LMFDB - Embedded newform 4925.2.a.r.1.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4925,2,Mod(1,4925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [49,-5,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$39.3263229955$
Analytic rank:	$1$
Dimension:	$49$
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	4925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.45182 q^{2} -0.257058 q^{3} +0.107768 q^{4} +0.373200 q^{6} -4.95623 q^{7} +2.74717 q^{8} -2.93392 q^{9} +4.28184 q^{11} -0.0277025 q^{12} -0.0395948 q^{13} +7.19553 q^{14} -4.20392 q^{16} +2.35942 q^{17} +4.25951 q^{18} -0.622913 q^{19} +1.27404 q^{21} -6.21644 q^{22} -4.49034 q^{23} -0.706181 q^{24} +0.0574844 q^{26} +1.52536 q^{27} -0.534122 q^{28} -0.366538 q^{29} +5.71743 q^{31} +0.608974 q^{32} -1.10068 q^{33} -3.42545 q^{34} -0.316182 q^{36} -0.447300 q^{37} +0.904355 q^{38} +0.0101782 q^{39} -1.30887 q^{41} -1.84967 q^{42} -11.6799 q^{43} +0.461444 q^{44} +6.51915 q^{46} -2.91457 q^{47} +1.08065 q^{48} +17.5642 q^{49} -0.606508 q^{51} -0.00426704 q^{52} +5.77092 q^{53} -2.21454 q^{54} -13.6156 q^{56} +0.160125 q^{57} +0.532146 q^{58} +1.75917 q^{59} +6.71285 q^{61} -8.30066 q^{62} +14.5412 q^{63} +7.52373 q^{64} +1.59798 q^{66} +10.5271 q^{67} +0.254270 q^{68} +1.15428 q^{69} -2.65514 q^{71} -8.05999 q^{72} -0.622508 q^{73} +0.649396 q^{74} -0.0671299 q^{76} -21.2218 q^{77} -0.0147768 q^{78} +14.8361 q^{79} +8.40966 q^{81} +1.90024 q^{82} +9.41793 q^{83} +0.137300 q^{84} +16.9570 q^{86} +0.0942214 q^{87} +11.7630 q^{88} -10.5409 q^{89} +0.196241 q^{91} -0.483914 q^{92} -1.46971 q^{93} +4.23142 q^{94} -0.156541 q^{96} +4.20242 q^{97} -25.5000 q^{98} -12.5626 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q - 5 q^{2} - 22 q^{3} + 49 q^{4} + 2 q^{6} - 32 q^{7} - 15 q^{8} + 51 q^{9} - 2 q^{11} - 44 q^{12} - 32 q^{13} - 8 q^{14} + 49 q^{16} - 14 q^{17} - 25 q^{18} + 4 q^{19} + 10 q^{21} - 38 q^{22} - 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) $ 49 * q - 5 * q^2 - 22 * q^3 + 49 * q^4 + 2 * q^6 - 32 * q^7 - 15 * q^8 + 51 * q^9 - 2 * q^11 - 44 * q^12 - 32 * q^13 - 8 * q^14 + 49 * q^16 - 14 * q^17 - 25 * q^18 + 4 * q^19 + 10 * q^21 - 38 * q^22 - 24 * q^23 - 2 * q^26 - 88 * q^27 - 80 * q^28 + 6 * q^29 - 2 * q^31 - 35 * q^32 - 48 * q^33 - 12 * q^34 + 43 * q^36 - 60 * q^37 - 34 * q^38 - 2 * q^41 - 36 * q^42 - 80 * q^43 - 8 * q^44 - 22 * q^46 - 48 * q^47 - 88 * q^48 + 65 * q^49 + 16 * q^51 - 96 * q^52 - 18 * q^53 + 42 * q^54 - 22 * q^56 - 40 * q^57 - 52 * q^58 + 8 * q^59 + 16 * q^61 - 38 * q^62 - 82 * q^63 + 55 * q^64 + 42 * q^66 - 146 * q^67 + 68 * q^68 - 6 * q^69 - 75 * q^72 - 78 * q^73 - 24 * q^74 - 4 * q^76 - 32 * q^77 + 38 * q^78 + 10 * q^79 + 41 * q^81 - 96 * q^82 - 64 * q^83 + 42 * q^84 - 28 * q^86 - 56 * q^87 - 72 * q^88 + 4 * q^89 - 60 * q^92 + 28 * q^93 - 8 * q^94 - 34 * q^96 - 124 * q^97 + 19 * q^98 + 22 * q^99

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 $

$2$	−1.45182	−1.02659	−0.513294	−	0.858213i	$-0.671575\pi$
$2$	−1.45182	−1.02659	−0.513294	+	0.858213i	$0.671575\pi$
$3$	−0.257058	−0.148412	−0.0742061	−	0.997243i	$-0.523642\pi$
$3$	−0.257058	−0.148412	−0.0742061	+	0.997243i	$0.523642\pi$
$4$	0.107768	0.0538838
$5$	0	0
$5$	0	0
$6$	0.373200	0.152358
$7$	−4.95623	−1.87328	−0.936640	−	0.350294i	$-0.886082\pi$
$7$	−4.95623	−1.87328	−0.936640	+	0.350294i	$0.886082\pi$
$8$	2.74717	0.971272
$9$	−2.93392	−0.977974
$10$	0	0
$11$	4.28184	1.29102	0.645512	−	0.763750i	$-0.276644\pi$
$11$	4.28184	1.29102	0.645512	+	0.763750i	$0.276644\pi$
$12$	−0.0277025	−0.00799702
$13$	−0.0395948	−0.0109816	−0.00549082	−	0.999985i	$-0.501748\pi$
$13$	−0.0395948	−0.0109816	−0.00549082	+	0.999985i	$0.501748\pi$
$14$	7.19553	1.92309
$15$	0	0
$16$	−4.20392	−1.05098
$17$	2.35942	0.572244	0.286122	−	0.958193i	$-0.407634\pi$
$17$	2.35942	0.572244	0.286122	+	0.958193i	$0.407634\pi$
$18$	4.25951	1.00398
$19$	−0.622913	−0.142906	−0.0714531	−	0.997444i	$-0.522764\pi$
$19$	−0.622913	−0.142906	−0.0714531	+	0.997444i	$0.522764\pi$
$20$	0	0
$21$	1.27404	0.278018
$22$	−6.21644	−1.32535
$23$	−4.49034	−0.936302	−0.468151	−	0.883649i	$-0.655080\pi$
$23$	−4.49034	−0.936302	−0.468151	+	0.883649i	$0.655080\pi$
$24$	−0.706181	−0.144149
$25$	0	0
$26$	0.0574844	0.0112736
$27$	1.52536	0.293556
$28$	−0.534122	−0.100939
$29$	−0.366538	−0.0680645	−0.0340322	−	0.999421i	$-0.510835\pi$
$29$	−0.366538	−0.0680645	−0.0340322	+	0.999421i	$0.510835\pi$
$30$	0	0
$31$	5.71743	1.02688	0.513441	−	0.858125i	$-0.328371\pi$
$31$	5.71743	1.02688	0.513441	+	0.858125i	$0.328371\pi$
$32$	0.608974	0.107652
$33$	−1.10068	−0.191604
$34$	−3.42545	−0.587459
$35$	0	0
$36$	−0.316182	−0.0526970
$37$	−0.447300	−0.0735356	−0.0367678	−	0.999324i	$-0.511706\pi$
$37$	−0.447300	−0.0735356	−0.0367678	+	0.999324i	$0.511706\pi$
$38$	0.904355	0.146706
$39$	0.0101782	0.00162981
$40$	0	0
$41$	−1.30887	−0.204411	−0.102205	−	0.994763i	$-0.532590\pi$
$41$	−1.30887	−0.204411	−0.102205	+	0.994763i	$0.532590\pi$
$42$	−1.84967	−0.285410
$43$	−11.6799	−1.78116	−0.890581	−	0.454825i	$-0.849702\pi$
$43$	−11.6799	−1.78116	−0.890581	+	0.454825i	$0.849702\pi$
$44$	0.461444	0.0695653
$45$	0	0
$46$	6.51915	0.961196
$47$	−2.91457	−0.425134	−0.212567	−	0.977146i	$-0.568182\pi$
$47$	−2.91457	−0.425134	−0.212567	+	0.977146i	$0.568182\pi$
$48$	1.08065	0.155978
$49$	17.5642	2.50918
$50$	0	0
$51$	−0.606508	−0.0849280
$52$	−0.00426704	−0.000591733 0
$53$	5.77092	0.792697	0.396348	−	0.918100i	$-0.370277\pi$
$53$	5.77092	0.792697	0.396348	+	0.918100i	$0.370277\pi$
$54$	−2.21454	−0.301361
$55$	0	0
$56$	−13.6156	−1.81946
$57$	0.160125	0.0212090
$58$	0.532146	0.0698742
$59$	1.75917	0.229024	0.114512	−	0.993422i	$-0.463470\pi$
$59$	1.75917	0.229024	0.114512	+	0.993422i	$0.463470\pi$
$60$	0	0
$61$	6.71285	0.859493	0.429746	−	0.902950i	$-0.358603\pi$
$61$	6.71285	0.859493	0.429746	+	0.902950i	$0.358603\pi$
$62$	−8.30066	−1.05418
$63$	14.5412	1.83202
$64$	7.52373	0.940466
$65$	0	0
$66$	1.59798	0.196698
$67$	10.5271	1.28609	0.643044	−	0.765829i	$-0.277671\pi$
$67$	10.5271	1.28609	0.643044	+	0.765829i	$0.277671\pi$
$68$	0.254270	0.0308347
$69$	1.15428	0.138959
$70$	0	0
$71$	−2.65514	−0.315107	−0.157554	−	0.987510i	$-0.550361\pi$
$71$	−2.65514	−0.315107	−0.157554	+	0.987510i	$0.550361\pi$
$72$	−8.05999	−0.949878
$73$	−0.622508	−0.0728590	−0.0364295	−	0.999336i	$-0.511598\pi$
$73$	−0.622508	−0.0728590	−0.0364295	+	0.999336i	$0.511598\pi$
$74$	0.649396	0.0754908
$75$	0	0
$76$	−0.0671299	−0.00770033
$77$	−21.2218	−2.41845
$78$	−0.0147768	−0.00167314
$79$	14.8361	1.66919	0.834594	−	0.550865i	$-0.185702\pi$
$79$	14.8361	1.66919	0.834594	+	0.550865i	$0.185702\pi$
$80$	0	0
$81$	8.40966	0.934407
$82$	1.90024	0.209846
$83$	9.41793	1.03375	0.516876	−	0.856060i	$-0.327095\pi$
$83$	9.41793	1.03375	0.516876	+	0.856060i	$0.327095\pi$
$84$	0.137300	0.0149807
$85$	0	0
$86$	16.9570	1.82852
$87$	0.0942214	0.0101016
$88$	11.7630	1.25393
$89$	−10.5409	−1.11733	−0.558667	−	0.829392i	$-0.688687\pi$
$89$	−10.5409	−1.11733	−0.558667	+	0.829392i	$0.688687\pi$
$90$	0	0
$91$	0.196241	0.0205717
$92$	−0.483914	−0.0504515
$93$	−1.46971	−0.152402
$94$	4.23142	0.436438
$95$	0	0
$96$	−0.156541	−0.0159769
$97$	4.20242	0.426691	0.213345	−	0.976977i	$-0.431564\pi$
$97$	4.20242	0.426691	0.213345	+	0.976977i	$0.431564\pi$
$98$	−25.5000	−2.57589
$99$	−12.5626	−1.26259
$100$	0	0
$101$	9.19741	0.915177	0.457588	−	0.889164i	$-0.348713\pi$
$101$	9.19741	0.915177	0.457588	+	0.889164i	$0.348713\pi$
$102$	0.880537	0.0871862
$103$	−12.3007	−1.21202	−0.606012	−	0.795456i	$-0.707232\pi$
$103$	−12.3007	−1.21202	−0.606012	+	0.795456i	$0.707232\pi$
$104$	−0.108774	−0.0106662
$105$	0	0
$106$	−8.37831	−0.813773
$107$	0.189655	0.0183346	0.00916732	−	0.999958i	$-0.497082\pi$
$107$	0.189655	0.0183346	0.00916732	+	0.999958i	$0.497082\pi$
$108$	0.164384	0.0158179
$109$	9.03968	0.865844	0.432922	−	0.901431i	$-0.357483\pi$
$109$	9.03968	0.865844	0.432922	+	0.901431i	$0.357483\pi$
$110$	0	0
$111$	0.114982	0.0109136
$112$	20.8356	1.96878
$113$	−2.29175	−0.215590	−0.107795	−	0.994173i	$-0.534379\pi$
$113$	−2.29175	−0.215590	−0.107795	+	0.994173i	$0.534379\pi$
$114$	−0.232471	−0.0217729
$115$	0	0
$116$	−0.0395010	−0.00366757
$117$	0.116168	0.0107398
$118$	−2.55399	−0.235114
$119$	−11.6938	−1.07197
$120$	0	0
$121$	7.33416	0.666742
$122$	−9.74583	−0.882346
$123$	0.336455	0.0303371
$124$	0.616155	0.0553323
$125$	0	0
$126$	−21.1111	−1.88073
$127$	10.6800	0.947694	0.473847	−	0.880607i	$-0.342865\pi$
$127$	10.6800	0.947694	0.473847	+	0.880607i	$0.342865\pi$
$128$	−12.1410	−1.07312
$129$	3.00240	0.264346
$130$	0	0
$131$	−11.3072	−0.987916	−0.493958	−	0.869486i	$-0.664450\pi$
$131$	−11.3072	−0.987916	−0.493958	+	0.869486i	$0.664450\pi$
$132$	−0.118618	−0.0103243
$133$	3.08730	0.267703
$134$	−15.2834	−1.32028
$135$	0	0
$136$	6.48174	0.555805
$137$	15.4681	1.32153	0.660764	−	0.750593i	$-0.270232\pi$
$137$	15.4681	1.32153	0.660764	+	0.750593i	$0.270232\pi$
$138$	−1.67580	−0.142653
$139$	10.7058	0.908053	0.454027	−	0.890988i	$-0.349987\pi$
$139$	10.7058	0.908053	0.454027	+	0.890988i	$0.349987\pi$
$140$	0	0
$141$	0.749213	0.0630951
$142$	3.85477	0.323485
$143$	−0.169539	−0.0141775
$144$	12.3340	1.02783
$145$	0	0
$146$	0.903766	0.0747962
$147$	−4.51502	−0.372392
$148$	−0.0482044	−0.00396238
$149$	−0.487807	−0.0399627	−0.0199813	−	0.999800i	$-0.506361\pi$
$149$	−0.487807	−0.0399627	−0.0199813	+	0.999800i	$0.506361\pi$
$150$	0	0
$151$	−3.38077	−0.275123	−0.137562	−	0.990493i	$-0.543927\pi$
$151$	−3.38077	−0.275123	−0.137562	+	0.990493i	$0.543927\pi$
$152$	−1.71125	−0.138801
$153$	−6.92236	−0.559640
$154$	30.8101	2.48275
$155$	0	0
$156$	0.00109688	8.78204e−5 0
$157$	−15.3820	−1.22762	−0.613808	−	0.789456i	$-0.710363\pi$
$157$	−15.3820	−1.22762	−0.613808	+	0.789456i	$0.710363\pi$
$158$	−21.5392	−1.71357
$159$	−1.48346	−0.117646
$160$	0	0
$161$	22.2552	1.75395
$162$	−12.2093	−0.959251
$163$	−15.6399	−1.22501	−0.612505	−	0.790467i	$-0.709838\pi$
$163$	−15.6399	−1.22501	−0.612505	+	0.790467i	$0.709838\pi$
$164$	−0.141054	−0.0110144
$165$	0	0
$166$	−13.6731	−1.06124
$167$	−21.8447	−1.69040	−0.845199	−	0.534452i	$-0.820518\pi$
$167$	−21.8447	−1.69040	−0.845199	+	0.534452i	$0.820518\pi$
$168$	3.50000	0.270031
$169$	−12.9984	−0.999879
$170$	0	0
$171$	1.82758	0.139758
$172$	−1.25871	−0.0959758
$173$	−14.1366	−1.07478	−0.537392	−	0.843333i	$-0.680590\pi$
$173$	−14.1366	−1.07478	−0.537392	+	0.843333i	$0.680590\pi$
$174$	−0.136792	−0.0103702
$175$	0	0
$176$	−18.0005	−1.35684
$177$	−0.452207	−0.0339900
$178$	15.3035	1.14704
$179$	−22.8075	−1.70471	−0.852356	−	0.522961i	$-0.824827\pi$
$179$	−22.8075	−1.70471	−0.852356	+	0.522961i	$0.824827\pi$
$180$	0	0
$181$	5.65631	0.420430	0.210215	−	0.977655i	$-0.432584\pi$
$181$	5.65631	0.420430	0.210215	+	0.977655i	$0.432584\pi$
$182$	−0.284906	−0.0211186
$183$	−1.72559	−0.127559
$184$	−12.3357	−0.909403
$185$	0	0
$186$	2.13375	0.156454
$187$	10.1027	0.738781
$188$	−0.314097	−0.0229078
$189$	−7.56003	−0.549911
$190$	0	0
$191$	19.8821	1.43862	0.719308	−	0.694692i	$-0.244459\pi$
$191$	19.8821	1.43862	0.719308	+	0.694692i	$0.244459\pi$
$192$	−1.93403	−0.139577
$193$	−13.4667	−0.969353	−0.484677	−	0.874693i	$-0.661063\pi$
$193$	−13.4667	−0.969353	−0.484677	+	0.874693i	$0.661063\pi$
$194$	−6.10113	−0.438036
$195$	0	0
$196$	1.89286	0.135204
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	18.2386	1.29616
$199$	−7.23463	−0.512849	−0.256425	−	0.966564i	$-0.582545\pi$
$199$	−7.23463	−0.512849	−0.256425	+	0.966564i	$0.582545\pi$
$200$	0	0
$201$	−2.70607	−0.190871
$202$	−13.3529	−0.939510
$203$	1.81665	0.127504
$204$	−0.0653619	−0.00457625
$205$	0	0
$206$	17.8583	1.24425
$207$	13.1743	0.915678
$208$	0.166454	0.0115415
$209$	−2.66722	−0.184495
$210$	0	0
$211$	−10.3252	−0.710814	−0.355407	−	0.934712i	$-0.615658\pi$
$211$	−10.3252	−0.710814	−0.355407	+	0.934712i	$0.615658\pi$
$212$	0.621918	0.0427135
$213$	0.682524	0.0467658
$214$	−0.275344	−0.0188221
$215$	0	0
$216$	4.19042	0.285122
$217$	−28.3369	−1.92364
$218$	−13.1239	−0.888866
$219$	0.160020	0.0108132
$220$	0	0
$221$	−0.0934210	−0.00628418
$222$	−0.166932	−0.0112038
$223$	25.7382	1.72356	0.861779	−	0.507284i	$-0.169351\pi$
$223$	25.7382	1.72356	0.861779	+	0.507284i	$0.169351\pi$
$224$	−3.01822	−0.201663
$225$	0	0
$226$	3.32720	0.221322
$227$	−26.6219	−1.76696	−0.883478	−	0.468473i	$-0.844804\pi$
$227$	−26.6219	−1.76696	−0.883478	+	0.468473i	$0.844804\pi$
$228$	0.0172563	0.00114282
$229$	−0.747925	−0.0494243	−0.0247121	−	0.999695i	$-0.507867\pi$
$229$	−0.747925	−0.0494243	−0.0247121	+	0.999695i	$0.507867\pi$
$230$	0	0
$231$	5.45522	0.358927
$232$	−1.00694	−0.0661091
$233$	−6.30576	−0.413104	−0.206552	−	0.978436i	$-0.566224\pi$
$233$	−6.30576	−0.413104	−0.206552	+	0.978436i	$0.566224\pi$
$234$	−0.168655	−0.0110253
$235$	0	0
$236$	0.189581	0.0123407
$237$	−3.81372	−0.247728
$238$	16.9773	1.10048
$239$	5.24997	0.339593	0.169796	−	0.985479i	$-0.445689\pi$
$239$	5.24997	0.339593	0.169796	+	0.985479i	$0.445689\pi$
$240$	0	0
$241$	−4.37801	−0.282012	−0.141006	−	0.990009i	$-0.545034\pi$
$241$	−4.37801	−0.282012	−0.141006	+	0.990009i	$0.545034\pi$
$242$	−10.6478	−0.684470
$243$	−6.73784	−0.432233
$244$	0.723429	0.0463128
$245$	0	0
$246$	−0.488470	−0.0311437
$247$	0.0246642	0.00156934
$248$	15.7068	0.997381
$249$	−2.42095	−0.153422
$250$	0	0
$251$	−13.0903	−0.826255	−0.413127	−	0.910673i	$-0.635564\pi$
$251$	−13.0903	−0.826255	−0.413127	+	0.910673i	$0.635564\pi$
$252$	1.56707	0.0987162
$253$	−19.2269	−1.20879
$254$	−15.5053	−0.972891
$255$	0	0
$256$	2.57905	0.161191
$257$	−24.5117	−1.52900	−0.764499	−	0.644625i	$-0.777013\pi$
$257$	−24.5117	−1.52900	−0.764499	+	0.644625i	$0.777013\pi$
$258$	−4.35892	−0.271375
$259$	2.21692	0.137753
$260$	0	0
$261$	1.07539	0.0665653
$262$	16.4160	1.01418
$263$	18.5522	1.14398	0.571988	−	0.820262i	$-0.306173\pi$
$263$	18.5522	1.14398	0.571988	+	0.820262i	$0.306173\pi$
$264$	−3.02376	−0.186099
$265$	0	0
$266$	−4.48219	−0.274821
$267$	2.70962	0.165826
$268$	1.13448	0.0692993
$269$	22.8633	1.39400	0.697000	−	0.717071i	$-0.254518\pi$
$269$	22.8633	1.39400	0.697000	+	0.717071i	$0.254518\pi$
$270$	0	0
$271$	−25.1564	−1.52815	−0.764073	−	0.645130i	$-0.776803\pi$
$271$	−25.1564	−1.52815	−0.764073	+	0.645130i	$0.776803\pi$
$272$	−9.91883	−0.601417
$273$	−0.0504453	−0.00305309
$274$	−22.4568	−1.35667
$275$	0	0
$276$	0.124394	0.00748762
$277$	−4.86686	−0.292421	−0.146211	−	0.989253i	$-0.546708\pi$
$277$	−4.86686	−0.292421	−0.146211	+	0.989253i	$0.546708\pi$
$278$	−15.5428	−0.932197
$279$	−16.7745	−1.00426
$280$	0	0
$281$	30.6294	1.82720	0.913599	−	0.406616i	$-0.133291\pi$
$281$	30.6294	1.82720	0.913599	+	0.406616i	$0.133291\pi$
$282$	−1.08772	−0.0647727
$283$	33.4865	1.99057	0.995283	−	0.0970176i	$-0.0309303\pi$
$283$	33.4865	1.99057	0.995283	+	0.0970176i	$0.0309303\pi$
$284$	−0.286138	−0.0169792
$285$	0	0
$286$	0.246139	0.0145545
$287$	6.48706	0.382919
$288$	−1.78668	−0.105281
$289$	−11.4331	−0.672537
$290$	0	0
$291$	−1.08026	−0.0633261
$292$	−0.0670862	−0.00392592
$293$	−15.1265	−0.883697	−0.441849	−	0.897090i	$-0.645677\pi$
$293$	−15.1265	−0.883697	−0.441849	+	0.897090i	$0.645677\pi$
$294$	6.55497	0.382294
$295$	0	0
$296$	−1.22881	−0.0714231
$297$	6.53135	0.378987
$298$	0.708205	0.0410252
$299$	0.177795	0.0102821
$300$	0	0
$301$	57.8881	3.33661
$302$	4.90826	0.282439
$303$	−2.36426	−0.135823
$304$	2.61868	0.150192
$305$	0	0
$306$	10.0500	0.574520
$307$	−16.5626	−0.945276	−0.472638	−	0.881257i	$-0.656698\pi$
$307$	−16.5626	−0.945276	−0.472638	+	0.881257i	$0.656698\pi$
$308$	−2.28702	−0.130315
$309$	3.16199	0.179879
$310$	0	0
$311$	−11.7823	−0.668112	−0.334056	−	0.942553i	$-0.608418\pi$
$311$	−11.7823	−0.668112	−0.334056	+	0.942553i	$0.608418\pi$
$312$	0.0279611	0.00158299
$313$	−11.5586	−0.653333	−0.326666	−	0.945140i	$-0.605925\pi$
$313$	−11.5586	−0.653333	−0.326666	+	0.945140i	$0.605925\pi$
$314$	22.3318	1.26026
$315$	0	0
$316$	1.59885	0.0899423
$317$	20.0168	1.12425	0.562127	−	0.827051i	$-0.309983\pi$
$317$	20.0168	1.12425	0.562127	+	0.827051i	$0.309983\pi$
$318$	2.15371	0.120774
$319$	−1.56946	−0.0878728
$320$	0	0
$321$	−0.0487523	−0.00272109
$322$	−32.3104	−1.80059
$323$	−1.46972	−0.0817772
$324$	0.906289	0.0503494
$325$	0	0
$326$	22.7062	1.25758
$327$	−2.32372	−0.128502
$328$	−3.59569	−0.198539
$329$	14.4453	0.796395
$330$	0	0
$331$	−14.1397	−0.777186	−0.388593	−	0.921410i	$-0.627039\pi$
$331$	−14.1397	−0.777186	−0.388593	+	0.921410i	$0.627039\pi$
$332$	1.01495	0.0557025
$333$	1.31234	0.0719159
$334$	31.7145	1.73534
$335$	0	0
$336$	−5.35595	−0.292191
$337$	24.6996	1.34547	0.672737	−	0.739882i	$-0.265118\pi$
$337$	24.6996	1.34547	0.672737	+	0.739882i	$0.265118\pi$
$338$	18.8713	1.02646
$339$	0.589112	0.0319962
$340$	0	0
$341$	24.4811	1.32573
$342$	−2.65331	−0.143474
$343$	−52.3588	−2.82711
$344$	−32.0866	−1.72999
$345$	0	0
$346$	20.5237	1.10336
$347$	−33.1998	−1.78226	−0.891130	−	0.453749i	$-0.850086\pi$
$347$	−33.1998	−1.78226	−0.891130	+	0.453749i	$0.850086\pi$
$348$	0.0101540	0.000544313 0
$349$	−3.85836	−0.206533	−0.103267	−	0.994654i	$-0.532929\pi$
$349$	−3.85836	−0.206533	−0.103267	+	0.994654i	$0.532929\pi$
$350$	0	0
$351$	−0.0603964	−0.00322372
$352$	2.60753	0.138982
$353$	28.5121	1.51754	0.758772	−	0.651356i	$-0.225800\pi$
$353$	28.5121	1.51754	0.758772	+	0.651356i	$0.225800\pi$
$354$	0.656522	0.0348937
$355$	0	0
$356$	−1.13597	−0.0602063
$357$	3.00599	0.159094
$358$	33.1123	1.75004
$359$	−13.5351	−0.714354	−0.357177	−	0.934037i	$-0.616261\pi$
$359$	−13.5351	−0.714354	−0.357177	+	0.934037i	$0.616261\pi$
$360$	0	0
$361$	−18.6120	−0.979578
$362$	−8.21192	−0.431609
$363$	−1.88530	−0.0989527
$364$	0.0211485	0.00110848
$365$	0	0
$366$	2.50524	0.130951
$367$	5.02879	0.262501	0.131250	−	0.991349i	$-0.458101\pi$
$367$	5.02879	0.262501	0.131250	+	0.991349i	$0.458101\pi$
$368$	18.8771	0.984035
$369$	3.84012	0.199909
$370$	0	0
$371$	−28.6020	−1.48494
$372$	−0.158387	−0.00821199
$373$	2.49864	0.129375	0.0646874	−	0.997906i	$-0.479395\pi$
$373$	2.49864	0.129375	0.0646874	+	0.997906i	$0.479395\pi$
$374$	−14.6672	−0.758424
$375$	0	0
$376$	−8.00683	−0.412921
$377$	0.0145130	0.000747459 0
$378$	10.9758	0.564533
$379$	30.1669	1.54957	0.774785	−	0.632225i	$-0.217858\pi$
$379$	30.1669	1.54957	0.774785	+	0.632225i	$0.217858\pi$
$380$	0	0
$381$	−2.74537	−0.140649
$382$	−28.8651	−1.47687
$383$	21.7802	1.11292	0.556459	−	0.830875i	$-0.312160\pi$
$383$	21.7802	1.11292	0.556459	+	0.830875i	$0.312160\pi$
$384$	3.12094	0.159265
$385$	0	0
$386$	19.5511	0.995127
$387$	34.2678	1.74193
$388$	0.452885	0.0229917
$389$	−13.3871	−0.678755	−0.339377	−	0.940650i	$-0.610216\pi$
$389$	−13.3871	−0.678755	−0.339377	+	0.940650i	$0.610216\pi$
$390$	0	0
$391$	−10.5946	−0.535793
$392$	48.2520	2.43709
$393$	2.90660	0.146619
$394$	−1.45182	−0.0731414
$395$	0	0
$396$	−1.35384	−0.0680330
$397$	15.9000	0.797996	0.398998	−	0.916952i	$-0.369358\pi$
$397$	15.9000	0.797996	0.398998	+	0.916952i	$0.369358\pi$
$398$	10.5033	0.526485
$399$	−0.793614	−0.0397304
$400$	0	0
$401$	−30.1704	−1.50664	−0.753319	−	0.657655i	$-0.771548\pi$
$401$	−30.1704	−1.50664	−0.753319	+	0.657655i	$0.771548\pi$
$402$	3.92871	0.195946
$403$	−0.226381	−0.0112768
$404$	0.991184	0.0493132
$405$	0	0
$406$	−2.63744	−0.130894
$407$	−1.91527	−0.0949362
$408$	−1.66618	−0.0824882
$409$	−10.5035	−0.519367	−0.259683	−	0.965694i	$-0.583618\pi$
$409$	−10.5035	−0.519367	−0.259683	+	0.965694i	$0.583618\pi$
$410$	0	0
$411$	−3.97619	−0.196131
$412$	−1.32562	−0.0653085
$413$	−8.71884	−0.429026
$414$	−19.1267	−0.940025
$415$	0	0
$416$	−0.0241122	−0.00118220
$417$	−2.75200	−0.134766
$418$	3.87230	0.189401
$419$	−12.8054	−0.625586	−0.312793	−	0.949821i	$-0.601265\pi$
$419$	−12.8054	−0.625586	−0.312793	+	0.949821i	$0.601265\pi$
$420$	0	0
$421$	−10.0841	−0.491469	−0.245734	−	0.969337i	$-0.579029\pi$
$421$	−10.0841	−0.491469	−0.245734	+	0.969337i	$0.579029\pi$
$422$	14.9902	0.729713
$423$	8.55112	0.415770
$424$	15.8537	0.769924
$425$	0	0
$426$	−0.990899	−0.0480092
$427$	−33.2705	−1.61007
$428$	0.0204387	0.000987941 0
$429$	0.0435812	0.00210412
$430$	0	0
$431$	16.4580	0.792753	0.396376	−	0.918088i	$-0.370268\pi$
$431$	16.4580	0.792753	0.396376	+	0.918088i	$0.370268\pi$
$432$	−6.41249	−0.308521
$433$	−33.8333	−1.62593	−0.812963	−	0.582315i	$-0.802147\pi$
$433$	−33.8333	−1.62593	−0.812963	+	0.582315i	$0.802147\pi$
$434$	41.1400	1.97478
$435$	0	0
$436$	0.974185	0.0466550
$437$	2.79710	0.133803
$438$	−0.232320	−0.0111007
$439$	−28.6744	−1.36856	−0.684278	−	0.729221i	$-0.739882\pi$
$439$	−28.6744	−1.36856	−0.684278	+	0.729221i	$0.739882\pi$
$440$	0	0
$441$	−51.5321	−2.45391
$442$	0.135630	0.00645126
$443$	18.6714	0.887104	0.443552	−	0.896249i	$-0.353718\pi$
$443$	18.6714	0.887104	0.443552	+	0.896249i	$0.353718\pi$
$444$	0.0123913	0.000588066 0
$445$	0	0
$446$	−37.3671	−1.76938
$447$	0.125394	0.00593095
$448$	−37.2893	−1.76175
$449$	−22.7683	−1.07450	−0.537250	−	0.843423i	$-0.680537\pi$
$449$	−22.7683	−1.07450	−0.537250	+	0.843423i	$0.680537\pi$
$450$	0	0
$451$	−5.60437	−0.263899
$452$	−0.246977	−0.0116168
$453$	0.869053	0.0408317
$454$	38.6500	1.81394
$455$	0	0
$456$	0.439890	0.0205997
$457$	4.08443	0.191062	0.0955309	−	0.995426i	$-0.469545\pi$
$457$	4.08443	0.191062	0.0955309	+	0.995426i	$0.469545\pi$
$458$	1.08585	0.0507384
$459$	3.59897	0.167985
$460$	0	0
$461$	−25.3244	−1.17947	−0.589737	−	0.807595i	$-0.700769\pi$
$461$	−25.3244	−1.17947	−0.589737	+	0.807595i	$0.700769\pi$
$462$	−7.91997	−0.368471
$463$	−29.3448	−1.36377	−0.681883	−	0.731461i	$-0.738839\pi$
$463$	−29.3448	−1.36377	−0.681883	+	0.731461i	$0.738839\pi$
$464$	1.54090	0.0715344
$465$	0	0
$466$	9.15480	0.424088
$467$	29.7840	1.37824	0.689119	−	0.724648i	$-0.257998\pi$
$467$	29.7840	1.37824	0.689119	+	0.724648i	$0.257998\pi$
$468$	0.0125192	0.000578699 0
$469$	−52.1746	−2.40920
$470$	0	0
$471$	3.95405	0.182193
$472$	4.83274	0.222445
$473$	−50.0113	−2.29952
$474$	5.53682	0.254315
$475$	0	0
$476$	−1.26022	−0.0577620
$477$	−16.9314	−0.775236
$478$	−7.62199	−0.348622
$479$	17.1420	0.783238	0.391619	−	0.920127i	$-0.371915\pi$
$479$	17.1420	0.783238	0.391619	+	0.920127i	$0.371915\pi$
$480$	0	0
$481$	0.0177108	0.000807541 0
$482$	6.35606	0.289510
$483$	−5.72086	−0.260308
$484$	0.790385	0.0359266
$485$	0	0
$486$	9.78211	0.443725
$487$	−16.0308	−0.726423	−0.363212	−	0.931707i	$-0.618320\pi$
$487$	−16.0308	−0.726423	−0.363212	+	0.931707i	$0.618320\pi$
$488$	18.4414	0.834801
$489$	4.02035	0.181807
$490$	0	0
$491$	−42.9924	−1.94022	−0.970109	−	0.242669i	$-0.921977\pi$
$491$	−42.9924	−1.94022	−0.970109	+	0.242669i	$0.921977\pi$
$492$	0.0362589	0.00163468
$493$	−0.864819	−0.0389495
$494$	−0.0358078	−0.00161107
$495$	0	0
$496$	−24.0356	−1.07923
$497$	13.1595	0.590284
$498$	3.51477	0.157501
$499$	−25.7746	−1.15383	−0.576915	−	0.816804i	$-0.695744\pi$
$499$	−25.7746	−1.15383	−0.576915	+	0.816804i	$0.695744\pi$
$500$	0	0
$501$	5.61536	0.250876
$502$	19.0047	0.848223
$503$	−31.3319	−1.39702	−0.698510	−	0.715600i	$-0.746153\pi$
$503$	−31.3319	−1.39702	−0.698510	+	0.715600i	$0.746153\pi$
$504$	39.9472	1.77939
$505$	0	0
$506$	27.9140	1.24093
$507$	3.34135	0.148394
$508$	1.15095	0.0510654
$509$	−0.670154	−0.0297041	−0.0148520	−	0.999890i	$-0.504728\pi$
$509$	−0.670154	−0.0297041	−0.0148520	+	0.999890i	$0.504728\pi$
$510$	0	0
$511$	3.08529	0.136485
$512$	20.5377	0.907647
$513$	−0.950167	−0.0419509
$514$	35.5865	1.56965
$515$	0	0
$516$	0.323561	0.0142440
$517$	−12.4797	−0.548858
$518$	−3.21856	−0.141415
$519$	3.63391	0.159511
$520$	0	0
$521$	−34.2371	−1.49995	−0.749976	−	0.661465i	$-0.769935\pi$
$521$	−34.2371	−1.49995	−0.749976	+	0.661465i	$0.769935\pi$
$522$	−1.56127	−0.0683351
$523$	−14.6338	−0.639893	−0.319947	−	0.947436i	$-0.603665\pi$
$523$	−14.6338	−0.639893	−0.319947	+	0.947436i	$0.603665\pi$
$524$	−1.21855	−0.0532327
$525$	0	0
$526$	−26.9343	−1.17439
$527$	13.4898	0.587627
$528$	4.62717	0.201372
$529$	−2.83681	−0.123339
$530$	0	0
$531$	−5.16126	−0.223980
$532$	0.332711	0.0144249
$533$	0.0518245	0.00224477
$534$	−3.93387	−0.170235
$535$	0	0
$536$	28.9197	1.24914
$537$	5.86284	0.253000
$538$	−33.1933	−1.43106
$539$	75.2072	3.23940
$540$	0	0
$541$	26.9024	1.15663	0.578313	−	0.815815i	$-0.303711\pi$
$541$	26.9024	1.15663	0.578313	+	0.815815i	$0.303711\pi$
$542$	36.5225	1.56878
$543$	−1.45400	−0.0623970
$544$	1.43683	0.0616035
$545$	0	0
$546$	0.0732372	0.00313426
$547$	−13.7772	−0.589071	−0.294535	−	0.955641i	$-0.595165\pi$
$547$	−13.7772	−0.589071	−0.294535	+	0.955641i	$0.595165\pi$
$548$	1.66696	0.0712090
$549$	−19.6950	−0.840562
$550$	0	0
$551$	0.228322	0.00972683
$552$	3.17100	0.134967
$553$	−73.5310	−3.12686
$554$	7.06578	0.300196
$555$	0	0
$556$	1.15374	0.0489294
$557$	−40.8747	−1.73192	−0.865959	−	0.500115i	$-0.833291\pi$
$557$	−40.8747	−1.73192	−0.865959	+	0.500115i	$0.833291\pi$
$558$	24.3535	1.03096
$559$	0.462462	0.0195601
$560$	0	0
$561$	−2.59697	−0.109644
$562$	−44.4683	−1.87578
$563$	36.5476	1.54030	0.770149	−	0.637864i	$-0.220182\pi$
$563$	36.5476	1.54030	0.770149	+	0.637864i	$0.220182\pi$
$564$	0.0807409	0.00339981
$565$	0	0
$566$	−48.6162	−2.04349
$567$	−41.6802	−1.75040
$568$	−7.29413	−0.306055
$569$	−0.561509	−0.0235397	−0.0117698	−	0.999931i	$-0.503747\pi$
$569$	−0.561509	−0.0235397	−0.0117698	+	0.999931i	$0.503747\pi$
$570$	0	0
$571$	−10.8440	−0.453807	−0.226904	−	0.973917i	$-0.572860\pi$
$571$	−10.8440	−0.453807	−0.226904	+	0.973917i	$0.572860\pi$
$572$	−0.0182708	−0.000763941 0
$573$	−5.11083	−0.213508
$574$	−9.41801	−0.393100
$575$	0	0
$576$	−22.0740	−0.919751
$577$	19.9454	0.830338	0.415169	−	0.909744i	$-0.363723\pi$
$577$	19.9454	0.830338	0.415169	+	0.909744i	$0.363723\pi$
$578$	16.5988	0.690418
$579$	3.46171	0.143864
$580$	0	0
$581$	−46.6775	−1.93651
$582$	1.56834	0.0650099
$583$	24.7102	1.02339
$584$	−1.71014	−0.0707659
$585$	0	0
$586$	21.9608	0.907194
$587$	34.0002	1.40334	0.701668	−	0.712504i	$-0.252439\pi$
$587$	34.0002	1.40334	0.701668	+	0.712504i	$0.252439\pi$
$588$	−0.486573	−0.0200659
$589$	−3.56147	−0.146748
$590$	0	0
$591$	−0.257058	−0.0105739
$592$	1.88041	0.0772845
$593$	−0.289835	−0.0119021	−0.00595105	−	0.999982i	$-0.501894\pi$
$593$	−0.289835	−0.0119021	−0.00595105	+	0.999982i	$0.501894\pi$
$594$	−9.48231	−0.389064
$595$	0	0
$596$	−0.0525698	−0.00215334
$597$	1.85972	0.0761131
$598$	−0.258125	−0.0105555
$599$	−23.3847	−0.955473	−0.477736	−	0.878503i	$-0.658543\pi$
$599$	−23.3847	−0.955473	−0.477736	+	0.878503i	$0.658543\pi$
$600$	0	0
$601$	47.2544	1.92755	0.963773	−	0.266725i	$-0.0859417\pi$
$601$	47.2544	1.92755	0.963773	+	0.266725i	$0.0859417\pi$
$602$	−84.0428	−3.42533
$603$	−30.8856	−1.25776
$604$	−0.364338	−0.0148247
$605$	0	0
$606$	3.43247	0.139435
$607$	−41.4883	−1.68396	−0.841980	−	0.539509i	$-0.818610\pi$
$607$	−41.4883	−1.68396	−0.841980	+	0.539509i	$0.818610\pi$
$608$	−0.379338	−0.0153842
$609$	−0.466983	−0.0189231
$610$	0	0
$611$	0.115402	0.00466867
$612$	−0.746007	−0.0301555
$613$	39.8431	1.60925	0.804625	−	0.593784i	$-0.202366\pi$
$613$	39.8431	1.60925	0.804625	+	0.593784i	$0.202366\pi$
$614$	24.0458	0.970409
$615$	0	0
$616$	−58.2999	−2.34897
$617$	−10.8488	−0.436758	−0.218379	−	0.975864i	$-0.570077\pi$
$617$	−10.8488	−0.436758	−0.218379	+	0.975864i	$0.570077\pi$
$618$	−4.59062	−0.184662
$619$	14.3789	0.577937	0.288969	−	0.957339i	$-0.406688\pi$
$619$	14.3789	0.577937	0.288969	+	0.957339i	$0.406688\pi$
$620$	0	0
$621$	−6.84939	−0.274857
$622$	17.1057	0.685876
$623$	52.2432	2.09308
$624$	−0.0427882	−0.00171290
$625$	0	0
$626$	16.7810	0.670704
$627$	0.685628	0.0273813
$628$	−1.65768	−0.0661486
$629$	−1.05537	−0.0420803
$630$	0	0
$631$	−29.2361	−1.16387	−0.581935	−	0.813236i	$-0.697704\pi$
$631$	−29.2361	−1.16387	−0.581935	+	0.813236i	$0.697704\pi$
$632$	40.7572	1.62124
$633$	2.65416	0.105494
$634$	−29.0607	−1.15415
$635$	0	0
$636$	−0.159869	−0.00633921
$637$	−0.695453	−0.0275548
$638$	2.27856	0.0902092
$639$	7.78997	0.308167
$640$	0	0
$641$	−5.02891	−0.198630	−0.0993149	−	0.995056i	$-0.531665\pi$
$641$	−5.02891	−0.198630	−0.0993149	+	0.995056i	$0.531665\pi$
$642$	0.0707793	0.00279344
$643$	21.5016	0.847940	0.423970	−	0.905676i	$-0.360636\pi$
$643$	21.5016	0.847940	0.423970	+	0.905676i	$0.360636\pi$
$644$	2.39839	0.0945098
$645$	0	0
$646$	2.13376	0.0839515
$647$	16.0881	0.632487	0.316243	−	0.948678i	$-0.397578\pi$
$647$	16.0881	0.632487	0.316243	+	0.948678i	$0.397578\pi$
$648$	23.1028	0.907563
$649$	7.53248	0.295676
$650$	0	0
$651$	7.28422	0.285491
$652$	−1.68547	−0.0660083
$653$	−29.9566	−1.17229	−0.586146	−	0.810205i	$-0.699356\pi$
$653$	−29.9566	−1.17229	−0.586146	+	0.810205i	$0.699356\pi$
$654$	3.37361	0.131919
$655$	0	0
$656$	5.50238	0.214832
$657$	1.82639	0.0712542
$658$	−20.9719	−0.817570
$659$	14.8597	0.578853	0.289426	−	0.957200i	$-0.406535\pi$
$659$	14.8597	0.578853	0.289426	+	0.957200i	$0.406535\pi$
$660$	0	0
$661$	10.7252	0.417161	0.208581	−	0.978005i	$-0.433116\pi$
$661$	10.7252	0.417161	0.208581	+	0.978005i	$0.433116\pi$
$662$	20.5282	0.797850
$663$	0.0240146	0.000932649 0
$664$	25.8727	1.00405
$665$	0	0
$666$	−1.90528	−0.0738280
$667$	1.64588	0.0637289
$668$	−2.35416	−0.0910851
$669$	−6.61620	−0.255797
$670$	0	0
$671$	28.7434	1.10963
$672$	0.775855	0.0299293
$673$	17.9563	0.692164	0.346082	−	0.938204i	$-0.387512\pi$
$673$	17.9563	0.692164	0.346082	+	0.938204i	$0.387512\pi$
$674$	−35.8593	−1.38125
$675$	0	0
$676$	−1.40081	−0.0538773
$677$	21.3175	0.819298	0.409649	−	0.912243i	$-0.365651\pi$
$677$	21.3175	0.819298	0.409649	+	0.912243i	$0.365651\pi$
$678$	−0.855281	−0.0328469
$679$	−20.8281	−0.799311
$680$	0	0
$681$	6.84335	0.262238
$682$	−35.5421	−1.36098
$683$	−2.40314	−0.0919538	−0.0459769	−	0.998943i	$-0.514640\pi$
$683$	−2.40314	−0.0919538	−0.0459769	+	0.998943i	$0.514640\pi$
$684$	0.196954	0.00753072
$685$	0	0
$686$	76.0152	2.90228
$687$	0.192260	0.00733517
$688$	49.1012	1.87197
$689$	−0.228499	−0.00870510
$690$	0	0
$691$	−3.06820	−0.116720	−0.0583599	−	0.998296i	$-0.518587\pi$
$691$	−3.06820	−0.116720	−0.0583599	+	0.998296i	$0.518587\pi$
$692$	−1.52346	−0.0579134
$693$	62.2631	2.36518
$694$	48.2000	1.82965
$695$	0	0
$696$	0.258842	0.00981140
$697$	−3.08818	−0.116973
$698$	5.60162	0.212024
$699$	1.62094	0.0613097
$700$	0	0
$701$	21.7249	0.820537	0.410268	−	0.911965i	$-0.365435\pi$
$701$	21.7249	0.820537	0.410268	+	0.911965i	$0.365435\pi$
$702$	0.0876844	0.00330943
$703$	0.278629	0.0105087
$704$	32.2154	1.21416
$705$	0	0
$706$	−41.3943	−1.55789
$707$	−45.5845	−1.71438
$708$	−0.0487333	−0.00183151
$709$	−0.452624	−0.0169986	−0.00849932	−	0.999964i	$-0.502705\pi$
$709$	−0.452624	−0.0169986	−0.00849932	+	0.999964i	$0.502705\pi$
$710$	0	0
$711$	−43.5279	−1.63242
$712$	−28.9577	−1.08524
$713$	−25.6732	−0.961471
$714$	−4.36414	−0.163324
$715$	0	0
$716$	−2.45791	−0.0918565
$717$	−1.34955	−0.0503997
$718$	19.6504	0.733348
$719$	−40.8858	−1.52478	−0.762392	−	0.647115i	$-0.775975\pi$
$719$	−40.8858	−1.52478	−0.762392	+	0.647115i	$0.775975\pi$
$720$	0	0
$721$	60.9651	2.27046
$722$	27.0212	1.00562
$723$	1.12540	0.0418541
$724$	0.609567	0.0226544
$725$	0	0
$726$	2.73711	0.101584
$727$	29.9609	1.11119	0.555594	−	0.831454i	$-0.312491\pi$
$727$	29.9609	1.11119	0.555594	+	0.831454i	$0.312491\pi$
$728$	0.539108	0.0199807
$729$	−23.4970	−0.870258
$730$	0	0
$731$	−27.5577	−1.01926
$732$	−0.185963	−0.00687338
$733$	32.6206	1.20487	0.602434	−	0.798169i	$-0.294198\pi$
$733$	32.6206	1.20487	0.602434	+	0.798169i	$0.294198\pi$
$734$	−7.30087	−0.269480
$735$	0	0
$736$	−2.73450	−0.100795
$737$	45.0753	1.66037
$738$	−5.57514	−0.205224
$739$	−36.0853	−1.32742	−0.663709	−	0.747991i	$-0.731019\pi$
$739$	−36.0853	−1.32742	−0.663709	+	0.747991i	$0.731019\pi$
$740$	0	0
$741$	−0.00634011	−0.000232910 0
$742$	41.5248	1.52442
$743$	−3.64167	−0.133600	−0.0668000	−	0.997766i	$-0.521279\pi$
$743$	−3.64167	−0.133600	−0.0668000	+	0.997766i	$0.521279\pi$
$744$	−4.03754	−0.148024
$745$	0	0
$746$	−3.62757	−0.132815
$747$	−27.6315	−1.01098
$748$	1.08874	0.0398083
$749$	−0.939974	−0.0343459
$750$	0	0
$751$	−53.2512	−1.94316	−0.971582	−	0.236704i	$-0.923933\pi$
$751$	−53.2512	−1.94316	−0.971582	+	0.236704i	$0.923933\pi$
$752$	12.2526	0.446807
$753$	3.36497	0.122626
$754$	−0.0210702	−0.000767333 0
$755$	0	0
$756$	−0.814727	−0.0296313
$757$	−32.1949	−1.17014	−0.585072	−	0.810982i	$-0.698934\pi$
$757$	−32.1949	−1.17014	−0.585072	+	0.810982i	$0.698934\pi$
$758$	−43.7968	−1.59077
$759$	4.94243	0.179399
$760$	0	0
$761$	−15.9438	−0.577961	−0.288981	−	0.957335i	$-0.593316\pi$
$761$	−15.9438	−0.577961	−0.288981	+	0.957335i	$0.593316\pi$
$762$	3.98576	0.144389
$763$	−44.8027	−1.62197
$764$	2.14264	0.0775181
$765$	0	0
$766$	−31.6209	−1.14251
$767$	−0.0696540	−0.00251506
$768$	−0.662964	−0.0239227
$769$	−11.4972	−0.414598	−0.207299	−	0.978278i	$-0.566467\pi$
$769$	−11.4972	−0.414598	−0.207299	+	0.978278i	$0.566467\pi$
$770$	0	0
$771$	6.30092	0.226922
$772$	−1.45127	−0.0522325
$773$	17.9213	0.644583	0.322292	−	0.946640i	$-0.395547\pi$
$773$	17.9213	0.644583	0.322292	+	0.946640i	$0.395547\pi$
$774$	−49.7505	−1.78824
$775$	0	0
$776$	11.5448	0.414433
$777$	−0.569876	−0.0204442
$778$	19.4356	0.696802
$779$	0.815312	0.0292116
$780$	0	0
$781$	−11.3689	−0.406811
$782$	15.3814	0.550039
$783$	−0.559103	−0.0199807
$784$	−73.8386	−2.63709
$785$	0	0
$786$	−4.21985	−0.150517
$787$	−15.1463	−0.539906	−0.269953	−	0.962873i	$-0.587008\pi$
$787$	−15.1463	−0.539906	−0.269953	+	0.962873i	$0.587008\pi$
$788$	0.107768	0.00383906
$789$	−4.76897	−0.169780
$790$	0	0
$791$	11.3584	0.403860
$792$	−34.5116	−1.22632
$793$	−0.265794	−0.00943864
$794$	−23.0838	−0.819214
$795$	0	0
$796$	−0.779659	−0.0276343
$797$	10.5416	0.373403	0.186701	−	0.982417i	$-0.440220\pi$
$797$	10.5416	0.373403	0.186701	+	0.982417i	$0.440220\pi$
$798$	1.15218	0.0407868
$799$	−6.87671	−0.243280
$800$	0	0
$801$	30.9262	1.09272
$802$	43.8019	1.54670
$803$	−2.66548	−0.0940627
$804$	−0.291626	−0.0102849
$805$	0	0
$806$	0.328663	0.0115767
$807$	−5.87718	−0.206887
$808$	25.2669	0.888885
$809$	−22.7520	−0.799917	−0.399959	−	0.916533i	$-0.630976\pi$
$809$	−22.7520	−0.799917	−0.399959	+	0.916533i	$0.630976\pi$
$810$	0	0
$811$	−46.8973	−1.64679	−0.823394	−	0.567470i	$-0.807922\pi$
$811$	−46.8973	−1.64679	−0.823394	+	0.567470i	$0.807922\pi$
$812$	0.195776	0.00687039
$813$	6.46665	0.226795
$814$	2.78061	0.0974604
$815$	0	0
$816$	2.54971	0.0892577
$817$	7.27554	0.254539
$818$	15.2492	0.533176
$819$	−0.575756	−0.0201186
$820$	0	0
$821$	−16.1605	−0.564007	−0.282003	−	0.959413i	$-0.590999\pi$
$821$	−16.1605	−0.564007	−0.282003	+	0.959413i	$0.590999\pi$
$822$	5.77269	0.201346
$823$	−36.7619	−1.28144	−0.640719	−	0.767775i	$-0.721364\pi$
$823$	−36.7619	−1.28144	−0.640719	+	0.767775i	$0.721364\pi$
$824$	−33.7921	−1.17720
$825$	0	0
$826$	12.6581	0.440433
$827$	11.0539	0.384381	0.192190	−	0.981358i	$-0.438441\pi$
$827$	11.0539	0.384381	0.192190	+	0.981358i	$0.438441\pi$
$828$	1.41977	0.0493403
$829$	44.8204	1.55668	0.778339	−	0.627845i	$-0.216063\pi$
$829$	44.8204	1.55668	0.778339	+	0.627845i	$0.216063\pi$
$830$	0	0
$831$	1.25106	0.0433989
$832$	−0.297901	−0.0103279
$833$	41.4415	1.43586
$834$	3.99540	0.138349
$835$	0	0
$836$	−0.287440	−0.00994131
$837$	8.72114	0.301447
$838$	18.5911	0.642219
$839$	46.6571	1.61078	0.805391	−	0.592744i	$-0.201955\pi$
$839$	46.6571	1.61078	0.805391	+	0.592744i	$0.201955\pi$
$840$	0	0
$841$	−28.8656	−0.995367
$842$	14.6403	0.504536
$843$	−7.87352	−0.271179
$844$	−1.11272	−0.0383014
$845$	0	0
$846$	−12.4147	−0.426825
$847$	−36.3498	−1.24899
$848$	−24.2605	−0.833109
$849$	−8.60796	−0.295424
$850$	0	0
$851$	2.00853	0.0688515
$852$	0.0735540	0.00251992
$853$	9.30749	0.318682	0.159341	−	0.987224i	$-0.449063\pi$
$853$	9.30749	0.318682	0.159341	+	0.987224i	$0.449063\pi$
$854$	48.3026	1.65288
$855$	0	0
$856$	0.521015	0.0178079
$857$	3.16972	0.108276	0.0541378	−	0.998533i	$-0.482759\pi$
$857$	3.16972	0.108276	0.0541378	+	0.998533i	$0.482759\pi$
$858$	−0.0632719	−0.00216007
$859$	23.8272	0.812973	0.406486	−	0.913657i	$-0.366754\pi$
$859$	23.8272	0.812973	0.406486	+	0.913657i	$0.366754\pi$
$860$	0	0
$861$	−1.66755	−0.0568298
$862$	−23.8939	−0.813831
$863$	1.45888	0.0496608	0.0248304	−	0.999692i	$-0.492095\pi$
$863$	1.45888	0.0496608	0.0248304	+	0.999692i	$0.492095\pi$
$864$	0.928904	0.0316020
$865$	0	0
$866$	49.1198	1.66916
$867$	2.93897	0.0998127
$868$	−3.05380	−0.103653
$869$	63.5257	2.15496
$870$	0	0
$871$	−0.416818	−0.0141233
$872$	24.8335	0.840970
$873$	−12.3296	−0.417292
$874$	−4.06087	−0.137361
$875$	0	0
$876$	0.0172450	0.000582655 0
$877$	31.0847	1.04966	0.524828	−	0.851208i	$-0.324130\pi$
$877$	31.0847	1.04966	0.524828	+	0.851208i	$0.324130\pi$
$878$	41.6300	1.40494
$879$	3.88837	0.131152
$880$	0	0
$881$	16.2242	0.546608	0.273304	−	0.961928i	$-0.411884\pi$
$881$	16.2242	0.546608	0.273304	+	0.961928i	$0.411884\pi$
$882$	74.8150	2.51915
$883$	−36.7133	−1.23550	−0.617751	−	0.786374i	$-0.711956\pi$
$883$	−36.7133	−1.23550	−0.617751	+	0.786374i	$0.711956\pi$
$884$	−0.0100678	−0.000338616 0
$885$	0	0
$886$	−27.1074	−0.910691
$887$	55.8502	1.87527	0.937633	−	0.347627i	$-0.113012\pi$
$887$	55.8502	1.87527	0.937633	+	0.347627i	$0.113012\pi$
$888$	0.315875	0.0106001
$889$	−52.9324	−1.77529
$890$	0	0
$891$	36.0088	1.20634
$892$	2.77375	0.0928719
$893$	1.81553	0.0607542
$894$	−0.182050	−0.00608865
$895$	0	0
$896$	60.1736	2.01026
$897$	−0.0457034	−0.00152599
$898$	33.0553	1.10307
$899$	−2.09566	−0.0698941
$900$	0	0
$901$	13.6160	0.453616
$902$	8.13651	0.270916
$903$	−14.8806	−0.495194
$904$	−6.29583	−0.209396
$905$	0	0
$906$	−1.26170	−0.0419173
$907$	43.3615	1.43979	0.719897	−	0.694081i	$-0.244189\pi$
$907$	43.3615	1.43979	0.719897	+	0.694081i	$0.244189\pi$
$908$	−2.86898	−0.0952104
$909$	−26.9845	−0.895019
$910$	0	0
$911$	−35.9819	−1.19214	−0.596068	−	0.802934i	$-0.703271\pi$
$911$	−35.9819	−1.19214	−0.596068	+	0.802934i	$0.703271\pi$
$912$	−0.673151	−0.0222903
$913$	40.3261	1.33460
$914$	−5.92984	−0.196142
$915$	0	0
$916$	−0.0806022	−0.00266317
$917$	56.0411	1.85064
$918$	−5.22504	−0.172452
$919$	41.1384	1.35703	0.678514	−	0.734587i	$-0.262624\pi$
$919$	41.1384	1.35703	0.678514	+	0.734587i	$0.262624\pi$
$920$	0	0
$921$	4.25753	0.140291
$922$	36.7663	1.21083
$923$	0.105130	0.00346039
$924$	0.587897	0.0193404
$925$	0	0
$926$	42.6032	1.40003
$927$	36.0893	1.18533
$928$	−0.223212	−0.00732730
$929$	−25.6149	−0.840398	−0.420199	−	0.907432i	$-0.638040\pi$
$929$	−25.6149	−0.840398	−0.420199	+	0.907432i	$0.638040\pi$
$930$	0	0
$931$	−10.9410	−0.358577
$932$	−0.679557	−0.0222596
$933$	3.02873	0.0991560
$934$	−43.2408	−1.41488
$935$	0	0
$936$	0.319134	0.0104312
$937$	4.63318	0.151359	0.0756797	−	0.997132i	$-0.475887\pi$
$937$	4.63318	0.151359	0.0756797	+	0.997132i	$0.475887\pi$
$938$	75.7479	2.47326
$939$	2.97124	0.0969626
$940$	0	0
$941$	24.1346	0.786766	0.393383	−	0.919375i	$-0.371305\pi$
$941$	24.1346	0.786766	0.393383	+	0.919375i	$0.371305\pi$
$942$	−5.74056	−0.187037
$943$	5.87727	0.191390
$944$	−7.39540	−0.240700
$945$	0	0
$946$	72.6072	2.36066
$947$	18.2542	0.593183	0.296592	−	0.955004i	$-0.404150\pi$
$947$	18.2542	0.593183	0.296592	+	0.955004i	$0.404150\pi$
$948$	−0.410996	−0.0133485
$949$	0.0246481	0.000800111 0
$950$	0	0
$951$	−5.14546	−0.166853
$952$	−32.1250	−1.04118
$953$	20.7642	0.672618	0.336309	−	0.941752i	$-0.390821\pi$
$953$	20.7642	0.672618	0.336309	+	0.941752i	$0.390821\pi$
$954$	24.5813	0.795849
$955$	0	0
$956$	0.565778	0.0182986
$957$	0.403441	0.0130414
$958$	−24.8870	−0.804063
$959$	−76.6635	−2.47559
$960$	0	0
$961$	1.68905	0.0544854
$962$	−0.0257128	−0.000829012 0
$963$	−0.556433	−0.0179308
$964$	−0.471808	−0.0151959
$965$	0	0
$966$	8.30564	0.267229
$967$	−15.2969	−0.491914	−0.245957	−	0.969281i	$-0.579102\pi$
$967$	−15.2969	−0.491914	−0.245957	+	0.969281i	$0.579102\pi$
$968$	20.1482	0.647588
$969$	0.377802	0.0121367
$970$	0	0
$971$	−7.87646	−0.252768	−0.126384	−	0.991981i	$-0.540337\pi$
$971$	−7.87646	−0.252768	−0.126384	+	0.991981i	$0.540337\pi$
$972$	−0.726122	−0.0232904
$973$	−53.0604	−1.70104
$974$	23.2737	0.745738
$975$	0	0
$976$	−28.2203	−0.903310
$977$	−22.9232	−0.733378	−0.366689	−	0.930344i	$-0.619509\pi$
$977$	−22.9232	−0.733378	−0.366689	+	0.930344i	$0.619509\pi$
$978$	−5.83681	−0.186641
$979$	−45.1345	−1.44250
$980$	0	0
$981$	−26.5217	−0.846773
$982$	62.4170	1.99181
$983$	10.5939	0.337893	0.168947	−	0.985625i	$-0.445963\pi$
$983$	10.5939	0.337893	0.168947	+	0.985625i	$0.445963\pi$
$984$	0.924299	0.0294656
$985$	0	0
$986$	1.25556	0.0399851
$987$	−3.71327	−0.118195
$988$	0.00265800	8.45622e−5 0
$989$	52.4466	1.66770
$990$	0	0
$991$	−5.87151	−0.186515	−0.0932573	−	0.995642i	$-0.529728\pi$
$991$	−5.87151	−0.186515	−0.0932573	+	0.995642i	$0.529728\pi$
$992$	3.48177	0.110546
$993$	3.63470	0.115344
$994$	−19.1052	−0.605979
$995$	0	0
$996$	−0.260900	−0.00826694
$997$	−13.2653	−0.420116	−0.210058	−	0.977689i	$-0.567365\pi$
$997$	−13.2653	−0.420116	−0.210058	+	0.977689i	$0.567365\pi$
$998$	37.4200	1.18451
$999$	−0.682293	−0.0215868

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.r.1.14		49
5.2	odd	4		985.2.b.a.789.26	✓	98
5.3	odd	4		985.2.b.a.789.73	yes	98
5.4	even	2		4925.2.a.s.1.36		49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.b.a.789.26	✓	98	5.2	odd	4
985.2.b.a.789.73	yes	98	5.3	odd	4
4925.2.a.r.1.14		49	1.1	even	1	trivial
4925.2.a.s.1.36		49	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q-1.45182 q^{2} -0.257058 q^{3} +0.107768 q^{4} +0.373200 q^{6} -4.95623 q^{7} +2.74717 q^{8} -2.93392 q^{9} +4.28184 q^{11} -0.0277025 q^{12} -0.0395948 q^{13} +7.19553 q^{14} -4.20392 q^{16} +2.35942 q^{17} +4.25951 q^{18} -0.622913 q^{19} +1.27404 q^{21} -6.21644 q^{22} -4.49034 q^{23} -0.706181 q^{24} +0.0574844 q^{26} +1.52536 q^{27} -0.534122 q^{28} -0.366538 q^{29} +5.71743 q^{31} +0.608974 q^{32} -1.10068 q^{33} -3.42545 q^{34} -0.316182 q^{36} -0.447300 q^{37} +0.904355 q^{38} +0.0101782 q^{39} -1.30887 q^{41} -1.84967 q^{42} -11.6799 q^{43} +0.461444 q^{44} +6.51915 q^{46} -2.91457 q^{47} +1.08065 q^{48} +17.5642 q^{49} -0.606508 q^{51} -0.00426704 q^{52} +5.77092 q^{53} -2.21454 q^{54} -13.6156 q^{56} +0.160125 q^{57} +0.532146 q^{58} +1.75917 q^{59} +6.71285 q^{61} -8.30066 q^{62} +14.5412 q^{63} +7.52373 q^{64} +1.59798 q^{66} +10.5271 q^{67} +0.254270 q^{68} +1.15428 q^{69} -2.65514 q^{71} -8.05999 q^{72} -0.622508 q^{73} +0.649396 q^{74} -0.0671299 q^{76} -21.2218 q^{77} -0.0147768 q^{78} +14.8361 q^{79} +8.40966 q^{81} +1.90024 q^{82} +9.41793 q^{83} +0.137300 q^{84} +16.9570 q^{86} +0.0942214 q^{87} +11.7630 q^{88} -10.5409 q^{89} +0.196241 q^{91} -0.483914 q^{92} -1.46971 q^{93} +4.23142 q^{94} -0.156541 q^{96} +4.20242 q^{97} -25.5000 q^{98} -12.5626 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 5 q^{2} - 22 q^{3} + 49 q^{4} + 2 q^{6} - 32 q^{7} - 15 q^{8} + 51 q^{9} - 2 q^{11} - 44 q^{12} - 32 q^{13} - 8 q^{14} + 49 q^{16} - 14 q^{17} - 25 q^{18} + 4 q^{19} + 10 q^{21} - 38 q^{22} - 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) \) 49 * q - 5 * q^2 - 22 * q^3 + 49 * q^4 + 2 * q^6 - 32 * q^7 - 15 * q^8 + 51 * q^9 - 2 * q^11 - 44 * q^12 - 32 * q^13 - 8 * q^14 + 49 * q^16 - 14 * q^17 - 25 * q^18 + 4 * q^19 + 10 * q^21 - 38 * q^22 - 24 * q^23 - 2 * q^26 - 88 * q^27 - 80 * q^28 + 6 * q^29 - 2 * q^31 - 35 * q^32 - 48 * q^33 - 12 * q^34 + 43 * q^36 - 60 * q^37 - 34 * q^38 - 2 * q^41 - 36 * q^42 - 80 * q^43 - 8 * q^44 - 22 * q^46 - 48 * q^47 - 88 * q^48 + 65 * q^49 + 16 * q^51 - 96 * q^52 - 18 * q^53 + 42 * q^54 - 22 * q^56 - 40 * q^57 - 52 * q^58 + 8 * q^59 + 16 * q^61 - 38 * q^62 - 82 * q^63 + 55 * q^64 + 42 * q^66 - 146 * q^67 + 68 * q^68 - 6 * q^69 - 75 * q^72 - 78 * q^73 - 24 * q^74 - 4 * q^76 - 32 * q^77 + 38 * q^78 + 10 * q^79 + 41 * q^81 - 96 * q^82 - 64 * q^83 + 42 * q^84 - 28 * q^86 - 56 * q^87 - 72 * q^88 + 4 * q^89 - 60 * q^92 + 28 * q^93 - 8 * q^94 - 34 * q^96 - 124 * q^97 + 19 * q^98 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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