LMFDB - Embedded newform 8025.2.a.bc.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8025,2,Mod(1,8025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8025 = 3 \cdot 5^{2} \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8025.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:	$64.0799476221$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 13x^{8} + 26x^{7} + 51x^{6} - 101x^{5} - 65x^{4} + 126x^{3} + 5x^{2} - 10x + 1 $ x^10 - 2x^9 - 13x^8 + 26x^7 + 51x^6 - 101x^5 - 65x^4 + 126x^3 + 5x^2 - 10*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1605)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.66746$ of defining polynomial
Character	$\chi$	$=$	8025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.66746 q^{2} +1.00000 q^{3} +5.11534 q^{4} +2.66746 q^{6} -3.56475 q^{7} +8.31005 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.66746 q^{2} +1.00000 q^{3} +5.11534 q^{4} +2.66746 q^{6} -3.56475 q^{7} +8.31005 q^{8} +1.00000 q^{9} -4.56949 q^{11} +5.11534 q^{12} -1.90555 q^{13} -9.50884 q^{14} +11.9361 q^{16} -6.37815 q^{17} +2.66746 q^{18} -4.34379 q^{19} -3.56475 q^{21} -12.1889 q^{22} -0.445344 q^{23} +8.31005 q^{24} -5.08298 q^{26} +1.00000 q^{27} -18.2349 q^{28} -10.1747 q^{29} -7.04713 q^{31} +15.2188 q^{32} -4.56949 q^{33} -17.0135 q^{34} +5.11534 q^{36} +6.31431 q^{37} -11.5869 q^{38} -1.90555 q^{39} +7.36848 q^{41} -9.50884 q^{42} +11.5958 q^{43} -23.3745 q^{44} -1.18794 q^{46} -7.64542 q^{47} +11.9361 q^{48} +5.70746 q^{49} -6.37815 q^{51} -9.74755 q^{52} -8.72007 q^{53} +2.66746 q^{54} -29.6233 q^{56} -4.34379 q^{57} -27.1407 q^{58} -0.246820 q^{59} +12.7177 q^{61} -18.7979 q^{62} -3.56475 q^{63} +16.7235 q^{64} -12.1889 q^{66} -0.459577 q^{67} -32.6264 q^{68} -0.445344 q^{69} -2.77480 q^{71} +8.31005 q^{72} +6.45033 q^{73} +16.8432 q^{74} -22.2200 q^{76} +16.2891 q^{77} -5.08298 q^{78} -7.51271 q^{79} +1.00000 q^{81} +19.6551 q^{82} +1.44626 q^{83} -18.2349 q^{84} +30.9313 q^{86} -10.1747 q^{87} -37.9727 q^{88} +9.45126 q^{89} +6.79282 q^{91} -2.27809 q^{92} -7.04713 q^{93} -20.3939 q^{94} +15.2188 q^{96} +8.67043 q^{97} +15.2244 q^{98} -4.56949 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{6} + 10 q^{9}+O(q^{10}) $ 10 * q - 2 * q^2 + 10 * q^3 + 10 * q^4 - 2 * q^6 + 10 * q^9	$ 10 q - 2 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{6} + 10 q^{9} - 14 q^{11} + 10 q^{12} + 3 q^{13} - 16 q^{14} + 10 q^{16} - 8 q^{17} - 2 q^{18} - 19 q^{19} - 5 q^{22} - 4 q^{23} - 22 q^{26} + 10 q^{27} - 25 q^{29} - 2 q^{31} + 13 q^{32} - 14 q^{33} - 37 q^{34} + 10 q^{36} + 10 q^{37} + 13 q^{38} + 3 q^{39} - 31 q^{41} - 16 q^{42} - 62 q^{44} + 2 q^{46} + q^{47} + 10 q^{48} + 26 q^{49} - 8 q^{51} + 30 q^{52} - 9 q^{53} - 2 q^{54} - 63 q^{56} - 19 q^{57} - 30 q^{58} - 65 q^{59} + 12 q^{61} + 39 q^{62} - 2 q^{64} - 5 q^{66} + 10 q^{67} - 22 q^{68} - 4 q^{69} - 45 q^{71} - q^{73} + 19 q^{74} - 39 q^{76} + q^{77} - 22 q^{78} - 47 q^{79} + 10 q^{81} + 23 q^{82} - q^{83} + 12 q^{86} - 25 q^{87} + 8 q^{88} - 34 q^{89} - 26 q^{91} - 14 q^{92} - 2 q^{93} - 64 q^{94} + 13 q^{96} - 5 q^{97} + 51 q^{98} - 14 q^{99}+O(q^{100}) $ 10 * q - 2 * q^2 + 10 * q^3 + 10 * q^4 - 2 * q^6 + 10 * q^9 - 14 * q^11 + 10 * q^12 + 3 * q^13 - 16 * q^14 + 10 * q^16 - 8 * q^17 - 2 * q^18 - 19 * q^19 - 5 * q^22 - 4 * q^23 - 22 * q^26 + 10 * q^27 - 25 * q^29 - 2 * q^31 + 13 * q^32 - 14 * q^33 - 37 * q^34 + 10 * q^36 + 10 * q^37 + 13 * q^38 + 3 * q^39 - 31 * q^41 - 16 * q^42 - 62 * q^44 + 2 * q^46 + q^47 + 10 * q^48 + 26 * q^49 - 8 * q^51 + 30 * q^52 - 9 * q^53 - 2 * q^54 - 63 * q^56 - 19 * q^57 - 30 * q^58 - 65 * q^59 + 12 * q^61 + 39 * q^62 - 2 * q^64 - 5 * q^66 + 10 * q^67 - 22 * q^68 - 4 * q^69 - 45 * q^71 - q^73 + 19 * q^74 - 39 * q^76 + q^77 - 22 * q^78 - 47 * q^79 + 10 * q^81 + 23 * q^82 - q^83 + 12 * q^86 - 25 * q^87 + 8 * q^88 - 34 * q^89 - 26 * q^91 - 14 * q^92 - 2 * q^93 - 64 * q^94 + 13 * q^96 - 5 * q^97 + 51 * q^98 - 14 * 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 $

$2$	2.66746	1.88618	0.943090	−	0.332539i	$-0.107905\pi$
$2$	2.66746	1.88618	0.943090	+	0.332539i	$0.107905\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.11534	2.55767
$5$	0	0
$5$	0	0
$6$	2.66746	1.08899
$7$	−3.56475	−1.34735	−0.673675	−	0.739028i	$-0.735285\pi$
$7$	−3.56475	−1.34735	−0.673675	+	0.739028i	$0.735285\pi$
$8$	8.31005	2.93805
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.56949	−1.37775	−0.688876	−	0.724879i	$-0.741896\pi$
$11$	−4.56949	−1.37775	−0.688876	+	0.724879i	$0.741896\pi$
$12$	5.11534	1.47667
$13$	−1.90555	−0.528505	−0.264253	−	0.964454i	$-0.585125\pi$
$13$	−1.90555	−0.528505	−0.264253	+	0.964454i	$0.585125\pi$
$14$	−9.50884	−2.54134
$15$	0	0
$16$	11.9361	2.98401
$17$	−6.37815	−1.54693	−0.773464	−	0.633840i	$-0.781478\pi$
$17$	−6.37815	−1.54693	−0.773464	+	0.633840i	$0.781478\pi$
$18$	2.66746	0.628726
$19$	−4.34379	−0.996534	−0.498267	−	0.867024i	$-0.666030\pi$
$19$	−4.34379	−0.996534	−0.498267	+	0.867024i	$0.666030\pi$
$20$	0	0
$21$	−3.56475	−0.777893
$22$	−12.1889	−2.59869
$23$	−0.445344	−0.0928607	−0.0464303	−	0.998922i	$-0.514785\pi$
$23$	−0.445344	−0.0928607	−0.0464303	+	0.998922i	$0.514785\pi$
$24$	8.31005	1.69628
$25$	0	0
$26$	−5.08298	−0.996855
$27$	1.00000	0.192450
$28$	−18.2349	−3.44608
$29$	−10.1747	−1.88940	−0.944700	−	0.327935i	$-0.893647\pi$
$29$	−10.1747	−1.88940	−0.944700	+	0.327935i	$0.893647\pi$
$30$	0	0
$31$	−7.04713	−1.26570	−0.632851	−	0.774273i	$-0.718116\pi$
$31$	−7.04713	−1.26570	−0.632851	+	0.774273i	$0.718116\pi$
$32$	15.2188	2.69034
$33$	−4.56949	−0.795446
$34$	−17.0135	−2.91778
$35$	0	0
$36$	5.11534	0.852557
$37$	6.31431	1.03807	0.519033	−	0.854754i	$-0.326292\pi$
$37$	6.31431	1.03807	0.519033	+	0.854754i	$0.326292\pi$
$38$	−11.5869	−1.87964
$39$	−1.90555	−0.305133
$40$	0	0
$41$	7.36848	1.15076	0.575382	−	0.817885i	$-0.304853\pi$
$41$	7.36848	1.15076	0.575382	+	0.817885i	$0.304853\pi$
$42$	−9.50884	−1.46725
$43$	11.5958	1.76834	0.884170	−	0.467165i	$-0.154725\pi$
$43$	11.5958	1.76834	0.884170	+	0.467165i	$0.154725\pi$
$44$	−23.3745	−3.52384
$45$	0	0
$46$	−1.18794	−0.175152
$47$	−7.64542	−1.11520	−0.557600	−	0.830110i	$-0.688278\pi$
$47$	−7.64542	−1.11520	−0.557600	+	0.830110i	$0.688278\pi$
$48$	11.9361	1.72282
$49$	5.70746	0.815352
$50$	0	0
$51$	−6.37815	−0.893119
$52$	−9.74755	−1.35174
$53$	−8.72007	−1.19779	−0.598897	−	0.800826i	$-0.704394\pi$
$53$	−8.72007	−1.19779	−0.598897	+	0.800826i	$0.704394\pi$
$54$	2.66746	0.362995
$55$	0	0
$56$	−29.6233	−3.95858
$57$	−4.34379	−0.575349
$58$	−27.1407	−3.56375
$59$	−0.246820	−0.0321332	−0.0160666	−	0.999871i	$-0.505114\pi$
$59$	−0.246820	−0.0321332	−0.0160666	+	0.999871i	$0.505114\pi$
$60$	0	0
$61$	12.7177	1.62833	0.814165	−	0.580633i	$-0.197195\pi$
$61$	12.7177	1.62833	0.814165	+	0.580633i	$0.197195\pi$
$62$	−18.7979	−2.38734
$63$	−3.56475	−0.449117
$64$	16.7235	2.09044
$65$	0	0
$66$	−12.1889	−1.50035
$67$	−0.459577	−0.0561463	−0.0280731	−	0.999606i	$-0.508937\pi$
$67$	−0.459577	−0.0561463	−0.0280731	+	0.999606i	$0.508937\pi$
$68$	−32.6264	−3.95653
$69$	−0.445344	−0.0536131
$70$	0	0
$71$	−2.77480	−0.329308	−0.164654	−	0.986351i	$-0.552651\pi$
$71$	−2.77480	−0.329308	−0.164654	+	0.986351i	$0.552651\pi$
$72$	8.31005	0.979349
$73$	6.45033	0.754954	0.377477	−	0.926019i	$-0.376792\pi$
$73$	6.45033	0.754954	0.377477	+	0.926019i	$0.376792\pi$
$74$	16.8432	1.95798
$75$	0	0
$76$	−22.2200	−2.54881
$77$	16.2891	1.85631
$78$	−5.08298	−0.575535
$79$	−7.51271	−0.845246	−0.422623	−	0.906306i	$-0.638891\pi$
$79$	−7.51271	−0.845246	−0.422623	+	0.906306i	$0.638891\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	19.6551	2.17055
$83$	1.44626	0.158747	0.0793736	−	0.996845i	$-0.474708\pi$
$83$	1.44626	0.158747	0.0793736	+	0.996845i	$0.474708\pi$
$84$	−18.2349	−1.98959
$85$	0	0
$86$	30.9313	3.33541
$87$	−10.1747	−1.09085
$88$	−37.9727	−4.04790
$89$	9.45126	1.00183	0.500916	−	0.865496i	$-0.332997\pi$
$89$	9.45126	1.00183	0.500916	+	0.865496i	$0.332997\pi$
$90$	0	0
$91$	6.79282	0.712081
$92$	−2.27809	−0.237507
$93$	−7.04713	−0.730754
$94$	−20.3939	−2.10347
$95$	0	0
$96$	15.2188	1.55327
$97$	8.67043	0.880349	0.440174	−	0.897912i	$-0.354917\pi$
$97$	8.67043	0.880349	0.440174	+	0.897912i	$0.354917\pi$
$98$	15.2244	1.53790
$99$	−4.56949	−0.459251
$100$	0	0
$101$	−0.172036	−0.0171182	−0.00855911	−	0.999963i	$-0.502724\pi$
$101$	−0.172036	−0.0171182	−0.00855911	+	0.999963i	$0.502724\pi$
$102$	−17.0135	−1.68458
$103$	8.91888	0.878804	0.439402	−	0.898291i	$-0.355190\pi$
$103$	8.91888	0.878804	0.439402	+	0.898291i	$0.355190\pi$
$104$	−15.8352	−1.55277
$105$	0	0
$106$	−23.2604	−2.25925
$107$	1.00000	0.0966736
$107$	1.00000	0.0966736
$108$	5.11534	0.492224
$109$	8.68017	0.831410	0.415705	−	0.909500i	$-0.363535\pi$
$109$	8.68017	0.831410	0.415705	+	0.909500i	$0.363535\pi$
$110$	0	0
$111$	6.31431	0.599328
$112$	−42.5491	−4.02051
$113$	10.8072	1.01666	0.508330	−	0.861163i	$-0.330263\pi$
$113$	10.8072	1.01666	0.508330	+	0.861163i	$0.330263\pi$
$114$	−11.5869	−1.08521
$115$	0	0
$116$	−52.0473	−4.83247
$117$	−1.90555	−0.176168
$118$	−0.658381	−0.0606089
$119$	22.7365	2.08425
$120$	0	0
$121$	9.88021	0.898201
$122$	33.9239	3.07132
$123$	7.36848	0.664394
$124$	−36.0485	−3.23725
$125$	0	0
$126$	−9.50884	−0.847114
$127$	−17.0227	−1.51052	−0.755259	−	0.655426i	$-0.772489\pi$
$127$	−17.0227	−1.51052	−0.755259	+	0.655426i	$0.772489\pi$
$128$	14.1717	1.25261
$129$	11.5958	1.02095
$130$	0	0
$131$	−19.1532	−1.67342	−0.836711	−	0.547645i	$-0.815524\pi$
$131$	−19.1532	−1.67342	−0.836711	+	0.547645i	$0.815524\pi$
$132$	−23.3745	−2.03449
$133$	15.4845	1.34268
$134$	−1.22590	−0.105902
$135$	0	0
$136$	−53.0027	−4.54495
$137$	5.55052	0.474213	0.237107	−	0.971484i	$-0.423801\pi$
$137$	5.55052	0.474213	0.237107	+	0.971484i	$0.423801\pi$
$138$	−1.18794	−0.101124
$139$	19.4492	1.64966	0.824829	−	0.565382i	$-0.191271\pi$
$139$	19.4492	1.64966	0.824829	+	0.565382i	$0.191271\pi$
$140$	0	0
$141$	−7.64542	−0.643861
$142$	−7.40166	−0.621133
$143$	8.70740	0.728149
$144$	11.9361	0.994671
$145$	0	0
$146$	17.2060	1.42398
$147$	5.70746	0.470744
$148$	32.2999	2.65503
$149$	−22.5132	−1.84435	−0.922176	−	0.386770i	$-0.873591\pi$
$149$	−22.5132	−1.84435	−0.922176	+	0.386770i	$0.873591\pi$
$150$	0	0
$151$	−19.7812	−1.60977	−0.804887	−	0.593428i	$-0.797774\pi$
$151$	−19.7812	−1.60977	−0.804887	+	0.593428i	$0.797774\pi$
$152$	−36.0971	−2.92786
$153$	−6.37815	−0.515643
$154$	43.4505	3.50134
$155$	0	0
$156$	−9.74755	−0.780429
$157$	3.68999	0.294493	0.147246	−	0.989100i	$-0.452959\pi$
$157$	3.68999	0.294493	0.147246	+	0.989100i	$0.452959\pi$
$158$	−20.0399	−1.59428
$159$	−8.72007	−0.691546
$160$	0	0
$161$	1.58754	0.125116
$162$	2.66746	0.209575
$163$	−2.32086	−0.181784	−0.0908919	−	0.995861i	$-0.528972\pi$
$163$	−2.32086	−0.181784	−0.0908919	+	0.995861i	$0.528972\pi$
$164$	37.6923	2.94328
$165$	0	0
$166$	3.85783	0.299426
$167$	−3.22100	−0.249249	−0.124624	−	0.992204i	$-0.539773\pi$
$167$	−3.22100	−0.249249	−0.124624	+	0.992204i	$0.539773\pi$
$168$	−29.6233	−2.28549
$169$	−9.36887	−0.720682
$170$	0	0
$171$	−4.34379	−0.332178
$172$	59.3164	4.52283
$173$	5.83210	0.443406	0.221703	−	0.975114i	$-0.428838\pi$
$173$	5.83210	0.443406	0.221703	+	0.975114i	$0.428838\pi$
$174$	−27.1407	−2.05753
$175$	0	0
$176$	−54.5416	−4.11123
$177$	−0.246820	−0.0185521
$178$	25.2109	1.88963
$179$	−13.5557	−1.01320	−0.506599	−	0.862182i	$-0.669098\pi$
$179$	−13.5557	−1.01320	−0.506599	+	0.862182i	$0.669098\pi$
$180$	0	0
$181$	−19.2093	−1.42782	−0.713909	−	0.700239i	$-0.753077\pi$
$181$	−19.2093	−1.42782	−0.713909	+	0.700239i	$0.753077\pi$
$182$	18.1196	1.34311
$183$	12.7177	0.940117
$184$	−3.70083	−0.272829
$185$	0	0
$186$	−18.7979	−1.37833
$187$	29.1449	2.13128
$188$	−39.1090	−2.85231
$189$	−3.56475	−0.259298
$190$	0	0
$191$	−11.3052	−0.818018	−0.409009	−	0.912530i	$-0.634126\pi$
$191$	−11.3052	−0.818018	−0.409009	+	0.912530i	$0.634126\pi$
$192$	16.7235	1.20692
$193$	14.4576	1.04068	0.520340	−	0.853959i	$-0.325805\pi$
$193$	14.4576	1.04068	0.520340	+	0.853959i	$0.325805\pi$
$194$	23.1280	1.66050
$195$	0	0
$196$	29.1956	2.08540
$197$	17.0418	1.21418	0.607091	−	0.794633i	$-0.292336\pi$
$197$	17.0418	1.21418	0.607091	+	0.794633i	$0.292336\pi$
$198$	−12.1889	−0.866229
$199$	15.0846	1.06932	0.534660	−	0.845067i	$-0.320440\pi$
$199$	15.0846	1.06932	0.534660	+	0.845067i	$0.320440\pi$
$200$	0	0
$201$	−0.459577	−0.0324161
$202$	−0.458899	−0.0322880
$203$	36.2704	2.54568
$204$	−32.6264	−2.28431
$205$	0	0
$206$	23.7908	1.65758
$207$	−0.445344	−0.0309536
$208$	−22.7448	−1.57707
$209$	19.8489	1.37298
$210$	0	0
$211$	0.838053	0.0576940	0.0288470	−	0.999584i	$-0.490816\pi$
$211$	0.838053	0.0576940	0.0288470	+	0.999584i	$0.490816\pi$
$212$	−44.6061	−3.06356
$213$	−2.77480	−0.190126
$214$	2.66746	0.182344
$215$	0	0
$216$	8.31005	0.565428
$217$	25.1213	1.70534
$218$	23.1540	1.56819
$219$	6.45033	0.435873
$220$	0	0
$221$	12.1539	0.817559
$222$	16.8432	1.13044
$223$	−0.966702	−0.0647352	−0.0323676	−	0.999476i	$-0.510305\pi$
$223$	−0.966702	−0.0647352	−0.0323676	+	0.999476i	$0.510305\pi$
$224$	−54.2514	−3.62482
$225$	0	0
$226$	28.8279	1.91760
$227$	14.4582	0.959626	0.479813	−	0.877371i	$-0.340705\pi$
$227$	14.4582	0.959626	0.479813	+	0.877371i	$0.340705\pi$
$228$	−22.2200	−1.47155
$229$	16.8814	1.11555	0.557776	−	0.829991i	$-0.311655\pi$
$229$	16.8814	1.11555	0.557776	+	0.829991i	$0.311655\pi$
$230$	0	0
$231$	16.2891	1.07174
$232$	−84.5526	−5.55115
$233$	−4.13726	−0.271041	−0.135521	−	0.990775i	$-0.543271\pi$
$233$	−4.13726	−0.271041	−0.135521	+	0.990775i	$0.543271\pi$
$234$	−5.08298	−0.332285
$235$	0	0
$236$	−1.26257	−0.0821861
$237$	−7.51271	−0.488003
$238$	60.6488	3.93127
$239$	−7.55978	−0.489002	−0.244501	−	0.969649i	$-0.578624\pi$
$239$	−7.55978	−0.489002	−0.244501	+	0.969649i	$0.578624\pi$
$240$	0	0
$241$	−5.11686	−0.329606	−0.164803	−	0.986326i	$-0.552699\pi$
$241$	−5.11686	−0.329606	−0.164803	+	0.986326i	$0.552699\pi$
$242$	26.3551	1.69417
$243$	1.00000	0.0641500
$244$	65.0552	4.16473
$245$	0	0
$246$	19.6551	1.25317
$247$	8.27732	0.526673
$248$	−58.5621	−3.71869
$249$	1.44626	0.0916527
$250$	0	0
$251$	6.96610	0.439696	0.219848	−	0.975534i	$-0.429444\pi$
$251$	6.96610	0.439696	0.219848	+	0.975534i	$0.429444\pi$
$252$	−18.2349	−1.14869
$253$	2.03499	0.127939
$254$	−45.4073	−2.84911
$255$	0	0
$256$	4.35534	0.272208
$257$	−22.2354	−1.38701	−0.693503	−	0.720453i	$-0.743934\pi$
$257$	−22.2354	−1.38701	−0.693503	+	0.720453i	$0.743934\pi$
$258$	30.9313	1.92570
$259$	−22.5090	−1.39864
$260$	0	0
$261$	−10.1747	−0.629800
$262$	−51.0903	−3.15637
$263$	19.8664	1.22501	0.612507	−	0.790465i	$-0.290161\pi$
$263$	19.8664	1.22501	0.612507	+	0.790465i	$0.290161\pi$
$264$	−37.9727	−2.33706
$265$	0	0
$266$	41.3044	2.53253
$267$	9.45126	0.578408
$268$	−2.35089	−0.143604
$269$	−24.5877	−1.49914	−0.749568	−	0.661928i	$-0.769739\pi$
$269$	−24.5877	−1.49914	−0.749568	+	0.661928i	$0.769739\pi$
$270$	0	0
$271$	−18.4353	−1.11986	−0.559931	−	0.828539i	$-0.689172\pi$
$271$	−18.4353	−1.11986	−0.559931	+	0.828539i	$0.689172\pi$
$272$	−76.1299	−4.61605
$273$	6.79282	0.411120
$274$	14.8058	0.894451
$275$	0	0
$276$	−2.27809	−0.137125
$277$	−21.1478	−1.27065	−0.635323	−	0.772247i	$-0.719133\pi$
$277$	−21.1478	−1.27065	−0.635323	+	0.772247i	$0.719133\pi$
$278$	51.8799	3.11155
$279$	−7.04713	−0.421901
$280$	0	0
$281$	−25.0179	−1.49244	−0.746222	−	0.665697i	$-0.768134\pi$
$281$	−25.0179	−1.49244	−0.746222	+	0.665697i	$0.768134\pi$
$282$	−20.3939	−1.21444
$283$	−25.8922	−1.53913	−0.769564	−	0.638569i	$-0.779527\pi$
$283$	−25.8922	−1.53913	−0.769564	+	0.638569i	$0.779527\pi$
$284$	−14.1940	−0.842261
$285$	0	0
$286$	23.2266	1.37342
$287$	−26.2668	−1.55048
$288$	15.2188	0.896778
$289$	23.6808	1.39299
$290$	0	0
$291$	8.67043	0.508270
$292$	32.9957	1.93092
$293$	16.9587	0.990737	0.495368	−	0.868683i	$-0.335033\pi$
$293$	16.9587	0.990737	0.495368	+	0.868683i	$0.335033\pi$
$294$	15.2244	0.887907
$295$	0	0
$296$	52.4723	3.04989
$297$	−4.56949	−0.265149
$298$	−60.0530	−3.47878
$299$	0.848627	0.0490773
$300$	0	0
$301$	−41.3361	−2.38257
$302$	−52.7657	−3.03632
$303$	−0.172036	−0.00988321
$304$	−51.8477	−2.97367
$305$	0	0
$306$	−17.0135	−0.972594
$307$	4.72397	0.269611	0.134806	−	0.990872i	$-0.456959\pi$
$307$	4.72397	0.269611	0.134806	+	0.990872i	$0.456959\pi$
$308$	83.3243	4.74784
$309$	8.91888	0.507378
$310$	0	0
$311$	−9.10457	−0.516273	−0.258136	−	0.966108i	$-0.583108\pi$
$311$	−9.10457	−0.516273	−0.258136	+	0.966108i	$0.583108\pi$
$312$	−15.8352	−0.896494
$313$	6.83646	0.386420	0.193210	−	0.981157i	$-0.438110\pi$
$313$	6.83646	0.386420	0.193210	+	0.981157i	$0.438110\pi$
$314$	9.84289	0.555466
$315$	0	0
$316$	−38.4301	−2.16186
$317$	−23.0994	−1.29739	−0.648696	−	0.761048i	$-0.724685\pi$
$317$	−23.0994	−1.29739	−0.648696	+	0.761048i	$0.724685\pi$
$318$	−23.2604	−1.30438
$319$	46.4933	2.60313
$320$	0	0
$321$	1.00000	0.0558146
$322$	4.23470	0.235991
$323$	27.7053	1.54157
$324$	5.11534	0.284186
$325$	0	0
$326$	−6.19080	−0.342877
$327$	8.68017	0.480015
$328$	61.2325	3.38100
$329$	27.2540	1.50256
$330$	0	0
$331$	−28.6058	−1.57232	−0.786158	−	0.618025i	$-0.787933\pi$
$331$	−28.6058	−1.57232	−0.786158	+	0.618025i	$0.787933\pi$
$332$	7.39810	0.406023
$333$	6.31431	0.346022
$334$	−8.59190	−0.470128
$335$	0	0
$336$	−42.5491	−2.32124
$337$	−21.4987	−1.17111	−0.585553	−	0.810634i	$-0.699123\pi$
$337$	−21.4987	−1.17111	−0.585553	+	0.810634i	$0.699123\pi$
$338$	−24.9911	−1.35934
$339$	10.8072	0.586968
$340$	0	0
$341$	32.2018	1.74382
$342$	−11.5869	−0.626547
$343$	4.60758	0.248786
$344$	96.3616	5.19547
$345$	0	0
$346$	15.5569	0.836343
$347$	17.3535	0.931583	0.465791	−	0.884895i	$-0.345770\pi$
$347$	17.3535	0.931583	0.465791	+	0.884895i	$0.345770\pi$
$348$	−52.0473	−2.79003
$349$	7.65449	0.409736	0.204868	−	0.978790i	$-0.434324\pi$
$349$	7.65449	0.409736	0.204868	+	0.978790i	$0.434324\pi$
$350$	0	0
$351$	−1.90555	−0.101711
$352$	−69.5423	−3.70662
$353$	−8.32649	−0.443174	−0.221587	−	0.975141i	$-0.571124\pi$
$353$	−8.32649	−0.443174	−0.221587	+	0.975141i	$0.571124\pi$
$354$	−0.658381	−0.0349926
$355$	0	0
$356$	48.3464	2.56236
$357$	22.7365	1.20334
$358$	−36.1592	−1.91107
$359$	15.6498	0.825965	0.412983	−	0.910739i	$-0.364487\pi$
$359$	15.6498	0.825965	0.412983	+	0.910739i	$0.364487\pi$
$360$	0	0
$361$	−0.131485	−0.00692025
$362$	−51.2401	−2.69312
$363$	9.88021	0.518577
$364$	34.7476	1.82127
$365$	0	0
$366$	33.9239	1.77323
$367$	−17.7116	−0.924539	−0.462269	−	0.886740i	$-0.652965\pi$
$367$	−17.7116	−0.924539	−0.462269	+	0.886740i	$0.652965\pi$
$368$	−5.31565	−0.277098
$369$	7.36848	0.383588
$370$	0	0
$371$	31.0849	1.61385
$372$	−36.0485	−1.86903
$373$	−20.4818	−1.06051	−0.530253	−	0.847839i	$-0.677903\pi$
$373$	−20.4818	−1.06051	−0.530253	+	0.847839i	$0.677903\pi$
$374$	77.7427	4.01998
$375$	0	0
$376$	−63.5339	−3.27651
$377$	19.3885	0.998558
$378$	−9.50884	−0.489082
$379$	−28.4865	−1.46325	−0.731627	−	0.681705i	$-0.761239\pi$
$379$	−28.4865	−1.46325	−0.731627	+	0.681705i	$0.761239\pi$
$380$	0	0
$381$	−17.0227	−0.872098
$382$	−30.1563	−1.54293
$383$	9.23642	0.471959	0.235980	−	0.971758i	$-0.424170\pi$
$383$	9.23642	0.471959	0.235980	+	0.971758i	$0.424170\pi$
$384$	14.1717	0.723196
$385$	0	0
$386$	38.5650	1.96291
$387$	11.5958	0.589447
$388$	44.3522	2.25164
$389$	−25.7824	−1.30722	−0.653610	−	0.756831i	$-0.726746\pi$
$389$	−25.7824	−1.30722	−0.653610	+	0.756831i	$0.726746\pi$
$390$	0	0
$391$	2.84047	0.143649
$392$	47.4293	2.39554
$393$	−19.1532	−0.966150
$394$	45.4584	2.29016
$395$	0	0
$396$	−23.3745	−1.17461
$397$	−4.19812	−0.210697	−0.105349	−	0.994435i	$-0.533596\pi$
$397$	−4.19812	−0.210697	−0.105349	+	0.994435i	$0.533596\pi$
$398$	40.2376	2.01693
$399$	15.4845	0.775197
$400$	0	0
$401$	7.77294	0.388162	0.194081	−	0.980986i	$-0.437828\pi$
$401$	7.77294	0.388162	0.194081	+	0.980986i	$0.437828\pi$
$402$	−1.22590	−0.0611425
$403$	13.4287	0.668930
$404$	−0.880023	−0.0437828
$405$	0	0
$406$	96.7499	4.80162
$407$	−28.8532	−1.43020
$408$	−53.0027	−2.62403
$409$	−4.75315	−0.235028	−0.117514	−	0.993071i	$-0.537493\pi$
$409$	−4.75315	−0.235028	−0.117514	+	0.993071i	$0.537493\pi$
$410$	0	0
$411$	5.55052	0.273787
$412$	45.6232	2.24769
$413$	0.879851	0.0432946
$414$	−1.18794	−0.0583840
$415$	0	0
$416$	−29.0003	−1.42186
$417$	19.4492	0.952430
$418$	52.9461	2.58968
$419$	−4.37380	−0.213674	−0.106837	−	0.994277i	$-0.534072\pi$
$419$	−4.37380	−0.213674	−0.106837	+	0.994277i	$0.534072\pi$
$420$	0	0
$421$	−17.1919	−0.837883	−0.418941	−	0.908013i	$-0.637599\pi$
$421$	−17.1919	−0.837883	−0.418941	+	0.908013i	$0.637599\pi$
$422$	2.23547	0.108821
$423$	−7.64542	−0.371733
$424$	−72.4642	−3.51917
$425$	0	0
$426$	−7.40166	−0.358611
$427$	−45.3353	−2.19393
$428$	5.11534	0.247259
$429$	8.70740	0.420397
$430$	0	0
$431$	−27.6615	−1.33241	−0.666203	−	0.745770i	$-0.732082\pi$
$431$	−27.6615	−1.33241	−0.666203	+	0.745770i	$0.732082\pi$
$432$	11.9361	0.574274
$433$	4.92873	0.236860	0.118430	−	0.992962i	$-0.462214\pi$
$433$	4.92873	0.236860	0.118430	+	0.992962i	$0.462214\pi$
$434$	67.0100	3.21658
$435$	0	0
$436$	44.4021	2.12647
$437$	1.93448	0.0925388
$438$	17.2060	0.822135
$439$	25.7745	1.23015	0.615074	−	0.788469i	$-0.289126\pi$
$439$	25.7745	1.23015	0.615074	+	0.788469i	$0.289126\pi$
$440$	0	0
$441$	5.70746	0.271784
$442$	32.4200	1.54206
$443$	−32.9659	−1.56626	−0.783128	−	0.621860i	$-0.786377\pi$
$443$	−32.9659	−1.56626	−0.783128	+	0.621860i	$0.786377\pi$
$444$	32.2999	1.53288
$445$	0	0
$446$	−2.57864	−0.122102
$447$	−22.5132	−1.06484
$448$	−59.6153	−2.81656
$449$	−10.2858	−0.485418	−0.242709	−	0.970099i	$-0.578036\pi$
$449$	−10.2858	−0.485418	−0.242709	+	0.970099i	$0.578036\pi$
$450$	0	0
$451$	−33.6702	−1.58547
$452$	55.2827	2.60028
$453$	−19.7812	−0.929404
$454$	38.5667	1.81003
$455$	0	0
$456$	−36.0971	−1.69040
$457$	0.460686	0.0215500	0.0107750	−	0.999942i	$-0.496570\pi$
$457$	0.460686	0.0215500	0.0107750	+	0.999942i	$0.496570\pi$
$458$	45.0304	2.10413
$459$	−6.37815	−0.297706
$460$	0	0
$461$	−10.5627	−0.491952	−0.245976	−	0.969276i	$-0.579108\pi$
$461$	−10.5627	−0.491952	−0.245976	+	0.969276i	$0.579108\pi$
$462$	43.4505	2.02150
$463$	−22.5754	−1.04917	−0.524585	−	0.851358i	$-0.675779\pi$
$463$	−22.5754	−1.04917	−0.524585	+	0.851358i	$0.675779\pi$
$464$	−121.446	−5.63800
$465$	0	0
$466$	−11.0360	−0.511232
$467$	−24.0576	−1.11325	−0.556627	−	0.830763i	$-0.687905\pi$
$467$	−24.0576	−1.11325	−0.556627	+	0.830763i	$0.687905\pi$
$468$	−9.74755	−0.450581
$469$	1.63828	0.0756487
$470$	0	0
$471$	3.68999	0.170026
$472$	−2.05108	−0.0944088
$473$	−52.9868	−2.43633
$474$	−20.0399	−0.920461
$475$	0	0
$476$	116.305	5.33084
$477$	−8.72007	−0.399264
$478$	−20.1654	−0.922345
$479$	−20.5022	−0.936771	−0.468386	−	0.883524i	$-0.655164\pi$
$479$	−20.5022	−0.936771	−0.468386	+	0.883524i	$0.655164\pi$
$480$	0	0
$481$	−12.0322	−0.548623
$482$	−13.6490	−0.621696
$483$	1.58754	0.0722357
$484$	50.5407	2.29730
$485$	0	0
$486$	2.66746	0.120998
$487$	26.0869	1.18211	0.591055	−	0.806631i	$-0.298711\pi$
$487$	26.0869	1.18211	0.591055	+	0.806631i	$0.298711\pi$
$488$	105.684	4.78411
$489$	−2.32086	−0.104953
$490$	0	0
$491$	37.5668	1.69537	0.847684	−	0.530502i	$-0.177996\pi$
$491$	37.5668	1.69537	0.847684	+	0.530502i	$0.177996\pi$
$492$	37.6923	1.69930
$493$	64.8959	2.92277
$494$	22.0794	0.993400
$495$	0	0
$496$	−84.1149	−3.77687
$497$	9.89146	0.443693
$498$	3.85783	0.172873
$499$	12.3413	0.552470	0.276235	−	0.961090i	$-0.410913\pi$
$499$	12.3413	0.552470	0.276235	+	0.961090i	$0.410913\pi$
$500$	0	0
$501$	−3.22100	−0.143904
$502$	18.5818	0.829345
$503$	36.3381	1.62024	0.810118	−	0.586267i	$-0.199403\pi$
$503$	36.3381	1.62024	0.810118	+	0.586267i	$0.199403\pi$
$504$	−29.6233	−1.31953
$505$	0	0
$506$	5.42827	0.241316
$507$	−9.36887	−0.416086
$508$	−87.0768	−3.86341
$509$	26.7093	1.18387	0.591935	−	0.805986i	$-0.298364\pi$
$509$	26.7093	1.18387	0.591935	+	0.805986i	$0.298364\pi$
$510$	0	0
$511$	−22.9938	−1.01719
$512$	−16.7257	−0.739178
$513$	−4.34379	−0.191783
$514$	−59.3121	−2.61614
$515$	0	0
$516$	59.3164	2.61126
$517$	34.9357	1.53647
$518$	−60.0418	−2.63808
$519$	5.83210	0.256001
$520$	0	0
$521$	22.1279	0.969442	0.484721	−	0.874669i	$-0.338921\pi$
$521$	22.1279	0.969442	0.484721	+	0.874669i	$0.338921\pi$
$522$	−27.1407	−1.18792
$523$	−15.5809	−0.681307	−0.340654	−	0.940189i	$-0.610648\pi$
$523$	−15.5809	−0.681307	−0.340654	+	0.940189i	$0.610648\pi$
$524$	−97.9751	−4.28006
$525$	0	0
$526$	52.9928	2.31060
$527$	44.9476	1.95795
$528$	−54.5416	−2.37362
$529$	−22.8017	−0.991377
$530$	0	0
$531$	−0.246820	−0.0107111
$532$	79.2087	3.43413
$533$	−14.0410	−0.608185
$534$	25.2109	1.09098
$535$	0	0
$536$	−3.81911	−0.164960
$537$	−13.5557	−0.584970
$538$	−65.5866	−2.82764
$539$	−26.0802	−1.12335
$540$	0	0
$541$	−0.217428	−0.00934794	−0.00467397	−	0.999989i	$-0.501488\pi$
$541$	−0.217428	−0.00934794	−0.00467397	+	0.999989i	$0.501488\pi$
$542$	−49.1753	−2.11226
$543$	−19.2093	−0.824351
$544$	−97.0680	−4.16175
$545$	0	0
$546$	18.1196	0.775447
$547$	25.5890	1.09411	0.547054	−	0.837097i	$-0.315749\pi$
$547$	25.5890	1.09411	0.547054	+	0.837097i	$0.315749\pi$
$548$	28.3928	1.21288
$549$	12.7177	0.542777
$550$	0	0
$551$	44.1969	1.88285
$552$	−3.70083	−0.157518
$553$	26.7809	1.13884
$554$	−56.4108	−2.39666
$555$	0	0
$556$	99.4892	4.21928
$557$	7.09985	0.300830	0.150415	−	0.988623i	$-0.451939\pi$
$557$	7.09985	0.300830	0.150415	+	0.988623i	$0.451939\pi$
$558$	−18.7979	−0.795780
$559$	−22.0964	−0.934577
$560$	0	0
$561$	29.1449	1.23050
$562$	−66.7343	−2.81502
$563$	−7.09071	−0.298838	−0.149419	−	0.988774i	$-0.547740\pi$
$563$	−7.09071	−0.298838	−0.149419	+	0.988774i	$0.547740\pi$
$564$	−39.1090	−1.64678
$565$	0	0
$566$	−69.0663	−2.90307
$567$	−3.56475	−0.149706
$568$	−23.0587	−0.967522
$569$	−8.07933	−0.338703	−0.169351	−	0.985556i	$-0.554167\pi$
$569$	−8.07933	−0.338703	−0.169351	+	0.985556i	$0.554167\pi$
$570$	0	0
$571$	1.11473	0.0466500	0.0233250	−	0.999728i	$-0.492575\pi$
$571$	1.11473	0.0466500	0.0233250	+	0.999728i	$0.492575\pi$
$572$	44.5413	1.86237
$573$	−11.3052	−0.472283
$574$	−70.0657	−2.92449
$575$	0	0
$576$	16.7235	0.696814
$577$	41.5500	1.72975	0.864874	−	0.501989i	$-0.167398\pi$
$577$	41.5500	1.72975	0.864874	+	0.501989i	$0.167398\pi$
$578$	63.1675	2.62742
$579$	14.4576	0.600837
$580$	0	0
$581$	−5.15554	−0.213888
$582$	23.1280	0.958688
$583$	39.8462	1.65026
$584$	53.6026	2.21809
$585$	0	0
$586$	45.2366	1.86871
$587$	18.3684	0.758144	0.379072	−	0.925367i	$-0.376243\pi$
$587$	18.3684	0.758144	0.379072	+	0.925367i	$0.376243\pi$
$588$	29.1956	1.20401
$589$	30.6113	1.26132
$590$	0	0
$591$	17.0418	0.701008
$592$	75.3680	3.09760
$593$	−35.1309	−1.44265	−0.721326	−	0.692595i	$-0.756467\pi$
$593$	−35.1309	−1.44265	−0.721326	+	0.692595i	$0.756467\pi$
$594$	−12.1889	−0.500118
$595$	0	0
$596$	−115.163	−4.71725
$597$	15.0846	0.617373
$598$	2.26368	0.0925687
$599$	−3.25795	−0.133116	−0.0665582	−	0.997783i	$-0.521202\pi$
$599$	−3.25795	−0.133116	−0.0665582	+	0.997783i	$0.521202\pi$
$600$	0	0
$601$	2.33783	0.0953621	0.0476811	−	0.998863i	$-0.484817\pi$
$601$	2.33783	0.0953621	0.0476811	+	0.998863i	$0.484817\pi$
$602$	−110.262	−4.49396
$603$	−0.459577	−0.0187154
$604$	−101.188	−4.11728
$605$	0	0
$606$	−0.458899	−0.0186415
$607$	24.6335	0.999844	0.499922	−	0.866070i	$-0.333362\pi$
$607$	24.6335	0.999844	0.499922	+	0.866070i	$0.333362\pi$
$608$	−66.1074	−2.68101
$609$	36.2704	1.46975
$610$	0	0
$611$	14.5688	0.589389
$612$	−32.6264	−1.31884
$613$	−11.4810	−0.463715	−0.231857	−	0.972750i	$-0.574480\pi$
$613$	−11.4810	−0.463715	−0.231857	+	0.972750i	$0.574480\pi$
$614$	12.6010	0.508536
$615$	0	0
$616$	135.363	5.45394
$617$	−9.82376	−0.395490	−0.197745	−	0.980254i	$-0.563362\pi$
$617$	−9.82376	−0.395490	−0.197745	+	0.980254i	$0.563362\pi$
$618$	23.7908	0.957005
$619$	12.6302	0.507649	0.253825	−	0.967250i	$-0.418311\pi$
$619$	12.6302	0.507649	0.253825	+	0.967250i	$0.418311\pi$
$620$	0	0
$621$	−0.445344	−0.0178710
$622$	−24.2861	−0.973783
$623$	−33.6914	−1.34982
$624$	−22.7448	−0.910520
$625$	0	0
$626$	18.2360	0.728857
$627$	19.8489	0.792688
$628$	18.8755	0.753216
$629$	−40.2736	−1.60581
$630$	0	0
$631$	−28.2014	−1.12268	−0.561339	−	0.827586i	$-0.689714\pi$
$631$	−28.2014	−1.12268	−0.561339	+	0.827586i	$0.689714\pi$
$632$	−62.4310	−2.48337
$633$	0.838053	0.0333096
$634$	−61.6167	−2.44711
$635$	0	0
$636$	−44.6061	−1.76875
$637$	−10.8759	−0.430918
$638$	124.019	4.90996
$639$	−2.77480	−0.109769
$640$	0	0
$641$	5.28911	0.208907	0.104454	−	0.994530i	$-0.466691\pi$
$641$	5.28911	0.208907	0.104454	+	0.994530i	$0.466691\pi$
$642$	2.66746	0.105276
$643$	−27.3867	−1.08003	−0.540014	−	0.841656i	$-0.681581\pi$
$643$	−27.3867	−1.08003	−0.540014	+	0.841656i	$0.681581\pi$
$644$	8.12082	0.320005
$645$	0	0
$646$	73.9029	2.90767
$647$	−8.32676	−0.327359	−0.163679	−	0.986514i	$-0.552336\pi$
$647$	−8.32676	−0.327359	−0.163679	+	0.986514i	$0.552336\pi$
$648$	8.31005	0.326450
$649$	1.12784	0.0442715
$650$	0	0
$651$	25.1213	0.984581
$652$	−11.8720	−0.464943
$653$	−8.30077	−0.324834	−0.162417	−	0.986722i	$-0.551929\pi$
$653$	−8.30077	−0.324834	−0.162417	+	0.986722i	$0.551929\pi$
$654$	23.1540	0.905394
$655$	0	0
$656$	87.9506	3.43389
$657$	6.45033	0.251651
$658$	72.6991	2.83410
$659$	46.9058	1.82719	0.913595	−	0.406625i	$-0.133294\pi$
$659$	46.9058	1.82719	0.913595	+	0.406625i	$0.133294\pi$
$660$	0	0
$661$	−9.65524	−0.375545	−0.187773	−	0.982213i	$-0.560127\pi$
$661$	−9.65524	−0.375545	−0.187773	+	0.982213i	$0.560127\pi$
$662$	−76.3048	−2.96567
$663$	12.1539	0.472018
$664$	12.0185	0.466407
$665$	0	0
$666$	16.8432	0.652660
$667$	4.53126	0.175451
$668$	−16.4765	−0.637496
$669$	−0.966702	−0.0373749
$670$	0	0
$671$	−58.1132	−2.24344
$672$	−54.2514	−2.09279
$673$	42.3850	1.63382	0.816910	−	0.576765i	$-0.195685\pi$
$673$	42.3850	1.63382	0.816910	+	0.576765i	$0.195685\pi$
$674$	−57.3468	−2.20892
$675$	0	0
$676$	−47.9250	−1.84327
$677$	−12.5719	−0.483176	−0.241588	−	0.970379i	$-0.577668\pi$
$677$	−12.5719	−0.483176	−0.241588	+	0.970379i	$0.577668\pi$
$678$	28.8279	1.10713
$679$	−30.9079	−1.18614
$680$	0	0
$681$	14.4582	0.554040
$682$	85.8970	3.28917
$683$	−7.21703	−0.276152	−0.138076	−	0.990422i	$-0.544092\pi$
$683$	−7.21703	−0.276152	−0.138076	+	0.990422i	$0.544092\pi$
$684$	−22.2200	−0.849602
$685$	0	0
$686$	12.2905	0.469254
$687$	16.8814	0.644064
$688$	138.408	5.27675
$689$	16.6165	0.633040
$690$	0	0
$691$	8.45731	0.321731	0.160866	−	0.986976i	$-0.448571\pi$
$691$	8.45731	0.321731	0.160866	+	0.986976i	$0.448571\pi$
$692$	29.8332	1.13409
$693$	16.2891	0.618771
$694$	46.2897	1.75713
$695$	0	0
$696$	−84.5526	−3.20496
$697$	−46.9973	−1.78015
$698$	20.4181	0.772835
$699$	−4.13726	−0.156486
$700$	0	0
$701$	25.7866	0.973948	0.486974	−	0.873417i	$-0.338101\pi$
$701$	25.7866	0.973948	0.486974	+	0.873417i	$0.338101\pi$
$702$	−5.08298	−0.191845
$703$	−27.4280	−1.03447
$704$	−76.4180	−2.88011
$705$	0	0
$706$	−22.2106	−0.835906
$707$	0.613266	0.0230642
$708$	−1.26257	−0.0474502
$709$	24.5139	0.920638	0.460319	−	0.887753i	$-0.347735\pi$
$709$	24.5139	0.920638	0.460319	+	0.887753i	$0.347735\pi$
$710$	0	0
$711$	−7.51271	−0.281749
$712$	78.5405	2.94343
$713$	3.13840	0.117534
$714$	60.6488	2.26972
$715$	0	0
$716$	−69.3419	−2.59143
$717$	−7.55978	−0.282325
$718$	41.7452	1.55792
$719$	−26.7310	−0.996897	−0.498449	−	0.866919i	$-0.666097\pi$
$719$	−26.7310	−0.996897	−0.498449	+	0.866919i	$0.666097\pi$
$720$	0	0
$721$	−31.7936	−1.18406
$722$	−0.350730	−0.0130528
$723$	−5.11686	−0.190298
$724$	−98.2623	−3.65189
$725$	0	0
$726$	26.3551	0.978128
$727$	44.9993	1.66893	0.834466	−	0.551060i	$-0.185777\pi$
$727$	44.9993	1.66893	0.834466	+	0.551060i	$0.185777\pi$
$728$	56.4487	2.09213
$729$	1.00000	0.0370370
$730$	0	0
$731$	−73.9596	−2.73549
$732$	65.0552	2.40451
$733$	−17.2924	−0.638709	−0.319355	−	0.947635i	$-0.603466\pi$
$733$	−17.2924	−0.638709	−0.319355	+	0.947635i	$0.603466\pi$
$734$	−47.2450	−1.74385
$735$	0	0
$736$	−6.77762	−0.249826
$737$	2.10003	0.0773556
$738$	19.6551	0.723516
$739$	−21.6445	−0.796207	−0.398104	−	0.917340i	$-0.630332\pi$
$739$	−21.6445	−0.796207	−0.398104	+	0.917340i	$0.630332\pi$
$740$	0	0
$741$	8.27732	0.304075
$742$	82.9177	3.04400
$743$	17.9480	0.658449	0.329225	−	0.944252i	$-0.393213\pi$
$743$	17.9480	0.658449	0.329225	+	0.944252i	$0.393213\pi$
$744$	−58.5621	−2.14699
$745$	0	0
$746$	−54.6344	−2.00031
$747$	1.44626	0.0529157
$748$	149.086	5.45112
$749$	−3.56475	−0.130253
$750$	0	0
$751$	−27.6964	−1.01066	−0.505329	−	0.862927i	$-0.668629\pi$
$751$	−27.6964	−1.01066	−0.505329	+	0.862927i	$0.668629\pi$
$752$	−91.2562	−3.32777
$753$	6.96610	0.253859
$754$	51.7180	1.88346
$755$	0	0
$756$	−18.2349	−0.663198
$757$	−19.9840	−0.726331	−0.363165	−	0.931725i	$-0.618304\pi$
$757$	−19.9840	−0.726331	−0.363165	+	0.931725i	$0.618304\pi$
$758$	−75.9867	−2.75996
$759$	2.03499	0.0738656
$760$	0	0
$761$	31.3101	1.13499	0.567495	−	0.823377i	$-0.307913\pi$
$761$	31.3101	1.13499	0.567495	+	0.823377i	$0.307913\pi$
$762$	−45.4073	−1.64493
$763$	−30.9427	−1.12020
$764$	−57.8302	−2.09222
$765$	0	0
$766$	24.6378	0.890200
$767$	0.470328	0.0169825
$768$	4.35534	0.157160
$769$	21.3203	0.768829	0.384415	−	0.923161i	$-0.374403\pi$
$769$	21.3203	0.768829	0.384415	+	0.923161i	$0.374403\pi$
$770$	0	0
$771$	−22.2354	−0.800789
$772$	73.9555	2.66172
$773$	−9.54666	−0.343370	−0.171685	−	0.985152i	$-0.554921\pi$
$773$	−9.54666	−0.343370	−0.171685	+	0.985152i	$0.554921\pi$
$774$	30.9313	1.11180
$775$	0	0
$776$	72.0518	2.58651
$777$	−22.5090	−0.807504
$778$	−68.7736	−2.46565
$779$	−32.0072	−1.14678
$780$	0	0
$781$	12.6794	0.453704
$782$	7.57684	0.270947
$783$	−10.1747	−0.363615
$784$	68.1246	2.43302
$785$	0	0
$786$	−51.0903	−1.82233
$787$	−14.1468	−0.504279	−0.252139	−	0.967691i	$-0.581134\pi$
$787$	−14.1468	−0.504279	−0.252139	+	0.967691i	$0.581134\pi$
$788$	87.1749	3.10548
$789$	19.8664	0.707262
$790$	0	0
$791$	−38.5251	−1.36980
$792$	−37.9727	−1.34930
$793$	−24.2342	−0.860581
$794$	−11.1983	−0.397413
$795$	0	0
$796$	77.1630	2.73497
$797$	23.0209	0.815441	0.407720	−	0.913107i	$-0.366324\pi$
$797$	23.0209	0.815441	0.407720	+	0.913107i	$0.366324\pi$
$798$	41.3044	1.46216
$799$	48.7636	1.72513
$800$	0	0
$801$	9.45126	0.333944
$802$	20.7340	0.732143
$803$	−29.4747	−1.04014
$804$	−2.35089	−0.0829096
$805$	0	0
$806$	35.8205	1.26172
$807$	−24.5877	−0.865526
$808$	−1.42963	−0.0502942
$809$	34.0933	1.19866	0.599329	−	0.800503i	$-0.295434\pi$
$809$	34.0933	1.19866	0.599329	+	0.800503i	$0.295434\pi$
$810$	0	0
$811$	8.03982	0.282316	0.141158	−	0.989987i	$-0.454917\pi$
$811$	8.03982	0.282316	0.141158	+	0.989987i	$0.454917\pi$
$812$	185.536	6.51102
$813$	−18.4353	−0.646553
$814$	−76.9647	−2.69761
$815$	0	0
$816$	−76.1299	−2.66508
$817$	−50.3696	−1.76221
$818$	−12.6788	−0.443305
$819$	6.79282	0.237360
$820$	0	0
$821$	−25.7752	−0.899559	−0.449780	−	0.893140i	$-0.648497\pi$
$821$	−25.7752	−0.899559	−0.449780	+	0.893140i	$0.648497\pi$
$822$	14.8058	0.516411
$823$	44.5116	1.55158	0.775789	−	0.630993i	$-0.217352\pi$
$823$	44.5116	1.55158	0.775789	+	0.630993i	$0.217352\pi$
$824$	74.1164	2.58197
$825$	0	0
$826$	2.34697	0.0816614
$827$	34.9797	1.21636	0.608181	−	0.793798i	$-0.291900\pi$
$827$	34.9797	1.21636	0.608181	+	0.793798i	$0.291900\pi$
$828$	−2.27809	−0.0791691
$829$	8.95653	0.311073	0.155537	−	0.987830i	$-0.450289\pi$
$829$	8.95653	0.311073	0.155537	+	0.987830i	$0.450289\pi$
$830$	0	0
$831$	−21.1478	−0.733607
$832$	−31.8676	−1.10481
$833$	−36.4030	−1.26129
$834$	51.8799	1.79645
$835$	0	0
$836$	101.534	3.51162
$837$	−7.04713	−0.243585
$838$	−11.6669	−0.403027
$839$	−13.8538	−0.478288	−0.239144	−	0.970984i	$-0.576867\pi$
$839$	−13.8538	−0.478288	−0.239144	+	0.970984i	$0.576867\pi$
$840$	0	0
$841$	74.5252	2.56984
$842$	−45.8588	−1.58040
$843$	−25.0179	−0.861663
$844$	4.28693	0.147562
$845$	0	0
$846$	−20.3939	−0.701155
$847$	−35.2205	−1.21019
$848$	−104.083	−3.57423
$849$	−25.8922	−0.888616
$850$	0	0
$851$	−2.81204	−0.0963956
$852$	−14.1940	−0.486280
$853$	−18.0848	−0.619211	−0.309606	−	0.950865i	$-0.600197\pi$
$853$	−18.0848	−0.619211	−0.309606	+	0.950865i	$0.600197\pi$
$854$	−120.930	−4.13815
$855$	0	0
$856$	8.31005	0.284032
$857$	4.13507	0.141251	0.0706257	−	0.997503i	$-0.477500\pi$
$857$	4.13507	0.141251	0.0706257	+	0.997503i	$0.477500\pi$
$858$	23.2266	0.792944
$859$	−35.2176	−1.20161	−0.600805	−	0.799395i	$-0.705153\pi$
$859$	−35.2176	−1.20161	−0.600805	+	0.799395i	$0.705153\pi$
$860$	0	0
$861$	−26.2668	−0.895171
$862$	−73.7859	−2.51316
$863$	−25.5384	−0.869337	−0.434668	−	0.900591i	$-0.643134\pi$
$863$	−25.5384	−0.869337	−0.434668	+	0.900591i	$0.643134\pi$
$864$	15.2188	0.517755
$865$	0	0
$866$	13.1472	0.446760
$867$	23.6808	0.804241
$868$	128.504	4.36171
$869$	34.3292	1.16454
$870$	0	0
$871$	0.875748	0.0296736
$872$	72.1327	2.44272
$873$	8.67043	0.293450
$874$	5.16015	0.174545
$875$	0	0
$876$	32.9957	1.11482
$877$	−47.5286	−1.60493	−0.802463	−	0.596702i	$-0.796477\pi$
$877$	−47.5286	−1.60493	−0.802463	+	0.596702i	$0.796477\pi$
$878$	68.7523	2.32028
$879$	16.9587	0.572002
$880$	0	0
$881$	9.40438	0.316842	0.158421	−	0.987372i	$-0.449360\pi$
$881$	9.40438	0.316842	0.158421	+	0.987372i	$0.449360\pi$
$882$	15.2244	0.512633
$883$	−24.8808	−0.837304	−0.418652	−	0.908147i	$-0.637497\pi$
$883$	−24.8808	−0.837304	−0.418652	+	0.908147i	$0.637497\pi$
$884$	62.1713	2.09105
$885$	0	0
$886$	−87.9352	−2.95424
$887$	−52.8888	−1.77583	−0.887916	−	0.460005i	$-0.847847\pi$
$887$	−52.8888	−1.77583	−0.887916	+	0.460005i	$0.847847\pi$
$888$	52.4723	1.76085
$889$	60.6816	2.03520
$890$	0	0
$891$	−4.56949	−0.153084
$892$	−4.94501	−0.165571
$893$	33.2101	1.11133
$894$	−60.0530	−2.00847
$895$	0	0
$896$	−50.5185	−1.68771
$897$	0.848627	0.0283348
$898$	−27.4370	−0.915586
$899$	71.7027	2.39142
$900$	0	0
$901$	55.6179	1.85290
$902$	−89.8139	−2.99048
$903$	−41.3361	−1.37558
$904$	89.8087	2.98699
$905$	0	0
$906$	−52.7657	−1.75302
$907$	−7.77264	−0.258086	−0.129043	−	0.991639i	$-0.541191\pi$
$907$	−7.77264	−0.258086	−0.129043	+	0.991639i	$0.541191\pi$
$908$	73.9588	2.45441
$909$	−0.172036	−0.00570608
$910$	0	0
$911$	15.4556	0.512068	0.256034	−	0.966668i	$-0.417584\pi$
$911$	15.4556	0.512068	0.256034	+	0.966668i	$0.417584\pi$
$912$	−51.8477	−1.71685
$913$	−6.60865	−0.218714
$914$	1.22886	0.0406472
$915$	0	0
$916$	86.3540	2.85322
$917$	68.2764	2.25468
$918$	−17.0135	−0.561528
$919$	19.0460	0.628268	0.314134	−	0.949379i	$-0.398286\pi$
$919$	19.0460	0.628268	0.314134	+	0.949379i	$0.398286\pi$
$920$	0	0
$921$	4.72397	0.155660
$922$	−28.1755	−0.927909
$923$	5.28752	0.174041
$924$	83.3243	2.74117
$925$	0	0
$926$	−60.2191	−1.97892
$927$	8.91888	0.292935
$928$	−154.848	−5.08312
$929$	−47.4390	−1.55642	−0.778211	−	0.628003i	$-0.783872\pi$
$929$	−47.4390	−1.55642	−0.778211	+	0.628003i	$0.783872\pi$
$930$	0	0
$931$	−24.7920	−0.812526
$932$	−21.1635	−0.693234
$933$	−9.10457	−0.298070
$934$	−64.1727	−2.09980
$935$	0	0
$936$	−15.8352	−0.517591
$937$	0.293062	0.00957392	0.00478696	−	0.999989i	$-0.498476\pi$
$937$	0.293062	0.00957392	0.00478696	+	0.999989i	$0.498476\pi$
$938$	4.37004	0.142687
$939$	6.83646	0.223099
$940$	0	0
$941$	33.2503	1.08393	0.541964	−	0.840402i	$-0.317681\pi$
$941$	33.2503	1.08393	0.541964	+	0.840402i	$0.317681\pi$
$942$	9.84289	0.320699
$943$	−3.28151	−0.106861
$944$	−2.94605	−0.0958858
$945$	0	0
$946$	−141.340	−4.59536
$947$	−27.2128	−0.884297	−0.442148	−	0.896942i	$-0.645784\pi$
$947$	−27.2128	−0.884297	−0.442148	+	0.896942i	$0.645784\pi$
$948$	−38.4301	−1.24815
$949$	−12.2914	−0.398997
$950$	0	0
$951$	−23.0994	−0.749050
$952$	188.942	6.12364
$953$	1.63480	0.0529563	0.0264782	−	0.999649i	$-0.491571\pi$
$953$	1.63480	0.0529563	0.0264782	+	0.999649i	$0.491571\pi$
$954$	−23.2604	−0.753084
$955$	0	0
$956$	−38.6709	−1.25071
$957$	46.4933	1.50292
$958$	−54.6889	−1.76692
$959$	−19.7862	−0.638931
$960$	0	0
$961$	18.6621	0.602003
$962$	−32.0955	−1.03480
$963$	1.00000	0.0322245
$964$	−26.1745	−0.843024
$965$	0	0
$966$	4.23470	0.136249
$967$	24.1815	0.777624	0.388812	−	0.921317i	$-0.372886\pi$
$967$	24.1815	0.777624	0.388812	+	0.921317i	$0.372886\pi$
$968$	82.1051	2.63896
$969$	27.7053	0.890024
$970$	0	0
$971$	−11.2709	−0.361700	−0.180850	−	0.983511i	$-0.557885\pi$
$971$	−11.2709	−0.361700	−0.180850	+	0.983511i	$0.557885\pi$
$972$	5.11534	0.164075
$973$	−69.3315	−2.22267
$974$	69.5858	2.22967
$975$	0	0
$976$	151.799	4.85896
$977$	−0.715120	−0.0228787	−0.0114394	−	0.999935i	$-0.503641\pi$
$977$	−0.715120	−0.0228787	−0.0114394	+	0.999935i	$0.503641\pi$
$978$	−6.19080	−0.197960
$979$	−43.1874	−1.38028
$980$	0	0
$981$	8.68017	0.277137
$982$	100.208	3.19777
$983$	53.6823	1.71220	0.856100	−	0.516810i	$-0.172881\pi$
$983$	53.6823	1.71220	0.856100	+	0.516810i	$0.172881\pi$
$984$	61.2325	1.95202
$985$	0	0
$986$	173.107	5.51286
$987$	27.2540	0.867506
$988$	42.3413	1.34706
$989$	−5.16411	−0.164209
$990$	0	0
$991$	−42.6156	−1.35373	−0.676865	−	0.736107i	$-0.736662\pi$
$991$	−42.6156	−1.35373	−0.676865	+	0.736107i	$0.736662\pi$
$992$	−107.249	−3.40516
$993$	−28.6058	−0.907777
$994$	26.3851	0.836884
$995$	0	0
$996$	7.39810	0.234418
$997$	41.9404	1.32827	0.664133	−	0.747614i	$-0.268801\pi$
$997$	41.9404	1.32827	0.664133	+	0.747614i	$0.268801\pi$
$998$	32.9198	1.04206
$999$	6.31431	0.199776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8025.2.a.bc.1.10		10
5.4	even	2		1605.2.a.k.1.1	✓	10
15.14	odd	2		4815.2.a.p.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1605.2.a.k.1.1	✓	10	5.4	even	2
4815.2.a.p.1.10		10	15.14	odd	2
8025.2.a.bc.1.10		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.66746 q^{2} +1.00000 q^{3} +5.11534 q^{4} +2.66746 q^{6} -3.56475 q^{7} +8.31005 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.66746 q^{2} +1.00000 q^{3} +5.11534 q^{4} +2.66746 q^{6} -3.56475 q^{7} +8.31005 q^{8} +1.00000 q^{9} -4.56949 q^{11} +5.11534 q^{12} -1.90555 q^{13} -9.50884 q^{14} +11.9361 q^{16} -6.37815 q^{17} +2.66746 q^{18} -4.34379 q^{19} -3.56475 q^{21} -12.1889 q^{22} -0.445344 q^{23} +8.31005 q^{24} -5.08298 q^{26} +1.00000 q^{27} -18.2349 q^{28} -10.1747 q^{29} -7.04713 q^{31} +15.2188 q^{32} -4.56949 q^{33} -17.0135 q^{34} +5.11534 q^{36} +6.31431 q^{37} -11.5869 q^{38} -1.90555 q^{39} +7.36848 q^{41} -9.50884 q^{42} +11.5958 q^{43} -23.3745 q^{44} -1.18794 q^{46} -7.64542 q^{47} +11.9361 q^{48} +5.70746 q^{49} -6.37815 q^{51} -9.74755 q^{52} -8.72007 q^{53} +2.66746 q^{54} -29.6233 q^{56} -4.34379 q^{57} -27.1407 q^{58} -0.246820 q^{59} +12.7177 q^{61} -18.7979 q^{62} -3.56475 q^{63} +16.7235 q^{64} -12.1889 q^{66} -0.459577 q^{67} -32.6264 q^{68} -0.445344 q^{69} -2.77480 q^{71} +8.31005 q^{72} +6.45033 q^{73} +16.8432 q^{74} -22.2200 q^{76} +16.2891 q^{77} -5.08298 q^{78} -7.51271 q^{79} +1.00000 q^{81} +19.6551 q^{82} +1.44626 q^{83} -18.2349 q^{84} +30.9313 q^{86} -10.1747 q^{87} -37.9727 q^{88} +9.45126 q^{89} +6.79282 q^{91} -2.27809 q^{92} -7.04713 q^{93} -20.3939 q^{94} +15.2188 q^{96} +8.67043 q^{97} +15.2244 q^{98} -4.56949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{6} + 10 q^{9}+O(q^{10}) \) 10 * q - 2 * q^2 + 10 * q^3 + 10 * q^4 - 2 * q^6 + 10 * q^9	\( 10 q - 2 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{6} + 10 q^{9} - 14 q^{11} + 10 q^{12} + 3 q^{13} - 16 q^{14} + 10 q^{16} - 8 q^{17} - 2 q^{18} - 19 q^{19} - 5 q^{22} - 4 q^{23} - 22 q^{26} + 10 q^{27} - 25 q^{29} - 2 q^{31} + 13 q^{32} - 14 q^{33} - 37 q^{34} + 10 q^{36} + 10 q^{37} + 13 q^{38} + 3 q^{39} - 31 q^{41} - 16 q^{42} - 62 q^{44} + 2 q^{46} + q^{47} + 10 q^{48} + 26 q^{49} - 8 q^{51} + 30 q^{52} - 9 q^{53} - 2 q^{54} - 63 q^{56} - 19 q^{57} - 30 q^{58} - 65 q^{59} + 12 q^{61} + 39 q^{62} - 2 q^{64} - 5 q^{66} + 10 q^{67} - 22 q^{68} - 4 q^{69} - 45 q^{71} - q^{73} + 19 q^{74} - 39 q^{76} + q^{77} - 22 q^{78} - 47 q^{79} + 10 q^{81} + 23 q^{82} - q^{83} + 12 q^{86} - 25 q^{87} + 8 q^{88} - 34 q^{89} - 26 q^{91} - 14 q^{92} - 2 q^{93} - 64 q^{94} + 13 q^{96} - 5 q^{97} + 51 q^{98} - 14 q^{99}+O(q^{100}) \) 10 * q - 2 * q^2 + 10 * q^3 + 10 * q^4 - 2 * q^6 + 10 * q^9 - 14 * q^11 + 10 * q^12 + 3 * q^13 - 16 * q^14 + 10 * q^16 - 8 * q^17 - 2 * q^18 - 19 * q^19 - 5 * q^22 - 4 * q^23 - 22 * q^26 + 10 * q^27 - 25 * q^29 - 2 * q^31 + 13 * q^32 - 14 * q^33 - 37 * q^34 + 10 * q^36 + 10 * q^37 + 13 * q^38 + 3 * q^39 - 31 * q^41 - 16 * q^42 - 62 * q^44 + 2 * q^46 + q^47 + 10 * q^48 + 26 * q^49 - 8 * q^51 + 30 * q^52 - 9 * q^53 - 2 * q^54 - 63 * q^56 - 19 * q^57 - 30 * q^58 - 65 * q^59 + 12 * q^61 + 39 * q^62 - 2 * q^64 - 5 * q^66 + 10 * q^67 - 22 * q^68 - 4 * q^69 - 45 * q^71 - q^73 + 19 * q^74 - 39 * q^76 + q^77 - 22 * q^78 - 47 * q^79 + 10 * q^81 + 23 * q^82 - q^83 + 12 * q^86 - 25 * q^87 + 8 * q^88 - 34 * q^89 - 26 * q^91 - 14 * q^92 - 2 * q^93 - 64 * q^94 + 13 * q^96 - 5 * q^97 + 51 * q^98 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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