LMFDB - Embedded newform 2500.4.a.g.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2500,4,Mod(1,2500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2500, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2500.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2500 = 2^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2500.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:	$147.504775014$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 100)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	2500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.95605 q^{3} -16.8592 q^{7} -2.43753 q^{9} +O(q^{10})$	$q-4.95605 q^{3} -16.8592 q^{7} -2.43753 q^{9} +59.1321 q^{11} -6.31508 q^{13} -131.580 q^{17} +82.1717 q^{19} +83.5549 q^{21} -159.450 q^{23} +145.894 q^{27} -50.0444 q^{29} -44.6792 q^{31} -293.062 q^{33} -75.0473 q^{37} +31.2979 q^{39} +353.628 q^{41} -79.3703 q^{43} +304.791 q^{47} -58.7686 q^{49} +652.120 q^{51} +339.282 q^{53} -407.248 q^{57} -786.798 q^{59} -834.413 q^{61} +41.0948 q^{63} -607.900 q^{67} +790.243 q^{69} -61.4511 q^{71} +39.5735 q^{73} -996.917 q^{77} +577.934 q^{79} -657.245 q^{81} -53.3674 q^{83} +248.023 q^{87} -1447.22 q^{89} +106.467 q^{91} +221.432 q^{93} -53.5327 q^{97} -144.136 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 328 q^{9}+O(q^{10}) $ 32 * q + 328 * q^9	$ 32 q + 328 q^{9} + 60 q^{11} + 128 q^{19} + 312 q^{21} + 848 q^{29} + 312 q^{31} + 628 q^{39} + 1352 q^{41} + 2796 q^{49} + 1664 q^{51} + 1616 q^{59} + 1204 q^{61} + 1644 q^{69} - 944 q^{71} + 184 q^{79} + 5048 q^{81} + 2392 q^{89} + 1588 q^{91} + 2540 q^{99}+O(q^{100}) $ 32 * q + 328 * q^9 + 60 * q^11 + 128 * q^19 + 312 * q^21 + 848 * q^29 + 312 * q^31 + 628 * q^39 + 1352 * q^41 + 2796 * q^49 + 1664 * q^51 + 1616 * q^59 + 1204 * q^61 + 1644 * q^69 - 944 * q^71 + 184 * q^79 + 5048 * q^81 + 2392 * q^89 + 1588 * q^91 + 2540 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−4.95605	−0.953793	−0.476896	−	0.878959i	$-0.658238\pi$
$3$	−4.95605	−0.953793	−0.476896	+	0.878959i	$0.658238\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−16.8592	−0.910309	−0.455155	−	0.890412i	$-0.650416\pi$
$7$	−16.8592	−0.910309	−0.455155	+	0.890412i	$0.650416\pi$
$8$	0	0
$9$	−2.43753	−0.0902790
$10$	0	0
$11$	59.1321	1.62082	0.810409	−	0.585865i	$-0.199245\pi$
$11$	59.1321	1.62082	0.810409	+	0.585865i	$0.199245\pi$
$12$	0	0
$13$	−6.31508	−0.134730	−0.0673649	−	0.997728i	$-0.521459\pi$
$13$	−6.31508	−0.134730	−0.0673649	+	0.997728i	$0.521459\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−131.580	−1.87723	−0.938616	−	0.344963i	$-0.887891\pi$
$17$	−131.580	−1.87723	−0.938616	+	0.344963i	$0.887891\pi$
$18$	0	0
$19$	82.1717	0.992184	0.496092	−	0.868270i	$-0.334768\pi$
$19$	82.1717	0.992184	0.496092	+	0.868270i	$0.334768\pi$
$20$	0	0
$21$	83.5549	0.868247
$22$	0	0
$23$	−159.450	−1.44555	−0.722775	−	0.691084i	$-0.757134\pi$
$23$	−159.450	−1.44555	−0.722775	+	0.691084i	$0.757134\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	145.894	1.03990
$28$	0	0
$29$	−50.0444	−0.320448	−0.160224	−	0.987081i	$-0.551222\pi$
$29$	−50.0444	−0.320448	−0.160224	+	0.987081i	$0.551222\pi$
$30$	0	0
$31$	−44.6792	−0.258859	−0.129429	−	0.991589i	$-0.541315\pi$
$31$	−44.6792	−0.258859	−0.129429	+	0.991589i	$0.541315\pi$
$32$	0	0
$33$	−293.062	−1.54592
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−75.0473	−0.333451	−0.166726	−	0.986003i	$-0.553319\pi$
$37$	−75.0473	−0.333451	−0.166726	+	0.986003i	$0.553319\pi$
$38$	0	0
$39$	31.2979	0.128504
$40$	0	0
$41$	353.628	1.34701	0.673504	−	0.739183i	$-0.264788\pi$
$41$	353.628	1.34701	0.673504	+	0.739183i	$0.264788\pi$
$42$	0	0
$43$	−79.3703	−0.281485	−0.140743	−	0.990046i	$-0.544949\pi$
$43$	−79.3703	−0.281485	−0.140743	+	0.990046i	$0.544949\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	304.791	0.945923	0.472961	−	0.881083i	$-0.343185\pi$
$47$	304.791	0.945923	0.472961	+	0.881083i	$0.343185\pi$
$48$	0	0
$49$	−58.7686	−0.171337
$50$	0	0
$51$	652.120	1.79049
$52$	0	0
$53$	339.282	0.879319	0.439660	−	0.898164i	$-0.355099\pi$
$53$	339.282	0.879319	0.439660	+	0.898164i	$0.355099\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−407.248	−0.946338
$58$	0	0
$59$	−786.798	−1.73614	−0.868070	−	0.496441i	$-0.834640\pi$
$59$	−786.798	−1.73614	−0.868070	+	0.496441i	$0.834640\pi$
$60$	0	0
$61$	−834.413	−1.75140	−0.875702	−	0.482851i	$-0.839601\pi$
$61$	−834.413	−1.75140	−0.875702	+	0.482851i	$0.839601\pi$
$62$	0	0
$63$	41.0948	0.0821818
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−607.900	−1.10846	−0.554229	−	0.832364i	$-0.686987\pi$
$67$	−607.900	−1.10846	−0.554229	+	0.832364i	$0.686987\pi$
$68$	0	0
$69$	790.243	1.37875
$70$	0	0
$71$	−61.4511	−0.102717	−0.0513585	−	0.998680i	$-0.516355\pi$
$71$	−61.4511	−0.102717	−0.0513585	+	0.998680i	$0.516355\pi$
$72$	0	0
$73$	39.5735	0.0634483	0.0317241	−	0.999497i	$-0.489900\pi$
$73$	39.5735	0.0634483	0.0317241	+	0.999497i	$0.489900\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−996.917	−1.47545
$78$	0	0
$79$	577.934	0.823072	0.411536	−	0.911394i	$-0.364992\pi$
$79$	577.934	0.823072	0.411536	+	0.911394i	$0.364992\pi$
$80$	0	0
$81$	−657.245	−0.901571
$82$	0	0
$83$	−53.3674	−0.0705763	−0.0352881	−	0.999377i	$-0.511235\pi$
$83$	−53.3674	−0.0705763	−0.0352881	+	0.999377i	$0.511235\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	248.023	0.305642
$88$	0	0
$89$	−1447.22	−1.72365	−0.861827	−	0.507202i	$-0.830680\pi$
$89$	−1447.22	−1.72365	−0.861827	+	0.507202i	$0.830680\pi$
$90$	0	0
$91$	106.467	0.122646
$92$	0	0
$93$	221.432	0.246898
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−53.5327	−0.0560353	−0.0280177	−	0.999607i	$-0.508919\pi$
$97$	−53.5327	−0.0560353	−0.0280177	+	0.999607i	$0.508919\pi$
$98$	0	0
$99$	−144.136	−0.146326
$100$	0	0
$101$	−47.2102	−0.0465108	−0.0232554	−	0.999730i	$-0.507403\pi$
$101$	−47.2102	−0.0465108	−0.0232554	+	0.999730i	$0.507403\pi$
$102$	0	0
$103$	147.154	0.140772	0.0703860	−	0.997520i	$-0.477577\pi$
$103$	147.154	0.140772	0.0703860	+	0.997520i	$0.477577\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1350.74	−1.22038	−0.610192	−	0.792254i	$-0.708908\pi$
$107$	−1350.74	−1.22038	−0.610192	+	0.792254i	$0.708908\pi$
$108$	0	0
$109$	272.793	0.239714	0.119857	−	0.992791i	$-0.461756\pi$
$109$	272.793	0.239714	0.119857	+	0.992791i	$0.461756\pi$
$110$	0	0
$111$	371.938	0.318043
$112$	0	0
$113$	632.450	0.526512	0.263256	−	0.964726i	$-0.415204\pi$
$113$	632.450	0.526512	0.263256	+	0.964726i	$0.415204\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	15.3932	0.0121633
$118$	0	0
$119$	2218.34	1.70886
$120$	0	0
$121$	2165.60	1.62705
$122$	0	0
$123$	−1752.60	−1.28477
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−916.288	−0.640216	−0.320108	−	0.947381i	$-0.603719\pi$
$127$	−916.288	−0.640216	−0.320108	+	0.947381i	$0.603719\pi$
$128$	0	0
$129$	393.363	0.268478
$130$	0	0
$131$	654.768	0.436698	0.218349	−	0.975871i	$-0.429933\pi$
$131$	654.768	0.436698	0.218349	+	0.975871i	$0.429933\pi$
$132$	0	0
$133$	−1385.35	−0.903194
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−568.184	−0.354330	−0.177165	−	0.984181i	$-0.556693\pi$
$137$	−568.184	−0.354330	−0.177165	+	0.984181i	$0.556693\pi$
$138$	0	0
$139$	2570.64	1.56862	0.784312	−	0.620367i	$-0.213016\pi$
$139$	2570.64	1.56862	0.784312	+	0.620367i	$0.213016\pi$
$140$	0	0
$141$	−1510.56	−0.902215
$142$	0	0
$143$	−373.424	−0.218373
$144$	0	0
$145$	0	0
$146$	0	0
$147$	291.260	0.163420
$148$	0	0
$149$	−3146.97	−1.73027	−0.865134	−	0.501541i	$-0.832767\pi$
$149$	−3146.97	−1.73027	−0.865134	+	0.501541i	$0.832767\pi$
$150$	0	0
$151$	−1560.39	−0.840946	−0.420473	−	0.907305i	$-0.638136\pi$
$151$	−1560.39	−0.840946	−0.420473	+	0.907305i	$0.638136\pi$
$152$	0	0
$153$	320.732	0.169475
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2777.66	−1.41198	−0.705992	−	0.708219i	$-0.749499\pi$
$157$	−2777.66	−1.41198	−0.705992	+	0.708219i	$0.749499\pi$
$158$	0	0
$159$	−1681.50	−0.838688
$160$	0	0
$161$	2688.19	1.31590
$162$	0	0
$163$	−543.340	−0.261090	−0.130545	−	0.991442i	$-0.541673\pi$
$163$	−543.340	−0.261090	−0.130545	+	0.991442i	$0.541673\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	482.073	0.223377	0.111688	−	0.993743i	$-0.464374\pi$
$167$	482.073	0.223377	0.111688	+	0.993743i	$0.464374\pi$
$168$	0	0
$169$	−2157.12	−0.981848
$170$	0	0
$171$	−200.296	−0.0895734
$172$	0	0
$173$	3520.67	1.54724	0.773618	−	0.633653i	$-0.218445\pi$
$173$	3520.67	1.54724	0.773618	+	0.633653i	$0.218445\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3899.41	1.65592
$178$	0	0
$179$	1517.69	0.633728	0.316864	−	0.948471i	$-0.397370\pi$
$179$	1517.69	0.633728	0.316864	+	0.948471i	$0.397370\pi$
$180$	0	0
$181$	3292.51	1.35210	0.676051	−	0.736855i	$-0.263690\pi$
$181$	3292.51	1.35210	0.676051	+	0.736855i	$0.263690\pi$
$182$	0	0
$183$	4135.40	1.67048
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7780.63	−3.04265
$188$	0	0
$189$	−2459.65	−0.946631
$190$	0	0
$191$	4207.60	1.59399	0.796993	−	0.603988i	$-0.206423\pi$
$191$	4207.60	1.59399	0.796993	+	0.603988i	$0.206423\pi$
$192$	0	0
$193$	1843.42	0.687524	0.343762	−	0.939057i	$-0.388299\pi$
$193$	1843.42	0.687524	0.343762	+	0.939057i	$0.388299\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3724.29	1.34693	0.673465	−	0.739220i	$-0.264805\pi$
$197$	3724.29	1.34693	0.673465	+	0.739220i	$0.264805\pi$
$198$	0	0
$199$	−324.095	−0.115450	−0.0577248	−	0.998333i	$-0.518385\pi$
$199$	−324.095	−0.115450	−0.0577248	+	0.998333i	$0.518385\pi$
$200$	0	0
$201$	3012.78	1.05724
$202$	0	0
$203$	843.706	0.291707
$204$	0	0
$205$	0	0
$206$	0	0
$207$	388.665	0.130503
$208$	0	0
$209$	4858.99	1.60815
$210$	0	0
$211$	3761.42	1.22724	0.613618	−	0.789603i	$-0.289713\pi$
$211$	3761.42	1.22724	0.613618	+	0.789603i	$0.289713\pi$
$212$	0	0
$213$	304.555	0.0979708
$214$	0	0
$215$	0	0
$216$	0	0
$217$	753.254	0.235641
$218$	0	0
$219$	−196.128	−0.0605165
$220$	0	0
$221$	830.941	0.252919
$222$	0	0
$223$	2769.86	0.831763	0.415882	−	0.909419i	$-0.363473\pi$
$223$	2769.86	0.831763	0.415882	+	0.909419i	$0.363473\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1766.29	−0.516445	−0.258222	−	0.966086i	$-0.583137\pi$
$227$	−1766.29	−0.516445	−0.258222	+	0.966086i	$0.583137\pi$
$228$	0	0
$229$	1076.11	0.310529	0.155265	−	0.987873i	$-0.450377\pi$
$229$	1076.11	0.310529	0.155265	+	0.987873i	$0.450377\pi$
$230$	0	0
$231$	4940.78	1.40727
$232$	0	0
$233$	3391.67	0.953629	0.476814	−	0.879004i	$-0.341791\pi$
$233$	3391.67	0.953629	0.476814	+	0.879004i	$0.341791\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2864.27	−0.785040
$238$	0	0
$239$	−4277.50	−1.15769	−0.578847	−	0.815436i	$-0.696497\pi$
$239$	−4277.50	−1.15769	−0.578847	+	0.815436i	$0.696497\pi$
$240$	0	0
$241$	2177.82	0.582098	0.291049	−	0.956708i	$-0.405996\pi$
$241$	2177.82	0.582098	0.291049	+	0.956708i	$0.405996\pi$
$242$	0	0
$243$	−681.796	−0.179989
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−518.921	−0.133677
$248$	0	0
$249$	264.492	0.0673152
$250$	0	0
$251$	3829.18	0.962931	0.481465	−	0.876465i	$-0.340105\pi$
$251$	3829.18	0.962931	0.481465	+	0.876465i	$0.340105\pi$
$252$	0	0
$253$	−9428.61	−2.34297
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1052.72	−0.255513	−0.127757	−	0.991806i	$-0.540778\pi$
$257$	−1052.72	−0.255513	−0.127757	+	0.991806i	$0.540778\pi$
$258$	0	0
$259$	1265.23	0.303544
$260$	0	0
$261$	121.985	0.0289298
$262$	0	0
$263$	−4044.38	−0.948240	−0.474120	−	0.880460i	$-0.657234\pi$
$263$	−4044.38	−0.948240	−0.474120	+	0.880460i	$0.657234\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7172.51	1.64401
$268$	0	0
$269$	−3642.86	−0.825684	−0.412842	−	0.910803i	$-0.635464\pi$
$269$	−3642.86	−0.825684	−0.412842	+	0.910803i	$0.635464\pi$
$270$	0	0
$271$	−2890.83	−0.647990	−0.323995	−	0.946059i	$-0.605026\pi$
$271$	−2890.83	−0.647990	−0.323995	+	0.946059i	$0.605026\pi$
$272$	0	0
$273$	−527.656	−0.116979
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4528.23	0.982220	0.491110	−	0.871098i	$-0.336591\pi$
$277$	4528.23	0.982220	0.491110	+	0.871098i	$0.336591\pi$
$278$	0	0
$279$	108.907	0.0233695
$280$	0	0
$281$	3025.28	0.642254	0.321127	−	0.947036i	$-0.395938\pi$
$281$	3025.28	0.642254	0.321127	+	0.947036i	$0.395938\pi$
$282$	0	0
$283$	−5704.24	−1.19817	−0.599085	−	0.800686i	$-0.704469\pi$
$283$	−5704.24	−1.19817	−0.599085	+	0.800686i	$0.704469\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5961.86	−1.22619
$288$	0	0
$289$	12400.4	2.52400
$290$	0	0
$291$	265.311	0.0534461
$292$	0	0
$293$	−6874.37	−1.37067	−0.685333	−	0.728230i	$-0.740343\pi$
$293$	−6874.37	−1.37067	−0.685333	+	0.728230i	$0.740343\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8627.02	1.68549
$298$	0	0
$299$	1006.94	0.194759
$300$	0	0
$301$	1338.12	0.256238
$302$	0	0
$303$	233.976	0.0443616
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4653.22	−0.865060	−0.432530	−	0.901620i	$-0.642379\pi$
$307$	−4653.22	−0.865060	−0.432530	+	0.901620i	$0.642379\pi$
$308$	0	0
$309$	−729.303	−0.134267
$310$	0	0
$311$	5194.35	0.947088	0.473544	−	0.880770i	$-0.342975\pi$
$311$	5194.35	0.947088	0.473544	+	0.880770i	$0.342975\pi$
$312$	0	0
$313$	4763.02	0.860134	0.430067	−	0.902797i	$-0.358490\pi$
$313$	4763.02	0.860134	0.430067	+	0.902797i	$0.358490\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6267.40	1.11045	0.555225	−	0.831700i	$-0.312632\pi$
$317$	6267.40	1.11045	0.555225	+	0.831700i	$0.312632\pi$
$318$	0	0
$319$	−2959.23	−0.519389
$320$	0	0
$321$	6694.35	1.16399
$322$	0	0
$323$	−10812.2	−1.86256
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1351.98	−0.228637
$328$	0	0
$329$	−5138.53	−0.861082
$330$	0	0
$331$	−9445.88	−1.56856	−0.784279	−	0.620409i	$-0.786967\pi$
$331$	−9445.88	−1.56856	−0.784279	+	0.620409i	$0.786967\pi$
$332$	0	0
$333$	182.930	0.0301036
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11024.7	1.78205	0.891027	−	0.453951i	$-0.149986\pi$
$337$	11024.7	1.78205	0.891027	+	0.453951i	$0.149986\pi$
$338$	0	0
$339$	−3134.45	−0.502183
$340$	0	0
$341$	−2641.97	−0.419563
$342$	0	0
$343$	6773.48	1.06628
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10088.3	1.56072	0.780359	−	0.625332i	$-0.215036\pi$
$347$	10088.3	1.56072	0.780359	+	0.625332i	$0.215036\pi$
$348$	0	0
$349$	7071.09	1.08455	0.542273	−	0.840202i	$-0.317564\pi$
$349$	7071.09	1.08455	0.542273	+	0.840202i	$0.317564\pi$
$350$	0	0
$351$	−921.332	−0.140106
$352$	0	0
$353$	10178.8	1.53474	0.767372	−	0.641203i	$-0.221564\pi$
$353$	10178.8	1.53474	0.767372	+	0.641203i	$0.221564\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−10994.2	−1.62990
$358$	0	0
$359$	−3407.04	−0.500883	−0.250441	−	0.968132i	$-0.580576\pi$
$359$	−3407.04	−0.500883	−0.250441	+	0.968132i	$0.580576\pi$
$360$	0	0
$361$	−106.806	−0.0155717
$362$	0	0
$363$	−10732.8	−1.55187
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6663.08	0.947711	0.473855	−	0.880603i	$-0.342862\pi$
$367$	6663.08	0.947711	0.473855	+	0.880603i	$0.342862\pi$
$368$	0	0
$369$	−861.979	−0.121607
$370$	0	0
$371$	−5720.00	−0.800452
$372$	0	0
$373$	6128.85	0.850777	0.425389	−	0.905011i	$-0.360137\pi$
$373$	6128.85	0.850777	0.425389	+	0.905011i	$0.360137\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	316.034	0.0431740
$378$	0	0
$379$	−11685.1	−1.58371	−0.791854	−	0.610711i	$-0.790884\pi$
$379$	−11685.1	−1.58371	−0.791854	+	0.610711i	$0.790884\pi$
$380$	0	0
$381$	4541.17	0.610634
$382$	0	0
$383$	−10245.3	−1.36686	−0.683432	−	0.730015i	$-0.739513\pi$
$383$	−10245.3	−1.36686	−0.683432	+	0.730015i	$0.739513\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	193.468	0.0254122
$388$	0	0
$389$	−4513.74	−0.588318	−0.294159	−	0.955756i	$-0.595040\pi$
$389$	−4513.74	−0.588318	−0.294159	+	0.955756i	$0.595040\pi$
$390$	0	0
$391$	20980.5	2.71363
$392$	0	0
$393$	−3245.07	−0.416519
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11765.6	1.48740	0.743698	−	0.668516i	$-0.233070\pi$
$397$	11765.6	1.48740	0.743698	+	0.668516i	$0.233070\pi$
$398$	0	0
$399$	6865.85	0.861460
$400$	0	0
$401$	6156.96	0.766743	0.383371	−	0.923594i	$-0.374763\pi$
$401$	6156.96	0.766743	0.383371	+	0.923594i	$0.374763\pi$
$402$	0	0
$403$	282.153	0.0348760
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4437.70	−0.540463
$408$	0	0
$409$	5731.19	0.692882	0.346441	−	0.938072i	$-0.387390\pi$
$409$	5731.19	0.692882	0.346441	+	0.938072i	$0.387390\pi$
$410$	0	0
$411$	2815.95	0.337958
$412$	0	0
$413$	13264.7	1.58043
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−12740.2	−1.49614
$418$	0	0
$419$	1566.31	0.182624	0.0913120	−	0.995822i	$-0.470894\pi$
$419$	1566.31	0.182624	0.0913120	+	0.995822i	$0.470894\pi$
$420$	0	0
$421$	15872.2	1.83744	0.918721	−	0.394907i	$-0.129223\pi$
$421$	15872.2	1.83744	0.918721	+	0.394907i	$0.129223\pi$
$422$	0	0
$423$	−742.939	−0.0853970
$424$	0	0
$425$	0	0
$426$	0	0
$427$	14067.5	1.59432
$428$	0	0
$429$	1850.71	0.208282
$430$	0	0
$431$	12464.3	1.39300	0.696501	−	0.717556i	$-0.254739\pi$
$431$	12464.3	1.39300	0.696501	+	0.717556i	$0.254739\pi$
$432$	0	0
$433$	14661.0	1.62717	0.813583	−	0.581448i	$-0.197514\pi$
$433$	14661.0	1.62717	0.813583	+	0.581448i	$0.197514\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13102.3	−1.43425
$438$	0	0
$439$	3884.90	0.422360	0.211180	−	0.977447i	$-0.432269\pi$
$439$	3884.90	0.422360	0.211180	+	0.977447i	$0.432269\pi$
$440$	0	0
$441$	143.250	0.0154681
$442$	0	0
$443$	4840.23	0.519111	0.259556	−	0.965728i	$-0.416424\pi$
$443$	4840.23	0.519111	0.259556	+	0.965728i	$0.416424\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	15596.6	1.65032
$448$	0	0
$449$	9780.24	1.02797	0.513985	−	0.857799i	$-0.328169\pi$
$449$	9780.24	1.02797	0.513985	+	0.857799i	$0.328169\pi$
$450$	0	0
$451$	20910.7	2.18325
$452$	0	0
$453$	7733.38	0.802088
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3537.61	−0.362106	−0.181053	−	0.983473i	$-0.557950\pi$
$457$	−3537.61	−0.362106	−0.181053	+	0.983473i	$0.557950\pi$
$458$	0	0
$459$	−19196.8	−1.95213
$460$	0	0
$461$	−11295.6	−1.14119	−0.570596	−	0.821231i	$-0.693288\pi$
$461$	−11295.6	−1.14119	−0.570596	+	0.821231i	$0.693288\pi$
$462$	0	0
$463$	1997.51	0.200502	0.100251	−	0.994962i	$-0.468035\pi$
$463$	1997.51	0.200502	0.100251	+	0.994962i	$0.468035\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11446.6	−1.13423	−0.567115	−	0.823639i	$-0.691941\pi$
$467$	−11446.6	−1.13423	−0.567115	+	0.823639i	$0.691941\pi$
$468$	0	0
$469$	10248.7	1.00904
$470$	0	0
$471$	13766.2	1.34674
$472$	0	0
$473$	−4693.33	−0.456236
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−827.010	−0.0793841
$478$	0	0
$479$	−4392.43	−0.418988	−0.209494	−	0.977810i	$-0.567182\pi$
$479$	−4392.43	−0.418988	−0.209494	+	0.977810i	$0.567182\pi$
$480$	0	0
$481$	473.930	0.0449258
$482$	0	0
$483$	−13322.8	−1.25509
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7498.81	−0.697749	−0.348874	−	0.937169i	$-0.613436\pi$
$487$	−7498.81	−0.697749	−0.348874	+	0.937169i	$0.613436\pi$
$488$	0	0
$489$	2692.82	0.249026
$490$	0	0
$491$	430.358	0.0395556	0.0197778	−	0.999804i	$-0.493704\pi$
$491$	430.358	0.0395556	0.0197778	+	0.999804i	$0.493704\pi$
$492$	0	0
$493$	6584.86	0.601556
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1036.01	0.0935043
$498$	0	0
$499$	−1063.14	−0.0953765	−0.0476883	−	0.998862i	$-0.515185\pi$
$499$	−1063.14	−0.0953765	−0.0476883	+	0.998862i	$0.515185\pi$
$500$	0	0
$501$	−2389.18	−0.213055
$502$	0	0
$503$	2735.01	0.242441	0.121221	−	0.992626i	$-0.461319\pi$
$503$	2735.01	0.242441	0.121221	+	0.992626i	$0.461319\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	10690.8	0.936480
$508$	0	0
$509$	9333.33	0.812756	0.406378	−	0.913705i	$-0.366792\pi$
$509$	9333.33	0.812756	0.406378	+	0.913705i	$0.366792\pi$
$510$	0	0
$511$	−667.176	−0.0577576
$512$	0	0
$513$	11988.4	1.03177
$514$	0	0
$515$	0	0
$516$	0	0
$517$	18022.9	1.53317
$518$	0	0
$519$	−17448.6	−1.47574
$520$	0	0
$521$	21936.7	1.84465	0.922325	−	0.386415i	$-0.126287\pi$
$521$	21936.7	1.84465	0.922325	+	0.386415i	$0.126287\pi$
$522$	0	0
$523$	10089.8	0.843588	0.421794	−	0.906692i	$-0.361401\pi$
$523$	10089.8	0.843588	0.421794	+	0.906692i	$0.361401\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5878.91	0.485938
$528$	0	0
$529$	13257.3	1.08961
$530$	0	0
$531$	1917.85	0.156737
$532$	0	0
$533$	−2233.19	−0.181482
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7521.74	−0.604445
$538$	0	0
$539$	−3475.11	−0.277706
$540$	0	0
$541$	−4929.86	−0.391777	−0.195889	−	0.980626i	$-0.562759\pi$
$541$	−4929.86	−0.391777	−0.195889	+	0.980626i	$0.562759\pi$
$542$	0	0
$543$	−16317.9	−1.28963
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16651.7	1.30160	0.650801	−	0.759248i	$-0.274433\pi$
$547$	16651.7	1.30160	0.650801	+	0.759248i	$0.274433\pi$
$548$	0	0
$549$	2033.91	0.158115
$550$	0	0
$551$	−4112.23	−0.317944
$552$	0	0
$553$	−9743.49	−0.749250
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13982.7	−1.06367	−0.531836	−	0.846847i	$-0.678498\pi$
$557$	−13982.7	−1.06367	−0.531836	+	0.846847i	$0.678498\pi$
$558$	0	0
$559$	501.230	0.0379244
$560$	0	0
$561$	38561.2	2.90206
$562$	0	0
$563$	−9916.61	−0.742336	−0.371168	−	0.928566i	$-0.621043\pi$
$563$	−9916.61	−0.742336	−0.371168	+	0.928566i	$0.621043\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	11080.6	0.820708
$568$	0	0
$569$	8291.19	0.610870	0.305435	−	0.952213i	$-0.401198\pi$
$569$	8291.19	0.610870	0.305435	+	0.952213i	$0.401198\pi$
$570$	0	0
$571$	5903.95	0.432702	0.216351	−	0.976316i	$-0.430585\pi$
$571$	5903.95	0.432702	0.216351	+	0.976316i	$0.430585\pi$
$572$	0	0
$573$	−20853.1	−1.52033
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6014.90	0.433975	0.216987	−	0.976174i	$-0.430377\pi$
$577$	6014.90	0.433975	0.216987	+	0.976174i	$0.430377\pi$
$578$	0	0
$579$	−9136.07	−0.655755
$580$	0	0
$581$	899.729	0.0642462
$582$	0	0
$583$	20062.4	1.42522
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8567.35	0.602406	0.301203	−	0.953560i	$-0.402612\pi$
$587$	8567.35	0.602406	0.301203	+	0.953560i	$0.402612\pi$
$588$	0	0
$589$	−3671.37	−0.256835
$590$	0	0
$591$	−18457.8	−1.28469
$592$	0	0
$593$	19704.6	1.36453	0.682267	−	0.731103i	$-0.260994\pi$
$593$	19704.6	1.36453	0.682267	+	0.731103i	$0.260994\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1606.23	0.110115
$598$	0	0
$599$	−2540.05	−0.173262	−0.0866308	−	0.996240i	$-0.527610\pi$
$599$	−2540.05	−0.173262	−0.0866308	+	0.996240i	$0.527610\pi$
$600$	0	0
$601$	−18593.4	−1.26197	−0.630983	−	0.775797i	$-0.717348\pi$
$601$	−18593.4	−1.26197	−0.630983	+	0.775797i	$0.717348\pi$
$602$	0	0
$603$	1481.78	0.100071
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−22878.6	−1.52984	−0.764922	−	0.644123i	$-0.777222\pi$
$607$	−22878.6	−1.52984	−0.764922	+	0.644123i	$0.777222\pi$
$608$	0	0
$609$	−4181.45	−0.278228
$610$	0	0
$611$	−1924.78	−0.127444
$612$	0	0
$613$	−14458.6	−0.952654	−0.476327	−	0.879268i	$-0.658032\pi$
$613$	−14458.6	−0.952654	−0.476327	+	0.879268i	$0.658032\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19540.3	−1.27498	−0.637489	−	0.770459i	$-0.720027\pi$
$617$	−19540.3	−1.27498	−0.637489	+	0.770459i	$0.720027\pi$
$618$	0	0
$619$	18567.0	1.20561	0.602804	−	0.797889i	$-0.294050\pi$
$619$	18567.0	1.20561	0.602804	+	0.797889i	$0.294050\pi$
$620$	0	0
$621$	−23262.8	−1.50323
$622$	0	0
$623$	24399.0	1.56906
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−24081.4	−1.53384
$628$	0	0
$629$	9874.75	0.625965
$630$	0	0
$631$	−6274.48	−0.395853	−0.197927	−	0.980217i	$-0.563421\pi$
$631$	−6274.48	−0.395853	−0.197927	+	0.980217i	$0.563421\pi$
$632$	0	0
$633$	−18641.8	−1.17053
$634$	0	0
$635$	0	0
$636$	0	0
$637$	371.128	0.0230842
$638$	0	0
$639$	149.789	0.00927319
$640$	0	0
$641$	6631.88	0.408648	0.204324	−	0.978903i	$-0.434500\pi$
$641$	6631.88	0.408648	0.204324	+	0.978903i	$0.434500\pi$
$642$	0	0
$643$	−12509.7	−0.767240	−0.383620	−	0.923491i	$-0.625323\pi$
$643$	−12509.7	−0.767240	−0.383620	+	0.923491i	$0.625323\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25315.1	−1.53824	−0.769120	−	0.639105i	$-0.779305\pi$
$647$	−25315.1	−1.53824	−0.769120	+	0.639105i	$0.779305\pi$
$648$	0	0
$649$	−46525.0	−2.81397
$650$	0	0
$651$	−3733.17	−0.224753
$652$	0	0
$653$	−907.343	−0.0543753	−0.0271876	−	0.999630i	$-0.508655\pi$
$653$	−907.343	−0.0543753	−0.0271876	+	0.999630i	$0.508655\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−96.4617	−0.00572805
$658$	0	0
$659$	11641.2	0.688130	0.344065	−	0.938946i	$-0.388196\pi$
$659$	11641.2	0.688130	0.344065	+	0.938946i	$0.388196\pi$
$660$	0	0
$661$	−19804.3	−1.16535	−0.582677	−	0.812704i	$-0.697995\pi$
$661$	−19804.3	−1.16535	−0.582677	+	0.812704i	$0.697995\pi$
$662$	0	0
$663$	−4118.19	−0.241233
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7979.58	0.463224
$668$	0	0
$669$	−13727.6	−0.793330
$670$	0	0
$671$	−49340.6	−2.83871
$672$	0	0
$673$	3948.28	0.226144	0.113072	−	0.993587i	$-0.463931\pi$
$673$	3948.28	0.226144	0.113072	+	0.993587i	$0.463931\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25081.6	−1.42388	−0.711939	−	0.702241i	$-0.752183\pi$
$677$	−25081.6	−1.42388	−0.711939	+	0.702241i	$0.752183\pi$
$678$	0	0
$679$	902.517	0.0510095
$680$	0	0
$681$	8753.84	0.492581
$682$	0	0
$683$	7372.10	0.413010	0.206505	−	0.978446i	$-0.433791\pi$
$683$	7372.10	0.413010	0.206505	+	0.978446i	$0.433791\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−5333.25	−0.296181
$688$	0	0
$689$	−2142.59	−0.118471
$690$	0	0
$691$	30127.0	1.65859	0.829294	−	0.558812i	$-0.188743\pi$
$691$	30127.0	1.65859	0.829294	+	0.558812i	$0.188743\pi$
$692$	0	0
$693$	2430.02	0.133202
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−46530.5	−2.52865
$698$	0	0
$699$	−16809.3	−0.909564
$700$	0	0
$701$	275.799	0.0148599	0.00742994	−	0.999972i	$-0.497635\pi$
$701$	275.799	0.0148599	0.00742994	+	0.999972i	$0.497635\pi$
$702$	0	0
$703$	−6166.76	−0.330845
$704$	0	0
$705$	0	0
$706$	0	0
$707$	795.924	0.0423392
$708$	0	0
$709$	−4467.14	−0.236625	−0.118313	−	0.992976i	$-0.537748\pi$
$709$	−4467.14	−0.236625	−0.118313	+	0.992976i	$0.537748\pi$
$710$	0	0
$711$	−1408.73	−0.0743061
$712$	0	0
$713$	7124.10	0.374193
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21199.5	1.10420
$718$	0	0
$719$	4761.45	0.246971	0.123486	−	0.992346i	$-0.460593\pi$
$719$	4761.45	0.246971	0.123486	+	0.992346i	$0.460593\pi$
$720$	0	0
$721$	−2480.89	−0.128146
$722$	0	0
$723$	−10793.4	−0.555201
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19899.2	1.01516	0.507578	−	0.861606i	$-0.330541\pi$
$727$	19899.2	1.01516	0.507578	+	0.861606i	$0.330541\pi$
$728$	0	0
$729$	21124.6	1.07324
$730$	0	0
$731$	10443.6	0.528413
$732$	0	0
$733$	23020.4	1.15999	0.579997	−	0.814618i	$-0.303054\pi$
$733$	23020.4	1.15999	0.579997	+	0.814618i	$0.303054\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−35946.4	−1.79661
$738$	0	0
$739$	−4820.32	−0.239944	−0.119972	−	0.992777i	$-0.538280\pi$
$739$	−4820.32	−0.239944	−0.119972	+	0.992777i	$0.538280\pi$
$740$	0	0
$741$	2571.80	0.127500
$742$	0	0
$743$	−11267.6	−0.556348	−0.278174	−	0.960531i	$-0.589729\pi$
$743$	−11267.6	−0.556348	−0.278174	+	0.960531i	$0.589729\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	130.085	0.00637156
$748$	0	0
$749$	22772.4	1.11093
$750$	0	0
$751$	21124.7	1.02643	0.513216	−	0.858259i	$-0.328454\pi$
$751$	21124.7	1.02643	0.513216	+	0.858259i	$0.328454\pi$
$752$	0	0
$753$	−18977.6	−0.918437
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25198.7	1.20986	0.604929	−	0.796279i	$-0.293201\pi$
$757$	25198.7	1.20986	0.604929	+	0.796279i	$0.293201\pi$
$758$	0	0
$759$	46728.7	2.23471
$760$	0	0
$761$	9269.83	0.441565	0.220783	−	0.975323i	$-0.429139\pi$
$761$	9269.83	0.441565	0.220783	+	0.975323i	$0.429139\pi$
$762$	0	0
$763$	−4599.06	−0.218214
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4968.69	0.233910
$768$	0	0
$769$	−7084.96	−0.332237	−0.166119	−	0.986106i	$-0.553123\pi$
$769$	−7084.96	−0.332237	−0.166119	+	0.986106i	$0.553123\pi$
$770$	0	0
$771$	5217.34	0.243707
$772$	0	0
$773$	13432.8	0.625024	0.312512	−	0.949914i	$-0.398830\pi$
$773$	13432.8	0.625024	0.312512	+	0.949914i	$0.398830\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6270.57	−0.289518
$778$	0	0
$779$	29058.2	1.33648
$780$	0	0
$781$	−3633.73	−0.166486
$782$	0	0
$783$	−7301.17	−0.333235
$784$	0	0
$785$	0	0
$786$	0	0
$787$	12623.8	0.571780	0.285890	−	0.958263i	$-0.407711\pi$
$787$	12623.8	0.571780	0.285890	+	0.958263i	$0.407711\pi$
$788$	0	0
$789$	20044.2	0.904424
$790$	0	0
$791$	−10662.6	−0.479289
$792$	0	0
$793$	5269.39	0.235967
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20294.5	0.901967	0.450983	−	0.892532i	$-0.351073\pi$
$797$	20294.5	0.901967	0.450983	+	0.892532i	$0.351073\pi$
$798$	0	0
$799$	−40104.6	−1.77572
$800$	0	0
$801$	3527.65	0.155610
$802$	0	0
$803$	2340.06	0.102838
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18054.2	0.787531
$808$	0	0
$809$	−42388.8	−1.84217	−0.921083	−	0.389366i	$-0.872694\pi$
$809$	−42388.8	−1.84217	−0.921083	+	0.389366i	$0.872694\pi$
$810$	0	0
$811$	−23265.0	−1.00733	−0.503666	−	0.863898i	$-0.668016\pi$
$811$	−23265.0	−1.00733	−0.503666	+	0.863898i	$0.668016\pi$
$812$	0	0
$813$	14327.1	0.618048
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6522.00	−0.279285
$818$	0	0
$819$	−259.517	−0.0110723
$820$	0	0
$821$	−20775.3	−0.883145	−0.441573	−	0.897225i	$-0.645579\pi$
$821$	−20775.3	−0.883145	−0.441573	+	0.897225i	$0.645579\pi$
$822$	0	0
$823$	−2474.84	−0.104821	−0.0524103	−	0.998626i	$-0.516690\pi$
$823$	−2474.84	−0.104821	−0.0524103	+	0.998626i	$0.516690\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10643.9	−0.447552	−0.223776	−	0.974641i	$-0.571838\pi$
$827$	−10643.9	−0.447552	−0.223776	+	0.974641i	$0.571838\pi$
$828$	0	0
$829$	−22637.2	−0.948400	−0.474200	−	0.880417i	$-0.657263\pi$
$829$	−22637.2	−0.948400	−0.474200	+	0.880417i	$0.657263\pi$
$830$	0	0
$831$	−22442.1	−0.936834
$832$	0	0
$833$	7732.80	0.321639
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6518.42	−0.269187
$838$	0	0
$839$	−33780.2	−1.39001	−0.695007	−	0.719003i	$-0.744599\pi$
$839$	−33780.2	−1.39001	−0.695007	+	0.719003i	$0.744599\pi$
$840$	0	0
$841$	−21884.6	−0.897313
$842$	0	0
$843$	−14993.5	−0.612577
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−36510.3	−1.48112
$848$	0	0
$849$	28270.5	1.14281
$850$	0	0
$851$	11966.3	0.482020
$852$	0	0
$853$	−37542.1	−1.50694	−0.753468	−	0.657485i	$-0.771620\pi$
$853$	−37542.1	−1.50694	−0.753468	+	0.657485i	$0.771620\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24120.1	0.961408	0.480704	−	0.876883i	$-0.340381\pi$
$857$	24120.1	0.961408	0.480704	+	0.876883i	$0.340381\pi$
$858$	0	0
$859$	−6703.01	−0.266244	−0.133122	−	0.991100i	$-0.542500\pi$
$859$	−6703.01	−0.266244	−0.133122	+	0.991100i	$0.542500\pi$
$860$	0	0
$861$	29547.3	1.16954
$862$	0	0
$863$	−31235.7	−1.23207	−0.616035	−	0.787719i	$-0.711262\pi$
$863$	−31235.7	−1.23207	−0.616035	+	0.787719i	$0.711262\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−61457.1	−2.40737
$868$	0	0
$869$	34174.5	1.33405
$870$	0	0
$871$	3838.93	0.149343
$872$	0	0
$873$	130.488	0.00505881
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24268.0	0.934403	0.467201	−	0.884151i	$-0.345262\pi$
$877$	24268.0	0.934403	0.467201	+	0.884151i	$0.345262\pi$
$878$	0	0
$879$	34069.8	1.30733
$880$	0	0
$881$	3250.81	0.124316	0.0621582	−	0.998066i	$-0.480202\pi$
$881$	3250.81	0.124316	0.0621582	+	0.998066i	$0.480202\pi$
$882$	0	0
$883$	12164.4	0.463607	0.231803	−	0.972763i	$-0.425537\pi$
$883$	12164.4	0.463607	0.231803	+	0.972763i	$0.425537\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12983.4	0.491476	0.245738	−	0.969336i	$-0.420970\pi$
$887$	12983.4	0.491476	0.245738	+	0.969336i	$0.420970\pi$
$888$	0	0
$889$	15447.9	0.582795
$890$	0	0
$891$	−38864.3	−1.46128
$892$	0	0
$893$	25045.2	0.938529
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4990.45	−0.185759
$898$	0	0
$899$	2235.94	0.0829509
$900$	0	0
$901$	−44642.8	−1.65069
$902$	0	0
$903$	−6631.78	−0.244398
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−34569.1	−1.26554	−0.632771	−	0.774339i	$-0.718083\pi$
$907$	−34569.1	−1.26554	−0.632771	+	0.774339i	$0.718083\pi$
$908$	0	0
$909$	115.076	0.00419895
$910$	0	0
$911$	9403.56	0.341991	0.170995	−	0.985272i	$-0.445302\pi$
$911$	9403.56	0.341991	0.170995	+	0.985272i	$0.445302\pi$
$912$	0	0
$913$	−3155.72	−0.114391
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11038.8	−0.397530
$918$	0	0
$919$	−40220.5	−1.44369	−0.721845	−	0.692055i	$-0.756705\pi$
$919$	−40220.5	−1.44369	−0.721845	+	0.692055i	$0.756705\pi$
$920$	0	0
$921$	23061.6	0.825088
$922$	0	0
$923$	388.069	0.0138391
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−358.693	−0.0127088
$928$	0	0
$929$	13218.6	0.466834	0.233417	−	0.972377i	$-0.425009\pi$
$929$	13218.6	0.466834	0.233417	+	0.972377i	$0.425009\pi$
$930$	0	0
$931$	−4829.12	−0.169998
$932$	0	0
$933$	−25743.5	−0.903326
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6749.65	0.235327	0.117664	−	0.993054i	$-0.462460\pi$
$937$	6749.65	0.235327	0.117664	+	0.993054i	$0.462460\pi$
$938$	0	0
$939$	−23605.8	−0.820390
$940$	0	0
$941$	24163.3	0.837089	0.418544	−	0.908196i	$-0.362540\pi$
$941$	24163.3	0.837089	0.418544	+	0.908196i	$0.362540\pi$
$942$	0	0
$943$	−56385.9	−1.94717
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2701.58	0.0927028	0.0463514	−	0.998925i	$-0.485241\pi$
$947$	2701.58	0.0927028	0.0463514	+	0.998925i	$0.485241\pi$
$948$	0	0
$949$	−249.910	−0.00854838
$950$	0	0
$951$	−31061.6	−1.05914
$952$	0	0
$953$	35584.4	1.20954	0.604770	−	0.796400i	$-0.293265\pi$
$953$	35584.4	1.20954	0.604770	+	0.796400i	$0.293265\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	14666.1	0.495389
$958$	0	0
$959$	9579.11	0.322550
$960$	0	0
$961$	−27794.8	−0.932992
$962$	0	0
$963$	3292.48	0.110175
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5885.66	−0.195729	−0.0978645	−	0.995200i	$-0.531201\pi$
$967$	−5885.66	−0.195729	−0.0978645	+	0.995200i	$0.531201\pi$
$968$	0	0
$969$	53585.8	1.77650
$970$	0	0
$971$	41347.7	1.36654	0.683271	−	0.730165i	$-0.260557\pi$
$971$	41347.7	1.36654	0.683271	+	0.730165i	$0.260557\pi$
$972$	0	0
$973$	−43338.8	−1.42793
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27595.1	−0.903630	−0.451815	−	0.892112i	$-0.649223\pi$
$977$	−27595.1	−0.903630	−0.451815	+	0.892112i	$0.649223\pi$
$978$	0	0
$979$	−85577.3	−2.79373
$980$	0	0
$981$	−664.941	−0.0216411
$982$	0	0
$983$	−56743.0	−1.84112	−0.920559	−	0.390603i	$-0.872267\pi$
$983$	−56743.0	−1.84112	−0.920559	+	0.390603i	$0.872267\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	25466.8	0.821294
$988$	0	0
$989$	12655.6	0.406901
$990$	0	0
$991$	17553.2	0.562659	0.281329	−	0.959611i	$-0.409225\pi$
$991$	17553.2	0.562659	0.281329	+	0.959611i	$0.409225\pi$
$992$	0	0
$993$	46814.3	1.49608
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−21257.8	−0.675269	−0.337634	−	0.941277i	$-0.609627\pi$
$997$	−21257.8	−0.675269	−0.337634	+	0.941277i	$0.609627\pi$
$998$	0	0
$999$	−10948.9	−0.346756

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2500.4.a.g.1.9		32
5.4	even	2	inner	2500.4.a.g.1.24		32
25.3	odd	20		500.4.i.a.49.6		32
25.4	even	10		500.4.g.b.201.12		64
25.6	even	5		500.4.g.b.301.5		64
25.8	odd	20		100.4.i.a.89.3	yes	32
25.17	odd	20		500.4.i.a.449.6		32
25.19	even	10		500.4.g.b.301.12		64
25.21	even	5		500.4.g.b.201.5		64
25.22	odd	20		100.4.i.a.9.3	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.i.a.9.3	✓	32	25.22	odd	20
100.4.i.a.89.3	yes	32	25.8	odd	20
500.4.g.b.201.5		64	25.21	even	5
500.4.g.b.201.12		64	25.4	even	10
500.4.g.b.301.5		64	25.6	even	5
500.4.g.b.301.12		64	25.19	even	10
500.4.i.a.49.6		32	25.3	odd	20
500.4.i.a.449.6		32	25.17	odd	20
2500.4.a.g.1.9		32	1.1	even	1	trivial
2500.4.a.g.1.24		32	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:	\( N \)	\(=\)	\( 2500 = 2^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2500.a (trivial)

\(f(q)\)	\(=\)	\(q-4.95605 q^{3} -16.8592 q^{7} -2.43753 q^{9} +O(q^{10})\)	\(q-4.95605 q^{3} -16.8592 q^{7} -2.43753 q^{9} +59.1321 q^{11} -6.31508 q^{13} -131.580 q^{17} +82.1717 q^{19} +83.5549 q^{21} -159.450 q^{23} +145.894 q^{27} -50.0444 q^{29} -44.6792 q^{31} -293.062 q^{33} -75.0473 q^{37} +31.2979 q^{39} +353.628 q^{41} -79.3703 q^{43} +304.791 q^{47} -58.7686 q^{49} +652.120 q^{51} +339.282 q^{53} -407.248 q^{57} -786.798 q^{59} -834.413 q^{61} +41.0948 q^{63} -607.900 q^{67} +790.243 q^{69} -61.4511 q^{71} +39.5735 q^{73} -996.917 q^{77} +577.934 q^{79} -657.245 q^{81} -53.3674 q^{83} +248.023 q^{87} -1447.22 q^{89} +106.467 q^{91} +221.432 q^{93} -53.5327 q^{97} -144.136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 328 q^{9}+O(q^{10}) \) 32 * q + 328 * q^9	\( 32 q + 328 q^{9} + 60 q^{11} + 128 q^{19} + 312 q^{21} + 848 q^{29} + 312 q^{31} + 628 q^{39} + 1352 q^{41} + 2796 q^{49} + 1664 q^{51} + 1616 q^{59} + 1204 q^{61} + 1644 q^{69} - 944 q^{71} + 184 q^{79} + 5048 q^{81} + 2392 q^{89} + 1588 q^{91} + 2540 q^{99}+O(q^{100}) \) 32 * q + 328 * q^9 + 60 * q^11 + 128 * q^19 + 312 * q^21 + 848 * q^29 + 312 * q^31 + 628 * q^39 + 1352 * q^41 + 2796 * q^49 + 1664 * q^51 + 1616 * q^59 + 1204 * q^61 + 1644 * q^69 - 944 * q^71 + 184 * q^79 + 5048 * q^81 + 2392 * q^89 + 1588 * q^91 + 2540 * q^99

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