LMFDB - Embedded newform 1550.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1550,2,Mod(1,1550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1550, 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("1550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1550 = 2 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1550.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:	$12.3768123133$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 310)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	1550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.90321 q^{3} +1.00000 q^{4} +2.90321 q^{6} +4.42864 q^{7} -1.00000 q^{8} +5.42864 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.90321 q^{3} +1.00000 q^{4} +2.90321 q^{6} +4.42864 q^{7} -1.00000 q^{8} +5.42864 q^{9} -2.28100 q^{11} -2.90321 q^{12} +4.90321 q^{13} -4.42864 q^{14} +1.00000 q^{16} +4.42864 q^{17} -5.42864 q^{18} +7.05086 q^{19} -12.8573 q^{21} +2.28100 q^{22} +0.622216 q^{23} +2.90321 q^{24} -4.90321 q^{26} -7.05086 q^{27} +4.42864 q^{28} -2.76986 q^{29} +1.00000 q^{31} -1.00000 q^{32} +6.62222 q^{33} -4.42864 q^{34} +5.42864 q^{36} -3.95407 q^{37} -7.05086 q^{38} -14.2351 q^{39} +3.67307 q^{41} +12.8573 q^{42} -7.76049 q^{43} -2.28100 q^{44} -0.622216 q^{46} -11.1842 q^{47} -2.90321 q^{48} +12.6128 q^{49} -12.8573 q^{51} +4.90321 q^{52} +0.0459330 q^{53} +7.05086 q^{54} -4.42864 q^{56} -20.4701 q^{57} +2.76986 q^{58} +2.19358 q^{59} -3.71900 q^{61} -1.00000 q^{62} +24.0415 q^{63} +1.00000 q^{64} -6.62222 q^{66} +8.29529 q^{67} +4.42864 q^{68} -1.80642 q^{69} +2.75557 q^{71} -5.42864 q^{72} +15.4795 q^{73} +3.95407 q^{74} +7.05086 q^{76} -10.1017 q^{77} +14.2351 q^{78} -13.4193 q^{79} +4.18421 q^{81} -3.67307 q^{82} -9.95407 q^{83} -12.8573 q^{84} +7.76049 q^{86} +8.04149 q^{87} +2.28100 q^{88} +14.8573 q^{89} +21.7146 q^{91} +0.622216 q^{92} -2.90321 q^{93} +11.1842 q^{94} +2.90321 q^{96} +12.2351 q^{97} -12.6128 q^{98} -12.3827 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^6 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} - 3 q^{8} + 3 q^{9} - 2 q^{12} + 8 q^{13} + 3 q^{16} - 3 q^{18} + 8 q^{19} - 12 q^{21} + 2 q^{23} + 2 q^{24} - 8 q^{26} - 8 q^{27} - 2 q^{29} + 3 q^{31} - 3 q^{32} + 20 q^{33} + 3 q^{36} + 8 q^{37} - 8 q^{38} - 16 q^{39} - 2 q^{41} + 12 q^{42} + 10 q^{43} - 2 q^{46} - 20 q^{47} - 2 q^{48} + 11 q^{49} - 12 q^{51} + 8 q^{52} + 20 q^{53} + 8 q^{54} - 8 q^{57} + 2 q^{58} + 20 q^{59} - 18 q^{61} - 3 q^{62} + 32 q^{63} + 3 q^{64} - 20 q^{66} + 12 q^{67} + 8 q^{69} + 8 q^{71} - 3 q^{72} + 20 q^{73} - 8 q^{74} + 8 q^{76} - 4 q^{77} + 16 q^{78} - q^{81} + 2 q^{82} - 10 q^{83} - 12 q^{84} - 10 q^{86} - 16 q^{87} + 18 q^{89} + 12 q^{91} + 2 q^{92} - 2 q^{93} + 20 q^{94} + 2 q^{96} + 10 q^{97} - 11 q^{98} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^6 - 3 * q^8 + 3 * q^9 - 2 * q^12 + 8 * q^13 + 3 * q^16 - 3 * q^18 + 8 * q^19 - 12 * q^21 + 2 * q^23 + 2 * q^24 - 8 * q^26 - 8 * q^27 - 2 * q^29 + 3 * q^31 - 3 * q^32 + 20 * q^33 + 3 * q^36 + 8 * q^37 - 8 * q^38 - 16 * q^39 - 2 * q^41 + 12 * q^42 + 10 * q^43 - 2 * q^46 - 20 * q^47 - 2 * q^48 + 11 * q^49 - 12 * q^51 + 8 * q^52 + 20 * q^53 + 8 * q^54 - 8 * q^57 + 2 * q^58 + 20 * q^59 - 18 * q^61 - 3 * q^62 + 32 * q^63 + 3 * q^64 - 20 * q^66 + 12 * q^67 + 8 * q^69 + 8 * q^71 - 3 * q^72 + 20 * q^73 - 8 * q^74 + 8 * q^76 - 4 * q^77 + 16 * q^78 - q^81 + 2 * q^82 - 10 * q^83 - 12 * q^84 - 10 * q^86 - 16 * q^87 + 18 * q^89 + 12 * q^91 + 2 * q^92 - 2 * q^93 + 20 * q^94 + 2 * q^96 + 10 * q^97 - 11 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.90321	−1.67617	−0.838085	−	0.545540i	$-0.816325\pi$
$3$	−2.90321	−1.67617	−0.838085	+	0.545540i	$0.816325\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.90321	1.18523
$7$	4.42864	1.67387	0.836934	−	0.547304i	$-0.184346\pi$
$7$	4.42864	1.67387	0.836934	+	0.547304i	$0.184346\pi$
$8$	−1.00000	−0.353553
$9$	5.42864	1.80955
$10$	0	0
$11$	−2.28100	−0.687746	−0.343873	−	0.939016i	$-0.611739\pi$
$11$	−2.28100	−0.687746	−0.343873	+	0.939016i	$0.611739\pi$
$12$	−2.90321	−0.838085
$13$	4.90321	1.35991	0.679953	−	0.733256i	$-0.262000\pi$
$13$	4.90321	1.35991	0.679953	+	0.733256i	$0.262000\pi$
$14$	−4.42864	−1.18360
$15$	0	0
$16$	1.00000	0.250000
$17$	4.42864	1.07410	0.537051	−	0.843550i	$-0.319538\pi$
$17$	4.42864	1.07410	0.537051	+	0.843550i	$0.319538\pi$
$18$	−5.42864	−1.27954
$19$	7.05086	1.61758	0.808789	−	0.588100i	$-0.200124\pi$
$19$	7.05086	1.61758	0.808789	+	0.588100i	$0.200124\pi$
$20$	0	0
$21$	−12.8573	−2.80569
$22$	2.28100	0.486310
$23$	0.622216	0.129741	0.0648705	−	0.997894i	$-0.479337\pi$
$23$	0.622216	0.129741	0.0648705	+	0.997894i	$0.479337\pi$
$24$	2.90321	0.592616
$25$	0	0
$26$	−4.90321	−0.961599
$27$	−7.05086	−1.35694
$28$	4.42864	0.836934
$29$	−2.76986	−0.514350	−0.257175	−	0.966365i	$-0.582792\pi$
$29$	−2.76986	−0.514350	−0.257175	+	0.966365i	$0.582792\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	6.62222	1.15278
$34$	−4.42864	−0.759505
$35$	0	0
$36$	5.42864	0.904773
$37$	−3.95407	−0.650045	−0.325022	−	0.945706i	$-0.605372\pi$
$37$	−3.95407	−0.650045	−0.325022	+	0.945706i	$0.605372\pi$
$38$	−7.05086	−1.14380
$39$	−14.2351	−2.27943
$40$	0	0
$41$	3.67307	0.573637	0.286819	−	0.957985i	$-0.407402\pi$
$41$	3.67307	0.573637	0.286819	+	0.957985i	$0.407402\pi$
$42$	12.8573	1.98392
$43$	−7.76049	−1.18346	−0.591732	−	0.806135i	$-0.701556\pi$
$43$	−7.76049	−1.18346	−0.591732	+	0.806135i	$0.701556\pi$
$44$	−2.28100	−0.343873
$45$	0	0
$46$	−0.622216	−0.0917407
$47$	−11.1842	−1.63138	−0.815692	−	0.578486i	$-0.803644\pi$
$47$	−11.1842	−1.63138	−0.815692	+	0.578486i	$0.803644\pi$
$48$	−2.90321	−0.419043
$49$	12.6128	1.80184
$50$	0	0
$51$	−12.8573	−1.80038
$52$	4.90321	0.679953
$53$	0.0459330	0.00630938	0.00315469	−	0.999995i	$-0.498996\pi$
$53$	0.0459330	0.00630938	0.00315469	+	0.999995i	$0.498996\pi$
$54$	7.05086	0.959500
$55$	0	0
$56$	−4.42864	−0.591802
$57$	−20.4701	−2.71133
$58$	2.76986	0.363700
$59$	2.19358	0.285579	0.142790	−	0.989753i	$-0.454393\pi$
$59$	2.19358	0.285579	0.142790	+	0.989753i	$0.454393\pi$
$60$	0	0
$61$	−3.71900	−0.476170	−0.238085	−	0.971244i	$-0.576520\pi$
$61$	−3.71900	−0.476170	−0.238085	+	0.971244i	$0.576520\pi$
$62$	−1.00000	−0.127000
$63$	24.0415	3.02894
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.62222	−0.815138
$67$	8.29529	1.01343	0.506715	−	0.862113i	$-0.330860\pi$
$67$	8.29529	1.01343	0.506715	+	0.862113i	$0.330860\pi$
$68$	4.42864	0.537051
$69$	−1.80642	−0.217468
$70$	0	0
$71$	2.75557	0.327026	0.163513	−	0.986541i	$-0.447717\pi$
$71$	2.75557	0.327026	0.163513	+	0.986541i	$0.447717\pi$
$72$	−5.42864	−0.639771
$73$	15.4795	1.81174	0.905869	−	0.423558i	$-0.139219\pi$
$73$	15.4795	1.81174	0.905869	+	0.423558i	$0.139219\pi$
$74$	3.95407	0.459651
$75$	0	0
$76$	7.05086	0.808789
$77$	−10.1017	−1.15120
$78$	14.2351	1.61180
$79$	−13.4193	−1.50979	−0.754893	−	0.655848i	$-0.772311\pi$
$79$	−13.4193	−1.50979	−0.754893	+	0.655848i	$0.772311\pi$
$80$	0	0
$81$	4.18421	0.464912
$82$	−3.67307	−0.405623
$83$	−9.95407	−1.09260	−0.546300	−	0.837589i	$-0.683964\pi$
$83$	−9.95407	−1.09260	−0.546300	+	0.837589i	$0.683964\pi$
$84$	−12.8573	−1.40284
$85$	0	0
$86$	7.76049	0.836835
$87$	8.04149	0.862138
$88$	2.28100	0.243155
$89$	14.8573	1.57487	0.787434	−	0.616399i	$-0.211409\pi$
$89$	14.8573	1.57487	0.787434	+	0.616399i	$0.211409\pi$
$90$	0	0
$91$	21.7146	2.27630
$92$	0.622216	0.0648705
$93$	−2.90321	−0.301049
$94$	11.1842	1.15356
$95$	0	0
$96$	2.90321	0.296308
$97$	12.2351	1.24228	0.621141	−	0.783699i	$-0.286669\pi$
$97$	12.2351	1.24228	0.621141	+	0.783699i	$0.286669\pi$
$98$	−12.6128	−1.27409
$99$	−12.3827	−1.24451
$100$	0	0
$101$	−13.6128	−1.35453	−0.677264	−	0.735740i	$-0.736835\pi$
$101$	−13.6128	−1.35453	−0.677264	+	0.735740i	$0.736835\pi$
$102$	12.8573	1.27306
$103$	11.6128	1.14425	0.572124	−	0.820167i	$-0.306120\pi$
$103$	11.6128	1.14425	0.572124	+	0.820167i	$0.306120\pi$
$104$	−4.90321	−0.480799
$105$	0	0
$106$	−0.0459330	−0.00446140
$107$	6.10171	0.589875	0.294937	−	0.955517i	$-0.404701\pi$
$107$	6.10171	0.589875	0.294937	+	0.955517i	$0.404701\pi$
$108$	−7.05086	−0.678469
$109$	7.80642	0.747720	0.373860	−	0.927485i	$-0.378034\pi$
$109$	7.80642	0.747720	0.373860	+	0.927485i	$0.378034\pi$
$110$	0	0
$111$	11.4795	1.08959
$112$	4.42864	0.418467
$113$	6.99063	0.657623	0.328812	−	0.944396i	$-0.393352\pi$
$113$	6.99063	0.657623	0.328812	+	0.944396i	$0.393352\pi$
$114$	20.4701	1.91720
$115$	0	0
$116$	−2.76986	−0.257175
$117$	26.6178	2.46081
$118$	−2.19358	−0.201935
$119$	19.6128	1.79791
$120$	0	0
$121$	−5.79706	−0.527005
$122$	3.71900	0.336703
$123$	−10.6637	−0.961514
$124$	1.00000	0.0898027
$125$	0	0
$126$	−24.0415	−2.14179
$127$	−15.2859	−1.35641	−0.678203	−	0.734875i	$-0.737241\pi$
$127$	−15.2859	−1.35641	−0.678203	+	0.734875i	$0.737241\pi$
$128$	−1.00000	−0.0883883
$129$	22.5303	1.98369
$130$	0	0
$131$	9.80642	0.856791	0.428396	−	0.903591i	$-0.359079\pi$
$131$	9.80642	0.856791	0.428396	+	0.903591i	$0.359079\pi$
$132$	6.62222	0.576390
$133$	31.2257	2.70761
$134$	−8.29529	−0.716604
$135$	0	0
$136$	−4.42864	−0.379753
$137$	20.0415	1.71226	0.856130	−	0.516761i	$-0.172862\pi$
$137$	20.0415	1.71226	0.856130	+	0.516761i	$0.172862\pi$
$138$	1.80642	0.153773
$139$	−2.01429	−0.170850	−0.0854249	−	0.996345i	$-0.527225\pi$
$139$	−2.01429	−0.170850	−0.0854249	+	0.996345i	$0.527225\pi$
$140$	0	0
$141$	32.4701	2.73448
$142$	−2.75557	−0.231242
$143$	−11.1842	−0.935270
$144$	5.42864	0.452387
$145$	0	0
$146$	−15.4795	−1.28109
$147$	−36.6178	−3.02018
$148$	−3.95407	−0.325022
$149$	18.5620	1.52066	0.760329	−	0.649538i	$-0.225038\pi$
$149$	18.5620	1.52066	0.760329	+	0.649538i	$0.225038\pi$
$150$	0	0
$151$	−4.29529	−0.349545	−0.174773	−	0.984609i	$-0.555919\pi$
$151$	−4.29529	−0.349545	−0.174773	+	0.984609i	$0.555919\pi$
$152$	−7.05086	−0.571900
$153$	24.0415	1.94364
$154$	10.1017	0.814019
$155$	0	0
$156$	−14.2351	−1.13972
$157$	−7.71456	−0.615689	−0.307844	−	0.951437i	$-0.599608\pi$
$157$	−7.71456	−0.615689	−0.307844	+	0.951437i	$0.599608\pi$
$158$	13.4193	1.06758
$159$	−0.133353	−0.0105756
$160$	0	0
$161$	2.75557	0.217169
$162$	−4.18421	−0.328742
$163$	−22.2766	−1.74483	−0.872417	−	0.488762i	$-0.837449\pi$
$163$	−22.2766	−1.74483	−0.872417	+	0.488762i	$0.837449\pi$
$164$	3.67307	0.286819
$165$	0	0
$166$	9.95407	0.772585
$167$	2.99063	0.231422	0.115711	−	0.993283i	$-0.463085\pi$
$167$	2.99063	0.231422	0.115711	+	0.993283i	$0.463085\pi$
$168$	12.8573	0.991961
$169$	11.0415	0.849345
$170$	0	0
$171$	38.2766	2.92708
$172$	−7.76049	−0.591732
$173$	−20.3684	−1.54858	−0.774291	−	0.632830i	$-0.781893\pi$
$173$	−20.3684	−1.54858	−0.774291	+	0.632830i	$0.781893\pi$
$174$	−8.04149	−0.609624
$175$	0	0
$176$	−2.28100	−0.171937
$177$	−6.36842	−0.478679
$178$	−14.8573	−1.11360
$179$	23.9956	1.79351	0.896756	−	0.442525i	$-0.145917\pi$
$179$	23.9956	1.79351	0.896756	+	0.442525i	$0.145917\pi$
$180$	0	0
$181$	−13.4336	−0.998509	−0.499254	−	0.866455i	$-0.666393\pi$
$181$	−13.4336	−0.998509	−0.499254	+	0.866455i	$0.666393\pi$
$182$	−21.7146	−1.60959
$183$	10.7971	0.798141
$184$	−0.622216	−0.0458703
$185$	0	0
$186$	2.90321	0.212874
$187$	−10.1017	−0.738710
$188$	−11.1842	−0.815692
$189$	−31.2257	−2.27134
$190$	0	0
$191$	−23.0923	−1.67090	−0.835452	−	0.549564i	$-0.814794\pi$
$191$	−23.0923	−1.67090	−0.835452	+	0.549564i	$0.814794\pi$
$192$	−2.90321	−0.209521
$193$	10.8573	0.781524	0.390762	−	0.920492i	$-0.372212\pi$
$193$	10.8573	0.781524	0.390762	+	0.920492i	$0.372212\pi$
$194$	−12.2351	−0.878426
$195$	0	0
$196$	12.6128	0.900918
$197$	0.903212	0.0643512	0.0321756	−	0.999482i	$-0.489756\pi$
$197$	0.903212	0.0643512	0.0321756	+	0.999482i	$0.489756\pi$
$198$	12.3827	0.880001
$199$	−4.85728	−0.344323	−0.172162	−	0.985069i	$-0.555075\pi$
$199$	−4.85728	−0.344323	−0.172162	+	0.985069i	$0.555075\pi$
$200$	0	0
$201$	−24.0830	−1.69868
$202$	13.6128	0.957797
$203$	−12.2667	−0.860954
$204$	−12.8573	−0.900190
$205$	0	0
$206$	−11.6128	−0.809105
$207$	3.37778	0.234772
$208$	4.90321	0.339977
$209$	−16.0830	−1.11248
$210$	0	0
$211$	10.4889	0.722083	0.361042	−	0.932550i	$-0.382421\pi$
$211$	10.4889	0.722083	0.361042	+	0.932550i	$0.382421\pi$
$212$	0.0459330	0.00315469
$213$	−8.00000	−0.548151
$214$	−6.10171	−0.417104
$215$	0	0
$216$	7.05086	0.479750
$217$	4.42864	0.300636
$218$	−7.80642	−0.528718
$219$	−44.9403	−3.03678
$220$	0	0
$221$	21.7146	1.46068
$222$	−11.4795	−0.770453
$223$	−17.6543	−1.18222	−0.591111	−	0.806590i	$-0.701310\pi$
$223$	−17.6543	−1.18222	−0.591111	+	0.806590i	$0.701310\pi$
$224$	−4.42864	−0.295901
$225$	0	0
$226$	−6.99063	−0.465010
$227$	19.5210	1.29565	0.647827	−	0.761788i	$-0.275678\pi$
$227$	19.5210	1.29565	0.647827	+	0.761788i	$0.275678\pi$
$228$	−20.4701	−1.35567
$229$	−19.8938	−1.31462	−0.657311	−	0.753619i	$-0.728306\pi$
$229$	−19.8938	−1.31462	−0.657311	+	0.753619i	$0.728306\pi$
$230$	0	0
$231$	29.3274	1.92960
$232$	2.76986	0.181850
$233$	−7.11108	−0.465862	−0.232931	−	0.972493i	$-0.574832\pi$
$233$	−7.11108	−0.465862	−0.232931	+	0.972493i	$0.574832\pi$
$234$	−26.6178	−1.74006
$235$	0	0
$236$	2.19358	0.142790
$237$	38.9590	2.53066
$238$	−19.6128	−1.27131
$239$	7.52098	0.486492	0.243246	−	0.969965i	$-0.421788\pi$
$239$	7.52098	0.486492	0.243246	+	0.969965i	$0.421788\pi$
$240$	0	0
$241$	10.5620	0.680358	0.340179	−	0.940361i	$-0.389512\pi$
$241$	10.5620	0.680358	0.340179	+	0.940361i	$0.389512\pi$
$242$	5.79706	0.372649
$243$	9.00492	0.577666
$244$	−3.71900	−0.238085
$245$	0	0
$246$	10.6637	0.679893
$247$	34.5718	2.19975
$248$	−1.00000	−0.0635001
$249$	28.8988	1.83138
$250$	0	0
$251$	−18.5763	−1.17252	−0.586262	−	0.810121i	$-0.699401\pi$
$251$	−18.5763	−1.17252	−0.586262	+	0.810121i	$0.699401\pi$
$252$	24.0415	1.51447
$253$	−1.41927	−0.0892288
$254$	15.2859	0.959124
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.75557	−0.546157	−0.273079	−	0.961992i	$-0.588042\pi$
$257$	−8.75557	−0.546157	−0.273079	+	0.961992i	$0.588042\pi$
$258$	−22.5303	−1.40268
$259$	−17.5111	−1.08809
$260$	0	0
$261$	−15.0366	−0.930740
$262$	−9.80642	−0.605843
$263$	4.79706	0.295799	0.147900	−	0.989002i	$-0.452749\pi$
$263$	4.79706	0.295799	0.147900	+	0.989002i	$0.452749\pi$
$264$	−6.62222	−0.407569
$265$	0	0
$266$	−31.2257	−1.91457
$267$	−43.1338	−2.63975
$268$	8.29529	0.506715
$269$	−0.106156	−0.00647245	−0.00323622	−	0.999995i	$-0.501030\pi$
$269$	−0.106156	−0.00647245	−0.00323622	+	0.999995i	$0.501030\pi$
$270$	0	0
$271$	19.0509	1.15726	0.578629	−	0.815591i	$-0.303588\pi$
$271$	19.0509	1.15726	0.578629	+	0.815591i	$0.303588\pi$
$272$	4.42864	0.268526
$273$	−63.0420	−3.81547
$274$	−20.0415	−1.21075
$275$	0	0
$276$	−1.80642	−0.108734
$277$	16.0459	0.964107	0.482053	−	0.876142i	$-0.339891\pi$
$277$	16.0459	0.964107	0.482053	+	0.876142i	$0.339891\pi$
$278$	2.01429	0.120809
$279$	5.42864	0.325004
$280$	0	0
$281$	13.2257	0.788979	0.394489	−	0.918900i	$-0.370921\pi$
$281$	13.2257	0.788979	0.394489	+	0.918900i	$0.370921\pi$
$282$	−32.4701	−1.93357
$283$	−8.17484	−0.485944	−0.242972	−	0.970033i	$-0.578122\pi$
$283$	−8.17484	−0.485944	−0.242972	+	0.970033i	$0.578122\pi$
$284$	2.75557	0.163513
$285$	0	0
$286$	11.1842	0.661336
$287$	16.2667	0.960193
$288$	−5.42864	−0.319886
$289$	2.61285	0.153697
$290$	0	0
$291$	−35.5210	−2.08228
$292$	15.4795	0.905869
$293$	−5.43801	−0.317692	−0.158846	−	0.987303i	$-0.550777\pi$
$293$	−5.43801	−0.317692	−0.158846	+	0.987303i	$0.550777\pi$
$294$	36.6178	2.13559
$295$	0	0
$296$	3.95407	0.229825
$297$	16.0830	0.933229
$298$	−18.5620	−1.07527
$299$	3.05086	0.176436
$300$	0	0
$301$	−34.3684	−1.98096
$302$	4.29529	0.247166
$303$	39.5210	2.27042
$304$	7.05086	0.404394
$305$	0	0
$306$	−24.0415	−1.37436
$307$	12.4701	0.711708	0.355854	−	0.934542i	$-0.384190\pi$
$307$	12.4701	0.711708	0.355854	+	0.934542i	$0.384190\pi$
$308$	−10.1017	−0.575598
$309$	−33.7146	−1.91795
$310$	0	0
$311$	19.8796	1.12727	0.563633	−	0.826025i	$-0.309403\pi$
$311$	19.8796	1.12727	0.563633	+	0.826025i	$0.309403\pi$
$312$	14.2351	0.805902
$313$	19.6543	1.11093	0.555464	−	0.831540i	$-0.312541\pi$
$313$	19.6543	1.11093	0.555464	+	0.831540i	$0.312541\pi$
$314$	7.71456	0.435358
$315$	0	0
$316$	−13.4193	−0.754893
$317$	10.5620	0.593221	0.296610	−	0.954999i	$-0.404144\pi$
$317$	10.5620	0.593221	0.296610	+	0.954999i	$0.404144\pi$
$318$	0.133353	0.00747807
$319$	6.31804	0.353742
$320$	0	0
$321$	−17.7146	−0.988730
$322$	−2.75557	−0.153562
$323$	31.2257	1.73744
$324$	4.18421	0.232456
$325$	0	0
$326$	22.2766	1.23378
$327$	−22.6637	−1.25331
$328$	−3.67307	−0.202811
$329$	−49.5308	−2.73072
$330$	0	0
$331$	−8.64941	−0.475415	−0.237707	−	0.971337i	$-0.576396\pi$
$331$	−8.64941	−0.475415	−0.237707	+	0.971337i	$0.576396\pi$
$332$	−9.95407	−0.546300
$333$	−21.4652	−1.17629
$334$	−2.99063	−0.163640
$335$	0	0
$336$	−12.8573	−0.701422
$337$	11.4795	0.625328	0.312664	−	0.949864i	$-0.398779\pi$
$337$	11.4795	0.625328	0.312664	+	0.949864i	$0.398779\pi$
$338$	−11.0415	−0.600578
$339$	−20.2953	−1.10229
$340$	0	0
$341$	−2.28100	−0.123523
$342$	−38.2766	−2.06976
$343$	24.8573	1.34217
$344$	7.76049	0.418418
$345$	0	0
$346$	20.3684	1.09501
$347$	3.85236	0.206805	0.103403	−	0.994640i	$-0.467027\pi$
$347$	3.85236	0.206805	0.103403	+	0.994640i	$0.467027\pi$
$348$	8.04149	0.431069
$349$	−7.33630	−0.392703	−0.196351	−	0.980534i	$-0.562909\pi$
$349$	−7.33630	−0.392703	−0.196351	+	0.980534i	$0.562909\pi$
$350$	0	0
$351$	−34.5718	−1.84531
$352$	2.28100	0.121578
$353$	−20.6035	−1.09661	−0.548306	−	0.836278i	$-0.684727\pi$
$353$	−20.6035	−1.09661	−0.548306	+	0.836278i	$0.684727\pi$
$354$	6.36842	0.338478
$355$	0	0
$356$	14.8573	0.787434
$357$	−56.9403	−3.01360
$358$	−23.9956	−1.26820
$359$	8.72393	0.460431	0.230216	−	0.973140i	$-0.426057\pi$
$359$	8.72393	0.460431	0.230216	+	0.973140i	$0.426057\pi$
$360$	0	0
$361$	30.7146	1.61656
$362$	13.4336	0.706052
$363$	16.8301	0.883350
$364$	21.7146	1.13815
$365$	0	0
$366$	−10.7971	−0.564371
$367$	−11.1111	−0.579994	−0.289997	−	0.957028i	$-0.593654\pi$
$367$	−11.1111	−0.579994	−0.289997	+	0.957028i	$0.593654\pi$
$368$	0.622216	0.0324352
$369$	19.9398	1.03802
$370$	0	0
$371$	0.203420	0.0105611
$372$	−2.90321	−0.150525
$373$	22.4415	1.16198	0.580990	−	0.813911i	$-0.302666\pi$
$373$	22.4415	1.16198	0.580990	+	0.813911i	$0.302666\pi$
$374$	10.1017	0.522347
$375$	0	0
$376$	11.1842	0.576781
$377$	−13.5812	−0.699468
$378$	31.2257	1.60608
$379$	−8.56199	−0.439800	−0.219900	−	0.975522i	$-0.570573\pi$
$379$	−8.56199	−0.439800	−0.219900	+	0.975522i	$0.570573\pi$
$380$	0	0
$381$	44.3783	2.27357
$382$	23.0923	1.18151
$383$	−35.6731	−1.82281	−0.911404	−	0.411512i	$-0.865001\pi$
$383$	−35.6731	−1.82281	−0.911404	+	0.411512i	$0.865001\pi$
$384$	2.90321	0.148154
$385$	0	0
$386$	−10.8573	−0.552621
$387$	−42.1289	−2.14153
$388$	12.2351	0.621141
$389$	8.30958	0.421312	0.210656	−	0.977560i	$-0.432440\pi$
$389$	8.30958	0.421312	0.210656	+	0.977560i	$0.432440\pi$
$390$	0	0
$391$	2.75557	0.139355
$392$	−12.6128	−0.637045
$393$	−28.4701	−1.43613
$394$	−0.903212	−0.0455032
$395$	0	0
$396$	−12.3827	−0.622254
$397$	7.53972	0.378408	0.189204	−	0.981938i	$-0.439409\pi$
$397$	7.53972	0.378408	0.189204	+	0.981938i	$0.439409\pi$
$398$	4.85728	0.243473
$399$	−90.6548	−4.53842
$400$	0	0
$401$	−11.4193	−0.570251	−0.285126	−	0.958490i	$-0.592035\pi$
$401$	−11.4193	−0.570251	−0.285126	+	0.958490i	$0.592035\pi$
$402$	24.0830	1.20115
$403$	4.90321	0.244246
$404$	−13.6128	−0.677264
$405$	0	0
$406$	12.2667	0.608786
$407$	9.01921	0.447066
$408$	12.8573	0.636530
$409$	2.94914	0.145826	0.0729129	−	0.997338i	$-0.476770\pi$
$409$	2.94914	0.145826	0.0729129	+	0.997338i	$0.476770\pi$
$410$	0	0
$411$	−58.1847	−2.87004
$412$	11.6128	0.572124
$413$	9.71456	0.478022
$414$	−3.37778	−0.166009
$415$	0	0
$416$	−4.90321	−0.240400
$417$	5.84791	0.286373
$418$	16.0830	0.786644
$419$	−1.15257	−0.0563065	−0.0281533	−	0.999604i	$-0.508963\pi$
$419$	−1.15257	−0.0563065	−0.0281533	+	0.999604i	$0.508963\pi$
$420$	0	0
$421$	−24.0098	−1.17017	−0.585084	−	0.810973i	$-0.698939\pi$
$421$	−24.0098	−1.17017	−0.585084	+	0.810973i	$0.698939\pi$
$422$	−10.4889	−0.510590
$423$	−60.7150	−2.95207
$424$	−0.0459330	−0.00223070
$425$	0	0
$426$	8.00000	0.387601
$427$	−16.4701	−0.797045
$428$	6.10171	0.294937
$429$	32.4701	1.56767
$430$	0	0
$431$	13.4608	0.648382	0.324191	−	0.945992i	$-0.394908\pi$
$431$	13.4608	0.648382	0.324191	+	0.945992i	$0.394908\pi$
$432$	−7.05086	−0.339234
$433$	−0.815792	−0.0392045	−0.0196022	−	0.999808i	$-0.506240\pi$
$433$	−0.815792	−0.0392045	−0.0196022	+	0.999808i	$0.506240\pi$
$434$	−4.42864	−0.212581
$435$	0	0
$436$	7.80642	0.373860
$437$	4.38715	0.209866
$438$	44.9403	2.14733
$439$	−5.37778	−0.256668	−0.128334	−	0.991731i	$-0.540963\pi$
$439$	−5.37778	−0.256668	−0.128334	+	0.991731i	$0.540963\pi$
$440$	0	0
$441$	68.4706	3.26050
$442$	−21.7146	−1.03286
$443$	−28.8573	−1.37105	−0.685525	−	0.728049i	$-0.740427\pi$
$443$	−28.8573	−1.37105	−0.685525	+	0.728049i	$0.740427\pi$
$444$	11.4795	0.544793
$445$	0	0
$446$	17.6543	0.835957
$447$	−53.8894	−2.54888
$448$	4.42864	0.209234
$449$	17.0509	0.804680	0.402340	−	0.915490i	$-0.368197\pi$
$449$	17.0509	0.804680	0.402340	+	0.915490i	$0.368197\pi$
$450$	0	0
$451$	−8.37826	−0.394517
$452$	6.99063	0.328812
$453$	12.4701	0.585898
$454$	−19.5210	−0.916165
$455$	0	0
$456$	20.4701	0.958602
$457$	29.7560	1.39193	0.695965	−	0.718076i	$-0.254977\pi$
$457$	29.7560	1.39193	0.695965	+	0.718076i	$0.254977\pi$
$458$	19.8938	0.929578
$459$	−31.2257	−1.45749
$460$	0	0
$461$	24.4844	1.14035	0.570176	−	0.821522i	$-0.306875\pi$
$461$	24.4844	1.14035	0.570176	+	0.821522i	$0.306875\pi$
$462$	−29.3274	−1.36443
$463$	21.0923	0.980244	0.490122	−	0.871654i	$-0.336952\pi$
$463$	21.0923	0.980244	0.490122	+	0.871654i	$0.336952\pi$
$464$	−2.76986	−0.128587
$465$	0	0
$466$	7.11108	0.329414
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	26.6178	1.23041
$469$	36.7368	1.69635
$470$	0	0
$471$	22.3970	1.03200
$472$	−2.19358	−0.100968
$473$	17.7017	0.813923
$474$	−38.9590	−1.78945
$475$	0	0
$476$	19.6128	0.898953
$477$	0.249353	0.0114171
$478$	−7.52098	−0.344002
$479$	−1.04101	−0.0475650	−0.0237825	−	0.999717i	$-0.507571\pi$
$479$	−1.04101	−0.0475650	−0.0237825	+	0.999717i	$0.507571\pi$
$480$	0	0
$481$	−19.3876	−0.884000
$482$	−10.5620	−0.481086
$483$	−8.00000	−0.364013
$484$	−5.79706	−0.263503
$485$	0	0
$486$	−9.00492	−0.408472
$487$	15.9398	0.722300	0.361150	−	0.932508i	$-0.382384\pi$
$487$	15.9398	0.722300	0.361150	+	0.932508i	$0.382384\pi$
$488$	3.71900	0.168351
$489$	64.6735	2.92464
$490$	0	0
$491$	−39.8751	−1.79954	−0.899769	−	0.436366i	$-0.856265\pi$
$491$	−39.8751	−1.79954	−0.899769	+	0.436366i	$0.856265\pi$
$492$	−10.6637	−0.480757
$493$	−12.2667	−0.552465
$494$	−34.5718	−1.55546
$495$	0	0
$496$	1.00000	0.0449013
$497$	12.2034	0.547398
$498$	−28.8988	−1.29498
$499$	18.4558	0.826197	0.413098	−	0.910686i	$-0.364447\pi$
$499$	18.4558	0.826197	0.413098	+	0.910686i	$0.364447\pi$
$500$	0	0
$501$	−8.68244	−0.387903
$502$	18.5763	0.829100
$503$	8.26671	0.368594	0.184297	−	0.982871i	$-0.440999\pi$
$503$	8.26671	0.368594	0.184297	+	0.982871i	$0.440999\pi$
$504$	−24.0415	−1.07089
$505$	0	0
$506$	1.41927	0.0630943
$507$	−32.0558	−1.42365
$508$	−15.2859	−0.678203
$509$	3.81087	0.168914	0.0844569	−	0.996427i	$-0.473084\pi$
$509$	3.81087	0.168914	0.0844569	+	0.996427i	$0.473084\pi$
$510$	0	0
$511$	68.5531	3.03261
$512$	−1.00000	−0.0441942
$513$	−49.7146	−2.19495
$514$	8.75557	0.386192
$515$	0	0
$516$	22.5303	0.991843
$517$	25.5111	1.12198
$518$	17.5111	0.769395
$519$	59.1338	2.59569
$520$	0	0
$521$	7.89829	0.346030	0.173015	−	0.984919i	$-0.444649\pi$
$521$	7.89829	0.346030	0.173015	+	0.984919i	$0.444649\pi$
$522$	15.0366	0.658133
$523$	−21.0335	−0.919731	−0.459865	−	0.887989i	$-0.652102\pi$
$523$	−21.0335	−0.919731	−0.459865	+	0.887989i	$0.652102\pi$
$524$	9.80642	0.428396
$525$	0	0
$526$	−4.79706	−0.209162
$527$	4.42864	0.192915
$528$	6.62222	0.288195
$529$	−22.6128	−0.983167
$530$	0	0
$531$	11.9081	0.516769
$532$	31.2257	1.35381
$533$	18.0098	0.780093
$534$	43.1338	1.86658
$535$	0	0
$536$	−8.29529	−0.358302
$537$	−69.6642	−3.00623
$538$	0.106156	0.00457671
$539$	−28.7699	−1.23921
$540$	0	0
$541$	7.21585	0.310234	0.155117	−	0.987896i	$-0.450425\pi$
$541$	7.21585	0.310234	0.155117	+	0.987896i	$0.450425\pi$
$542$	−19.0509	−0.818304
$543$	39.0005	1.67367
$544$	−4.42864	−0.189876
$545$	0	0
$546$	63.0420	2.69795
$547$	−6.27655	−0.268366	−0.134183	−	0.990957i	$-0.542841\pi$
$547$	−6.27655	−0.268366	−0.134183	+	0.990957i	$0.542841\pi$
$548$	20.0415	0.856130
$549$	−20.1891	−0.861651
$550$	0	0
$551$	−19.5299	−0.832001
$552$	1.80642	0.0768865
$553$	−59.4291	−2.52718
$554$	−16.0459	−0.681726
$555$	0	0
$556$	−2.01429	−0.0854249
$557$	−8.81135	−0.373349	−0.186674	−	0.982422i	$-0.559771\pi$
$557$	−8.81135	−0.373349	−0.186674	+	0.982422i	$0.559771\pi$
$558$	−5.42864	−0.229813
$559$	−38.0513	−1.60940
$560$	0	0
$561$	29.3274	1.23820
$562$	−13.2257	−0.557892
$563$	26.5433	1.11866	0.559332	−	0.828943i	$-0.311058\pi$
$563$	26.5433	1.11866	0.559332	+	0.828943i	$0.311058\pi$
$564$	32.4701	1.36724
$565$	0	0
$566$	8.17484	0.343614
$567$	18.5303	0.778202
$568$	−2.75557	−0.115621
$569$	5.97142	0.250335	0.125167	−	0.992136i	$-0.460053\pi$
$569$	5.97142	0.250335	0.125167	+	0.992136i	$0.460053\pi$
$570$	0	0
$571$	−28.2908	−1.18393	−0.591967	−	0.805962i	$-0.701649\pi$
$571$	−28.2908	−1.18393	−0.591967	+	0.805962i	$0.701649\pi$
$572$	−11.1842	−0.467635
$573$	67.0420	2.80072
$574$	−16.2667	−0.678959
$575$	0	0
$576$	5.42864	0.226193
$577$	−37.9496	−1.57986	−0.789932	−	0.613195i	$-0.789884\pi$
$577$	−37.9496	−1.57986	−0.789932	+	0.613195i	$0.789884\pi$
$578$	−2.61285	−0.108680
$579$	−31.5210	−1.30997
$580$	0	0
$581$	−44.0830	−1.82887
$582$	35.5210	1.47239
$583$	−0.104773	−0.00433925
$584$	−15.4795	−0.640546
$585$	0	0
$586$	5.43801	0.224642
$587$	15.2904	0.631101	0.315550	−	0.948909i	$-0.397811\pi$
$587$	15.2904	0.631101	0.315550	+	0.948909i	$0.397811\pi$
$588$	−36.6178	−1.51009
$589$	7.05086	0.290525
$590$	0	0
$591$	−2.62222	−0.107864
$592$	−3.95407	−0.162511
$593$	−5.59994	−0.229962	−0.114981	−	0.993368i	$-0.536681\pi$
$593$	−5.59994	−0.229962	−0.114981	+	0.993368i	$0.536681\pi$
$594$	−16.0830	−0.659892
$595$	0	0
$596$	18.5620	0.760329
$597$	14.1017	0.577145
$598$	−3.05086	−0.124759
$599$	−16.1334	−0.659191	−0.329595	−	0.944122i	$-0.606912\pi$
$599$	−16.1334	−0.659191	−0.329595	+	0.944122i	$0.606912\pi$
$600$	0	0
$601$	28.6923	1.17038	0.585191	−	0.810895i	$-0.301019\pi$
$601$	28.6923	1.17038	0.585191	+	0.810895i	$0.301019\pi$
$602$	34.3684	1.40075
$603$	45.0321	1.83385
$604$	−4.29529	−0.174773
$605$	0	0
$606$	−39.5210	−1.60543
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	−7.05086	−0.285950
$609$	35.6128	1.44311
$610$	0	0
$611$	−54.8385	−2.21853
$612$	24.0415	0.971820
$613$	−5.02366	−0.202904	−0.101452	−	0.994840i	$-0.532349\pi$
$613$	−5.02366	−0.202904	−0.101452	+	0.994840i	$0.532349\pi$
$614$	−12.4701	−0.503253
$615$	0	0
$616$	10.1017	0.407010
$617$	−6.20342	−0.249740	−0.124870	−	0.992173i	$-0.539851\pi$
$617$	−6.20342	−0.249740	−0.124870	+	0.992173i	$0.539851\pi$
$618$	33.7146	1.35620
$619$	8.38271	0.336929	0.168465	−	0.985708i	$-0.446119\pi$
$619$	8.38271	0.336929	0.168465	+	0.985708i	$0.446119\pi$
$620$	0	0
$621$	−4.38715	−0.176050
$622$	−19.8796	−0.797097
$623$	65.7975	2.63612
$624$	−14.2351	−0.569859
$625$	0	0
$626$	−19.6543	−0.785545
$627$	46.6923	1.86471
$628$	−7.71456	−0.307844
$629$	−17.5111	−0.698215
$630$	0	0
$631$	−31.6040	−1.25813	−0.629067	−	0.777351i	$-0.716563\pi$
$631$	−31.6040	−1.25813	−0.629067	+	0.777351i	$0.716563\pi$
$632$	13.4193	0.533790
$633$	−30.4514	−1.21033
$634$	−10.5620	−0.419470
$635$	0	0
$636$	−0.133353	−0.00528779
$637$	61.8435	2.45033
$638$	−6.31804	−0.250134
$639$	14.9590	0.591768
$640$	0	0
$641$	−33.4005	−1.31924	−0.659621	−	0.751598i	$-0.729283\pi$
$641$	−33.4005	−1.31924	−0.659621	+	0.751598i	$0.729283\pi$
$642$	17.7146	0.699138
$643$	−28.9131	−1.14022	−0.570110	−	0.821568i	$-0.693099\pi$
$643$	−28.9131	−1.14022	−0.570110	+	0.821568i	$0.693099\pi$
$644$	2.75557	0.108585
$645$	0	0
$646$	−31.2257	−1.22856
$647$	16.2636	0.639390	0.319695	−	0.947521i	$-0.396420\pi$
$647$	16.2636	0.639390	0.319695	+	0.947521i	$0.396420\pi$
$648$	−4.18421	−0.164371
$649$	−5.00354	−0.196406
$650$	0	0
$651$	−12.8573	−0.503916
$652$	−22.2766	−0.872417
$653$	5.52987	0.216401	0.108200	−	0.994129i	$-0.465491\pi$
$653$	5.52987	0.216401	0.108200	+	0.994129i	$0.465491\pi$
$654$	22.6637	0.886221
$655$	0	0
$656$	3.67307	0.143409
$657$	84.0326	3.27842
$658$	49.5308	1.93091
$659$	44.9688	1.75174	0.875869	−	0.482550i	$-0.160289\pi$
$659$	44.9688	1.75174	0.875869	+	0.482550i	$0.160289\pi$
$660$	0	0
$661$	48.4800	1.88565	0.942826	−	0.333285i	$-0.108157\pi$
$661$	48.4800	1.88565	0.942826	+	0.333285i	$0.108157\pi$
$662$	8.64941	0.336169
$663$	−63.0420	−2.44835
$664$	9.95407	0.386293
$665$	0	0
$666$	21.4652	0.831760
$667$	−1.72345	−0.0667322
$668$	2.99063	0.115711
$669$	51.2543	1.98160
$670$	0	0
$671$	8.48303	0.327484
$672$	12.8573	0.495980
$673$	26.9175	1.03759	0.518797	−	0.854898i	$-0.326380\pi$
$673$	26.9175	1.03759	0.518797	+	0.854898i	$0.326380\pi$
$674$	−11.4795	−0.442174
$675$	0	0
$676$	11.0415	0.424673
$677$	3.77923	0.145247	0.0726237	−	0.997359i	$-0.476863\pi$
$677$	3.77923	0.145247	0.0726237	+	0.997359i	$0.476863\pi$
$678$	20.2953	0.779436
$679$	54.1847	2.07942
$680$	0	0
$681$	−56.6735	−2.17174
$682$	2.28100	0.0873439
$683$	30.1017	1.15181	0.575905	−	0.817517i	$-0.304650\pi$
$683$	30.1017	1.15181	0.575905	+	0.817517i	$0.304650\pi$
$684$	38.2766	1.46354
$685$	0	0
$686$	−24.8573	−0.949055
$687$	57.7560	2.20353
$688$	−7.76049	−0.295866
$689$	0.225219	0.00858016
$690$	0	0
$691$	24.7368	0.941033	0.470517	−	0.882391i	$-0.344068\pi$
$691$	24.7368	0.941033	0.470517	+	0.882391i	$0.344068\pi$
$692$	−20.3684	−0.774291
$693$	−54.8385	−2.08314
$694$	−3.85236	−0.146233
$695$	0	0
$696$	−8.04149	−0.304812
$697$	16.2667	0.616145
$698$	7.33630	0.277683
$699$	20.6450	0.780864
$700$	0	0
$701$	−35.1240	−1.32661	−0.663307	−	0.748347i	$-0.730848\pi$
$701$	−35.1240	−1.32661	−0.663307	+	0.748347i	$0.730848\pi$
$702$	34.5718	1.30483
$703$	−27.8796	−1.05150
$704$	−2.28100	−0.0859683
$705$	0	0
$706$	20.6035	0.775422
$707$	−60.2864	−2.26730
$708$	−6.36842	−0.239340
$709$	−4.96343	−0.186406	−0.0932029	−	0.995647i	$-0.529711\pi$
$709$	−4.96343	−0.186406	−0.0932029	+	0.995647i	$0.529711\pi$
$710$	0	0
$711$	−72.8484	−2.73203
$712$	−14.8573	−0.556800
$713$	0.622216	0.0233022
$714$	56.9403	2.13094
$715$	0	0
$716$	23.9956	0.896756
$717$	−21.8350	−0.815443
$718$	−8.72393	−0.325574
$719$	−0.470127	−0.0175328	−0.00876638	−	0.999962i	$-0.502790\pi$
$719$	−0.470127	−0.0175328	−0.00876638	+	0.999962i	$0.502790\pi$
$720$	0	0
$721$	51.4291	1.91532
$722$	−30.7146	−1.14308
$723$	−30.6637	−1.14040
$724$	−13.4336	−0.499254
$725$	0	0
$726$	−16.8301	−0.624623
$727$	38.8385	1.44044	0.720221	−	0.693745i	$-0.244040\pi$
$727$	38.8385	1.44044	0.720221	+	0.693745i	$0.244040\pi$
$728$	−21.7146	−0.804795
$729$	−38.6958	−1.43318
$730$	0	0
$731$	−34.3684	−1.27116
$732$	10.7971	0.399071
$733$	−2.09187	−0.0772648	−0.0386324	−	0.999253i	$-0.512300\pi$
$733$	−2.09187	−0.0772648	−0.0386324	+	0.999253i	$0.512300\pi$
$734$	11.1111	0.410117
$735$	0	0
$736$	−0.622216	−0.0229352
$737$	−18.9215	−0.696983
$738$	−19.9398	−0.733993
$739$	−11.7288	−0.431453	−0.215726	−	0.976454i	$-0.569212\pi$
$739$	−11.7288	−0.431453	−0.215726	+	0.976454i	$0.569212\pi$
$740$	0	0
$741$	−100.369	−3.68716
$742$	−0.203420	−0.00746780
$743$	−38.1245	−1.39865	−0.699325	−	0.714803i	$-0.746516\pi$
$743$	−38.1245	−1.39865	−0.699325	+	0.714803i	$0.746516\pi$
$744$	2.90321	0.106437
$745$	0	0
$746$	−22.4415	−0.821643
$747$	−54.0370	−1.97711
$748$	−10.1017	−0.369355
$749$	27.0223	0.987372
$750$	0	0
$751$	7.09234	0.258803	0.129402	−	0.991592i	$-0.458694\pi$
$751$	7.09234	0.258803	0.129402	+	0.991592i	$0.458694\pi$
$752$	−11.1842	−0.407846
$753$	53.9309	1.96535
$754$	13.5812	0.494598
$755$	0	0
$756$	−31.2257	−1.13567
$757$	−1.76049	−0.0639861	−0.0319931	−	0.999488i	$-0.510185\pi$
$757$	−1.76049	−0.0639861	−0.0319931	+	0.999488i	$0.510185\pi$
$758$	8.56199	0.310986
$759$	4.12045	0.149563
$760$	0	0
$761$	−4.00984	−0.145357	−0.0726784	−	0.997355i	$-0.523155\pi$
$761$	−4.00984	−0.145357	−0.0726784	+	0.997355i	$0.523155\pi$
$762$	−44.3783	−1.60765
$763$	34.5718	1.25158
$764$	−23.0923	−0.835452
$765$	0	0
$766$	35.6731	1.28892
$767$	10.7556	0.388361
$768$	−2.90321	−0.104761
$769$	32.1432	1.15911	0.579557	−	0.814932i	$-0.303226\pi$
$769$	32.1432	1.15911	0.579557	+	0.814932i	$0.303226\pi$
$770$	0	0
$771$	25.4193	0.915453
$772$	10.8573	0.390762
$773$	−23.6588	−0.850947	−0.425474	−	0.904971i	$-0.639892\pi$
$773$	−23.6588	−0.850947	−0.425474	+	0.904971i	$0.639892\pi$
$774$	42.1289	1.51429
$775$	0	0
$776$	−12.2351	−0.439213
$777$	50.8385	1.82382
$778$	−8.30958	−0.297913
$779$	25.8983	0.927903
$780$	0	0
$781$	−6.28544	−0.224911
$782$	−2.75557	−0.0985389
$783$	19.5299	0.697941
$784$	12.6128	0.450459
$785$	0	0
$786$	28.4701	1.01550
$787$	11.3733	0.405416	0.202708	−	0.979239i	$-0.435026\pi$
$787$	11.3733	0.405416	0.202708	+	0.979239i	$0.435026\pi$
$788$	0.903212	0.0321756
$789$	−13.9269	−0.495810
$790$	0	0
$791$	30.9590	1.10077
$792$	12.3827	0.440000
$793$	−18.2351	−0.647546
$794$	−7.53972	−0.267575
$795$	0	0
$796$	−4.85728	−0.172162
$797$	−22.4429	−0.794969	−0.397485	−	0.917609i	$-0.630117\pi$
$797$	−22.4429	−0.794969	−0.397485	+	0.917609i	$0.630117\pi$
$798$	90.6548	3.20915
$799$	−49.5308	−1.75227
$800$	0	0
$801$	80.6548	2.84980
$802$	11.4193	0.403228
$803$	−35.3087	−1.24602
$804$	−24.0830	−0.849341
$805$	0	0
$806$	−4.90321	−0.172708
$807$	0.308193	0.0108489
$808$	13.6128	0.478898
$809$	−0.967881	−0.0340289	−0.0170144	−	0.999855i	$-0.505416\pi$
$809$	−0.967881	−0.0340289	−0.0170144	+	0.999855i	$0.505416\pi$
$810$	0	0
$811$	−8.82870	−0.310018	−0.155009	−	0.987913i	$-0.549541\pi$
$811$	−8.82870	−0.310018	−0.155009	+	0.987913i	$0.549541\pi$
$812$	−12.2667	−0.430477
$813$	−55.3087	−1.93976
$814$	−9.01921	−0.316123
$815$	0	0
$816$	−12.8573	−0.450095
$817$	−54.7181	−1.91434
$818$	−2.94914	−0.103114
$819$	117.881	4.11908
$820$	0	0
$821$	6.35413	0.221761	0.110880	−	0.993834i	$-0.464633\pi$
$821$	6.35413	0.221761	0.110880	+	0.993834i	$0.464633\pi$
$822$	58.1847	2.02942
$823$	−1.10123	−0.0383866	−0.0191933	−	0.999816i	$-0.506110\pi$
$823$	−1.10123	−0.0383866	−0.0191933	+	0.999816i	$0.506110\pi$
$824$	−11.6128	−0.404553
$825$	0	0
$826$	−9.71456	−0.338013
$827$	27.1985	0.945784	0.472892	−	0.881120i	$-0.343210\pi$
$827$	27.1985	0.945784	0.472892	+	0.881120i	$0.343210\pi$
$828$	3.37778	0.117386
$829$	−46.3738	−1.61063	−0.805315	−	0.592848i	$-0.798004\pi$
$829$	−46.3738	−1.61063	−0.805315	+	0.592848i	$0.798004\pi$
$830$	0	0
$831$	−46.5847	−1.61601
$832$	4.90321	0.169988
$833$	55.8578	1.93536
$834$	−5.84791	−0.202497
$835$	0	0
$836$	−16.0830	−0.556241
$837$	−7.05086	−0.243713
$838$	1.15257	0.0398147
$839$	2.03164	0.0701401	0.0350701	−	0.999385i	$-0.488835\pi$
$839$	2.03164	0.0701401	0.0350701	+	0.999385i	$0.488835\pi$
$840$	0	0
$841$	−21.3279	−0.735444
$842$	24.0098	0.827434
$843$	−38.3970	−1.32246
$844$	10.4889	0.361042
$845$	0	0
$846$	60.7150	2.08743
$847$	−25.6731	−0.882137
$848$	0.0459330	0.00157734
$849$	23.7333	0.814525
$850$	0	0
$851$	−2.46028	−0.0843374
$852$	−8.00000	−0.274075
$853$	11.8983	0.407390	0.203695	−	0.979034i	$-0.434705\pi$
$853$	11.8983	0.407390	0.203695	+	0.979034i	$0.434705\pi$
$854$	16.4701	0.563596
$855$	0	0
$856$	−6.10171	−0.208552
$857$	35.2070	1.20265	0.601323	−	0.799006i	$-0.294640\pi$
$857$	35.2070	1.20265	0.601323	+	0.799006i	$0.294640\pi$
$858$	−32.4701	−1.10851
$859$	−22.4844	−0.767158	−0.383579	−	0.923508i	$-0.625309\pi$
$859$	−22.4844	−0.767158	−0.383579	+	0.923508i	$0.625309\pi$
$860$	0	0
$861$	−47.2257	−1.60945
$862$	−13.4608	−0.458475
$863$	−17.1842	−0.584957	−0.292479	−	0.956272i	$-0.594480\pi$
$863$	−17.1842	−0.584957	−0.292479	+	0.956272i	$0.594480\pi$
$864$	7.05086	0.239875
$865$	0	0
$866$	0.815792	0.0277217
$867$	−7.58565	−0.257622
$868$	4.42864	0.150318
$869$	30.6093	1.03835
$870$	0	0
$871$	40.6735	1.37817
$872$	−7.80642	−0.264359
$873$	66.4197	2.24797
$874$	−4.38715	−0.148398
$875$	0	0
$876$	−44.9403	−1.51839
$877$	8.58073	0.289751	0.144875	−	0.989450i	$-0.453722\pi$
$877$	8.58073	0.289751	0.144875	+	0.989450i	$0.453722\pi$
$878$	5.37778	0.181491
$879$	15.7877	0.532505
$880$	0	0
$881$	54.1659	1.82490	0.912449	−	0.409191i	$-0.134189\pi$
$881$	54.1659	1.82490	0.912449	+	0.409191i	$0.134189\pi$
$882$	−68.4706	−2.30553
$883$	40.2306	1.35387	0.676934	−	0.736044i	$-0.263308\pi$
$883$	40.2306	1.35387	0.676934	+	0.736044i	$0.263308\pi$
$884$	21.7146	0.730340
$885$	0	0
$886$	28.8573	0.969479
$887$	−13.1655	−0.442053	−0.221027	−	0.975268i	$-0.570941\pi$
$887$	−13.1655	−0.442053	−0.221027	+	0.975268i	$0.570941\pi$
$888$	−11.4795	−0.385227
$889$	−67.6958	−2.27045
$890$	0	0
$891$	−9.54416	−0.319742
$892$	−17.6543	−0.591111
$893$	−78.8582	−2.63889
$894$	53.8894	1.80233
$895$	0	0
$896$	−4.42864	−0.147950
$897$	−8.85728	−0.295736
$898$	−17.0509	−0.568994
$899$	−2.76986	−0.0923800
$900$	0	0
$901$	0.203420	0.00677692
$902$	8.37826	0.278966
$903$	99.7788	3.32043
$904$	−6.99063	−0.232505
$905$	0	0
$906$	−12.4701	−0.414292
$907$	48.4068	1.60732	0.803661	−	0.595087i	$-0.202882\pi$
$907$	48.4068	1.60732	0.803661	+	0.595087i	$0.202882\pi$
$908$	19.5210	0.647827
$909$	−73.8992	−2.45108
$910$	0	0
$911$	−18.7556	−0.621400	−0.310700	−	0.950508i	$-0.600563\pi$
$911$	−18.7556	−0.621400	−0.310700	+	0.950508i	$0.600563\pi$
$912$	−20.4701	−0.677834
$913$	22.7052	0.751432
$914$	−29.7560	−0.984242
$915$	0	0
$916$	−19.8938	−0.657311
$917$	43.4291	1.43416
$918$	31.2257	1.03060
$919$	43.3461	1.42986	0.714929	−	0.699197i	$-0.246459\pi$
$919$	43.3461	1.42986	0.714929	+	0.699197i	$0.246459\pi$
$920$	0	0
$921$	−36.2034	−1.19294
$922$	−24.4844	−0.806351
$923$	13.5111	0.444725
$924$	29.3274	0.964801
$925$	0	0
$926$	−21.0923	−0.693137
$927$	63.0420	2.07057
$928$	2.76986	0.0909251
$929$	−10.8287	−0.355278	−0.177639	−	0.984096i	$-0.556846\pi$
$929$	−10.8287	−0.355278	−0.177639	+	0.984096i	$0.556846\pi$
$930$	0	0
$931$	88.9314	2.91461
$932$	−7.11108	−0.232931
$933$	−57.7146	−1.88949
$934$	8.00000	0.261768
$935$	0	0
$936$	−26.6178	−0.870029
$937$	−31.2070	−1.01949	−0.509743	−	0.860326i	$-0.670260\pi$
$937$	−31.2070	−1.01949	−0.509743	+	0.860326i	$0.670260\pi$
$938$	−36.7368	−1.19950
$939$	−57.0607	−1.86211
$940$	0	0
$941$	4.69673	0.153109	0.0765545	−	0.997065i	$-0.475608\pi$
$941$	4.69673	0.153109	0.0765545	+	0.997065i	$0.475608\pi$
$942$	−22.3970	−0.729734
$943$	2.28544	0.0744242
$944$	2.19358	0.0713948
$945$	0	0
$946$	−17.7017	−0.575530
$947$	−16.4715	−0.535252	−0.267626	−	0.963523i	$-0.586239\pi$
$947$	−16.4715	−0.535252	−0.267626	+	0.963523i	$0.586239\pi$
$948$	38.9590	1.26533
$949$	75.8992	2.46379
$950$	0	0
$951$	−30.6637	−0.994339
$952$	−19.6128	−0.635656
$953$	−15.3876	−0.498454	−0.249227	−	0.968445i	$-0.580177\pi$
$953$	−15.3876	−0.498454	−0.249227	+	0.968445i	$0.580177\pi$
$954$	−0.249353	−0.00807312
$955$	0	0
$956$	7.52098	0.243246
$957$	−18.3426	−0.592932
$958$	1.04101	0.0336335
$959$	88.7565	2.86610
$960$	0	0
$961$	1.00000	0.0322581
$962$	19.3876	0.625082
$963$	33.1240	1.06741
$964$	10.5620	0.340179
$965$	0	0
$966$	8.00000	0.257396
$967$	−12.2636	−0.394372	−0.197186	−	0.980366i	$-0.563180\pi$
$967$	−12.2636	−0.394372	−0.197186	+	0.980366i	$0.563180\pi$
$968$	5.79706	0.186324
$969$	−90.6548	−2.91225
$970$	0	0
$971$	−19.5496	−0.627375	−0.313688	−	0.949526i	$-0.601564\pi$
$971$	−19.5496	−0.627375	−0.313688	+	0.949526i	$0.601564\pi$
$972$	9.00492	0.288833
$973$	−8.92056	−0.285980
$974$	−15.9398	−0.510743
$975$	0	0
$976$	−3.71900	−0.119042
$977$	−44.8385	−1.43451	−0.717256	−	0.696810i	$-0.754602\pi$
$977$	−44.8385	−1.43451	−0.717256	+	0.696810i	$0.754602\pi$
$978$	−64.6735	−2.06803
$979$	−33.8894	−1.08311
$980$	0	0
$981$	42.3783	1.35303
$982$	39.8751	1.27247
$983$	−21.0005	−0.669811	−0.334906	−	0.942252i	$-0.608704\pi$
$983$	−21.0005	−0.669811	−0.334906	+	0.942252i	$0.608704\pi$
$984$	10.6637	0.339946
$985$	0	0
$986$	12.2667	0.390652
$987$	143.798	4.57716
$988$	34.5718	1.09988
$989$	−4.82870	−0.153544
$990$	0	0
$991$	29.1240	0.925154	0.462577	−	0.886579i	$-0.346925\pi$
$991$	29.1240	0.925154	0.462577	+	0.886579i	$0.346925\pi$
$992$	−1.00000	−0.0317500
$993$	25.1111	0.796876
$994$	−12.2034	−0.387069
$995$	0	0
$996$	28.8988	0.915692
$997$	−57.2168	−1.81207	−0.906037	−	0.423198i	$-0.860907\pi$
$997$	−57.2168	−1.81207	−0.906037	+	0.423198i	$0.860907\pi$
$998$	−18.4558	−0.584209
$999$	27.8796	0.882070

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1550.2.a.k.1.1		3
5.2	odd	4		1550.2.b.j.249.3		6
5.3	odd	4		1550.2.b.j.249.4		6
5.4	even	2		310.2.a.e.1.3	✓	3
15.14	odd	2		2790.2.a.bi.1.1		3
20.19	odd	2		2480.2.a.u.1.1		3
40.19	odd	2		9920.2.a.bx.1.3		3
40.29	even	2		9920.2.a.bw.1.1		3
155.154	odd	2		9610.2.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.a.e.1.3	✓	3	5.4	even	2
1550.2.a.k.1.1		3	1.1	even	1	trivial
1550.2.b.j.249.3		6	5.2	odd	4
1550.2.b.j.249.4		6	5.3	odd	4
2480.2.a.u.1.1		3	20.19	odd	2
2790.2.a.bi.1.1		3	15.14	odd	2
9610.2.a.u.1.1		3	155.154	odd	2
9920.2.a.bw.1.1		3	40.29	even	2
9920.2.a.bx.1.3		3	40.19	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:	\( N \)	\(=\)	\( 1550 = 2 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.90321 q^{3} +1.00000 q^{4} +2.90321 q^{6} +4.42864 q^{7} -1.00000 q^{8} +5.42864 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.90321 q^{3} +1.00000 q^{4} +2.90321 q^{6} +4.42864 q^{7} -1.00000 q^{8} +5.42864 q^{9} -2.28100 q^{11} -2.90321 q^{12} +4.90321 q^{13} -4.42864 q^{14} +1.00000 q^{16} +4.42864 q^{17} -5.42864 q^{18} +7.05086 q^{19} -12.8573 q^{21} +2.28100 q^{22} +0.622216 q^{23} +2.90321 q^{24} -4.90321 q^{26} -7.05086 q^{27} +4.42864 q^{28} -2.76986 q^{29} +1.00000 q^{31} -1.00000 q^{32} +6.62222 q^{33} -4.42864 q^{34} +5.42864 q^{36} -3.95407 q^{37} -7.05086 q^{38} -14.2351 q^{39} +3.67307 q^{41} +12.8573 q^{42} -7.76049 q^{43} -2.28100 q^{44} -0.622216 q^{46} -11.1842 q^{47} -2.90321 q^{48} +12.6128 q^{49} -12.8573 q^{51} +4.90321 q^{52} +0.0459330 q^{53} +7.05086 q^{54} -4.42864 q^{56} -20.4701 q^{57} +2.76986 q^{58} +2.19358 q^{59} -3.71900 q^{61} -1.00000 q^{62} +24.0415 q^{63} +1.00000 q^{64} -6.62222 q^{66} +8.29529 q^{67} +4.42864 q^{68} -1.80642 q^{69} +2.75557 q^{71} -5.42864 q^{72} +15.4795 q^{73} +3.95407 q^{74} +7.05086 q^{76} -10.1017 q^{77} +14.2351 q^{78} -13.4193 q^{79} +4.18421 q^{81} -3.67307 q^{82} -9.95407 q^{83} -12.8573 q^{84} +7.76049 q^{86} +8.04149 q^{87} +2.28100 q^{88} +14.8573 q^{89} +21.7146 q^{91} +0.622216 q^{92} -2.90321 q^{93} +11.1842 q^{94} +2.90321 q^{96} +12.2351 q^{97} -12.6128 q^{98} -12.3827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^6 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} - 3 q^{8} + 3 q^{9} - 2 q^{12} + 8 q^{13} + 3 q^{16} - 3 q^{18} + 8 q^{19} - 12 q^{21} + 2 q^{23} + 2 q^{24} - 8 q^{26} - 8 q^{27} - 2 q^{29} + 3 q^{31} - 3 q^{32} + 20 q^{33} + 3 q^{36} + 8 q^{37} - 8 q^{38} - 16 q^{39} - 2 q^{41} + 12 q^{42} + 10 q^{43} - 2 q^{46} - 20 q^{47} - 2 q^{48} + 11 q^{49} - 12 q^{51} + 8 q^{52} + 20 q^{53} + 8 q^{54} - 8 q^{57} + 2 q^{58} + 20 q^{59} - 18 q^{61} - 3 q^{62} + 32 q^{63} + 3 q^{64} - 20 q^{66} + 12 q^{67} + 8 q^{69} + 8 q^{71} - 3 q^{72} + 20 q^{73} - 8 q^{74} + 8 q^{76} - 4 q^{77} + 16 q^{78} - q^{81} + 2 q^{82} - 10 q^{83} - 12 q^{84} - 10 q^{86} - 16 q^{87} + 18 q^{89} + 12 q^{91} + 2 q^{92} - 2 q^{93} + 20 q^{94} + 2 q^{96} + 10 q^{97} - 11 q^{98} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^6 - 3 * q^8 + 3 * q^9 - 2 * q^12 + 8 * q^13 + 3 * q^16 - 3 * q^18 + 8 * q^19 - 12 * q^21 + 2 * q^23 + 2 * q^24 - 8 * q^26 - 8 * q^27 - 2 * q^29 + 3 * q^31 - 3 * q^32 + 20 * q^33 + 3 * q^36 + 8 * q^37 - 8 * q^38 - 16 * q^39 - 2 * q^41 + 12 * q^42 + 10 * q^43 - 2 * q^46 - 20 * q^47 - 2 * q^48 + 11 * q^49 - 12 * q^51 + 8 * q^52 + 20 * q^53 + 8 * q^54 - 8 * q^57 + 2 * q^58 + 20 * q^59 - 18 * q^61 - 3 * q^62 + 32 * q^63 + 3 * q^64 - 20 * q^66 + 12 * q^67 + 8 * q^69 + 8 * q^71 - 3 * q^72 + 20 * q^73 - 8 * q^74 + 8 * q^76 - 4 * q^77 + 16 * q^78 - q^81 + 2 * q^82 - 10 * q^83 - 12 * q^84 - 10 * q^86 - 16 * q^87 + 18 * q^89 + 12 * q^91 + 2 * q^92 - 2 * q^93 + 20 * q^94 + 2 * q^96 + 10 * q^97 - 11 * q^98 - 4 * q^99

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