LMFDB - Embedded newform 475.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.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:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.78165$ of defining polynomial
Character	$\chi$	$=$	475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.95594 q^{2} -0.296842 q^{3} +1.82571 q^{4} +0.580605 q^{6} +3.56331 q^{7} +0.340899 q^{8} -2.91188 q^{9} +5.56331 q^{11} -0.541947 q^{12} -5.26647 q^{13} -6.96962 q^{14} -4.31820 q^{16} -1.40632 q^{17} +5.69548 q^{18} +1.00000 q^{19} -1.05774 q^{21} -10.8815 q^{22} +6.96962 q^{23} -0.101193 q^{24} +10.3009 q^{26} +1.75489 q^{27} +6.50557 q^{28} +1.40632 q^{29} +1.75489 q^{31} +7.76435 q^{32} -1.65142 q^{33} +2.75067 q^{34} -5.31626 q^{36} -3.61504 q^{37} -1.95594 q^{38} +1.56331 q^{39} +4.34858 q^{41} +2.06888 q^{42} +3.56331 q^{43} +10.1570 q^{44} -13.6322 q^{46} +8.26046 q^{47} +1.28182 q^{48} +5.69716 q^{49} +0.417453 q^{51} -9.61504 q^{52} +7.61504 q^{53} -3.43247 q^{54} +1.21473 q^{56} -0.296842 q^{57} -2.75067 q^{58} +9.47519 q^{59} +9.21473 q^{61} -3.43247 q^{62} -10.3759 q^{63} -6.55023 q^{64} +3.23009 q^{66} +4.76090 q^{67} -2.56753 q^{68} -2.06888 q^{69} -14.0689 q^{71} -0.992660 q^{72} -6.59368 q^{73} +7.07082 q^{74} +1.82571 q^{76} +19.8238 q^{77} -3.05774 q^{78} +5.47519 q^{79} +8.21473 q^{81} -8.50557 q^{82} +4.15699 q^{83} -1.93112 q^{84} -6.96962 q^{86} -0.417453 q^{87} +1.89653 q^{88} -9.23009 q^{89} -18.7660 q^{91} +12.7245 q^{92} -0.520926 q^{93} -16.1570 q^{94} -2.30478 q^{96} -11.5116 q^{97} -11.1433 q^{98} -16.1997 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9} + 4 q^{11} - 6 q^{12} - 2 q^{13} - 8 q^{14} + 4 q^{16} - 4 q^{17} + 34 q^{18} + 4 q^{19} - 4 q^{21} - 4 q^{22} + 8 q^{23} - 24 q^{24} + 4 q^{26}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 - 2 * q^3 + 8 * q^4 - 4 * q^7 + 12 * q^8 + 8 * q^9 + 4 * q^11 - 6 * q^12 - 2 * q^13 - 8 * q^14 + 4 * q^16 - 4 * q^17 + 34 * q^18 + 4 * q^19 - 4 * q^21 - 4 * q^22 + 8 * q^23 - 24 * q^24 + 4 * q^26 + 4 * q^27 + 8 * q^28 + 4 * q^29 + 4 * q^31 + 6 * q^32 - 8 * q^33 - 4 * q^34 + 40 * q^36 + 6 * q^37 + 2 * q^38 - 12 * q^39 + 16 * q^41 - 28 * q^42 - 4 * q^43 + 24 * q^44 + 12 * q^47 - 38 * q^48 + 20 * q^49 - 36 * q^51 - 18 * q^52 + 10 * q^53 - 20 * q^54 - 12 * q^56 - 2 * q^57 + 4 * q^58 + 20 * q^61 - 20 * q^62 - 20 * q^63 - 4 * q^64 - 28 * q^66 + 18 * q^67 - 4 * q^68 + 28 * q^69 - 20 * q^71 + 52 * q^72 - 28 * q^73 + 32 * q^74 + 8 * q^76 + 40 * q^77 - 12 * q^78 - 16 * q^79 + 16 * q^81 - 16 * q^82 - 44 * q^84 - 8 * q^86 + 36 * q^87 + 12 * q^88 + 4 * q^89 - 36 * q^91 + 28 * q^92 + 40 * q^93 - 48 * q^94 - 52 * q^96 - 30 * q^97 - 38 * q^98 - 4 * 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 $

$2$	−1.95594	−1.38306	−0.691530	−	0.722348i	$-0.743063\pi$
$2$	−1.95594	−1.38306	−0.691530	+	0.722348i	$0.743063\pi$
$3$	−0.296842	−0.171382	−0.0856908	−	0.996322i	$-0.527310\pi$
$3$	−0.296842	−0.171382	−0.0856908	+	0.996322i	$0.527310\pi$
$4$	1.82571	0.912855
$5$	0	0
$5$	0	0
$6$	0.580605	0.237031
$7$	3.56331	1.34680	0.673402	−	0.739277i	$-0.264832\pi$
$7$	3.56331	1.34680	0.673402	+	0.739277i	$0.264832\pi$
$8$	0.340899	0.120526
$9$	−2.91188	−0.970628
$10$	0	0
$11$	5.56331	1.67740	0.838700	−	0.544594i	$-0.183316\pi$
$11$	5.56331	1.67740	0.838700	+	0.544594i	$0.183316\pi$
$12$	−0.541947	−0.156447
$13$	−5.26647	−1.46065	−0.730327	−	0.683097i	$-0.760632\pi$
$13$	−5.26647	−1.46065	−0.730327	+	0.683097i	$0.760632\pi$
$14$	−6.96962	−1.86271
$15$	0	0
$16$	−4.31820	−1.07955
$17$	−1.40632	−0.341082	−0.170541	−	0.985351i	$-0.554552\pi$
$17$	−1.40632	−0.341082	−0.170541	+	0.985351i	$0.554552\pi$
$18$	5.69548	1.34244
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.05774	−0.230817
$22$	−10.8815	−2.31995
$23$	6.96962	1.45327	0.726633	−	0.687025i	$-0.241084\pi$
$23$	6.96962	1.45327	0.726633	+	0.687025i	$0.241084\pi$
$24$	−0.101193	−0.0206560
$25$	0	0
$26$	10.3009	2.02017
$27$	1.75489	0.337730
$28$	6.50557	1.22944
$29$	1.40632	0.261146	0.130573	−	0.991439i	$-0.458318\pi$
$29$	1.40632	0.261146	0.130573	+	0.991439i	$0.458318\pi$
$30$	0	0
$31$	1.75489	0.315188	0.157594	−	0.987504i	$-0.449626\pi$
$31$	1.75489	0.315188	0.157594	+	0.987504i	$0.449626\pi$
$32$	7.76435	1.37256
$33$	−1.65142	−0.287476
$34$	2.75067	0.471737
$35$	0	0
$36$	−5.31626	−0.886043
$37$	−3.61504	−0.594309	−0.297155	−	0.954829i	$-0.596038\pi$
$37$	−3.61504	−0.594309	−0.297155	+	0.954829i	$0.596038\pi$
$38$	−1.95594	−0.317296
$39$	1.56331	0.250329
$40$	0	0
$41$	4.34858	0.679134	0.339567	−	0.940582i	$-0.389720\pi$
$41$	4.34858	0.679134	0.339567	+	0.940582i	$0.389720\pi$
$42$	2.06888	0.319234
$43$	3.56331	0.543399	0.271700	−	0.962382i	$-0.412414\pi$
$43$	3.56331	0.543399	0.271700	+	0.962382i	$0.412414\pi$
$44$	10.1570	1.53122
$45$	0	0
$46$	−13.6322	−2.00996
$47$	8.26046	1.20491	0.602456	−	0.798152i	$-0.294189\pi$
$47$	8.26046	1.20491	0.602456	+	0.798152i	$0.294189\pi$
$48$	1.28182	0.185015
$49$	5.69716	0.813879
$50$	0	0
$51$	0.417453	0.0584552
$52$	−9.61504	−1.33337
$53$	7.61504	1.04601	0.523003	−	0.852331i	$-0.324812\pi$
$53$	7.61504	1.04601	0.523003	+	0.852331i	$0.324812\pi$
$54$	−3.43247	−0.467100
$55$	0	0
$56$	1.21473	0.162325
$57$	−0.296842	−0.0393177
$58$	−2.75067	−0.361181
$59$	9.47519	1.23356	0.616782	−	0.787134i	$-0.288436\pi$
$59$	9.47519	1.23356	0.616782	+	0.787134i	$0.288436\pi$
$60$	0	0
$61$	9.21473	1.17983	0.589913	−	0.807467i	$-0.299162\pi$
$61$	9.21473	1.17983	0.589913	+	0.807467i	$0.299162\pi$
$62$	−3.43247	−0.435924
$63$	−10.3759	−1.30725
$64$	−6.55023	−0.818779
$65$	0	0
$66$	3.23009	0.397596
$67$	4.76090	0.581636	0.290818	−	0.956778i	$-0.406073\pi$
$67$	4.76090	0.581636	0.290818	+	0.956778i	$0.406073\pi$
$68$	−2.56753	−0.311358
$69$	−2.06888	−0.249063
$70$	0	0
$71$	−14.0689	−1.66967	−0.834834	−	0.550502i	$-0.814437\pi$
$71$	−14.0689	−1.66967	−0.834834	+	0.550502i	$0.814437\pi$
$72$	−0.992660	−0.116986
$73$	−6.59368	−0.771732	−0.385866	−	0.922555i	$-0.626097\pi$
$73$	−6.59368	−0.771732	−0.385866	+	0.922555i	$0.626097\pi$
$74$	7.07082	0.821966
$75$	0	0
$76$	1.82571	0.209423
$77$	19.8238	2.25913
$78$	−3.05774	−0.346221
$79$	5.47519	0.616007	0.308004	−	0.951385i	$-0.400339\pi$
$79$	5.47519	0.616007	0.308004	+	0.951385i	$0.400339\pi$
$80$	0	0
$81$	8.21473	0.912748
$82$	−8.50557	−0.939283
$83$	4.15699	0.456289	0.228144	−	0.973627i	$-0.426734\pi$
$83$	4.15699	0.456289	0.228144	+	0.973627i	$0.426734\pi$
$84$	−1.93112	−0.210703
$85$	0	0
$86$	−6.96962	−0.751554
$87$	−0.417453	−0.0447557
$88$	1.89653	0.202171
$89$	−9.23009	−0.978387	−0.489194	−	0.872175i	$-0.662709\pi$
$89$	−9.23009	−0.978387	−0.489194	+	0.872175i	$0.662709\pi$
$90$	0	0
$91$	−18.7660	−1.96721
$92$	12.7245	1.32662
$93$	−0.520926	−0.0540175
$94$	−16.1570	−1.66647
$95$	0	0
$96$	−2.30478	−0.235231
$97$	−11.5116	−1.16882	−0.584411	−	0.811457i	$-0.698675\pi$
$97$	−11.5116	−1.16882	−0.584411	+	0.811457i	$0.698675\pi$
$98$	−11.1433	−1.12564
$99$	−16.1997	−1.62813
$100$	0	0
$101$	−11.8511	−1.17923	−0.589616	−	0.807684i	$-0.700721\pi$
$101$	−11.8511	−1.17923	−0.589616	+	0.807684i	$0.700721\pi$
$102$	−0.816515	−0.0808470
$103$	−1.35458	−0.133471	−0.0667354	−	0.997771i	$-0.521258\pi$
$103$	−1.35458	−0.133471	−0.0667354	+	0.997771i	$0.521258\pi$
$104$	−1.79533	−0.176047
$105$	0	0
$106$	−14.8946	−1.44669
$107$	7.06287	0.682794	0.341397	−	0.939919i	$-0.389100\pi$
$107$	7.06287	0.682794	0.341397	+	0.939919i	$0.389100\pi$
$108$	3.20393	0.308298
$109$	10.1844	0.975484	0.487742	−	0.872988i	$-0.337821\pi$
$109$	10.1844	0.975484	0.487742	+	0.872988i	$0.337821\pi$
$110$	0	0
$111$	1.07310	0.101854
$112$	−15.3871	−1.45394
$113$	−1.86015	−0.174988	−0.0874940	−	0.996165i	$-0.527886\pi$
$113$	−1.86015	−0.174988	−0.0874940	+	0.996165i	$0.527886\pi$
$114$	0.580605	0.0543787
$115$	0	0
$116$	2.56753	0.238389
$117$	15.3353	1.41775
$118$	−18.5329	−1.70609
$119$	−5.01114	−0.459370
$120$	0	0
$121$	19.9504	1.81367
$122$	−18.0235	−1.63177
$123$	−1.29084	−0.116391
$124$	3.20393	0.287721
$125$	0	0
$126$	20.2947	1.80800
$127$	4.89053	0.433964	0.216982	−	0.976176i	$-0.430379\pi$
$127$	4.89053	0.433964	0.216982	+	0.976176i	$0.430379\pi$
$128$	−2.71684	−0.240137
$129$	−1.05774	−0.0931287
$130$	0	0
$131$	2.81263	0.245741	0.122870	−	0.992423i	$-0.460790\pi$
$131$	2.81263	0.245741	0.122870	+	0.992423i	$0.460790\pi$
$132$	−3.01502	−0.262424
$133$	3.56331	0.308978
$134$	−9.31204	−0.804438
$135$	0	0
$136$	−0.479412	−0.0411093
$137$	−9.23009	−0.788579	−0.394290	−	0.918986i	$-0.629009\pi$
$137$	−9.23009	−0.788579	−0.394290	+	0.918986i	$0.629009\pi$
$138$	4.04660	0.344470
$139$	−3.67878	−0.312030	−0.156015	−	0.987755i	$-0.549865\pi$
$139$	−3.67878	−0.312030	−0.156015	+	0.987755i	$0.549865\pi$
$140$	0	0
$141$	−2.45205	−0.206500
$142$	27.5179	2.30925
$143$	−29.2990	−2.45010
$144$	12.5741	1.04784
$145$	0	0
$146$	12.8969	1.06735
$147$	−1.69115	−0.139484
$148$	−6.60002	−0.542519
$149$	7.09925	0.581593	0.290797	−	0.956785i	$-0.406080\pi$
$149$	7.09925	0.581593	0.290797	+	0.956785i	$0.406080\pi$
$150$	0	0
$151$	−18.3567	−1.49385	−0.746924	−	0.664910i	$-0.768470\pi$
$151$	−18.3567	−1.49385	−0.746924	+	0.664910i	$0.768470\pi$
$152$	0.340899	0.0276506
$153$	4.09503	0.331064
$154$	−38.7742	−3.12451
$155$	0	0
$156$	2.85415	0.228515
$157$	−17.2301	−1.37511	−0.687555	−	0.726132i	$-0.741316\pi$
$157$	−17.2301	−1.37511	−0.687555	+	0.726132i	$0.741316\pi$
$158$	−10.7092	−0.851975
$159$	−2.26046	−0.179266
$160$	0	0
$161$	24.8349	1.95726
$162$	−16.0675	−1.26238
$163$	−10.8662	−0.851103	−0.425551	−	0.904934i	$-0.639920\pi$
$163$	−10.8662	−0.851103	−0.425551	+	0.904934i	$0.639920\pi$
$164$	7.93925	0.619951
$165$	0	0
$166$	−8.13083	−0.631075
$167$	−2.82977	−0.218974	−0.109487	−	0.993988i	$-0.534921\pi$
$167$	−2.82977	−0.218974	−0.109487	+	0.993988i	$0.534921\pi$
$168$	−0.360582	−0.0278195
$169$	14.7357	1.13351
$170$	0	0
$171$	−2.91188	−0.222677
$172$	6.50557	0.496045
$173$	9.26647	0.704516	0.352258	−	0.935903i	$-0.385414\pi$
$173$	9.26647	0.704516	0.352258	+	0.935903i	$0.385414\pi$
$174$	0.816515	0.0618998
$175$	0	0
$176$	−24.0235	−1.81084
$177$	−2.81263	−0.211410
$178$	18.0535	1.35317
$179$	−3.59067	−0.268379	−0.134190	−	0.990956i	$-0.542843\pi$
$179$	−3.59067	−0.268379	−0.134190	+	0.990956i	$0.542843\pi$
$180$	0	0
$181$	−19.7630	−1.46897	−0.734487	−	0.678623i	$-0.762577\pi$
$181$	−19.7630	−1.46897	−0.734487	+	0.678623i	$0.762577\pi$
$182$	36.7053	2.72078
$183$	−2.73532	−0.202200
$184$	2.37594	0.175157
$185$	0	0
$186$	1.01890	0.0747095
$187$	−7.82377	−0.572131
$188$	15.0812	1.09991
$189$	6.25323	0.454855
$190$	0	0
$191$	8.31398	0.601579	0.300789	−	0.953691i	$-0.402750\pi$
$191$	8.31398	0.601579	0.300789	+	0.953691i	$0.402750\pi$
$192$	1.94438	0.140324
$193$	22.2514	1.60169	0.800847	−	0.598869i	$-0.204383\pi$
$193$	22.2514	1.60169	0.800847	+	0.598869i	$0.204383\pi$
$194$	22.5160	1.61655
$195$	0	0
$196$	10.4014	0.742954
$197$	8.81263	0.627874	0.313937	−	0.949444i	$-0.398352\pi$
$197$	8.81263	0.627874	0.313937	+	0.949444i	$0.398352\pi$
$198$	31.6857	2.25180
$199$	21.0659	1.49332	0.746660	−	0.665206i	$-0.231656\pi$
$199$	21.0659	1.49332	0.746660	+	0.665206i	$0.231656\pi$
$200$	0	0
$201$	−1.41323	−0.0996818
$202$	23.1801	1.63095
$203$	5.01114	0.351713
$204$	0.762149	0.0533611
$205$	0	0
$206$	2.64948	0.184598
$207$	−20.2947	−1.41058
$208$	22.7417	1.57685
$209$	5.56331	0.384822
$210$	0	0
$211$	5.34556	0.368004	0.184002	−	0.982926i	$-0.441095\pi$
$211$	5.34556	0.368004	0.184002	+	0.982926i	$0.441095\pi$
$212$	13.9029	0.954853
$213$	4.17623	0.286151
$214$	−13.8146	−0.944345
$215$	0	0
$216$	0.598242	0.0407052
$217$	6.25323	0.424497
$218$	−19.9200	−1.34915
$219$	1.95728	0.132261
$220$	0	0
$221$	7.40632	0.498203
$222$	−2.09891	−0.140870
$223$	3.10947	0.208226	0.104113	−	0.994565i	$-0.466800\pi$
$223$	3.10947	0.208226	0.104113	+	0.994565i	$0.466800\pi$
$224$	27.6668	1.84856
$225$	0	0
$226$	3.63834	0.242019
$227$	−14.4692	−0.960354	−0.480177	−	0.877172i	$-0.659428\pi$
$227$	−14.4692	−0.960354	−0.480177	+	0.877172i	$0.659428\pi$
$228$	−0.541947	−0.0358913
$229$	−5.21473	−0.344599	−0.172299	−	0.985045i	$-0.555120\pi$
$229$	−5.21473	−0.344599	−0.172299	+	0.985045i	$0.555120\pi$
$230$	0	0
$231$	−5.88452	−0.387173
$232$	0.479412	0.0314750
$233$	3.18737	0.208811	0.104406	−	0.994535i	$-0.466706\pi$
$233$	3.18737	0.208811	0.104406	+	0.994535i	$0.466706\pi$
$234$	−29.9950	−1.96084
$235$	0	0
$236$	17.2990	1.12607
$237$	−1.62527	−0.105572
$238$	9.80150	0.635337
$239$	16.5209	1.06865	0.534325	−	0.845279i	$-0.320566\pi$
$239$	16.5209	1.06865	0.534325	+	0.845279i	$0.320566\pi$
$240$	0	0
$241$	−12.2271	−0.787615	−0.393807	−	0.919193i	$-0.628842\pi$
$241$	−12.2271	−0.787615	−0.393807	+	0.919193i	$0.628842\pi$
$242$	−39.0218	−2.50842
$243$	−7.70316	−0.494158
$244$	16.8234	1.07701
$245$	0	0
$246$	2.52481	0.160976
$247$	−5.26647	−0.335097
$248$	0.598242	0.0379884
$249$	−1.23397	−0.0781996
$250$	0	0
$251$	4.52093	0.285358	0.142679	−	0.989769i	$-0.454428\pi$
$251$	4.52093	0.285358	0.142679	+	0.989769i	$0.454428\pi$
$252$	−18.9435	−1.19333
$253$	38.7742	2.43771
$254$	−9.56559	−0.600198
$255$	0	0
$256$	18.4144	1.15090
$257$	−15.9290	−0.993625	−0.496813	−	0.867858i	$-0.665496\pi$
$257$	−15.9290	−0.993625	−0.496813	+	0.867858i	$0.665496\pi$
$258$	2.06888	0.128803
$259$	−12.8815	−0.800418
$260$	0	0
$261$	−4.09503	−0.253476
$262$	−5.50135	−0.339874
$263$	−0.854147	−0.0526689	−0.0263345	−	0.999653i	$-0.508383\pi$
$263$	−0.854147	−0.0526689	−0.0263345	+	0.999653i	$0.508383\pi$
$264$	−0.562969	−0.0346483
$265$	0	0
$266$	−6.96962	−0.427335
$267$	2.73988	0.167678
$268$	8.69202	0.530950
$269$	10.3913	0.633569	0.316784	−	0.948498i	$-0.397397\pi$
$269$	10.3913	0.633569	0.316784	+	0.948498i	$0.397397\pi$
$270$	0	0
$271$	−8.19971	−0.498097	−0.249048	−	0.968491i	$-0.580118\pi$
$271$	−8.19971	−0.498097	−0.249048	+	0.968491i	$0.580118\pi$
$272$	6.07276	0.368215
$273$	5.57054	0.337145
$274$	18.0535	1.09065
$275$	0	0
$276$	−3.77717	−0.227359
$277$	−14.6484	−0.880137	−0.440069	−	0.897964i	$-0.645046\pi$
$277$	−14.6484	−0.880137	−0.440069	+	0.897964i	$0.645046\pi$
$278$	7.19549	0.431557
$279$	−5.11005	−0.305931
$280$	0	0
$281$	−31.6129	−1.88587	−0.942935	−	0.332977i	$-0.891947\pi$
$281$	−31.6129	−1.88587	−0.942935	+	0.332977i	$0.891947\pi$
$282$	4.79607	0.285602
$283$	−24.7326	−1.47020	−0.735101	−	0.677957i	$-0.762865\pi$
$283$	−24.7326	−1.47020	−0.735101	+	0.677957i	$0.762865\pi$
$284$	−25.6857	−1.52417
$285$	0	0
$286$	57.3071	3.38864
$287$	15.4953	0.914660
$288$	−22.6089	−1.33224
$289$	−15.0223	−0.883663
$290$	0	0
$291$	3.41712	0.200315
$292$	−12.0382	−0.704480
$293$	−30.2857	−1.76931	−0.884655	−	0.466245i	$-0.845606\pi$
$293$	−30.2857	−1.76931	−0.884655	+	0.466245i	$0.845606\pi$
$294$	3.30780	0.192915
$295$	0	0
$296$	−1.23237	−0.0716298
$297$	9.76302	0.566508
$298$	−13.8857	−0.804379
$299$	−36.7053	−2.12272
$300$	0	0
$301$	12.6972	0.731852
$302$	35.9046	2.06608
$303$	3.51791	0.202099
$304$	−4.31820	−0.247666
$305$	0	0
$306$	−8.00965	−0.457881
$307$	23.0629	1.31627	0.658134	−	0.752901i	$-0.271346\pi$
$307$	23.0629	1.31627	0.658134	+	0.752901i	$0.271346\pi$
$308$	36.1925	2.06226
$309$	0.402096	0.0228744
$310$	0	0
$311$	−10.3152	−0.584921	−0.292460	−	0.956278i	$-0.594474\pi$
$311$	−10.3152	−0.584921	−0.292460	+	0.956278i	$0.594474\pi$
$312$	0.532930	0.0301712
$313$	4.95038	0.279812	0.139906	−	0.990165i	$-0.455320\pi$
$313$	4.95038	0.279812	0.139906	+	0.990165i	$0.455320\pi$
$314$	33.7011	1.90186
$315$	0	0
$316$	9.99612	0.562326
$317$	14.2433	0.799985	0.399992	−	0.916518i	$-0.369013\pi$
$317$	14.2433	0.799985	0.399992	+	0.916518i	$0.369013\pi$
$318$	4.42134	0.247936
$319$	7.82377	0.438047
$320$	0	0
$321$	−2.09656	−0.117018
$322$	−48.5756	−2.70702
$323$	−1.40632	−0.0782495
$324$	14.9977	0.833207
$325$	0	0
$326$	21.2536	1.17713
$327$	−3.02314	−0.167180
$328$	1.48243	0.0818534
$329$	29.4346	1.62278
$330$	0	0
$331$	2.04272	0.112278	0.0561390	−	0.998423i	$-0.482121\pi$
$331$	2.04272	0.112278	0.0561390	+	0.998423i	$0.482121\pi$
$332$	7.58946	0.416526
$333$	10.5266	0.576854
$334$	5.53487	0.302855
$335$	0	0
$336$	4.56753	0.249179
$337$	−34.6951	−1.88996	−0.944980	−	0.327128i	$-0.893919\pi$
$337$	−34.6951	−1.88996	−0.944980	+	0.327128i	$0.893919\pi$
$338$	−28.8221	−1.56772
$339$	0.552170	0.0299898
$340$	0	0
$341$	9.76302	0.528697
$342$	5.69548	0.307976
$343$	−4.64243	−0.250668
$344$	1.21473	0.0654938
$345$	0	0
$346$	−18.1247	−0.974388
$347$	−7.35280	−0.394719	−0.197359	−	0.980331i	$-0.563237\pi$
$347$	−7.35280	−0.394719	−0.197359	+	0.980331i	$0.563237\pi$
$348$	−0.762149	−0.0408555
$349$	−31.7630	−1.70024	−0.850118	−	0.526593i	$-0.823469\pi$
$349$	−31.7630	−1.70024	−0.850118	+	0.526593i	$0.823469\pi$
$350$	0	0
$351$	−9.24209	−0.493306
$352$	43.1955	2.30233
$353$	7.52179	0.400345	0.200172	−	0.979761i	$-0.435850\pi$
$353$	7.52179	0.400345	0.200172	+	0.979761i	$0.435850\pi$
$354$	5.50135	0.292393
$355$	0	0
$356$	−16.8515	−0.893126
$357$	1.48751	0.0787276
$358$	7.02314	0.371185
$359$	−30.3982	−1.60436	−0.802178	−	0.597085i	$-0.796326\pi$
$359$	−30.3982	−1.60436	−0.802178	+	0.597085i	$0.796326\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	38.6553	2.03168
$363$	−5.92211	−0.310830
$364$	−34.2613	−1.79578
$365$	0	0
$366$	5.35012	0.279655
$367$	−3.91577	−0.204401	−0.102201	−	0.994764i	$-0.532588\pi$
$367$	−3.91577	−0.204401	−0.102201	+	0.994764i	$0.532588\pi$
$368$	−30.0962	−1.56887
$369$	−12.6626	−0.659186
$370$	0	0
$371$	27.1347	1.40877
$372$	−0.951060	−0.0493102
$373$	−26.7759	−1.38641	−0.693203	−	0.720743i	$-0.743801\pi$
$373$	−26.7759	−1.38641	−0.693203	+	0.720743i	$0.743801\pi$
$374$	15.3028	0.791291
$375$	0	0
$376$	2.81599	0.145223
$377$	−7.40632	−0.381445
$378$	−12.2310	−0.629092
$379$	14.9504	0.767950	0.383975	−	0.923344i	$-0.374555\pi$
$379$	14.9504	0.767950	0.383975	+	0.923344i	$0.374555\pi$
$380$	0	0
$381$	−1.45171	−0.0743735
$382$	−16.2617	−0.832019
$383$	27.9910	1.43027	0.715136	−	0.698985i	$-0.246365\pi$
$383$	27.9910	1.43027	0.715136	+	0.698985i	$0.246365\pi$
$384$	0.806471	0.0411551
$385$	0	0
$386$	−43.5225	−2.21524
$387$	−10.3759	−0.527439
$388$	−21.0168	−1.06697
$389$	−35.2036	−1.78489	−0.892447	−	0.451152i	$-0.851013\pi$
$389$	−35.2036	−1.78489	−0.892447	+	0.451152i	$0.851013\pi$
$390$	0	0
$391$	−9.80150	−0.495683
$392$	1.94216	0.0980937
$393$	−0.834907	−0.0421155
$394$	−17.2370	−0.868388
$395$	0	0
$396$	−29.5760	−1.48625
$397$	35.9735	1.80546	0.902730	−	0.430208i	$-0.141560\pi$
$397$	35.9735	1.80546	0.902730	+	0.430208i	$0.141560\pi$
$398$	−41.2036	−2.06535
$399$	−1.05774	−0.0529532
$400$	0	0
$401$	23.2421	1.16065	0.580327	−	0.814383i	$-0.302925\pi$
$401$	23.2421	1.16065	0.580327	+	0.814383i	$0.302925\pi$
$402$	2.76420	0.137866
$403$	−9.24209	−0.460381
$404$	−21.6367	−1.07647
$405$	0	0
$406$	−9.80150	−0.486440
$407$	−20.1116	−0.996895
$408$	0.142310	0.00704538
$409$	−31.8926	−1.57699	−0.788495	−	0.615041i	$-0.789139\pi$
$409$	−31.8926	−1.57699	−0.788495	+	0.615041i	$0.789139\pi$
$410$	0	0
$411$	2.73988	0.135148
$412$	−2.47307	−0.121840
$413$	33.7630	1.66137
$414$	39.6953	1.95092
$415$	0	0
$416$	−40.8907	−2.00483
$417$	1.09202	0.0534763
$418$	−10.8815	−0.532232
$419$	31.8238	1.55469	0.777346	−	0.629073i	$-0.216565\pi$
$419$	31.8238	1.55469	0.777346	+	0.629073i	$0.216565\pi$
$420$	0	0
$421$	0.348578	0.0169887	0.00849433	−	0.999964i	$-0.497296\pi$
$421$	0.348578	0.0169887	0.00849433	+	0.999964i	$0.497296\pi$
$422$	−10.4556	−0.508971
$423$	−24.0535	−1.16952
$424$	2.59596	0.126071
$425$	0	0
$426$	−8.16847	−0.395763
$427$	32.8349	1.58899
$428$	12.8948	0.623292
$429$	8.69716	0.419903
$430$	0	0
$431$	29.2764	1.41019	0.705097	−	0.709111i	$-0.250904\pi$
$431$	29.2764	1.41019	0.705097	+	0.709111i	$0.250904\pi$
$432$	−7.57799	−0.364596
$433$	0.883290	0.0424482	0.0212241	−	0.999775i	$-0.493244\pi$
$433$	0.883290	0.0424482	0.0212241	+	0.999775i	$0.493244\pi$
$434$	−12.2310	−0.587105
$435$	0	0
$436$	18.5937	0.890476
$437$	6.96962	0.333402
$438$	−3.82833	−0.182925
$439$	−13.8584	−0.661424	−0.330712	−	0.943732i	$-0.607289\pi$
$439$	−13.8584	−0.661424	−0.330712	+	0.943732i	$0.607289\pi$
$440$	0	0
$441$	−16.5895	−0.789974
$442$	−14.4863	−0.689044
$443$	13.5753	0.644982	0.322491	−	0.946572i	$-0.395480\pi$
$443$	13.5753	0.644982	0.322491	+	0.946572i	$0.395480\pi$
$444$	1.95916	0.0929778
$445$	0	0
$446$	−6.08195	−0.287989
$447$	−2.10735	−0.0996745
$448$	−23.3405	−1.10273
$449$	−15.6334	−0.737785	−0.368893	−	0.929472i	$-0.620263\pi$
$449$	−15.6334	−0.737785	−0.368893	+	0.929472i	$0.620263\pi$
$450$	0	0
$451$	24.1925	1.13918
$452$	−3.39609	−0.159739
$453$	5.44904	0.256018
$454$	28.3009	1.32823
$455$	0	0
$456$	−0.101193	−0.00473880
$457$	−5.30284	−0.248057	−0.124028	−	0.992279i	$-0.539581\pi$
$457$	−5.30284	−0.248057	−0.124028	+	0.992279i	$0.539581\pi$
$458$	10.1997	0.476601
$459$	−2.46794	−0.115193
$460$	0	0
$461$	0.374734	0.0174531	0.00872656	−	0.999962i	$-0.497222\pi$
$461$	0.374734	0.0174531	0.00872656	+	0.999962i	$0.497222\pi$
$462$	11.5098	0.535484
$463$	−6.65564	−0.309314	−0.154657	−	0.987968i	$-0.549427\pi$
$463$	−6.65564	−0.309314	−0.154657	+	0.987968i	$0.549427\pi$
$464$	−6.07276	−0.281921
$465$	0	0
$466$	−6.23431	−0.288799
$467$	0.854147	0.0395252	0.0197626	−	0.999805i	$-0.493709\pi$
$467$	0.854147	0.0395252	0.0197626	+	0.999805i	$0.493709\pi$
$468$	27.9979	1.29420
$469$	16.9645	0.783349
$470$	0	0
$471$	5.11461	0.235669
$472$	3.23009	0.148677
$473$	19.8238	0.911498
$474$	3.17893	0.146013
$475$	0	0
$476$	−9.14889	−0.419339
$477$	−22.1741	−1.01528
$478$	−32.3140	−1.47801
$479$	−17.0731	−0.780090	−0.390045	−	0.920796i	$-0.627540\pi$
$479$	−17.0731	−0.780090	−0.390045	+	0.920796i	$0.627540\pi$
$480$	0	0
$481$	19.0385	0.868081
$482$	23.9154	1.08932
$483$	−7.37204	−0.335439
$484$	36.4236	1.65562
$485$	0	0
$486$	15.0669	0.683450
$487$	−12.8259	−0.581197	−0.290598	−	0.956845i	$-0.593854\pi$
$487$	−12.8259	−0.581197	−0.290598	+	0.956845i	$0.593854\pi$
$488$	3.14129	0.142200
$489$	3.22553	0.145863
$490$	0	0
$491$	−10.4054	−0.469591	−0.234796	−	0.972045i	$-0.575442\pi$
$491$	−10.4054	−0.469591	−0.234796	+	0.972045i	$0.575442\pi$
$492$	−2.35670	−0.106248
$493$	−1.97773	−0.0890723
$494$	10.3009	0.463460
$495$	0	0
$496$	−7.57799	−0.340262
$497$	−50.1317	−2.24872
$498$	2.41357	0.108155
$499$	36.1612	1.61880	0.809399	−	0.587258i	$-0.199793\pi$
$499$	36.1612	1.61880	0.809399	+	0.587258i	$0.199793\pi$
$500$	0	0
$501$	0.839995	0.0375282
$502$	−8.84267	−0.394668
$503$	−33.2536	−1.48270	−0.741352	−	0.671117i	$-0.765815\pi$
$503$	−33.2536	−1.48270	−0.741352	+	0.671117i	$0.765815\pi$
$504$	−3.53715	−0.157557
$505$	0	0
$506$	−75.8400	−3.37150
$507$	−4.37416	−0.194263
$508$	8.92869	0.396146
$509$	7.29084	0.323161	0.161580	−	0.986860i	$-0.448341\pi$
$509$	7.29084	0.323161	0.161580	+	0.986860i	$0.448341\pi$
$510$	0	0
$511$	−23.4953	−1.03937
$512$	−30.5839	−1.35163
$513$	1.75489	0.0774805
$514$	31.1563	1.37424
$515$	0	0
$516$	−1.93112	−0.0850130
$517$	45.9555	2.02112
$518$	25.1955	1.10703
$519$	−2.75067	−0.120741
$520$	0	0
$521$	−9.87849	−0.432785	−0.216392	−	0.976306i	$-0.569429\pi$
$521$	−9.87849	−0.432785	−0.216392	+	0.976306i	$0.569429\pi$
$522$	8.00965	0.350573
$523$	11.1925	0.489414	0.244707	−	0.969597i	$-0.421308\pi$
$523$	11.1925	0.489414	0.244707	+	0.969597i	$0.421308\pi$
$524$	5.13505	0.224326
$525$	0	0
$526$	1.67066	0.0728443
$527$	−2.46794	−0.107505
$528$	7.13117	0.310344
$529$	25.5756	1.11198
$530$	0	0
$531$	−27.5907	−1.19733
$532$	6.50557	0.282052
$533$	−22.9016	−0.991980
$534$	−5.35904	−0.231908
$535$	0	0
$536$	1.62299	0.0701023
$537$	1.06586	0.0459953
$538$	−20.3248	−0.876263
$539$	31.6950	1.36520
$540$	0	0
$541$	2.22587	0.0956974	0.0478487	−	0.998855i	$-0.484763\pi$
$541$	2.22587	0.0956974	0.0478487	+	0.998855i	$0.484763\pi$
$542$	16.0382	0.688898
$543$	5.86649	0.251755
$544$	−10.9191	−0.468154
$545$	0	0
$546$	−10.8957	−0.466291
$547$	34.9675	1.49510	0.747552	−	0.664204i	$-0.231229\pi$
$547$	34.9675	1.49510	0.747552	+	0.664204i	$0.231229\pi$
$548$	−16.8515	−0.719859
$549$	−26.8322	−1.14517
$550$	0	0
$551$	1.40632	0.0599111
$552$	−0.705278	−0.0300186
$553$	19.5098	0.829641
$554$	28.6514	1.21728
$555$	0	0
$556$	−6.71640	−0.284839
$557$	8.88539	0.376486	0.188243	−	0.982122i	$-0.439721\pi$
$557$	8.88539	0.376486	0.188243	+	0.982122i	$0.439721\pi$
$558$	9.99497	0.423121
$559$	−18.7660	−0.793719
$560$	0	0
$561$	2.32242	0.0980527
$562$	61.8331	2.60827
$563$	29.6767	1.25072	0.625362	−	0.780335i	$-0.284952\pi$
$563$	29.6767	1.25072	0.625362	+	0.780335i	$0.284952\pi$
$564$	−4.47674	−0.188505
$565$	0	0
$566$	48.3756	2.03338
$567$	29.2716	1.22929
$568$	−4.79607	−0.201239
$569$	−22.1152	−0.927116	−0.463558	−	0.886067i	$-0.653427\pi$
$569$	−22.1152	−0.927116	−0.463558	+	0.886067i	$0.653427\pi$
$570$	0	0
$571$	−33.2656	−1.39212	−0.696060	−	0.717983i	$-0.745065\pi$
$571$	−33.2656	−1.39212	−0.696060	+	0.717983i	$0.745065\pi$
$572$	−53.4914	−2.23659
$573$	−2.46794	−0.103100
$574$	−30.3080	−1.26503
$575$	0	0
$576$	19.0735	0.794730
$577$	−0.934140	−0.0388887	−0.0194444	−	0.999811i	$-0.506190\pi$
$577$	−0.934140	−0.0388887	−0.0194444	+	0.999811i	$0.506190\pi$
$578$	29.3827	1.22216
$579$	−6.60516	−0.274501
$580$	0	0
$581$	14.8126	0.614532
$582$	−6.68368	−0.277047
$583$	42.3648	1.75457
$584$	−2.24778	−0.0930139
$585$	0	0
$586$	59.2371	2.44706
$587$	−1.57531	−0.0650200	−0.0325100	−	0.999471i	$-0.510350\pi$
$587$	−1.57531	−0.0650200	−0.0325100	+	0.999471i	$0.510350\pi$
$588$	−3.08756	−0.127329
$589$	1.75489	0.0723092
$590$	0	0
$591$	−2.61596	−0.107606
$592$	15.6105	0.641587
$593$	−26.0162	−1.06836	−0.534180	−	0.845371i	$-0.679379\pi$
$593$	−26.0162	−1.06836	−0.534180	+	0.845371i	$0.679379\pi$
$594$	−19.0959	−0.783514
$595$	0	0
$596$	12.9612	0.530911
$597$	−6.25323	−0.255928
$598$	71.7934	2.93585
$599$	19.5359	0.798217	0.399109	−	0.916904i	$-0.369320\pi$
$599$	19.5359	0.798217	0.399109	+	0.916904i	$0.369320\pi$
$600$	0	0
$601$	43.5299	1.77562	0.887811	−	0.460208i	$-0.152225\pi$
$601$	43.5299	1.77562	0.887811	+	0.460208i	$0.152225\pi$
$602$	−24.8349	−1.01220
$603$	−13.8632	−0.564552
$604$	−33.5140	−1.36367
$605$	0	0
$606$	−6.88083	−0.279515
$607$	12.5847	0.510796	0.255398	−	0.966836i	$-0.417794\pi$
$607$	12.5847	0.510796	0.255398	+	0.966836i	$0.417794\pi$
$608$	7.76435	0.314886
$609$	−1.48751	−0.0602771
$610$	0	0
$611$	−43.5034	−1.75996
$612$	7.47634	0.302213
$613$	18.2412	0.736756	0.368378	−	0.929676i	$-0.379913\pi$
$613$	18.2412	0.736756	0.368378	+	0.929676i	$0.379913\pi$
$614$	−45.1097	−1.82048
$615$	0	0
$616$	6.75791	0.272284
$617$	−26.7314	−1.07617	−0.538084	−	0.842892i	$-0.680852\pi$
$617$	−26.7314	−1.07617	−0.538084	+	0.842892i	$0.680852\pi$
$618$	−0.786477	−0.0316367
$619$	2.92690	0.117642	0.0588211	−	0.998269i	$-0.481266\pi$
$619$	2.92690	0.117642	0.0588211	+	0.998269i	$0.481266\pi$
$620$	0	0
$621$	12.2310	0.490811
$622$	20.1759	0.808980
$623$	−32.8896	−1.31770
$624$	−6.75067	−0.270243
$625$	0	0
$626$	−9.68267	−0.386997
$627$	−1.65142	−0.0659514
$628$	−31.4572	−1.25528
$629$	5.08389	0.202708
$630$	0	0
$631$	−13.2493	−0.527447	−0.263724	−	0.964598i	$-0.584951\pi$
$631$	−13.2493	−0.527447	−0.263724	+	0.964598i	$0.584951\pi$
$632$	1.86649	0.0742450
$633$	−1.58679	−0.0630691
$634$	−27.8591	−1.10643
$635$	0	0
$636$	−4.12695	−0.163644
$637$	−30.0039	−1.18880
$638$	−15.3028	−0.605845
$639$	40.9669	1.62063
$640$	0	0
$641$	6.93026	0.273729	0.136864	−	0.990590i	$-0.456298\pi$
$641$	6.93026	0.273729	0.136864	+	0.990590i	$0.456298\pi$
$642$	4.10074	0.161843
$643$	−14.4794	−0.571012	−0.285506	−	0.958377i	$-0.592162\pi$
$643$	−14.4794	−0.571012	−0.285506	+	0.958377i	$0.592162\pi$
$644$	45.3414	1.78670
$645$	0	0
$646$	2.75067	0.108224
$647$	−6.35549	−0.249860	−0.124930	−	0.992166i	$-0.539871\pi$
$647$	−6.35549	−0.249860	−0.124930	+	0.992166i	$0.539871\pi$
$648$	2.80040	0.110010
$649$	52.7134	2.06918
$650$	0	0
$651$	−1.85622	−0.0727510
$652$	−19.8385	−0.776934
$653$	34.4030	1.34629	0.673146	−	0.739509i	$-0.264942\pi$
$653$	34.4030	1.34629	0.673146	+	0.739509i	$0.264942\pi$
$654$	5.91309	0.231220
$655$	0	0
$656$	−18.7780	−0.733159
$657$	19.2000	0.749065
$658$	−57.5723	−2.24440
$659$	−19.6214	−0.764341	−0.382170	−	0.924092i	$-0.624823\pi$
$659$	−19.6214	−0.764341	−0.382170	+	0.924092i	$0.624823\pi$
$660$	0	0
$661$	39.8054	1.54825	0.774126	−	0.633032i	$-0.218190\pi$
$661$	39.8054	1.54825	0.774126	+	0.633032i	$0.218190\pi$
$662$	−3.99544	−0.155287
$663$	−2.19850	−0.0853828
$664$	1.41712	0.0549947
$665$	0	0
$666$	−20.5894	−0.797823
$667$	9.80150	0.379515
$668$	−5.16635	−0.199892
$669$	−0.923022	−0.0356861
$670$	0	0
$671$	51.2644	1.97904
$672$	−8.21266	−0.316810
$673$	8.90374	0.343214	0.171607	−	0.985166i	$-0.445104\pi$
$673$	8.90374	0.343214	0.171607	+	0.985166i	$0.445104\pi$
$674$	67.8615	2.61393
$675$	0	0
$676$	26.9030	1.03473
$677$	−22.2695	−0.855886	−0.427943	−	0.903806i	$-0.640762\pi$
$677$	−22.2695	−0.855886	−0.427943	+	0.903806i	$0.640762\pi$
$678$	−1.08001	−0.0414776
$679$	−41.0193	−1.57417
$680$	0	0
$681$	4.29506	0.164587
$682$	−19.0959	−0.731220
$683$	−15.4054	−0.589472	−0.294736	−	0.955579i	$-0.595232\pi$
$683$	−15.4054	−0.589472	−0.294736	+	0.955579i	$0.595232\pi$
$684$	−5.31626	−0.203272
$685$	0	0
$686$	9.08033	0.346689
$687$	1.54795	0.0590580
$688$	−15.3871	−0.586627
$689$	−40.1044	−1.52785
$690$	0	0
$691$	−17.8773	−0.680084	−0.340042	−	0.940410i	$-0.610441\pi$
$691$	−17.8773	−0.680084	−0.340042	+	0.940410i	$0.610441\pi$
$692$	16.9179	0.643122
$693$	−57.7245	−2.19277
$694$	14.3817	0.545920
$695$	0	0
$696$	−0.142310	−0.00539423
$697$	−6.11548	−0.231640
$698$	62.1266	2.35153
$699$	−0.946144	−0.0357864
$700$	0	0
$701$	35.3609	1.33556	0.667782	−	0.744357i	$-0.267244\pi$
$701$	35.3609	1.33556	0.667782	+	0.744357i	$0.267244\pi$
$702$	18.0770	0.682272
$703$	−3.61504	−0.136344
$704$	−36.4409	−1.37342
$705$	0	0
$706$	−14.7122	−0.553701
$707$	−42.2292	−1.58819
$708$	−5.13505	−0.192987
$709$	6.90410	0.259289	0.129644	−	0.991561i	$-0.458616\pi$
$709$	6.90410	0.259289	0.129644	+	0.991561i	$0.458616\pi$
$710$	0	0
$711$	−15.9431	−0.597914
$712$	−3.14653	−0.117921
$713$	12.2310	0.458053
$714$	−2.90949	−0.108885
$715$	0	0
$716$	−6.55552	−0.244991
$717$	−4.90410	−0.183147
$718$	59.4572	2.21892
$719$	−38.9431	−1.45233	−0.726167	−	0.687518i	$-0.758700\pi$
$719$	−38.9431	−1.45233	−0.726167	+	0.687518i	$0.758700\pi$
$720$	0	0
$721$	−4.82678	−0.179759
$722$	−1.95594	−0.0727926
$723$	3.62951	0.134983
$724$	−36.0816	−1.34096
$725$	0	0
$726$	11.5833	0.429897
$727$	6.18347	0.229332	0.114666	−	0.993404i	$-0.463420\pi$
$727$	6.18347	0.229332	0.114666	+	0.993404i	$0.463420\pi$
$728$	−6.39733	−0.237101
$729$	−22.3576	−0.828058
$730$	0	0
$731$	−5.01114	−0.185344
$732$	−4.99390	−0.184580
$733$	0.688715	0.0254383	0.0127191	−	0.999919i	$-0.495951\pi$
$733$	0.688715	0.0254383	0.0127191	+	0.999919i	$0.495951\pi$
$734$	7.65902	0.282699
$735$	0	0
$736$	54.1146	1.99469
$737$	26.4863	0.975636
$738$	24.7672	0.911695
$739$	−26.2532	−0.965741	−0.482870	−	0.875692i	$-0.660406\pi$
$739$	−26.2532	−0.965741	−0.482870	+	0.875692i	$0.660406\pi$
$740$	0	0
$741$	1.56331	0.0574295
$742$	−53.0740	−1.94841
$743$	32.5688	1.19483	0.597416	−	0.801931i	$-0.296194\pi$
$743$	32.5688	1.19483	0.597416	+	0.801931i	$0.296194\pi$
$744$	−0.177583	−0.00651052
$745$	0	0
$746$	52.3722	1.91748
$747$	−12.1047	−0.442887
$748$	−14.2839	−0.522273
$749$	25.1672	0.919589
$750$	0	0
$751$	45.4833	1.65971	0.829855	−	0.557979i	$-0.188423\pi$
$751$	45.4833	1.65971	0.829855	+	0.557979i	$0.188423\pi$
$752$	−35.6703	−1.30076
$753$	−1.34200	−0.0489052
$754$	14.4863	0.527561
$755$	0	0
$756$	11.4166	0.415217
$757$	−2.74947	−0.0999311	−0.0499656	−	0.998751i	$-0.515911\pi$
$757$	−2.74947	−0.0999311	−0.0499656	+	0.998751i	$0.515911\pi$
$758$	−29.2421	−1.06212
$759$	−11.5098	−0.417779
$760$	0	0
$761$	−33.2978	−1.20704	−0.603521	−	0.797347i	$-0.706236\pi$
$761$	−33.2978	−1.20704	−0.603521	+	0.797347i	$0.706236\pi$
$762$	2.83947	0.102863
$763$	36.2900	1.31379
$764$	15.1789	0.549154
$765$	0	0
$766$	−54.7488	−1.97815
$767$	−49.9008	−1.80181
$768$	−5.46617	−0.197244
$769$	−19.1540	−0.690710	−0.345355	−	0.938472i	$-0.612241\pi$
$769$	−19.1540	−0.690710	−0.345355	+	0.938472i	$0.612241\pi$
$770$	0	0
$771$	4.72840	0.170289
$772$	40.6247	1.46212
$773$	−0.569309	−0.0204766	−0.0102383	−	0.999948i	$-0.503259\pi$
$773$	−0.569309	−0.0204766	−0.0102383	+	0.999948i	$0.503259\pi$
$774$	20.2947	0.729479
$775$	0	0
$776$	−3.92429	−0.140874
$777$	3.82377	0.137177
$778$	68.8562	2.46862
$779$	4.34858	0.155804
$780$	0	0
$781$	−78.2695	−2.80070
$782$	19.1712	0.685559
$783$	2.46794	0.0881969
$784$	−24.6015	−0.878624
$785$	0	0
$786$	1.63303	0.0582483
$787$	−15.9991	−0.570307	−0.285153	−	0.958482i	$-0.592044\pi$
$787$	−15.9991	−0.570307	−0.285153	+	0.958482i	$0.592044\pi$
$788$	16.0893	0.573158
$789$	0.253546	0.00902649
$790$	0	0
$791$	−6.62828	−0.235675
$792$	−5.52247	−0.196232
$793$	−48.5290	−1.72332
$794$	−70.3621	−2.49706
$795$	0	0
$796$	38.4602	1.36318
$797$	35.7528	1.26643	0.633214	−	0.773976i	$-0.281735\pi$
$797$	35.7528	1.26643	0.633214	+	0.773976i	$0.281735\pi$
$798$	2.06888	0.0732374
$799$	−11.6168	−0.410974
$800$	0	0
$801$	26.8769	0.949650
$802$	−45.4602	−1.60526
$803$	−36.6827	−1.29450
$804$	−2.58016	−0.0909951
$805$	0	0
$806$	18.0770	0.636735
$807$	−3.08457	−0.108582
$808$	−4.04004	−0.142128
$809$	23.2036	0.815796	0.407898	−	0.913028i	$-0.366262\pi$
$809$	23.2036	0.815796	0.407898	+	0.913028i	$0.366262\pi$
$810$	0	0
$811$	−21.7549	−0.763918	−0.381959	−	0.924179i	$-0.624750\pi$
$811$	−21.7549	−0.763918	−0.381959	+	0.924179i	$0.624750\pi$
$812$	9.14889	0.321063
$813$	2.43402	0.0853647
$814$	39.3371	1.37877
$815$	0	0
$816$	−1.80265	−0.0631053
$817$	3.56331	0.124664
$818$	62.3802	2.18107
$819$	54.6445	1.90943
$820$	0	0
$821$	52.2532	1.82365	0.911825	−	0.410579i	$-0.134673\pi$
$821$	52.2532	1.82365	0.911825	+	0.410579i	$0.134673\pi$
$822$	−5.35904	−0.186918
$823$	9.44783	0.329331	0.164665	−	0.986349i	$-0.447346\pi$
$823$	9.44783	0.329331	0.164665	+	0.986349i	$0.447346\pi$
$824$	−0.461775	−0.0160867
$825$	0	0
$826$	−66.0385	−2.29777
$827$	50.2216	1.74638	0.873188	−	0.487383i	$-0.162048\pi$
$827$	50.2216	1.74638	0.873188	+	0.487383i	$0.162048\pi$
$828$	−37.0523	−1.28766
$829$	44.2066	1.53536	0.767680	−	0.640834i	$-0.221411\pi$
$829$	44.2066	1.53536	0.767680	+	0.640834i	$0.221411\pi$
$830$	0	0
$831$	4.34826	0.150839
$832$	34.4966	1.19595
$833$	−8.01200	−0.277599
$834$	−2.13592	−0.0739609
$835$	0	0
$836$	10.1570	0.351287
$837$	3.07965	0.106448
$838$	−62.2455	−2.15023
$839$	−30.4033	−1.04964	−0.524819	−	0.851214i	$-0.675867\pi$
$839$	−30.4033	−1.04964	−0.524819	+	0.851214i	$0.675867\pi$
$840$	0	0
$841$	−27.0223	−0.931803
$842$	−0.681799	−0.0234963
$843$	9.38404	0.323204
$844$	9.75945	0.335934
$845$	0	0
$846$	47.0473	1.61752
$847$	71.0893	2.44266
$848$	−32.8833	−1.12922
$849$	7.34168	0.251966
$850$	0	0
$851$	−25.1955	−0.863690
$852$	7.62459	0.261214
$853$	−3.93925	−0.134877	−0.0674386	−	0.997723i	$-0.521483\pi$
$853$	−3.93925	−0.134877	−0.0674386	+	0.997723i	$0.521483\pi$
$854$	−64.2232	−2.19767
$855$	0	0
$856$	2.40773	0.0822945
$857$	−27.4388	−0.937292	−0.468646	−	0.883386i	$-0.655258\pi$
$857$	−27.4388	−0.937292	−0.468646	+	0.883386i	$0.655258\pi$
$858$	−17.0111	−0.580751
$859$	32.4517	1.10724	0.553619	−	0.832770i	$-0.313246\pi$
$859$	32.4517	1.10724	0.553619	+	0.832770i	$0.313246\pi$
$860$	0	0
$861$	−4.59966	−0.156756
$862$	−57.2629	−1.95038
$863$	2.10861	0.0717778	0.0358889	−	0.999356i	$-0.488574\pi$
$863$	2.10861	0.0717778	0.0358889	+	0.999356i	$0.488574\pi$
$864$	13.6256	0.463553
$865$	0	0
$866$	−1.72766	−0.0587084
$867$	4.45924	0.151444
$868$	11.4166	0.387504
$869$	30.4602	1.03329
$870$	0	0
$871$	−25.0731	−0.849569
$872$	3.47184	0.117571
$873$	33.5204	1.13449
$874$	−13.6322	−0.461115
$875$	0	0
$876$	3.57343	0.120735
$877$	37.6613	1.27173	0.635866	−	0.771799i	$-0.280643\pi$
$877$	37.6613	1.27173	0.635866	+	0.771799i	$0.280643\pi$
$878$	27.1062	0.914789
$879$	8.99007	0.303227
$880$	0	0
$881$	−39.8818	−1.34365	−0.671827	−	0.740708i	$-0.734490\pi$
$881$	−39.8818	−1.34365	−0.671827	+	0.740708i	$0.734490\pi$
$882$	32.4480	1.09258
$883$	−36.0458	−1.21304	−0.606518	−	0.795070i	$-0.707434\pi$
$883$	−36.0458	−1.21304	−0.606518	+	0.795070i	$0.707434\pi$
$884$	13.5218	0.454787
$885$	0	0
$886$	−26.5525	−0.892050
$887$	26.2057	0.879902	0.439951	−	0.898022i	$-0.354996\pi$
$887$	26.2057	0.879902	0.439951	+	0.898022i	$0.354996\pi$
$888$	0.365818	0.0122760
$889$	17.4264	0.584464
$890$	0	0
$891$	45.7011	1.53104
$892$	5.67700	0.190080
$893$	8.26046	0.276426
$894$	4.12186	0.137856
$895$	0	0
$896$	−9.68093	−0.323417
$897$	10.8957	0.363796
$898$	30.5780	1.02040
$899$	2.46794	0.0823103
$900$	0	0
$901$	−10.7092	−0.356774
$902$	−47.3191	−1.57555
$903$	−3.76905	−0.125426
$904$	−0.634123	−0.0210906
$905$	0	0
$906$	−10.6580	−0.354088
$907$	22.8036	0.757182	0.378591	−	0.925564i	$-0.376409\pi$
$907$	22.8036	0.757182	0.378591	+	0.925564i	$0.376409\pi$
$908$	−26.4166	−0.876664
$909$	34.5091	1.14460
$910$	0	0
$911$	4.44146	0.147152	0.0735761	−	0.997290i	$-0.476559\pi$
$911$	4.44146	0.147152	0.0735761	+	0.997290i	$0.476559\pi$
$912$	1.28182	0.0424454
$913$	23.1266	0.765379
$914$	10.3721	0.343077
$915$	0	0
$916$	−9.52059	−0.314569
$917$	10.0223	0.330965
$918$	4.82714	0.159319
$919$	37.0111	1.22088	0.610442	−	0.792061i	$-0.290992\pi$
$919$	37.0111	1.22088	0.610442	+	0.792061i	$0.290992\pi$
$920$	0	0
$921$	−6.84602	−0.225584
$922$	−0.732959	−0.0241387
$923$	74.0932	2.43881
$924$	−10.7434	−0.353433
$925$	0	0
$926$	13.0181	0.427800
$927$	3.94438	0.129550
$928$	10.9191	0.358438
$929$	−3.04185	−0.0997999	−0.0499000	−	0.998754i	$-0.515890\pi$
$929$	−3.04185	−0.0997999	−0.0499000	+	0.998754i	$0.515890\pi$
$930$	0	0
$931$	5.69716	0.186717
$932$	5.81921	0.190615
$933$	3.06198	0.100245
$934$	−1.67066	−0.0546657
$935$	0	0
$936$	5.22781	0.170876
$937$	−38.6099	−1.26133	−0.630666	−	0.776055i	$-0.717218\pi$
$937$	−38.6099	−1.26133	−0.630666	+	0.776055i	$0.717218\pi$
$938$	−33.1817	−1.08342
$939$	−1.46948	−0.0479547
$940$	0	0
$941$	2.69716	0.0879248	0.0439624	−	0.999033i	$-0.486002\pi$
$941$	2.69716	0.0879248	0.0439624	+	0.999033i	$0.486002\pi$
$942$	−10.0039	−0.325944
$943$	30.3080	0.986963
$944$	−40.9158	−1.33170
$945$	0	0
$946$	−38.7742	−1.26066
$947$	20.1877	0.656012	0.328006	−	0.944676i	$-0.393623\pi$
$947$	20.1877	0.656012	0.328006	+	0.944676i	$0.393623\pi$
$948$	−2.96727	−0.0963723
$949$	34.7254	1.12723
$950$	0	0
$951$	−4.22801	−0.137103
$952$	−1.70829	−0.0553661
$953$	−5.18559	−0.167978	−0.0839888	−	0.996467i	$-0.526766\pi$
$953$	−5.18559	−0.167978	−0.0839888	+	0.996467i	$0.526766\pi$
$954$	43.3713	1.40420
$955$	0	0
$956$	30.1624	0.975523
$957$	−2.32242	−0.0750732
$958$	33.3940	1.07891
$959$	−32.8896	−1.06206
$960$	0	0
$961$	−27.9203	−0.900656
$962$	−37.2382	−1.20061
$963$	−20.5663	−0.662739
$964$	−22.3231	−0.718979
$965$	0	0
$966$	14.4193	0.463933
$967$	−58.0054	−1.86533	−0.932665	−	0.360744i	$-0.882523\pi$
$967$	−58.0054	−1.86533	−0.932665	+	0.360744i	$0.882523\pi$
$968$	6.80107	0.218595
$969$	0.417453	0.0134105
$970$	0	0
$971$	−24.3221	−0.780533	−0.390267	−	0.920702i	$-0.627617\pi$
$971$	−24.3221	−0.780533	−0.390267	+	0.920702i	$0.627617\pi$
$972$	−14.0637	−0.451095
$973$	−13.1086	−0.420244
$974$	25.0867	0.803830
$975$	0	0
$976$	−39.7911	−1.27368
$977$	36.1134	1.15537	0.577685	−	0.816260i	$-0.303956\pi$
$977$	36.1134	1.15537	0.577685	+	0.816260i	$0.303956\pi$
$978$	−6.30895	−0.201738
$979$	−51.3498	−1.64115
$980$	0	0
$981$	−29.6557	−0.946832
$982$	20.3525	0.649473
$983$	−21.1603	−0.674909	−0.337455	−	0.941342i	$-0.609566\pi$
$983$	−21.1603	−0.674909	−0.337455	+	0.941342i	$0.609566\pi$
$984$	−0.440046	−0.0140282
$985$	0	0
$986$	3.86832	0.123192
$987$	−8.73741	−0.278115
$988$	−9.61504	−0.305895
$989$	24.8349	0.789704
$990$	0	0
$991$	20.2652	0.643746	0.321873	−	0.946783i	$-0.395688\pi$
$991$	20.2652	0.643746	0.321873	+	0.946783i	$0.395688\pi$
$992$	13.6256	0.432614
$993$	−0.606364	−0.0192424
$994$	98.0548	3.11011
$995$	0	0
$996$	−2.25287	−0.0713849
$997$	−24.2875	−0.769193	−0.384597	−	0.923085i	$-0.625659\pi$
$997$	−24.2875	−0.769193	−0.384597	+	0.923085i	$0.625659\pi$
$998$	−70.7293	−2.23890
$999$	−6.34402	−0.200716

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.i.1.1		4
3.2	odd	2		4275.2.a.bo.1.4		4
4.3	odd	2		7600.2.a.cf.1.2		4
5.2	odd	4		475.2.b.e.324.3		8
5.3	odd	4		475.2.b.e.324.6		8
5.4	even	2		95.2.a.b.1.4	✓	4
15.14	odd	2		855.2.a.m.1.1		4
19.18	odd	2		9025.2.a.bf.1.4		4
20.19	odd	2		1520.2.a.t.1.3		4
35.34	odd	2		4655.2.a.y.1.4		4
40.19	odd	2		6080.2.a.ch.1.2		4
40.29	even	2		6080.2.a.cc.1.3		4
95.94	odd	2		1805.2.a.p.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.4	✓	4	5.4	even	2
475.2.a.i.1.1		4	1.1	even	1	trivial
475.2.b.e.324.3		8	5.2	odd	4
475.2.b.e.324.6		8	5.3	odd	4
855.2.a.m.1.1		4	15.14	odd	2
1520.2.a.t.1.3		4	20.19	odd	2
1805.2.a.p.1.1		4	95.94	odd	2
4275.2.a.bo.1.4		4	3.2	odd	2
4655.2.a.y.1.4		4	35.34	odd	2
6080.2.a.cc.1.3		4	40.29	even	2
6080.2.a.ch.1.2		4	40.19	odd	2
7600.2.a.cf.1.2		4	4.3	odd	2
9025.2.a.bf.1.4		4	19.18	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

\(f(q)\)	\(=\)	\(q-1.95594 q^{2} -0.296842 q^{3} +1.82571 q^{4} +0.580605 q^{6} +3.56331 q^{7} +0.340899 q^{8} -2.91188 q^{9} +5.56331 q^{11} -0.541947 q^{12} -5.26647 q^{13} -6.96962 q^{14} -4.31820 q^{16} -1.40632 q^{17} +5.69548 q^{18} +1.00000 q^{19} -1.05774 q^{21} -10.8815 q^{22} +6.96962 q^{23} -0.101193 q^{24} +10.3009 q^{26} +1.75489 q^{27} +6.50557 q^{28} +1.40632 q^{29} +1.75489 q^{31} +7.76435 q^{32} -1.65142 q^{33} +2.75067 q^{34} -5.31626 q^{36} -3.61504 q^{37} -1.95594 q^{38} +1.56331 q^{39} +4.34858 q^{41} +2.06888 q^{42} +3.56331 q^{43} +10.1570 q^{44} -13.6322 q^{46} +8.26046 q^{47} +1.28182 q^{48} +5.69716 q^{49} +0.417453 q^{51} -9.61504 q^{52} +7.61504 q^{53} -3.43247 q^{54} +1.21473 q^{56} -0.296842 q^{57} -2.75067 q^{58} +9.47519 q^{59} +9.21473 q^{61} -3.43247 q^{62} -10.3759 q^{63} -6.55023 q^{64} +3.23009 q^{66} +4.76090 q^{67} -2.56753 q^{68} -2.06888 q^{69} -14.0689 q^{71} -0.992660 q^{72} -6.59368 q^{73} +7.07082 q^{74} +1.82571 q^{76} +19.8238 q^{77} -3.05774 q^{78} +5.47519 q^{79} +8.21473 q^{81} -8.50557 q^{82} +4.15699 q^{83} -1.93112 q^{84} -6.96962 q^{86} -0.417453 q^{87} +1.89653 q^{88} -9.23009 q^{89} -18.7660 q^{91} +12.7245 q^{92} -0.520926 q^{93} -16.1570 q^{94} -2.30478 q^{96} -11.5116 q^{97} -11.1433 q^{98} -16.1997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9} + 4 q^{11} - 6 q^{12} - 2 q^{13} - 8 q^{14} + 4 q^{16} - 4 q^{17} + 34 q^{18} + 4 q^{19} - 4 q^{21} - 4 q^{22} + 8 q^{23} - 24 q^{24} + 4 q^{26}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^3 + 8 * q^4 - 4 * q^7 + 12 * q^8 + 8 * q^9 + 4 * q^11 - 6 * q^12 - 2 * q^13 - 8 * q^14 + 4 * q^16 - 4 * q^17 + 34 * q^18 + 4 * q^19 - 4 * q^21 - 4 * q^22 + 8 * q^23 - 24 * q^24 + 4 * q^26 + 4 * q^27 + 8 * q^28 + 4 * q^29 + 4 * q^31 + 6 * q^32 - 8 * q^33 - 4 * q^34 + 40 * q^36 + 6 * q^37 + 2 * q^38 - 12 * q^39 + 16 * q^41 - 28 * q^42 - 4 * q^43 + 24 * q^44 + 12 * q^47 - 38 * q^48 + 20 * q^49 - 36 * q^51 - 18 * q^52 + 10 * q^53 - 20 * q^54 - 12 * q^56 - 2 * q^57 + 4 * q^58 + 20 * q^61 - 20 * q^62 - 20 * q^63 - 4 * q^64 - 28 * q^66 + 18 * q^67 - 4 * q^68 + 28 * q^69 - 20 * q^71 + 52 * q^72 - 28 * q^73 + 32 * q^74 + 8 * q^76 + 40 * q^77 - 12 * q^78 - 16 * q^79 + 16 * q^81 - 16 * q^82 - 44 * q^84 - 8 * q^86 + 36 * q^87 + 12 * q^88 + 4 * q^89 - 36 * q^91 + 28 * q^92 + 40 * q^93 - 48 * q^94 - 52 * q^96 - 30 * q^97 - 38 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more