LMFDB - Embedded newform 775.2.a.k.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [775,2,Mod(1,775)] 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("775.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,4,3] 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:	$6.18840615665$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.144209.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - x^{2} + 6x - 1 $ x^5 - 6x^3 - x^2 + 6x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.93413$ of defining polynomial
Character	$\chi$	$=$	775.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.23959 q^{2} -0.740841 q^{3} +3.01578 q^{4} -1.65918 q^{6} +3.67497 q^{7} +2.27494 q^{8} -2.45115 q^{9} +5.02619 q^{11} -2.23422 q^{12} -2.67247 q^{13} +8.23043 q^{14} -0.936618 q^{16} +3.30547 q^{17} -5.48959 q^{18} +2.88050 q^{19} -2.72256 q^{21} +11.2566 q^{22} +6.01040 q^{23} -1.68537 q^{24} -5.98526 q^{26} +4.03844 q^{27} +11.0829 q^{28} -5.13050 q^{29} -1.00000 q^{31} -6.64753 q^{32} -3.72360 q^{33} +7.40291 q^{34} -7.39215 q^{36} -7.90540 q^{37} +6.45115 q^{38} +1.97988 q^{39} -9.55651 q^{41} -6.09744 q^{42} +5.25000 q^{43} +15.1579 q^{44} +13.4609 q^{46} -1.06691 q^{47} +0.693885 q^{48} +6.50538 q^{49} -2.44883 q^{51} -8.05960 q^{52} -0.521408 q^{53} +9.04446 q^{54} +8.36034 q^{56} -2.13399 q^{57} -11.4902 q^{58} -8.09515 q^{59} +0.598687 q^{61} -2.23959 q^{62} -9.00791 q^{63} -13.0145 q^{64} -8.33936 q^{66} +12.4705 q^{67} +9.96858 q^{68} -4.45275 q^{69} -8.80048 q^{71} -5.57624 q^{72} +9.87423 q^{73} -17.7049 q^{74} +8.68697 q^{76} +18.4711 q^{77} +4.43412 q^{78} -2.40291 q^{79} +4.36163 q^{81} -21.4027 q^{82} -2.91400 q^{83} -8.21067 q^{84} +11.7579 q^{86} +3.80088 q^{87} +11.4343 q^{88} -7.65918 q^{89} -9.82125 q^{91} +18.1261 q^{92} +0.740841 q^{93} -2.38946 q^{94} +4.92476 q^{96} -14.5224 q^{97} +14.5694 q^{98} -12.3200 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 3 q^{3} + 6 q^{4} - q^{6} + 2 q^{7} + 9 q^{8} + 6 q^{9} - 2 q^{11} + 5 q^{12} + 4 q^{13} + 2 q^{14} + 4 q^{16} + 19 q^{17} + 5 q^{18} + 8 q^{19} - 15 q^{21} - 10 q^{22} + 12 q^{23} + 26 q^{24}+ \cdots - 33 q^{99}+O(q^{100}) $ 5 * q + 4 * q^2 + 3 * q^3 + 6 * q^4 - q^6 + 2 * q^7 + 9 * q^8 + 6 * q^9 - 2 * q^11 + 5 * q^12 + 4 * q^13 + 2 * q^14 + 4 * q^16 + 19 * q^17 + 5 * q^18 + 8 * q^19 - 15 * q^21 - 10 * q^22 + 12 * q^23 + 26 * q^24 - 16 * q^26 + 6 * q^27 + 6 * q^28 + 6 * q^29 - 5 * q^31 + 7 * q^32 + 3 * q^33 + 31 * q^34 - 13 * q^36 + 16 * q^37 + 14 * q^38 - 13 * q^39 - 2 * q^41 - 22 * q^42 + q^43 - 4 * q^44 - 11 * q^46 + 8 * q^47 + 31 * q^48 - 9 * q^49 + 5 * q^51 - 26 * q^52 + 3 * q^53 + 10 * q^54 - 9 * q^56 + 14 * q^57 + 5 * q^58 - 4 * q^59 - 5 * q^61 - 4 * q^62 - 26 * q^63 - 5 * q^64 - 25 * q^66 + 13 * q^67 + 30 * q^68 + 10 * q^69 + 6 * q^71 + 2 * q^72 + 19 * q^73 + 4 * q^74 - 5 * q^76 + 23 * q^77 - 38 * q^78 - 6 * q^79 - 7 * q^81 - 27 * q^82 + 18 * q^83 - 15 * q^84 - 7 * q^86 - 5 * q^87 - q^88 - 31 * q^89 + 8 * q^91 - 16 * q^92 - 3 * q^93 - 14 * q^94 + 10 * q^96 - q^97 + 10 * q^98 - 33 * 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 $

$2$	2.23959	1.58363	0.791816	−	0.610759i	$-0.209136\pi$
$2$	2.23959	1.58363	0.791816	+	0.610759i	$0.209136\pi$
$3$	−0.740841	−0.427725	−0.213862	−	0.976864i	$-0.568604\pi$
$3$	−0.740841	−0.427725	−0.213862	+	0.976864i	$0.568604\pi$
$4$	3.01578	1.50789
$5$	0	0
$5$	0	0
$6$	−1.65918	−0.677359
$7$	3.67497	1.38901	0.694503	−	0.719489i	$-0.255624\pi$
$7$	3.67497	1.38901	0.694503	+	0.719489i	$0.255624\pi$
$8$	2.27494	0.804314
$9$	−2.45115	−0.817052
$10$	0	0
$11$	5.02619	1.51545	0.757726	−	0.652573i	$-0.226310\pi$
$11$	5.02619	1.51545	0.757726	+	0.652573i	$0.226310\pi$
$12$	−2.23422	−0.644962
$13$	−2.67247	−0.741211	−0.370605	−	0.928790i	$-0.620850\pi$
$13$	−2.67247	−0.741211	−0.370605	+	0.928790i	$0.620850\pi$
$14$	8.23043	2.19968
$15$	0	0
$16$	−0.936618	−0.234155
$17$	3.30547	0.801694	0.400847	−	0.916145i	$-0.368716\pi$
$17$	3.30547	0.801694	0.400847	+	0.916145i	$0.368716\pi$
$18$	−5.48959	−1.29391
$19$	2.88050	0.660832	0.330416	−	0.943835i	$-0.392811\pi$
$19$	2.88050	0.660832	0.330416	+	0.943835i	$0.392811\pi$
$20$	0	0
$21$	−2.72256	−0.594112
$22$	11.2566	2.39992
$23$	6.01040	1.25326	0.626628	−	0.779319i	$-0.284435\pi$
$23$	6.01040	1.25326	0.626628	+	0.779319i	$0.284435\pi$
$24$	−1.68537	−0.344025
$25$	0	0
$26$	−5.98526	−1.17381
$27$	4.03844	0.777198
$28$	11.0829	2.09447
$29$	−5.13050	−0.952710	−0.476355	−	0.879253i	$-0.658042\pi$
$29$	−5.13050	−0.952710	−0.476355	+	0.879253i	$0.658042\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−6.64753	−1.17513
$33$	−3.72360	−0.648196
$34$	7.40291	1.26959
$35$	0	0
$36$	−7.39215	−1.23203
$37$	−7.90540	−1.29964	−0.649820	−	0.760088i	$-0.725156\pi$
$37$	−7.90540	−1.29964	−0.649820	+	0.760088i	$0.725156\pi$
$38$	6.45115	1.04652
$39$	1.97988	0.317034
$40$	0	0
$41$	−9.55651	−1.49248	−0.746238	−	0.665679i	$-0.768142\pi$
$41$	−9.55651	−1.49248	−0.746238	+	0.665679i	$0.768142\pi$
$42$	−6.09744	−0.940856
$43$	5.25000	0.800617	0.400309	−	0.916380i	$-0.368903\pi$
$43$	5.25000	0.800617	0.400309	+	0.916380i	$0.368903\pi$
$44$	15.1579	2.28514
$45$	0	0
$46$	13.4609	1.98470
$47$	−1.06691	−0.155625	−0.0778127	−	0.996968i	$-0.524794\pi$
$47$	−1.06691	−0.155625	−0.0778127	+	0.996968i	$0.524794\pi$
$48$	0.693885	0.100154
$49$	6.50538	0.929339
$50$	0	0
$51$	−2.44883	−0.342904
$52$	−8.05960	−1.11767
$53$	−0.521408	−0.0716210	−0.0358105	−	0.999359i	$-0.511401\pi$
$53$	−0.521408	−0.0716210	−0.0358105	+	0.999359i	$0.511401\pi$
$54$	9.04446	1.23080
$55$	0	0
$56$	8.36034	1.11720
$57$	−2.13399	−0.282654
$58$	−11.4902	−1.50874
$59$	−8.09515	−1.05390	−0.526950	−	0.849897i	$-0.676664\pi$
$59$	−8.09515	−1.05390	−0.526950	+	0.849897i	$0.676664\pi$
$60$	0	0
$61$	0.598687	0.0766541	0.0383270	−	0.999265i	$-0.487797\pi$
$61$	0.598687	0.0766541	0.0383270	+	0.999265i	$0.487797\pi$
$62$	−2.23959	−0.284429
$63$	−9.00791	−1.13489
$64$	−13.0145	−1.62682
$65$	0	0
$66$	−8.33936	−1.02650
$67$	12.4705	1.52351	0.761755	−	0.647865i	$-0.224338\pi$
$67$	12.4705	1.52351	0.761755	+	0.647865i	$0.224338\pi$
$68$	9.96858	1.20887
$69$	−4.45275	−0.536048
$70$	0	0
$71$	−8.80048	−1.04442	−0.522212	−	0.852815i	$-0.674893\pi$
$71$	−8.80048	−1.04442	−0.522212	+	0.852815i	$0.674893\pi$
$72$	−5.57624	−0.657166
$73$	9.87423	1.15569	0.577845	−	0.816146i	$-0.303894\pi$
$73$	9.87423	1.15569	0.577845	+	0.816146i	$0.303894\pi$
$74$	−17.7049	−2.05815
$75$	0	0
$76$	8.68697	0.996464
$77$	18.4711	2.10497
$78$	4.43412	0.502065
$79$	−2.40291	−0.270349	−0.135174	−	0.990822i	$-0.543159\pi$
$79$	−2.40291	−0.270349	−0.135174	+	0.990822i	$0.543159\pi$
$80$	0	0
$81$	4.36163	0.484625
$82$	−21.4027	−2.36353
$83$	−2.91400	−0.319853	−0.159927	−	0.987129i	$-0.551126\pi$
$83$	−2.91400	−0.319853	−0.159927	+	0.987129i	$0.551126\pi$
$84$	−8.21067	−0.895857
$85$	0	0
$86$	11.7579	1.26788
$87$	3.80088	0.407497
$88$	11.4343	1.21890
$89$	−7.65918	−0.811872	−0.405936	−	0.913902i	$-0.633054\pi$
$89$	−7.65918	−0.811872	−0.405936	+	0.913902i	$0.633054\pi$
$90$	0	0
$91$	−9.82125	−1.02955
$92$	18.1261	1.88977
$93$	0.740841	0.0768216
$94$	−2.38946	−0.246454
$95$	0	0
$96$	4.92476	0.502631
$97$	−14.5224	−1.47452	−0.737261	−	0.675608i	$-0.763881\pi$
$97$	−14.5224	−1.47452	−0.737261	+	0.675608i	$0.763881\pi$
$98$	14.5694	1.47173
$99$	−12.3200	−1.23820
$100$	0	0
$101$	−14.6832	−1.46103	−0.730516	−	0.682896i	$-0.760720\pi$
$101$	−14.6832	−1.46103	−0.730516	+	0.682896i	$0.760720\pi$
$102$	−5.48438	−0.543034
$103$	−16.2414	−1.60031	−0.800156	−	0.599792i	$-0.795250\pi$
$103$	−16.2414	−1.60031	−0.800156	+	0.599792i	$0.795250\pi$
$104$	−6.07972	−0.596166
$105$	0	0
$106$	−1.16774	−0.113421
$107$	7.72859	0.747151	0.373576	−	0.927600i	$-0.378132\pi$
$107$	7.72859	0.747151	0.373576	+	0.927600i	$0.378132\pi$
$108$	12.1791	1.17193
$109$	−6.60865	−0.632994	−0.316497	−	0.948594i	$-0.602507\pi$
$109$	−6.60865	−0.632994	−0.316497	+	0.948594i	$0.602507\pi$
$110$	0	0
$111$	5.85664	0.555888
$112$	−3.44204	−0.325242
$113$	11.3537	1.06807	0.534034	−	0.845463i	$-0.320676\pi$
$113$	11.3537	1.06807	0.534034	+	0.845463i	$0.320676\pi$
$114$	−4.77928	−0.447620
$115$	0	0
$116$	−15.4725	−1.43658
$117$	6.55065	0.605607
$118$	−18.1299	−1.66899
$119$	12.1475	1.11356
$120$	0	0
$121$	14.2626	1.29660
$122$	1.34082	0.121392
$123$	7.07985	0.638369
$124$	−3.01578	−0.270825
$125$	0	0
$126$	−20.1741	−1.79725
$127$	−11.1669	−0.990901	−0.495451	−	0.868636i	$-0.664997\pi$
$127$	−11.1669	−0.990901	−0.495451	+	0.868636i	$0.664997\pi$
$128$	−15.8522	−1.40115
$129$	−3.88941	−0.342444
$130$	0	0
$131$	−1.93468	−0.169034	−0.0845170	−	0.996422i	$-0.526935\pi$
$131$	−1.93468	−0.169034	−0.0845170	+	0.996422i	$0.526935\pi$
$132$	−11.2296	−0.977410
$133$	10.5857	0.917901
$134$	27.9288	2.41268
$135$	0	0
$136$	7.51975	0.644813
$137$	16.5372	1.41287	0.706434	−	0.707779i	$-0.250303\pi$
$137$	16.5372	1.41287	0.706434	+	0.707779i	$0.250303\pi$
$138$	−9.97236	−0.848904
$139$	−22.3954	−1.89956	−0.949778	−	0.312924i	$-0.898691\pi$
$139$	−22.3954	−1.89956	−0.949778	+	0.312924i	$0.898691\pi$
$140$	0	0
$141$	0.790414	0.0665648
$142$	−19.7095	−1.65399
$143$	−13.4324	−1.12327
$144$	2.29580	0.191316
$145$	0	0
$146$	22.1143	1.83019
$147$	−4.81945	−0.397501
$148$	−23.8410	−1.95972
$149$	−4.86954	−0.398928	−0.199464	−	0.979905i	$-0.563920\pi$
$149$	−4.86954	−0.398928	−0.199464	+	0.979905i	$0.563920\pi$
$150$	0	0
$151$	19.5409	1.59022	0.795109	−	0.606466i	$-0.207414\pi$
$151$	19.5409	1.59022	0.795109	+	0.606466i	$0.207414\pi$
$152$	6.55297	0.531516
$153$	−8.10222	−0.655025
$154$	41.3677	3.33350
$155$	0	0
$156$	5.97088	0.478053
$157$	−14.9019	−1.18930	−0.594649	−	0.803985i	$-0.702709\pi$
$157$	−14.9019	−1.18930	−0.594649	+	0.803985i	$0.702709\pi$
$158$	−5.38154	−0.428133
$159$	0.386281	0.0306340
$160$	0	0
$161$	22.0880	1.74078
$162$	9.76827	0.767468
$163$	−10.7995	−0.845878	−0.422939	−	0.906158i	$-0.639002\pi$
$163$	−10.7995	−0.845878	−0.422939	+	0.906158i	$0.639002\pi$
$164$	−28.8204	−2.25049
$165$	0	0
$166$	−6.52618	−0.506530
$167$	16.2405	1.25673	0.628364	−	0.777920i	$-0.283725\pi$
$167$	16.2405	1.25673	0.628364	+	0.777920i	$0.283725\pi$
$168$	−6.19368	−0.477853
$169$	−5.85789	−0.450607
$170$	0	0
$171$	−7.06056	−0.539934
$172$	15.8329	1.20724
$173$	−3.97252	−0.302025	−0.151013	−	0.988532i	$-0.548253\pi$
$173$	−3.97252	−0.302025	−0.151013	+	0.988532i	$0.548253\pi$
$174$	8.51244	0.645326
$175$	0	0
$176$	−4.70762	−0.354850
$177$	5.99722	0.450779
$178$	−17.1535	−1.28571
$179$	24.8049	1.85401	0.927003	−	0.375053i	$-0.122376\pi$
$179$	24.8049	1.85401	0.927003	+	0.375053i	$0.122376\pi$
$180$	0	0
$181$	−15.9359	−1.18451	−0.592253	−	0.805752i	$-0.701761\pi$
$181$	−15.9359	−1.18451	−0.592253	+	0.805752i	$0.701761\pi$
$182$	−21.9956	−1.63042
$183$	−0.443532	−0.0327868
$184$	13.6733	1.00801
$185$	0	0
$186$	1.65918	0.121657
$187$	16.6139	1.21493
$188$	−3.21758	−0.234666
$189$	14.8411	1.07953
$190$	0	0
$191$	22.5120	1.62891	0.814456	−	0.580226i	$-0.197036\pi$
$191$	22.5120	1.62891	0.814456	+	0.580226i	$0.197036\pi$
$192$	9.64170	0.695830
$193$	19.0305	1.36984	0.684922	−	0.728616i	$-0.259836\pi$
$193$	19.0305	1.36984	0.684922	+	0.728616i	$0.259836\pi$
$194$	−32.5242	−2.33510
$195$	0	0
$196$	19.6188	1.40134
$197$	−1.27674	−0.0909642	−0.0454821	−	0.998965i	$-0.514482\pi$
$197$	−1.27674	−0.0909642	−0.0454821	+	0.998965i	$0.514482\pi$
$198$	−27.5917	−1.96086
$199$	9.21392	0.653157	0.326579	−	0.945170i	$-0.394104\pi$
$199$	9.21392	0.653157	0.326579	+	0.945170i	$0.394104\pi$
$200$	0	0
$201$	−9.23863	−0.651643
$202$	−32.8844	−2.31374
$203$	−18.8544	−1.32332
$204$	−7.38513	−0.517062
$205$	0	0
$206$	−36.3741	−2.53431
$207$	−14.7324	−1.02397
$208$	2.50309	0.173558
$209$	14.4779	1.00146
$210$	0	0
$211$	2.86821	0.197456	0.0987279	−	0.995114i	$-0.468523\pi$
$211$	2.86821	0.197456	0.0987279	+	0.995114i	$0.468523\pi$
$212$	−1.57245	−0.107997
$213$	6.51976	0.446726
$214$	17.3089	1.18321
$215$	0	0
$216$	9.18721	0.625111
$217$	−3.67497	−0.249473
$218$	−14.8007	−1.00243
$219$	−7.31523	−0.494317
$220$	0	0
$221$	−8.83378	−0.594224
$222$	13.1165	0.880322
$223$	9.10462	0.609691	0.304845	−	0.952402i	$-0.401395\pi$
$223$	9.10462	0.609691	0.304845	+	0.952402i	$0.401395\pi$
$224$	−24.4294	−1.63226
$225$	0	0
$226$	25.4277	1.69143
$227$	4.62499	0.306971	0.153486	−	0.988151i	$-0.450950\pi$
$227$	4.62499	0.306971	0.153486	+	0.988151i	$0.450950\pi$
$228$	−6.43566	−0.426212
$229$	26.1553	1.72839	0.864196	−	0.503156i	$-0.167828\pi$
$229$	26.1553	1.72839	0.864196	+	0.503156i	$0.167828\pi$
$230$	0	0
$231$	−13.6841	−0.900349
$232$	−11.6716	−0.766278
$233$	−4.76330	−0.312054	−0.156027	−	0.987753i	$-0.549869\pi$
$233$	−4.76330	−0.312054	−0.156027	+	0.987753i	$0.549869\pi$
$234$	14.6708	0.959060
$235$	0	0
$236$	−24.4132	−1.58917
$237$	1.78017	0.115635
$238$	27.2054	1.76347
$239$	0.674275	0.0436152	0.0218076	−	0.999762i	$-0.493058\pi$
$239$	0.674275	0.0436152	0.0218076	+	0.999762i	$0.493058\pi$
$240$	0	0
$241$	4.14684	0.267122	0.133561	−	0.991041i	$-0.457359\pi$
$241$	4.14684	0.267122	0.133561	+	0.991041i	$0.457359\pi$
$242$	31.9423	2.05333
$243$	−15.3466	−0.984484
$244$	1.80551	0.115586
$245$	0	0
$246$	15.8560	1.01094
$247$	−7.69806	−0.489816
$248$	−2.27494	−0.144459
$249$	2.15881	0.136809
$250$	0	0
$251$	23.0876	1.45728	0.728638	−	0.684899i	$-0.240154\pi$
$251$	23.0876	1.45728	0.728638	+	0.684899i	$0.240154\pi$
$252$	−27.1659	−1.71129
$253$	30.2094	1.89925
$254$	−25.0093	−1.56922
$255$	0	0
$256$	−9.47347	−0.592092
$257$	27.3951	1.70886	0.854429	−	0.519568i	$-0.173907\pi$
$257$	27.3951	1.70886	0.854429	+	0.519568i	$0.173907\pi$
$258$	−8.71071	−0.542305
$259$	−29.0521	−1.80521
$260$	0	0
$261$	12.5756	0.778413
$262$	−4.33291	−0.267688
$263$	−13.1328	−0.809802	−0.404901	−	0.914361i	$-0.632694\pi$
$263$	−13.1328	−0.809802	−0.404901	+	0.914361i	$0.632694\pi$
$264$	−8.47099	−0.521353
$265$	0	0
$266$	23.7078	1.45362
$267$	5.67424	0.347258
$268$	37.6082	2.29729
$269$	23.5415	1.43535	0.717676	−	0.696377i	$-0.245206\pi$
$269$	23.5415	1.43535	0.717676	+	0.696377i	$0.245206\pi$
$270$	0	0
$271$	21.2089	1.28835	0.644173	−	0.764880i	$-0.277202\pi$
$271$	21.2089	1.28835	0.644173	+	0.764880i	$0.277202\pi$
$272$	−3.09596	−0.187720
$273$	7.27598	0.440362
$274$	37.0366	2.23746
$275$	0	0
$276$	−13.4285	−0.808303
$277$	−3.16271	−0.190029	−0.0950144	−	0.995476i	$-0.530290\pi$
$277$	−3.16271	−0.190029	−0.0950144	+	0.995476i	$0.530290\pi$
$278$	−50.1567	−3.00820
$279$	2.45115	0.146747
$280$	0	0
$281$	21.9912	1.31189	0.655943	−	0.754810i	$-0.272271\pi$
$281$	21.9912	1.31189	0.655943	+	0.754810i	$0.272271\pi$
$282$	1.77021	0.105414
$283$	15.6629	0.931065	0.465532	−	0.885031i	$-0.345863\pi$
$283$	15.6629	0.931065	0.465532	+	0.885031i	$0.345863\pi$
$284$	−26.5403	−1.57488
$285$	0	0
$286$	−30.0830	−1.77885
$287$	−35.1198	−2.07306
$288$	16.2941	0.960140
$289$	−6.07387	−0.357287
$290$	0	0
$291$	10.7588	0.630690
$292$	29.7785	1.74266
$293$	29.1815	1.70480	0.852401	−	0.522889i	$-0.175146\pi$
$293$	29.1815	1.70480	0.852401	+	0.522889i	$0.175146\pi$
$294$	−10.7936	−0.629496
$295$	0	0
$296$	−17.9843	−1.04532
$297$	20.2979	1.17781
$298$	−10.9058	−0.631756
$299$	−16.0626	−0.928927
$300$	0	0
$301$	19.2936	1.11206
$302$	43.7638	2.51832
$303$	10.8779	0.624919
$304$	−2.69793	−0.154737
$305$	0	0
$306$	−18.1457	−1.03732
$307$	−19.3160	−1.10243	−0.551213	−	0.834365i	$-0.685835\pi$
$307$	−19.3160	−1.10243	−0.551213	+	0.834365i	$0.685835\pi$
$308$	55.7047	3.17407
$309$	12.0323	0.684493
$310$	0	0
$311$	6.15914	0.349253	0.174626	−	0.984635i	$-0.444128\pi$
$311$	6.15914	0.349253	0.174626	+	0.984635i	$0.444128\pi$
$312$	4.50411	0.254995
$313$	−20.1839	−1.14086	−0.570430	−	0.821346i	$-0.693223\pi$
$313$	−20.1839	−1.14086	−0.570430	+	0.821346i	$0.693223\pi$
$314$	−33.3741	−1.88341
$315$	0	0
$316$	−7.24666	−0.407656
$317$	−25.2088	−1.41587	−0.707933	−	0.706280i	$-0.750372\pi$
$317$	−25.2088	−1.41587	−0.707933	+	0.706280i	$0.750372\pi$
$318$	0.865112	0.0485131
$319$	−25.7869	−1.44379
$320$	0	0
$321$	−5.72565	−0.319575
$322$	49.4682	2.75676
$323$	9.52141	0.529785
$324$	13.1537	0.730762
$325$	0	0
$326$	−24.1864	−1.33956
$327$	4.89596	0.270747
$328$	−21.7405	−1.20042
$329$	−3.92087	−0.216165
$330$	0	0
$331$	−17.6922	−0.972454	−0.486227	−	0.873833i	$-0.661627\pi$
$331$	−17.6922	−0.972454	−0.486227	+	0.873833i	$0.661627\pi$
$332$	−8.78800	−0.482304
$333$	19.3774	1.06187
$334$	36.3721	1.99019
$335$	0	0
$336$	2.55000	0.139114
$337$	2.89468	0.157683	0.0788415	−	0.996887i	$-0.474878\pi$
$337$	2.89468	0.157683	0.0788415	+	0.996887i	$0.474878\pi$
$338$	−13.1193	−0.713595
$339$	−8.41129	−0.456839
$340$	0	0
$341$	−5.02619	−0.272183
$342$	−15.8128	−0.855057
$343$	−1.81773	−0.0981480
$344$	11.9434	0.643947
$345$	0	0
$346$	−8.89684	−0.478297
$347$	6.89681	0.370240	0.185120	−	0.982716i	$-0.440733\pi$
$347$	6.89681	0.370240	0.185120	+	0.982716i	$0.440733\pi$
$348$	11.4626	0.614462
$349$	−14.6691	−0.785220	−0.392610	−	0.919705i	$-0.628428\pi$
$349$	−14.6691	−0.785220	−0.392610	+	0.919705i	$0.628428\pi$
$350$	0	0
$351$	−10.7926	−0.576067
$352$	−33.4117	−1.78085
$353$	−10.5568	−0.561883	−0.280942	−	0.959725i	$-0.590647\pi$
$353$	−10.5568	−0.561883	−0.280942	+	0.959725i	$0.590647\pi$
$354$	13.4313	0.713868
$355$	0	0
$356$	−23.0984	−1.22421
$357$	−8.99935	−0.476296
$358$	55.5530	2.93607
$359$	20.4230	1.07788	0.538942	−	0.842343i	$-0.318824\pi$
$359$	20.4230	1.07788	0.538942	+	0.842343i	$0.318824\pi$
$360$	0	0
$361$	−10.7027	−0.563301
$362$	−35.6899	−1.87582
$363$	−10.5663	−0.554586
$364$	−29.6188	−1.55244
$365$	0	0
$366$	−0.993332	−0.0519223
$367$	−15.0168	−0.783872	−0.391936	−	0.919993i	$-0.628194\pi$
$367$	−15.0168	−0.783872	−0.391936	+	0.919993i	$0.628194\pi$
$368$	−5.62945	−0.293456
$369$	23.4245	1.21943
$370$	0	0
$371$	−1.91616	−0.0994820
$372$	2.23422	0.115839
$373$	19.2471	0.996578	0.498289	−	0.867011i	$-0.333962\pi$
$373$	19.2471	0.996578	0.498289	+	0.867011i	$0.333962\pi$
$374$	37.2084	1.92400
$375$	0	0
$376$	−2.42717	−0.125172
$377$	13.7111	0.706159
$378$	33.2381	1.70958
$379$	3.24968	0.166925	0.0834624	−	0.996511i	$-0.473402\pi$
$379$	3.24968	0.166925	0.0834624	+	0.996511i	$0.473402\pi$
$380$	0	0
$381$	8.27289	0.423833
$382$	50.4177	2.57960
$383$	−2.01897	−0.103164	−0.0515822	−	0.998669i	$-0.516426\pi$
$383$	−2.01897	−0.103164	−0.0515822	+	0.998669i	$0.516426\pi$
$384$	11.7440	0.599307
$385$	0	0
$386$	42.6206	2.16933
$387$	−12.8686	−0.654146
$388$	−43.7963	−2.22342
$389$	−15.2990	−0.775689	−0.387844	−	0.921725i	$-0.626780\pi$
$389$	−15.2990	−0.775689	−0.387844	+	0.921725i	$0.626780\pi$
$390$	0	0
$391$	19.8672	1.00473
$392$	14.7994	0.747480
$393$	1.43329	0.0723000
$394$	−2.85939	−0.144054
$395$	0	0
$396$	−37.1543	−1.86708
$397$	14.1815	0.711751	0.355875	−	0.934533i	$-0.384183\pi$
$397$	14.1815	0.711751	0.355875	+	0.934533i	$0.384183\pi$
$398$	20.6354	1.03436
$399$	−7.84235	−0.392609
$400$	0	0
$401$	−6.59958	−0.329567	−0.164784	−	0.986330i	$-0.552693\pi$
$401$	−6.59958	−0.329567	−0.164784	+	0.986330i	$0.552693\pi$
$402$	−20.6908	−1.03196
$403$	2.67247	0.133125
$404$	−44.2813	−2.20308
$405$	0	0
$406$	−42.2262	−2.09565
$407$	−39.7340	−1.96954
$408$	−5.57094	−0.275803
$409$	−29.1922	−1.44346	−0.721730	−	0.692175i	$-0.756653\pi$
$409$	−29.1922	−1.44346	−0.721730	+	0.692175i	$0.756653\pi$
$410$	0	0
$411$	−12.2514	−0.604318
$412$	−48.9805	−2.41310
$413$	−29.7494	−1.46387
$414$	−32.9947	−1.62160
$415$	0	0
$416$	17.7653	0.871018
$417$	16.5915	0.812487
$418$	32.4247	1.58594
$419$	−28.0209	−1.36891	−0.684454	−	0.729056i	$-0.739960\pi$
$419$	−28.0209	−1.36891	−0.684454	+	0.729056i	$0.739960\pi$
$420$	0	0
$421$	−30.7552	−1.49892	−0.749459	−	0.662051i	$-0.769686\pi$
$421$	−30.7552	−1.49892	−0.749459	+	0.662051i	$0.769686\pi$
$422$	6.42363	0.312697
$423$	2.61517	0.127154
$424$	−1.18617	−0.0576057
$425$	0	0
$426$	14.6016	0.707450
$427$	2.20016	0.106473
$428$	23.3078	1.12662
$429$	9.95123	0.480450
$430$	0	0
$431$	−26.9481	−1.29804	−0.649021	−	0.760770i	$-0.724821\pi$
$431$	−26.9481	−1.29804	−0.649021	+	0.760770i	$0.724821\pi$
$432$	−3.78247	−0.181984
$433$	−9.45517	−0.454386	−0.227193	−	0.973850i	$-0.572955\pi$
$433$	−9.45517	−0.454386	−0.227193	+	0.973850i	$0.572955\pi$
$434$	−8.23043	−0.395073
$435$	0	0
$436$	−19.9303	−0.954486
$437$	17.3130	0.828192
$438$	−16.3831	−0.782817
$439$	−20.3379	−0.970673	−0.485337	−	0.874327i	$-0.661303\pi$
$439$	−20.3379	−0.970673	−0.485337	+	0.874327i	$0.661303\pi$
$440$	0	0
$441$	−15.9457	−0.759318
$442$	−19.7841	−0.941033
$443$	13.6821	0.650056	0.325028	−	0.945704i	$-0.394626\pi$
$443$	13.6821	0.650056	0.325028	+	0.945704i	$0.394626\pi$
$444$	17.6624	0.838219
$445$	0	0
$446$	20.3907	0.965526
$447$	3.60755	0.170631
$448$	−47.8280	−2.25966
$449$	41.1092	1.94006	0.970031	−	0.242980i	$-0.0781250\pi$
$449$	41.1092	1.94006	0.970031	+	0.242980i	$0.0781250\pi$
$450$	0	0
$451$	−48.0328	−2.26178
$452$	34.2403	1.61053
$453$	−14.4767	−0.680175
$454$	10.3581	0.486130
$455$	0	0
$456$	−4.85471	−0.227343
$457$	−16.6907	−0.780760	−0.390380	−	0.920654i	$-0.627656\pi$
$457$	−16.6907	−0.780760	−0.390380	+	0.920654i	$0.627656\pi$
$458$	58.5773	2.73714
$459$	13.3489	0.623075
$460$	0	0
$461$	2.70160	0.125826	0.0629129	−	0.998019i	$-0.479961\pi$
$461$	2.70160	0.125826	0.0629129	+	0.998019i	$0.479961\pi$
$462$	−30.6469	−1.42582
$463$	13.6205	0.633000	0.316500	−	0.948593i	$-0.397492\pi$
$463$	13.6205	0.633000	0.316500	+	0.948593i	$0.397492\pi$
$464$	4.80532	0.223081
$465$	0	0
$466$	−10.6678	−0.494179
$467$	−27.5703	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$467$	−27.5703	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$468$	19.7553	0.913190
$469$	45.8286	2.11617
$470$	0	0
$471$	11.0399	0.508692
$472$	−18.4160	−0.847665
$473$	26.3875	1.21330
$474$	3.98687	0.183123
$475$	0	0
$476$	36.6342	1.67913
$477$	1.27805	0.0585180
$478$	1.51010	0.0690704
$479$	5.41728	0.247522	0.123761	−	0.992312i	$-0.460504\pi$
$479$	5.41728	0.247522	0.123761	+	0.992312i	$0.460504\pi$
$480$	0	0
$481$	21.1270	0.963307
$482$	9.28724	0.423022
$483$	−16.3637	−0.744575
$484$	43.0128	1.95513
$485$	0	0
$486$	−34.3701	−1.55906
$487$	1.23959	0.0561714	0.0280857	−	0.999606i	$-0.491059\pi$
$487$	1.23959	0.0561714	0.0280857	+	0.999606i	$0.491059\pi$
$488$	1.36198	0.0616539
$489$	8.00067	0.361803
$490$	0	0
$491$	19.0450	0.859487	0.429743	−	0.902951i	$-0.358604\pi$
$491$	19.0450	0.859487	0.429743	+	0.902951i	$0.358604\pi$
$492$	21.3513	0.962591
$493$	−16.9587	−0.763782
$494$	−17.2405	−0.775689
$495$	0	0
$496$	0.936618	0.0420554
$497$	−32.3415	−1.45071
$498$	4.83486	0.216655
$499$	3.81671	0.170859	0.0854296	−	0.996344i	$-0.472774\pi$
$499$	3.81671	0.170859	0.0854296	+	0.996344i	$0.472774\pi$
$500$	0	0
$501$	−12.0316	−0.537533
$502$	51.7068	2.30779
$503$	30.4237	1.35653	0.678263	−	0.734819i	$-0.262733\pi$
$503$	30.4237	1.35653	0.678263	+	0.734819i	$0.262733\pi$
$504$	−20.4925	−0.912808
$505$	0	0
$506$	67.6568	3.00771
$507$	4.33976	0.192736
$508$	−33.6769	−1.49417
$509$	9.97990	0.442351	0.221176	−	0.975234i	$-0.429011\pi$
$509$	9.97990	0.442351	0.221176	+	0.975234i	$0.429011\pi$
$510$	0	0
$511$	36.2875	1.60526
$512$	10.4877	0.463495
$513$	11.6327	0.513597
$514$	61.3539	2.70620
$515$	0	0
$516$	−11.7296	−0.516368
$517$	−5.36251	−0.235843
$518$	−65.0649	−2.85879
$519$	2.94301	0.129184
$520$	0	0
$521$	8.27726	0.362633	0.181317	−	0.983425i	$-0.441964\pi$
$521$	8.27726	0.362633	0.181317	+	0.983425i	$0.441964\pi$
$522$	28.1644	1.23272
$523$	−28.3888	−1.24136	−0.620678	−	0.784066i	$-0.713143\pi$
$523$	−28.3888	−1.24136	−0.620678	+	0.784066i	$0.713143\pi$
$524$	−5.83459	−0.254885
$525$	0	0
$526$	−29.4121	−1.28243
$527$	−3.30547	−0.143988
$528$	3.48760	0.151778
$529$	13.1250	0.570650
$530$	0	0
$531$	19.8425	0.861090
$532$	31.9243	1.38409
$533$	25.5395	1.10624
$534$	12.7080	0.549928
$535$	0	0
$536$	28.3696	1.22538
$537$	−18.3765	−0.793004
$538$	52.7235	2.27307
$539$	32.6972	1.40837
$540$	0	0
$541$	−3.10137	−0.133338	−0.0666691	−	0.997775i	$-0.521237\pi$
$541$	−3.10137	−0.133338	−0.0666691	+	0.997775i	$0.521237\pi$
$542$	47.4992	2.04027
$543$	11.8060	0.506642
$544$	−21.9732	−0.942093
$545$	0	0
$546$	16.2952	0.697372
$547$	35.4405	1.51533	0.757663	−	0.652646i	$-0.226341\pi$
$547$	35.4405	1.51533	0.757663	+	0.652646i	$0.226341\pi$
$548$	49.8726	2.13045
$549$	−1.46748	−0.0626303
$550$	0	0
$551$	−14.7784	−0.629582
$552$	−10.1298	−0.431151
$553$	−8.83061	−0.375516
$554$	−7.08319	−0.300936
$555$	0	0
$556$	−67.5398	−2.86432
$557$	11.6471	0.493505	0.246752	−	0.969079i	$-0.420637\pi$
$557$	11.6471	0.493505	0.246752	+	0.969079i	$0.420637\pi$
$558$	5.48959	0.232393
$559$	−14.0305	−0.593426
$560$	0	0
$561$	−12.3083	−0.519655
$562$	49.2514	2.07755
$563$	−31.6878	−1.33548	−0.667741	−	0.744394i	$-0.732739\pi$
$563$	−31.6878	−1.33548	−0.667741	+	0.744394i	$0.732739\pi$
$564$	2.38372	0.100373
$565$	0	0
$566$	35.0786	1.47446
$567$	16.0288	0.673147
$568$	−20.0206	−0.840045
$569$	−29.7931	−1.24899	−0.624495	−	0.781028i	$-0.714695\pi$
$569$	−29.7931	−1.24899	−0.624495	+	0.781028i	$0.714695\pi$
$570$	0	0
$571$	35.6580	1.49224	0.746121	−	0.665810i	$-0.231914\pi$
$571$	35.6580	1.49224	0.746121	+	0.665810i	$0.231914\pi$
$572$	−40.5091	−1.69377
$573$	−16.6778	−0.696725
$574$	−78.6542	−3.28296
$575$	0	0
$576$	31.9006	1.32919
$577$	6.76735	0.281728	0.140864	−	0.990029i	$-0.455012\pi$
$577$	6.76735	0.281728	0.140864	+	0.990029i	$0.455012\pi$
$578$	−13.6030	−0.565811
$579$	−14.0986	−0.585916
$580$	0	0
$581$	−10.7089	−0.444278
$582$	24.0953	0.998781
$583$	−2.62070	−0.108538
$584$	22.4633	0.929538
$585$	0	0
$586$	65.3547	2.69978
$587$	1.89702	0.0782982	0.0391491	−	0.999233i	$-0.487535\pi$
$587$	1.89702	0.0782982	0.0391491	+	0.999233i	$0.487535\pi$
$588$	−14.5344	−0.599389
$589$	−2.88050	−0.118689
$590$	0	0
$591$	0.945864	0.0389076
$592$	7.40434	0.304317
$593$	24.2080	0.994103	0.497051	−	0.867721i	$-0.334416\pi$
$593$	24.2080	0.994103	0.497051	+	0.867721i	$0.334416\pi$
$594$	45.4592	1.86521
$595$	0	0
$596$	−14.6855	−0.601541
$597$	−6.82605	−0.279372
$598$	−35.9738	−1.47108
$599$	43.2713	1.76802	0.884009	−	0.467469i	$-0.154834\pi$
$599$	43.2713	1.76802	0.884009	+	0.467469i	$0.154834\pi$
$600$	0	0
$601$	4.08692	0.166709	0.0833545	−	0.996520i	$-0.473437\pi$
$601$	4.08692	0.166709	0.0833545	+	0.996520i	$0.473437\pi$
$602$	43.2098	1.76110
$603$	−30.5671	−1.24479
$604$	58.9312	2.39788
$605$	0	0
$606$	24.3621	0.989642
$607$	0.641570	0.0260405	0.0130203	−	0.999915i	$-0.495855\pi$
$607$	0.641570	0.0260405	0.0130203	+	0.999915i	$0.495855\pi$
$608$	−19.1482	−0.776563
$609$	13.9681	0.566017
$610$	0	0
$611$	2.85130	0.115351
$612$	−24.4345	−0.987707
$613$	−16.9855	−0.686038	−0.343019	−	0.939328i	$-0.611450\pi$
$613$	−16.9855	−0.686038	−0.343019	+	0.939328i	$0.611450\pi$
$614$	−43.2601	−1.74584
$615$	0	0
$616$	42.0206	1.69306
$617$	12.1444	0.488915	0.244457	−	0.969660i	$-0.421390\pi$
$617$	12.1444	0.488915	0.244457	+	0.969660i	$0.421390\pi$
$618$	26.9474	1.08399
$619$	7.09579	0.285204	0.142602	−	0.989780i	$-0.454453\pi$
$619$	7.09579	0.285204	0.142602	+	0.989780i	$0.454453\pi$
$620$	0	0
$621$	24.2726	0.974028
$622$	13.7940	0.553088
$623$	−28.1472	−1.12770
$624$	−1.85439	−0.0742350
$625$	0	0
$626$	−45.2037	−1.80670
$627$	−10.7258	−0.428349
$628$	−44.9408	−1.79333
$629$	−26.1311	−1.04191
$630$	0	0
$631$	−34.4872	−1.37291	−0.686457	−	0.727171i	$-0.740835\pi$
$631$	−34.4872	−1.37291	−0.686457	+	0.727171i	$0.740835\pi$
$632$	−5.46648	−0.217445
$633$	−2.12489	−0.0844567
$634$	−56.4574	−2.24221
$635$	0	0
$636$	1.16494	0.0461928
$637$	−17.3854	−0.688836
$638$	−57.7521	−2.28643
$639$	21.5713	0.853349
$640$	0	0
$641$	−20.5382	−0.811211	−0.405605	−	0.914048i	$-0.632939\pi$
$641$	−20.5382	−0.811211	−0.405605	+	0.914048i	$0.632939\pi$
$642$	−12.8231	−0.506089
$643$	−32.3455	−1.27558	−0.637792	−	0.770209i	$-0.720152\pi$
$643$	−32.3455	−1.27558	−0.637792	+	0.770209i	$0.720152\pi$
$644$	66.6127	2.62491
$645$	0	0
$646$	21.3241	0.838985
$647$	23.1202	0.908948	0.454474	−	0.890760i	$-0.349827\pi$
$647$	23.1202	0.908948	0.454474	+	0.890760i	$0.349827\pi$
$648$	9.92245	0.389790
$649$	−40.6877	−1.59713
$650$	0	0
$651$	2.72256	0.106706
$652$	−32.5688	−1.27549
$653$	−7.05761	−0.276186	−0.138093	−	0.990419i	$-0.544097\pi$
$653$	−7.05761	−0.276186	−0.138093	+	0.990419i	$0.544097\pi$
$654$	10.9650	0.428764
$655$	0	0
$656$	8.95080	0.349470
$657$	−24.2033	−0.944259
$658$	−8.78117	−0.342326
$659$	−13.8853	−0.540896	−0.270448	−	0.962735i	$-0.587172\pi$
$659$	−13.8853	−0.540896	−0.270448	+	0.962735i	$0.587172\pi$
$660$	0	0
$661$	−10.1962	−0.396584	−0.198292	−	0.980143i	$-0.563539\pi$
$661$	−10.1962	−0.396584	−0.198292	+	0.980143i	$0.563539\pi$
$662$	−39.6234	−1.54001
$663$	6.54442	0.254164
$664$	−6.62919	−0.257262
$665$	0	0
$666$	43.3974	1.68162
$667$	−30.8364	−1.19399
$668$	48.9778	1.89501
$669$	−6.74508	−0.260780
$670$	0	0
$671$	3.00912	0.116166
$672$	18.0983	0.698158
$673$	23.9562	0.923445	0.461722	−	0.887025i	$-0.347232\pi$
$673$	23.9562	0.923445	0.461722	+	0.887025i	$0.347232\pi$
$674$	6.48290	0.249712
$675$	0	0
$676$	−17.6661	−0.679466
$677$	9.99786	0.384249	0.192124	−	0.981371i	$-0.438462\pi$
$677$	9.99786	0.384249	0.192124	+	0.981371i	$0.438462\pi$
$678$	−18.8379	−0.723465
$679$	−53.3692	−2.04812
$680$	0	0
$681$	−3.42638	−0.131299
$682$	−11.2566	−0.431038
$683$	37.6974	1.44245	0.721226	−	0.692700i	$-0.243579\pi$
$683$	37.6974	1.44245	0.721226	+	0.692700i	$0.243579\pi$
$684$	−21.2931	−0.814162
$685$	0	0
$686$	−4.07097	−0.155430
$687$	−19.3769	−0.739275
$688$	−4.91724	−0.187468
$689$	1.39345	0.0530862
$690$	0	0
$691$	−32.1264	−1.22214	−0.611072	−	0.791575i	$-0.709261\pi$
$691$	−32.1264	−1.22214	−0.611072	+	0.791575i	$0.709261\pi$
$692$	−11.9803	−0.455422
$693$	−45.2754	−1.71987
$694$	15.4461	0.586324
$695$	0	0
$696$	8.64679	0.327756
$697$	−31.5887	−1.19651
$698$	−32.8529	−1.24350
$699$	3.52884	0.133473
$700$	0	0
$701$	−27.8813	−1.05306	−0.526530	−	0.850156i	$-0.676507\pi$
$701$	−27.8813	−1.05306	−0.526530	+	0.850156i	$0.676507\pi$
$702$	−24.1711	−0.912279
$703$	−22.7715	−0.858844
$704$	−65.4135	−2.46536
$705$	0	0
$706$	−23.6430	−0.889816
$707$	−53.9602	−2.02938
$708$	18.0863	0.679725
$709$	30.4246	1.14262	0.571310	−	0.820734i	$-0.306435\pi$
$709$	30.4246	1.14262	0.571310	+	0.820734i	$0.306435\pi$
$710$	0	0
$711$	5.88991	0.220889
$712$	−17.4242	−0.653000
$713$	−6.01040	−0.225091
$714$	−20.1549	−0.754278
$715$	0	0
$716$	74.8063	2.79564
$717$	−0.499530	−0.0186553
$718$	45.7392	1.70697
$719$	25.7714	0.961111	0.480556	−	0.876964i	$-0.340435\pi$
$719$	25.7714	0.961111	0.480556	+	0.876964i	$0.340435\pi$
$720$	0	0
$721$	−59.6866	−2.22284
$722$	−23.9697	−0.892061
$723$	−3.07215	−0.114254
$724$	−48.0592	−1.78611
$725$	0	0
$726$	−23.6642	−0.878261
$727$	2.49622	0.0925795	0.0462898	−	0.998928i	$-0.485260\pi$
$727$	2.49622	0.0925795	0.0462898	+	0.998928i	$0.485260\pi$
$728$	−22.3428	−0.828078
$729$	−1.71550	−0.0635370
$730$	0	0
$731$	17.3537	0.641850
$732$	−1.33760	−0.0494390
$733$	45.2444	1.67114	0.835569	−	0.549385i	$-0.185138\pi$
$733$	45.2444	1.67114	0.835569	+	0.549385i	$0.185138\pi$
$734$	−33.6316	−1.24136
$735$	0	0
$736$	−39.9543	−1.47274
$737$	62.6789	2.30881
$738$	52.4613	1.93113
$739$	8.57134	0.315302	0.157651	−	0.987495i	$-0.449608\pi$
$739$	8.57134	0.315302	0.157651	+	0.987495i	$0.449608\pi$
$740$	0	0
$741$	5.70304	0.209506
$742$	−4.29142	−0.157543
$743$	−5.94817	−0.218217	−0.109109	−	0.994030i	$-0.534800\pi$
$743$	−5.94817	−0.218217	−0.109109	+	0.994030i	$0.534800\pi$
$744$	1.68537	0.0617887
$745$	0	0
$746$	43.1057	1.57821
$747$	7.14267	0.261337
$748$	50.1039	1.83198
$749$	28.4023	1.03780
$750$	0	0
$751$	46.7694	1.70664	0.853319	−	0.521388i	$-0.174586\pi$
$751$	46.7694	1.70664	0.853319	+	0.521388i	$0.174586\pi$
$752$	0.999291	0.0364404
$753$	−17.1042	−0.623313
$754$	30.7074	1.11830
$755$	0	0
$756$	44.7576	1.62782
$757$	−45.3088	−1.64678	−0.823388	−	0.567479i	$-0.807919\pi$
$757$	−45.3088	−1.64678	−0.823388	+	0.567479i	$0.807919\pi$
$758$	7.27796	0.264348
$759$	−22.3804	−0.812356
$760$	0	0
$761$	−34.2873	−1.24291	−0.621457	−	0.783448i	$-0.713459\pi$
$761$	−34.2873	−1.24291	−0.621457	+	0.783448i	$0.713459\pi$
$762$	18.5279	0.671195
$763$	−24.2866	−0.879233
$764$	67.8913	2.45622
$765$	0	0
$766$	−4.52167	−0.163375
$767$	21.6341	0.781161
$768$	7.01833	0.253252
$769$	38.2808	1.38044	0.690220	−	0.723600i	$-0.257514\pi$
$769$	38.2808	1.38044	0.690220	+	0.723600i	$0.257514\pi$
$770$	0	0
$771$	−20.2954	−0.730921
$772$	57.3918	2.06558
$773$	−3.17845	−0.114321	−0.0571604	−	0.998365i	$-0.518205\pi$
$773$	−3.17845	−0.114321	−0.0571604	+	0.998365i	$0.518205\pi$
$774$	−28.8204	−1.03593
$775$	0	0
$776$	−33.0375	−1.18598
$777$	21.5230	0.772132
$778$	−34.2635	−1.22841
$779$	−27.5275	−0.986276
$780$	0	0
$781$	−44.2329	−1.58278
$782$	44.4945	1.59112
$783$	−20.7192	−0.740444
$784$	−6.09305	−0.217609
$785$	0	0
$786$	3.20999	0.114497
$787$	9.42277	0.335886	0.167943	−	0.985797i	$-0.446288\pi$
$787$	9.42277	0.335886	0.167943	+	0.985797i	$0.446288\pi$
$788$	−3.85038	−0.137164
$789$	9.72930	0.346372
$790$	0	0
$791$	41.7245	1.48355
$792$	−28.0272	−0.995903
$793$	−1.59998	−0.0568168
$794$	31.7609	1.12715
$795$	0	0
$796$	27.7872	0.984891
$797$	−8.28600	−0.293505	−0.146753	−	0.989173i	$-0.546882\pi$
$797$	−8.28600	−0.293505	−0.146753	+	0.989173i	$0.546882\pi$
$798$	−17.5637	−0.621748
$799$	−3.52665	−0.124764
$800$	0	0
$801$	18.7738	0.663341
$802$	−14.7804	−0.521914
$803$	49.6297	1.75139
$804$	−27.8617	−0.982607
$805$	0	0
$806$	5.98526	0.210822
$807$	−17.4405	−0.613936
$808$	−33.4034	−1.17513
$809$	−41.8918	−1.47284	−0.736418	−	0.676526i	$-0.763484\pi$
$809$	−41.8918	−1.47284	−0.736418	+	0.676526i	$0.763484\pi$
$810$	0	0
$811$	−33.5938	−1.17964	−0.589820	−	0.807535i	$-0.700801\pi$
$811$	−33.5938	−1.17964	−0.589820	+	0.807535i	$0.700801\pi$
$812$	−56.8608	−1.99542
$813$	−15.7124	−0.551057
$814$	−88.9881	−3.11903
$815$	0	0
$816$	2.29362	0.0802926
$817$	15.1226	0.529074
$818$	−65.3786	−2.28591
$819$	24.0734	0.841193
$820$	0	0
$821$	−30.9932	−1.08167	−0.540836	−	0.841128i	$-0.681892\pi$
$821$	−30.9932	−1.08167	−0.540836	+	0.841128i	$0.681892\pi$
$822$	−27.4382	−0.957018
$823$	9.48741	0.330710	0.165355	−	0.986234i	$-0.447123\pi$
$823$	9.48741	0.330710	0.165355	+	0.986234i	$0.447123\pi$
$824$	−36.9482	−1.28715
$825$	0	0
$826$	−66.6266	−2.31824
$827$	−4.97893	−0.173134	−0.0865671	−	0.996246i	$-0.527590\pi$
$827$	−4.97893	−0.173134	−0.0865671	+	0.996246i	$0.527590\pi$
$828$	−44.4298	−1.54404
$829$	−28.0734	−0.975030	−0.487515	−	0.873115i	$-0.662096\pi$
$829$	−28.0734	−0.975030	−0.487515	+	0.873115i	$0.662096\pi$
$830$	0	0
$831$	2.34306	0.0812800
$832$	34.7810	1.20581
$833$	21.5033	0.745046
$834$	37.1581	1.28668
$835$	0	0
$836$	43.6623	1.51009
$837$	−4.03844	−0.139589
$838$	−62.7554	−2.16785
$839$	0.672364	0.0232126	0.0116063	−	0.999933i	$-0.496306\pi$
$839$	0.672364	0.0232126	0.0116063	+	0.999933i	$0.496306\pi$
$840$	0	0
$841$	−2.67797	−0.0923438
$842$	−68.8792	−2.37373
$843$	−16.2920	−0.561126
$844$	8.64990	0.297742
$845$	0	0
$846$	5.85693	0.201365
$847$	52.4144	1.80098
$848$	0.488361	0.0167704
$849$	−11.6037	−0.398239
$850$	0	0
$851$	−47.5146	−1.62878
$852$	19.6622	0.673615
$853$	22.9459	0.785652	0.392826	−	0.919613i	$-0.371498\pi$
$853$	22.9459	0.785652	0.392826	+	0.919613i	$0.371498\pi$
$854$	4.92746	0.168614
$855$	0	0
$856$	17.5821	0.600944
$857$	−30.7670	−1.05098	−0.525491	−	0.850799i	$-0.676118\pi$
$857$	−30.7670	−1.05098	−0.525491	+	0.850799i	$0.676118\pi$
$858$	22.2867	0.760856
$859$	7.20222	0.245737	0.122868	−	0.992423i	$-0.460791\pi$
$859$	7.20222	0.245737	0.122868	+	0.992423i	$0.460791\pi$
$860$	0	0
$861$	26.0182	0.886698
$862$	−60.3527	−2.05562
$863$	15.3436	0.522303	0.261152	−	0.965298i	$-0.415898\pi$
$863$	15.3436	0.522303	0.261152	+	0.965298i	$0.415898\pi$
$864$	−26.8456	−0.913307
$865$	0	0
$866$	−21.1757	−0.719581
$867$	4.49977	0.152820
$868$	−11.0829	−0.376178
$869$	−12.0775	−0.409700
$870$	0	0
$871$	−33.3270	−1.12924
$872$	−15.0343	−0.509125
$873$	35.5966	1.20476
$874$	38.7740	1.31155
$875$	0	0
$876$	−22.0611	−0.745377
$877$	−2.70184	−0.0912348	−0.0456174	−	0.998959i	$-0.514526\pi$
$877$	−2.70184	−0.0912348	−0.0456174	+	0.998959i	$0.514526\pi$
$878$	−45.5486	−1.53719
$879$	−21.6188	−0.729186
$880$	0	0
$881$	21.5020	0.724420	0.362210	−	0.932097i	$-0.382022\pi$
$881$	21.5020	0.724420	0.362210	+	0.932097i	$0.382022\pi$
$882$	−35.7119	−1.20248
$883$	15.4417	0.519653	0.259827	−	0.965655i	$-0.416335\pi$
$883$	15.4417	0.519653	0.259827	+	0.965655i	$0.416335\pi$
$884$	−26.6408	−0.896026
$885$	0	0
$886$	30.6424	1.02945
$887$	41.1908	1.38305	0.691526	−	0.722352i	$-0.256939\pi$
$887$	41.1908	1.38305	0.691526	+	0.722352i	$0.256939\pi$
$888$	13.3235	0.447108
$889$	−41.0379	−1.37637
$890$	0	0
$891$	21.9223	0.734426
$892$	27.4576	0.919347
$893$	−3.07325	−0.102842
$894$	8.07946	0.270217
$895$	0	0
$896$	−58.2564	−1.94621
$897$	11.8999	0.397325
$898$	92.0679	3.07235
$899$	5.13050	0.171112
$900$	0	0
$901$	−1.72350	−0.0574181
$902$	−107.574	−3.58182
$903$	−14.2935	−0.475657
$904$	25.8290	0.859061
$905$	0	0
$906$	−32.4220	−1.07715
$907$	−36.5292	−1.21293	−0.606466	−	0.795109i	$-0.707413\pi$
$907$	−36.5292	−1.21293	−0.606466	+	0.795109i	$0.707413\pi$
$908$	13.9480	0.462880
$909$	35.9908	1.19374
$910$	0	0
$911$	−32.2943	−1.06996	−0.534979	−	0.844865i	$-0.679680\pi$
$911$	−32.2943	−1.06996	−0.534979	+	0.844865i	$0.679680\pi$
$912$	1.99874	0.0661848
$913$	−14.6463	−0.484723
$914$	−37.3805	−1.23644
$915$	0	0
$916$	78.8787	2.60623
$917$	−7.10990	−0.234789
$918$	29.8962	0.986721
$919$	9.06037	0.298874	0.149437	−	0.988771i	$-0.452254\pi$
$919$	9.06037	0.298874	0.149437	+	0.988771i	$0.452254\pi$
$920$	0	0
$921$	14.3101	0.471534
$922$	6.05048	0.199262
$923$	23.5190	0.774139
$924$	−41.2683	−1.35763
$925$	0	0
$926$	30.5045	1.00244
$927$	39.8102	1.30754
$928$	34.1051	1.11956
$929$	−23.6578	−0.776188	−0.388094	−	0.921620i	$-0.626866\pi$
$929$	−23.6578	−0.776188	−0.388094	+	0.921620i	$0.626866\pi$
$930$	0	0
$931$	18.7387	0.614138
$932$	−14.3651	−0.470543
$933$	−4.56294	−0.149384
$934$	−61.7463	−2.02040
$935$	0	0
$936$	14.9023	0.487098
$937$	35.7122	1.16667	0.583334	−	0.812232i	$-0.301748\pi$
$937$	35.7122	1.16667	0.583334	+	0.812232i	$0.301748\pi$
$938$	102.637	3.35123
$939$	14.9530	0.487974
$940$	0	0
$941$	−41.9221	−1.36662	−0.683311	−	0.730128i	$-0.739460\pi$
$941$	−41.9221	−1.36662	−0.683311	+	0.730128i	$0.739460\pi$
$942$	24.7249	0.805582
$943$	−57.4385	−1.87045
$944$	7.58207	0.246775
$945$	0	0
$946$	59.0972	1.92142
$947$	42.0893	1.36772	0.683859	−	0.729614i	$-0.260300\pi$
$947$	42.0893	1.36772	0.683859	+	0.729614i	$0.260300\pi$
$948$	5.36862	0.174365
$949$	−26.3886	−0.856611
$950$	0	0
$951$	18.6757	0.605600
$952$	27.6348	0.895650
$953$	34.6630	1.12284	0.561422	−	0.827530i	$-0.310254\pi$
$953$	34.6630	1.12284	0.561422	+	0.827530i	$0.310254\pi$
$954$	2.86232	0.0926710
$955$	0	0
$956$	2.03347	0.0657670
$957$	19.1040	0.617543
$958$	12.1325	0.391984
$959$	60.7736	1.96248
$960$	0	0
$961$	1.00000	0.0322581
$962$	47.3158	1.52552
$963$	−18.9440	−0.610461
$964$	12.5060	0.402790
$965$	0	0
$966$	−36.6481	−1.17913
$967$	−25.6742	−0.825628	−0.412814	−	0.910815i	$-0.635454\pi$
$967$	−25.6742	−0.825628	−0.412814	+	0.910815i	$0.635454\pi$
$968$	32.4465	1.04287
$969$	−7.05385	−0.226602
$970$	0	0
$971$	22.5664	0.724192	0.362096	−	0.932141i	$-0.382061\pi$
$971$	22.5664	0.724192	0.362096	+	0.932141i	$0.382061\pi$
$972$	−46.2820	−1.48449
$973$	−82.3025	−2.63850
$974$	2.77619	0.0889548
$975$	0	0
$976$	−0.560741	−0.0179489
$977$	19.5130	0.624275	0.312138	−	0.950037i	$-0.398955\pi$
$977$	19.5130	0.624275	0.312138	+	0.950037i	$0.398955\pi$
$978$	17.9183	0.572963
$979$	−38.4965	−1.23035
$980$	0	0
$981$	16.1988	0.517189
$982$	42.6530	1.36111
$983$	−39.3717	−1.25576	−0.627882	−	0.778309i	$-0.716078\pi$
$983$	−39.3717	−1.25576	−0.627882	+	0.778309i	$0.716078\pi$
$984$	16.1063	0.513449
$985$	0	0
$986$	−37.9806	−1.20955
$987$	2.90474	0.0924590
$988$	−23.2157	−0.738589
$989$	31.5546	1.00338
$990$	0	0
$991$	28.1540	0.894342	0.447171	−	0.894448i	$-0.352432\pi$
$991$	28.1540	0.894342	0.447171	+	0.894448i	$0.352432\pi$
$992$	6.64753	0.211059
$993$	13.1071	0.415942
$994$	−72.4318	−2.29740
$995$	0	0
$996$	6.51051	0.206293
$997$	−38.7855	−1.22835	−0.614175	−	0.789170i	$-0.710511\pi$
$997$	−38.7855	−1.22835	−0.614175	+	0.789170i	$0.710511\pi$
$998$	8.54787	0.270578
$999$	−31.9255	−1.01008

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.a.k.1.4	yes	5
3.2	odd	2		6975.2.a.bp.1.2		5
5.2	odd	4		775.2.b.g.249.9		10
5.3	odd	4		775.2.b.g.249.2		10
5.4	even	2		775.2.a.h.1.2	✓	5
15.14	odd	2		6975.2.a.by.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
775.2.a.h.1.2	✓	5	5.4	even	2
775.2.a.k.1.4	yes	5	1.1	even	1	trivial
775.2.b.g.249.2		10	5.3	odd	4
775.2.b.g.249.9		10	5.2	odd	4
6975.2.a.bp.1.2		5	3.2	odd	2
6975.2.a.by.1.4		5	15.14	odd	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:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23959 q^{2} -0.740841 q^{3} +3.01578 q^{4} -1.65918 q^{6} +3.67497 q^{7} +2.27494 q^{8} -2.45115 q^{9} +5.02619 q^{11} -2.23422 q^{12} -2.67247 q^{13} +8.23043 q^{14} -0.936618 q^{16} +3.30547 q^{17} -5.48959 q^{18} +2.88050 q^{19} -2.72256 q^{21} +11.2566 q^{22} +6.01040 q^{23} -1.68537 q^{24} -5.98526 q^{26} +4.03844 q^{27} +11.0829 q^{28} -5.13050 q^{29} -1.00000 q^{31} -6.64753 q^{32} -3.72360 q^{33} +7.40291 q^{34} -7.39215 q^{36} -7.90540 q^{37} +6.45115 q^{38} +1.97988 q^{39} -9.55651 q^{41} -6.09744 q^{42} +5.25000 q^{43} +15.1579 q^{44} +13.4609 q^{46} -1.06691 q^{47} +0.693885 q^{48} +6.50538 q^{49} -2.44883 q^{51} -8.05960 q^{52} -0.521408 q^{53} +9.04446 q^{54} +8.36034 q^{56} -2.13399 q^{57} -11.4902 q^{58} -8.09515 q^{59} +0.598687 q^{61} -2.23959 q^{62} -9.00791 q^{63} -13.0145 q^{64} -8.33936 q^{66} +12.4705 q^{67} +9.96858 q^{68} -4.45275 q^{69} -8.80048 q^{71} -5.57624 q^{72} +9.87423 q^{73} -17.7049 q^{74} +8.68697 q^{76} +18.4711 q^{77} +4.43412 q^{78} -2.40291 q^{79} +4.36163 q^{81} -21.4027 q^{82} -2.91400 q^{83} -8.21067 q^{84} +11.7579 q^{86} +3.80088 q^{87} +11.4343 q^{88} -7.65918 q^{89} -9.82125 q^{91} +18.1261 q^{92} +0.740841 q^{93} -2.38946 q^{94} +4.92476 q^{96} -14.5224 q^{97} +14.5694 q^{98} -12.3200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 3 q^{3} + 6 q^{4} - q^{6} + 2 q^{7} + 9 q^{8} + 6 q^{9} - 2 q^{11} + 5 q^{12} + 4 q^{13} + 2 q^{14} + 4 q^{16} + 19 q^{17} + 5 q^{18} + 8 q^{19} - 15 q^{21} - 10 q^{22} + 12 q^{23} + 26 q^{24}+ \cdots - 33 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 + 3 * q^3 + 6 * q^4 - q^6 + 2 * q^7 + 9 * q^8 + 6 * q^9 - 2 * q^11 + 5 * q^12 + 4 * q^13 + 2 * q^14 + 4 * q^16 + 19 * q^17 + 5 * q^18 + 8 * q^19 - 15 * q^21 - 10 * q^22 + 12 * q^23 + 26 * q^24 - 16 * q^26 + 6 * q^27 + 6 * q^28 + 6 * q^29 - 5 * q^31 + 7 * q^32 + 3 * q^33 + 31 * q^34 - 13 * q^36 + 16 * q^37 + 14 * q^38 - 13 * q^39 - 2 * q^41 - 22 * q^42 + q^43 - 4 * q^44 - 11 * q^46 + 8 * q^47 + 31 * q^48 - 9 * q^49 + 5 * q^51 - 26 * q^52 + 3 * q^53 + 10 * q^54 - 9 * q^56 + 14 * q^57 + 5 * q^58 - 4 * q^59 - 5 * q^61 - 4 * q^62 - 26 * q^63 - 5 * q^64 - 25 * q^66 + 13 * q^67 + 30 * q^68 + 10 * q^69 + 6 * q^71 + 2 * q^72 + 19 * q^73 + 4 * q^74 - 5 * q^76 + 23 * q^77 - 38 * q^78 - 6 * q^79 - 7 * q^81 - 27 * q^82 + 18 * q^83 - 15 * q^84 - 7 * q^86 - 5 * q^87 - q^88 - 31 * q^89 + 8 * q^91 - 16 * q^92 - 3 * q^93 - 14 * q^94 + 10 * q^96 - q^97 + 10 * q^98 - 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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