LMFDB - Embedded newform 7600.2.a.bw.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.24698 q^{3} +1.35690 q^{7} +2.04892 q^{9} +O(q^{10})$	$q+2.24698 q^{3} +1.35690 q^{7} +2.04892 q^{9} -4.85086 q^{11} -0.198062 q^{13} +1.13706 q^{17} +1.00000 q^{19} +3.04892 q^{21} +2.55496 q^{23} -2.13706 q^{27} -10.2349 q^{29} -2.51573 q^{31} -10.8998 q^{33} +0.137063 q^{37} -0.445042 q^{39} -11.7506 q^{41} -7.59179 q^{43} +2.69202 q^{47} -5.15883 q^{49} +2.55496 q^{51} -12.8780 q^{53} +2.24698 q^{57} -5.82371 q^{59} -7.58211 q^{61} +2.78017 q^{63} +8.01507 q^{67} +5.74094 q^{69} +8.82371 q^{71} +11.9705 q^{73} -6.58211 q^{77} -10.7409 q^{79} -10.9487 q^{81} +3.77479 q^{83} -22.9976 q^{87} +9.36658 q^{89} -0.268750 q^{91} -5.65279 q^{93} -0.198062 q^{97} -9.93900 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 - 3 * q^9	$ 3 q + 2 q^{3} - 3 q^{9} - q^{11} - 5 q^{13} - 2 q^{17} + 3 q^{19} + 8 q^{23} - q^{27} - 7 q^{29} + 5 q^{31} - 10 q^{33} - 5 q^{37} - q^{39} + q^{41} + 5 q^{43} + 3 q^{47} - 7 q^{49} + 8 q^{51} - 19 q^{53} + 2 q^{57} - 10 q^{59} - 17 q^{61} + 7 q^{63} - q^{67} + 3 q^{69} + 19 q^{71} + q^{73} - 14 q^{77} - 18 q^{79} - q^{81} + 13 q^{83} - 28 q^{87} + 2 q^{89} + 7 q^{91} + q^{93} - 5 q^{97} - 20 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 3 * q^9 - q^11 - 5 * q^13 - 2 * q^17 + 3 * q^19 + 8 * q^23 - q^27 - 7 * q^29 + 5 * q^31 - 10 * q^33 - 5 * q^37 - q^39 + q^41 + 5 * q^43 + 3 * q^47 - 7 * q^49 + 8 * q^51 - 19 * q^53 + 2 * q^57 - 10 * q^59 - 17 * q^61 + 7 * q^63 - q^67 + 3 * q^69 + 19 * q^71 + q^73 - 14 * q^77 - 18 * q^79 - q^81 + 13 * q^83 - 28 * q^87 + 2 * q^89 + 7 * q^91 + q^93 - 5 * q^97 - 20 * 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
$3$	2.24698	1.29729	0.648647	−	0.761089i	$-0.275335\pi$
$3$	2.24698	1.29729	0.648647	+	0.761089i	$0.275335\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.35690	0.512858	0.256429	−	0.966563i	$-0.417454\pi$
$7$	1.35690	0.512858	0.256429	+	0.966563i	$0.417454\pi$
$8$	0	0
$9$	2.04892	0.682972
$10$	0	0
$11$	−4.85086	−1.46259	−0.731294	−	0.682062i	$-0.761083\pi$
$11$	−4.85086	−1.46259	−0.731294	+	0.682062i	$0.761083\pi$
$12$	0	0
$13$	−0.198062	−0.0549326	−0.0274663	−	0.999623i	$-0.508744\pi$
$13$	−0.198062	−0.0549326	−0.0274663	+	0.999623i	$0.508744\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.13706	0.275778	0.137889	−	0.990448i	$-0.455968\pi$
$17$	1.13706	0.275778	0.137889	+	0.990448i	$0.455968\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.04892	0.665328
$22$	0	0
$23$	2.55496	0.532746	0.266373	−	0.963870i	$-0.414175\pi$
$23$	2.55496	0.532746	0.266373	+	0.963870i	$0.414175\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.13706	−0.411278
$28$	0	0
$29$	−10.2349	−1.90057	−0.950286	−	0.311377i	$-0.899210\pi$
$29$	−10.2349	−1.90057	−0.950286	+	0.311377i	$0.899210\pi$
$30$	0	0
$31$	−2.51573	−0.451838	−0.225919	−	0.974146i	$-0.572539\pi$
$31$	−2.51573	−0.451838	−0.225919	+	0.974146i	$0.572539\pi$
$32$	0	0
$33$	−10.8998	−1.89741
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.137063	0.0225331	0.0112665	−	0.999937i	$-0.496414\pi$
$37$	0.137063	0.0225331	0.0112665	+	0.999937i	$0.496414\pi$
$38$	0	0
$39$	−0.445042	−0.0712637
$40$	0	0
$41$	−11.7506	−1.83514	−0.917570	−	0.397575i	$-0.869852\pi$
$41$	−11.7506	−1.83514	−0.917570	+	0.397575i	$0.869852\pi$
$42$	0	0
$43$	−7.59179	−1.15774	−0.578869	−	0.815421i	$-0.696506\pi$
$43$	−7.59179	−1.15774	−0.578869	+	0.815421i	$0.696506\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.69202	0.392672	0.196336	−	0.980537i	$-0.437096\pi$
$47$	2.69202	0.392672	0.196336	+	0.980537i	$0.437096\pi$
$48$	0	0
$49$	−5.15883	−0.736976
$50$	0	0
$51$	2.55496	0.357766
$52$	0	0
$53$	−12.8780	−1.76893	−0.884465	−	0.466607i	$-0.845476\pi$
$53$	−12.8780	−1.76893	−0.884465	+	0.466607i	$0.845476\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.24698	0.297620
$58$	0	0
$59$	−5.82371	−0.758182	−0.379091	−	0.925359i	$-0.623763\pi$
$59$	−5.82371	−0.758182	−0.379091	+	0.925359i	$0.623763\pi$
$60$	0	0
$61$	−7.58211	−0.970789	−0.485395	−	0.874295i	$-0.661324\pi$
$61$	−7.58211	−0.970789	−0.485395	+	0.874295i	$0.661324\pi$
$62$	0	0
$63$	2.78017	0.350268
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.01507	0.979196	0.489598	−	0.871948i	$-0.337144\pi$
$67$	8.01507	0.979196	0.489598	+	0.871948i	$0.337144\pi$
$68$	0	0
$69$	5.74094	0.691128
$70$	0	0
$71$	8.82371	1.04718	0.523591	−	0.851970i	$-0.324592\pi$
$71$	8.82371	1.04718	0.523591	+	0.851970i	$0.324592\pi$
$72$	0	0
$73$	11.9705	1.40104	0.700518	−	0.713635i	$-0.252952\pi$
$73$	11.9705	1.40104	0.700518	+	0.713635i	$0.252952\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.58211	−0.750101
$78$	0	0
$79$	−10.7409	−1.20845	−0.604225	−	0.796814i	$-0.706517\pi$
$79$	−10.7409	−1.20845	−0.604225	+	0.796814i	$0.706517\pi$
$80$	0	0
$81$	−10.9487	−1.21652
$82$	0	0
$83$	3.77479	0.414337	0.207169	−	0.978305i	$-0.433575\pi$
$83$	3.77479	0.414337	0.207169	+	0.978305i	$0.433575\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−22.9976	−2.46560
$88$	0	0
$89$	9.36658	0.992856	0.496428	−	0.868078i	$-0.334645\pi$
$89$	9.36658	0.992856	0.496428	+	0.868078i	$0.334645\pi$
$90$	0	0
$91$	−0.268750	−0.0281726
$92$	0	0
$93$	−5.65279	−0.586167
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.198062	−0.0201102	−0.0100551	−	0.999949i	$-0.503201\pi$
$97$	−0.198062	−0.0201102	−0.0100551	+	0.999949i	$0.503201\pi$
$98$	0	0
$99$	−9.93900	−0.998907
$100$	0	0
$101$	11.5090	1.14519	0.572595	−	0.819838i	$-0.305937\pi$
$101$	11.5090	1.14519	0.572595	+	0.819838i	$0.305937\pi$
$102$	0	0
$103$	15.1564	1.49341	0.746704	−	0.665156i	$-0.231635\pi$
$103$	15.1564	1.49341	0.746704	+	0.665156i	$0.231635\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.65279	0.256455	0.128228	−	0.991745i	$-0.459071\pi$
$107$	2.65279	0.256455	0.128228	+	0.991745i	$0.459071\pi$
$108$	0	0
$109$	−2.49934	−0.239393	−0.119696	−	0.992811i	$-0.538192\pi$
$109$	−2.49934	−0.239393	−0.119696	+	0.992811i	$0.538192\pi$
$110$	0	0
$111$	0.307979	0.0292320
$112$	0	0
$113$	−8.52781	−0.802229	−0.401114	−	0.916028i	$-0.631377\pi$
$113$	−8.52781	−0.802229	−0.401114	+	0.916028i	$0.631377\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.405813	−0.0375174
$118$	0	0
$119$	1.54288	0.141435
$120$	0	0
$121$	12.5308	1.13916
$122$	0	0
$123$	−26.4034	−2.38072
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.4088	1.81099	0.905494	−	0.424359i	$-0.139501\pi$
$127$	20.4088	1.81099	0.905494	+	0.424359i	$0.139501\pi$
$128$	0	0
$129$	−17.0586	−1.50193
$130$	0	0
$131$	13.2131	1.15444	0.577218	−	0.816590i	$-0.304138\pi$
$131$	13.2131	1.15444	0.577218	+	0.816590i	$0.304138\pi$
$132$	0	0
$133$	1.35690	0.117658
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.86054	0.586136	0.293068	−	0.956092i	$-0.405324\pi$
$137$	6.86054	0.586136	0.293068	+	0.956092i	$0.405324\pi$
$138$	0	0
$139$	−4.28621	−0.363551	−0.181776	−	0.983340i	$-0.558184\pi$
$139$	−4.28621	−0.363551	−0.181776	+	0.983340i	$0.558184\pi$
$140$	0	0
$141$	6.04892	0.509411
$142$	0	0
$143$	0.960771	0.0803437
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−11.5918	−0.956075
$148$	0	0
$149$	−15.3545	−1.25789	−0.628945	−	0.777450i	$-0.716513\pi$
$149$	−15.3545	−1.25789	−0.628945	+	0.777450i	$0.716513\pi$
$150$	0	0
$151$	10.2295	0.832467	0.416233	−	0.909258i	$-0.363350\pi$
$151$	10.2295	0.832467	0.416233	+	0.909258i	$0.363350\pi$
$152$	0	0
$153$	2.32975	0.188349
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.17092	−0.253067	−0.126533	−	0.991962i	$-0.540385\pi$
$157$	−3.17092	−0.253067	−0.126533	+	0.991962i	$0.540385\pi$
$158$	0	0
$159$	−28.9366	−2.29482
$160$	0	0
$161$	3.46681	0.273223
$162$	0	0
$163$	−4.63773	−0.363255	−0.181627	−	0.983367i	$-0.558136\pi$
$163$	−4.63773	−0.363255	−0.181627	+	0.983367i	$0.558136\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.6286	−1.51891	−0.759454	−	0.650560i	$-0.774534\pi$
$167$	−19.6286	−1.51891	−0.759454	+	0.650560i	$0.774534\pi$
$168$	0	0
$169$	−12.9608	−0.996982
$170$	0	0
$171$	2.04892	0.156685
$172$	0	0
$173$	−20.5646	−1.56350	−0.781751	−	0.623591i	$-0.785673\pi$
$173$	−20.5646	−1.56350	−0.781751	+	0.623591i	$0.785673\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−13.0858	−0.983585
$178$	0	0
$179$	−6.92154	−0.517340	−0.258670	−	0.965966i	$-0.583284\pi$
$179$	−6.92154	−0.517340	−0.258670	+	0.965966i	$0.583284\pi$
$180$	0	0
$181$	−17.6461	−1.31162	−0.655812	−	0.754925i	$-0.727673\pi$
$181$	−17.6461	−1.31162	−0.655812	+	0.754925i	$0.727673\pi$
$182$	0	0
$183$	−17.0368	−1.25940
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.51573	−0.403350
$188$	0	0
$189$	−2.89977	−0.210927
$190$	0	0
$191$	−6.92931	−0.501387	−0.250694	−	0.968066i	$-0.580659\pi$
$191$	−6.92931	−0.501387	−0.250694	+	0.968066i	$0.580659\pi$
$192$	0	0
$193$	21.0368	1.51426	0.757132	−	0.653262i	$-0.226600\pi$
$193$	21.0368	1.51426	0.757132	+	0.653262i	$0.226600\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.5646	−1.53642	−0.768209	−	0.640199i	$-0.778852\pi$
$197$	−21.5646	−1.53642	−0.768209	+	0.640199i	$0.778852\pi$
$198$	0	0
$199$	−21.9909	−1.55889	−0.779447	−	0.626468i	$-0.784500\pi$
$199$	−21.9909	−1.55889	−0.779447	+	0.626468i	$0.784500\pi$
$200$	0	0
$201$	18.0097	1.27031
$202$	0	0
$203$	−13.8877	−0.974725
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.23490	0.363851
$208$	0	0
$209$	−4.85086	−0.335541
$210$	0	0
$211$	20.6233	1.41976	0.709882	−	0.704321i	$-0.248748\pi$
$211$	20.6233	1.41976	0.709882	+	0.704321i	$0.248748\pi$
$212$	0	0
$213$	19.8267	1.35850
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.41358	−0.231729
$218$	0	0
$219$	26.8974	1.81756
$220$	0	0
$221$	−0.225209	−0.0151492
$222$	0	0
$223$	7.97716	0.534190	0.267095	−	0.963670i	$-0.413936\pi$
$223$	7.97716	0.534190	0.267095	+	0.963670i	$0.413936\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.7942	1.31379	0.656893	−	0.753984i	$-0.271871\pi$
$227$	19.7942	1.31379	0.656893	+	0.753984i	$0.271871\pi$
$228$	0	0
$229$	−4.03385	−0.266564	−0.133282	−	0.991078i	$-0.542552\pi$
$229$	−4.03385	−0.266564	−0.133282	+	0.991078i	$0.542552\pi$
$230$	0	0
$231$	−14.7899	−0.973101
$232$	0	0
$233$	−26.8159	−1.75677	−0.878385	−	0.477953i	$-0.841379\pi$
$233$	−26.8159	−1.75677	−0.878385	+	0.477953i	$0.841379\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−24.1347	−1.56772
$238$	0	0
$239$	−3.36227	−0.217487	−0.108744	−	0.994070i	$-0.534683\pi$
$239$	−3.36227	−0.217487	−0.108744	+	0.994070i	$0.534683\pi$
$240$	0	0
$241$	27.7506	1.78758	0.893788	−	0.448491i	$-0.148038\pi$
$241$	27.7506	1.78758	0.893788	+	0.448491i	$0.148038\pi$
$242$	0	0
$243$	−18.1903	−1.16691
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.198062	−0.0126024
$248$	0	0
$249$	8.48188	0.537517
$250$	0	0
$251$	5.59419	0.353102	0.176551	−	0.984291i	$-0.443506\pi$
$251$	5.59419	0.353102	0.176551	+	0.984291i	$0.443506\pi$
$252$	0	0
$253$	−12.3937	−0.779187
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.4668	−0.652902	−0.326451	−	0.945214i	$-0.605853\pi$
$257$	−10.4668	−0.652902	−0.326451	+	0.945214i	$0.605853\pi$
$258$	0	0
$259$	0.185981	0.0115563
$260$	0	0
$261$	−20.9705	−1.29804
$262$	0	0
$263$	15.4795	0.954506	0.477253	−	0.878766i	$-0.341633\pi$
$263$	15.4795	0.954506	0.477253	+	0.878766i	$0.341633\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	21.0465	1.28803
$268$	0	0
$269$	9.13036	0.556688	0.278344	−	0.960481i	$-0.410215\pi$
$269$	9.13036	0.556688	0.278344	+	0.960481i	$0.410215\pi$
$270$	0	0
$271$	−7.44265	−0.452109	−0.226054	−	0.974115i	$-0.572583\pi$
$271$	−7.44265	−0.452109	−0.226054	+	0.974115i	$0.572583\pi$
$272$	0	0
$273$	−0.603875	−0.0365482
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.4155	0.685891	0.342946	−	0.939355i	$-0.388575\pi$
$277$	11.4155	0.685891	0.342946	+	0.939355i	$0.388575\pi$
$278$	0	0
$279$	−5.15452	−0.308593
$280$	0	0
$281$	21.5060	1.28294	0.641471	−	0.767147i	$-0.278324\pi$
$281$	21.5060	1.28294	0.641471	+	0.767147i	$0.278324\pi$
$282$	0	0
$283$	−5.45712	−0.324392	−0.162196	−	0.986759i	$-0.551858\pi$
$283$	−5.45712	−0.324392	−0.162196	+	0.986759i	$0.551858\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.9444	−0.941167
$288$	0	0
$289$	−15.7071	−0.923946
$290$	0	0
$291$	−0.445042	−0.0260888
$292$	0	0
$293$	−7.39075	−0.431772	−0.215886	−	0.976419i	$-0.569264\pi$
$293$	−7.39075	−0.431772	−0.215886	+	0.976419i	$0.569264\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	10.3666	0.601530
$298$	0	0
$299$	−0.506041	−0.0292651
$300$	0	0
$301$	−10.3013	−0.593756
$302$	0	0
$303$	25.8605	1.48565
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−32.2131	−1.83850	−0.919250	−	0.393674i	$-0.871204\pi$
$307$	−32.2131	−1.83850	−0.919250	+	0.393674i	$0.871204\pi$
$308$	0	0
$309$	34.0562	1.93739
$310$	0	0
$311$	−14.8442	−0.841735	−0.420867	−	0.907122i	$-0.638274\pi$
$311$	−14.8442	−0.841735	−0.420867	+	0.907122i	$0.638274\pi$
$312$	0	0
$313$	13.1491	0.743234	0.371617	−	0.928386i	$-0.378804\pi$
$313$	13.1491	0.743234	0.371617	+	0.928386i	$0.378804\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.13467	0.288392	0.144196	−	0.989549i	$-0.453940\pi$
$317$	5.13467	0.288392	0.144196	+	0.989549i	$0.453940\pi$
$318$	0	0
$319$	49.6480	2.77975
$320$	0	0
$321$	5.96077	0.332698
$322$	0	0
$323$	1.13706	0.0632679
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.61596	−0.310563
$328$	0	0
$329$	3.65279	0.201385
$330$	0	0
$331$	−2.00969	−0.110462	−0.0552312	−	0.998474i	$-0.517590\pi$
$331$	−2.00969	−0.110462	−0.0552312	+	0.998474i	$0.517590\pi$
$332$	0	0
$333$	0.280831	0.0153895
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.31873	0.0718359	0.0359180	−	0.999355i	$-0.488564\pi$
$337$	1.31873	0.0718359	0.0359180	+	0.999355i	$0.488564\pi$
$338$	0	0
$339$	−19.1618	−1.04073
$340$	0	0
$341$	12.2034	0.660853
$342$	0	0
$343$	−16.4983	−0.890823
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0151	0.645003	0.322501	−	0.946569i	$-0.395476\pi$
$347$	12.0151	0.645003	0.322501	+	0.946569i	$0.395476\pi$
$348$	0	0
$349$	−10.5579	−0.565154	−0.282577	−	0.959245i	$-0.591189\pi$
$349$	−10.5579	−0.565154	−0.282577	+	0.959245i	$0.591189\pi$
$350$	0	0
$351$	0.423272	0.0225926
$352$	0	0
$353$	1.01102	0.0538110	0.0269055	−	0.999638i	$-0.491435\pi$
$353$	1.01102	0.0538110	0.0269055	+	0.999638i	$0.491435\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.46681	0.183483
$358$	0	0
$359$	5.14244	0.271408	0.135704	−	0.990749i	$-0.456670\pi$
$359$	5.14244	0.271408	0.135704	+	0.990749i	$0.456670\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	28.1564	1.47783
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−29.5308	−1.54149	−0.770747	−	0.637141i	$-0.780117\pi$
$367$	−29.5308	−1.54149	−0.770747	+	0.637141i	$0.780117\pi$
$368$	0	0
$369$	−24.0761	−1.25335
$370$	0	0
$371$	−17.4741	−0.907210
$372$	0	0
$373$	−8.78986	−0.455121	−0.227561	−	0.973764i	$-0.573075\pi$
$373$	−8.78986	−0.455121	−0.227561	+	0.973764i	$0.573075\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.02715	0.104403
$378$	0	0
$379$	−19.1511	−0.983724	−0.491862	−	0.870673i	$-0.663684\pi$
$379$	−19.1511	−0.983724	−0.491862	+	0.870673i	$0.663684\pi$
$380$	0	0
$381$	45.8582	2.34938
$382$	0	0
$383$	−6.99894	−0.357629	−0.178814	−	0.983883i	$-0.557226\pi$
$383$	−6.99894	−0.357629	−0.178814	+	0.983883i	$0.557226\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−15.5550	−0.790703
$388$	0	0
$389$	8.08575	0.409964	0.204982	−	0.978766i	$-0.434286\pi$
$389$	8.08575	0.409964	0.204982	+	0.978766i	$0.434286\pi$
$390$	0	0
$391$	2.90515	0.146920
$392$	0	0
$393$	29.6896	1.49764
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.7006	0.888370	0.444185	−	0.895935i	$-0.353493\pi$
$397$	17.7006	0.888370	0.444185	+	0.895935i	$0.353493\pi$
$398$	0	0
$399$	3.04892	0.152637
$400$	0	0
$401$	−15.5418	−0.776121	−0.388061	−	0.921634i	$-0.626855\pi$
$401$	−15.5418	−0.776121	−0.388061	+	0.921634i	$0.626855\pi$
$402$	0	0
$403$	0.498271	0.0248207
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.664874	−0.0329566
$408$	0	0
$409$	13.1661	0.651023	0.325512	−	0.945538i	$-0.394463\pi$
$409$	13.1661	0.651023	0.325512	+	0.945538i	$0.394463\pi$
$410$	0	0
$411$	15.4155	0.760391
$412$	0	0
$413$	−7.90217	−0.388840
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.63102	−0.471633
$418$	0	0
$419$	25.1739	1.22983	0.614913	−	0.788595i	$-0.289191\pi$
$419$	25.1739	1.22983	0.614913	+	0.788595i	$0.289191\pi$
$420$	0	0
$421$	26.9420	1.31307	0.656536	−	0.754295i	$-0.272021\pi$
$421$	26.9420	1.31307	0.656536	+	0.754295i	$0.272021\pi$
$422$	0	0
$423$	5.51573	0.268184
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.2881	−0.497877
$428$	0	0
$429$	2.15883	0.104229
$430$	0	0
$431$	−18.9487	−0.912726	−0.456363	−	0.889794i	$-0.650848\pi$
$431$	−18.9487	−0.912726	−0.456363	+	0.889794i	$0.650848\pi$
$432$	0	0
$433$	3.68904	0.177284	0.0886419	−	0.996064i	$-0.471747\pi$
$433$	3.68904	0.177284	0.0886419	+	0.996064i	$0.471747\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.55496	0.122220
$438$	0	0
$439$	25.6926	1.22624	0.613121	−	0.789989i	$-0.289914\pi$
$439$	25.6926	1.22624	0.613121	+	0.789989i	$0.289914\pi$
$440$	0	0
$441$	−10.5700	−0.503334
$442$	0	0
$443$	−27.3653	−1.30016	−0.650081	−	0.759865i	$-0.725265\pi$
$443$	−27.3653	−1.30016	−0.650081	+	0.759865i	$0.725265\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−34.5013	−1.63185
$448$	0	0
$449$	7.55794	0.356681	0.178341	−	0.983969i	$-0.442927\pi$
$449$	7.55794	0.356681	0.178341	+	0.983969i	$0.442927\pi$
$450$	0	0
$451$	57.0006	2.68405
$452$	0	0
$453$	22.9855	1.07995
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.85623	0.367499	0.183750	−	0.982973i	$-0.441176\pi$
$457$	7.85623	0.367499	0.183750	+	0.982973i	$0.441176\pi$
$458$	0	0
$459$	−2.42998	−0.113422
$460$	0	0
$461$	3.87907	0.180666	0.0903331	−	0.995912i	$-0.471207\pi$
$461$	3.87907	0.180666	0.0903331	+	0.995912i	$0.471207\pi$
$462$	0	0
$463$	−13.0954	−0.608597	−0.304298	−	0.952577i	$-0.598422\pi$
$463$	−13.0954	−0.608597	−0.304298	+	0.952577i	$0.598422\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.44026	0.298020	0.149010	−	0.988836i	$-0.452391\pi$
$467$	6.44026	0.298020	0.149010	+	0.988836i	$0.452391\pi$
$468$	0	0
$469$	10.8756	0.502189
$470$	0	0
$471$	−7.12498	−0.328302
$472$	0	0
$473$	36.8267	1.69329
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−26.3860	−1.20813
$478$	0	0
$479$	−5.69096	−0.260026	−0.130013	−	0.991512i	$-0.541502\pi$
$479$	−5.69096	−0.260026	−0.130013	+	0.991512i	$0.541502\pi$
$480$	0	0
$481$	−0.0271471	−0.00123780
$482$	0	0
$483$	7.78986	0.354451
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−12.9632	−0.587417	−0.293709	−	0.955895i	$-0.594890\pi$
$487$	−12.9632	−0.587417	−0.293709	+	0.955895i	$0.594890\pi$
$488$	0	0
$489$	−10.4209	−0.471248
$490$	0	0
$491$	19.6045	0.884737	0.442369	−	0.896833i	$-0.354138\pi$
$491$	19.6045	0.884737	0.442369	+	0.896833i	$0.354138\pi$
$492$	0	0
$493$	−11.6377	−0.524137
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.9729	0.537056
$498$	0	0
$499$	−5.49827	−0.246136	−0.123068	−	0.992398i	$-0.539273\pi$
$499$	−5.49827	−0.246136	−0.123068	+	0.992398i	$0.539273\pi$
$500$	0	0
$501$	−44.1051	−1.97047
$502$	0	0
$503$	−35.6939	−1.59151	−0.795757	−	0.605616i	$-0.792927\pi$
$503$	−35.6939	−1.59151	−0.795757	+	0.605616i	$0.792927\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−29.1226	−1.29338
$508$	0	0
$509$	−16.4450	−0.728914	−0.364457	−	0.931220i	$-0.618745\pi$
$509$	−16.4450	−0.728914	−0.364457	+	0.931220i	$0.618745\pi$
$510$	0	0
$511$	16.2427	0.718533
$512$	0	0
$513$	−2.13706	−0.0943537
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−13.0586	−0.574317
$518$	0	0
$519$	−46.2083	−2.02832
$520$	0	0
$521$	−26.5435	−1.16289	−0.581445	−	0.813586i	$-0.697513\pi$
$521$	−26.5435	−1.16289	−0.581445	+	0.813586i	$0.697513\pi$
$522$	0	0
$523$	−24.1685	−1.05682	−0.528408	−	0.848991i	$-0.677211\pi$
$523$	−24.1685	−1.05682	−0.528408	+	0.848991i	$0.677211\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.86054	−0.124607
$528$	0	0
$529$	−16.4722	−0.716182
$530$	0	0
$531$	−11.9323	−0.517818
$532$	0	0
$533$	2.32736	0.100809
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−15.5526	−0.671143
$538$	0	0
$539$	25.0248	1.07789
$540$	0	0
$541$	9.80386	0.421501	0.210750	−	0.977540i	$-0.432409\pi$
$541$	9.80386	0.421501	0.210750	+	0.977540i	$0.432409\pi$
$542$	0	0
$543$	−39.6504	−1.70156
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.1739	0.905331	0.452665	−	0.891681i	$-0.350473\pi$
$547$	21.1739	0.905331	0.452665	+	0.891681i	$0.350473\pi$
$548$	0	0
$549$	−15.5351	−0.663022
$550$	0	0
$551$	−10.2349	−0.436021
$552$	0	0
$553$	−14.5743	−0.619764
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.4077	−1.03419	−0.517094	−	0.855928i	$-0.672986\pi$
$557$	−24.4077	−1.03419	−0.517094	+	0.855928i	$0.672986\pi$
$558$	0	0
$559$	1.50365	0.0635975
$560$	0	0
$561$	−12.3937	−0.523264
$562$	0	0
$563$	14.4849	0.610464	0.305232	−	0.952278i	$-0.401266\pi$
$563$	14.4849	0.610464	0.305232	+	0.952278i	$0.401266\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−14.8562	−0.623903
$568$	0	0
$569$	0.811626	0.0340251	0.0170126	−	0.999855i	$-0.494584\pi$
$569$	0.811626	0.0340251	0.0170126	+	0.999855i	$0.494584\pi$
$570$	0	0
$571$	37.6588	1.57597	0.787985	−	0.615694i	$-0.211124\pi$
$571$	37.6588	1.57597	0.787985	+	0.615694i	$0.211124\pi$
$572$	0	0
$573$	−15.5700	−0.650447
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.89307	−0.370223	−0.185112	−	0.982718i	$-0.559265\pi$
$577$	−8.89307	−0.370223	−0.185112	+	0.982718i	$0.559265\pi$
$578$	0	0
$579$	47.2693	1.96445
$580$	0	0
$581$	5.12200	0.212496
$582$	0	0
$583$	62.4693	2.58721
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.3327	0.839222	0.419611	−	0.907704i	$-0.362167\pi$
$587$	20.3327	0.839222	0.419611	+	0.907704i	$0.362167\pi$
$588$	0	0
$589$	−2.51573	−0.103659
$590$	0	0
$591$	−48.4553	−1.99319
$592$	0	0
$593$	−17.0049	−0.698308	−0.349154	−	0.937065i	$-0.613531\pi$
$593$	−17.0049	−0.698308	−0.349154	+	0.937065i	$0.613531\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−49.4131	−2.02234
$598$	0	0
$599$	12.4004	0.506668	0.253334	−	0.967379i	$-0.418473\pi$
$599$	12.4004	0.506668	0.253334	+	0.967379i	$0.418473\pi$
$600$	0	0
$601$	−9.73125	−0.396946	−0.198473	−	0.980106i	$-0.563598\pi$
$601$	−9.73125	−0.396946	−0.198473	+	0.980106i	$0.563598\pi$
$602$	0	0
$603$	16.4222	0.668764
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.4282	−0.950920	−0.475460	−	0.879737i	$-0.657718\pi$
$607$	−23.4282	−0.950920	−0.475460	+	0.879737i	$0.657718\pi$
$608$	0	0
$609$	−31.2054	−1.26450
$610$	0	0
$611$	−0.533188	−0.0215705
$612$	0	0
$613$	−28.1424	−1.13666	−0.568331	−	0.822800i	$-0.692411\pi$
$613$	−28.1424	−1.13666	−0.568331	+	0.822800i	$0.692411\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.4805	−1.46865	−0.734326	−	0.678797i	$-0.762502\pi$
$617$	−36.4805	−1.46865	−0.734326	+	0.678797i	$0.762502\pi$
$618$	0	0
$619$	38.2097	1.53578	0.767888	−	0.640584i	$-0.221308\pi$
$619$	38.2097	1.53578	0.767888	+	0.640584i	$0.221308\pi$
$620$	0	0
$621$	−5.46011	−0.219107
$622$	0	0
$623$	12.7095	0.509195
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−10.8998	−0.435295
$628$	0	0
$629$	0.155850	0.00621413
$630$	0	0
$631$	12.5235	0.498553	0.249276	−	0.968432i	$-0.419807\pi$
$631$	12.5235	0.498553	0.249276	+	0.968432i	$0.419807\pi$
$632$	0	0
$633$	46.3400	1.84185
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.02177	0.0404840
$638$	0	0
$639$	18.0790	0.715196
$640$	0	0
$641$	37.3564	1.47549	0.737745	−	0.675080i	$-0.235891\pi$
$641$	37.3564	1.47549	0.737745	+	0.675080i	$0.235891\pi$
$642$	0	0
$643$	−14.5483	−0.573727	−0.286864	−	0.957971i	$-0.592613\pi$
$643$	−14.5483	−0.573727	−0.286864	+	0.957971i	$0.592613\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.4403	−0.921532	−0.460766	−	0.887522i	$-0.652425\pi$
$647$	−23.4403	−0.921532	−0.460766	+	0.887522i	$0.652425\pi$
$648$	0	0
$649$	28.2500	1.10891
$650$	0	0
$651$	−7.67025	−0.300621
$652$	0	0
$653$	−27.1903	−1.06404	−0.532019	−	0.846732i	$-0.678567\pi$
$653$	−27.1903	−1.06404	−0.532019	+	0.846732i	$0.678567\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	24.5265	0.956869
$658$	0	0
$659$	−2.71486	−0.105756	−0.0528779	−	0.998601i	$-0.516839\pi$
$659$	−2.71486	−0.105756	−0.0528779	+	0.998601i	$0.516839\pi$
$660$	0	0
$661$	−9.41311	−0.366128	−0.183064	−	0.983101i	$-0.558601\pi$
$661$	−9.41311	−0.366128	−0.183064	+	0.983101i	$0.558601\pi$
$662$	0	0
$663$	−0.506041	−0.0196530
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−26.1497	−1.01252
$668$	0	0
$669$	17.9245	0.693002
$670$	0	0
$671$	36.7797	1.41986
$672$	0	0
$673$	44.8001	1.72692	0.863459	−	0.504419i	$-0.168293\pi$
$673$	44.8001	1.72692	0.863459	+	0.504419i	$0.168293\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.9197	1.26521	0.632604	−	0.774475i	$-0.281986\pi$
$677$	32.9197	1.26521	0.632604	+	0.774475i	$0.281986\pi$
$678$	0	0
$679$	−0.268750	−0.0103137
$680$	0	0
$681$	44.4771	1.70437
$682$	0	0
$683$	20.3448	0.778473	0.389236	−	0.921138i	$-0.372739\pi$
$683$	20.3448	0.778473	0.389236	+	0.921138i	$0.372739\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−9.06398	−0.345813
$688$	0	0
$689$	2.55065	0.0971719
$690$	0	0
$691$	46.1473	1.75553	0.877764	−	0.479094i	$-0.159035\pi$
$691$	46.1473	1.75553	0.877764	+	0.479094i	$0.159035\pi$
$692$	0	0
$693$	−13.4862	−0.512298
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−13.3612	−0.506092
$698$	0	0
$699$	−60.2549	−2.27905
$700$	0	0
$701$	13.1933	0.498303	0.249152	−	0.968464i	$-0.419848\pi$
$701$	13.1933	0.498303	0.249152	+	0.968464i	$0.419848\pi$
$702$	0	0
$703$	0.137063	0.00516944
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.6165	0.587321
$708$	0	0
$709$	41.1860	1.54677	0.773386	−	0.633935i	$-0.218562\pi$
$709$	41.1860	1.54677	0.773386	+	0.633935i	$0.218562\pi$
$710$	0	0
$711$	−22.0073	−0.825338
$712$	0	0
$713$	−6.42758	−0.240715
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.55496	−0.282145
$718$	0	0
$719$	35.6256	1.32861	0.664306	−	0.747461i	$-0.268727\pi$
$719$	35.6256	1.32861	0.664306	+	0.747461i	$0.268727\pi$
$720$	0	0
$721$	20.5657	0.765907
$722$	0	0
$723$	62.3551	2.31901
$724$	0	0
$725$	0	0
$726$	0	0
$727$	20.0116	0.742189	0.371095	−	0.928595i	$-0.378983\pi$
$727$	20.0116	0.742189	0.371095	+	0.928595i	$0.378983\pi$
$728$	0	0
$729$	−8.02715	−0.297302
$730$	0	0
$731$	−8.63235	−0.319279
$732$	0	0
$733$	18.9952	0.701604	0.350802	−	0.936450i	$-0.385909\pi$
$733$	18.9952	0.701604	0.350802	+	0.936450i	$0.385909\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−38.8799	−1.43216
$738$	0	0
$739$	−48.1704	−1.77198	−0.885989	−	0.463706i	$-0.846519\pi$
$739$	−48.1704	−1.77198	−0.885989	+	0.463706i	$0.846519\pi$
$740$	0	0
$741$	−0.445042	−0.0163490
$742$	0	0
$743$	−13.2446	−0.485897	−0.242948	−	0.970039i	$-0.578115\pi$
$743$	−13.2446	−0.485897	−0.242948	+	0.970039i	$0.578115\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.73423	0.282981
$748$	0	0
$749$	3.59956	0.131525
$750$	0	0
$751$	−12.0562	−0.439937	−0.219969	−	0.975507i	$-0.570596\pi$
$751$	−12.0562	−0.439937	−0.219969	+	0.975507i	$0.570596\pi$
$752$	0	0
$753$	12.5700	0.458077
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−15.0054	−0.545380	−0.272690	−	0.962102i	$-0.587913\pi$
$757$	−15.0054	−0.545380	−0.272690	+	0.962102i	$0.587913\pi$
$758$	0	0
$759$	−27.8485	−1.01084
$760$	0	0
$761$	44.3967	1.60938	0.804690	−	0.593695i	$-0.202332\pi$
$761$	44.3967	1.60938	0.804690	+	0.593695i	$0.202332\pi$
$762$	0	0
$763$	−3.39134	−0.122775
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.15346	0.0416489
$768$	0	0
$769$	−39.7211	−1.43238	−0.716190	−	0.697906i	$-0.754115\pi$
$769$	−39.7211	−1.43238	−0.716190	+	0.697906i	$0.754115\pi$
$770$	0	0
$771$	−23.5187	−0.847006
$772$	0	0
$773$	1.72779	0.0621444	0.0310722	−	0.999517i	$-0.490108\pi$
$773$	1.72779	0.0621444	0.0310722	+	0.999517i	$0.490108\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.417895	0.0149919
$778$	0	0
$779$	−11.7506	−0.421010
$780$	0	0
$781$	−42.8025	−1.53159
$782$	0	0
$783$	21.8726	0.781664
$784$	0	0
$785$	0	0
$786$	0	0
$787$	42.6329	1.51970	0.759850	−	0.650098i	$-0.225272\pi$
$787$	42.6329	1.51970	0.759850	+	0.650098i	$0.225272\pi$
$788$	0	0
$789$	34.7821	1.23828
$790$	0	0
$791$	−11.5714	−0.411430
$792$	0	0
$793$	1.50173	0.0533280
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38.5864	−1.36680	−0.683401	−	0.730044i	$-0.739500\pi$
$797$	−38.5864	−1.36680	−0.683401	+	0.730044i	$0.739500\pi$
$798$	0	0
$799$	3.06100	0.108290
$800$	0	0
$801$	19.1914	0.678093
$802$	0	0
$803$	−58.0670	−2.04914
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20.5157	0.722188
$808$	0	0
$809$	−8.38298	−0.294730	−0.147365	−	0.989082i	$-0.547079\pi$
$809$	−8.38298	−0.294730	−0.147365	+	0.989082i	$0.547079\pi$
$810$	0	0
$811$	0.340765	0.0119659	0.00598295	−	0.999982i	$-0.498096\pi$
$811$	0.340765	0.0119659	0.00598295	+	0.999982i	$0.498096\pi$
$812$	0	0
$813$	−16.7235	−0.586518
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7.59179	−0.265603
$818$	0	0
$819$	−0.550646	−0.0192411
$820$	0	0
$821$	−33.7506	−1.17791	−0.588953	−	0.808168i	$-0.700460\pi$
$821$	−33.7506	−1.17791	−0.588953	+	0.808168i	$0.700460\pi$
$822$	0	0
$823$	37.2669	1.29904	0.649522	−	0.760343i	$-0.274969\pi$
$823$	37.2669	1.29904	0.649522	+	0.760343i	$0.274969\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21.1691	−0.736122	−0.368061	−	0.929802i	$-0.619978\pi$
$827$	−21.1691	−0.736122	−0.368061	+	0.929802i	$0.619978\pi$
$828$	0	0
$829$	−31.0374	−1.07797	−0.538987	−	0.842314i	$-0.681193\pi$
$829$	−31.0374	−1.07797	−0.538987	+	0.842314i	$0.681193\pi$
$830$	0	0
$831$	25.6504	0.889803
$832$	0	0
$833$	−5.86592	−0.203242
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.37627	0.185831
$838$	0	0
$839$	−33.2403	−1.14758	−0.573791	−	0.819002i	$-0.694528\pi$
$839$	−33.2403	−1.14758	−0.573791	+	0.819002i	$0.694528\pi$
$840$	0	0
$841$	75.7531	2.61218
$842$	0	0
$843$	48.3236	1.66435
$844$	0	0
$845$	0	0
$846$	0	0
$847$	17.0030	0.584229
$848$	0	0
$849$	−12.2620	−0.420832
$850$	0	0
$851$	0.350191	0.0120044
$852$	0	0
$853$	24.3086	0.832310	0.416155	−	0.909294i	$-0.363377\pi$
$853$	24.3086	0.832310	0.416155	+	0.909294i	$0.363377\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	57.3889	1.96037	0.980185	−	0.198087i	$-0.0634727\pi$
$857$	57.3889	1.96037	0.980185	+	0.198087i	$0.0634727\pi$
$858$	0	0
$859$	13.8135	0.471312	0.235656	−	0.971837i	$-0.424276\pi$
$859$	13.8135	0.471312	0.235656	+	0.971837i	$0.424276\pi$
$860$	0	0
$861$	−35.8267	−1.22097
$862$	0	0
$863$	−9.01938	−0.307023	−0.153512	−	0.988147i	$-0.549058\pi$
$863$	−9.01938	−0.307023	−0.153512	+	0.988147i	$0.549058\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−35.2935	−1.19863
$868$	0	0
$869$	52.1027	1.76746
$870$	0	0
$871$	−1.58748	−0.0537898
$872$	0	0
$873$	−0.405813	−0.0137347
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.37675	0.0802570	0.0401285	−	0.999195i	$-0.487223\pi$
$877$	2.37675	0.0802570	0.0401285	+	0.999195i	$0.487223\pi$
$878$	0	0
$879$	−16.6069	−0.560135
$880$	0	0
$881$	−7.99330	−0.269301	−0.134650	−	0.990893i	$-0.542991\pi$
$881$	−7.99330	−0.269301	−0.134650	+	0.990893i	$0.542991\pi$
$882$	0	0
$883$	−1.76377	−0.0593557	−0.0296779	−	0.999560i	$-0.509448\pi$
$883$	−1.76377	−0.0593557	−0.0296779	+	0.999560i	$0.509448\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.9197	1.54183	0.770917	−	0.636936i	$-0.219798\pi$
$887$	45.9197	1.54183	0.770917	+	0.636936i	$0.219798\pi$
$888$	0	0
$889$	27.6926	0.928780
$890$	0	0
$891$	53.1105	1.77927
$892$	0	0
$893$	2.69202	0.0900851
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.13706	−0.0379654
$898$	0	0
$899$	25.7482	0.858752
$900$	0	0
$901$	−14.6431	−0.487833
$902$	0	0
$903$	−23.1468	−0.770276
$904$	0	0
$905$	0	0
$906$	0	0
$907$	55.0549	1.82807	0.914034	−	0.405638i	$-0.132951\pi$
$907$	55.0549	1.82807	0.914034	+	0.405638i	$0.132951\pi$
$908$	0	0
$909$	23.5810	0.782134
$910$	0	0
$911$	−51.8998	−1.71952	−0.859758	−	0.510702i	$-0.829386\pi$
$911$	−51.8998	−1.71952	−0.859758	+	0.510702i	$0.829386\pi$
$912$	0	0
$913$	−18.3110	−0.606004
$914$	0	0
$915$	0	0
$916$	0	0
$917$	17.9288	0.592062
$918$	0	0
$919$	−11.4614	−0.378078	−0.189039	−	0.981970i	$-0.560537\pi$
$919$	−11.4614	−0.378078	−0.189039	+	0.981970i	$0.560537\pi$
$920$	0	0
$921$	−72.3822	−2.38508
$922$	0	0
$923$	−1.74764	−0.0575244
$924$	0	0
$925$	0	0
$926$	0	0
$927$	31.0543	1.01996
$928$	0	0
$929$	36.5295	1.19849	0.599246	−	0.800565i	$-0.295467\pi$
$929$	36.5295	1.19849	0.599246	+	0.800565i	$0.295467\pi$
$930$	0	0
$931$	−5.15883	−0.169074
$932$	0	0
$933$	−33.3545	−1.09198
$934$	0	0
$935$	0	0
$936$	0	0
$937$	48.7845	1.59372	0.796860	−	0.604164i	$-0.206493\pi$
$937$	48.7845	1.59372	0.796860	+	0.604164i	$0.206493\pi$
$938$	0	0
$939$	29.5459	0.964193
$940$	0	0
$941$	−18.1817	−0.592705	−0.296353	−	0.955079i	$-0.595770\pi$
$941$	−18.1817	−0.592705	−0.296353	+	0.955079i	$0.595770\pi$
$942$	0	0
$943$	−30.0224	−0.977663
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.55150	−0.245391	−0.122695	−	0.992444i	$-0.539154\pi$
$947$	−7.55150	−0.245391	−0.122695	+	0.992444i	$0.539154\pi$
$948$	0	0
$949$	−2.37090	−0.0769626
$950$	0	0
$951$	11.5375	0.374129
$952$	0	0
$953$	−13.8592	−0.448944	−0.224472	−	0.974481i	$-0.572066\pi$
$953$	−13.8592	−0.448944	−0.224472	+	0.974481i	$0.572066\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	111.558	3.60616
$958$	0	0
$959$	9.30904	0.300605
$960$	0	0
$961$	−24.6711	−0.795842
$962$	0	0
$963$	5.43535	0.175152
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.89977	0.157566	0.0787830	−	0.996892i	$-0.474897\pi$
$967$	4.89977	0.157566	0.0787830	+	0.996892i	$0.474897\pi$
$968$	0	0
$969$	2.55496	0.0820771
$970$	0	0
$971$	−14.5133	−0.465755	−0.232878	−	0.972506i	$-0.574814\pi$
$971$	−14.5133	−0.465755	−0.232878	+	0.972506i	$0.574814\pi$
$972$	0	0
$973$	−5.81594	−0.186450
$974$	0	0
$975$	0	0
$976$	0	0
$977$	19.6644	0.629120	0.314560	−	0.949238i	$-0.398143\pi$
$977$	19.6644	0.629120	0.314560	+	0.949238i	$0.398143\pi$
$978$	0	0
$979$	−45.4359	−1.45214
$980$	0	0
$981$	−5.12093	−0.163499
$982$	0	0
$983$	−15.4397	−0.492449	−0.246224	−	0.969213i	$-0.579190\pi$
$983$	−15.4397	−0.492449	−0.246224	+	0.969213i	$0.579190\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.20775	0.261256
$988$	0	0
$989$	−19.3967	−0.616780
$990$	0	0
$991$	−11.9377	−0.379213	−0.189606	−	0.981860i	$-0.560721\pi$
$991$	−11.9377	−0.379213	−0.189606	+	0.981860i	$0.560721\pi$
$992$	0	0
$993$	−4.51573	−0.143302
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−17.9390	−0.568134	−0.284067	−	0.958804i	$-0.591684\pi$
$997$	−17.9390	−0.568134	−0.284067	+	0.958804i	$0.591684\pi$
$998$	0	0
$999$	−0.292913	−0.00926736

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bw.1.3		3
4.3	odd	2		475.2.a.d.1.2	✓	3
5.4	even	2		7600.2.a.bn.1.1		3
12.11	even	2		4275.2.a.bn.1.2		3
20.3	even	4		475.2.b.c.324.5		6
20.7	even	4		475.2.b.c.324.2		6
20.19	odd	2		475.2.a.h.1.2	yes	3
60.59	even	2		4275.2.a.z.1.2		3
76.75	even	2		9025.2.a.be.1.2		3
380.379	even	2		9025.2.a.w.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.d.1.2	✓	3	4.3	odd	2
475.2.a.h.1.2	yes	3	20.19	odd	2
475.2.b.c.324.2		6	20.7	even	4
475.2.b.c.324.5		6	20.3	even	4
4275.2.a.z.1.2		3	60.59	even	2
4275.2.a.bn.1.2		3	12.11	even	2
7600.2.a.bn.1.1		3	5.4	even	2
7600.2.a.bw.1.3		3	1.1	even	1	trivial
9025.2.a.w.1.2		3	380.379	even	2
9025.2.a.be.1.2		3	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+2.24698 q^{3} +1.35690 q^{7} +2.04892 q^{9} +O(q^{10})\)	\(q+2.24698 q^{3} +1.35690 q^{7} +2.04892 q^{9} -4.85086 q^{11} -0.198062 q^{13} +1.13706 q^{17} +1.00000 q^{19} +3.04892 q^{21} +2.55496 q^{23} -2.13706 q^{27} -10.2349 q^{29} -2.51573 q^{31} -10.8998 q^{33} +0.137063 q^{37} -0.445042 q^{39} -11.7506 q^{41} -7.59179 q^{43} +2.69202 q^{47} -5.15883 q^{49} +2.55496 q^{51} -12.8780 q^{53} +2.24698 q^{57} -5.82371 q^{59} -7.58211 q^{61} +2.78017 q^{63} +8.01507 q^{67} +5.74094 q^{69} +8.82371 q^{71} +11.9705 q^{73} -6.58211 q^{77} -10.7409 q^{79} -10.9487 q^{81} +3.77479 q^{83} -22.9976 q^{87} +9.36658 q^{89} -0.268750 q^{91} -5.65279 q^{93} -0.198062 q^{97} -9.93900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 - 3 * q^9	\( 3 q + 2 q^{3} - 3 q^{9} - q^{11} - 5 q^{13} - 2 q^{17} + 3 q^{19} + 8 q^{23} - q^{27} - 7 q^{29} + 5 q^{31} - 10 q^{33} - 5 q^{37} - q^{39} + q^{41} + 5 q^{43} + 3 q^{47} - 7 q^{49} + 8 q^{51} - 19 q^{53} + 2 q^{57} - 10 q^{59} - 17 q^{61} + 7 q^{63} - q^{67} + 3 q^{69} + 19 q^{71} + q^{73} - 14 q^{77} - 18 q^{79} - q^{81} + 13 q^{83} - 28 q^{87} + 2 q^{89} + 7 q^{91} + q^{93} - 5 q^{97} - 20 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 3 * q^9 - q^11 - 5 * q^13 - 2 * q^17 + 3 * q^19 + 8 * q^23 - q^27 - 7 * q^29 + 5 * q^31 - 10 * q^33 - 5 * q^37 - q^39 + q^41 + 5 * q^43 + 3 * q^47 - 7 * q^49 + 8 * q^51 - 19 * q^53 + 2 * q^57 - 10 * q^59 - 17 * q^61 + 7 * q^63 - q^67 + 3 * q^69 + 19 * q^71 + q^73 - 14 * q^77 - 18 * q^79 - q^81 + 13 * q^83 - 28 * q^87 + 2 * q^89 + 7 * q^91 + q^93 - 5 * q^97 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more