LMFDB - Embedded newform 475.2.a.j.1.6

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:	$6$
Coefficient field:	6.6.66064384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 13x^{2} - 1 $ x^6 - 9x^4 + 13x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.6
Root			$1.30397$ 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)

$f(q)$	$=$	\(q+2.41987 q^{2} -0.537080 q^{3} +3.85577 q^{4} -1.29966 q^{6} +3.18676 q^{7} +4.49073 q^{8} -2.71155 q^{9} +4.15544 q^{11} -2.07086 q^{12} -2.07086 q^{13} +7.71155 q^{14} +3.15544 q^{16} -5.79470 q^{17} -6.56159 q^{18} -1.00000 q^{19} -1.71155 q^{21} +10.0556 q^{22} +2.60794 q^{23} -2.41188 q^{24} -5.01121 q^{26} +3.06756 q^{27} +12.2874 q^{28} +6.00000 q^{29} +2.59933 q^{31} -1.34571 q^{32} -2.23180 q^{33} -14.0224 q^{34} -10.4551 q^{36} -4.30266 q^{37} -2.41987 q^{38} +1.11222 q^{39} -0.599328 q^{41} -4.14172 q^{42} -3.18676 q^{43} +16.0224 q^{44} +6.31087 q^{46} -11.7086 q^{47} -1.69472 q^{48} +3.15544 q^{49} +3.11222 q^{51} -7.98476 q^{52} -11.7503 q^{53} +7.42309 q^{54} +14.3109 q^{56} +0.537080 q^{57} +14.5192 q^{58} -1.71155 q^{59} -8.75476 q^{61} +6.29004 q^{62} -8.64104 q^{63} -9.56732 q^{64} -5.40067 q^{66} -4.76228 q^{67} -22.3430 q^{68} -1.40067 q^{69} +13.7115 q^{71} -12.1768 q^{72} +2.72714 q^{73} -10.4119 q^{74} -3.85577 q^{76} +13.2424 q^{77} +2.69142 q^{78} -1.40067 q^{79} +6.48711 q^{81} -1.45030 q^{82} +7.07154 q^{83} -6.59933 q^{84} -7.71155 q^{86} -3.22248 q^{87} +18.6609 q^{88} +16.5353 q^{89} -6.59933 q^{91} +10.0556 q^{92} -1.39605 q^{93} -28.3333 q^{94} +0.722754 q^{96} -2.07086 q^{97} +7.63575 q^{98} -11.2677 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51}+ \cdots - 30 q^{99}+O(q^{100}) $ 6 * q + 8 * q^4 + 14 * q^9 + 2 * q^11 + 16 * q^14 - 4 * q^16 - 6 * q^19 + 20 * q^21 + 8 * q^24 + 8 * q^26 + 36 * q^29 - 8 * q^34 - 32 * q^36 - 8 * q^39 + 12 * q^41 + 20 * q^44 - 8 * q^46 - 4 * q^49 + 4 * q^51 - 16 * q^54 + 40 * q^56 + 20 * q^59 - 14 * q^61 - 12 * q^64 - 48 * q^66 - 24 * q^69 + 52 * q^71 - 40 * q^74 - 8 * q^76 - 24 * q^79 + 38 * q^81 - 24 * q^84 - 16 * q^86 + 24 * q^89 - 24 * q^91 - 48 * q^94 - 64 * q^96 - 30 * 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 $

$2$	2.41987	1.71111	0.855553	−	0.517715i	$-0.173217\pi$
$2$	2.41987	1.71111	0.855553	+	0.517715i	$0.173217\pi$
$3$	−0.537080	−0.310083	−0.155042	−	0.987908i	$-0.549551\pi$
$3$	−0.537080	−0.310083	−0.155042	+	0.987908i	$0.549551\pi$
$4$	3.85577	1.92789
$5$	0	0
$5$	0	0
$6$	−1.29966	−0.530586
$7$	3.18676	1.20448	0.602241	−	0.798314i	$-0.294275\pi$
$7$	3.18676	1.20448	0.602241	+	0.798314i	$0.294275\pi$
$8$	4.49073	1.58771
$9$	−2.71155	−0.903848
$10$	0	0
$11$	4.15544	1.25291	0.626456	−	0.779457i	$-0.284505\pi$
$11$	4.15544	1.25291	0.626456	+	0.779457i	$0.284505\pi$
$12$	−2.07086	−0.597805
$13$	−2.07086	−0.574353	−0.287176	−	0.957878i	$-0.592717\pi$
$13$	−2.07086	−0.574353	−0.287176	+	0.957878i	$0.592717\pi$
$14$	7.71155	2.06100
$15$	0	0
$16$	3.15544	0.788859
$17$	−5.79470	−1.40542	−0.702710	−	0.711476i	$-0.748027\pi$
$17$	−5.79470	−1.40542	−0.702710	+	0.711476i	$0.748027\pi$
$18$	−6.56159	−1.54658
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.71155	−0.373490
$22$	10.0556	2.14386
$23$	2.60794	0.543793	0.271896	−	0.962327i	$-0.412349\pi$
$23$	2.60794	0.543793	0.271896	+	0.962327i	$0.412349\pi$
$24$	−2.41188	−0.492323
$25$	0	0
$26$	−5.01121	−0.982779
$27$	3.06756	0.590352
$28$	12.2874	2.32210
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	2.59933	0.466853	0.233427	−	0.972374i	$-0.425006\pi$
$31$	2.59933	0.466853	0.233427	+	0.972374i	$0.425006\pi$
$32$	−1.34571	−0.237890
$33$	−2.23180	−0.388507
$34$	−14.0224	−2.40482
$35$	0	0
$36$	−10.4551	−1.74252
$37$	−4.30266	−0.707353	−0.353677	−	0.935368i	$-0.615069\pi$
$37$	−4.30266	−0.707353	−0.353677	+	0.935368i	$0.615069\pi$
$38$	−2.41987	−0.392555
$39$	1.11222	0.178097
$40$	0	0
$41$	−0.599328	−0.0935993	−0.0467997	−	0.998904i	$-0.514902\pi$
$41$	−0.599328	−0.0935993	−0.0467997	+	0.998904i	$0.514902\pi$
$42$	−4.14172	−0.639081
$43$	−3.18676	−0.485976	−0.242988	−	0.970029i	$-0.578128\pi$
$43$	−3.18676	−0.485976	−0.242988	+	0.970029i	$0.578128\pi$
$44$	16.0224	2.41547
$45$	0	0
$46$	6.31087	0.930487
$47$	−11.7086	−1.70787	−0.853937	−	0.520376i	$-0.825792\pi$
$47$	−11.7086	−1.70787	−0.853937	+	0.520376i	$0.825792\pi$
$48$	−1.69472	−0.244612
$49$	3.15544	0.450777
$50$	0	0
$51$	3.11222	0.435798
$52$	−7.98476	−1.10729
$53$	−11.7503	−1.61403	−0.807017	−	0.590529i	$-0.798919\pi$
$53$	−11.7503	−1.61403	−0.807017	+	0.590529i	$0.798919\pi$
$54$	7.42309	1.01015
$55$	0	0
$56$	14.3109	1.91237
$57$	0.537080	0.0711380
$58$	14.5192	1.90647
$59$	−1.71155	−0.222824	−0.111412	−	0.993774i	$-0.535537\pi$
$59$	−1.71155	−0.222824	−0.111412	+	0.993774i	$0.535537\pi$
$60$	0	0
$61$	−8.75476	−1.12093	−0.560466	−	0.828177i	$-0.689378\pi$
$61$	−8.75476	−1.12093	−0.560466	+	0.828177i	$0.689378\pi$
$62$	6.29004	0.798836
$63$	−8.64104	−1.08867
$64$	−9.56732	−1.19591
$65$	0	0
$66$	−5.40067	−0.664777
$67$	−4.76228	−0.581805	−0.290902	−	0.956753i	$-0.593956\pi$
$67$	−4.76228	−0.581805	−0.290902	+	0.956753i	$0.593956\pi$
$68$	−22.3430	−2.70949
$69$	−1.40067	−0.168621
$70$	0	0
$71$	13.7115	1.62726	0.813631	−	0.581382i	$-0.197488\pi$
$71$	13.7115	1.62726	0.813631	+	0.581382i	$0.197488\pi$
$72$	−12.1768	−1.43505
$73$	2.72714	0.319188	0.159594	−	0.987183i	$-0.448982\pi$
$73$	2.72714	0.319188	0.159594	+	0.987183i	$0.448982\pi$
$74$	−10.4119	−1.21036
$75$	0	0
$76$	−3.85577	−0.442287
$77$	13.2424	1.50911
$78$	2.69142	0.304743
$79$	−1.40067	−0.157588	−0.0787939	−	0.996891i	$-0.525107\pi$
$79$	−1.40067	−0.157588	−0.0787939	+	0.996891i	$0.525107\pi$
$80$	0	0
$81$	6.48711	0.720790
$82$	−1.45030	−0.160158
$83$	7.07154	0.776203	0.388101	−	0.921617i	$-0.373131\pi$
$83$	7.07154	0.776203	0.388101	+	0.921617i	$0.373131\pi$
$84$	−6.59933	−0.720046
$85$	0	0
$86$	−7.71155	−0.831557
$87$	−3.22248	−0.345486
$88$	18.6609	1.98926
$89$	16.5353	1.75274	0.876370	−	0.481639i	$-0.159958\pi$
$89$	16.5353	1.75274	0.876370	+	0.481639i	$0.159958\pi$
$90$	0	0
$91$	−6.59933	−0.691798
$92$	10.0556	1.04837
$93$	−1.39605	−0.144763
$94$	−28.3333	−2.92236
$95$	0	0
$96$	0.722754	0.0737658
$97$	−2.07086	−0.210264	−0.105132	−	0.994458i	$-0.533526\pi$
$97$	−2.07086	−0.210264	−0.105132	+	0.994458i	$0.533526\pi$
$98$	7.63575	0.771327
$99$	−11.2677	−1.13244
$100$	0	0
$101$	−1.71155	−0.170305	−0.0851525	−	0.996368i	$-0.527138\pi$
$101$	−1.71155	−0.170305	−0.0851525	+	0.996368i	$0.527138\pi$
$102$	7.53116	0.745696
$103$	−5.75296	−0.566856	−0.283428	−	0.958994i	$-0.591472\pi$
$103$	−5.75296	−0.566856	−0.283428	+	0.958994i	$0.591472\pi$
$104$	−9.29966	−0.911907
$105$	0	0
$106$	−28.4343	−2.76178
$107$	15.4324	1.49191	0.745955	−	0.665996i	$-0.231993\pi$
$107$	15.4324	1.49191	0.745955	+	0.665996i	$0.231993\pi$
$108$	11.8278	1.13813
$109$	11.7115	1.12176	0.560881	−	0.827896i	$-0.310462\pi$
$109$	11.7115	1.12176	0.560881	+	0.827896i	$0.310462\pi$
$110$	0	0
$111$	2.31087	0.219338
$112$	10.0556	0.950167
$113$	10.5927	0.996477	0.498239	−	0.867040i	$-0.333980\pi$
$113$	10.5927	0.996477	0.498239	+	0.867040i	$0.333980\pi$
$114$	1.29966	0.121725
$115$	0	0
$116$	23.1346	2.14800
$117$	5.61523	0.519128
$118$	−4.14172	−0.381276
$119$	−18.4663	−1.69280
$120$	0	0
$121$	6.26765	0.569787
$122$	−21.1854	−1.91804
$123$	0.321887	0.0290236
$124$	10.0224	0.900040
$125$	0	0
$126$	−20.9102	−1.86283
$127$	6.07484	0.539055	0.269528	−	0.962993i	$-0.413132\pi$
$127$	6.07484	0.539055	0.269528	+	0.962993i	$0.413132\pi$
$128$	−20.4602	−1.80845
$129$	1.71155	0.150693
$130$	0	0
$131$	−13.5785	−1.18636	−0.593181	−	0.805069i	$-0.702128\pi$
$131$	−13.5785	−1.18636	−0.593181	+	0.805069i	$0.702128\pi$
$132$	−8.60532	−0.748997
$133$	−3.18676	−0.276327
$134$	−11.5241	−0.995530
$135$	0	0
$136$	−26.0224	−2.23140
$137$	7.94302	0.678618	0.339309	−	0.940675i	$-0.389807\pi$
$137$	7.94302	0.678618	0.339309	+	0.940675i	$0.389807\pi$
$138$	−3.38944	−0.288529
$139$	3.26765	0.277159	0.138579	−	0.990351i	$-0.455746\pi$
$139$	3.26765	0.277159	0.138579	+	0.990351i	$0.455746\pi$
$140$	0	0
$141$	6.28845	0.529583
$142$	33.1802	2.78442
$143$	−8.60532	−0.719613
$144$	−8.55611	−0.713009
$145$	0	0
$146$	6.59933	0.546164
$147$	−1.69472	−0.139778
$148$	−16.5901	−1.36370
$149$	8.44389	0.691751	0.345875	−	0.938280i	$-0.387582\pi$
$149$	8.44389	0.691751	0.345875	+	0.938280i	$0.387582\pi$
$150$	0	0
$151$	0.887783	0.0722468	0.0361234	−	0.999347i	$-0.488499\pi$
$151$	0.887783	0.0722468	0.0361234	+	0.999347i	$0.488499\pi$
$152$	−4.49073	−0.364246
$153$	15.7126	1.27029
$154$	32.0448	2.58225
$155$	0	0
$156$	4.28845	0.343351
$157$	−4.14172	−0.330545	−0.165273	−	0.986248i	$-0.552850\pi$
$157$	−4.14172	−0.330545	−0.165273	+	0.986248i	$0.552850\pi$
$158$	−3.38944	−0.269650
$159$	6.31087	0.500485
$160$	0	0
$161$	8.31087	0.654989
$162$	15.6980	1.23335
$163$	24.7126	1.93564	0.967819	−	0.251647i	$-0.0809723\pi$
$163$	24.7126	1.93564	0.967819	+	0.251647i	$0.0809723\pi$
$164$	−2.31087	−0.180449
$165$	0	0
$166$	17.1122	1.32817
$167$	3.60464	0.278935	0.139468	−	0.990227i	$-0.455461\pi$
$167$	3.60464	0.278935	0.139468	+	0.990227i	$0.455461\pi$
$168$	−7.68608	−0.592994
$169$	−8.71155	−0.670119
$170$	0	0
$171$	2.71155	0.207357
$172$	−12.2874	−0.936907
$173$	22.4205	1.70460	0.852300	−	0.523054i	$-0.175207\pi$
$173$	22.4205	1.70460	0.852300	+	0.523054i	$0.175207\pi$
$174$	−7.79798	−0.591164
$175$	0	0
$176$	13.1122	0.988371
$177$	0.919237	0.0690941
$178$	40.0133	2.99912
$179$	−5.13464	−0.383781	−0.191890	−	0.981416i	$-0.561462\pi$
$179$	−5.13464	−0.383781	−0.191890	+	0.981416i	$0.561462\pi$
$180$	0	0
$181$	20.8462	1.54948	0.774742	−	0.632277i	$-0.217880\pi$
$181$	20.8462	1.54948	0.774742	+	0.632277i	$0.217880\pi$
$182$	−15.9695	−1.18374
$183$	4.70201	0.347583
$184$	11.7115	0.863387
$185$	0	0
$186$	−3.37825	−0.247706
$187$	−24.0795	−1.76087
$188$	−45.1457	−3.29259
$189$	9.77557	0.711068
$190$	0	0
$191$	−5.26765	−0.381154	−0.190577	−	0.981672i	$-0.561036\pi$
$191$	−5.26765	−0.381154	−0.190577	+	0.981672i	$0.561036\pi$
$192$	5.13842	0.370833
$193$	−2.07086	−0.149064	−0.0745318	−	0.997219i	$-0.523746\pi$
$193$	−2.07086	−0.149064	−0.0745318	+	0.997219i	$0.523746\pi$
$194$	−5.01121	−0.359784
$195$	0	0
$196$	12.1666	0.869046
$197$	−10.4318	−0.743232	−0.371616	−	0.928387i	$-0.621196\pi$
$197$	−10.4318	−0.743232	−0.371616	+	0.928387i	$0.621196\pi$
$198$	−27.2663	−1.93773
$199$	−2.73235	−0.193691	−0.0968455	−	0.995299i	$-0.530875\pi$
$199$	−2.73235	−0.193691	−0.0968455	+	0.995299i	$0.530875\pi$
$200$	0	0
$201$	2.55773	0.180408
$202$	−4.14172	−0.291410
$203$	19.1206	1.34200
$204$	12.0000	0.840168
$205$	0	0
$206$	−13.9214	−0.969951
$207$	−7.07154	−0.491506
$208$	−6.53446	−0.453083
$209$	−4.15544	−0.287438
$210$	0	0
$211$	−15.7340	−1.08317	−0.541585	−	0.840646i	$-0.682176\pi$
$211$	−15.7340	−1.08317	−0.541585	+	0.840646i	$0.682176\pi$
$212$	−45.3066	−3.11167
$213$	−7.36420	−0.504586
$214$	37.3445	2.55282
$215$	0	0
$216$	13.7756	0.937309
$217$	8.28343	0.562316
$218$	28.3404	1.91946
$219$	−1.46469	−0.0989748
$220$	0	0
$221$	12.0000	0.807207
$222$	5.59201	0.375311
$223$	−18.8219	−1.26041	−0.630203	−	0.776430i	$-0.717028\pi$
$223$	−18.8219	−1.26041	−0.630203	+	0.776430i	$0.717028\pi$
$224$	−4.28845	−0.286534
$225$	0	0
$226$	25.6330	1.70508
$227$	14.4418	0.958533	0.479267	−	0.877669i	$-0.340903\pi$
$227$	14.4418	0.958533	0.479267	+	0.877669i	$0.340903\pi$
$228$	2.07086	0.137146
$229$	−4.17785	−0.276080	−0.138040	−	0.990427i	$-0.544080\pi$
$229$	−4.17785	−0.276080	−0.138040	+	0.990427i	$0.544080\pi$
$230$	0	0
$231$	−7.11222	−0.467950
$232$	26.9444	1.76898
$233$	−12.0847	−0.791697	−0.395849	−	0.918316i	$-0.629550\pi$
$233$	−12.0847	−0.791697	−0.395849	+	0.918316i	$0.629550\pi$
$234$	13.5881	0.888283
$235$	0	0
$236$	−6.59933	−0.429580
$237$	0.752273	0.0488654
$238$	−44.6861	−2.89657
$239$	−11.3541	−0.734435	−0.367218	−	0.930135i	$-0.619690\pi$
$239$	−11.3541	−0.734435	−0.367218	+	0.930135i	$0.619690\pi$
$240$	0	0
$241$	−3.40067	−0.219057	−0.109528	−	0.993984i	$-0.534934\pi$
$241$	−3.40067	−0.219057	−0.109528	+	0.993984i	$0.534934\pi$
$242$	15.1669	0.974966
$243$	−12.6868	−0.813857
$244$	−33.7564	−2.16103
$245$	0	0
$246$	0.778925	0.0496625
$247$	2.07086	0.131766
$248$	11.6729	0.741229
$249$	−3.79798	−0.240687
$250$	0	0
$251$	3.04322	0.192086	0.0960432	−	0.995377i	$-0.469381\pi$
$251$	3.04322	0.192086	0.0960432	+	0.995377i	$0.469381\pi$
$252$	−33.3179	−2.09883
$253$	10.8371	0.681324
$254$	14.7003	0.922381
$255$	0	0
$256$	−30.3765	−1.89853
$257$	−17.2881	−1.07840	−0.539201	−	0.842177i	$-0.681274\pi$
$257$	−17.2881	−1.07840	−0.539201	+	0.842177i	$0.681274\pi$
$258$	4.14172	0.257852
$259$	−13.7115	−0.851994
$260$	0	0
$261$	−16.2693	−1.00704
$262$	−32.8583	−2.02999
$263$	1.19336	0.0735859	0.0367930	−	0.999323i	$-0.488286\pi$
$263$	1.19336	0.0735859	0.0367930	+	0.999323i	$0.488286\pi$
$264$	−10.0224	−0.616837
$265$	0	0
$266$	−7.71155	−0.472825
$267$	−8.88078	−0.543495
$268$	−18.3623	−1.12165
$269$	22.1089	1.34800	0.674000	−	0.738731i	$-0.264575\pi$
$269$	22.1089	1.34800	0.674000	+	0.738731i	$0.264575\pi$
$270$	0	0
$271$	−4.08644	−0.248234	−0.124117	−	0.992268i	$-0.539610\pi$
$271$	−4.08644	−0.248234	−0.124117	+	0.992268i	$0.539610\pi$
$272$	−18.2848	−1.10868
$273$	3.54437	0.214515
$274$	19.2211	1.16119
$275$	0	0
$276$	−5.40067	−0.325082
$277$	10.0199	0.602037	0.301019	−	0.953618i	$-0.402673\pi$
$277$	10.0199	0.602037	0.301019	+	0.953618i	$0.402673\pi$
$278$	7.90730	0.474248
$279$	−7.04820	−0.421964
$280$	0	0
$281$	0.599328	0.0357529	0.0178765	−	0.999840i	$-0.494309\pi$
$281$	0.599328	0.0357529	0.0178765	+	0.999840i	$0.494309\pi$
$282$	15.2172	0.906174
$283$	−5.41856	−0.322100	−0.161050	−	0.986946i	$-0.551488\pi$
$283$	−5.41856	−0.322100	−0.161050	+	0.986946i	$0.551488\pi$
$284$	52.8686	3.13717
$285$	0	0
$286$	−20.8238	−1.23133
$287$	−1.90991	−0.112739
$288$	3.64895	0.215017
$289$	16.5785	0.975207
$290$	0	0
$291$	1.11222	0.0651993
$292$	10.5152	0.615358
$293$	−3.46691	−0.202539	−0.101269	−	0.994859i	$-0.532290\pi$
$293$	−3.46691	−0.202539	−0.101269	+	0.994859i	$0.532290\pi$
$294$	−4.10101	−0.239176
$295$	0	0
$296$	−19.3221	−1.12307
$297$	12.7470	0.739658
$298$	20.4331	1.18366
$299$	−5.40067	−0.312329
$300$	0	0
$301$	−10.1554	−0.585350
$302$	2.14832	0.123622
$303$	0.919237	0.0528088
$304$	−3.15544	−0.180977
$305$	0	0
$306$	38.0224	2.17360
$307$	16.5901	0.946846	0.473423	−	0.880835i	$-0.343018\pi$
$307$	16.5901	0.946846	0.473423	+	0.880835i	$0.343018\pi$
$308$	51.0596	2.90939
$309$	3.08980	0.175772
$310$	0	0
$311$	−4.15544	−0.235633	−0.117817	−	0.993035i	$-0.537589\pi$
$311$	−4.15544	−0.235633	−0.117817	+	0.993035i	$0.537589\pi$
$312$	4.99466	0.282767
$313$	0.919237	0.0519583	0.0259792	−	0.999662i	$-0.491730\pi$
$313$	0.919237	0.0519583	0.0259792	+	0.999662i	$0.491730\pi$
$314$	−10.0224	−0.565598
$315$	0	0
$316$	−5.40067	−0.303812
$317$	26.7292	1.50126	0.750630	−	0.660723i	$-0.229750\pi$
$317$	26.7292	1.50126	0.750630	+	0.660723i	$0.229750\pi$
$318$	15.2715	0.856383
$319$	24.9326	1.39596
$320$	0	0
$321$	−8.28845	−0.462616
$322$	20.1112	1.12076
$323$	5.79470	0.322426
$324$	25.0128	1.38960
$325$	0	0
$326$	59.8012	3.31208
$327$	−6.29004	−0.347840
$328$	−2.69142	−0.148609
$329$	−37.3125	−2.05710
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	27.2663	1.49643
$333$	11.6669	0.639340
$334$	8.72275	0.477288
$335$	0	0
$336$	−5.40067	−0.294631
$337$	−22.5040	−1.22587	−0.612935	−	0.790133i	$-0.710011\pi$
$337$	−22.5040	−1.22587	−0.612935	+	0.790133i	$0.710011\pi$
$338$	−21.0808	−1.14664
$339$	−5.68913	−0.308991
$340$	0	0
$341$	10.8013	0.584926
$342$	6.56159	0.354810
$343$	−12.2517	−0.661530
$344$	−14.3109	−0.771591
$345$	0	0
$346$	54.2547	2.91675
$347$	14.4543	0.775946	0.387973	−	0.921671i	$-0.373175\pi$
$347$	14.4543	0.775946	0.387973	+	0.921671i	$0.373175\pi$
$348$	−12.4252	−0.666058
$349$	−13.3541	−0.714828	−0.357414	−	0.933946i	$-0.616342\pi$
$349$	−13.3541	−0.714828	−0.357414	+	0.933946i	$0.616342\pi$
$350$	0	0
$351$	−6.35248	−0.339070
$352$	−5.59201	−0.298055
$353$	−17.6410	−0.938937	−0.469469	−	0.882949i	$-0.655554\pi$
$353$	−17.6410	−0.938937	−0.469469	+	0.882949i	$0.655554\pi$
$354$	2.22443	0.118227
$355$	0	0
$356$	63.7564	3.37908
$357$	9.91789	0.524910
$358$	−12.4252	−0.656690
$359$	12.4663	0.657947	0.328973	−	0.944339i	$-0.393297\pi$
$359$	12.4663	0.657947	0.328973	+	0.944339i	$0.393297\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	50.4451	2.65133
$363$	−3.36623	−0.176681
$364$	−25.4455	−1.33371
$365$	0	0
$366$	11.3783	0.594751
$367$	16.4291	0.857594	0.428797	−	0.903401i	$-0.358938\pi$
$367$	16.4291	0.857594	0.428797	+	0.903401i	$0.358938\pi$
$368$	8.22918	0.428976
$369$	1.62511	0.0845996
$370$	0	0
$371$	−37.4455	−1.94407
$372$	−5.38284	−0.279087
$373$	5.29334	0.274079	0.137039	−	0.990566i	$-0.456241\pi$
$373$	5.29334	0.274079	0.137039	+	0.990566i	$0.456241\pi$
$374$	−58.2693	−3.01303
$375$	0	0
$376$	−52.5801	−2.71161
$377$	−12.4252	−0.639928
$378$	23.6556	1.21671
$379$	14.5353	0.746629	0.373314	−	0.927705i	$-0.378221\pi$
$379$	14.5353	0.746629	0.373314	+	0.927705i	$0.378221\pi$
$380$	0	0
$381$	−3.26268	−0.167152
$382$	−12.7470	−0.652195
$383$	−0.453598	−0.0231778	−0.0115889	−	0.999933i	$-0.503689\pi$
$383$	−0.453598	−0.0231778	−0.0115889	+	0.999933i	$0.503689\pi$
$384$	10.9888	0.560769
$385$	0	0
$386$	−5.01121	−0.255064
$387$	8.64104	0.439249
$388$	−7.98476	−0.405365
$389$	16.1554	0.819113	0.409557	−	0.912285i	$-0.365683\pi$
$389$	16.1554	0.819113	0.409557	+	0.912285i	$0.365683\pi$
$390$	0	0
$391$	−15.1122	−0.764258
$392$	14.1702	0.715704
$393$	7.29276	0.367871
$394$	−25.2435	−1.27175
$395$	0	0
$396$	−43.4455	−2.18322
$397$	−32.7563	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$397$	−32.7563	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$398$	−6.61192	−0.331426
$399$	1.71155	0.0856844
$400$	0	0
$401$	12.0864	0.603568	0.301784	−	0.953376i	$-0.402418\pi$
$401$	12.0864	0.603568	0.301784	+	0.953376i	$0.402418\pi$
$402$	6.18936	0.308697
$403$	−5.38284	−0.268138
$404$	−6.59933	−0.328329
$405$	0	0
$406$	46.2693	2.29631
$407$	−17.8794	−0.886251
$408$	13.9761	0.691921
$409$	−19.1346	−0.946147	−0.473073	−	0.881023i	$-0.656855\pi$
$409$	−19.1346	−0.946147	−0.473073	+	0.881023i	$0.656855\pi$
$410$	0	0
$411$	−4.26604	−0.210428
$412$	−22.1821	−1.09283
$413$	−5.45428	−0.268388
$414$	−17.1122	−0.841020
$415$	0	0
$416$	2.78678	0.136633
$417$	−1.75499	−0.0859423
$418$	−10.0556	−0.491836
$419$	8.04484	0.393016	0.196508	−	0.980502i	$-0.437040\pi$
$419$	8.04484	0.393016	0.196508	+	0.980502i	$0.437040\pi$
$420$	0	0
$421$	−29.3591	−1.43087	−0.715437	−	0.698678i	$-0.753772\pi$
$421$	−29.3591	−1.43087	−0.715437	+	0.698678i	$0.753772\pi$
$422$	−38.0742	−1.85342
$423$	31.7484	1.54366
$424$	−52.7676	−2.56262
$425$	0	0
$426$	−17.8204	−0.863401
$427$	−27.8993	−1.35014
$428$	59.5040	2.87623
$429$	4.62175	0.223140
$430$	0	0
$431$	−32.0448	−1.54355	−0.771773	−	0.635898i	$-0.780630\pi$
$431$	−32.0448	−1.54355	−0.771773	+	0.635898i	$0.780630\pi$
$432$	9.67948	0.465704
$433$	−0.482831	−0.0232034	−0.0116017	−	0.999933i	$-0.503693\pi$
$433$	−0.482831	−0.0232034	−0.0116017	+	0.999933i	$0.503693\pi$
$434$	20.0448	0.962183
$435$	0	0
$436$	45.1571	2.16263
$437$	−2.60794	−0.124755
$438$	−3.54437	−0.169356
$439$	−27.3591	−1.30578	−0.652889	−	0.757454i	$-0.726443\pi$
$439$	−27.3591	−1.30578	−0.652889	+	0.757454i	$0.726443\pi$
$440$	0	0
$441$	−8.55611	−0.407434
$442$	29.0384	1.38122
$443$	−23.3815	−1.11089	−0.555444	−	0.831554i	$-0.687452\pi$
$443$	−23.3815	−1.11089	−0.555444	+	0.831554i	$0.687452\pi$
$444$	8.91020	0.422859
$445$	0	0
$446$	−45.5465	−2.15669
$447$	−4.53505	−0.214500
$448$	−30.4887	−1.44046
$449$	23.1346	1.09179	0.545895	−	0.837853i	$-0.316190\pi$
$449$	23.1346	1.09179	0.545895	+	0.837853i	$0.316190\pi$
$450$	0	0
$451$	−2.49047	−0.117272
$452$	40.8430	1.92109
$453$	−0.476811	−0.0224025
$454$	34.9472	1.64015
$455$	0	0
$456$	2.41188	0.112947
$457$	−21.2503	−0.994049	−0.497025	−	0.867736i	$-0.665574\pi$
$457$	−21.2503	−0.994049	−0.497025	+	0.867736i	$0.665574\pi$
$458$	−10.1099	−0.472403
$459$	−17.7756	−0.829692
$460$	0	0
$461$	31.5785	1.47076	0.735379	−	0.677656i	$-0.237004\pi$
$461$	31.5785	1.47076	0.735379	+	0.677656i	$0.237004\pi$
$462$	−17.2106	−0.800712
$463$	15.6119	0.725547	0.362774	−	0.931877i	$-0.381830\pi$
$463$	15.6119	0.725547	0.362774	+	0.931877i	$0.381830\pi$
$464$	18.9326	0.878925
$465$	0	0
$466$	−29.2435	−1.35468
$467$	29.8264	1.38020	0.690101	−	0.723713i	$-0.257566\pi$
$467$	29.8264	1.38020	0.690101	+	0.723713i	$0.257566\pi$
$468$	21.6510	1.00082
$469$	−15.1762	−0.700774
$470$	0	0
$471$	2.22443	0.102496
$472$	−7.68608	−0.353781
$473$	−13.2424	−0.608885
$474$	1.82040	0.0836139
$475$	0	0
$476$	−71.2019	−3.26353
$477$	31.8616	1.45884
$478$	−27.4754	−1.25670
$479$	4.53531	0.207223	0.103612	−	0.994618i	$-0.466960\pi$
$479$	4.53531	0.207223	0.103612	+	0.994618i	$0.466960\pi$
$480$	0	0
$481$	8.91020	0.406270
$482$	−8.22918	−0.374829
$483$	−4.46360	−0.203101
$484$	24.1666	1.09848
$485$	0	0
$486$	−30.7003	−1.39260
$487$	−39.2550	−1.77881	−0.889407	−	0.457116i	$-0.848882\pi$
$487$	−39.2550	−1.77881	−0.889407	+	0.457116i	$0.848882\pi$
$488$	−39.3153	−1.77972
$489$	−13.2726	−0.600209
$490$	0	0
$491$	38.8910	1.75513	0.877564	−	0.479460i	$-0.159168\pi$
$491$	38.8910	1.75513	0.877564	+	0.479460i	$0.159168\pi$
$492$	1.24112	0.0559542
$493$	−34.7682	−1.56588
$494$	5.01121	0.225465
$495$	0	0
$496$	8.20202	0.368281
$497$	43.6954	1.96001
$498$	−9.19063	−0.411842
$499$	4.73235	0.211849	0.105924	−	0.994374i	$-0.466220\pi$
$499$	4.73235	0.211849	0.105924	+	0.994374i	$0.466220\pi$
$500$	0	0
$501$	−1.93598	−0.0864931
$502$	7.36420	0.328680
$503$	−1.85567	−0.0827400	−0.0413700	−	0.999144i	$-0.513172\pi$
$503$	−1.85567	−0.0827400	−0.0413700	+	0.999144i	$0.513172\pi$
$504$	−38.8046	−1.72849
$505$	0	0
$506$	26.2244	1.16582
$507$	4.67880	0.207793
$508$	23.4232	1.03924
$509$	22.8878	1.01448	0.507242	−	0.861804i	$-0.330665\pi$
$509$	22.8878	1.01448	0.507242	+	0.861804i	$0.330665\pi$
$510$	0	0
$511$	8.69074	0.384456
$512$	−32.5867	−1.44014
$513$	−3.06756	−0.135436
$514$	−41.8350	−1.84526
$515$	0	0
$516$	6.59933	0.290519
$517$	−48.6543	−2.13982
$518$	−33.1802	−1.45785
$519$	−12.0416	−0.528568
$520$	0	0
$521$	−29.7340	−1.30267	−0.651334	−	0.758791i	$-0.725790\pi$
$521$	−29.7340	−1.30267	−0.651334	+	0.758791i	$0.725790\pi$
$522$	−39.3695	−1.72316
$523$	−30.6497	−1.34022	−0.670109	−	0.742263i	$-0.733752\pi$
$523$	−30.6497	−1.34022	−0.670109	+	0.742263i	$0.733752\pi$
$524$	−52.3557	−2.28717
$525$	0	0
$526$	2.88778	0.125913
$527$	−15.0623	−0.656125
$528$	−7.04231	−0.306477
$529$	−16.1987	−0.704289
$530$	0	0
$531$	4.64093	0.201399
$532$	−12.2874	−0.532727
$533$	1.24112	0.0537590
$534$	−21.4903	−0.929978
$535$	0	0
$536$	−21.3861	−0.923739
$537$	2.75771	0.119004
$538$	53.5006	2.30657
$539$	13.1122	0.564783
$540$	0	0
$541$	2.21946	0.0954220	0.0477110	−	0.998861i	$-0.484807\pi$
$541$	2.21946	0.0954220	0.0477110	+	0.998861i	$0.484807\pi$
$542$	−9.88865	−0.424754
$543$	−11.1961	−0.480469
$544$	7.79798	0.334336
$545$	0	0
$546$	8.57691	0.367058
$547$	14.4297	0.616970	0.308485	−	0.951229i	$-0.400178\pi$
$547$	14.4297	0.616970	0.308485	+	0.951229i	$0.400178\pi$
$548$	30.6265	1.30830
$549$	23.7389	1.01315
$550$	0	0
$551$	−6.00000	−0.255609
$552$	−6.29004	−0.267722
$553$	−4.46360	−0.189812
$554$	24.2469	1.03015
$555$	0	0
$556$	12.5993	0.534331
$557$	−19.4610	−0.824588	−0.412294	−	0.911051i	$-0.635272\pi$
$557$	−19.4610	−0.824588	−0.412294	+	0.911051i	$0.635272\pi$
$558$	−17.0557	−0.722026
$559$	6.59933	0.279122
$560$	0	0
$561$	12.9326	0.546016
$562$	1.45030	0.0611771
$563$	12.3649	0.521118	0.260559	−	0.965458i	$-0.416093\pi$
$563$	12.3649	0.521118	0.260559	+	0.965458i	$0.416093\pi$
$564$	24.2469	1.02098
$565$	0	0
$566$	−13.1122	−0.551148
$567$	20.6729	0.868179
$568$	61.5748	2.58362
$569$	15.0898	0.632597	0.316299	−	0.948660i	$-0.397560\pi$
$569$	15.0898	0.632597	0.316299	+	0.948660i	$0.397560\pi$
$570$	0	0
$571$	17.1571	0.718000	0.359000	−	0.933337i	$-0.383118\pi$
$571$	17.1571	0.718000	0.359000	+	0.933337i	$0.383118\pi$
$572$	−33.1802	−1.38733
$573$	2.82915	0.118190
$574$	−4.62175	−0.192908
$575$	0	0
$576$	25.9422	1.08093
$577$	22.5165	0.937374	0.468687	−	0.883364i	$-0.344727\pi$
$577$	22.5165	0.937374	0.468687	+	0.883364i	$0.344727\pi$
$578$	40.1179	1.66868
$579$	1.11222	0.0462222
$580$	0	0
$581$	22.5353	0.934922
$582$	2.69142	0.111563
$583$	−48.8278	−2.02224
$584$	12.2469	0.506778
$585$	0	0
$586$	−8.38946	−0.346566
$587$	−7.49544	−0.309370	−0.154685	−	0.987964i	$-0.549436\pi$
$587$	−7.49544	−0.309370	−0.154685	+	0.987964i	$0.549436\pi$
$588$	−6.53446	−0.269477
$589$	−2.59933	−0.107103
$590$	0	0
$591$	5.60269	0.230464
$592$	−13.5768	−0.558002
$593$	−27.8094	−1.14199	−0.570997	−	0.820952i	$-0.693443\pi$
$593$	−27.8094	−1.14199	−0.570997	+	0.820952i	$0.693443\pi$
$594$	30.8462	1.26563
$595$	0	0
$596$	32.5577	1.33362
$597$	1.46749	0.0600603
$598$	−13.0689	−0.534428
$599$	−45.4903	−1.85869	−0.929343	−	0.369219i	$-0.879625\pi$
$599$	−45.4903	−1.85869	−0.929343	+	0.369219i	$0.879625\pi$
$600$	0	0
$601$	−16.5993	−0.677101	−0.338550	−	0.940948i	$-0.609937\pi$
$601$	−16.5993	−0.677101	−0.338550	+	0.940948i	$0.609937\pi$
$602$	−24.5748	−1.00160
$603$	12.9131	0.525863
$604$	3.42309	0.139284
$605$	0	0
$606$	2.22443	0.0903614
$607$	5.08417	0.206360	0.103180	−	0.994663i	$-0.467098\pi$
$607$	5.08417	0.206360	0.103180	+	0.994663i	$0.467098\pi$
$608$	1.34571	0.0545758
$609$	−10.2693	−0.416132
$610$	0	0
$611$	24.2469	0.980923
$612$	60.5842	2.44897
$613$	−4.63706	−0.187289	−0.0936445	−	0.995606i	$-0.529852\pi$
$613$	−4.63706	−0.187289	−0.0936445	+	0.995606i	$0.529852\pi$
$614$	40.1458	1.62015
$615$	0	0
$616$	59.4679	2.39603
$617$	−40.2874	−1.62191	−0.810955	−	0.585108i	$-0.801052\pi$
$617$	−40.2874	−1.62191	−0.810955	+	0.585108i	$0.801052\pi$
$618$	7.47691	0.300766
$619$	−43.3815	−1.74365	−0.871825	−	0.489818i	$-0.837063\pi$
$619$	−43.3815	−1.74365	−0.871825	+	0.489818i	$0.837063\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	−10.0556	−0.403194
$623$	52.6940	2.11114
$624$	3.50953	0.140494
$625$	0	0
$626$	2.22443	0.0889062
$627$	2.23180	0.0891296
$628$	−15.9695	−0.637253
$629$	24.9326	0.994129
$630$	0	0
$631$	7.53369	0.299911	0.149956	−	0.988693i	$-0.452087\pi$
$631$	7.53369	0.299911	0.149956	+	0.988693i	$0.452087\pi$
$632$	−6.29004	−0.250204
$633$	8.45040	0.335873
$634$	64.6812	2.56882
$635$	0	0
$636$	24.3333	0.964878
$637$	−6.53446	−0.258905
$638$	60.3337	2.38863
$639$	−37.1795	−1.47080
$640$	0	0
$641$	−23.3075	−0.920591	−0.460296	−	0.887766i	$-0.652257\pi$
$641$	−23.3075	−0.920591	−0.460296	+	0.887766i	$0.652257\pi$
$642$	−20.0570	−0.791586
$643$	−1.87419	−0.0739110	−0.0369555	−	0.999317i	$-0.511766\pi$
$643$	−1.87419	−0.0739110	−0.0369555	+	0.999317i	$0.511766\pi$
$644$	32.0448	1.26274
$645$	0	0
$646$	14.0224	0.551705
$647$	47.0371	1.84922	0.924609	−	0.380917i	$-0.124392\pi$
$647$	47.0371	1.84922	0.924609	+	0.380917i	$0.124392\pi$
$648$	29.1319	1.14441
$649$	−7.11222	−0.279179
$650$	0	0
$651$	−4.44887	−0.174365
$652$	95.2861	3.73169
$653$	−24.1630	−0.945571	−0.472785	−	0.881178i	$-0.656751\pi$
$653$	−24.1630	−0.945571	−0.472785	+	0.881178i	$0.656751\pi$
$654$	−15.2211	−0.595191
$655$	0	0
$656$	−1.89114	−0.0738367
$657$	−7.39477	−0.288497
$658$	−90.2914	−3.51992
$659$	10.2885	0.400781	0.200391	−	0.979716i	$-0.435779\pi$
$659$	10.2885	0.400781	0.200391	+	0.979716i	$0.435779\pi$
$660$	0	0
$661$	5.93598	0.230883	0.115441	−	0.993314i	$-0.463172\pi$
$661$	5.93598	0.230883	0.115441	+	0.993314i	$0.463172\pi$
$662$	19.3590	0.752407
$663$	−6.44496	−0.250302
$664$	31.7564	1.23239
$665$	0	0
$666$	28.2323	1.09398
$667$	15.6476	0.605879
$668$	13.8987	0.537755
$669$	10.1089	0.390831
$670$	0	0
$671$	−36.3799	−1.40443
$672$	2.30324	0.0888496
$673$	40.7053	1.56907	0.784537	−	0.620082i	$-0.212901\pi$
$673$	40.7053	1.56907	0.784537	+	0.620082i	$0.212901\pi$
$674$	−54.4567	−2.09759
$675$	0	0
$676$	−33.5897	−1.29191
$677$	−18.8761	−0.725469	−0.362734	−	0.931893i	$-0.618157\pi$
$677$	−18.8761	−0.725469	−0.362734	+	0.931893i	$0.618157\pi$
$678$	−13.7669	−0.528716
$679$	−6.59933	−0.253259
$680$	0	0
$681$	−7.75638	−0.297225
$682$	26.1379	1.00087
$683$	19.6576	0.752179	0.376089	−	0.926583i	$-0.377269\pi$
$683$	19.6576	0.752179	0.376089	+	0.926583i	$0.377269\pi$
$684$	10.4551	0.399761
$685$	0	0
$686$	−29.6475	−1.13195
$687$	2.24384	0.0856079
$688$	−10.0556	−0.383367
$689$	24.3333	0.927025
$690$	0	0
$691$	−48.2451	−1.83533	−0.917665	−	0.397354i	$-0.869928\pi$
$691$	−48.2451	−1.83533	−0.917665	+	0.397354i	$0.869928\pi$
$692$	86.4483	3.28627
$693$	−35.9073	−1.36401
$694$	34.9775	1.32773
$695$	0	0
$696$	−14.4713	−0.548533
$697$	3.47293	0.131546
$698$	−32.3152	−1.22315
$699$	6.49047	0.245492
$700$	0	0
$701$	0.512889	0.0193715	0.00968577	−	0.999953i	$-0.496917\pi$
$701$	0.512889	0.0193715	0.00968577	+	0.999953i	$0.496917\pi$
$702$	−15.3722	−0.580185
$703$	4.30266	0.162278
$704$	−39.7564	−1.49838
$705$	0	0
$706$	−42.6890	−1.60662
$707$	−5.45428	−0.205129
$708$	3.54437	0.133205
$709$	−8.84618	−0.332225	−0.166113	−	0.986107i	$-0.553122\pi$
$709$	−8.84618	−0.332225	−0.166113	+	0.986107i	$0.553122\pi$
$710$	0	0
$711$	3.79798	0.142436
$712$	74.2556	2.78285
$713$	6.77889	0.253871
$714$	24.0000	0.898177
$715$	0	0
$716$	−19.7980	−0.739885
$717$	6.09806	0.227736
$718$	30.1669	1.12582
$719$	7.84456	0.292553	0.146276	−	0.989244i	$-0.453271\pi$
$719$	7.84456	0.292553	0.146276	+	0.989244i	$0.453271\pi$
$720$	0	0
$721$	−18.3333	−0.682767
$722$	2.41987	0.0900582
$723$	1.82643	0.0679258
$724$	80.3781	2.98723
$725$	0	0
$726$	−8.14584	−0.302321
$727$	2.91130	0.107974	0.0539870	−	0.998542i	$-0.482807\pi$
$727$	2.91130	0.107974	0.0539870	+	0.998542i	$0.482807\pi$
$728$	−29.6358	−1.09838
$729$	−12.6475	−0.468427
$730$	0	0
$731$	18.4663	0.683001
$732$	18.1299	0.670100
$733$	33.7775	1.24760	0.623800	−	0.781584i	$-0.285588\pi$
$733$	33.7775	1.24760	0.623800	+	0.781584i	$0.285588\pi$
$734$	39.7564	1.46743
$735$	0	0
$736$	−3.50953	−0.129363
$737$	−19.7893	−0.728950
$738$	3.93254	0.144759
$739$	−35.8030	−1.31703	−0.658517	−	0.752566i	$-0.728816\pi$
$739$	−35.8030	−1.31703	−0.658517	+	0.752566i	$0.728816\pi$
$740$	0	0
$741$	−1.11222	−0.0408583
$742$	−90.6133	−3.32652
$743$	5.66948	0.207993	0.103996	−	0.994578i	$-0.466837\pi$
$743$	5.66948	0.207993	0.103996	+	0.994578i	$0.466837\pi$
$744$	−6.26927	−0.229843
$745$	0	0
$746$	12.8092	0.468978
$747$	−19.1748	−0.701569
$748$	−92.8451	−3.39475
$749$	49.1795	1.79698
$750$	0	0
$751$	−27.4679	−1.00232	−0.501159	−	0.865355i	$-0.667093\pi$
$751$	−27.4679	−1.00232	−0.501159	+	0.865355i	$0.667093\pi$
$752$	−36.9457	−1.34727
$753$	−1.63445	−0.0595628
$754$	−30.0673	−1.09498
$755$	0	0
$756$	37.6924	1.37086
$757$	41.6370	1.51332	0.756662	−	0.653806i	$-0.226829\pi$
$757$	41.6370	1.51332	0.756662	+	0.653806i	$0.226829\pi$
$758$	35.1736	1.27756
$759$	−5.82040	−0.211267
$760$	0	0
$761$	−9.46967	−0.343275	−0.171638	−	0.985160i	$-0.554906\pi$
$761$	−9.46967	−0.343275	−0.171638	+	0.985160i	$0.554906\pi$
$762$	−7.89526	−0.286015
$763$	37.3219	1.35114
$764$	−20.3109	−0.734822
$765$	0	0
$766$	−1.09765	−0.0396597
$767$	3.54437	0.127980
$768$	16.3146	0.588703
$769$	1.90858	0.0688253	0.0344127	−	0.999408i	$-0.489044\pi$
$769$	1.90858	0.0688253	0.0344127	+	0.999408i	$0.489044\pi$
$770$	0	0
$771$	9.28510	0.334395
$772$	−7.98476	−0.287378
$773$	28.4007	1.02150	0.510751	−	0.859729i	$-0.329367\pi$
$773$	28.4007	1.02150	0.510751	+	0.859729i	$0.329367\pi$
$774$	20.9102	0.751602
$775$	0	0
$776$	−9.29966	−0.333838
$777$	7.36420	0.264189
$778$	39.0941	1.40159
$779$	0.599328	0.0214732
$780$	0	0
$781$	56.9775	2.03881
$782$	−36.5696	−1.30773
$783$	18.4053	0.657753
$784$	9.95678	0.355599
$785$	0	0
$786$	17.6475	0.629466
$787$	−15.6708	−0.558605	−0.279303	−	0.960203i	$-0.590103\pi$
$787$	−15.6708	−0.558605	−0.279303	+	0.960203i	$0.590103\pi$
$788$	−40.2225	−1.43287
$789$	−0.640931	−0.0228178
$790$	0	0
$791$	33.7564	1.20024
$792$	−50.6000	−1.79799
$793$	18.1299	0.643811
$794$	−79.2659	−2.81304
$795$	0	0
$796$	−10.5353	−0.373414
$797$	36.9225	1.30786	0.653932	−	0.756554i	$-0.273118\pi$
$797$	36.9225	1.30786	0.653932	+	0.756554i	$0.273118\pi$
$798$	4.14172	0.146615
$799$	67.8478	2.40028
$800$	0	0
$801$	−44.8362	−1.58421
$802$	29.2476	1.03277
$803$	11.3325	0.399914
$804$	9.86201	0.347806
$805$	0	0
$806$	−13.0258	−0.458813
$807$	−11.8742	−0.417993
$808$	−7.68608	−0.270396
$809$	43.0465	1.51343	0.756716	−	0.653743i	$-0.226802\pi$
$809$	43.0465	1.51343	0.756716	+	0.653743i	$0.226802\pi$
$810$	0	0
$811$	32.2469	1.13234	0.566170	−	0.824288i	$-0.308425\pi$
$811$	32.2469	1.13234	0.566170	+	0.824288i	$0.308425\pi$
$812$	73.7245	2.58722
$813$	2.19475	0.0769731
$814$	−43.2659	−1.51647
$815$	0	0
$816$	9.82040	0.343783
$817$	3.18676	0.111491
$818$	−46.3033	−1.61896
$819$	17.8944	0.625280
$820$	0	0
$821$	5.18121	0.180826	0.0904128	−	0.995904i	$-0.471181\pi$
$821$	5.18121	0.180826	0.0904128	+	0.995904i	$0.471181\pi$
$822$	−10.3233	−0.360065
$823$	25.8517	0.901133	0.450567	−	0.892743i	$-0.351222\pi$
$823$	25.8517	0.901133	0.450567	+	0.892743i	$0.351222\pi$
$824$	−25.8350	−0.900004
$825$	0	0
$826$	−13.1987	−0.459240
$827$	9.97816	0.346974	0.173487	−	0.984836i	$-0.444496\pi$
$827$	9.97816	0.346974	0.173487	+	0.984836i	$0.444496\pi$
$828$	−27.2663	−0.947568
$829$	21.9808	0.763425	0.381713	−	0.924281i	$-0.375334\pi$
$829$	21.9808	0.763425	0.381713	+	0.924281i	$0.375334\pi$
$830$	0	0
$831$	−5.38149	−0.186682
$832$	19.8126	0.686877
$833$	−18.2848	−0.633531
$834$	−4.24685	−0.147056
$835$	0	0
$836$	−16.0224	−0.554147
$837$	7.97359	0.275607
$838$	19.4675	0.672492
$839$	−16.1089	−0.556140	−0.278070	−	0.960561i	$-0.589695\pi$
$839$	−16.1089	−0.556140	−0.278070	+	0.960561i	$0.589695\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−71.0451	−2.44838
$843$	−0.321887	−0.0110864
$844$	−60.6666	−2.08823
$845$	0	0
$846$	76.8270	2.64137
$847$	19.9735	0.686298
$848$	−37.0775	−1.27324
$849$	2.91020	0.0998779
$850$	0	0
$851$	−11.2211	−0.384653
$852$	−28.3947	−0.972785
$853$	−44.9615	−1.53945	−0.769727	−	0.638373i	$-0.779608\pi$
$853$	−44.9615	−1.53945	−0.769727	+	0.638373i	$0.779608\pi$
$854$	−67.5128	−2.31024
$855$	0	0
$856$	69.3029	2.36872
$857$	46.2431	1.57963	0.789817	−	0.613343i	$-0.210176\pi$
$857$	46.2431	1.57963	0.789817	+	0.613343i	$0.210176\pi$
$858$	11.1840	0.381816
$859$	−10.7772	−0.367713	−0.183856	−	0.982953i	$-0.558858\pi$
$859$	−10.7772	−0.367713	−0.183856	+	0.982953i	$0.558858\pi$
$860$	0	0
$861$	1.02578	0.0349584
$862$	−77.5443	−2.64117
$863$	−22.7966	−0.776007	−0.388003	−	0.921658i	$-0.626835\pi$
$863$	−22.7966	−0.776007	−0.388003	+	0.921658i	$0.626835\pi$
$864$	−4.12804	−0.140439
$865$	0	0
$866$	−1.16839	−0.0397034
$867$	−8.90400	−0.302396
$868$	31.9390	1.08408
$869$	−5.82040	−0.197444
$870$	0	0
$871$	9.86201	0.334161
$872$	52.5934	1.78104
$873$	5.61523	0.190047
$874$	−6.31087	−0.213468
$875$	0	0
$876$	−5.64752	−0.190812
$877$	4.94644	0.167029	0.0835146	−	0.996507i	$-0.473385\pi$
$877$	4.94644	0.167029	0.0835146	+	0.996507i	$0.473385\pi$
$878$	−66.2054	−2.23432
$879$	1.86201	0.0628039
$880$	0	0
$881$	2.53033	0.0852490	0.0426245	−	0.999091i	$-0.486428\pi$
$881$	2.53033	0.0852490	0.0426245	+	0.999091i	$0.486428\pi$
$882$	−20.7047	−0.697163
$883$	−29.7430	−1.00093	−0.500465	−	0.865757i	$-0.666838\pi$
$883$	−29.7430	−1.00093	−0.500465	+	0.865757i	$0.666838\pi$
$884$	46.2693	1.55620
$885$	0	0
$886$	−56.5801	−1.90085
$887$	−45.5450	−1.52925	−0.764626	−	0.644474i	$-0.777076\pi$
$887$	−45.5450	−1.52925	−0.764626	+	0.644474i	$0.777076\pi$
$888$	10.3775	0.348246
$889$	19.3591	0.649282
$890$	0	0
$891$	26.9568	0.903086
$892$	−72.5729	−2.42992
$893$	11.7086	0.391813
$894$	−10.9742	−0.367033
$895$	0	0
$896$	−65.2019	−2.17824
$897$	2.90059	0.0968480
$898$	55.9828	1.86817
$899$	15.5960	0.520155
$900$	0	0
$901$	68.0897	2.26840
$902$	−6.02662	−0.200664
$903$	5.45428	0.181507
$904$	47.5689	1.58212
$905$	0	0
$906$	−1.15382	−0.0383331
$907$	−33.2034	−1.10250	−0.551250	−	0.834340i	$-0.685849\pi$
$907$	−33.2034	−1.10250	−0.551250	+	0.834340i	$0.685849\pi$
$908$	55.6841	1.84794
$909$	4.64093	0.153930
$910$	0	0
$911$	55.7788	1.84803	0.924017	−	0.382351i	$-0.124886\pi$
$911$	55.7788	1.84803	0.924017	+	0.382351i	$0.124886\pi$
$912$	1.69472	0.0561179
$913$	29.3853	0.972513
$914$	−51.4231	−1.70092
$915$	0	0
$916$	−16.1089	−0.532252
$917$	−43.2715	−1.42895
$918$	−43.0146	−1.41969
$919$	−28.5769	−0.942665	−0.471333	−	0.881956i	$-0.656227\pi$
$919$	−28.5769	−0.942665	−0.471333	+	0.881956i	$0.656227\pi$
$920$	0	0
$921$	−8.91020	−0.293601
$922$	76.4159	2.51662
$923$	−28.3947	−0.934622
$924$	−27.4231	−0.902153
$925$	0	0
$926$	37.7788	1.24149
$927$	15.5994	0.512352
$928$	−8.07426	−0.265051
$929$	−54.4937	−1.78788	−0.893940	−	0.448186i	$-0.852070\pi$
$929$	−54.4937	−1.78788	−0.893940	+	0.448186i	$0.852070\pi$
$930$	0	0
$931$	−3.15544	−0.103415
$932$	−46.5960	−1.52630
$933$	2.23180	0.0730659
$934$	72.1761	2.36167
$935$	0	0
$936$	25.2165	0.824226
$937$	37.1484	1.21359	0.606793	−	0.794860i	$-0.292456\pi$
$937$	37.1484	1.21359	0.606793	+	0.794860i	$0.292456\pi$
$938$	−36.7245	−1.19910
$939$	−0.493704	−0.0161114
$940$	0	0
$941$	32.3973	1.05612	0.528061	−	0.849206i	$-0.322919\pi$
$941$	32.3973	1.05612	0.528061	+	0.849206i	$0.322919\pi$
$942$	5.38284	0.175382
$943$	−1.56301	−0.0508986
$944$	−5.40067	−0.175777
$945$	0	0
$946$	−32.0448	−1.04187
$947$	−35.4662	−1.15250	−0.576249	−	0.817275i	$-0.695484\pi$
$947$	−35.4662	−1.15250	−0.576249	+	0.817275i	$0.695484\pi$
$948$	2.90059	0.0942069
$949$	−5.64752	−0.183326
$950$	0	0
$951$	−14.3557	−0.465516
$952$	−82.9272	−2.68769
$953$	14.3411	0.464553	0.232277	−	0.972650i	$-0.425383\pi$
$953$	14.3411	0.464553	0.232277	+	0.972650i	$0.425383\pi$
$954$	77.1009	2.49623
$955$	0	0
$956$	−43.7788	−1.41591
$957$	−13.3908	−0.432864
$958$	10.9749	0.354581
$959$	25.3125	0.817383
$960$	0	0
$961$	−24.2435	−0.782048
$962$	21.5615	0.695172
$963$	−41.8458	−1.34846
$964$	−13.1122	−0.422316
$965$	0	0
$966$	−10.8013	−0.347528
$967$	27.9351	0.898331	0.449165	−	0.893449i	$-0.351721\pi$
$967$	27.9351	0.898331	0.449165	+	0.893449i	$0.351721\pi$
$968$	28.1463	0.904657
$969$	−3.11222	−0.0999788
$970$	0	0
$971$	−42.5993	−1.36708	−0.683539	−	0.729914i	$-0.739560\pi$
$971$	−42.5993	−1.36708	−0.683539	+	0.729914i	$0.739560\pi$
$972$	−48.9173	−1.56902
$973$	10.4132	0.333833
$974$	−94.9920	−3.04374
$975$	0	0
$976$	−27.6251	−0.884258
$977$	−39.5597	−1.26563	−0.632813	−	0.774304i	$-0.718100\pi$
$977$	−39.5597	−1.26563	−0.632813	+	0.774304i	$0.718100\pi$
$978$	−32.1180	−1.02702
$979$	68.7114	2.19603
$980$	0	0
$981$	−31.7564	−1.01390
$982$	94.1112	3.00321
$983$	39.7689	1.26843	0.634215	−	0.773157i	$-0.281323\pi$
$983$	39.7689	1.26843	0.634215	+	0.773157i	$0.281323\pi$
$984$	1.44551	0.0460811
$985$	0	0
$986$	−84.1345	−2.67939
$987$	20.0398	0.637874
$988$	7.98476	0.254029
$989$	−8.31087	−0.264270
$990$	0	0
$991$	55.2019	1.75355	0.876773	−	0.480905i	$-0.159692\pi$
$991$	55.2019	1.75355	0.876773	+	0.480905i	$0.159692\pi$
$992$	−3.49794	−0.111060
$993$	−4.29664	−0.136350
$994$	105.737	3.35378
$995$	0	0
$996$	−14.6442	−0.464018
$997$	11.6543	0.369097	0.184548	−	0.982823i	$-0.440918\pi$
$997$	11.6543	0.369097	0.184548	+	0.982823i	$0.440918\pi$
$998$	11.4517	0.362496
$999$	−13.1987	−0.417587

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.j.1.6		6
3.2	odd	2		4275.2.a.br.1.1		6
4.3	odd	2		7600.2.a.ck.1.4		6
5.2	odd	4		95.2.b.b.39.6	yes	6
5.3	odd	4		95.2.b.b.39.1	✓	6
5.4	even	2	inner	475.2.a.j.1.1		6
15.2	even	4		855.2.c.d.514.1		6
15.8	even	4		855.2.c.d.514.6		6
15.14	odd	2		4275.2.a.br.1.6		6
19.18	odd	2		9025.2.a.bx.1.1		6
20.3	even	4		1520.2.d.h.609.4		6
20.7	even	4		1520.2.d.h.609.3		6
20.19	odd	2		7600.2.a.ck.1.3		6
95.18	even	4		1805.2.b.e.1084.6		6
95.37	even	4		1805.2.b.e.1084.1		6
95.94	odd	2		9025.2.a.bx.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.1	✓	6	5.3	odd	4
95.2.b.b.39.6	yes	6	5.2	odd	4
475.2.a.j.1.1		6	5.4	even	2	inner
475.2.a.j.1.6		6	1.1	even	1	trivial
855.2.c.d.514.1		6	15.2	even	4
855.2.c.d.514.6		6	15.8	even	4
1520.2.d.h.609.3		6	20.7	even	4
1520.2.d.h.609.4		6	20.3	even	4
1805.2.b.e.1084.1		6	95.37	even	4
1805.2.b.e.1084.6		6	95.18	even	4
4275.2.a.br.1.1		6	3.2	odd	2
4275.2.a.br.1.6		6	15.14	odd	2
7600.2.a.ck.1.3		6	20.19	odd	2
7600.2.a.ck.1.4		6	4.3	odd	2
9025.2.a.bx.1.1		6	19.18	odd	2
9025.2.a.bx.1.6		6	95.94	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)

\(f(q)\)	\(=\)	\(q+2.41987 q^{2} -0.537080 q^{3} +3.85577 q^{4} -1.29966 q^{6} +3.18676 q^{7} +4.49073 q^{8} -2.71155 q^{9} +4.15544 q^{11} -2.07086 q^{12} -2.07086 q^{13} +7.71155 q^{14} +3.15544 q^{16} -5.79470 q^{17} -6.56159 q^{18} -1.00000 q^{19} -1.71155 q^{21} +10.0556 q^{22} +2.60794 q^{23} -2.41188 q^{24} -5.01121 q^{26} +3.06756 q^{27} +12.2874 q^{28} +6.00000 q^{29} +2.59933 q^{31} -1.34571 q^{32} -2.23180 q^{33} -14.0224 q^{34} -10.4551 q^{36} -4.30266 q^{37} -2.41987 q^{38} +1.11222 q^{39} -0.599328 q^{41} -4.14172 q^{42} -3.18676 q^{43} +16.0224 q^{44} +6.31087 q^{46} -11.7086 q^{47} -1.69472 q^{48} +3.15544 q^{49} +3.11222 q^{51} -7.98476 q^{52} -11.7503 q^{53} +7.42309 q^{54} +14.3109 q^{56} +0.537080 q^{57} +14.5192 q^{58} -1.71155 q^{59} -8.75476 q^{61} +6.29004 q^{62} -8.64104 q^{63} -9.56732 q^{64} -5.40067 q^{66} -4.76228 q^{67} -22.3430 q^{68} -1.40067 q^{69} +13.7115 q^{71} -12.1768 q^{72} +2.72714 q^{73} -10.4119 q^{74} -3.85577 q^{76} +13.2424 q^{77} +2.69142 q^{78} -1.40067 q^{79} +6.48711 q^{81} -1.45030 q^{82} +7.07154 q^{83} -6.59933 q^{84} -7.71155 q^{86} -3.22248 q^{87} +18.6609 q^{88} +16.5353 q^{89} -6.59933 q^{91} +10.0556 q^{92} -1.39605 q^{93} -28.3333 q^{94} +0.722754 q^{96} -2.07086 q^{97} +7.63575 q^{98} -11.2677 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51}+ \cdots - 30 q^{99}+O(q^{100}) \) 6 * q + 8 * q^4 + 14 * q^9 + 2 * q^11 + 16 * q^14 - 4 * q^16 - 6 * q^19 + 20 * q^21 + 8 * q^24 + 8 * q^26 + 36 * q^29 - 8 * q^34 - 32 * q^36 - 8 * q^39 + 12 * q^41 + 20 * q^44 - 8 * q^46 - 4 * q^49 + 4 * q^51 - 16 * q^54 + 40 * q^56 + 20 * q^59 - 14 * q^61 - 12 * q^64 - 48 * q^66 - 24 * q^69 + 52 * q^71 - 40 * q^74 - 8 * q^76 - 24 * q^79 + 38 * q^81 - 24 * q^84 - 16 * q^86 + 24 * q^89 - 24 * q^91 - 48 * q^94 - 64 * q^96 - 30 * q^99

Introduction

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