LMFDB - Embedded newform 425.2.a.h.1.4

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			1.4
Root			$0.796815$ 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)

$f(q)$	$=$	\(q+2.31627 q^{2} +0.203185 q^{3} +3.36509 q^{4} +0.470630 q^{6} +0.683735 q^{7} +3.16190 q^{8} -2.95872 q^{9} +3.68135 q^{11} +0.683735 q^{12} +4.43927 q^{13} +1.58371 q^{14} +0.593630 q^{16} -1.00000 q^{17} -6.85317 q^{18} -1.03890 q^{19} +0.138925 q^{21} +8.52699 q^{22} -4.52699 q^{23} +0.642450 q^{24} +10.2825 q^{26} -1.21072 q^{27} +2.30083 q^{28} -3.69127 q^{29} -10.8921 q^{31} -4.94880 q^{32} +0.747995 q^{33} -2.31627 q^{34} -9.95633 q^{36} +0.308729 q^{37} -2.40637 q^{38} +0.901992 q^{39} -6.15198 q^{41} +0.321786 q^{42} +7.88454 q^{43} +12.3881 q^{44} -10.4857 q^{46} +4.43927 q^{47} +0.120617 q^{48} -6.53251 q^{49} -0.203185 q^{51} +14.9385 q^{52} +11.4603 q^{53} -2.80435 q^{54} +2.16190 q^{56} -0.211089 q^{57} -8.54996 q^{58} -2.00000 q^{59} +9.94089 q^{61} -25.2289 q^{62} -2.02298 q^{63} -12.6500 q^{64} +1.73255 q^{66} +9.16944 q^{67} -3.36509 q^{68} -0.919815 q^{69} -9.37262 q^{71} -9.35517 q^{72} -2.26946 q^{73} +0.715099 q^{74} -3.49599 q^{76} +2.51707 q^{77} +2.08925 q^{78} +7.42696 q^{79} +8.63015 q^{81} -14.2496 q^{82} -8.92344 q^{83} +0.467493 q^{84} +18.2627 q^{86} -0.750010 q^{87} +11.6401 q^{88} +11.5523 q^{89} +3.03528 q^{91} -15.2337 q^{92} -2.21310 q^{93} +10.2825 q^{94} -1.00552 q^{96} -12.5500 q^{97} -15.1310 q^{98} -10.8921 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.31627	1.63785	0.818924	−	0.573903i	$-0.194571\pi$
$2$	2.31627	1.63785	0.818924	+	0.573903i	$0.194571\pi$
$3$	0.203185	0.117309	0.0586544	−	0.998278i	$-0.481319\pi$
$3$	0.203185	0.117309	0.0586544	+	0.998278i	$0.481319\pi$
$4$	3.36509	1.68254
$5$	0	0
$5$	0	0
$6$	0.470630	0.192134
$7$	0.683735	0.258427	0.129214	−	0.991617i	$-0.458755\pi$
$7$	0.683735	0.258427	0.129214	+	0.991617i	$0.458755\pi$
$8$	3.16190	1.11790
$9$	−2.95872	−0.986239
$10$	0	0
$11$	3.68135	1.10997	0.554985	−	0.831861i	$-0.312724\pi$
$11$	3.68135	1.10997	0.554985	+	0.831861i	$0.312724\pi$
$12$	0.683735	0.197377
$13$	4.43927	1.23123	0.615615	−	0.788047i	$-0.288908\pi$
$13$	4.43927	1.23123	0.615615	+	0.788047i	$0.288908\pi$
$14$	1.58371	0.423264
$15$	0	0
$16$	0.593630	0.148408
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−6.85317	−1.61531
$19$	−1.03890	−0.238340	−0.119170	−	0.992874i	$-0.538023\pi$
$19$	−1.03890	−0.238340	−0.119170	+	0.992874i	$0.538023\pi$
$20$	0	0
$21$	0.138925	0.0303158
$22$	8.52699	1.81796
$23$	−4.52699	−0.943942	−0.471971	−	0.881614i	$-0.656457\pi$
$23$	−4.52699	−0.943942	−0.471971	+	0.881614i	$0.656457\pi$
$24$	0.642450	0.131140
$25$	0	0
$26$	10.2825	2.01657
$27$	−1.21072	−0.233003
$28$	2.30083	0.434815
$29$	−3.69127	−0.685452	−0.342726	−	0.939435i	$-0.611350\pi$
$29$	−3.69127	−0.685452	−0.342726	+	0.939435i	$0.611350\pi$
$30$	0	0
$31$	−10.8921	−1.95627	−0.978137	−	0.207962i	$-0.933317\pi$
$31$	−10.8921	−1.95627	−0.978137	+	0.207962i	$0.933317\pi$
$32$	−4.94880	−0.874832
$33$	0.747995	0.130209
$34$	−2.31627	−0.397236
$35$	0	0
$36$	−9.95633	−1.65939
$37$	0.308729	0.0507548	0.0253774	−	0.999678i	$-0.491921\pi$
$37$	0.308729	0.0507548	0.0253774	+	0.999678i	$0.491921\pi$
$38$	−2.40637	−0.390365
$39$	0.901992	0.144434
$40$	0	0
$41$	−6.15198	−0.960778	−0.480389	−	0.877056i	$-0.659505\pi$
$41$	−6.15198	−0.960778	−0.480389	+	0.877056i	$0.659505\pi$
$42$	0.321786	0.0496527
$43$	7.88454	1.20238	0.601190	−	0.799106i	$-0.294693\pi$
$43$	7.88454	1.20238	0.601190	+	0.799106i	$0.294693\pi$
$44$	12.3881	1.86757
$45$	0	0
$46$	−10.4857	−1.54603
$47$	4.43927	0.647533	0.323767	−	0.946137i	$-0.395051\pi$
$47$	4.43927	0.647533	0.323767	+	0.946137i	$0.395051\pi$
$48$	0.120617	0.0174095
$49$	−6.53251	−0.933215
$50$	0	0
$51$	−0.203185	−0.0284516
$52$	14.9385	2.07160
$53$	11.4603	1.57420	0.787100	−	0.616826i	$-0.211582\pi$
$53$	11.4603	1.57420	0.787100	+	0.616826i	$0.211582\pi$
$54$	−2.80435	−0.381624
$55$	0	0
$56$	2.16190	0.288896
$57$	−0.211089	−0.0279594
$58$	−8.54996	−1.12267
$59$	−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
$61$	9.94089	1.27280	0.636400	−	0.771359i	$-0.280423\pi$
$61$	9.94089	1.27280	0.636400	+	0.771359i	$0.280423\pi$
$62$	−25.2289	−3.20408
$63$	−2.02298	−0.254871
$64$	−12.6500	−1.58125
$65$	0	0
$66$	1.73255	0.213263
$67$	9.16944	1.12023	0.560113	−	0.828417i	$-0.310758\pi$
$67$	9.16944	1.12023	0.560113	+	0.828417i	$0.310758\pi$
$68$	−3.36509	−0.408077
$69$	−0.919815	−0.110733
$70$	0	0
$71$	−9.37262	−1.11233	−0.556163	−	0.831073i	$-0.687727\pi$
$71$	−9.37262	−1.11233	−0.556163	+	0.831073i	$0.687727\pi$
$72$	−9.35517	−1.10252
$73$	−2.26946	−0.265620	−0.132810	−	0.991141i	$-0.542400\pi$
$73$	−2.26946	−0.265620	−0.132810	+	0.991141i	$0.542400\pi$
$74$	0.715099	0.0831286
$75$	0	0
$76$	−3.49599	−0.401018
$77$	2.51707	0.286846
$78$	2.08925	0.236561
$79$	7.42696	0.835599	0.417799	−	0.908539i	$-0.362802\pi$
$79$	7.42696	0.835599	0.417799	+	0.908539i	$0.362802\pi$
$80$	0	0
$81$	8.63015	0.958905
$82$	−14.2496	−1.57361
$83$	−8.92344	−0.979474	−0.489737	−	0.871870i	$-0.662907\pi$
$83$	−8.92344	−0.979474	−0.489737	+	0.871870i	$0.662907\pi$
$84$	0.467493	0.0510077
$85$	0	0
$86$	18.2627	1.96932
$87$	−0.750010	−0.0804096
$88$	11.6401	1.24084
$89$	11.5523	1.22455	0.612273	−	0.790646i	$-0.290255\pi$
$89$	11.5523	1.22455	0.612273	+	0.790646i	$0.290255\pi$
$90$	0	0
$91$	3.03528	0.318184
$92$	−15.2337	−1.58822
$93$	−2.21310	−0.229488
$94$	10.2825	1.06056
$95$	0	0
$96$	−1.00552	−0.102626
$97$	−12.5500	−1.27426	−0.637128	−	0.770758i	$-0.719878\pi$
$97$	−12.5500	−1.27426	−0.637128	+	0.770758i	$0.719878\pi$
$98$	−15.1310	−1.52846
$99$	−10.8921	−1.09469
$100$	0	0
$101$	−6.78653	−0.675285	−0.337642	−	0.941274i	$-0.609629\pi$
$101$	−6.78653	−0.675285	−0.337642	+	0.941274i	$0.609629\pi$
$102$	−0.470630	−0.0465993
$103$	8.74799	0.861966	0.430983	−	0.902360i	$-0.358167\pi$
$103$	8.74799	0.861966	0.430983	+	0.902360i	$0.358167\pi$
$104$	14.0365	1.37639
$105$	0	0
$106$	26.5452	2.57830
$107$	8.10078	0.783132	0.391566	−	0.920150i	$-0.371934\pi$
$107$	8.10078	0.783132	0.391566	+	0.920150i	$0.371934\pi$
$108$	−4.07418	−0.392038
$109$	−3.11308	−0.298179	−0.149090	−	0.988824i	$-0.547634\pi$
$109$	−3.11308	−0.298179	−0.149090	+	0.988824i	$0.547634\pi$
$110$	0	0
$111$	0.0627291	0.00595399
$112$	0.405885	0.0383526
$113$	−6.93287	−0.652190	−0.326095	−	0.945337i	$-0.605733\pi$
$113$	−6.93287	−0.652190	−0.326095	+	0.945337i	$0.605733\pi$
$114$	−0.488938	−0.0457932
$115$	0	0
$116$	−12.4214	−1.15330
$117$	−13.1345	−1.21429
$118$	−4.63253	−0.426459
$119$	−0.683735	−0.0626778
$120$	0	0
$121$	2.55235	0.232031
$122$	23.0257	2.08465
$123$	−1.24999	−0.112708
$124$	−36.6528	−3.29151
$125$	0	0
$126$	−4.68575	−0.417440
$127$	21.1496	1.87672	0.938362	−	0.345655i	$-0.112343\pi$
$127$	21.1496	1.87672	0.938362	+	0.345655i	$0.112343\pi$
$128$	−19.4031	−1.71501
$129$	1.60202	0.141050
$130$	0	0
$131$	8.71186	0.761159	0.380580	−	0.924748i	$-0.375725\pi$
$131$	8.71186	0.761159	0.380580	+	0.924748i	$0.375725\pi$
$132$	2.51707	0.219083
$133$	−0.710332	−0.0615936
$134$	21.2388	1.83476
$135$	0	0
$136$	−3.16190	−0.271131
$137$	0.243985	0.0208450	0.0104225	−	0.999946i	$-0.496682\pi$
$137$	0.243985	0.0208450	0.0104225	+	0.999946i	$0.496682\pi$
$138$	−2.13054	−0.181363
$139$	12.6884	1.07622	0.538108	−	0.842876i	$-0.319139\pi$
$139$	12.6884	1.07622	0.538108	+	0.842876i	$0.319139\pi$
$140$	0	0
$141$	0.901992	0.0759614
$142$	−21.7095	−1.82182
$143$	16.3425	1.36663
$144$	−1.75638	−0.146365
$145$	0	0
$146$	−5.25667	−0.435045
$147$	−1.32731	−0.109474
$148$	1.03890	0.0853971
$149$	12.0103	0.983923	0.491961	−	0.870617i	$-0.336280\pi$
$149$	12.0103	0.983923	0.491961	+	0.870617i	$0.336280\pi$
$150$	0	0
$151$	8.95633	0.728856	0.364428	−	0.931232i	$-0.381265\pi$
$151$	8.95633	0.728856	0.364428	+	0.931232i	$0.381265\pi$
$152$	−3.28490	−0.266441
$153$	2.95872	0.239198
$154$	5.83020	0.469811
$155$	0	0
$156$	3.03528	0.243017
$157$	−19.9127	−1.58920	−0.794602	−	0.607131i	$-0.792320\pi$
$157$	−19.9127	−1.58920	−0.794602	+	0.607131i	$0.792320\pi$
$158$	17.2028	1.36858
$159$	2.32857	0.184667
$160$	0	0
$161$	−3.09526	−0.243940
$162$	19.9897	1.57054
$163$	22.4011	1.75459	0.877296	−	0.479951i	$-0.159345\pi$
$163$	22.4011	1.75459	0.877296	+	0.479951i	$0.159345\pi$
$164$	−20.7019	−1.61655
$165$	0	0
$166$	−20.6690	−1.60423
$167$	−4.58133	−0.354514	−0.177257	−	0.984165i	$-0.556722\pi$
$167$	−4.58133	−0.354514	−0.177257	+	0.984165i	$0.556722\pi$
$168$	0.439266	0.0338901
$169$	6.70708	0.515929
$170$	0	0
$171$	3.07381	0.235060
$172$	26.5321	2.02306
$173$	−8.09764	−0.615652	−0.307826	−	0.951443i	$-0.599601\pi$
$173$	−8.09764	−0.615652	−0.307826	+	0.951443i	$0.599601\pi$
$174$	−1.73722	−0.131699
$175$	0	0
$176$	2.18536	0.164728
$177$	−0.406370	−0.0305446
$178$	26.7583	2.00562
$179$	4.40637	0.329348	0.164674	−	0.986348i	$-0.447343\pi$
$179$	4.40637	0.329348	0.164674	+	0.986348i	$0.447343\pi$
$180$	0	0
$181$	−5.93727	−0.441314	−0.220657	−	0.975351i	$-0.570820\pi$
$181$	−5.93727	−0.441314	−0.220657	+	0.975351i	$0.570820\pi$
$182$	7.03051	0.521136
$183$	2.01984	0.149311
$184$	−14.3139	−1.05523
$185$	0	0
$186$	−5.12614	−0.375867
$187$	−3.68135	−0.269207
$188$	14.9385	1.08950
$189$	−0.827812	−0.0602144
$190$	0	0
$191$	−9.89759	−0.716165	−0.358082	−	0.933690i	$-0.616569\pi$
$191$	−9.89759	−0.716165	−0.358082	+	0.933690i	$0.616569\pi$
$192$	−2.57029	−0.185494
$193$	−8.47578	−0.610100	−0.305050	−	0.952336i	$-0.598673\pi$
$193$	−8.47578	−0.610100	−0.305050	+	0.952336i	$0.598673\pi$
$194$	−29.0690	−2.08704
$195$	0	0
$196$	−21.9824	−1.57017
$197$	13.5381	0.964553	0.482276	−	0.876019i	$-0.339810\pi$
$197$	13.5381	0.964553	0.482276	+	0.876019i	$0.339810\pi$
$198$	−25.2289	−1.79294
$199$	−9.15388	−0.648901	−0.324451	−	0.945903i	$-0.605179\pi$
$199$	−9.15388	−0.648901	−0.324451	+	0.945903i	$0.605179\pi$
$200$	0	0
$201$	1.86309	0.131412
$202$	−15.7194	−1.10601
$203$	−2.52385	−0.177139
$204$	−0.683735	−0.0478710
$205$	0	0
$206$	20.2627	1.41177
$207$	13.3941	0.930952
$208$	2.63528	0.182724
$209$	−3.82456	−0.264550
$210$	0	0
$211$	7.72465	0.531787	0.265893	−	0.964002i	$-0.414333\pi$
$211$	7.72465	0.531787	0.265893	+	0.964002i	$0.414333\pi$
$212$	38.5650	2.64866
$213$	−1.90438	−0.130486
$214$	18.7636	1.28265
$215$	0	0
$216$	−3.82818	−0.260475
$217$	−7.44729	−0.505555
$218$	−7.21072	−0.488372
$219$	−0.461120	−0.0311596
$220$	0	0
$221$	−4.43927	−0.298617
$222$	0.145297	0.00975172
$223$	−13.7194	−0.918719	−0.459359	−	0.888250i	$-0.651921\pi$
$223$	−13.7194	−0.918719	−0.459359	+	0.888250i	$0.651921\pi$
$224$	−3.38366	−0.226080
$225$	0	0
$226$	−16.0584	−1.06819
$227$	−1.80044	−0.119499	−0.0597496	−	0.998213i	$-0.519030\pi$
$227$	−1.80044	−0.119499	−0.0597496	+	0.998213i	$0.519030\pi$
$228$	−0.710332	−0.0470429
$229$	−3.24838	−0.214659	−0.107330	−	0.994223i	$-0.534230\pi$
$229$	−3.24838	−0.214659	−0.107330	+	0.994223i	$0.534230\pi$
$230$	0	0
$231$	0.511430	0.0336496
$232$	−11.6714	−0.766267
$233$	5.38254	0.352622	0.176311	−	0.984335i	$-0.443584\pi$
$233$	5.38254	0.352622	0.176311	+	0.984335i	$0.443584\pi$
$234$	−30.4230	−1.98882
$235$	0	0
$236$	−6.73017	−0.438097
$237$	1.50905	0.0980231
$238$	−1.58371	−0.102657
$239$	−3.62071	−0.234204	−0.117102	−	0.993120i	$-0.537361\pi$
$239$	−3.62071	−0.234204	−0.117102	+	0.993120i	$0.537361\pi$
$240$	0	0
$241$	−7.66781	−0.493927	−0.246964	−	0.969025i	$-0.579433\pi$
$241$	−7.66781	−0.493927	−0.246964	+	0.969025i	$0.579433\pi$
$242$	5.91191	0.380032
$243$	5.38568	0.345491
$244$	33.4520	2.14154
$245$	0	0
$246$	−2.89531	−0.184598
$247$	−4.61196	−0.293452
$248$	−34.4397	−2.18692
$249$	−1.81311	−0.114901
$250$	0	0
$251$	−6.04595	−0.381617	−0.190809	−	0.981627i	$-0.561111\pi$
$251$	−6.04595	−0.381617	−0.190809	+	0.981627i	$0.561111\pi$
$252$	−6.80749	−0.428831
$253$	−16.6654	−1.04775
$254$	48.9881	3.07379
$255$	0	0
$256$	−19.6428	−1.22768
$257$	−6.13054	−0.382412	−0.191206	−	0.981550i	$-0.561240\pi$
$257$	−6.13054	−0.382412	−0.191206	+	0.981550i	$0.561240\pi$
$258$	3.71070	0.231018
$259$	0.211089	0.0131164
$260$	0	0
$261$	10.9214	0.676019
$262$	20.1790	1.24666
$263$	−12.3012	−0.758525	−0.379263	−	0.925289i	$-0.623822\pi$
$263$	−12.3012	−0.758525	−0.379263	+	0.925289i	$0.623822\pi$
$264$	2.36509	0.145561
$265$	0	0
$266$	−1.64532	−0.100881
$267$	2.34726	0.143650
$268$	30.8559	1.88483
$269$	9.92508	0.605143	0.302572	−	0.953127i	$-0.402155\pi$
$269$	9.92508	0.605143	0.302572	+	0.953127i	$0.402155\pi$
$270$	0	0
$271$	20.3040	1.23338	0.616689	−	0.787207i	$-0.288474\pi$
$271$	20.3040	1.23338	0.616689	+	0.787207i	$0.288474\pi$
$272$	−0.593630	−0.0359941
$273$	0.616723	0.0373258
$274$	0.565133	0.0341410
$275$	0	0
$276$	−3.09526	−0.186313
$277$	18.1671	1.09155	0.545776	−	0.837931i	$-0.316235\pi$
$277$	18.1671	1.09155	0.545776	+	0.837931i	$0.316235\pi$
$278$	29.3897	1.76268
$279$	32.2265	1.92935
$280$	0	0
$281$	10.6983	0.638208	0.319104	−	0.947720i	$-0.396618\pi$
$281$	10.6983	0.638208	0.319104	+	0.947720i	$0.396618\pi$
$282$	2.08925	0.124413
$283$	10.0773	0.599034	0.299517	−	0.954091i	$-0.403174\pi$
$283$	10.0773	0.599034	0.299517	+	0.954091i	$0.403174\pi$
$284$	−31.5397	−1.87154
$285$	0	0
$286$	37.8536	2.23833
$287$	−4.20632	−0.248291
$288$	14.6421	0.862793
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.54996	−0.149481
$292$	−7.63693	−0.446918
$293$	−20.5349	−1.19966	−0.599831	−	0.800127i	$-0.704765\pi$
$293$	−20.5349	−1.19966	−0.599831	+	0.800127i	$0.704765\pi$
$294$	−3.07439	−0.179302
$295$	0	0
$296$	0.976172	0.0567388
$297$	−4.45709	−0.258627
$298$	27.8191	1.61151
$299$	−20.0965	−1.16221
$300$	0	0
$301$	5.39093	0.310728
$302$	20.7452	1.19375
$303$	−1.37892	−0.0792169
$304$	−0.616723	−0.0353715
$305$	0	0
$306$	6.85317	0.391770
$307$	13.1177	0.748669	0.374335	−	0.927294i	$-0.377871\pi$
$307$	13.1177	0.748669	0.374335	+	0.927294i	$0.377871\pi$
$308$	8.47015	0.482631
$309$	1.77746	0.101116
$310$	0	0
$311$	−12.3646	−0.701132	−0.350566	−	0.936538i	$-0.614011\pi$
$311$	−12.3646	−0.701132	−0.350566	+	0.936538i	$0.614011\pi$
$312$	2.85201	0.161463
$313$	−5.16000	−0.291661	−0.145830	−	0.989310i	$-0.546585\pi$
$313$	−5.16000	−0.291661	−0.145830	+	0.989310i	$0.546585\pi$
$314$	−46.1230	−2.60287
$315$	0	0
$316$	24.9924	1.40593
$317$	−23.7655	−1.33480	−0.667400	−	0.744699i	$-0.732593\pi$
$317$	−23.7655	−1.33480	−0.667400	+	0.744699i	$0.732593\pi$
$318$	5.39358	0.302457
$319$	−13.5889	−0.760830
$320$	0	0
$321$	1.64596	0.0918683
$322$	−7.16944	−0.399537
$323$	1.03890	0.0578060
$324$	29.0412	1.61340
$325$	0	0
$326$	51.8869	2.87375
$327$	−0.632531	−0.0349790
$328$	−19.4520	−1.07405
$329$	3.03528	0.167340
$330$	0	0
$331$	20.8341	1.14514	0.572572	−	0.819854i	$-0.305946\pi$
$331$	20.8341	1.14514	0.572572	+	0.819854i	$0.305946\pi$
$332$	−30.0281	−1.64801
$333$	−0.913442	−0.0500563
$334$	−10.6116	−0.580639
$335$	0	0
$336$	0.0824698	0.00449910
$337$	14.3080	0.779406	0.389703	−	0.920941i	$-0.372578\pi$
$337$	14.3080	0.779406	0.389703	+	0.920941i	$0.372578\pi$
$338$	15.5354	0.845013
$339$	−1.40865	−0.0765076
$340$	0	0
$341$	−40.0975	−2.17140
$342$	7.11976	0.384993
$343$	−9.25264	−0.499596
$344$	24.9301	1.34414
$345$	0	0
$346$	−18.7563	−1.00834
$347$	−21.9988	−1.18096	−0.590478	−	0.807054i	$-0.701061\pi$
$347$	−21.9988	−1.18096	−0.590478	+	0.807054i	$0.701061\pi$
$348$	−2.52385	−0.135293
$349$	−6.06199	−0.324491	−0.162246	−	0.986750i	$-0.551874\pi$
$349$	−6.06199	−0.324491	−0.162246	+	0.986750i	$0.551874\pi$
$350$	0	0
$351$	−5.37471	−0.286881
$352$	−18.2183	−0.971036
$353$	−20.8785	−1.11125	−0.555626	−	0.831432i	$-0.687521\pi$
$353$	−20.8785	−1.11125	−0.555626	+	0.831432i	$0.687521\pi$
$354$	−0.941260	−0.0500274
$355$	0	0
$356$	38.8746	2.06035
$357$	−0.138925	−0.00735267
$358$	10.2063	0.539421
$359$	18.4842	0.975557	0.487779	−	0.872967i	$-0.337807\pi$
$359$	18.4842	0.975557	0.487779	+	0.872967i	$0.337807\pi$
$360$	0	0
$361$	−17.9207	−0.943194
$362$	−13.7523	−0.722805
$363$	0.518598	0.0272193
$364$	10.2140	0.535358
$365$	0	0
$366$	4.67848	0.244548
$367$	23.6044	1.23214	0.616070	−	0.787691i	$-0.288724\pi$
$367$	23.6044	1.23214	0.616070	+	0.787691i	$0.288724\pi$
$368$	−2.68736	−0.140088
$369$	18.2020	0.947556
$370$	0	0
$371$	7.83583	0.406816
$372$	−7.44729	−0.386124
$373$	18.0123	0.932643	0.466321	−	0.884615i	$-0.345579\pi$
$373$	18.0123	0.932643	0.466321	+	0.884615i	$0.345579\pi$
$374$	−8.52699	−0.440920
$375$	0	0
$376$	14.0365	0.723878
$377$	−16.3865	−0.843949
$378$	−1.91743	−0.0986221
$379$	−5.74523	−0.295112	−0.147556	−	0.989054i	$-0.547141\pi$
$379$	−5.74523	−0.295112	−0.147556	+	0.989054i	$0.547141\pi$
$380$	0	0
$381$	4.29728	0.220156
$382$	−22.9255	−1.17297
$383$	35.3281	1.80518	0.902591	−	0.430499i	$-0.141663\pi$
$383$	35.3281	1.80518	0.902591	+	0.430499i	$0.141663\pi$
$384$	−3.94242	−0.201186
$385$	0	0
$386$	−19.6322	−0.999251
$387$	−23.3281	−1.18583
$388$	−42.2317	−2.14399
$389$	−18.2627	−0.925955	−0.462977	−	0.886370i	$-0.653219\pi$
$389$	−18.2627	−0.925955	−0.462977	+	0.886370i	$0.653219\pi$
$390$	0	0
$391$	4.52699	0.228940
$392$	−20.6551	−1.04324
$393$	1.77012	0.0892907
$394$	31.3579	1.57979
$395$	0	0
$396$	−36.6528	−1.84187
$397$	−4.64092	−0.232921	−0.116461	−	0.993195i	$-0.537155\pi$
$397$	−4.64092	−0.232921	−0.116461	+	0.993195i	$0.537155\pi$
$398$	−21.2028	−1.06280
$399$	−0.144329	−0.00722548
$400$	0	0
$401$	−27.4049	−1.36853	−0.684267	−	0.729232i	$-0.739878\pi$
$401$	−27.4049	−1.36853	−0.684267	+	0.729232i	$0.739878\pi$
$402$	4.31541	0.215233
$403$	−48.3528	−2.40862
$404$	−22.8372	−1.13620
$405$	0	0
$406$	−5.84590	−0.290127
$407$	1.13654	0.0563363
$408$	−0.642450	−0.0318060
$409$	1.38254	0.0683623	0.0341811	−	0.999416i	$-0.489118\pi$
$409$	1.38254	0.0683623	0.0341811	+	0.999416i	$0.489118\pi$
$410$	0	0
$411$	0.0495740	0.00244531
$412$	29.4378	1.45029
$413$	−1.36747	−0.0672888
$414$	31.0242	1.52476
$415$	0	0
$416$	−21.9690	−1.07712
$417$	2.57809	0.126250
$418$	−8.85869	−0.433293
$419$	−33.2300	−1.62339	−0.811695	−	0.584081i	$-0.801455\pi$
$419$	−33.2300	−1.62339	−0.811695	+	0.584081i	$0.801455\pi$
$420$	0	0
$421$	4.61109	0.224731	0.112365	−	0.993667i	$-0.464157\pi$
$421$	4.61109	0.224731	0.112365	+	0.993667i	$0.464157\pi$
$422$	17.8923	0.870986
$423$	−13.1345	−0.638622
$424$	36.2365	1.75980
$425$	0	0
$426$	−4.41104	−0.213715
$427$	6.79693	0.328927
$428$	27.2598	1.31765
$429$	3.32055	0.160318
$430$	0	0
$431$	7.33812	0.353465	0.176732	−	0.984259i	$-0.443447\pi$
$431$	7.33812	0.353465	0.176732	+	0.984259i	$0.443447\pi$
$432$	−0.718721	−0.0345795
$433$	−15.3487	−0.737610	−0.368805	−	0.929507i	$-0.620233\pi$
$433$	−15.3487	−0.737610	−0.368805	+	0.929507i	$0.620233\pi$
$434$	−17.2499	−0.828021
$435$	0	0
$436$	−10.4758	−0.501699
$437$	4.70309	0.224979
$438$	−1.06808	−0.0510347
$439$	−5.60355	−0.267443	−0.133721	−	0.991019i	$-0.542693\pi$
$439$	−5.60355	−0.267443	−0.133721	+	0.991019i	$0.542693\pi$
$440$	0	0
$441$	19.3278	0.920373
$442$	−10.2825	−0.489089
$443$	−29.6338	−1.40794	−0.703971	−	0.710228i	$-0.748592\pi$
$443$	−29.6338	−1.40794	−0.703971	+	0.710228i	$0.748592\pi$
$444$	0.211089	0.0100178
$445$	0	0
$446$	−31.7778	−1.50472
$447$	2.44031	0.115423
$448$	−8.64923	−0.408638
$449$	−34.4203	−1.62439	−0.812197	−	0.583383i	$-0.801729\pi$
$449$	−34.4203	−1.62439	−0.812197	+	0.583383i	$0.801729\pi$
$450$	0	0
$451$	−22.6476	−1.06643
$452$	−23.3297	−1.09734
$453$	1.81979	0.0855013
$454$	−4.17029	−0.195721
$455$	0	0
$456$	−0.667442	−0.0312558
$457$	9.32504	0.436207	0.218103	−	0.975926i	$-0.430013\pi$
$457$	9.32504	0.436207	0.218103	+	0.975926i	$0.430013\pi$
$458$	−7.52412	−0.351579
$459$	1.21072	0.0565116
$460$	0	0
$461$	19.1715	0.892904	0.446452	−	0.894808i	$-0.352687\pi$
$461$	19.1715	0.892904	0.446452	+	0.894808i	$0.352687\pi$
$462$	1.18461	0.0551129
$463$	19.6385	0.912680	0.456340	−	0.889805i	$-0.349160\pi$
$463$	19.6385	0.912680	0.456340	+	0.889805i	$0.349160\pi$
$464$	−2.19125	−0.101726
$465$	0	0
$466$	12.4674	0.577541
$467$	−21.1027	−0.976515	−0.488258	−	0.872699i	$-0.662367\pi$
$467$	−21.1027	−0.976515	−0.488258	+	0.872699i	$0.662367\pi$
$468$	−44.1988	−2.04309
$469$	6.26946	0.289497
$470$	0	0
$471$	−4.04595	−0.186428
$472$	−6.32380	−0.291077
$473$	29.0257	1.33461
$474$	3.49535	0.160547
$475$	0	0
$476$	−2.30083	−0.105458
$477$	−33.9079	−1.55254
$478$	−8.38653	−0.383591
$479$	−28.6762	−1.31025	−0.655125	−	0.755521i	$-0.727384\pi$
$479$	−28.6762	−1.31025	−0.655125	+	0.755521i	$0.727384\pi$
$480$	0	0
$481$	1.37053	0.0624909
$482$	−17.7607	−0.808977
$483$	−0.628909	−0.0286164
$484$	8.58886	0.390403
$485$	0	0
$486$	12.4747	0.565862
$487$	39.2139	1.77695	0.888475	−	0.458925i	$-0.151766\pi$
$487$	39.2139	1.77695	0.888475	+	0.458925i	$0.151766\pi$
$488$	31.4321	1.42287
$489$	4.55157	0.205829
$490$	0	0
$491$	−36.3499	−1.64045	−0.820224	−	0.572042i	$-0.806151\pi$
$491$	−36.3499	−1.64045	−0.820224	+	0.572042i	$0.806151\pi$
$492$	−4.20632	−0.189636
$493$	3.69127	0.166246
$494$	−10.6825	−0.480629
$495$	0	0
$496$	−6.46586	−0.290326
$497$	−6.40839	−0.287455
$498$	−4.19964	−0.188190
$499$	−8.52061	−0.381435	−0.190718	−	0.981645i	$-0.561081\pi$
$499$	−8.52061	−0.381435	−0.190718	+	0.981645i	$0.561081\pi$
$500$	0	0
$501$	−0.930856	−0.0415876
$502$	−14.0040	−0.625030
$503$	18.6790	0.832854	0.416427	−	0.909169i	$-0.363282\pi$
$503$	18.6790	0.832854	0.416427	+	0.909169i	$0.363282\pi$
$504$	−6.39645	−0.284921
$505$	0	0
$506$	−38.6016	−1.71605
$507$	1.36278	0.0605231
$508$	71.1702	3.15767
$509$	3.53567	0.156716	0.0783579	−	0.996925i	$-0.475032\pi$
$509$	3.53567	0.156716	0.0783579	+	0.996925i	$0.475032\pi$
$510$	0	0
$511$	−1.55171	−0.0686436
$512$	−6.69175	−0.295737
$513$	1.25782	0.0555341
$514$	−14.1999	−0.626333
$515$	0	0
$516$	5.39093	0.237322
$517$	16.3425	0.718742
$518$	0.488938	0.0214827
$519$	−1.64532	−0.0722215
$520$	0	0
$521$	−4.64206	−0.203373	−0.101686	−	0.994817i	$-0.532424\pi$
$521$	−4.64206	−0.203373	−0.101686	+	0.994817i	$0.532424\pi$
$522$	25.2969	1.10722
$523$	18.3331	0.801649	0.400824	−	0.916155i	$-0.368724\pi$
$523$	18.3331	0.801649	0.400824	+	0.916155i	$0.368724\pi$
$524$	29.3162	1.28068
$525$	0	0
$526$	−28.4929	−1.24235
$527$	10.8921	0.474466
$528$	0.444032	0.0193240
$529$	−2.50639	−0.108974
$530$	0	0
$531$	5.91743	0.256795
$532$	−2.39033	−0.103634
$533$	−27.3103	−1.18294
$534$	5.43688	0.235277
$535$	0	0
$536$	28.9928	1.25230
$537$	0.895308	0.0386354
$538$	22.9891	0.991132
$539$	−24.0485	−1.03584
$540$	0	0
$541$	22.4428	0.964891	0.482445	−	0.875926i	$-0.339749\pi$
$541$	22.4428	0.964891	0.482445	+	0.875926i	$0.339749\pi$
$542$	47.0294	2.02008
$543$	−1.20636	−0.0517700
$544$	4.94880	0.212178
$545$	0	0
$546$	1.42849	0.0611339
$547$	−25.9778	−1.11073	−0.555365	−	0.831607i	$-0.687422\pi$
$547$	−25.9778	−1.11073	−0.555365	+	0.831607i	$0.687422\pi$
$548$	0.821029	0.0350726
$549$	−29.4123	−1.25529
$550$	0	0
$551$	3.83486	0.163371
$552$	−2.90836	−0.123788
$553$	5.07807	0.215942
$554$	42.0797	1.78780
$555$	0	0
$556$	42.6976	1.81078
$557$	−4.86144	−0.205986	−0.102993	−	0.994682i	$-0.532842\pi$
$557$	−4.86144	−0.205986	−0.102993	+	0.994682i	$0.532842\pi$
$558$	74.6452	3.15998
$559$	35.0015	1.48041
$560$	0	0
$561$	−0.747995	−0.0315804
$562$	24.7802	1.04529
$563$	−1.53365	−0.0646357	−0.0323179	−	0.999478i	$-0.510289\pi$
$563$	−1.53365	−0.0646357	−0.0323179	+	0.999478i	$0.510289\pi$
$564$	3.03528	0.127808
$565$	0	0
$566$	23.3417	0.981127
$567$	5.90073	0.247807
$568$	−29.6353	−1.24347
$569$	−16.5770	−0.694946	−0.347473	−	0.937690i	$-0.612960\pi$
$569$	−16.5770	−0.694946	−0.347473	+	0.937690i	$0.612960\pi$
$570$	0	0
$571$	−14.2904	−0.598036	−0.299018	−	0.954248i	$-0.596659\pi$
$571$	−14.2904	−0.598036	−0.299018	+	0.954248i	$0.596659\pi$
$572$	54.9939	2.29941
$573$	−2.01104	−0.0840125
$574$	−9.74296	−0.406663
$575$	0	0
$576$	37.4277	1.55949
$577$	−28.5424	−1.18824	−0.594119	−	0.804377i	$-0.702499\pi$
$577$	−28.5424	−1.18824	−0.594119	+	0.804377i	$0.702499\pi$
$578$	2.31627	0.0963439
$579$	−1.72215	−0.0715702
$580$	0	0
$581$	−6.10126	−0.253123
$582$	−5.90639	−0.244828
$583$	42.1895	1.74731
$584$	−7.17581	−0.296937
$585$	0	0
$586$	−47.5643	−1.96486
$587$	33.5281	1.38385	0.691927	−	0.721967i	$-0.256762\pi$
$587$	33.5281	1.38385	0.691927	+	0.721967i	$0.256762\pi$
$588$	−4.46650	−0.184195
$589$	11.3158	0.466259
$590$	0	0
$591$	2.75075	0.113151
$592$	0.183271	0.00753239
$593$	−42.4729	−1.74415	−0.872077	−	0.489368i	$-0.837227\pi$
$593$	−42.4729	−1.74415	−0.872077	+	0.489368i	$0.837227\pi$
$594$	−10.3238	−0.423591
$595$	0	0
$596$	40.4157	1.65549
$597$	−1.85993	−0.0761219
$598$	−46.5488	−1.90352
$599$	−7.00705	−0.286300	−0.143150	−	0.989701i	$-0.545723\pi$
$599$	−7.00705	−0.286300	−0.143150	+	0.989701i	$0.545723\pi$
$600$	0	0
$601$	39.3146	1.60368	0.801839	−	0.597541i	$-0.203855\pi$
$601$	39.3146	1.60368	0.801839	+	0.597541i	$0.203855\pi$
$602$	12.4868	0.508925
$603$	−27.1298	−1.10481
$604$	30.1388	1.22633
$605$	0	0
$606$	−3.19394	−0.129745
$607$	18.7928	0.762777	0.381389	−	0.924415i	$-0.375446\pi$
$607$	18.7928	0.762777	0.381389	+	0.924415i	$0.375446\pi$
$608$	5.14131	0.208508
$609$	−0.512808	−0.0207800
$610$	0	0
$611$	19.7071	0.797263
$612$	9.95633	0.402461
$613$	1.83560	0.0741392	0.0370696	−	0.999313i	$-0.488198\pi$
$613$	1.83560	0.0741392	0.0370696	+	0.999313i	$0.488198\pi$
$614$	30.3842	1.22621
$615$	0	0
$616$	7.95872	0.320666
$617$	−29.0011	−1.16754	−0.583771	−	0.811918i	$-0.698423\pi$
$617$	−29.0011	−1.16754	−0.583771	+	0.811918i	$0.698423\pi$
$618$	4.11707	0.165613
$619$	27.7941	1.11714	0.558569	−	0.829458i	$-0.311351\pi$
$619$	27.7941	1.11714	0.558569	+	0.829458i	$0.311351\pi$
$620$	0	0
$621$	5.48092	0.219942
$622$	−28.6397	−1.14835
$623$	7.89874	0.316456
$624$	0.535450	0.0214351
$625$	0	0
$626$	−11.9519	−0.477695
$627$	−0.777092	−0.0310341
$628$	−67.0078	−2.67390
$629$	−0.308729	−0.0123098
$630$	0	0
$631$	−14.1785	−0.564437	−0.282219	−	0.959350i	$-0.591070\pi$
$631$	−14.1785	−0.564437	−0.282219	+	0.959350i	$0.591070\pi$
$632$	23.4833	0.934116
$633$	1.56953	0.0623833
$634$	−55.0471	−2.18620
$635$	0	0
$636$	7.83583	0.310711
$637$	−28.9995	−1.14900
$638$	−31.4754	−1.24612
$639$	27.7309	1.09702
$640$	0	0
$641$	−27.8734	−1.10093	−0.550466	−	0.834857i	$-0.685550\pi$
$641$	−27.8734	−1.10093	−0.550466	+	0.834857i	$0.685550\pi$
$642$	3.81247	0.150466
$643$	−25.8277	−1.01854	−0.509272	−	0.860605i	$-0.670085\pi$
$643$	−25.8277	−1.01854	−0.509272	+	0.860605i	$0.670085\pi$
$644$	−10.4158	−0.410440
$645$	0	0
$646$	2.40637	0.0946774
$647$	−4.40912	−0.173340	−0.0866702	−	0.996237i	$-0.527623\pi$
$647$	−4.40912	−0.173340	−0.0866702	+	0.996237i	$0.527623\pi$
$648$	27.2877	1.07196
$649$	−7.36270	−0.289011
$650$	0	0
$651$	−1.51318	−0.0593060
$652$	75.3817	2.95217
$653$	30.9245	1.21017	0.605084	−	0.796161i	$-0.293139\pi$
$653$	30.9245	1.21017	0.605084	+	0.796161i	$0.293139\pi$
$654$	−1.46511	−0.0572903
$655$	0	0
$656$	−3.65200	−0.142587
$657$	6.71469	0.261965
$658$	7.03051	0.274078
$659$	39.7992	1.55036	0.775179	−	0.631742i	$-0.217660\pi$
$659$	39.7992	1.55036	0.775179	+	0.631742i	$0.217660\pi$
$660$	0	0
$661$	−13.0969	−0.509409	−0.254704	−	0.967019i	$-0.581978\pi$
$661$	−13.0969	−0.509409	−0.254704	+	0.967019i	$0.581978\pi$
$662$	48.2573	1.87557
$663$	−0.901992	−0.0350305
$664$	−28.2150	−1.09496
$665$	0	0
$666$	−2.11578	−0.0819846
$667$	16.7103	0.647027
$668$	−15.4166	−0.596485
$669$	−2.78757	−0.107774
$670$	0	0
$671$	36.5959	1.41277
$672$	−0.687509	−0.0265212
$673$	22.2830	0.858946	0.429473	−	0.903080i	$-0.358699\pi$
$673$	22.2830	0.858946	0.429473	+	0.903080i	$0.358699\pi$
$674$	33.1411	1.27655
$675$	0	0
$676$	22.5699	0.868073
$677$	−10.5576	−0.405762	−0.202881	−	0.979203i	$-0.565030\pi$
$677$	−10.5576	−0.405762	−0.202881	+	0.979203i	$0.565030\pi$
$678$	−3.26282	−0.125308
$679$	−8.58084	−0.329303
$680$	0	0
$681$	−0.365822	−0.0140183
$682$	−92.8766	−3.55643
$683$	1.80882	0.0692128	0.0346064	−	0.999401i	$-0.488982\pi$
$683$	1.80882	0.0692128	0.0346064	+	0.999401i	$0.488982\pi$
$684$	10.3436	0.395499
$685$	0	0
$686$	−21.4316	−0.818261
$687$	−0.660022	−0.0251814
$688$	4.68050	0.178442
$689$	50.8755	1.93820
$690$	0	0
$691$	3.84649	0.146327	0.0731636	−	0.997320i	$-0.476690\pi$
$691$	3.84649	0.146327	0.0731636	+	0.997320i	$0.476690\pi$
$692$	−27.2493	−1.03586
$693$	−7.44729	−0.282899
$694$	−50.9550	−1.93423
$695$	0	0
$696$	−2.37146	−0.0898899
$697$	6.15198	0.233023
$698$	−14.0412	−0.531467
$699$	1.09365	0.0413657
$700$	0	0
$701$	43.9484	1.65991	0.829955	−	0.557830i	$-0.188366\pi$
$701$	43.9484	1.65991	0.829955	+	0.557830i	$0.188366\pi$
$702$	−12.4493	−0.469867
$703$	−0.320739	−0.0120969
$704$	−46.5690	−1.75514
$705$	0	0
$706$	−48.3602	−1.82006
$707$	−4.64018	−0.174512
$708$	−1.36747	−0.0513926
$709$	45.9706	1.72646	0.863232	−	0.504808i	$-0.168437\pi$
$709$	45.9706	1.72646	0.863232	+	0.504808i	$0.168437\pi$
$710$	0	0
$711$	−21.9743	−0.824100
$712$	36.5274	1.36892
$713$	49.3083	1.84661
$714$	−0.321786	−0.0120425
$715$	0	0
$716$	14.8278	0.554141
$717$	−0.735674	−0.0274742
$718$	42.8142	1.59781
$719$	6.24809	0.233014	0.116507	−	0.993190i	$-0.462830\pi$
$719$	6.24809	0.233014	0.116507	+	0.993190i	$0.462830\pi$
$720$	0	0
$721$	5.98131	0.222755
$722$	−41.5091	−1.54481
$723$	−1.55798	−0.0579420
$724$	−19.9794	−0.742529
$725$	0	0
$726$	1.20121	0.0445811
$727$	49.9606	1.85294	0.926469	−	0.376372i	$-0.122829\pi$
$727$	49.9606	1.85294	0.926469	+	0.376372i	$0.122829\pi$
$728$	9.59725	0.355698
$729$	−24.7962	−0.918376
$730$	0	0
$731$	−7.88454	−0.291620
$732$	6.79693	0.251222
$733$	−37.4111	−1.38181	−0.690906	−	0.722945i	$-0.742788\pi$
$733$	−37.4111	−1.38181	−0.690906	+	0.722945i	$0.742788\pi$
$734$	54.6741	2.01806
$735$	0	0
$736$	22.4031	0.825790
$737$	33.7559	1.24342
$738$	42.1606	1.55195
$739$	38.9348	1.43224	0.716120	−	0.697978i	$-0.245916\pi$
$739$	38.9348	1.43224	0.716120	+	0.697978i	$0.245916\pi$
$740$	0	0
$741$	−0.937080	−0.0344245
$742$	18.1499	0.666303
$743$	13.0253	0.477850	0.238925	−	0.971038i	$-0.423205\pi$
$743$	13.0253	0.477850	0.238925	+	0.971038i	$0.423205\pi$
$744$	−6.99762	−0.256545
$745$	0	0
$746$	41.7213	1.52753
$747$	26.4019	0.965996
$748$	−12.3881	−0.452952
$749$	5.53878	0.202383
$750$	0	0
$751$	−42.3558	−1.54559	−0.772793	−	0.634659i	$-0.781141\pi$
$751$	−42.3558	−1.54559	−0.772793	+	0.634659i	$0.781141\pi$
$752$	2.63528	0.0960989
$753$	−1.22845	−0.0447671
$754$	−37.9556	−1.38226
$755$	0	0
$756$	−2.78566	−0.101313
$757$	4.20834	0.152955	0.0764773	−	0.997071i	$-0.475633\pi$
$757$	4.20834	0.152955	0.0764773	+	0.997071i	$0.475633\pi$
$758$	−13.3075	−0.483349
$759$	−3.38616	−0.122910
$760$	0	0
$761$	31.8563	1.15479	0.577395	−	0.816465i	$-0.304069\pi$
$761$	31.8563	1.15479	0.577395	+	0.816465i	$0.304069\pi$
$762$	9.95364	0.360582
$763$	−2.12852	−0.0770576
$764$	−33.3062	−1.20498
$765$	0	0
$766$	81.8293	2.95661
$767$	−8.87853	−0.320585
$768$	−3.99113	−0.144017
$769$	−7.43010	−0.267936	−0.133968	−	0.990986i	$-0.542772\pi$
$769$	−7.43010	−0.267936	−0.133968	+	0.990986i	$0.542772\pi$
$770$	0	0
$771$	−1.24563	−0.0448604
$772$	−28.5217	−1.02652
$773$	7.33384	0.263780	0.131890	−	0.991264i	$-0.457895\pi$
$773$	7.33384	0.263780	0.131890	+	0.991264i	$0.457895\pi$
$774$	−54.0341	−1.94221
$775$	0	0
$776$	−39.6817	−1.42449
$777$	0.0428901	0.00153867
$778$	−42.3012	−1.51657
$779$	6.39130	0.228992
$780$	0	0
$781$	−34.5039	−1.23465
$782$	10.4857	0.374968
$783$	4.46910	0.159713
$784$	−3.87789	−0.138496
$785$	0	0
$786$	4.10007	0.146244
$787$	−35.7257	−1.27348	−0.636742	−	0.771077i	$-0.719718\pi$
$787$	−35.7257	−1.27348	−0.636742	+	0.771077i	$0.719718\pi$
$788$	45.5570	1.62290
$789$	−2.49942	−0.0889817
$790$	0	0
$791$	−4.74024	−0.168544
$792$	−34.4397	−1.22376
$793$	44.1303	1.56711
$794$	−10.7496	−0.381489
$795$	0	0
$796$	−30.8036	−1.09180
$797$	28.2025	0.998985	0.499492	−	0.866318i	$-0.333520\pi$
$797$	28.2025	0.998985	0.499492	+	0.866318i	$0.333520\pi$
$798$	−0.334304	−0.0118342
$799$	−4.43927	−0.157050
$800$	0	0
$801$	−34.1801	−1.20769
$802$	−63.4769	−2.24145
$803$	−8.35468	−0.294830
$804$	6.26946	0.221107
$805$	0	0
$806$	−111.998	−3.94496
$807$	2.01663	0.0709886
$808$	−21.4583	−0.754901
$809$	−41.9318	−1.47424	−0.737121	−	0.675761i	$-0.763815\pi$
$809$	−41.9318	−1.47424	−0.737121	+	0.675761i	$0.763815\pi$
$810$	0	0
$811$	4.46604	0.156824	0.0784119	−	0.996921i	$-0.475015\pi$
$811$	4.46604	0.156824	0.0784119	+	0.996921i	$0.475015\pi$
$812$	−8.49297	−0.298045
$813$	4.12546	0.144686
$814$	2.63253	0.0922702
$815$	0	0
$816$	−0.120617	−0.00422243
$817$	−8.19125	−0.286576
$818$	3.20233	0.111967
$819$	−8.98053	−0.313805
$820$	0	0
$821$	19.6083	0.684334	0.342167	−	0.939639i	$-0.388839\pi$
$821$	19.6083	0.684334	0.342167	+	0.939639i	$0.388839\pi$
$822$	0.114827	0.00400504
$823$	46.9128	1.63528	0.817638	−	0.575732i	$-0.195283\pi$
$823$	46.9128	1.63528	0.817638	+	0.575732i	$0.195283\pi$
$824$	27.6603	0.963592
$825$	0	0
$826$	−3.16742	−0.110209
$827$	14.0134	0.487295	0.243648	−	0.969864i	$-0.421656\pi$
$827$	14.0134	0.487295	0.243648	+	0.969864i	$0.421656\pi$
$828$	45.0722	1.56637
$829$	−40.5206	−1.40734	−0.703669	−	0.710528i	$-0.748456\pi$
$829$	−40.5206	−1.40734	−0.703669	+	0.710528i	$0.748456\pi$
$830$	0	0
$831$	3.69127	0.128049
$832$	−56.1566	−1.94688
$833$	6.53251	0.226338
$834$	5.97154	0.206778
$835$	0	0
$836$	−12.8700	−0.445117
$837$	13.1873	0.455818
$838$	−76.9694	−2.65887
$839$	8.68840	0.299957	0.149978	−	0.988689i	$-0.452080\pi$
$839$	8.68840	0.299957	0.149978	+	0.988689i	$0.452080\pi$
$840$	0	0
$841$	−15.3745	−0.530156
$842$	10.6805	0.368074
$843$	2.17374	0.0748675
$844$	25.9941	0.894754
$845$	0	0
$846$	−30.4230	−1.04597
$847$	1.74513	0.0599633
$848$	6.80321	0.233623
$849$	2.04756	0.0702720
$850$	0	0
$851$	−1.39761	−0.0479096
$852$	−6.40839	−0.219548
$853$	−23.1523	−0.792721	−0.396361	−	0.918095i	$-0.629727\pi$
$853$	−23.1523	−0.792721	−0.396361	+	0.918095i	$0.629727\pi$
$854$	15.7435	0.538731
$855$	0	0
$856$	25.6139	0.875464
$857$	43.6043	1.48950	0.744748	−	0.667346i	$-0.232570\pi$
$857$	43.6043	1.48950	0.744748	+	0.667346i	$0.232570\pi$
$858$	7.69127	0.262576
$859$	−14.2532	−0.486314	−0.243157	−	0.969987i	$-0.578183\pi$
$859$	−14.2532	−0.486314	−0.243157	+	0.969987i	$0.578183\pi$
$860$	0	0
$861$	−0.854661	−0.0291268
$862$	16.9970	0.578921
$863$	−24.1162	−0.820926	−0.410463	−	0.911877i	$-0.634633\pi$
$863$	−24.1162	−0.820926	−0.410463	+	0.911877i	$0.634633\pi$
$864$	5.99161	0.203839
$865$	0	0
$866$	−35.5516	−1.20809
$867$	0.203185	0.00690052
$868$	−25.0608	−0.850617
$869$	27.3413	0.927489
$870$	0	0
$871$	40.7056	1.37926
$872$	−9.84325	−0.333335
$873$	37.1318	1.25672
$874$	10.8936	0.368482
$875$	0	0
$876$	−1.55171	−0.0524274
$877$	−1.56540	−0.0528599	−0.0264299	−	0.999651i	$-0.508414\pi$
$877$	−1.56540	−0.0528599	−0.0264299	+	0.999651i	$0.508414\pi$
$878$	−12.9793	−0.438030
$879$	−4.17238	−0.140731
$880$	0	0
$881$	−39.0973	−1.31722	−0.658610	−	0.752484i	$-0.728855\pi$
$881$	−39.0973	−1.31722	−0.658610	+	0.752484i	$0.728855\pi$
$882$	44.7684	1.50743
$883$	−30.9249	−1.04071	−0.520354	−	0.853951i	$-0.674200\pi$
$883$	−30.9249	−1.04071	−0.520354	+	0.853951i	$0.674200\pi$
$884$	−14.9385	−0.502436
$885$	0	0
$886$	−68.6397	−2.30599
$887$	−45.1839	−1.51713	−0.758564	−	0.651599i	$-0.774099\pi$
$887$	−45.1839	−1.51713	−0.758564	+	0.651599i	$0.774099\pi$
$888$	0.198343	0.00665597
$889$	14.4607	0.484997
$890$	0	0
$891$	31.7706	1.06436
$892$	−46.1670	−1.54578
$893$	−4.61196	−0.154333
$894$	5.65241	0.189045
$895$	0	0
$896$	−13.2666	−0.443206
$897$	−4.08330	−0.136338
$898$	−79.7265	−2.66051
$899$	40.2056	1.34093
$900$	0	0
$901$	−11.4603	−0.381799
$902$	−52.4579	−1.74666
$903$	1.09536	0.0364511
$904$	−21.9211	−0.729083
$905$	0	0
$906$	4.21512	0.140038
$907$	−33.9862	−1.12849	−0.564246	−	0.825606i	$-0.690833\pi$
$907$	−33.9862	−1.12849	−0.564246	+	0.825606i	$0.690833\pi$
$908$	−6.05862	−0.201062
$909$	20.0794	0.665992
$910$	0	0
$911$	22.1997	0.735509	0.367754	−	0.929923i	$-0.380127\pi$
$911$	22.1997	0.735509	0.367754	+	0.929923i	$0.380127\pi$
$912$	−0.125309	−0.00414939
$913$	−32.8503	−1.08719
$914$	21.5993	0.714440
$915$	0	0
$916$	−10.9311	−0.361173
$917$	5.95660	0.196704
$918$	2.80435	0.0925574
$919$	−8.84668	−0.291825	−0.145913	−	0.989297i	$-0.546612\pi$
$919$	−8.84668	−0.291825	−0.145913	+	0.989297i	$0.546612\pi$
$920$	0	0
$921$	2.66533	0.0878256
$922$	44.4062	1.46244
$923$	−41.6076	−1.36953
$924$	1.72101	0.0566169
$925$	0	0
$926$	45.4881	1.49483
$927$	−25.8828	−0.850104
$928$	18.2673	0.599655
$929$	13.9941	0.459131	0.229566	−	0.973293i	$-0.426269\pi$
$929$	13.9941	0.459131	0.229566	+	0.973293i	$0.426269\pi$
$930$	0	0
$931$	6.78663	0.222423
$932$	18.1127	0.593302
$933$	−2.51230	−0.0822490
$934$	−48.8794	−1.59938
$935$	0	0
$936$	−41.5301	−1.35745
$937$	5.19129	0.169592	0.0847960	−	0.996398i	$-0.472976\pi$
$937$	5.19129	0.169592	0.0847960	+	0.996398i	$0.472976\pi$
$938$	14.5217	0.474151
$939$	−1.04843	−0.0342144
$940$	0	0
$941$	−26.2746	−0.856527	−0.428264	−	0.903654i	$-0.640875\pi$
$941$	−26.2746	−0.856527	−0.428264	+	0.903654i	$0.640875\pi$
$942$	−9.37150	−0.305340
$943$	27.8499	0.906919
$944$	−1.18726	−0.0386420
$945$	0	0
$946$	67.2313	2.18588
$947$	−13.9416	−0.453040	−0.226520	−	0.974007i	$-0.572735\pi$
$947$	−13.9416	−0.453040	−0.226520	+	0.974007i	$0.572735\pi$
$948$	5.07807	0.164928
$949$	−10.0747	−0.327040
$950$	0	0
$951$	−4.82878	−0.156584
$952$	−2.16190	−0.0700676
$953$	−19.3298	−0.626154	−0.313077	−	0.949728i	$-0.601360\pi$
$953$	−19.3298	−0.626154	−0.313077	+	0.949728i	$0.601360\pi$
$954$	−78.5397	−2.54282
$955$	0	0
$956$	−12.1840	−0.394059
$957$	−2.76105	−0.0892521
$958$	−66.4217	−2.14599
$959$	0.166821	0.00538692
$960$	0	0
$961$	87.6372	2.82701
$962$	3.17451	0.102350
$963$	−23.9679	−0.772355
$964$	−25.8028	−0.831053
$965$	0	0
$966$	−1.45672	−0.0468692
$967$	−27.0663	−0.870393	−0.435197	−	0.900335i	$-0.643321\pi$
$967$	−27.0663	−0.870393	−0.435197	+	0.900335i	$0.643321\pi$
$968$	8.07027	0.259388
$969$	0.211089	0.00678115
$970$	0	0
$971$	9.36344	0.300487	0.150244	−	0.988649i	$-0.451994\pi$
$971$	9.36344	0.300487	0.150244	+	0.988649i	$0.451994\pi$
$972$	18.1233	0.581304
$973$	8.67550	0.278124
$974$	90.8297	2.91037
$975$	0	0
$976$	5.90121	0.188893
$977$	−14.1127	−0.451506	−0.225753	−	0.974185i	$-0.572484\pi$
$977$	−14.1127	−0.451506	−0.225753	+	0.974185i	$0.572484\pi$
$978$	10.5426	0.337116
$979$	42.5282	1.35921
$980$	0	0
$981$	9.21072	0.294076
$982$	−84.1961	−2.68680
$983$	48.9970	1.56276	0.781381	−	0.624054i	$-0.214515\pi$
$983$	48.9970	1.56276	0.781381	+	0.624054i	$0.214515\pi$
$984$	−3.95234	−0.125996
$985$	0	0
$986$	8.54996	0.272286
$987$	0.616723	0.0196305
$988$	−15.5196	−0.493745
$989$	−35.6932	−1.13498
$990$	0	0
$991$	−20.7542	−0.659279	−0.329639	−	0.944107i	$-0.606927\pi$
$991$	−20.7542	−0.659279	−0.329639	+	0.944107i	$0.606927\pi$
$992$	53.9026	1.71141
$993$	4.23317	0.134336
$994$	−14.8435	−0.470808
$995$	0	0
$996$	−6.10126	−0.193326
$997$	9.54268	0.302220	0.151110	−	0.988517i	$-0.451715\pi$
$997$	9.54268	0.302220	0.151110	+	0.988517i	$0.451715\pi$
$998$	−19.7360	−0.624732
$999$	−0.373785	−0.0118260

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.a.h.1.4		4
3.2	odd	2		3825.2.a.bh.1.1		4
4.3	odd	2		6800.2.a.bt.1.3		4
5.2	odd	4		85.2.b.a.69.8	yes	8
5.3	odd	4		85.2.b.a.69.1	✓	8
5.4	even	2		425.2.a.g.1.1		4
15.2	even	4		765.2.b.c.154.1		8
15.8	even	4		765.2.b.c.154.8		8
15.14	odd	2		3825.2.a.bj.1.4		4
17.16	even	2		7225.2.a.w.1.4		4
20.3	even	4		1360.2.e.d.1089.4		8
20.7	even	4		1360.2.e.d.1089.5		8
20.19	odd	2		6800.2.a.bw.1.2		4
85.33	odd	4		1445.2.b.e.579.1		8
85.67	odd	4		1445.2.b.e.579.8		8
85.84	even	2		7225.2.a.v.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.b.a.69.1	✓	8	5.3	odd	4
85.2.b.a.69.8	yes	8	5.2	odd	4
425.2.a.g.1.1		4	5.4	even	2
425.2.a.h.1.4		4	1.1	even	1	trivial
765.2.b.c.154.1		8	15.2	even	4
765.2.b.c.154.8		8	15.8	even	4
1360.2.e.d.1089.4		8	20.3	even	4
1360.2.e.d.1089.5		8	20.7	even	4
1445.2.b.e.579.1		8	85.33	odd	4
1445.2.b.e.579.8		8	85.67	odd	4
3825.2.a.bh.1.1		4	3.2	odd	2
3825.2.a.bj.1.4		4	15.14	odd	2
6800.2.a.bt.1.3		4	4.3	odd	2
6800.2.a.bw.1.2		4	20.19	odd	2
7225.2.a.v.1.1		4	85.84	even	2
7225.2.a.w.1.4		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)

\(f(q)\)	\(=\)	\(q+2.31627 q^{2} +0.203185 q^{3} +3.36509 q^{4} +0.470630 q^{6} +0.683735 q^{7} +3.16190 q^{8} -2.95872 q^{9} +3.68135 q^{11} +0.683735 q^{12} +4.43927 q^{13} +1.58371 q^{14} +0.593630 q^{16} -1.00000 q^{17} -6.85317 q^{18} -1.03890 q^{19} +0.138925 q^{21} +8.52699 q^{22} -4.52699 q^{23} +0.642450 q^{24} +10.2825 q^{26} -1.21072 q^{27} +2.30083 q^{28} -3.69127 q^{29} -10.8921 q^{31} -4.94880 q^{32} +0.747995 q^{33} -2.31627 q^{34} -9.95633 q^{36} +0.308729 q^{37} -2.40637 q^{38} +0.901992 q^{39} -6.15198 q^{41} +0.321786 q^{42} +7.88454 q^{43} +12.3881 q^{44} -10.4857 q^{46} +4.43927 q^{47} +0.120617 q^{48} -6.53251 q^{49} -0.203185 q^{51} +14.9385 q^{52} +11.4603 q^{53} -2.80435 q^{54} +2.16190 q^{56} -0.211089 q^{57} -8.54996 q^{58} -2.00000 q^{59} +9.94089 q^{61} -25.2289 q^{62} -2.02298 q^{63} -12.6500 q^{64} +1.73255 q^{66} +9.16944 q^{67} -3.36509 q^{68} -0.919815 q^{69} -9.37262 q^{71} -9.35517 q^{72} -2.26946 q^{73} +0.715099 q^{74} -3.49599 q^{76} +2.51707 q^{77} +2.08925 q^{78} +7.42696 q^{79} +8.63015 q^{81} -14.2496 q^{82} -8.92344 q^{83} +0.467493 q^{84} +18.2627 q^{86} -0.750010 q^{87} +11.6401 q^{88} +11.5523 q^{89} +3.03528 q^{91} -15.2337 q^{92} -2.21310 q^{93} +10.2825 q^{94} -1.00552 q^{96} -12.5500 q^{97} -15.1310 q^{98} -10.8921 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} + 10 q^{7} + 4 q^{9} - 2 q^{11} + 10 q^{12} + 6 q^{13} - 6 q^{14} - 4 q^{16} - 4 q^{17} - 4 q^{18} + 4 q^{19} + 12 q^{21} + 12 q^{22} + 4 q^{23} - 6 q^{24} + 10 q^{27}+ \cdots - 12 q^{99}+O(q^{100}) \) 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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