LMFDB - Embedded newform 425.2.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(1,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, 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("425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.6224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 2x + 5 $ x^4 - 6x^2 - 2x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.87228$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.57942 q^{2} +2.87228 q^{3} +0.494582 q^{4} +4.53654 q^{6} +1.42058 q^{7} -2.37769 q^{8} +5.24997 q^{9} +0.0740063 q^{11} +1.42058 q^{12} -5.70167 q^{13} +2.24369 q^{14} -4.74455 q^{16} -1.00000 q^{17} +8.29193 q^{18} -4.90340 q^{19} +4.08029 q^{21} +0.116887 q^{22} +3.88311 q^{23} -6.82940 q^{24} -9.00536 q^{26} +6.46254 q^{27} +0.702591 q^{28} +5.91424 q^{29} +0.388531 q^{31} -2.73827 q^{32} +0.212567 q^{33} -1.57942 q^{34} +2.59654 q^{36} +9.91424 q^{37} -7.74455 q^{38} -16.3768 q^{39} -6.61055 q^{41} +6.44450 q^{42} +6.94628 q^{43} +0.0366022 q^{44} +6.13308 q^{46} -5.70167 q^{47} -13.6277 q^{48} -4.98197 q^{49} -2.87228 q^{51} -2.81994 q^{52} -0.0216729 q^{53} +10.2071 q^{54} -3.37769 q^{56} -14.0839 q^{57} +9.34109 q^{58} -2.00000 q^{59} -3.47337 q^{61} +0.613655 q^{62} +7.45798 q^{63} +5.16421 q^{64} +0.335733 q^{66} -6.71251 q^{67} -0.494582 q^{68} +11.1534 q^{69} +3.84023 q^{71} -12.4828 q^{72} +13.5356 q^{73} +15.6588 q^{74} -2.42513 q^{76} +0.105132 q^{77} -25.8659 q^{78} -1.06000 q^{79} +2.81228 q^{81} -10.4409 q^{82} -11.8497 q^{83} +2.01803 q^{84} +10.9711 q^{86} +16.9873 q^{87} -0.175964 q^{88} -1.99452 q^{89} -8.09965 q^{91} +1.92052 q^{92} +1.11597 q^{93} -9.00536 q^{94} -7.86508 q^{96} +5.34109 q^{97} -7.86864 q^{98} +0.388531 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} + 10 q^{7} + 4 q^{9} - 2 q^{11} + 10 q^{12} + 6 q^{13} - 6 q^{14} - 4 q^{16} - 4 q^{17} - 4 q^{18} + 4 q^{19} + 12 q^{21} + 12 q^{22} + 4 q^{23} - 6 q^{24} + 10 q^{27}+ \cdots - 12 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^3 + 4 * q^4 + 10 * q^7 + 4 * q^9 - 2 * q^11 + 10 * q^12 + 6 * q^13 - 6 * q^14 - 4 * q^16 - 4 * q^17 - 4 * q^18 + 4 * q^19 + 12 * q^21 + 12 * q^22 + 4 * q^23 - 6 * q^24 + 10 * q^27 + 8 * q^28 - 4 * q^29 - 12 * q^31 + 2 * q^32 + 2 * q^33 - 2 * q^34 + 12 * q^37 - 16 * q^38 - 22 * q^39 - 6 * q^41 - 18 * q^42 + 18 * q^43 + 16 * q^44 - 4 * q^46 + 6 * q^47 - 28 * q^48 + 8 * q^49 - 4 * q^51 + 2 * q^52 + 8 * q^53 + 10 * q^54 - 4 * q^56 - 16 * q^57 + 12 * q^58 - 8 * q^59 + 6 * q^61 - 14 * q^62 + 16 * q^63 - 24 * q^64 + 12 * q^66 + 6 * q^67 - 4 * q^68 + 4 * q^69 - 10 * q^71 - 22 * q^72 + 2 * q^73 + 20 * q^74 - 12 * q^76 - 18 * q^77 - 30 * q^78 - 12 * q^79 - 4 * q^81 - 34 * q^82 - 14 * q^83 + 36 * q^84 + 20 * q^86 + 4 * q^87 + 14 * q^88 + 24 * q^89 + 18 * q^91 - 22 * q^92 - 18 * q^93 + 8 * q^96 - 4 * q^97 - 60 * q^98 - 12 * q^99

Coefficient data

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

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

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

$2$	1.57942	1.11682	0.558411	−	0.829565i	$-0.311411\pi$
$2$	1.57942	1.11682	0.558411	+	0.829565i	$0.311411\pi$
$3$	2.87228	1.65831	0.829155	−	0.559019i	$-0.188822\pi$
$3$	2.87228	1.65831	0.829155	+	0.559019i	$0.188822\pi$
$4$	0.494582	0.247291
$5$	0	0
$5$	0	0
$6$	4.53654	1.85204
$7$	1.42058	0.536927	0.268464	−	0.963290i	$-0.413484\pi$
$7$	1.42058	0.536927	0.268464	+	0.963290i	$0.413484\pi$
$8$	−2.37769	−0.840642
$9$	5.24997	1.74999
$10$	0	0
$11$	0.0740063	0.0223137	0.0111569	−	0.999938i	$-0.496449\pi$
$11$	0.0740063	0.0223137	0.0111569	+	0.999938i	$0.496449\pi$
$12$	1.42058	0.410085
$13$	−5.70167	−1.58136	−0.790680	−	0.612230i	$-0.790273\pi$
$13$	−5.70167	−1.58136	−0.790680	+	0.612230i	$0.790273\pi$
$14$	2.24369	0.599652
$15$	0	0
$16$	−4.74455	−1.18614
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	8.29193	1.95443
$19$	−4.90340	−1.12492	−0.562459	−	0.826825i	$-0.690144\pi$
$19$	−4.90340	−1.12492	−0.562459	+	0.826825i	$0.690144\pi$
$20$	0	0
$21$	4.08029	0.890391
$22$	0.116887	0.0249205
$23$	3.88311	0.809685	0.404842	−	0.914386i	$-0.367326\pi$
$23$	3.88311	0.809685	0.404842	+	0.914386i	$0.367326\pi$
$24$	−6.82940	−1.39404
$25$	0	0
$26$	−9.00536	−1.76610
$27$	6.46254	1.24372
$28$	0.702591	0.132777
$29$	5.91424	1.09825	0.549123	−	0.835741i	$-0.314962\pi$
$29$	5.91424	1.09825	0.549123	+	0.835741i	$0.314962\pi$
$30$	0	0
$31$	0.388531	0.0697822	0.0348911	−	0.999391i	$-0.488892\pi$
$31$	0.388531	0.0697822	0.0348911	+	0.999391i	$0.488892\pi$
$32$	−2.73827	−0.484063
$33$	0.212567	0.0370031
$34$	−1.57942	−0.270869
$35$	0	0
$36$	2.59654	0.432757
$37$	9.91424	1.62989	0.814945	−	0.579538i	$-0.196767\pi$
$37$	9.91424	1.62989	0.814945	+	0.579538i	$0.196767\pi$
$38$	−7.74455	−1.25633
$39$	−16.3768	−2.62238
$40$	0	0
$41$	−6.61055	−1.03239	−0.516197	−	0.856470i	$-0.672653\pi$
$41$	−6.61055	−1.03239	−0.516197	+	0.856470i	$0.672653\pi$
$42$	6.44450	0.994408
$43$	6.94628	1.05930	0.529649	−	0.848217i	$-0.322324\pi$
$43$	6.94628	1.05930	0.529649	+	0.848217i	$0.322324\pi$
$44$	0.0366022	0.00551798
$45$	0	0
$46$	6.13308	0.904274
$47$	−5.70167	−0.831674	−0.415837	−	0.909439i	$-0.636511\pi$
$47$	−5.70167	−0.831674	−0.415837	+	0.909439i	$0.636511\pi$
$48$	−13.6277	−1.96698
$49$	−4.98197	−0.711709
$50$	0	0
$51$	−2.87228	−0.402199
$52$	−2.81994	−0.391056
$53$	−0.0216729	−0.00297700	−0.00148850	−	0.999999i	$-0.500474\pi$
$53$	−0.0216729	−0.00297700	−0.00148850	+	0.999999i	$0.500474\pi$
$54$	10.2071	1.38901
$55$	0	0
$56$	−3.37769	−0.451363
$57$	−14.0839	−1.86546
$58$	9.34109	1.22655
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	−3.47337	−0.444720	−0.222360	−	0.974965i	$-0.571376\pi$
$61$	−3.47337	−0.444720	−0.222360	+	0.974965i	$0.571376\pi$
$62$	0.613655	0.0779343
$63$	7.45798	0.939617
$64$	5.16421	0.645526
$65$	0	0
$66$	0.335733	0.0413258
$67$	−6.71251	−0.820063	−0.410032	−	0.912071i	$-0.634482\pi$
$67$	−6.71251	−0.820063	−0.410032	+	0.912071i	$0.634482\pi$
$68$	−0.494582	−0.0599769
$69$	11.1534	1.34271
$70$	0	0
$71$	3.84023	0.455752	0.227876	−	0.973690i	$-0.426822\pi$
$71$	3.84023	0.455752	0.227876	+	0.973690i	$0.426822\pi$
$72$	−12.4828	−1.47112
$73$	13.5356	1.58422	0.792112	−	0.610375i	$-0.208981\pi$
$73$	13.5356	1.58422	0.792112	+	0.610375i	$0.208981\pi$
$74$	15.6588	1.82030
$75$	0	0
$76$	−2.42513	−0.278182
$77$	0.105132	0.0119808
$78$	−25.8659	−2.92873
$79$	−1.06000	−0.119259	−0.0596295	−	0.998221i	$-0.518992\pi$
$79$	−1.06000	−0.119259	−0.0596295	+	0.998221i	$0.518992\pi$
$80$	0	0
$81$	2.81228	0.312476
$82$	−10.4409	−1.15300
$83$	−11.8497	−1.30067	−0.650336	−	0.759647i	$-0.725372\pi$
$83$	−11.8497	−1.30067	−0.650336	+	0.759647i	$0.725372\pi$
$84$	2.01803	0.220186
$85$	0	0
$86$	10.9711	1.18305
$87$	16.9873	1.82123
$88$	−0.175964	−0.0187579
$89$	−1.99452	−0.211419	−0.105710	−	0.994397i	$-0.533711\pi$
$89$	−1.99452	−0.211419	−0.105710	+	0.994397i	$0.533711\pi$
$90$	0	0
$91$	−8.09965	−0.849075
$92$	1.92052	0.200228
$93$	1.11597	0.115720
$94$	−9.00536	−0.928832
$95$	0	0
$96$	−7.86508	−0.802726
$97$	5.34109	0.542306	0.271153	−	0.962536i	$-0.412595\pi$
$97$	5.34109	0.542306	0.271153	+	0.962536i	$0.412595\pi$
$98$	−7.86864	−0.794852
$99$	0.388531	0.0390488
$100$	0	0
$101$	11.4305	1.13738	0.568688	−	0.822553i	$-0.307451\pi$
$101$	11.4305	1.13738	0.568688	+	0.822553i	$0.307451\pi$
$102$	−4.53654	−0.449185
$103$	8.21257	0.809208	0.404604	−	0.914492i	$-0.367409\pi$
$103$	8.21257	0.809208	0.404604	+	0.914492i	$0.367409\pi$
$104$	13.5568	1.32936
$105$	0	0
$106$	−0.0342307	−0.00332478
$107$	6.34882	0.613764	0.306882	−	0.951748i	$-0.400714\pi$
$107$	6.34882	0.613764	0.306882	+	0.951748i	$0.400714\pi$
$108$	3.19625	0.307560
$109$	0.292852	0.0280501	0.0140251	−	0.999902i	$-0.495536\pi$
$109$	0.292852	0.0280501	0.0140251	+	0.999902i	$0.495536\pi$
$110$	0	0
$111$	28.4764	2.70286
$112$	−6.73999	−0.636870
$113$	8.62311	0.811194	0.405597	−	0.914052i	$-0.367064\pi$
$113$	8.62311	0.811194	0.405597	+	0.914052i	$0.367064\pi$
$114$	−22.2445	−2.08339
$115$	0	0
$116$	2.92507	0.271586
$117$	−29.9336	−2.76736
$118$	−3.15885	−0.290796
$119$	−1.42058	−0.130224
$120$	0	0
$121$	−10.9945	−0.999502
$122$	−5.48593	−0.496673
$123$	−18.9873	−1.71203
$124$	0.192160	0.0172565
$125$	0	0
$126$	11.7793	1.04938
$127$	17.2640	1.53193	0.765965	−	0.642882i	$-0.222261\pi$
$127$	17.2640	1.53193	0.765965	+	0.642882i	$0.222261\pi$
$128$	13.6330	1.20500
$129$	19.9516	1.75664
$130$	0	0
$131$	−14.7188	−1.28599	−0.642993	−	0.765872i	$-0.722308\pi$
$131$	−14.7188	−1.28599	−0.642993	+	0.765872i	$0.722308\pi$
$132$	0.105132	0.00915052
$133$	−6.96565	−0.603999
$134$	−10.6019	−0.915865
$135$	0	0
$136$	2.37769	0.203886
$137$	−1.36230	−0.116389	−0.0581946	−	0.998305i	$-0.518534\pi$
$137$	−1.36230	−0.116389	−0.0581946	+	0.998305i	$0.518534\pi$
$138$	17.6159	1.49957
$139$	−13.7454	−1.16587	−0.582933	−	0.812520i	$-0.698095\pi$
$139$	−13.7454	−1.16587	−0.582933	+	0.812520i	$0.698095\pi$
$140$	0	0
$141$	−16.3768	−1.37917
$142$	6.06536	0.508993
$143$	−0.421960	−0.0352860
$144$	−24.9088	−2.07573
$145$	0	0
$146$	21.3785	1.76930
$147$	−14.3096	−1.18023
$148$	4.90340	0.403057
$149$	−17.3628	−1.42241	−0.711207	−	0.702983i	$-0.751851\pi$
$149$	−17.3628	−1.42241	−0.711207	+	0.702983i	$0.751851\pi$
$150$	0	0
$151$	−3.59654	−0.292682	−0.146341	−	0.989234i	$-0.546750\pi$
$151$	−3.59654	−0.292682	−0.146341	+	0.989234i	$0.546750\pi$
$152$	11.6588	0.945653
$153$	−5.24997	−0.424435
$154$	0.166047	0.0133805
$155$	0	0
$156$	−8.09965	−0.648491
$157$	5.19308	0.414453	0.207226	−	0.978293i	$-0.433556\pi$
$157$	5.19308	0.414453	0.207226	+	0.978293i	$0.433556\pi$
$158$	−1.67418	−0.133191
$159$	−0.0622505	−0.00493678
$160$	0	0
$161$	5.51625	0.434742
$162$	4.44178	0.348979
$163$	6.56687	0.514357	0.257178	−	0.966364i	$-0.417207\pi$
$163$	6.56687	0.514357	0.257178	+	0.966364i	$0.417207\pi$
$164$	−3.26946	−0.255302
$165$	0	0
$166$	−18.7157	−1.45262
$167$	−0.897123	−0.0694214	−0.0347107	−	0.999397i	$-0.511051\pi$
$167$	−0.897123	−0.0694214	−0.0347107	+	0.999397i	$0.511051\pi$
$168$	−9.70167	−0.748500
$169$	19.5091	1.50070
$170$	0	0
$171$	−25.7427	−1.96859
$172$	3.43550	0.261955
$173$	−3.83031	−0.291213	−0.145607	−	0.989343i	$-0.546513\pi$
$173$	−3.83031	−0.291213	−0.145607	+	0.989343i	$0.546513\pi$
$174$	26.8302	2.03399
$175$	0	0
$176$	−0.351127	−0.0264672
$177$	−5.74455	−0.431787
$178$	−3.15020	−0.236117
$179$	9.74455	0.728342	0.364171	−	0.931332i	$-0.381352\pi$
$179$	9.74455	0.728342	0.364171	+	0.931332i	$0.381352\pi$
$180$	0	0
$181$	22.4764	1.67066	0.835330	−	0.549749i	$-0.185277\pi$
$181$	22.4764	1.67066	0.835330	+	0.549749i	$0.185277\pi$
$182$	−12.7928	−0.948265
$183$	−9.97649	−0.737483
$184$	−9.23286	−0.680655
$185$	0	0
$186$	1.76259	0.129239
$187$	−0.0740063	−0.00541188
$188$	−2.81994	−0.205665
$189$	9.18052	0.667785
$190$	0	0
$191$	−5.47655	−0.396269	−0.198135	−	0.980175i	$-0.563488\pi$
$191$	−5.47655	−0.396269	−0.198135	+	0.980175i	$0.563488\pi$
$192$	14.8330	1.07048
$193$	2.14484	0.154389	0.0771944	−	0.997016i	$-0.475404\pi$
$193$	2.14484	0.154389	0.0771944	+	0.997016i	$0.475404\pi$
$194$	8.43585	0.605659
$195$	0	0
$196$	−2.46399	−0.175999
$197$	9.78513	0.697162	0.348581	−	0.937279i	$-0.386664\pi$
$197$	9.78513	0.697162	0.348581	+	0.937279i	$0.386664\pi$
$198$	0.613655	0.0436106
$199$	−1.47257	−0.104388	−0.0521939	−	0.998637i	$-0.516621\pi$
$199$	−1.47257	−0.104388	−0.0521939	+	0.998637i	$0.516621\pi$
$200$	0	0
$201$	−19.2802	−1.35992
$202$	18.0536	1.27025
$203$	8.40162	0.589678
$204$	−1.42058	−0.0994602
$205$	0	0
$206$	12.9711	0.903741
$207$	20.3862	1.41694
$208$	27.0519	1.87571
$209$	−0.362883	−0.0251011
$210$	0	0
$211$	−4.87591	−0.335672	−0.167836	−	0.985815i	$-0.553678\pi$
$211$	−4.87591	−0.335672	−0.167836	+	0.985815i	$0.553678\pi$
$212$	−0.0107190	−0.000736184 0
$213$	11.0302	0.755777
$214$	10.0275	0.685465
$215$	0	0
$216$	−15.3659	−1.04552
$217$	0.551937	0.0374680
$218$	0.462537	0.0313270
$219$	38.8781	2.62714
$220$	0	0
$221$	5.70167	0.383536
$222$	44.9764	3.01862
$223$	20.0536	1.34289	0.671444	−	0.741055i	$-0.265674\pi$
$223$	20.0536	1.34289	0.671444	+	0.741055i	$0.265674\pi$
$224$	−3.88992	−0.259906
$225$	0	0
$226$	13.6195	0.905959
$227$	−14.1308	−0.937893	−0.468946	−	0.883227i	$-0.655366\pi$
$227$	−14.1308	−0.937893	−0.468946	+	0.883227i	$0.655366\pi$
$228$	−6.96565	−0.461312
$229$	11.2156	0.741149	0.370575	−	0.928803i	$-0.379161\pi$
$229$	11.2156	0.741149	0.370575	+	0.928803i	$0.379161\pi$
$230$	0	0
$231$	0.301967	0.0198680
$232$	−14.0623	−0.923232
$233$	−13.8285	−0.905934	−0.452967	−	0.891527i	$-0.649634\pi$
$233$	−13.8285	−0.905934	−0.452967	+	0.891527i	$0.649634\pi$
$234$	−47.2779	−3.09065
$235$	0	0
$236$	−0.989164	−0.0643891
$237$	−3.04460	−0.197768
$238$	−2.24369	−0.145437
$239$	−16.2851	−1.05339	−0.526697	−	0.850053i	$-0.676570\pi$
$239$	−16.2851	−1.05339	−0.526697	+	0.850053i	$0.676570\pi$
$240$	0	0
$241$	4.94081	0.318265	0.159133	−	0.987257i	$-0.449130\pi$
$241$	4.94081	0.318265	0.159133	+	0.987257i	$0.449130\pi$
$242$	−17.3650	−1.11627
$243$	−11.3100	−0.725535
$244$	−1.71787	−0.109975
$245$	0	0
$246$	−29.9890	−1.91203
$247$	27.9576	1.77890
$248$	−0.923808	−0.0586618
$249$	−34.0356	−2.15692
$250$	0	0
$251$	12.9160	0.815248	0.407624	−	0.913150i	$-0.366357\pi$
$251$	12.9160	0.815248	0.407624	+	0.913150i	$0.366357\pi$
$252$	3.68858	0.232359
$253$	0.287375	0.0180671
$254$	27.2672	1.71089
$255$	0	0
$256$	11.2039	0.700245
$257$	13.6159	0.849337	0.424669	−	0.905349i	$-0.360391\pi$
$257$	13.6159	0.849337	0.424669	+	0.905349i	$0.360391\pi$
$258$	31.5121	1.96186
$259$	14.0839	0.875132
$260$	0	0
$261$	31.0496	1.92192
$262$	−23.2472	−1.43622
$263$	12.6719	0.781385	0.390692	−	0.920521i	$-0.372236\pi$
$263$	12.6719	0.781385	0.390692	+	0.920521i	$0.372236\pi$
$264$	−0.505418	−0.0311063
$265$	0	0
$266$	−11.0017	−0.674559
$267$	−5.72882	−0.350598
$268$	−3.31988	−0.202794
$269$	−25.8967	−1.57895	−0.789474	−	0.613784i	$-0.789646\pi$
$269$	−25.8967	−1.57895	−0.789474	+	0.613784i	$0.789646\pi$
$270$	0	0
$271$	21.2211	1.28909	0.644545	−	0.764566i	$-0.277047\pi$
$271$	21.2211	1.28909	0.644545	+	0.764566i	$0.277047\pi$
$272$	4.74455	0.287681
$273$	−23.2644	−1.40803
$274$	−2.15165	−0.129986
$275$	0	0
$276$	5.51625	0.332040
$277$	−2.05908	−0.123718	−0.0618590	−	0.998085i	$-0.519703\pi$
$277$	−2.05908	−0.123718	−0.0618590	+	0.998085i	$0.519703\pi$
$278$	−21.7097	−1.30206
$279$	2.03978	0.122118
$280$	0	0
$281$	−21.7336	−1.29652	−0.648259	−	0.761420i	$-0.724503\pi$
$281$	−21.7336	−1.29652	−0.648259	+	0.761420i	$0.724503\pi$
$282$	−25.8659	−1.54029
$283$	5.32226	0.316375	0.158188	−	0.987409i	$-0.449435\pi$
$283$	5.32226	0.316375	0.158188	+	0.987409i	$0.449435\pi$
$284$	1.89931	0.112703
$285$	0	0
$286$	−0.666453	−0.0394082
$287$	−9.39078	−0.554321
$288$	−14.3759	−0.847105
$289$	1.00000	0.0588235
$290$	0	0
$291$	15.3411	0.899311
$292$	6.69447	0.391764
$293$	−23.3285	−1.36287	−0.681434	−	0.731880i	$-0.738643\pi$
$293$	−23.3285	−1.36287	−0.681434	+	0.731880i	$0.738643\pi$
$294$	−22.6009	−1.31811
$295$	0	0
$296$	−23.5730	−1.37015
$297$	0.478268	0.0277519
$298$	−27.4232	−1.58858
$299$	−22.1402	−1.28040
$300$	0	0
$301$	9.86772	0.568766
$302$	−5.68046	−0.326874
$303$	32.8315	1.88612
$304$	23.2644	1.33431
$305$	0	0
$306$	−8.29193	−0.474018
$307$	−17.4588	−0.996425	−0.498213	−	0.867055i	$-0.666010\pi$
$307$	−17.4588	−0.996425	−0.498213	+	0.867055i	$0.666010\pi$
$308$	0.0519961	0.00296275
$309$	23.5888	1.34192
$310$	0	0
$311$	2.98997	0.169545	0.0847727	−	0.996400i	$-0.472984\pi$
$311$	2.98997	0.169545	0.0847727	+	0.996400i	$0.472984\pi$
$312$	38.9390	2.20448
$313$	−7.76028	−0.438637	−0.219319	−	0.975653i	$-0.570383\pi$
$313$	−7.76028	−0.438637	−0.219319	+	0.975653i	$0.570383\pi$
$314$	8.20208	0.462870
$315$	0	0
$316$	−0.524255	−0.0294916
$317$	−6.88951	−0.386953	−0.193477	−	0.981105i	$-0.561976\pi$
$317$	−6.88951	−0.386953	−0.193477	+	0.981105i	$0.561976\pi$
$318$	−0.0983199	−0.00551351
$319$	0.437691	0.0245060
$320$	0	0
$321$	18.2356	1.01781
$322$	8.71251	0.485529
$323$	4.90340	0.272833
$324$	1.39090	0.0772724
$325$	0	0
$326$	10.3719	0.574445
$327$	0.841151	0.0465158
$328$	15.7179	0.867874
$329$	−8.09965	−0.446548
$330$	0	0
$331$	15.8565	0.871552	0.435776	−	0.900055i	$-0.356474\pi$
$331$	15.8565	0.871552	0.435776	+	0.900055i	$0.356474\pi$
$332$	−5.86064	−0.321644
$333$	52.0495	2.85229
$334$	−1.41694	−0.0775314
$335$	0	0
$336$	−19.3591	−1.05613
$337$	−19.1787	−1.04473	−0.522365	−	0.852722i	$-0.674950\pi$
$337$	−19.1787	−1.04473	−0.522365	+	0.852722i	$0.674950\pi$
$338$	30.8131	1.67601
$339$	24.7679	1.34521
$340$	0	0
$341$	0.0287537	0.00155710
$342$	−40.6587	−2.19857
$343$	−17.0213	−0.919063
$344$	−16.5161	−0.890490
$345$	0	0
$346$	−6.04969	−0.325233
$347$	33.5775	1.80253	0.901266	−	0.433265i	$-0.142639\pi$
$347$	33.5775	1.80253	0.901266	+	0.433265i	$0.142639\pi$
$348$	8.40162	0.450374
$349$	8.61649	0.461230	0.230615	−	0.973045i	$-0.425926\pi$
$349$	8.61649	0.461230	0.230615	+	0.973045i	$0.425926\pi$
$350$	0	0
$351$	−36.8473	−1.96676
$352$	−0.202649	−0.0108013
$353$	−0.596657	−0.0317569	−0.0158784	−	0.999874i	$-0.505054\pi$
$353$	−0.596657	−0.0317569	−0.0158784	+	0.999874i	$0.505054\pi$
$354$	−9.07309	−0.482229
$355$	0	0
$356$	−0.986455	−0.0522820
$357$	−4.08029	−0.215952
$358$	15.3908	0.813428
$359$	31.5514	1.66522	0.832608	−	0.553862i	$-0.186847\pi$
$359$	31.5514	1.66522	0.832608	+	0.553862i	$0.186847\pi$
$360$	0	0
$361$	5.04335	0.265439
$362$	35.4998	1.86583
$363$	−31.5793	−1.65748
$364$	−4.00594	−0.209968
$365$	0	0
$366$	−15.7571	−0.823637
$367$	1.37723	0.0718908	0.0359454	−	0.999354i	$-0.488556\pi$
$367$	1.37723	0.0718908	0.0359454	+	0.999354i	$0.488556\pi$
$368$	−18.4236	−0.960398
$369$	−34.7052	−1.80668
$370$	0	0
$371$	−0.0307879	−0.00159843
$372$	0.551937	0.0286166
$373$	−28.5627	−1.47892	−0.739459	−	0.673201i	$-0.764919\pi$
$373$	−28.5627	−1.47892	−0.739459	+	0.673201i	$0.764919\pi$
$374$	−0.116887	−0.00604410
$375$	0	0
$376$	13.5568	0.699140
$377$	−33.7210	−1.73672
$378$	14.4999	0.745797
$379$	−24.2405	−1.24515	−0.622576	−	0.782559i	$-0.713914\pi$
$379$	−24.2405	−1.24515	−0.622576	+	0.782559i	$0.713914\pi$
$380$	0	0
$381$	49.5869	2.54041
$382$	−8.64979	−0.442562
$383$	−24.4678	−1.25025	−0.625123	−	0.780527i	$-0.714951\pi$
$383$	−24.4678	−1.25025	−0.625123	+	0.780527i	$0.714951\pi$
$384$	39.1578	1.99826
$385$	0	0
$386$	3.38761	0.172425
$387$	36.4678	1.85376
$388$	2.64161	0.134107
$389$	−10.9711	−0.556258	−0.278129	−	0.960544i	$-0.589714\pi$
$389$	−10.9711	−0.556258	−0.278129	+	0.960544i	$0.589714\pi$
$390$	0	0
$391$	−3.88311	−0.196377
$392$	11.8456	0.598293
$393$	−42.2764	−2.13256
$394$	15.4549	0.778605
$395$	0	0
$396$	0.192160	0.00965642
$397$	−26.8550	−1.34782	−0.673908	−	0.738815i	$-0.735386\pi$
$397$	−26.8550	−1.34782	−0.673908	+	0.738815i	$0.735386\pi$
$398$	−2.32582	−0.116583
$399$	−20.0073	−1.00162
$400$	0	0
$401$	−17.5080	−0.874308	−0.437154	−	0.899387i	$-0.644014\pi$
$401$	−17.5080	−0.874308	−0.437154	+	0.899387i	$0.644014\pi$
$402$	−30.4516	−1.51879
$403$	−2.21528	−0.110351
$404$	5.65331	0.281263
$405$	0	0
$406$	13.2697	0.658565
$407$	0.733716	0.0363690
$408$	6.82940	0.338105
$409$	−17.8285	−0.881561	−0.440781	−	0.897615i	$-0.645298\pi$
$409$	−17.8285	−0.881561	−0.440781	+	0.897615i	$0.645298\pi$
$410$	0	0
$411$	−3.91290	−0.193009
$412$	4.06179	0.200110
$413$	−2.84115	−0.139804
$414$	32.1985	1.58247
$415$	0	0
$416$	15.6127	0.765477
$417$	−39.4805	−1.93337
$418$	−0.573146	−0.0280335
$419$	34.7826	1.69924	0.849622	−	0.527393i	$-0.176830\pi$
$419$	34.7826	1.69924	0.849622	+	0.527393i	$0.176830\pi$
$420$	0	0
$421$	−17.0676	−0.831824	−0.415912	−	0.909405i	$-0.636538\pi$
$421$	−17.0676	−0.831824	−0.415912	+	0.909405i	$0.636538\pi$
$422$	−7.70114	−0.374886
$423$	−29.9336	−1.45542
$424$	0.0515315	0.00250259
$425$	0	0
$426$	17.4214	0.844068
$427$	−4.93419	−0.238782
$428$	3.14001	0.151778
$429$	−1.21198	−0.0585152
$430$	0	0
$431$	−22.5970	−1.08846	−0.544229	−	0.838937i	$-0.683178\pi$
$431$	−22.5970	−1.08846	−0.544229	+	0.838937i	$0.683178\pi$
$432$	−30.6618	−1.47522
$433$	13.3513	0.641625	0.320812	−	0.947143i	$-0.396044\pi$
$433$	13.3513	0.641625	0.320812	+	0.947143i	$0.396044\pi$
$434$	0.871743	0.0418450
$435$	0	0
$436$	0.144839	0.00693654
$437$	−19.0405	−0.910829
$438$	61.4049	2.93404
$439$	5.73280	0.273611	0.136806	−	0.990598i	$-0.456316\pi$
$439$	5.73280	0.273611	0.136806	+	0.990598i	$0.456316\pi$
$440$	0	0
$441$	−26.1552	−1.24548
$442$	9.00536	0.428341
$443$	−38.8153	−1.84417	−0.922086	−	0.386985i	$-0.873517\pi$
$443$	−38.8153	−1.84417	−0.922086	+	0.386985i	$0.873517\pi$
$444$	14.0839	0.668393
$445$	0	0
$446$	31.6731	1.49977
$447$	−49.8707	−2.35880
$448$	7.33615	0.346600
$449$	−25.3848	−1.19798	−0.598992	−	0.800755i	$-0.704432\pi$
$449$	−25.3848	−1.19798	−0.598992	+	0.800755i	$0.704432\pi$
$450$	0	0
$451$	−0.489222	−0.0230366
$452$	4.26483	0.200601
$453$	−10.3303	−0.485358
$454$	−22.3185	−1.04746
$455$	0	0
$456$	33.4873	1.56818
$457$	8.90110	0.416376	0.208188	−	0.978089i	$-0.433243\pi$
$457$	8.90110	0.416376	0.208188	+	0.978089i	$0.433243\pi$
$458$	17.7142	0.827732
$459$	−6.46254	−0.301645
$460$	0	0
$461$	−13.9124	−0.647965	−0.323983	−	0.946063i	$-0.605022\pi$
$461$	−13.9124	−0.647965	−0.323983	+	0.946063i	$0.605022\pi$
$462$	0.476934	0.0221890
$463$	37.5085	1.74317	0.871583	−	0.490247i	$-0.163094\pi$
$463$	37.5085	1.74317	0.871583	+	0.490247i	$0.163094\pi$
$464$	−28.0604	−1.30267
$465$	0	0
$466$	−21.8410	−1.01177
$467$	−11.2108	−0.518776	−0.259388	−	0.965773i	$-0.583521\pi$
$467$	−11.2108	−0.518776	−0.259388	+	0.965773i	$0.583521\pi$
$468$	−14.8046	−0.684344
$469$	−9.53562	−0.440314
$470$	0	0
$471$	14.9160	0.687291
$472$	4.75539	0.218885
$473$	0.514069	0.0236369
$474$	−4.80872	−0.220872
$475$	0	0
$476$	−0.702591	−0.0322032
$477$	−0.113782	−0.00520972
$478$	−25.7210	−1.17645
$479$	5.16559	0.236022	0.118011	−	0.993012i	$-0.462348\pi$
$479$	5.16559	0.236022	0.118011	+	0.993012i	$0.462348\pi$
$480$	0	0
$481$	−56.5277	−2.57744
$482$	7.80363	0.355446
$483$	15.8442	0.720936
$484$	−5.43769	−0.247168
$485$	0	0
$486$	−17.8632	−0.810293
$487$	34.0560	1.54322	0.771612	−	0.636094i	$-0.219451\pi$
$487$	34.0560	1.54322	0.771612	+	0.636094i	$0.219451\pi$
$488$	8.25862	0.373850
$489$	18.8619	0.852963
$490$	0	0
$491$	−18.3051	−0.826099	−0.413050	−	0.910708i	$-0.635536\pi$
$491$	−18.3051	−0.826099	−0.413050	+	0.910708i	$0.635536\pi$
$492$	−9.39078	−0.423369
$493$	−5.91424	−0.266364
$494$	44.1569	1.98671
$495$	0	0
$496$	−1.84341	−0.0827713
$497$	5.45534	0.244705
$498$	−53.7566	−2.40889
$499$	40.7792	1.82553	0.912764	−	0.408488i	$-0.133944\pi$
$499$	40.7792	1.82553	0.912764	+	0.408488i	$0.133944\pi$
$500$	0	0
$501$	−2.57678	−0.115122
$502$	20.3998	0.910487
$503$	10.7274	0.478313	0.239156	−	0.970981i	$-0.423129\pi$
$503$	10.7274	0.478313	0.239156	+	0.970981i	$0.423129\pi$
$504$	−17.7328	−0.789882
$505$	0	0
$506$	0.453887	0.0201777
$507$	56.0354	2.48862
$508$	8.53845	0.378832
$509$	−21.5278	−0.954205	−0.477102	−	0.878848i	$-0.658313\pi$
$509$	−21.5278	−0.954205	−0.477102	+	0.878848i	$0.658313\pi$
$510$	0	0
$511$	19.2284	0.850613
$512$	−9.57031	−0.422952
$513$	−31.6884	−1.39908
$514$	21.5053	0.948558
$515$	0	0
$516$	9.86772	0.434402
$517$	−0.421960	−0.0185578
$518$	22.2445	0.977367
$519$	−11.0017	−0.482922
$520$	0	0
$521$	−20.5451	−0.900098	−0.450049	−	0.893004i	$-0.648594\pi$
$521$	−20.5451	−0.900098	−0.450049	+	0.893004i	$0.648594\pi$
$522$	49.0405	2.14644
$523$	20.0508	0.876762	0.438381	−	0.898789i	$-0.355552\pi$
$523$	20.0508	0.876762	0.438381	+	0.898789i	$0.355552\pi$
$524$	−7.27964	−0.318013
$525$	0	0
$526$	20.0144	0.872667
$527$	−0.388531	−0.0169247
$528$	−1.00853	−0.0438908
$529$	−7.92144	−0.344410
$530$	0	0
$531$	−10.4999	−0.455659
$532$	−3.44508	−0.149363
$533$	37.6912	1.63259
$534$	−9.04824	−0.391556
$535$	0	0
$536$	15.9603	0.689380
$537$	27.9890	1.20782
$538$	−40.9018	−1.76340
$539$	−0.368697	−0.0158809
$540$	0	0
$541$	−8.55768	−0.367924	−0.183962	−	0.982933i	$-0.558892\pi$
$541$	−8.55768	−0.367924	−0.183962	+	0.982933i	$0.558892\pi$
$542$	33.5171	1.43968
$543$	64.5585	2.77047
$544$	2.73827	0.117403
$545$	0	0
$546$	−36.7444	−1.57252
$547$	11.2911	0.482771	0.241385	−	0.970429i	$-0.422398\pi$
$547$	11.2911	0.482771	0.241385	+	0.970429i	$0.422398\pi$
$548$	−0.673769	−0.0287820
$549$	−18.2351	−0.778256
$550$	0	0
$551$	−28.9999	−1.23544
$552$	−26.5193	−1.12874
$553$	−1.50580	−0.0640333
$554$	−3.25216	−0.138171
$555$	0	0
$556$	−6.79820	−0.288308
$557$	−22.4662	−0.951922	−0.475961	−	0.879466i	$-0.657900\pi$
$557$	−22.4662	−0.951922	−0.475961	+	0.879466i	$0.657900\pi$
$558$	3.22167	0.136384
$559$	−39.6054	−1.67513
$560$	0	0
$561$	−0.212567	−0.00897457
$562$	−34.3266	−1.44798
$563$	6.32795	0.266691	0.133346	−	0.991070i	$-0.457428\pi$
$563$	6.32795	0.266691	0.133346	+	0.991070i	$0.457428\pi$
$564$	−8.09965	−0.341057
$565$	0	0
$566$	8.40610	0.353335
$567$	3.99506	0.167777
$568$	−9.13090	−0.383124
$569$	−16.6885	−0.699620	−0.349810	−	0.936821i	$-0.613754\pi$
$569$	−16.6885	−0.699620	−0.349810	+	0.936821i	$0.613754\pi$
$570$	0	0
$571$	−6.20629	−0.259725	−0.129863	−	0.991532i	$-0.541454\pi$
$571$	−6.20629	−0.259725	−0.129863	+	0.991532i	$0.541454\pi$
$572$	−0.208694	−0.00872591
$573$	−15.7302	−0.657137
$574$	−14.8320	−0.619077
$575$	0	0
$576$	27.1119	1.12966
$577$	23.9273	0.996105	0.498052	−	0.867147i	$-0.334049\pi$
$577$	23.9273	0.996105	0.498052	+	0.867147i	$0.334049\pi$
$578$	1.57942	0.0656954
$579$	6.16057	0.256025
$580$	0	0
$581$	−16.8334	−0.698366
$582$	24.2301	1.00437
$583$	−0.00160393	−6.64279e−5 0
$584$	−32.1836	−1.33177
$585$	0	0
$586$	−36.8457	−1.52208
$587$	−26.1140	−1.07784	−0.538920	−	0.842357i	$-0.681168\pi$
$587$	−26.1140	−1.07784	−0.538920	+	0.842357i	$0.681168\pi$
$588$	−7.07726	−0.291861
$589$	−1.90512	−0.0784992
$590$	0	0
$591$	28.1056	1.15611
$592$	−47.0386	−1.93328
$593$	29.8969	1.22772	0.613860	−	0.789415i	$-0.289616\pi$
$593$	29.8969	1.22772	0.613860	+	0.789415i	$0.289616\pi$
$594$	0.755389	0.0309940
$595$	0	0
$596$	−8.58731	−0.351750
$597$	−4.22963	−0.173107
$598$	−34.9688	−1.42998
$599$	15.8194	0.646362	0.323181	−	0.946337i	$-0.395248\pi$
$599$	15.8194	0.646362	0.323181	+	0.946337i	$0.395248\pi$
$600$	0	0
$601$	32.4048	1.32182	0.660910	−	0.750466i	$-0.270171\pi$
$601$	32.4048	1.32182	0.660910	+	0.750466i	$0.270171\pi$
$602$	15.5853	0.635210
$603$	−35.2405	−1.43510
$604$	−1.77878	−0.0723777
$605$	0	0
$606$	51.8549	2.10646
$607$	−20.4218	−0.828895	−0.414447	−	0.910073i	$-0.636025\pi$
$607$	−20.4218	−0.828895	−0.414447	+	0.910073i	$0.636025\pi$
$608$	13.4269	0.544531
$609$	24.1318	0.977869
$610$	0	0
$611$	32.5091	1.31518
$612$	−2.59654	−0.104959
$613$	12.0930	0.488433	0.244217	−	0.969721i	$-0.421469\pi$
$613$	12.0930	0.488433	0.244217	+	0.969721i	$0.421469\pi$
$614$	−27.5748	−1.11283
$615$	0	0
$616$	−0.249971	−0.0100716
$617$	−3.79742	−0.152878	−0.0764392	−	0.997074i	$-0.524355\pi$
$617$	−3.79742	−0.152878	−0.0764392	+	0.997074i	$0.524355\pi$
$618$	37.2567	1.49868
$619$	−0.765306	−0.0307602	−0.0153801	−	0.999882i	$-0.504896\pi$
$619$	−0.765306	−0.0307602	−0.0153801	+	0.999882i	$0.504896\pi$
$620$	0	0
$621$	25.0948	1.00702
$622$	4.72242	0.189352
$623$	−2.83337	−0.113517
$624$	77.7005	3.11051
$625$	0	0
$626$	−12.2568	−0.489880
$627$	−1.04230	−0.0416254
$628$	2.56840	0.102490
$629$	−9.91424	−0.395307
$630$	0	0
$631$	41.7318	1.66132	0.830658	−	0.556784i	$-0.187965\pi$
$631$	41.7318	1.66132	0.830658	+	0.556784i	$0.187965\pi$
$632$	2.52035	0.100254
$633$	−14.0050	−0.556648
$634$	−10.8815	−0.432158
$635$	0	0
$636$	−0.0307879	−0.00122082
$637$	28.4055	1.12547
$638$	0.691300	0.0273688
$639$	20.1611	0.797561
$640$	0	0
$641$	22.6429	0.894342	0.447171	−	0.894448i	$-0.352431\pi$
$641$	22.6429	0.894342	0.447171	+	0.894448i	$0.352431\pi$
$642$	28.8017	1.13671
$643$	−24.8814	−0.981226	−0.490613	−	0.871377i	$-0.663227\pi$
$643$	−24.8814	−0.981226	−0.490613	+	0.871377i	$0.663227\pi$
$644$	2.72824	0.107508
$645$	0	0
$646$	7.74455	0.304705
$647$	−35.6376	−1.40106	−0.700529	−	0.713624i	$-0.747053\pi$
$647$	−35.6376	−1.40106	−0.700529	+	0.713624i	$0.747053\pi$
$648$	−6.68674	−0.262680
$649$	−0.148013	−0.00581000
$650$	0	0
$651$	1.58532	0.0621335
$652$	3.24785	0.127196
$653$	−8.31930	−0.325559	−0.162780	−	0.986662i	$-0.552046\pi$
$653$	−8.31930	−0.325559	−0.162780	+	0.986662i	$0.552046\pi$
$654$	1.32853	0.0519498
$655$	0	0
$656$	31.3641	1.22456
$657$	71.0616	2.77238
$658$	−12.7928	−0.498715
$659$	−3.44669	−0.134264	−0.0671320	−	0.997744i	$-0.521385\pi$
$659$	−3.44669	−0.134264	−0.0671320	+	0.997744i	$0.521385\pi$
$660$	0	0
$661$	−36.6867	−1.42695	−0.713473	−	0.700682i	$-0.752879\pi$
$661$	−36.6867	−1.42695	−0.713473	+	0.700682i	$0.752879\pi$
$662$	25.0441	0.973368
$663$	16.3768	0.636021
$664$	28.1749	1.09340
$665$	0	0
$666$	82.2082	3.18550
$667$	22.9657	0.889234
$668$	−0.443700	−0.0171673
$669$	57.5995	2.22692
$670$	0	0
$671$	−0.257052	−0.00992336
$672$	−11.1729	−0.431005
$673$	41.5075	1.60000	0.799998	−	0.600002i	$-0.204834\pi$
$673$	41.5075	1.60000	0.799998	+	0.600002i	$0.204834\pi$
$674$	−30.2913	−1.16678
$675$	0	0
$676$	9.64882	0.371109
$677$	26.7378	1.02762	0.513809	−	0.857905i	$-0.328234\pi$
$677$	26.7378	1.02762	0.513809	+	0.857905i	$0.328234\pi$
$678$	39.1191	1.50236
$679$	7.58742	0.291179
$680$	0	0
$681$	−40.5875	−1.55532
$682$	0.0454143	0.00173901
$683$	37.8270	1.44741	0.723704	−	0.690110i	$-0.242438\pi$
$683$	37.8270	1.44741	0.723704	+	0.690110i	$0.242438\pi$
$684$	−12.7319	−0.486815
$685$	0	0
$686$	−26.8838	−1.02643
$687$	32.2144	1.22905
$688$	−32.9570	−1.25647
$689$	0.123572	0.00470770
$690$	0	0
$691$	33.0739	1.25819	0.629095	−	0.777328i	$-0.283426\pi$
$691$	33.0739	1.25819	0.629095	+	0.777328i	$0.283426\pi$
$692$	−1.89440	−0.0720144
$693$	0.551937	0.0209664
$694$	53.0331	2.01311
$695$	0	0
$696$	−40.3907	−1.53100
$697$	6.61055	0.250392
$698$	13.6091	0.515112
$699$	−39.7192	−1.50232
$700$	0	0
$701$	−24.7292	−0.934008	−0.467004	−	0.884255i	$-0.654667\pi$
$701$	−24.7292	−0.934008	−0.467004	+	0.884255i	$0.654667\pi$
$702$	−58.1975	−2.19652
$703$	−48.6135	−1.83349
$704$	0.382184	0.0144041
$705$	0	0
$706$	−0.942375	−0.0354668
$707$	16.2379	0.610688
$708$	−2.84115	−0.106777
$709$	40.5902	1.52440	0.762199	−	0.647343i	$-0.224120\pi$
$709$	40.5902	1.52440	0.762199	+	0.647343i	$0.224120\pi$
$710$	0	0
$711$	−5.56495	−0.208702
$712$	4.74237	0.177728
$713$	1.50871	0.0565016
$714$	−6.44450	−0.241179
$715$	0	0
$716$	4.81948	0.180112
$717$	−46.7752	−1.74685
$718$	49.8330	1.85975
$719$	32.1253	1.19807	0.599036	−	0.800722i	$-0.295551\pi$
$719$	32.1253	1.19807	0.599036	+	0.800722i	$0.295551\pi$
$720$	0	0
$721$	11.6666	0.434486
$722$	7.96558	0.296448
$723$	14.1914	0.527782
$724$	11.1164	0.413139
$725$	0	0
$726$	−49.8771	−1.85111
$727$	−11.3089	−0.419425	−0.209713	−	0.977763i	$-0.567253\pi$
$727$	−11.3089	−0.419425	−0.209713	+	0.977763i	$0.567253\pi$
$728$	19.2585	0.713768
$729$	−40.9222	−1.51564
$730$	0	0
$731$	−6.94628	−0.256918
$732$	−4.93419	−0.182373
$733$	−32.5450	−1.20208	−0.601039	−	0.799220i	$-0.705246\pi$
$733$	−32.5450	−1.20208	−0.601039	+	0.799220i	$0.705246\pi$
$734$	2.17523	0.0802892
$735$	0	0
$736$	−10.6330	−0.391938
$737$	−0.496768	−0.0182987
$738$	−54.8142	−2.01774
$739$	−29.6821	−1.09187	−0.545936	−	0.837827i	$-0.683826\pi$
$739$	−29.6821	−1.09187	−0.545936	+	0.837827i	$0.683826\pi$
$740$	0	0
$741$	80.3019	2.94996
$742$	−0.0486272	−0.00178516
$743$	−27.9705	−1.02614	−0.513069	−	0.858348i	$-0.671491\pi$
$743$	−27.9705	−1.02614	−0.513069	+	0.858348i	$0.671491\pi$
$744$	−2.65343	−0.0972795
$745$	0	0
$746$	−45.1126	−1.65169
$747$	−62.2105	−2.27616
$748$	−0.0366022	−0.00133831
$749$	9.01898	0.329546
$750$	0	0
$751$	−52.7167	−1.92366	−0.961829	−	0.273650i	$-0.911769\pi$
$751$	−52.7167	−1.92366	−0.961829	+	0.273650i	$0.911769\pi$
$752$	27.0519	0.986481
$753$	37.0982	1.35193
$754$	−53.2598	−1.93961
$755$	0	0
$756$	4.54052	0.165137
$757$	−7.80911	−0.283827	−0.141913	−	0.989879i	$-0.545325\pi$
$757$	−7.80911	−0.283827	−0.141913	+	0.989879i	$0.545325\pi$
$758$	−38.2861	−1.39061
$759$	0.825420	0.0299608
$760$	0	0
$761$	19.2266	0.696963	0.348481	−	0.937316i	$-0.386697\pi$
$761$	19.2266	0.696963	0.348481	+	0.937316i	$0.386697\pi$
$762$	78.3188	2.83719
$763$	0.416018	0.0150609
$764$	−2.70860	−0.0979937
$765$	0	0
$766$	−38.6450	−1.39630
$767$	11.4033	0.411751
$768$	32.1807	1.16122
$769$	−1.45851	−0.0525953	−0.0262977	−	0.999654i	$-0.508372\pi$
$769$	−1.45851	−0.0525953	−0.0262977	+	0.999654i	$0.508372\pi$
$770$	0	0
$771$	39.1087	1.40846
$772$	1.06080	0.0381790
$773$	−18.8055	−0.676388	−0.338194	−	0.941076i	$-0.609816\pi$
$773$	−18.8055	−0.676388	−0.338194	+	0.941076i	$0.609816\pi$
$774$	57.5981	2.07032
$775$	0	0
$776$	−12.6995	−0.455885
$777$	40.4529	1.45124
$778$	−17.3281	−0.621241
$779$	32.4142	1.16136
$780$	0	0
$781$	0.284201	0.0101695
$782$	−6.13308	−0.219319
$783$	38.2210	1.36591
$784$	23.6372	0.844186
$785$	0	0
$786$	−66.7724	−2.38169
$787$	19.0449	0.678877	0.339439	−	0.940628i	$-0.389763\pi$
$787$	19.0449	0.678877	0.339439	+	0.940628i	$0.389763\pi$
$788$	4.83955	0.172402
$789$	36.3973	1.29578
$790$	0	0
$791$	12.2498	0.435552
$792$	−0.923808	−0.0328261
$793$	19.8040	0.703262
$794$	−42.4155	−1.50527
$795$	0	0
$796$	−0.728307	−0.0258142
$797$	49.6667	1.75929	0.879643	−	0.475634i	$-0.157781\pi$
$797$	49.6667	1.75929	0.879643	+	0.475634i	$0.157781\pi$
$798$	−31.6000	−1.11863
$799$	5.70167	0.201711
$800$	0	0
$801$	−10.4712	−0.369981
$802$	−27.6526	−0.976447
$803$	1.00172	0.0353500
$804$	−9.53562	−0.336296
$805$	0	0
$806$	−3.49886	−0.123242
$807$	−74.3824	−2.61838
$808$	−27.1782	−0.956126
$809$	38.2625	1.34524	0.672619	−	0.739989i	$-0.265169\pi$
$809$	38.2625	1.34524	0.672619	+	0.739989i	$0.265169\pi$
$810$	0	0
$811$	−31.4356	−1.10385	−0.551927	−	0.833893i	$-0.686107\pi$
$811$	−31.4356	−1.10385	−0.551927	+	0.833893i	$0.686107\pi$
$812$	4.15529	0.145822
$813$	60.9529	2.13771
$814$	1.15885	0.0406176
$815$	0	0
$816$	13.6277	0.477064
$817$	−34.0604	−1.19162
$818$	−28.1587	−0.984547
$819$	−42.5229	−1.48587
$820$	0	0
$821$	42.9887	1.50031	0.750157	−	0.661259i	$-0.229978\pi$
$821$	42.9887	1.50031	0.750157	+	0.661259i	$0.229978\pi$
$822$	−6.18014	−0.215557
$823$	12.7450	0.444263	0.222132	−	0.975017i	$-0.428699\pi$
$823$	12.7450	0.444263	0.222132	+	0.975017i	$0.428699\pi$
$824$	−19.5270	−0.680254
$825$	0	0
$826$	−4.48738	−0.156136
$827$	−12.8443	−0.446639	−0.223319	−	0.974745i	$-0.571689\pi$
$827$	−12.8443	−0.446639	−0.223319	+	0.974745i	$0.571689\pi$
$828$	10.0827	0.350397
$829$	−17.2491	−0.599087	−0.299543	−	0.954083i	$-0.596834\pi$
$829$	−17.2491	−0.599087	−0.299543	+	0.954083i	$0.596834\pi$
$830$	0	0
$831$	−5.91424	−0.205163
$832$	−29.4446	−1.02081
$833$	4.98197	0.172615
$834$	−62.3564	−2.15923
$835$	0	0
$836$	−0.179475	−0.00620728
$837$	2.51090	0.0867892
$838$	54.9366	1.89775
$839$	−17.7454	−0.612638	−0.306319	−	0.951929i	$-0.599097\pi$
$839$	−17.7454	−0.612638	−0.306319	+	0.951929i	$0.599097\pi$
$840$	0	0
$841$	5.97821	0.206145
$842$	−26.9570	−0.929000
$843$	−62.4249	−2.15003
$844$	−2.41154	−0.0830086
$845$	0	0
$846$	−47.2779	−1.62545
$847$	−15.6185	−0.536660
$848$	0.102828	0.00353113
$849$	15.2870	0.524648
$850$	0	0
$851$	38.4981	1.31970
$852$	5.45534	0.186897
$853$	−45.1570	−1.54615	−0.773073	−	0.634317i	$-0.781282\pi$
$853$	−45.1570	−1.54615	−0.773073	+	0.634317i	$0.781282\pi$
$854$	−7.79318	−0.266677
$855$	0	0
$856$	−15.0956	−0.515956
$857$	−40.5102	−1.38380	−0.691900	−	0.721993i	$-0.743226\pi$
$857$	−40.5102	−1.38380	−0.691900	+	0.721993i	$0.743226\pi$
$858$	−1.91424	−0.0653510
$859$	−25.4439	−0.868135	−0.434068	−	0.900880i	$-0.642922\pi$
$859$	−25.4439	−0.868135	−0.434068	+	0.900880i	$0.642922\pi$
$860$	0	0
$861$	−26.9729	−0.919235
$862$	−35.6902	−1.21561
$863$	21.6953	0.738517	0.369259	−	0.929327i	$-0.379612\pi$
$863$	21.6953	0.738517	0.369259	+	0.929327i	$0.379612\pi$
$864$	−17.6962	−0.602037
$865$	0	0
$866$	21.0874	0.716581
$867$	2.87228	0.0975476
$868$	0.272978	0.00926548
$869$	−0.0784464	−0.00266111
$870$	0	0
$871$	38.2725	1.29681
$872$	−0.696312	−0.0235801
$873$	28.0406	0.949030
$874$	−30.0730	−1.01723
$875$	0	0
$876$	19.2284	0.649667
$877$	15.4643	0.522191	0.261095	−	0.965313i	$-0.415916\pi$
$877$	15.4643	0.522191	0.261095	+	0.965313i	$0.415916\pi$
$878$	9.05452	0.305575
$879$	−67.0060	−2.26006
$880$	0	0
$881$	−13.2839	−0.447544	−0.223772	−	0.974641i	$-0.571837\pi$
$881$	−13.2839	−0.447544	−0.223772	+	0.974641i	$0.571837\pi$
$882$	−41.3101	−1.39098
$883$	−30.1936	−1.01609	−0.508047	−	0.861329i	$-0.669632\pi$
$883$	−30.1936	−1.01609	−0.508047	+	0.861329i	$0.669632\pi$
$884$	2.81994	0.0948449
$885$	0	0
$886$	−61.3059	−2.05961
$887$	39.7258	1.33386	0.666930	−	0.745120i	$-0.267608\pi$
$887$	39.7258	1.33386	0.666930	+	0.745120i	$0.267608\pi$
$888$	−67.7082	−2.27214
$889$	24.5248	0.822535
$890$	0	0
$891$	0.208126	0.00697250
$892$	9.91815	0.332084
$893$	27.9576	0.935565
$894$	−78.7669	−2.63436
$895$	0	0
$896$	19.3667	0.646997
$897$	−63.5929	−2.12330
$898$	−40.0934	−1.33794
$899$	2.29786	0.0766381
$900$	0	0
$901$	0.0216729	0.000722028 0
$902$	−0.772690	−0.0257278
$903$	28.3428	0.943190
$904$	−20.5031	−0.681923
$905$	0	0
$906$	−16.3159	−0.542058
$907$	−20.4051	−0.677541	−0.338771	−	0.940869i	$-0.610011\pi$
$907$	−20.4051	−0.677541	−0.338771	+	0.940869i	$0.610011\pi$
$908$	−6.98883	−0.231932
$909$	60.0098	1.99040
$910$	0	0
$911$	−44.1137	−1.46155	−0.730776	−	0.682618i	$-0.760841\pi$
$911$	−44.1137	−1.46155	−0.730776	+	0.682618i	$0.760841\pi$
$912$	66.8219	2.21269
$913$	−0.876951	−0.0290228
$914$	14.0586	0.465018
$915$	0	0
$916$	5.54704	0.183279
$917$	−20.9091	−0.690481
$918$	−10.2071	−0.336884
$919$	38.1261	1.25766	0.628832	−	0.777541i	$-0.283533\pi$
$919$	38.1261	1.25766	0.628832	+	0.777541i	$0.283533\pi$
$920$	0	0
$921$	−50.1464	−1.65238
$922$	−21.9736	−0.723661
$923$	−21.8957	−0.720707
$924$	0.149347	0.00491316
$925$	0	0
$926$	59.2418	1.94681
$927$	43.1157	1.41611
$928$	−16.1948	−0.531620
$929$	11.6168	0.381134	0.190567	−	0.981674i	$-0.438967\pi$
$929$	11.6168	0.381134	0.190567	+	0.981674i	$0.438967\pi$
$930$	0	0
$931$	24.4286	0.800614
$932$	−6.83931	−0.224029
$933$	8.58801	0.281159
$934$	−17.7067	−0.579380
$935$	0	0
$936$	71.1730	2.32636
$937$	−39.8889	−1.30311	−0.651557	−	0.758600i	$-0.725884\pi$
$937$	−39.8889	−1.30311	−0.651557	+	0.758600i	$0.725884\pi$
$938$	−15.0608	−0.491752
$939$	−22.2897	−0.727396
$940$	0	0
$941$	−21.8113	−0.711028	−0.355514	−	0.934671i	$-0.615694\pi$
$941$	−21.8113	−0.711028	−0.355514	+	0.934671i	$0.615694\pi$
$942$	23.5586	0.767582
$943$	−25.6695	−0.835914
$944$	9.48910	0.308844
$945$	0	0
$946$	0.811933	0.0263982
$947$	18.2678	0.593625	0.296812	−	0.954936i	$-0.404076\pi$
$947$	18.2678	0.593625	0.296812	+	0.954936i	$0.404076\pi$
$948$	−1.50580	−0.0489063
$949$	−77.1757	−2.50523
$950$	0	0
$951$	−19.7886	−0.641688
$952$	3.37769	0.109472
$953$	−27.5942	−0.893865	−0.446932	−	0.894568i	$-0.647484\pi$
$953$	−27.5942	−0.893865	−0.446932	+	0.894568i	$0.647484\pi$
$954$	−0.179710	−0.00581832
$955$	0	0
$956$	−8.05430	−0.260495
$957$	1.25717	0.0406385
$958$	8.15866	0.263594
$959$	−1.93525	−0.0624925
$960$	0	0
$961$	−30.8490	−0.995130
$962$	−89.2813	−2.87854
$963$	33.3311	1.07408
$964$	2.44363	0.0787041
$965$	0	0
$966$	25.0247	0.805157
$967$	36.3289	1.16826	0.584129	−	0.811661i	$-0.301436\pi$
$967$	36.3289	1.16826	0.584129	+	0.811661i	$0.301436\pi$
$968$	26.1416	0.840223
$969$	14.0839	0.452441
$970$	0	0
$971$	45.2409	1.45185	0.725925	−	0.687774i	$-0.241412\pi$
$971$	45.2409	1.45185	0.725925	+	0.687774i	$0.241412\pi$
$972$	−5.59370	−0.179418
$973$	−19.5263	−0.625985
$974$	53.7888	1.72351
$975$	0	0
$976$	16.4796	0.527499
$977$	10.8393	0.346780	0.173390	−	0.984853i	$-0.444528\pi$
$977$	10.8393	0.346780	0.173390	+	0.984853i	$0.444528\pi$
$978$	29.7909	0.952607
$979$	−0.147607	−0.00471755
$980$	0	0
$981$	1.53746	0.0490874
$982$	−28.9116	−0.922606
$983$	−3.69018	−0.117699	−0.0588493	−	0.998267i	$-0.518743\pi$
$983$	−3.69018	−0.117699	−0.0588493	+	0.998267i	$0.518743\pi$
$984$	45.1461	1.43920
$985$	0	0
$986$	−9.34109	−0.297481
$987$	−23.2644	−0.740515
$988$	13.8273	0.439905
$989$	26.9732	0.857698
$990$	0	0
$991$	36.6378	1.16384	0.581919	−	0.813247i	$-0.302302\pi$
$991$	36.6378	1.16384	0.581919	+	0.813247i	$0.302302\pi$
$992$	−1.06390	−0.0337790
$993$	45.5442	1.44530
$994$	8.61630	0.273292
$995$	0	0
$996$	−16.8334	−0.533386
$997$	32.6021	1.03252	0.516259	−	0.856432i	$-0.327324\pi$
$997$	32.6021	1.03252	0.516259	+	0.856432i	$0.327324\pi$
$998$	64.4077	2.03879
$999$	64.0711	2.02712

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.a.h.1.3		4
3.2	odd	2		3825.2.a.bh.1.2		4
4.3	odd	2		6800.2.a.bt.1.1		4
5.2	odd	4		85.2.b.a.69.6	yes	8
5.3	odd	4		85.2.b.a.69.3	✓	8
5.4	even	2		425.2.a.g.1.2		4
15.2	even	4		765.2.b.c.154.3		8
15.8	even	4		765.2.b.c.154.6		8
15.14	odd	2		3825.2.a.bj.1.3		4
17.16	even	2		7225.2.a.w.1.3		4
20.3	even	4		1360.2.e.d.1089.1		8
20.7	even	4		1360.2.e.d.1089.8		8
20.19	odd	2		6800.2.a.bw.1.4		4
85.33	odd	4		1445.2.b.e.579.3		8
85.67	odd	4		1445.2.b.e.579.6		8
85.84	even	2		7225.2.a.v.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.b.a.69.3	✓	8	5.3	odd	4
85.2.b.a.69.6	yes	8	5.2	odd	4
425.2.a.g.1.2		4	5.4	even	2
425.2.a.h.1.3		4	1.1	even	1	trivial
765.2.b.c.154.3		8	15.2	even	4
765.2.b.c.154.6		8	15.8	even	4
1360.2.e.d.1089.1		8	20.3	even	4
1360.2.e.d.1089.8		8	20.7	even	4
1445.2.b.e.579.3		8	85.33	odd	4
1445.2.b.e.579.6		8	85.67	odd	4
3825.2.a.bh.1.2		4	3.2	odd	2
3825.2.a.bj.1.3		4	15.14	odd	2
6800.2.a.bt.1.1		4	4.3	odd	2
6800.2.a.bw.1.4		4	20.19	odd	2
7225.2.a.v.1.2		4	85.84	even	2
7225.2.a.w.1.3		4	17.16	even	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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q+1.57942 q^{2} +2.87228 q^{3} +0.494582 q^{4} +4.53654 q^{6} +1.42058 q^{7} -2.37769 q^{8} +5.24997 q^{9} +0.0740063 q^{11} +1.42058 q^{12} -5.70167 q^{13} +2.24369 q^{14} -4.74455 q^{16} -1.00000 q^{17} +8.29193 q^{18} -4.90340 q^{19} +4.08029 q^{21} +0.116887 q^{22} +3.88311 q^{23} -6.82940 q^{24} -9.00536 q^{26} +6.46254 q^{27} +0.702591 q^{28} +5.91424 q^{29} +0.388531 q^{31} -2.73827 q^{32} +0.212567 q^{33} -1.57942 q^{34} +2.59654 q^{36} +9.91424 q^{37} -7.74455 q^{38} -16.3768 q^{39} -6.61055 q^{41} +6.44450 q^{42} +6.94628 q^{43} +0.0366022 q^{44} +6.13308 q^{46} -5.70167 q^{47} -13.6277 q^{48} -4.98197 q^{49} -2.87228 q^{51} -2.81994 q^{52} -0.0216729 q^{53} +10.2071 q^{54} -3.37769 q^{56} -14.0839 q^{57} +9.34109 q^{58} -2.00000 q^{59} -3.47337 q^{61} +0.613655 q^{62} +7.45798 q^{63} +5.16421 q^{64} +0.335733 q^{66} -6.71251 q^{67} -0.494582 q^{68} +11.1534 q^{69} +3.84023 q^{71} -12.4828 q^{72} +13.5356 q^{73} +15.6588 q^{74} -2.42513 q^{76} +0.105132 q^{77} -25.8659 q^{78} -1.06000 q^{79} +2.81228 q^{81} -10.4409 q^{82} -11.8497 q^{83} +2.01803 q^{84} +10.9711 q^{86} +16.9873 q^{87} -0.175964 q^{88} -1.99452 q^{89} -8.09965 q^{91} +1.92052 q^{92} +1.11597 q^{93} -9.00536 q^{94} -7.86508 q^{96} +5.34109 q^{97} -7.86864 q^{98} +0.388531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} + 10 q^{7} + 4 q^{9} - 2 q^{11} + 10 q^{12} + 6 q^{13} - 6 q^{14} - 4 q^{16} - 4 q^{17} - 4 q^{18} + 4 q^{19} + 12 q^{21} + 12 q^{22} + 4 q^{23} - 6 q^{24} + 10 q^{27}+ \cdots - 12 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^3 + 4 * q^4 + 10 * q^7 + 4 * q^9 - 2 * q^11 + 10 * q^12 + 6 * q^13 - 6 * q^14 - 4 * q^16 - 4 * q^17 - 4 * q^18 + 4 * q^19 + 12 * q^21 + 12 * q^22 + 4 * q^23 - 6 * q^24 + 10 * q^27 + 8 * q^28 - 4 * q^29 - 12 * q^31 + 2 * q^32 + 2 * q^33 - 2 * q^34 + 12 * q^37 - 16 * q^38 - 22 * q^39 - 6 * q^41 - 18 * q^42 + 18 * q^43 + 16 * q^44 - 4 * q^46 + 6 * q^47 - 28 * q^48 + 8 * q^49 - 4 * q^51 + 2 * q^52 + 8 * q^53 + 10 * q^54 - 4 * q^56 - 16 * q^57 + 12 * q^58 - 8 * q^59 + 6 * q^61 - 14 * q^62 + 16 * q^63 - 24 * q^64 + 12 * q^66 + 6 * q^67 - 4 * q^68 + 4 * q^69 - 10 * q^71 - 22 * q^72 + 2 * q^73 + 20 * q^74 - 12 * q^76 - 18 * q^77 - 30 * q^78 - 12 * q^79 - 4 * q^81 - 34 * q^82 - 14 * q^83 + 36 * q^84 + 20 * q^86 + 4 * q^87 + 14 * q^88 + 24 * q^89 + 18 * q^91 - 22 * q^92 - 18 * q^93 + 8 * q^96 - 4 * q^97 - 60 * q^98 - 12 * q^99

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