$200$	0	0
$201$	−134.567	−0.0472219
$202$	0	0
$203$	− 1139.44i	− 0.393956i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1334.97i	− 0.448246i
$208$	0	0
$209$	1612.93	0.533822
$210$	0	0
$211$	5327.41	1.73817	0.869085	−	0.494664i	$-0.164709\pi$
$211$	5327.41	1.73817	0.869085	+	0.494664i	$0.164709\pi$
$212$	0	0
$213$	2801.82i	0.901304i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 322.720i	− 0.100957i
$218$	0	0
$219$	−814.778	−0.251405
$220$	0	0
$221$	−385.888	−0.117455
$222$	0	0
$223$	− 2788.84i	− 0.837466i	−0.908109	−	0.418733i	$-0.862474\pi$
$223$	− 2788.84i	− 0.837466i	0.908109	−	0.418733i	$-0.137526\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4200.49i	1.22818i	0.789236	+	0.614089i	$0.210477\pi$
$227$	4200.49i	1.22818i	−0.789236	+	0.614089i	$0.789523\pi$
$228$	0	0
$229$	2719.69	0.784814	0.392407	−	0.919792i	$-0.371643\pi$
$229$	2719.69	0.784814	0.392407	+	0.919792i	$0.371643\pi$
$230$	0	0
$231$	419.936	0.119609
$232$	0	0
$233$	− 3695.17i	− 1.03897i	−0.854481	−	0.519483i	$-0.826125\pi$
$233$	− 3695.17i	− 1.03897i	0.854481	−	0.519483i	$-0.173875\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1072.64i	− 0.293989i
$238$	0	0
$239$	2166.80	0.586437	0.293219	−	0.956045i	$-0.405274\pi$
$239$	2166.80	0.586437	0.293219	+	0.956045i	$0.405274\pi$
$240$	0	0
$241$	−186.228	−0.0497761	−0.0248880	−	0.999690i	$-0.507923\pi$
$241$	−186.228	−0.0497761	−0.0248880	+	0.999690i	$0.507923\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2948.57i	− 0.759568i
$248$	0	0
$249$	1471.69	0.374557
$250$	0	0
$251$	−1211.86	−0.304748	−0.152374	−	0.988323i	$-0.548692\pi$
$251$	−1211.86	−0.304748	−0.152374	+	0.988323i	$0.548692\pi$
$252$	0	0
$253$	2966.15i	0.737076i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1716.96i	0.416735i	0.978051	+	0.208368i	$0.0668151\pi$
$257$	1716.96i	0.416735i	−0.978051	+	0.208368i	$0.933185\pi$
$258$	0	0
$259$	−2684.73	−0.644097
$260$	0	0
$261$	1465.00	0.347436
$262$	0	0
$263$	4564.91i	1.07028i	0.844762	+	0.535142i	$0.179742\pi$
$263$	4564.91i	1.07028i	−0.844762	+	0.535142i	$0.820258\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2579.92i	− 0.591343i
$268$	0	0
$269$	−2567.99	−0.582055	−0.291028	−	0.956715i	$-0.593997\pi$
$269$	−2567.99	−0.582055	−0.291028	+	0.956715i	$0.593997\pi$
$270$	0	0
$271$	2870.51	0.643435	0.321718	−	0.946836i	$-0.395740\pi$
$271$	2870.51	0.643435	0.321718	+	0.946836i	$0.395740\pi$
$272$	0	0
$273$	− 767.677i	− 0.170190i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6162.16i	1.33664i	0.743875	+	0.668319i	$0.232986\pi$
$277$	6162.16i	1.33664i	−0.743875	+	0.668319i	$0.767014\pi$
$278$	0	0
$279$	414.926	0.0890358
$280$	0	0
$281$	−2231.84	−0.473810	−0.236905	−	0.971533i	$-0.576133\pi$
$281$	−2231.84	−0.473810	−0.236905	+	0.971533i	$0.576133\pi$
$282$	0	0
$283$	− 856.119i	− 0.179827i	−0.995950	−	0.0899134i	$-0.971341\pi$
$283$	− 856.119i	− 0.179827i	0.995950	−	0.0899134i	$-0.0286590\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2417.95i	0.497306i
$288$	0	0
$289$	4801.57	0.977319
$290$	0	0
$291$	−547.207	−0.110233
$292$	0	0
$293$	− 5027.35i	− 1.00239i	−0.865334	−	0.501196i	$-0.832894\pi$
$293$	− 5027.35i	− 1.00239i	0.865334	−	0.501196i	$-0.167106\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	539.917i	0.105485i
$298$	0	0
$299$	5422.37	1.04877
$300$	0	0
$301$	971.398	0.186015
$302$	0	0
$303$	− 3617.74i	− 0.685919i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8468.94i	1.57442i	0.616683	+	0.787212i	$0.288476\pi$
$307$	8468.94i	1.57442i	−0.616683	+	0.787212i	$0.711524\pi$
$308$	0	0
$309$	4118.96	0.758315
$310$	0	0
$311$	3578.29	0.652432	0.326216	−	0.945295i	$-0.394226\pi$
$311$	3578.29	0.652432	0.326216	+	0.945295i	$0.394226\pi$
$312$	0	0
$313$	− 9131.00i	− 1.64893i	−0.565914	−	0.824464i	$-0.691476\pi$
$313$	− 9131.00i	− 1.64893i	0.565914	−	0.824464i	$-0.308524\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2093.09i	0.370850i	0.982658	+	0.185425i	$0.0593662\pi$
$317$	2093.09i	0.370850i	−0.982658	+	0.185425i	$0.940634\pi$
$318$	0	0
$319$	−3255.05	−0.571309
$320$	0	0
$321$	551.530	0.0958984
$322$	0	0
$323$	− 851.441i	− 0.146673i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1532.80i	0.259217i
$328$	0	0
$329$	396.699	0.0664763
$330$	0	0
$331$	7327.60	1.21680	0.608401	−	0.793630i	$-0.291811\pi$
$331$	7327.60	1.21680	0.608401	+	0.793630i	$0.291811\pi$
$332$	0	0
$333$	− 3451.80i	− 0.568040i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3093.40i	− 0.500024i	−0.968243	−	0.250012i	$-0.919565\pi$
$337$	− 3093.40i	− 0.500024i	0.968243	−	0.250012i	$-0.0804345\pi$
$338$	0	0
$339$	1900.58	0.304499
$340$	0	0
$341$	−921.917	−0.146406
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1643.49i	0.254257i	0.991886	+	0.127129i	$0.0405761\pi$
$347$	1643.49i	0.254257i	−0.991886	+	0.127129i	$0.959424\pi$
$348$	0	0
$349$	4989.76	0.765317	0.382659	−	0.923890i	$-0.375009\pi$
$349$	4989.76	0.765317	0.382659	+	0.923890i	$0.375009\pi$
$350$	0	0
$351$	987.014	0.150094
$352$	0	0
$353$	8085.60i	1.21913i	0.792736	+	0.609565i	$0.208656\pi$
$353$	8085.60i	1.21913i	−0.792736	+	0.609565i	$0.791344\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 221.677i	− 0.0328639i
$358$	0	0
$359$	5666.37	0.833035	0.416518	−	0.909128i	$-0.363250\pi$
$359$	5666.37	0.833035	0.416518	+	0.909128i	$0.363250\pi$
$360$	0	0
$361$	−353.130	−0.0514841
$362$	0	0
$363$	2793.37i	0.403895i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 8341.18i	− 1.18639i	−0.805058	−	0.593196i	$-0.797866\pi$
$367$	− 8341.18i	− 1.18639i	0.805058	−	0.593196i	$-0.202134\pi$
$368$	0	0
$369$	−3108.79	−0.438583
$370$	0	0
$371$	1178.29	0.164889
$372$	0	0
$373$	692.645i	0.0961495i	0.998844	+	0.0480748i	$0.0153086\pi$
$373$	692.645i	0.0961495i	−0.998844	+	0.0480748i	$0.984691\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5950.49i	0.812907i
$378$	0	0
$379$	3371.55	0.456952	0.228476	−	0.973550i	$-0.426626\pi$
$379$	3371.55	0.456952	0.228476	+	0.973550i	$0.426626\pi$
$380$	0	0
$381$	8198.85	1.10247
$382$	0	0
$383$	− 2578.69i	− 0.344034i	−0.985094	−	0.172017i	$-0.944972\pi$
$383$	− 2578.69i	− 0.344034i	0.985094	−	0.172017i	$-0.0550283\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1248.94i	0.164050i
$388$	0	0
$389$	8996.49	1.17260	0.586298	−	0.810095i	$-0.300585\pi$
$389$	8996.49	1.17260	0.586298	+	0.810095i	$0.300585\pi$
$390$	0	0
$391$	1565.78	0.202519
$392$	0	0
$393$	− 4484.71i	− 0.575633i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9233.08i	1.16724i	0.812026	+	0.583621i	$0.198365\pi$
$397$	9233.08i	1.16724i	−0.812026	+	0.583621i	$0.801635\pi$
$398$	0	0
$399$	1693.84	0.212526
$400$	0	0
$401$	−5543.70	−0.690372	−0.345186	−	0.938534i	$-0.612184\pi$
$401$	−5543.70	−0.690372	−0.345186	+	0.938534i	$0.612184\pi$
$402$	0	0
$403$	1685.34i	0.208320i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7669.48i	0.934059i
$408$	0	0
$409$	3948.78	0.477395	0.238697	−	0.971094i	$-0.423280\pi$
$409$	3948.78	0.477395	0.238697	+	0.971094i	$0.423280\pi$
$410$	0	0
$411$	3796.19	0.455602
$412$	0	0
$413$	− 4230.04i	− 0.503987i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6589.62i	0.773849i
$418$	0	0
$419$	7348.44	0.856790	0.428395	−	0.903592i	$-0.359079\pi$
$419$	7348.44	0.856790	0.428395	+	0.903592i	$0.359079\pi$
$420$	0	0
$421$	−6555.05	−0.758844	−0.379422	−	0.925224i	$-0.623877\pi$
$421$	−6555.05	−0.758844	−0.379422	+	0.925224i	$0.623877\pi$
$422$	0	0
$423$	510.041i	0.0586266i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2537.99i	0.287639i
$428$	0	0
$429$	−2193.03	−0.246807
$430$	0	0
$431$	1239.26	0.138498	0.0692492	−	0.997599i	$-0.477940\pi$
$431$	1239.26	0.138498	0.0692492	+	0.997599i	$0.477940\pi$
$432$	0	0
$433$	2158.52i	0.239565i	0.992800	+	0.119783i	$0.0382198\pi$
$433$	2158.52i	0.239565i	−0.992800	+	0.119783i	$0.961780\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11964.2i	1.30966i
$438$	0	0
$439$	−684.773	−0.0744475	−0.0372237	−	0.999307i	$-0.511851\pi$
$439$	−684.773	−0.0744475	−0.0372237	+	0.999307i	$0.511851\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	7350.87i	0.788376i	0.919030	+	0.394188i	$0.128974\pi$
$443$	7350.87i	0.788376i	−0.919030	+	0.394188i	$0.871026\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1936.84i	− 0.204943i
$448$	0	0
$449$	−12893.8	−1.35522	−0.677612	−	0.735419i	$-0.736985\pi$
$449$	−12893.8	−1.35522	−0.677612	+	0.735419i	$0.736985\pi$
$450$	0	0
$451$	6907.36	0.721186
$452$	0	0
$453$	233.958i	0.0242656i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4235.78i	0.433569i	0.976219	+	0.216785i	$0.0695570\pi$
$457$	4235.78i	0.433569i	−0.976219	+	0.216785i	$0.930443\pi$
$458$	0	0
$459$	285.014	0.0289832
$460$	0	0
$461$	−15353.1	−1.55112	−0.775561	−	0.631273i	$-0.782533\pi$
$461$	−15353.1	−1.55112	−0.775561	+	0.631273i	$0.782533\pi$
$462$	0	0
$463$	− 13021.3i	− 1.30703i	−0.756916	−	0.653513i	$-0.773295\pi$
$463$	− 13021.3i	− 1.30703i	0.756916	−	0.653513i	$-0.226705\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2072.38i	− 0.205350i	−0.994715	−	0.102675i	$-0.967260\pi$
$467$	− 2072.38i	− 0.205350i	0.994715	−	0.102675i	$-0.0327401\pi$
$468$	0	0
$469$	313.989	0.0309140
$470$	0	0
$471$	4161.93	0.407158
$472$	0	0
$473$	− 2775.00i	− 0.269756i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1514.94i	0.145418i
$478$	0	0
$479$	10459.0	0.997675	0.498837	−	0.866696i	$-0.333761\pi$
$479$	10459.0	0.997675	0.498837	+	0.866696i	$0.333761\pi$
$480$	0	0
$481$	14020.5	1.32906
$482$	0	0
$483$	3114.93i	0.293446i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 7065.60i	− 0.657439i	−0.944428	−	0.328720i	$-0.893383\pi$
$487$	− 7065.60i	− 0.657439i	0.944428	−	0.328720i	$-0.106617\pi$
$488$	0	0
$489$	−2166.25	−0.200330
$490$	0	0
$491$	1904.09	0.175011	0.0875057	−	0.996164i	$-0.472110\pi$
$491$	1904.09	0.175011	0.0875057	+	0.996164i	$0.472110\pi$
$492$	0	0
$493$	1718.29i	0.156973i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6537.59i	− 0.590042i
$498$	0	0
$499$	9795.02	0.878728	0.439364	−	0.898309i	$-0.355204\pi$
$499$	9795.02	0.878728	0.439364	+	0.898309i	$0.355204\pi$
$500$	0	0
$501$	3634.41	0.324099
$502$	0	0
$503$	− 8554.22i	− 0.758278i	−0.925340	−	0.379139i	$-0.876220\pi$
$503$	− 8554.22i	− 0.758278i	0.925340	−	0.379139i	$-0.123780\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2581.96i	− 0.226172i
$508$	0	0
$509$	−10200.4	−0.888257	−0.444129	−	0.895963i	$-0.646487\pi$
$509$	−10200.4	−0.888257	−0.444129	+	0.895963i	$0.646487\pi$
$510$	0	0
$511$	1901.15	0.164583
$512$	0	0
$513$	2177.79i	0.187431i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1133.25i	− 0.0964030i
$518$	0	0
$519$	2651.35	0.224241
$520$	0	0
$521$	−13236.4	−1.11304	−0.556522	−	0.830833i	$-0.687865\pi$
$521$	−13236.4	−1.11304	−0.556522	+	0.830833i	$0.687865\pi$
$522$	0	0
$523$	6256.94i	0.523130i	0.965186	+	0.261565i	$0.0842385\pi$
$523$	6256.94i	0.523130i	−0.965186	+	0.261565i	$0.915761\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	486.665i	0.0402267i
$528$	0	0
$529$	−9834.86	−0.808322
$530$	0	0
$531$	5438.63	0.444475
$532$	0	0
$533$	− 12627.2i	− 1.02616i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 8801.15i	− 0.707258i
$538$	0	0
$539$	−979.850	−0.0783026
$540$	0	0
$541$	−5225.05	−0.415236	−0.207618	−	0.978210i	$-0.566571\pi$
$541$	−5225.05	−0.415236	−0.207618	+	0.978210i	$0.566571\pi$
$542$	0	0
$543$	− 5187.66i	− 0.409989i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13863.0i	1.08362i	0.840501	+	0.541810i	$0.182261\pi$
$547$	13863.0i	1.08362i	−0.840501	+	0.541810i	$0.817739\pi$
$548$	0	0
$549$	−3263.13	−0.253674
$550$	0	0
$551$	−13129.4	−1.01512
$552$	0	0
$553$	2502.83i	0.192461i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14624.8i	1.11252i	0.831008	+	0.556260i	$0.187764\pi$
$557$	14624.8i	1.11252i	−0.831008	+	0.556260i	$0.812236\pi$
$558$	0	0
$559$	−5072.92	−0.383832
$560$	0	0
$561$	−633.266	−0.0476587
$562$	0	0
$563$	13971.3i	1.04586i	0.852375	+	0.522931i	$0.175162\pi$
$563$	13971.3i	1.04586i	−0.852375	+	0.522931i	$0.824838\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	567.000i	0.0419961i
$568$	0	0
$569$	18720.5	1.37927	0.689633	−	0.724159i	$-0.257772\pi$
$569$	18720.5	1.37927	0.689633	+	0.724159i	$0.257772\pi$
$570$	0	0
$571$	6065.05	0.444509	0.222254	−	0.974989i	$-0.428658\pi$
$571$	6065.05	0.444509	0.222254	+	0.974989i	$0.428658\pi$
$572$	0	0
$573$	− 6208.89i	− 0.452670i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 5144.33i	− 0.371163i	−0.982629	−	0.185582i	$-0.940583\pi$
$577$	− 5144.33i	− 0.371163i	0.982629	−	0.185582i	$-0.0594169\pi$
$578$	0	0
$579$	−13762.0	−0.987787
$580$	0	0
$581$	−3433.95	−0.245205
$582$	0	0
$583$	− 3366.03i	− 0.239119i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12292.2i	− 0.864313i	−0.901799	−	0.432156i	$-0.857753\pi$
$587$	− 12292.2i	− 0.864313i	0.901799	−	0.432156i	$-0.142247\pi$
$588$	0	0
$589$	−3718.61	−0.260141
$590$	0	0
$591$	9300.91	0.647358
$592$	0	0
$593$	28477.2i	1.97204i	0.166633	+	0.986019i	$0.446711\pi$
$593$	28477.2i	1.97204i	−0.166633	+	0.986019i	$0.553289\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 14655.8i	− 1.00473i
$598$	0	0
$599$	−21061.1	−1.43662	−0.718308	−	0.695725i	$-0.755083\pi$
$599$	−21061.1	−1.43662	−0.718308	+	0.695725i	$0.755083\pi$
$600$	0	0
$601$	3865.61	0.262365	0.131183	−	0.991358i	$-0.458123\pi$
$601$	3865.61	0.262365	0.131183	+	0.991358i	$0.458123\pi$
$602$	0	0
$603$	403.700i	0.0272636i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15750.3i	1.05319i	0.850117	+	0.526594i	$0.176531\pi$
$607$	15750.3i	1.05319i	−0.850117	+	0.526594i	$0.823469\pi$
$608$	0	0
$609$	−3418.32	−0.227451
$610$	0	0
$611$	−2071.68	−0.137170
$612$	0	0
$613$	2431.42i	0.160202i	0.996787	+	0.0801012i	$0.0255244\pi$
$613$	2431.42i	0.160202i	−0.996787	+	0.0801012i	$0.974476\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12650.1i	0.825402i	0.910867	+	0.412701i	$0.135415\pi$
$617$	12650.1i	0.825402i	−0.910867	+	0.412701i	$0.864585\pi$
$618$	0	0
$619$	−8081.54	−0.524757	−0.262379	−	0.964965i	$-0.584507\pi$
$619$	−8081.54	−0.524757	−0.262379	+	0.964965i	$0.584507\pi$
$620$	0	0
$621$	−4004.92	−0.258795
$622$	0	0
$623$	6019.82i	0.387125i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4838.80i	− 0.308202i
$628$	0	0
$629$	4048.60	0.256643
$630$	0	0
$631$	−6549.12	−0.413180	−0.206590	−	0.978428i	$-0.566237\pi$
$631$	−6549.12	−0.413180	−0.206590	+	0.978428i	$0.566237\pi$
$632$	0	0
$633$	− 15982.2i	− 1.00353i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1791.25i	0.111416i
$638$	0	0
$639$	8405.47	0.520368
$640$	0	0
$641$	12606.6	0.776804	0.388402	−	0.921490i	$-0.373027\pi$
$641$	12606.6	0.776804	0.388402	+	0.921490i	$0.373027\pi$
$642$	0	0
$643$	1350.32i	0.0828169i	0.999142	+	0.0414085i	$0.0131845\pi$
$643$	1350.32i	0.0828169i	−0.999142	+	0.0414085i	$0.986816\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8112.68i	0.492956i	0.969148	+	0.246478i	$0.0792732\pi$
$647$	8112.68i	0.492956i	−0.969148	+	0.246478i	$0.920727\pi$
$648$	0	0
$649$	−12084.0	−0.730875
$650$	0	0
$651$	−968.161	−0.0582876
$652$	0	0
$653$	22135.9i	1.32656i	0.748369	+	0.663282i	$0.230837\pi$
$653$	22135.9i	1.32656i	−0.748369	+	0.663282i	$0.769163\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2444.33i	0.145149i
$658$	0	0
$659$	−5810.81	−0.343486	−0.171743	−	0.985142i	$-0.554940\pi$
$659$	−5810.81	−0.343486	−0.171743	+	0.985142i	$0.554940\pi$
$660$	0	0
$661$	28036.4	1.64976	0.824878	−	0.565311i	$-0.191244\pi$
$661$	28036.4	1.64976	0.824878	+	0.565311i	$0.191244\pi$
$662$	0	0
$663$	1157.66i	0.0678129i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 24144.8i	− 1.40163i
$668$	0	0
$669$	−8366.53	−0.483511
$670$	0	0
$671$	7250.28	0.417130
$672$	0	0
$673$	− 26742.4i	− 1.53171i	−0.643012	−	0.765856i	$-0.722316\pi$
$673$	− 26742.4i	− 1.53171i	0.643012	−	0.765856i	$-0.277684\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6808.36i	0.386509i	0.981149	+	0.193254i	$0.0619043\pi$
$677$	6808.36i	0.386509i	−0.981149	+	0.193254i	$0.938096\pi$
$678$	0	0
$679$	1276.82	0.0721645
$680$	0	0
$681$	12601.5	0.709089
$682$	0	0
$683$	11679.6i	0.654333i	0.944967	+	0.327166i	$0.106094\pi$
$683$	11679.6i	0.654333i	−0.944967	+	0.327166i	$0.893906\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 8159.08i	− 0.453112i
$688$	0	0
$689$	−6153.38	−0.340240
$690$	0	0
$691$	4375.33	0.240876	0.120438	−	0.992721i	$-0.461570\pi$
$691$	4375.33	0.240876	0.120438	+	0.992721i	$0.461570\pi$
$692$	0	0
$693$	− 1259.81i	− 0.0690564i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 3646.28i	− 0.198153i
$698$	0	0
$699$	−11085.5	−0.599847
$700$	0	0
$701$	−19397.6	−1.04513	−0.522566	−	0.852599i	$-0.675025\pi$
$701$	−19397.6	−1.04513	−0.522566	+	0.852599i	$0.675025\pi$
$702$	0	0
$703$	30935.4i	1.65967i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8441.39i	0.449040i
$708$	0	0
$709$	−24658.1	−1.30614	−0.653072	−	0.757296i	$-0.726520\pi$
$709$	−24658.1	−1.30614	−0.653072	+	0.757296i	$0.726520\pi$
$710$	0	0
$711$	−3217.92	−0.169735
$712$	0	0
$713$	− 6838.46i	− 0.359190i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6500.40i	− 0.338580i
$718$	0	0
$719$	24003.3	1.24502	0.622512	−	0.782610i	$-0.286112\pi$
$719$	24003.3	1.24502	0.622512	+	0.782610i	$0.286112\pi$
$720$	0	0
$721$	−9610.90	−0.496434
$722$	0	0
$723$	558.685i	0.0287382i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 31621.4i	− 1.61317i	−0.591121	−	0.806583i	$-0.701314\pi$
$727$	− 31621.4i	− 1.61317i	0.591121	−	0.806583i	$-0.298686\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−1464.88	−0.0741182
$732$	0	0
$733$	− 6032.39i	− 0.303972i	−0.988383	−	0.151986i	$-0.951433\pi$
$733$	− 6032.39i	− 0.303972i	0.988383	−	0.151986i	$-0.0485668\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 896.974i	− 0.0448310i
$738$	0	0
$739$	−6739.75	−0.335488	−0.167744	−	0.985831i	$-0.553648\pi$
$739$	−6739.75	−0.335488	−0.167744	+	0.985831i	$0.553648\pi$
$740$	0	0
$741$	−8845.72	−0.438537
$742$	0	0
$743$	27295.0i	1.34772i	0.738860	+	0.673859i	$0.235365\pi$
$743$	27295.0i	1.34772i	−0.738860	+	0.673859i	$0.764635\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4415.07i	− 0.216250i
$748$	0	0
$749$	−1286.90	−0.0627803
$750$	0	0
$751$	1444.72	0.0701980	0.0350990	−	0.999384i	$-0.488825\pi$
$751$	1444.72	0.0701980	0.0350990	+	0.999384i	$0.488825\pi$
$752$	0	0
$753$	3635.58i	0.175947i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 35256.0i	− 1.69273i	−0.532600	−	0.846367i	$-0.678785\pi$
$757$	− 35256.0i	− 1.69273i	0.532600	−	0.846367i	$-0.321215\pi$
$758$	0	0
$759$	8898.45	0.425551
$760$	0	0
$761$	−25120.4	−1.19660	−0.598300	−	0.801272i	$-0.704157\pi$
$761$	−25120.4	−1.19660	−0.598300	+	0.801272i	$0.704157\pi$
$762$	0	0
$763$	− 3576.54i	− 0.169698i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22090.5i	1.03995i
$768$	0	0
$769$	10086.4	0.472984	0.236492	−	0.971633i	$-0.424002\pi$
$769$	10086.4	0.472984	0.236492	+	0.971633i	$0.424002\pi$
$770$	0	0
$771$	5150.88	0.240602
$772$	0	0
$773$	15884.5i	0.739102i	0.929210	+	0.369551i	$0.120489\pi$
$773$	15884.5i	0.739102i	−0.929210	+	0.369551i	$0.879511\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8054.19i	0.371869i
$778$	0	0
$779$	27861.3	1.28143
$780$	0	0
$781$	−18676.0	−0.855670
$782$	0	0
$783$	− 4394.99i	− 0.200593i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 12518.9i	− 0.567026i	−0.958968	−	0.283513i	$-0.908500\pi$
$787$	− 12518.9i	− 0.567026i	0.958968	−	0.283513i	$-0.0914999\pi$
$788$	0	0
$789$	13694.7	0.617929
$790$	0	0
$791$	−4434.68	−0.199341
$792$	0	0
$793$	− 13254.1i	− 0.593528i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20432.7i	0.908109i	0.890974	+	0.454054i	$0.150023\pi$
$797$	20432.7i	0.908109i	−0.890974	+	0.454054i	$0.849977\pi$
$798$	0	0
$799$	−598.225	−0.0264877
$800$	0	0
$801$	−7739.76	−0.341412
$802$	0	0
$803$	− 5431.02i	− 0.238676i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7703.96i	0.336050i
$808$	0	0
$809$	−21835.3	−0.948933	−0.474467	−	0.880273i	$-0.657359\pi$
$809$	−21835.3	−0.948933	−0.474467	+	0.880273i	$0.657359\pi$
$810$	0	0
$811$	3246.73	0.140577	0.0702885	−	0.997527i	$-0.477608\pi$
$811$	3246.73	0.140577	0.0702885	+	0.997527i	$0.477608\pi$
$812$	0	0
$813$	− 8611.53i	− 0.371488i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 11193.1i	− 0.479312i
$818$	0	0
$819$	−2303.03	−0.0982594
$820$	0	0
$821$	−30160.9	−1.28212	−0.641061	−	0.767490i	$-0.721506\pi$
$821$	−30160.9	−1.28212	−0.641061	+	0.767490i	$0.721506\pi$
$822$	0	0
$823$	− 4496.24i	− 0.190436i	−0.995456	−	0.0952181i	$-0.969645\pi$
$823$	− 4496.24i	− 0.190436i	0.995456	−	0.0952181i	$-0.0303548\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 31187.2i	− 1.31135i	−0.755044	−	0.655674i	$-0.772384\pi$
$827$	− 31187.2i	− 1.31135i	0.755044	−	0.655674i	$-0.227616\pi$
$828$	0	0
$829$	−19563.4	−0.819622	−0.409811	−	0.912171i	$-0.634405\pi$
$829$	−19563.4	−0.819622	−0.409811	+	0.912171i	$0.634405\pi$
$830$	0	0
$831$	18486.5	0.771708
$832$	0	0
$833$	517.247i	0.0215145i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1244.78i	− 0.0514048i
$838$	0	0
$839$	1162.00	0.0478151	0.0239075	−	0.999714i	$-0.492389\pi$
$839$	1162.00	0.0478151	0.0239075	+	0.999714i	$0.492389\pi$
$840$	0	0
$841$	2107.43	0.0864090
$842$	0	0
$843$	6695.53i	0.273554i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 6517.86i	− 0.264411i
$848$	0	0
$849$	−2568.36	−0.103823
$850$	0	0
$851$	−56889.5	−2.29160
$852$	0	0
$853$	− 37961.5i	− 1.52377i	−0.647712	−	0.761885i	$-0.724274\pi$
$853$	− 37961.5i	− 1.52377i	0.647712	−	0.761885i	$-0.275726\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 26046.8i	− 1.03820i	−0.854712	−	0.519102i	$-0.826266\pi$
$857$	− 26046.8i	− 1.03820i	0.854712	−	0.519102i	$-0.173734\pi$
$858$	0	0
$859$	−8914.15	−0.354071	−0.177035	−	0.984204i	$-0.556651\pi$
$859$	−8914.15	−0.354071	−0.177035	+	0.984204i	$0.556651\pi$
$860$	0	0
$861$	7253.84	0.287120
$862$	0	0
$863$	− 20101.9i	− 0.792904i	−0.918055	−	0.396452i	$-0.870241\pi$
$863$	− 20101.9i	− 0.792904i	0.918055	−	0.396452i	$-0.129759\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 14404.7i	− 0.564256i
$868$	0	0
$869$	7149.84	0.279104
$870$	0	0
$871$	−1639.74	−0.0637894
$872$	0	0
$873$	1641.62i	0.0636431i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7566.36i	− 0.291332i	−0.989334	−	0.145666i	$-0.953468\pi$
$877$	− 7566.36i	− 0.291332i	0.989334	−	0.145666i	$-0.0465324\pi$
$878$	0	0
$879$	−15082.0	−0.578731
$880$	0	0
$881$	−39005.3	−1.49163	−0.745813	−	0.666155i	$-0.767939\pi$
$881$	−39005.3	−1.49163	−0.745813	+	0.666155i	$0.767939\pi$
$882$	0	0
$883$	3484.16i	0.132787i	0.997794	+	0.0663937i	$0.0211493\pi$
$883$	3484.16i	0.132787i	−0.997794	+	0.0663937i	$0.978851\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7471.48i	0.282827i	0.989951	+	0.141414i	$0.0451648\pi$
$887$	7471.48i	0.282827i	−0.989951	+	0.141414i	$0.954835\pi$
$888$	0	0
$889$	−19130.7	−0.721734
$890$	0	0
$891$	1619.75	0.0609020
$892$	0	0
$893$	− 4571.05i	− 0.171292i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 16267.1i	− 0.605510i
$898$	0	0
$899$	7504.50	0.278409
$900$	0	0
$901$	−1776.87	−0.0657005
$902$	0	0
$903$	− 2914.19i	− 0.107396i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 37334.6i	− 1.36679i	−0.730050	−	0.683394i	$-0.760503\pi$
$907$	− 37334.6i	− 1.36679i	0.730050	−	0.683394i	$-0.239497\pi$
$908$	0	0
$909$	−10853.2	−0.396016
$910$	0	0
$911$	−18545.6	−0.674473	−0.337236	−	0.941420i	$-0.609492\pi$
$911$	−18545.6	−0.674473	−0.337236	+	0.941420i	$0.609492\pi$
$912$	0	0
$913$	9809.77i	0.355592i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10464.3i	0.376840i
$918$	0	0
$919$	14692.9	0.527392	0.263696	−	0.964606i	$-0.415058\pi$
$919$	14692.9	0.527392	0.263696	+	0.964606i	$0.415058\pi$
$920$	0	0
$921$	25406.8	0.908994
$922$	0	0
$923$	34141.2i	1.21752i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 12356.9i	− 0.437813i
$928$	0	0
$929$	−37775.8	−1.33410	−0.667052	−	0.745011i	$-0.732444\pi$
$929$	−37775.8	−1.33410	−0.667052	+	0.745011i	$0.732444\pi$
$930$	0	0
$931$	−3952.29	−0.139131
$932$	0	0
$933$	− 10734.9i	− 0.376682i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 7701.47i	− 0.268512i	−0.990947	−	0.134256i	$-0.957135\pi$
$937$	− 7701.47i	− 0.268512i	0.990947	−	0.134256i	$-0.0428645\pi$
$938$	0	0
$939$	−27393.0	−0.952010
$940$	0	0
$941$	3370.73	0.116772	0.0583861	−	0.998294i	$-0.481405\pi$
$941$	3370.73	0.116772	0.0583861	+	0.998294i	$0.481405\pi$
$942$	0	0
$943$	51236.4i	1.76934i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 37917.4i	− 1.30111i	−0.759459	−	0.650555i	$-0.774536\pi$
$947$	− 37917.4i	− 1.30111i	0.759459	−	0.650555i	$-0.225464\pi$
$948$	0	0
$949$	−9928.36	−0.339608
$950$	0	0
$951$	6279.26	0.214111
$952$	0	0
$953$	39134.6i	1.33021i	0.746748	+	0.665107i	$0.231614\pi$
$953$	39134.6i	1.33021i	−0.746748	+	0.665107i	$0.768386\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9765.14i	0.329845i
$958$	0	0
$959$	−8857.78	−0.298261
$960$	0	0
$961$	−27665.5	−0.928654
$962$	0	0
$963$	− 1654.59i	− 0.0553670i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 3073.70i	− 0.102217i	−0.998693	−	0.0511084i	$-0.983725\pi$
$967$	− 3073.70i	− 0.102217i	0.998693	−	0.0511084i	$-0.0162754\pi$
$968$	0	0
$969$	−2554.32	−0.0846818
$970$	0	0
$971$	4162.99	0.137587	0.0687934	−	0.997631i	$-0.478085\pi$
$971$	4162.99	0.137587	0.0687934	+	0.997631i	$0.478085\pi$
$972$	0	0
$973$	− 15375.8i	− 0.506603i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 20800.0i	− 0.681116i	−0.940224	−	0.340558i	$-0.889384\pi$
$977$	− 20800.0i	− 0.681116i	0.940224	−	0.340558i	$-0.110616\pi$
$978$	0	0
$979$	17196.8	0.561403
$980$	0	0
$981$	4598.40	0.149659
$982$	0	0
$983$	60972.1i	1.97834i	0.146777	+	0.989170i	$0.453110\pi$
$983$	60972.1i	1.97834i	−0.146777	+	0.989170i	$0.546890\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1190.10i	− 0.0383801i
$988$	0	0
$989$	20583.9	0.661811
$990$	0	0
$991$	37755.6	1.21024	0.605119	−	0.796135i	$-0.293126\pi$
$991$	37755.6	1.21024	0.605119	+	0.796135i	$0.293126\pi$
$992$	0	0
$993$	− 21982.8i	− 0.702520i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	32529.9i	1.03333i	0.856187	+	0.516666i	$0.172827\pi$
$997$	32529.9i	1.03333i	−0.856187	+	0.516666i	$0.827173\pi$
$998$	0	0
$999$	−10355.4	−0.327958

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.4.k.q.1849.2		6
5.2	odd	4		2100.4.a.w.1.2	✓	3
5.3	odd	4		2100.4.a.ba.1.2	yes	3
5.4	even	2	inner	2100.4.k.q.1849.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.4.a.w.1.2	✓	3	5.2	odd	4
2100.4.a.ba.1.2	yes	3	5.3	odd	4
2100.4.k.q.1849.2		6	1.1	even	1	trivial
2100.4.k.q.1849.5		6	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2100.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} +19.9969 q^{11} -36.5561i q^{13} -10.5561i q^{17} +80.6590 q^{19} +21.0000 q^{21} +148.330i q^{23} +27.0000i q^{27} -162.777 q^{29} -46.1029 q^{31} -59.9908i q^{33} +383.533i q^{37} -109.668 q^{39} +345.421 q^{41} -138.771i q^{43} -56.6713i q^{47} -49.0000 q^{49} -31.6682 q^{51} -168.327i q^{53} -241.977i q^{57} -604.292 q^{59} +362.570 q^{61} -63.0000i q^{63} -44.8556i q^{67} +444.991 q^{69} -933.941 q^{71} -271.593i q^{73} +139.979i q^{77} +357.547 q^{79} +81.0000 q^{81} +490.564i q^{83} +488.332i q^{87} +859.974 q^{89} +255.892 q^{91} +138.309i q^{93} -182.402i q^{97} -179.972 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 54 q^{9} + 16 q^{11} - 204 q^{19} + 126 q^{21} - 246 q^{29} + 286 q^{31} - 282 q^{39} + 134 q^{41} - 294 q^{49} + 186 q^{51} - 82 q^{59} + 1234 q^{61} - 210 q^{69} + 1508 q^{71} - 860 q^{79} + 486 q^{81}+ \cdots - 144 q^{99}+O(q^{100}) \) 6 * q - 54 * q^9 + 16 * q^11 - 204 * q^19 + 126 * q^21 - 246 * q^29 + 286 * q^31 - 282 * q^39 + 134 * q^41 - 294 * q^49 + 186 * q^51 - 82 * q^59 + 1234 * q^61 - 210 * q^69 + 1508 * q^71 - 860 * q^79 + 486 * q^81 + 424 * q^89 + 658 * q^91 - 144 * q^99

Introduction

L-functions

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Database

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