LMFDB - Embedded newform 1260.4.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,4,Mod(1,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1260.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1260.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:	$74.3424066072$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 420)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.72015$ of defining polynomial
Character	$\chi$	$=$	1260.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} +7.00000 q^{7} +56.6418 q^{11} +46.6418 q^{13} +66.0000 q^{17} +52.6418 q^{19} -12.0000 q^{23} +25.0000 q^{25} -54.0000 q^{29} -65.9255 q^{31} +35.0000 q^{35} +140.716 q^{37} -270.000 q^{41} +146.716 q^{43} -291.851 q^{47} +49.0000 q^{49} -374.642 q^{53} +283.209 q^{55} -77.2837 q^{59} +758.000 q^{61} +233.209 q^{65} +356.000 q^{67} -820.344 q^{71} -401.209 q^{73} +396.493 q^{77} +707.851 q^{79} +900.986 q^{83} +330.000 q^{85} +636.269 q^{89} +326.493 q^{91} +263.209 q^{95} +120.074 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{5} + 14 q^{7} - 12 q^{11} - 32 q^{13} + 132 q^{17} - 20 q^{19} - 24 q^{23} + 50 q^{25} - 108 q^{29} + 244 q^{31} + 70 q^{35} + 532 q^{37} - 540 q^{41} + 544 q^{43} + 168 q^{47} + 98 q^{49} - 624 q^{53}+ \cdots + 616 q^{97}+O(q^{100}) $ 2 * q + 10 * q^5 + 14 * q^7 - 12 * q^11 - 32 * q^13 + 132 * q^17 - 20 * q^19 - 24 * q^23 + 50 * q^25 - 108 * q^29 + 244 * q^31 + 70 * q^35 + 532 * q^37 - 540 * q^41 + 544 * q^43 + 168 * q^47 + 98 * q^49 - 624 * q^53 - 60 * q^55 + 96 * q^59 + 1516 * q^61 - 160 * q^65 + 712 * q^67 - 12 * q^71 - 176 * q^73 - 84 * q^77 + 664 * q^79 + 48 * q^83 + 660 * q^85 - 732 * q^89 - 224 * q^91 - 100 * q^95 + 616 * q^97

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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	56.6418	1.55256	0.776280	−	0.630388i	$-0.217104\pi$
$11$	56.6418	1.55256	0.776280	+	0.630388i	$0.217104\pi$
$12$	0	0
$13$	46.6418	0.995086	0.497543	−	0.867439i	$-0.334236\pi$
$13$	46.6418	0.995086	0.497543	+	0.867439i	$0.334236\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	66.0000	0.941609	0.470804	−	0.882238i	$-0.343964\pi$
$17$	66.0000	0.941609	0.470804	+	0.882238i	$0.343964\pi$
$18$	0	0
$19$	52.6418	0.635625	0.317812	−	0.948154i	$-0.397052\pi$
$19$	52.6418	0.635625	0.317812	+	0.948154i	$0.397052\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−12.0000	−0.108790	−0.0543951	−	0.998519i	$-0.517323\pi$
$23$	−12.0000	−0.108790	−0.0543951	+	0.998519i	$0.517323\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−54.0000	−0.345778	−0.172889	−	0.984941i	$-0.555310\pi$
$29$	−54.0000	−0.345778	−0.172889	+	0.984941i	$0.555310\pi$
$30$	0	0
$31$	−65.9255	−0.381954	−0.190977	−	0.981595i	$-0.561166\pi$
$31$	−65.9255	−0.381954	−0.190977	+	0.981595i	$0.561166\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	140.716	0.625233	0.312616	−	0.949879i	$-0.398795\pi$
$37$	140.716	0.625233	0.312616	+	0.949879i	$0.398795\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−270.000	−1.02846	−0.514231	−	0.857652i	$-0.671922\pi$
$41$	−270.000	−1.02846	−0.514231	+	0.857652i	$0.671922\pi$
$42$	0	0
$43$	146.716	0.520326	0.260163	−	0.965565i	$-0.416224\pi$
$43$	146.716	0.520326	0.260163	+	0.965565i	$0.416224\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−291.851	−0.905763	−0.452881	−	0.891571i	$-0.649604\pi$
$47$	−291.851	−0.905763	−0.452881	+	0.891571i	$0.649604\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−374.642	−0.970963	−0.485481	−	0.874247i	$-0.661356\pi$
$53$	−374.642	−0.970963	−0.485481	+	0.874247i	$0.661356\pi$
$54$	0	0
$55$	283.209	0.694326
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−77.2837	−0.170534	−0.0852668	−	0.996358i	$-0.527174\pi$
$59$	−77.2837	−0.170534	−0.0852668	+	0.996358i	$0.527174\pi$
$60$	0	0
$61$	758.000	1.59102	0.795508	−	0.605943i	$-0.207204\pi$
$61$	758.000	1.59102	0.795508	+	0.605943i	$0.207204\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	233.209	0.445016
$66$	0	0
$67$	356.000	0.649139	0.324570	−	0.945862i	$-0.394781\pi$
$67$	356.000	0.649139	0.324570	+	0.945862i	$0.394781\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−820.344	−1.37122	−0.685612	−	0.727967i	$-0.740465\pi$
$71$	−820.344	−1.37122	−0.685612	+	0.727967i	$0.740465\pi$
$72$	0	0
$73$	−401.209	−0.643260	−0.321630	−	0.946865i	$-0.604231\pi$
$73$	−401.209	−0.643260	−0.321630	+	0.946865i	$0.604231\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	396.493	0.586812
$78$	0	0
$79$	707.851	1.00809	0.504047	−	0.863676i	$-0.331844\pi$
$79$	707.851	1.00809	0.504047	+	0.863676i	$0.331844\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	900.986	1.19152	0.595759	−	0.803163i	$-0.296851\pi$
$83$	900.986	1.19152	0.595759	+	0.803163i	$0.296851\pi$
$84$	0	0
$85$	330.000	0.421100
$86$	0	0
$87$	0	0
$88$	0	0
$89$	636.269	0.757803	0.378901	−	0.925437i	$-0.376302\pi$
$89$	636.269	0.757803	0.378901	+	0.925437i	$0.376302\pi$
$90$	0	0
$91$	326.493	0.376107
$92$	0	0
$93$	0	0
$94$	0	0
$95$	263.209	0.284260
$96$	0	0
$97$	120.074	0.125688	0.0628439	−	0.998023i	$-0.479983\pi$
$97$	120.074	0.125688	0.0628439	+	0.998023i	$0.479983\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1668.27	−1.64355	−0.821777	−	0.569809i	$-0.807017\pi$
$101$	−1668.27	−1.64355	−0.821777	+	0.569809i	$0.807017\pi$
$102$	0	0
$103$	−1450.27	−1.38737	−0.693686	−	0.720278i	$-0.744014\pi$
$103$	−1450.27	−1.38737	−0.693686	+	0.720278i	$0.744014\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−257.284	−0.232454	−0.116227	−	0.993223i	$-0.537080\pi$
$107$	−257.284	−0.232454	−0.116227	+	0.993223i	$0.537080\pi$
$108$	0	0
$109$	2042.54	1.79486	0.897430	−	0.441157i	$-0.145432\pi$
$109$	2042.54	1.79486	0.897430	+	0.441157i	$0.145432\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−85.2092	−0.0709363	−0.0354682	−	0.999371i	$-0.511292\pi$
$113$	−85.2092	−0.0709363	−0.0354682	+	0.999371i	$0.511292\pi$
$114$	0	0
$115$	−60.0000	−0.0486524
$116$	0	0
$117$	0	0
$118$	0	0
$119$	462.000	0.355895
$120$	0	0
$121$	1877.30	1.41044
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	526.418	0.367812	0.183906	−	0.982944i	$-0.441126\pi$
$127$	526.418	0.367812	0.183906	+	0.982944i	$0.441126\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1656.54	1.10483	0.552414	−	0.833570i	$-0.313707\pi$
$131$	1656.54	1.10483	0.552414	+	0.833570i	$0.313707\pi$
$132$	0	0
$133$	368.493	0.240243
$134$	0	0
$135$	0	0
$136$	0	0
$137$	177.805	0.110883	0.0554413	−	0.998462i	$-0.482343\pi$
$137$	177.805	0.110883	0.0554413	+	0.998462i	$0.482343\pi$
$138$	0	0
$139$	−963.060	−0.587667	−0.293833	−	0.955857i	$-0.594931\pi$
$139$	−963.060	−0.587667	−0.293833	+	0.955857i	$0.594931\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2641.88	1.54493
$144$	0	0
$145$	−270.000	−0.154636
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−852.716	−0.468841	−0.234420	−	0.972135i	$-0.575319\pi$
$149$	−852.716	−0.468841	−0.234420	+	0.972135i	$0.575319\pi$
$150$	0	0
$151$	−3166.81	−1.70670	−0.853348	−	0.521341i	$-0.825432\pi$
$151$	−3166.81	−1.70670	−0.853348	+	0.521341i	$0.825432\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−329.628	−0.170815
$156$	0	0
$157$	−1327.18	−0.674653	−0.337327	−	0.941388i	$-0.609523\pi$
$157$	−1327.18	−0.674653	−0.337327	+	0.941388i	$0.609523\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−84.0000	−0.0411188
$162$	0	0
$163$	1759.55	0.845514	0.422757	−	0.906243i	$-0.361062\pi$
$163$	1759.55	0.845514	0.422757	+	0.906243i	$0.361062\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3186.27	1.47641	0.738206	−	0.674575i	$-0.235673\pi$
$167$	3186.27	1.47641	0.738206	+	0.674575i	$0.235673\pi$
$168$	0	0
$169$	−21.5388	−0.00980375
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−570.986	−0.250932	−0.125466	−	0.992098i	$-0.540043\pi$
$173$	−570.986	−0.250932	−0.125466	+	0.992098i	$0.540043\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1865.24	0.778851	0.389426	−	0.921058i	$-0.372673\pi$
$179$	1865.24	0.778851	0.389426	+	0.921058i	$0.372673\pi$
$180$	0	0
$181$	3651.52	1.49953	0.749767	−	0.661702i	$-0.230166\pi$
$181$	3651.52	1.49953	0.749767	+	0.661702i	$0.230166\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	703.582	0.279613
$186$	0	0
$187$	3738.36	1.46190
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1437.63	−0.544623	−0.272312	−	0.962209i	$-0.587788\pi$
$191$	−1437.63	−0.544623	−0.272312	+	0.962209i	$0.587788\pi$
$192$	0	0
$193$	−1347.73	−0.502652	−0.251326	−	0.967903i	$-0.580867\pi$
$193$	−1347.73	−0.502652	−0.251326	+	0.967903i	$0.580867\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1275.09	−0.461149	−0.230574	−	0.973055i	$-0.574061\pi$
$197$	−1275.09	−0.461149	−0.230574	+	0.973055i	$0.574061\pi$
$198$	0	0
$199$	790.372	0.281548	0.140774	−	0.990042i	$-0.455041\pi$
$199$	790.372	0.281548	0.140774	+	0.990042i	$0.455041\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−378.000	−0.130692
$204$	0	0
$205$	−1350.00	−0.459942
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2981.73	0.986845
$210$	0	0
$211$	3343.55	1.09090	0.545449	−	0.838144i	$-0.316359\pi$
$211$	3343.55	1.09090	0.545449	+	0.838144i	$0.316359\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	733.582	0.232697
$216$	0	0
$217$	−461.479	−0.144365
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3078.36	0.936982
$222$	0	0
$223$	−3091.85	−0.928456	−0.464228	−	0.885716i	$-0.653668\pi$
$223$	−3091.85	−0.928456	−0.464228	+	0.885716i	$0.653668\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6024.63	1.76154	0.880768	−	0.473548i	$-0.157027\pi$
$227$	6024.63	1.76154	0.880768	+	0.473548i	$0.157027\pi$
$228$	0	0
$229$	1470.30	0.424280	0.212140	−	0.977239i	$-0.431957\pi$
$229$	1470.30	0.424280	0.212140	+	0.977239i	$0.431957\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2599.03	0.730765	0.365382	−	0.930857i	$-0.380938\pi$
$233$	2599.03	0.730765	0.365382	+	0.930857i	$0.380938\pi$
$234$	0	0
$235$	−1459.26	−0.405069
$236$	0	0
$237$	0	0
$238$	0	0
$239$	395.060	0.106922	0.0534609	−	0.998570i	$-0.482975\pi$
$239$	395.060	0.106922	0.0534609	+	0.998570i	$0.482975\pi$
$240$	0	0
$241$	4140.66	1.10674	0.553368	−	0.832937i	$-0.313342\pi$
$241$	4140.66	1.10674	0.553368	+	0.832937i	$0.313342\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	2455.31	0.632501
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1013.28	0.254812	0.127406	−	0.991851i	$-0.459335\pi$
$251$	1013.28	0.254812	0.127406	+	0.991851i	$0.459335\pi$
$252$	0	0
$253$	−679.702	−0.168903
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6981.50	−1.69453	−0.847264	−	0.531172i	$-0.821752\pi$
$257$	−6981.50	−1.69453	−0.847264	+	0.531172i	$0.821752\pi$
$258$	0	0
$259$	985.014	0.236316
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6093.58	1.42869	0.714347	−	0.699792i	$-0.246724\pi$
$263$	6093.58	1.42869	0.714347	+	0.699792i	$0.246724\pi$
$264$	0	0
$265$	−1873.21	−0.434228
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−706.298	−0.160088	−0.0800441	−	0.996791i	$-0.525506\pi$
$269$	−706.298	−0.160088	−0.0800441	+	0.996791i	$0.525506\pi$
$270$	0	0
$271$	3520.73	0.789186	0.394593	−	0.918856i	$-0.370886\pi$
$271$	3520.73	0.789186	0.394593	+	0.918856i	$0.370886\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1416.05	0.310512
$276$	0	0
$277$	1369.55	0.297070	0.148535	−	0.988907i	$-0.452544\pi$
$277$	1369.55	0.297070	0.148535	+	0.988907i	$0.452544\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7204.66	−1.52952	−0.764758	−	0.644318i	$-0.777141\pi$
$281$	−7204.66	−1.52952	−0.764758	+	0.644318i	$0.777141\pi$
$282$	0	0
$283$	−189.192	−0.0397395	−0.0198698	−	0.999803i	$-0.506325\pi$
$283$	−189.192	−0.0397395	−0.0198698	+	0.999803i	$0.506325\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1890.00	−0.388722
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1816.48	0.362183	0.181092	−	0.983466i	$-0.442037\pi$
$293$	1816.48	0.362183	0.181092	+	0.983466i	$0.442037\pi$
$294$	0	0
$295$	−386.418	−0.0762649
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−559.702	−0.108256
$300$	0	0
$301$	1027.01	0.196665
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3790.00	0.711524
$306$	0	0
$307$	1766.18	0.328342	0.164171	−	0.986432i	$-0.447505\pi$
$307$	1766.18	0.328342	0.164171	+	0.986432i	$0.447505\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2542.03	0.463489	0.231745	−	0.972777i	$-0.425557\pi$
$311$	2542.03	0.463489	0.231745	+	0.972777i	$0.425557\pi$
$312$	0	0
$313$	−8114.97	−1.46545	−0.732724	−	0.680525i	$-0.761752\pi$
$313$	−8114.97	−1.46545	−0.732724	+	0.680525i	$0.761752\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4997.15	−0.885388	−0.442694	−	0.896673i	$-0.645977\pi$
$317$	−4997.15	−0.885388	−0.442694	+	0.896673i	$0.645977\pi$
$318$	0	0
$319$	−3058.66	−0.536840
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3474.36	0.598510
$324$	0	0
$325$	1166.05	0.199017
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2042.96	−0.342346
$330$	0	0
$331$	2593.64	0.430693	0.215346	−	0.976538i	$-0.430912\pi$
$331$	2593.64	0.430693	0.215346	+	0.976538i	$0.430912\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1780.00	0.290304
$336$	0	0
$337$	9363.97	1.51361	0.756807	−	0.653638i	$-0.226758\pi$
$337$	9363.97	1.51361	0.756807	+	0.653638i	$0.226758\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3734.14	−0.593006
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11371.3	−1.75921	−0.879605	−	0.475705i	$-0.842193\pi$
$347$	−11371.3	−1.75921	−0.879605	+	0.475705i	$0.842193\pi$
$348$	0	0
$349$	−10479.9	−1.60738	−0.803692	−	0.595046i	$-0.797134\pi$
$349$	−10479.9	−1.60738	−0.803692	+	0.595046i	$0.797134\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4551.49	−0.686264	−0.343132	−	0.939287i	$-0.611488\pi$
$353$	−4551.49	−0.686264	−0.343132	+	0.939287i	$0.611488\pi$
$354$	0	0
$355$	−4101.72	−0.613230
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6705.27	−0.985768	−0.492884	−	0.870095i	$-0.664057\pi$
$359$	−6705.27	−0.985768	−0.492884	+	0.870095i	$0.664057\pi$
$360$	0	0
$361$	−4087.84	−0.595981
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2006.05	−0.287675
$366$	0	0
$367$	5880.57	0.836413	0.418206	−	0.908352i	$-0.362659\pi$
$367$	5880.57	0.836413	0.418206	+	0.908352i	$0.362659\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2622.49	−0.366989
$372$	0	0
$373$	13540.7	1.87965	0.939827	−	0.341652i	$-0.110986\pi$
$373$	13540.7	1.87965	0.939827	+	0.341652i	$0.110986\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2518.66	−0.344078
$378$	0	0
$379$	10982.5	1.48847	0.744236	−	0.667917i	$-0.232814\pi$
$379$	10982.5	1.48847	0.744236	+	0.667917i	$0.232814\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11647.9	−1.55399	−0.776996	−	0.629505i	$-0.783258\pi$
$383$	−11647.9	−1.55399	−0.776996	+	0.629505i	$0.783258\pi$
$384$	0	0
$385$	1982.46	0.262430
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5323.17	0.693819	0.346909	−	0.937899i	$-0.387231\pi$
$389$	5323.17	0.693819	0.346909	+	0.937899i	$0.387231\pi$
$390$	0	0
$391$	−792.000	−0.102438
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3539.26	0.450834
$396$	0	0
$397$	−3272.35	−0.413689	−0.206845	−	0.978374i	$-0.566319\pi$
$397$	−3272.35	−0.413689	−0.206845	+	0.978374i	$0.566319\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8255.73	−1.02811	−0.514054	−	0.857758i	$-0.671857\pi$
$401$	−8255.73	−1.02811	−0.514054	+	0.857758i	$0.671857\pi$
$402$	0	0
$403$	−3074.89	−0.380077
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7970.43	0.970712
$408$	0	0
$409$	15949.6	1.92826	0.964131	−	0.265427i	$-0.0855128\pi$
$409$	15949.6	1.92826	0.964131	+	0.265427i	$0.0855128\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−540.986	−0.0644556
$414$	0	0
$415$	4504.93	0.532863
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5743.16	0.669623	0.334811	−	0.942285i	$-0.391327\pi$
$419$	5743.16	0.669623	0.334811	+	0.942285i	$0.391327\pi$
$420$	0	0
$421$	3396.92	0.393244	0.196622	−	0.980479i	$-0.437003\pi$
$421$	3396.92	0.393244	0.196622	+	0.980479i	$0.437003\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1650.00	0.188322
$426$	0	0
$427$	5306.00	0.601347
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15383.0	−1.71919	−0.859596	−	0.510974i	$-0.829285\pi$
$431$	−15383.0	−1.71919	−0.859596	+	0.510974i	$0.829285\pi$
$432$	0	0
$433$	−17906.1	−1.98732	−0.993662	−	0.112409i	$-0.964143\pi$
$433$	−17906.1	−1.98732	−0.993662	+	0.112409i	$0.964143\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−631.702	−0.0691497
$438$	0	0
$439$	13900.3	1.51122	0.755609	−	0.655023i	$-0.227341\pi$
$439$	13900.3	1.51122	0.755609	+	0.655023i	$0.227341\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4219.35	0.452522	0.226261	−	0.974067i	$-0.427350\pi$
$443$	4219.35	0.452522	0.226261	+	0.974067i	$0.427350\pi$
$444$	0	0
$445$	3181.35	0.338900
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6709.89	0.705254	0.352627	−	0.935764i	$-0.385288\pi$
$449$	6709.89	0.705254	0.352627	+	0.935764i	$0.385288\pi$
$450$	0	0
$451$	−15293.3	−1.59675
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1632.46	0.168200
$456$	0	0
$457$	−17330.9	−1.77398	−0.886988	−	0.461793i	$-0.847206\pi$
$457$	−17330.9	−1.77398	−0.886988	+	0.461793i	$0.847206\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4891.43	0.494179	0.247090	−	0.968993i	$-0.420526\pi$
$461$	4891.43	0.494179	0.247090	+	0.968993i	$0.420526\pi$
$462$	0	0
$463$	−16496.6	−1.65585	−0.827927	−	0.560835i	$-0.810480\pi$
$463$	−16496.6	−1.65585	−0.827927	+	0.560835i	$0.810480\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6046.48	−0.599138	−0.299569	−	0.954075i	$-0.596843\pi$
$467$	−6046.48	−0.599138	−0.299569	+	0.954075i	$0.596843\pi$
$468$	0	0
$469$	2492.00	0.245352
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8310.28	0.807838
$474$	0	0
$475$	1316.05	0.127125
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16334.5	1.55813	0.779063	−	0.626945i	$-0.215695\pi$
$479$	16334.5	1.55813	0.779063	+	0.626945i	$0.215695\pi$
$480$	0	0
$481$	6563.27	0.622161
$482$	0	0
$483$	0	0
$484$	0	0
$485$	600.372	0.0562093
$486$	0	0
$487$	−14770.5	−1.37437	−0.687184	−	0.726483i	$-0.741153\pi$
$487$	−14770.5	−1.37437	−0.687184	+	0.726483i	$0.741153\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6343.84	−0.583082	−0.291541	−	0.956558i	$-0.594168\pi$
$491$	−6343.84	−0.583082	−0.291541	+	0.956558i	$0.594168\pi$
$492$	0	0
$493$	−3564.00	−0.325587
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5742.41	−0.518274
$498$	0	0
$499$	−2904.24	−0.260544	−0.130272	−	0.991478i	$-0.541585\pi$
$499$	−2904.24	−0.260544	−0.130272	+	0.991478i	$0.541585\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5224.11	0.463085	0.231542	−	0.972825i	$-0.425623\pi$
$503$	5224.11	0.463085	0.231542	+	0.972825i	$0.425623\pi$
$504$	0	0
$505$	−8341.35	−0.735020
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8188.94	−0.713101	−0.356550	−	0.934276i	$-0.616047\pi$
$509$	−8188.94	−0.713101	−0.356550	+	0.934276i	$0.616047\pi$
$510$	0	0
$511$	−2808.46	−0.243129
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7251.35	−0.620452
$516$	0	0
$517$	−16531.0	−1.40625
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14331.3	−1.20512	−0.602559	−	0.798074i	$-0.705852\pi$
$521$	−14331.3	−1.20512	−0.602559	+	0.798074i	$0.705852\pi$
$522$	0	0
$523$	−14696.2	−1.22872	−0.614360	−	0.789026i	$-0.710586\pi$
$523$	−14696.2	−1.22872	−0.614360	+	0.789026i	$0.710586\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4351.08	−0.359651
$528$	0	0
$529$	−12023.0	−0.988165
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12593.3	−1.02341
$534$	0	0
$535$	−1286.42	−0.103956
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2775.45	0.221794
$540$	0	0
$541$	988.510	0.0785571	0.0392785	−	0.999228i	$-0.487494\pi$
$541$	988.510	0.0785571	0.0392785	+	0.999228i	$0.487494\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10212.7	0.802686
$546$	0	0
$547$	19765.7	1.54501	0.772507	−	0.635007i	$-0.219003\pi$
$547$	19765.7	1.54501	0.772507	+	0.635007i	$0.219003\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2842.66	−0.219785
$552$	0	0
$553$	4954.96	0.381024
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4856.65	0.369448	0.184724	−	0.982790i	$-0.440861\pi$
$557$	4856.65	0.369448	0.184724	+	0.982790i	$0.440861\pi$
$558$	0	0
$559$	6843.12	0.517769
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1008.35	−0.0754833	−0.0377416	−	0.999288i	$-0.512016\pi$
$563$	−1008.35	−0.0754833	−0.0377416	+	0.999288i	$0.512016\pi$
$564$	0	0
$565$	−426.046	−0.0317237
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18032.3	1.32857	0.664284	−	0.747480i	$-0.268737\pi$
$569$	18032.3	1.32857	0.664284	+	0.747480i	$0.268737\pi$
$570$	0	0
$571$	−12329.6	−0.903640	−0.451820	−	0.892109i	$-0.649225\pi$
$571$	−12329.6	−0.903640	−0.451820	+	0.892109i	$0.649225\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−300.000	−0.0217580
$576$	0	0
$577$	−19227.0	−1.38723	−0.693613	−	0.720348i	$-0.743982\pi$
$577$	−19227.0	−1.38723	−0.693613	+	0.720348i	$0.743982\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6306.90	0.450352
$582$	0	0
$583$	−21220.4	−1.50748
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5388.00	0.378853	0.189426	−	0.981895i	$-0.439337\pi$
$587$	5388.00	0.378853	0.189426	+	0.981895i	$0.439337\pi$
$588$	0	0
$589$	−3470.44	−0.242779
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14921.6	−1.03332	−0.516660	−	0.856191i	$-0.672825\pi$
$593$	−14921.6	−1.03332	−0.516660	+	0.856191i	$0.672825\pi$
$594$	0	0
$595$	2310.00	0.159161
$596$	0	0
$597$	0	0
$598$	0	0
$599$	25631.9	1.74840	0.874199	−	0.485568i	$-0.161387\pi$
$599$	25631.9	1.74840	0.874199	+	0.485568i	$0.161387\pi$
$600$	0	0
$601$	−11924.1	−0.809310	−0.404655	−	0.914470i	$-0.632608\pi$
$601$	−11924.1	−0.809310	−0.404655	+	0.914470i	$0.632608\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9386.49	0.630769
$606$	0	0
$607$	−9262.72	−0.619377	−0.309689	−	0.950838i	$-0.600225\pi$
$607$	−9262.72	−0.619377	−0.309689	+	0.950838i	$0.600225\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13612.5	−0.901312
$612$	0	0
$613$	−17791.9	−1.17228	−0.586142	−	0.810209i	$-0.699354\pi$
$613$	−17791.9	−1.17228	−0.586142	+	0.810209i	$0.699354\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2381.69	−0.155402	−0.0777012	−	0.996977i	$-0.524758\pi$
$617$	−2381.69	−0.155402	−0.0777012	+	0.996977i	$0.524758\pi$
$618$	0	0
$619$	−1439.49	−0.0934698	−0.0467349	−	0.998907i	$-0.514882\pi$
$619$	−1439.49	−0.0934698	−0.0467349	+	0.998907i	$0.514882\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4453.89	0.286422
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9287.28	0.588725
$630$	0	0
$631$	12146.4	0.766306	0.383153	−	0.923685i	$-0.374838\pi$
$631$	12146.4	0.766306	0.383153	+	0.923685i	$0.374838\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2632.09	0.164490
$636$	0	0
$637$	2285.45	0.142155
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10331.8	−0.636635	−0.318317	−	0.947984i	$-0.603118\pi$
$641$	−10331.8	−0.636635	−0.318317	+	0.947984i	$0.603118\pi$
$642$	0	0
$643$	14572.9	0.893776	0.446888	−	0.894590i	$-0.352532\pi$
$643$	14572.9	0.893776	0.446888	+	0.894590i	$0.352532\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5365.80	−0.326045	−0.163023	−	0.986622i	$-0.552124\pi$
$647$	−5365.80	−0.326045	−0.163023	+	0.986622i	$0.552124\pi$
$648$	0	0
$649$	−4377.49	−0.264763
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−31532.8	−1.88970	−0.944851	−	0.327501i	$-0.893793\pi$
$653$	−31532.8	−1.88970	−0.944851	+	0.327501i	$0.893793\pi$
$654$	0	0
$655$	8282.69	0.494094
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4830.39	0.285532	0.142766	−	0.989756i	$-0.454400\pi$
$659$	4830.39	0.285532	0.142766	+	0.989756i	$0.454400\pi$
$660$	0	0
$661$	−24168.9	−1.42218	−0.711090	−	0.703101i	$-0.751798\pi$
$661$	−24168.9	−1.42218	−0.711090	+	0.703101i	$0.751798\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1842.46	0.107440
$666$	0	0
$667$	648.000	0.0376172
$668$	0	0
$669$	0	0
$670$	0	0
$671$	42934.5	2.47015
$672$	0	0
$673$	5383.64	0.308357	0.154178	−	0.988043i	$-0.450727\pi$
$673$	5383.64	0.308357	0.154178	+	0.988043i	$0.450727\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27370.4	1.55381	0.776905	−	0.629617i	$-0.216788\pi$
$677$	27370.4	1.55381	0.776905	+	0.629617i	$0.216788\pi$
$678$	0	0
$679$	840.521	0.0475055
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−33313.3	−1.86632	−0.933160	−	0.359461i	$-0.882960\pi$
$683$	−33313.3	−1.86632	−0.933160	+	0.359461i	$0.882960\pi$
$684$	0	0
$685$	889.025	0.0495882
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−17474.0	−0.966191
$690$	0	0
$691$	20185.5	1.11128	0.555639	−	0.831423i	$-0.312474\pi$
$691$	20185.5	1.11128	0.555639	+	0.831423i	$0.312474\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4815.30	−0.262813
$696$	0	0
$697$	−17820.0	−0.968408
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12215.2	−0.658148	−0.329074	−	0.944304i	$-0.606736\pi$
$701$	−12215.2	−0.658148	−0.329074	+	0.944304i	$0.606736\pi$
$702$	0	0
$703$	7407.57	0.397413
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−11677.9	−0.621205
$708$	0	0
$709$	−15402.3	−0.815859	−0.407929	−	0.913014i	$-0.633749\pi$
$709$	−15402.3	−0.815859	−0.407929	+	0.913014i	$0.633749\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	791.106	0.0415528
$714$	0	0
$715$	13209.4	0.690914
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20646.6	1.07091	0.535457	−	0.844563i	$-0.320140\pi$
$719$	20646.6	1.07091	0.535457	+	0.844563i	$0.320140\pi$
$720$	0	0
$721$	−10151.9	−0.524377
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1350.00	−0.0691555
$726$	0	0
$727$	17329.1	0.884045	0.442023	−	0.897004i	$-0.354261\pi$
$727$	17329.1	0.884045	0.442023	+	0.897004i	$0.354261\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9683.28	0.489944
$732$	0	0
$733$	−29890.5	−1.50618	−0.753091	−	0.657917i	$-0.771438\pi$
$733$	−29890.5	−1.50618	−0.753091	+	0.657917i	$0.771438\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20164.5	1.00783
$738$	0	0
$739$	−14143.0	−0.704006	−0.352003	−	0.935999i	$-0.614499\pi$
$739$	−14143.0	−0.704006	−0.352003	+	0.935999i	$0.614499\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−25944.6	−1.28104	−0.640522	−	0.767940i	$-0.721282\pi$
$743$	−25944.6	−1.28104	−0.640522	+	0.767940i	$0.721282\pi$
$744$	0	0
$745$	−4263.58	−0.209672
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1800.99	−0.0878593
$750$	0	0
$751$	6551.50	0.318332	0.159166	−	0.987252i	$-0.449119\pi$
$751$	6551.50	0.318332	0.159166	+	0.987252i	$0.449119\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15834.0	−0.763258
$756$	0	0
$757$	−14503.3	−0.696344	−0.348172	−	0.937431i	$-0.613197\pi$
$757$	−14503.3	−0.696344	−0.348172	+	0.937431i	$0.613197\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34740.1	1.65483	0.827417	−	0.561589i	$-0.189810\pi$
$761$	34740.1	1.65483	0.827417	+	0.561589i	$0.189810\pi$
$762$	0	0
$763$	14297.8	0.678393
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3604.65	−0.169696
$768$	0	0
$769$	8998.86	0.421986	0.210993	−	0.977488i	$-0.432330\pi$
$769$	8998.86	0.421986	0.210993	+	0.977488i	$0.432330\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−16966.6	−0.789450	−0.394725	−	0.918799i	$-0.629160\pi$
$773$	−16966.6	−0.789450	−0.394725	+	0.918799i	$0.629160\pi$
$774$	0	0
$775$	−1648.14	−0.0763908
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14213.3	−0.653715
$780$	0	0
$781$	−46465.8	−2.12891
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6635.90	−0.301714
$786$	0	0
$787$	14915.1	0.675562	0.337781	−	0.941225i	$-0.390324\pi$
$787$	14915.1	0.675562	0.337781	+	0.941225i	$0.390324\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−596.464	−0.0268114
$792$	0	0
$793$	35354.5	1.58320
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12855.4	0.571344	0.285672	−	0.958327i	$-0.407783\pi$
$797$	12855.4	0.571344	0.285672	+	0.958327i	$0.407783\pi$
$798$	0	0
$799$	−19262.2	−0.852874
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−22725.2	−0.998700
$804$	0	0
$805$	−420.000	−0.0183889
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−29152.0	−1.26691	−0.633454	−	0.773780i	$-0.718363\pi$
$809$	−29152.0	−1.26691	−0.633454	+	0.773780i	$0.718363\pi$
$810$	0	0
$811$	36924.7	1.59877	0.799385	−	0.600820i	$-0.205159\pi$
$811$	36924.7	1.59877	0.799385	+	0.600820i	$0.205159\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8797.77	0.378126
$816$	0	0
$817$	7723.42	0.330732
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−37844.0	−1.60873	−0.804363	−	0.594138i	$-0.797493\pi$
$821$	−37844.0	−1.60873	−0.804363	+	0.594138i	$0.797493\pi$
$822$	0	0
$823$	7667.85	0.324769	0.162384	−	0.986728i	$-0.448082\pi$
$823$	7667.85	0.324769	0.162384	+	0.986728i	$0.448082\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23869.5	−1.00366	−0.501829	−	0.864967i	$-0.667339\pi$
$827$	−23869.5	−1.00366	−0.501829	+	0.864967i	$0.667339\pi$
$828$	0	0
$829$	33433.6	1.40072	0.700359	−	0.713791i	$-0.253023\pi$
$829$	33433.6	1.40072	0.700359	+	0.713791i	$0.253023\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3234.00	0.134516
$834$	0	0
$835$	15931.3	0.660272
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31273.0	1.28685	0.643423	−	0.765511i	$-0.277514\pi$
$839$	31273.0	1.28685	0.643423	+	0.765511i	$0.277514\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−107.694	−0.00438437
$846$	0	0
$847$	13141.1	0.533097
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1688.60	−0.0680192
$852$	0	0
$853$	−9972.48	−0.400294	−0.200147	−	0.979766i	$-0.564142\pi$
$853$	−9972.48	−0.400294	−0.200147	+	0.979766i	$0.564142\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25152.9	1.00257	0.501287	−	0.865281i	$-0.332860\pi$
$857$	25152.9	1.00257	0.501287	+	0.865281i	$0.332860\pi$
$858$	0	0
$859$	−7874.63	−0.312781	−0.156390	−	0.987695i	$-0.549986\pi$
$859$	−7874.63	−0.312781	−0.156390	+	0.987695i	$0.549986\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1959.42	0.0772878	0.0386439	−	0.999253i	$-0.487696\pi$
$863$	1959.42	0.0772878	0.0386439	+	0.999253i	$0.487696\pi$
$864$	0	0
$865$	−2854.93	−0.112220
$866$	0	0
$867$	0	0
$868$	0	0
$869$	40094.0	1.56513
$870$	0	0
$871$	16604.5	0.645949
$872$	0	0
$873$	0	0
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−31261.9	−1.20369	−0.601846	−	0.798612i	$-0.705568\pi$
$877$	−31261.9	−1.20369	−0.601846	+	0.798612i	$0.705568\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42459.0	−1.62370	−0.811850	−	0.583866i	$-0.801539\pi$
$881$	−42459.0	−1.62370	−0.811850	+	0.583866i	$0.801539\pi$
$882$	0	0
$883$	34673.0	1.32145	0.660723	−	0.750629i	$-0.270250\pi$
$883$	34673.0	1.32145	0.660723	+	0.750629i	$0.270250\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38271.3	−1.44873	−0.724366	−	0.689415i	$-0.757868\pi$
$887$	−38271.3	−1.44873	−0.724366	+	0.689415i	$0.757868\pi$
$888$	0	0
$889$	3684.93	0.139020
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−15363.6	−0.575725
$894$	0	0
$895$	9326.19	0.348313
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3559.98	0.132071
$900$	0	0
$901$	−24726.4	−0.914267
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18257.6	0.670612
$906$	0	0
$907$	17.6607	0.000646542 0	0.000323271	−	1.00000i	$-0.499897\pi$
$907$	17.6607	0.000646542 0	0.000323271		1.00000i	$0.499897\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20346.3	0.739959	0.369980	−	0.929040i	$-0.379365\pi$
$911$	20346.3	0.739959	0.369980	+	0.929040i	$0.379365\pi$
$912$	0	0
$913$	51033.5	1.84990
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11595.8	0.417586
$918$	0	0
$919$	38074.1	1.36665	0.683323	−	0.730116i	$-0.260534\pi$
$919$	38074.1	1.36665	0.683323	+	0.730116i	$0.260534\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−38262.3	−1.36449
$924$	0	0
$925$	3517.91	0.125047
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−5169.42	−0.182565	−0.0912826	−	0.995825i	$-0.529097\pi$
$929$	−5169.42	−0.182565	−0.0912826	+	0.995825i	$0.529097\pi$
$930$	0	0
$931$	2579.45	0.0908035
$932$	0	0
$933$	0	0
$934$	0	0
$935$	18691.8	0.653783
$936$	0	0
$937$	28753.8	1.00250	0.501252	−	0.865302i	$-0.332873\pi$
$937$	28753.8	1.00250	0.501252	+	0.865302i	$0.332873\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12332.7	0.427242	0.213621	−	0.976917i	$-0.431474\pi$
$941$	12332.7	0.427242	0.213621	+	0.976917i	$0.431474\pi$
$942$	0	0
$943$	3240.00	0.111886
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18195.0	0.624350	0.312175	−	0.950025i	$-0.398942\pi$
$947$	18195.0	0.624350	0.312175	+	0.950025i	$0.398942\pi$
$948$	0	0
$949$	−18713.1	−0.640099
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1331.58	−0.0452614	−0.0226307	−	0.999744i	$-0.507204\pi$
$953$	−1331.58	−0.0452614	−0.0226307	+	0.999744i	$0.507204\pi$
$954$	0	0
$955$	−7188.14	−0.243563
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1244.64	0.0419097
$960$	0	0
$961$	−25444.8	−0.854111
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6738.65	−0.224793
$966$	0	0
$967$	19516.2	0.649018	0.324509	−	0.945883i	$-0.394801\pi$
$967$	19516.2	0.649018	0.324509	+	0.945883i	$0.394801\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	16350.7	0.540390	0.270195	−	0.962806i	$-0.412912\pi$
$971$	16350.7	0.540390	0.270195	+	0.962806i	$0.412912\pi$
$972$	0	0
$973$	−6741.42	−0.222117
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−22315.8	−0.730754	−0.365377	−	0.930860i	$-0.619060\pi$
$977$	−22315.8	−0.730754	−0.365377	+	0.930860i	$0.619060\pi$
$978$	0	0
$979$	36039.5	1.17653
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4913.28	0.159420	0.0797098	−	0.996818i	$-0.474601\pi$
$983$	4913.28	0.159420	0.0797098	+	0.996818i	$0.474601\pi$
$984$	0	0
$985$	−6375.44	−0.206232
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1760.60	−0.0566064
$990$	0	0
$991$	−30336.3	−0.972418	−0.486209	−	0.873843i	$-0.661621\pi$
$991$	−30336.3	−0.972418	−0.486209	+	0.873843i	$0.661621\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3951.86	0.125912
$996$	0	0
$997$	34353.9	1.09127	0.545637	−	0.838022i	$-0.316288\pi$
$997$	34353.9	1.09127	0.545637	+	0.838022i	$0.316288\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1260.4.a.o.1.2		2
3.2	odd	2		420.4.a.g.1.1	✓	2
12.11	even	2		1680.4.a.bi.1.2		2
15.2	even	4		2100.4.k.m.1849.3		4
15.8	even	4		2100.4.k.m.1849.1		4
15.14	odd	2		2100.4.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.4.a.g.1.1	✓	2	3.2	odd	2
1260.4.a.o.1.2		2	1.1	even	1	trivial
1680.4.a.bi.1.2		2	12.11	even	2
2100.4.a.r.1.1		2	15.14	odd	2
2100.4.k.m.1849.1		4	15.8	even	4
2100.4.k.m.1849.3		4	15.2	even	4

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:	\( N \)	\(=\)	\( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1260.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +7.00000 q^{7} +56.6418 q^{11} +46.6418 q^{13} +66.0000 q^{17} +52.6418 q^{19} -12.0000 q^{23} +25.0000 q^{25} -54.0000 q^{29} -65.9255 q^{31} +35.0000 q^{35} +140.716 q^{37} -270.000 q^{41} +146.716 q^{43} -291.851 q^{47} +49.0000 q^{49} -374.642 q^{53} +283.209 q^{55} -77.2837 q^{59} +758.000 q^{61} +233.209 q^{65} +356.000 q^{67} -820.344 q^{71} -401.209 q^{73} +396.493 q^{77} +707.851 q^{79} +900.986 q^{83} +330.000 q^{85} +636.269 q^{89} +326.493 q^{91} +263.209 q^{95} +120.074 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{5} + 14 q^{7} - 12 q^{11} - 32 q^{13} + 132 q^{17} - 20 q^{19} - 24 q^{23} + 50 q^{25} - 108 q^{29} + 244 q^{31} + 70 q^{35} + 532 q^{37} - 540 q^{41} + 544 q^{43} + 168 q^{47} + 98 q^{49} - 624 q^{53}+ \cdots + 616 q^{97}+O(q^{100}) \) 2 * q + 10 * q^5 + 14 * q^7 - 12 * q^11 - 32 * q^13 + 132 * q^17 - 20 * q^19 - 24 * q^23 + 50 * q^25 - 108 * q^29 + 244 * q^31 + 70 * q^35 + 532 * q^37 - 540 * q^41 + 544 * q^43 + 168 * q^47 + 98 * q^49 - 624 * q^53 - 60 * q^55 + 96 * q^59 + 1516 * q^61 - 160 * q^65 + 712 * q^67 - 12 * q^71 - 176 * q^73 - 84 * q^77 + 664 * q^79 + 48 * q^83 + 660 * q^85 - 732 * q^89 - 224 * q^91 - 100 * q^95 + 616 * q^97

Introduction

L-functions

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