LMFDB - Embedded newform 862.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [862,2,Mod(1,862)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(862, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("862.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$862 = 2 \cdot 431$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	862.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:	$6.88310465423$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.11017801.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} - 6x^{4} + 13x^{3} + 3x^{2} - 10x - 1$ x^6 - 2x^5 - 6x^4 + 13x^3 + 3x^2 - 10*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.74564$ of defining polynomial
Character	$\chi$	$=$	862.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.74564 q^{3} +1.00000 q^{4} +2.20488 q^{5} +1.74564 q^{6} +0.156662 q^{7} -1.00000 q^{8} +0.0472515 q^{9} -2.20488 q^{10} +0.652346 q^{11} -1.74564 q^{12} -6.50126 q^{13} -0.156662 q^{14} -3.84892 q^{15} +1.00000 q^{16} -0.464467 q^{17} -0.0472515 q^{18} +5.83491 q^{19} +2.20488 q^{20} -0.273476 q^{21} -0.652346 q^{22} -8.89828 q^{23} +1.74564 q^{24} -0.138513 q^{25} +6.50126 q^{26} +5.15443 q^{27} +0.156662 q^{28} +0.0633705 q^{29} +3.84892 q^{30} +1.92051 q^{31} -1.00000 q^{32} -1.13876 q^{33} +0.464467 q^{34} +0.345421 q^{35} +0.0472515 q^{36} -2.74863 q^{37} -5.83491 q^{38} +11.3489 q^{39} -2.20488 q^{40} +0.948733 q^{41} +0.273476 q^{42} -5.66437 q^{43} +0.652346 q^{44} +0.104184 q^{45} +8.89828 q^{46} +13.3339 q^{47} -1.74564 q^{48} -6.97546 q^{49} +0.138513 q^{50} +0.810792 q^{51} -6.50126 q^{52} -12.5025 q^{53} -5.15443 q^{54} +1.43834 q^{55} -0.156662 q^{56} -10.1856 q^{57} -0.0633705 q^{58} -11.7901 q^{59} -3.84892 q^{60} -0.890281 q^{61} -1.92051 q^{62} +0.00740252 q^{63} +1.00000 q^{64} -14.3345 q^{65} +1.13876 q^{66} -11.0627 q^{67} -0.464467 q^{68} +15.5332 q^{69} -0.345421 q^{70} -0.438915 q^{71} -0.0472515 q^{72} +3.32578 q^{73} +2.74863 q^{74} +0.241794 q^{75} +5.83491 q^{76} +0.102198 q^{77} -11.3489 q^{78} -6.31109 q^{79} +2.20488 q^{80} -9.13952 q^{81} -0.948733 q^{82} -10.4609 q^{83} -0.273476 q^{84} -1.02409 q^{85} +5.66437 q^{86} -0.110622 q^{87} -0.652346 q^{88} -15.7312 q^{89} -0.104184 q^{90} -1.01850 q^{91} -8.89828 q^{92} -3.35252 q^{93} -13.3339 q^{94} +12.8653 q^{95} +1.74564 q^{96} -2.07690 q^{97} +6.97546 q^{98} +0.0308243 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} - 6 q^{8} - 2 q^{9} + 2 q^{10} - 6 q^{11} - 2 q^{12} - 9 q^{13} - 3 q^{14} - 3 q^{15} + 6 q^{16} - 17 q^{17} + 2 q^{18} + q^{19} - 2 q^{20}+ \cdots - 4 q^{99}+O(q^{100})$ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 2 * q^5 + 2 * q^6 + 3 * q^7 - 6 * q^8 - 2 * q^9 + 2 * q^10 - 6 * q^11 - 2 * q^12 - 9 * q^13 - 3 * q^14 - 3 * q^15 + 6 * q^16 - 17 * q^17 + 2 * q^18 + q^19 - 2 * q^20 - 8 * q^21 + 6 * q^22 - 20 * q^23 + 2 * q^24 + 10 * q^25 + 9 * q^26 + 7 * q^27 + 3 * q^28 + q^29 + 3 * q^30 + 8 * q^31 - 6 * q^32 - 5 * q^33 + 17 * q^34 - 14 * q^35 - 2 * q^36 - 3 * q^37 - q^38 - 13 * q^39 + 2 * q^40 - 19 * q^41 + 8 * q^42 - 21 * q^43 - 6 * q^44 - 16 * q^45 + 20 * q^46 - 10 * q^47 - 2 * q^48 - 7 * q^49 - 10 * q^50 - 2 * q^51 - 9 * q^52 - q^53 - 7 * q^54 + 3 * q^55 - 3 * q^56 - 11 * q^57 - q^58 + 11 * q^59 - 3 * q^60 - 2 * q^61 - 8 * q^62 + 6 * q^64 - 30 * q^65 + 5 * q^66 + 4 * q^67 - 17 * q^68 + 16 * q^69 + 14 * q^70 - 4 * q^71 + 2 * q^72 - 39 * q^73 + 3 * q^74 - 21 * q^75 + q^76 - 33 * q^77 + 13 * q^78 - 16 * q^79 - 2 * q^80 - 22 * q^81 + 19 * q^82 - 20 * q^83 - 8 * q^84 - 16 * q^85 + 21 * q^86 + q^87 + 6 * q^88 - 17 * q^89 + 16 * q^90 + 20 * q^91 - 20 * q^92 - 5 * q^93 + 10 * q^94 - 16 * q^95 + 2 * q^96 - 29 * q^97 + 7 * q^98 - 4 * 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

)

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.74564	−1.00784	−0.503922	−	0.863749i	$-0.668110\pi$
$3$	−1.74564	−1.00784	−0.503922	+	0.863749i	$0.668110\pi$
$4$	1.00000	0.500000
$5$	2.20488	0.986051	0.493026	−	0.870015i	$-0.335891\pi$
$5$	2.20488	0.986051	0.493026	+	0.870015i	$0.335891\pi$
$6$	1.74564	0.712654
$7$	0.156662	0.0592128	0.0296064	−	0.999562i	$-0.490575\pi$
$7$	0.156662	0.0592128	0.0296064	+	0.999562i	$0.490575\pi$
$8$	−1.00000	−0.353553
$9$	0.0472515	0.0157505
$10$	−2.20488	−0.697244
$11$	0.652346	0.196690	0.0983449	−	0.995152i	$-0.468645\pi$
$11$	0.652346	0.196690	0.0983449	+	0.995152i	$0.468645\pi$
$12$	−1.74564	−0.503922
$13$	−6.50126	−1.80313	−0.901563	−	0.432647i	$-0.857579\pi$
$13$	−6.50126	−1.80313	−0.901563	+	0.432647i	$0.857579\pi$
$14$	−0.156662	−0.0418698
$15$	−3.84892	−0.993786
$16$	1.00000	0.250000
$17$	−0.464467	−0.112650	−0.0563249	−	0.998412i	$-0.517938\pi$
$17$	−0.464467	−0.112650	−0.0563249	+	0.998412i	$0.517938\pi$
$18$	−0.0472515	−0.0111373
$19$	5.83491	1.33862	0.669311	−	0.742983i	$-0.266590\pi$
$19$	5.83491	1.33862	0.669311	+	0.742983i	$0.266590\pi$
$20$	2.20488	0.493026
$21$	−0.273476	−0.0596773
$22$	−0.652346	−0.139081
$23$	−8.89828	−1.85542	−0.927710	−	0.373301i	$-0.878226\pi$
$23$	−8.89828	−1.85542	−0.927710	+	0.373301i	$0.878226\pi$
$24$	1.74564	0.356327
$25$	−0.138513	−0.0277026
$26$	6.50126	1.27500
$27$	5.15443	0.991970
$28$	0.156662	0.0296064
$29$	0.0633705	0.0117676	0.00588381	−	0.999983i	$-0.498127\pi$
$29$	0.0633705	0.0117676	0.00588381	+	0.999983i	$0.498127\pi$
$30$	3.84892	0.702713
$31$	1.92051	0.344934	0.172467	−	0.985015i	$-0.444826\pi$
$31$	1.92051	0.344934	0.172467	+	0.985015i	$0.444826\pi$
$32$	−1.00000	−0.176777
$33$	−1.13876	−0.198233
$34$	0.464467	0.0796555
$35$	0.345421	0.0583868
$36$	0.0472515	0.00787525
$37$	−2.74863	−0.451872	−0.225936	−	0.974142i	$-0.572544\pi$
$37$	−2.74863	−0.451872	−0.225936	+	0.974142i	$0.572544\pi$
$38$	−5.83491	−0.946548
$39$	11.3489	1.81727
$40$	−2.20488	−0.348622
$41$	0.948733	0.148167	0.0740836	−	0.997252i	$-0.476397\pi$
$41$	0.948733	0.148167	0.0740836	+	0.997252i	$0.476397\pi$
$42$	0.273476	0.0421982
$43$	−5.66437	−0.863808	−0.431904	−	0.901920i	$-0.642158\pi$
$43$	−5.66437	−0.863808	−0.431904	+	0.901920i	$0.642158\pi$
$44$	0.652346	0.0983449
$45$	0.104184	0.0155308
$46$	8.89828	1.31198
$47$	13.3339	1.94496	0.972478	−	0.232995i	$-0.0748526\pi$
$47$	13.3339	1.94496	0.972478	+	0.232995i	$0.0748526\pi$
$48$	−1.74564	−0.251961
$49$	−6.97546	−0.996494
$50$	0.138513	0.0195887
$51$	0.810792	0.113534
$52$	−6.50126	−0.901563
$53$	−12.5025	−1.71735	−0.858677	−	0.512518i	$-0.828713\pi$
$53$	−12.5025	−1.71735	−0.858677	+	0.512518i	$0.828713\pi$
$54$	−5.15443	−0.701429
$55$	1.43834	0.193946
$56$	−0.156662	−0.0209349
$57$	−10.1856	−1.34912
$58$	−0.0633705	−0.00832096
$59$	−11.7901	−1.53494	−0.767472	−	0.641082i	$-0.778486\pi$
$59$	−11.7901	−1.53494	−0.767472	+	0.641082i	$0.778486\pi$
$60$	−3.84892	−0.496893
$61$	−0.890281	−0.113989	−0.0569944	−	0.998374i	$-0.518152\pi$
$61$	−0.890281	−0.113989	−0.0569944	+	0.998374i	$0.518152\pi$
$62$	−1.92051	−0.243905
$63$	0.00740252	0.000932630 0
$64$	1.00000	0.125000
$65$	−14.3345	−1.77798
$66$	1.13876	0.140172
$67$	−11.0627	−1.35152	−0.675762	−	0.737120i	$-0.736185\pi$
$67$	−11.0627	−1.35152	−0.675762	+	0.737120i	$0.736185\pi$
$68$	−0.464467	−0.0563249
$69$	15.5332	1.86998
$70$	−0.345421	−0.0412857
$71$	−0.438915	−0.0520897	−0.0260448	−	0.999661i	$-0.508291\pi$
$71$	−0.438915	−0.0520897	−0.0260448	+	0.999661i	$0.508291\pi$
$72$	−0.0472515	−0.00556864
$73$	3.32578	0.389253	0.194627	−	0.980877i	$-0.437650\pi$
$73$	3.32578	0.389253	0.194627	+	0.980877i	$0.437650\pi$
$74$	2.74863	0.319522
$75$	0.241794	0.0279199
$76$	5.83491	0.669311
$77$	0.102198	0.0116465
$78$	−11.3489	−1.28500
$79$	−6.31109	−0.710053	−0.355027	−	0.934856i	$-0.615528\pi$
$79$	−6.31109	−0.710053	−0.355027	+	0.934856i	$0.615528\pi$
$80$	2.20488	0.246513
$81$	−9.13952	−1.01550
$82$	−0.948733	−0.104770
$83$	−10.4609	−1.14823	−0.574117	−	0.818773i	$-0.694654\pi$
$83$	−10.4609	−1.14823	−0.574117	+	0.818773i	$0.694654\pi$
$84$	−0.273476	−0.0298386
$85$	−1.02409	−0.111079
$86$	5.66437	0.610804
$87$	−0.110622	−0.0118599
$88$	−0.652346	−0.0695403
$89$	−15.7312	−1.66750	−0.833750	−	0.552142i	$-0.813811\pi$
$89$	−15.7312	−1.66750	−0.833750	+	0.552142i	$0.813811\pi$
$90$	−0.104184	−0.0109819
$91$	−1.01850	−0.106768
$92$	−8.89828	−0.927710
$93$	−3.35252	−0.347640
$94$	−13.3339	−1.37529
$95$	12.8653	1.31995
$96$	1.74564	0.178163
$97$	−2.07690	−0.210878	−0.105439	−	0.994426i	$-0.533625\pi$
$97$	−2.07690	−0.210878	−0.105439	+	0.994426i	$0.533625\pi$
$98$	6.97546	0.704628
$99$	0.0308243	0.00309796
$100$	−0.138513	−0.0138513
$101$	−8.98620	−0.894160	−0.447080	−	0.894494i	$-0.647536\pi$
$101$	−8.98620	−0.894160	−0.447080	+	0.894494i	$0.647536\pi$
$102$	−0.810792	−0.0802803
$103$	13.3915	1.31950	0.659751	−	0.751484i	$-0.270662\pi$
$103$	13.3915	1.31950	0.659751	+	0.751484i	$0.270662\pi$
$104$	6.50126	0.637501
$105$	−0.602980	−0.0588448
$106$	12.5025	1.21435
$107$	10.8383	1.04778	0.523888	−	0.851787i	$-0.324481\pi$
$107$	10.8383	1.04778	0.523888	+	0.851787i	$0.324481\pi$
$108$	5.15443	0.495985
$109$	14.8775	1.42501	0.712504	−	0.701669i	$-0.247561\pi$
$109$	14.8775	1.42501	0.712504	+	0.701669i	$0.247561\pi$
$110$	−1.43834	−0.137141
$111$	4.79812	0.455417
$112$	0.156662	0.0148032
$113$	−5.37446	−0.505587	−0.252793	−	0.967520i	$-0.581349\pi$
$113$	−5.37446	−0.505587	−0.252793	+	0.967520i	$0.581349\pi$
$114$	10.1856	0.953973
$115$	−19.6196	−1.82954
$116$	0.0633705	0.00588381
$117$	−0.307194	−0.0284001
$118$	11.7901	1.08537
$119$	−0.0727645	−0.00667031
$120$	3.84892	0.351357
$121$	−10.5744	−0.961313
$122$	0.890281	0.0806023
$123$	−1.65614	−0.149329
$124$	1.92051	0.172467
$125$	−11.3298	−1.01337
$126$	−0.00740252	−0.000659469 0
$127$	0.993847	0.0881897	0.0440948	−	0.999027i	$-0.485960\pi$
$127$	0.993847	0.0881897	0.0440948	+	0.999027i	$0.485960\pi$
$128$	−1.00000	−0.0883883
$129$	9.88793	0.870584
$130$	14.3345	1.25722
$131$	12.1959	1.06556	0.532782	−	0.846252i	$-0.321146\pi$
$131$	12.1959	1.06556	0.532782	+	0.846252i	$0.321146\pi$
$132$	−1.13876	−0.0991163
$133$	0.914111	0.0792635
$134$	11.0627	0.955672
$135$	11.3649	0.978134
$136$	0.464467	0.0398277
$137$	−14.1885	−1.21220	−0.606102	−	0.795387i	$-0.707268\pi$
$137$	−14.1885	−1.21220	−0.606102	+	0.795387i	$0.707268\pi$
$138$	−15.5332	−1.32227
$139$	10.4139	0.883292	0.441646	−	0.897189i	$-0.354395\pi$
$139$	10.4139	0.883292	0.441646	+	0.897189i	$0.354395\pi$
$140$	0.345421	0.0291934
$141$	−23.2762	−1.96021
$142$	0.438915	0.0368330
$143$	−4.24107	−0.354656
$144$	0.0472515	0.00393762
$145$	0.139724	0.0116035
$146$	−3.32578	−0.275244
$147$	12.1766	1.00431
$148$	−2.74863	−0.225936
$149$	−16.5280	−1.35402	−0.677012	−	0.735972i	$-0.736726\pi$
$149$	−16.5280	−1.35402	−0.677012	+	0.735972i	$0.736726\pi$
$150$	−0.241794	−0.0197424
$151$	1.02621	0.0835116	0.0417558	−	0.999128i	$-0.486705\pi$
$151$	1.02621	0.0835116	0.0417558	+	0.999128i	$0.486705\pi$
$152$	−5.83491	−0.473274
$153$	−0.0219468	−0.00177429
$154$	−0.102198	−0.00823535
$155$	4.23449	0.340123
$156$	11.3489	0.908635
$157$	23.6676	1.88888	0.944441	−	0.328682i	$-0.106604\pi$
$157$	23.6676	1.88888	0.944441	+	0.328682i	$0.106604\pi$
$158$	6.31109	0.502084
$159$	21.8249	1.73083
$160$	−2.20488	−0.174311
$161$	−1.39403	−0.109865
$162$	9.13952	0.718069
$163$	24.9066	1.95083	0.975417	−	0.220367i	$-0.0707254\pi$
$163$	24.9066	1.95083	0.975417	+	0.220367i	$0.0707254\pi$
$164$	0.948733	0.0740836
$165$	−2.51083	−0.195468
$166$	10.4609	0.811924
$167$	−20.9722	−1.62288	−0.811439	−	0.584437i	$-0.801315\pi$
$167$	−20.9722	−1.62288	−0.811439	+	0.584437i	$0.801315\pi$
$168$	0.273476	0.0210991
$169$	29.2664	2.25126
$170$	1.02409	0.0785444
$171$	0.275708	0.0210839
$172$	−5.66437	−0.431904
$173$	−19.9984	−1.52045	−0.760226	−	0.649659i	$-0.774912\pi$
$173$	−19.9984	−1.52045	−0.760226	+	0.649659i	$0.774912\pi$
$174$	0.110622	0.00838623
$175$	−0.0216998	−0.00164035
$176$	0.652346	0.0491724
$177$	20.5813	1.54699
$178$	15.7312	1.17910
$179$	10.8889	0.813871	0.406935	−	0.913457i	$-0.366597\pi$
$179$	10.8889	0.813871	0.406935	+	0.913457i	$0.366597\pi$
$180$	0.104184	0.00776540
$181$	0.428557	0.0318544	0.0159272	−	0.999873i	$-0.494930\pi$
$181$	0.428557	0.0318544	0.0159272	+	0.999873i	$0.494930\pi$
$182$	1.01850	0.0754965
$183$	1.55411	0.114883
$184$	8.89828	0.655990
$185$	−6.06040	−0.445569
$186$	3.35252	0.245818
$187$	−0.302993	−0.0221571
$188$	13.3339	0.972478
$189$	0.807505	0.0587373
$190$	−12.8653	−0.933345
$191$	18.9964	1.37453	0.687266	−	0.726406i	$-0.258811\pi$
$191$	18.9964	1.37453	0.687266	+	0.726406i	$0.258811\pi$
$192$	−1.74564	−0.125981
$193$	4.16406	0.299736	0.149868	−	0.988706i	$-0.452115\pi$
$193$	4.16406	0.299736	0.149868	+	0.988706i	$0.452115\pi$
$194$	2.07690	0.149113
$195$	25.0228	1.79192
$196$	−6.97546	−0.498247
$197$	14.3473	1.02220	0.511101	−	0.859520i	$-0.329238\pi$
$197$	14.3473	1.02220	0.511101	+	0.859520i	$0.329238\pi$
$198$	−0.0308243	−0.00219059
$199$	−9.42430	−0.668071	−0.334036	−	0.942560i	$-0.608411\pi$
$199$	−9.42430	−0.668071	−0.334036	+	0.942560i	$0.608411\pi$
$200$	0.138513	0.00979435
$201$	19.3115	1.36213
$202$	8.98620	0.632267
$203$	0.00992777	0.000696793 0
$204$	0.810792	0.0567668
$205$	2.09184	0.146100
$206$	−13.3915	−0.933029
$207$	−0.420457	−0.0292238
$208$	−6.50126	−0.450782
$209$	3.80638	0.263293
$210$	0.602980	0.0416096
$211$	−0.781632	−0.0538098	−0.0269049	−	0.999638i	$-0.508565\pi$
$211$	−0.781632	−0.0538098	−0.0269049	+	0.999638i	$0.508565\pi$
$212$	−12.5025	−0.858677
$213$	0.766187	0.0524983
$214$	−10.8383	−0.740890
$215$	−12.4892	−0.851759
$216$	−5.15443	−0.350715
$217$	0.300872	0.0204245
$218$	−14.8775	−1.00763
$219$	−5.80561	−0.392307
$220$	1.43834	0.0969731
$221$	3.01962	0.203122
$222$	−4.79812	−0.322028
$223$	−1.70500	−0.114175	−0.0570876	−	0.998369i	$-0.518181\pi$
$223$	−1.70500	−0.114175	−0.0570876	+	0.998369i	$0.518181\pi$
$224$	−0.156662	−0.0104674
$225$	−0.00654495	−0.000436330 0
$226$	5.37446	0.357504
$227$	−20.5767	−1.36573	−0.682863	−	0.730547i	$-0.739265\pi$
$227$	−20.5767	−1.36573	−0.682863	+	0.730547i	$0.739265\pi$
$228$	−10.1856	−0.674561
$229$	20.5148	1.35566	0.677828	−	0.735221i	$-0.262921\pi$
$229$	20.5148	1.35566	0.677828	+	0.735221i	$0.262921\pi$
$230$	19.6196	1.29368
$231$	−0.178401	−0.0117379
$232$	−0.0633705	−0.00416048
$233$	−13.9158	−0.911652	−0.455826	−	0.890069i	$-0.650656\pi$
$233$	−13.9158	−0.911652	−0.455826	+	0.890069i	$0.650656\pi$
$234$	0.307194	0.0200819
$235$	29.3997	1.91783
$236$	−11.7901	−0.767472
$237$	11.0169	0.715623
$238$	0.0727645	0.00471662
$239$	−29.5089	−1.90877	−0.954387	−	0.298572i	$-0.903490\pi$
$239$	−29.5089	−1.90877	−0.954387	+	0.298572i	$0.903490\pi$
$240$	−3.84892	−0.248447
$241$	−8.99428	−0.579372	−0.289686	−	0.957122i	$-0.593551\pi$
$241$	−8.99428	−0.579372	−0.289686	+	0.957122i	$0.593551\pi$
$242$	10.5744	0.679751
$243$	0.491006	0.0314981
$244$	−0.890281	−0.0569944
$245$	−15.3800	−0.982594
$246$	1.65614	0.105592
$247$	−37.9343	−2.41370
$248$	−1.92051	−0.121953
$249$	18.2610	1.15724
$250$	11.3298	0.716559
$251$	−5.35945	−0.338285	−0.169143	−	0.985592i	$-0.554100\pi$
$251$	−5.35945	−0.338285	−0.169143	+	0.985592i	$0.554100\pi$
$252$	0.00740252	0.000466315 0
$253$	−5.80476	−0.364942
$254$	−0.993847	−0.0623595
$255$	1.78770	0.111950
$256$	1.00000	0.0625000
$257$	−10.2059	−0.636626	−0.318313	−	0.947986i	$-0.603116\pi$
$257$	−10.2059	−0.636626	−0.318313	+	0.947986i	$0.603116\pi$
$258$	−9.88793	−0.615596
$259$	−0.430607	−0.0267566
$260$	−14.3345	−0.888988
$261$	0.00299435	0.000185346 0
$262$	−12.1959	−0.753468
$263$	19.0954	1.17747	0.588736	−	0.808325i	$-0.299626\pi$
$263$	19.0954	1.17747	0.588736	+	0.808325i	$0.299626\pi$
$264$	1.13876	0.0700858
$265$	−27.5665	−1.69340
$266$	−0.914111	−0.0560477
$267$	27.4609	1.68058
$268$	−11.0627	−0.675762
$269$	18.4709	1.12619	0.563095	−	0.826392i	$-0.309611\pi$
$269$	18.4709	1.12619	0.563095	+	0.826392i	$0.309611\pi$
$270$	−11.3649	−0.691645
$271$	−6.62313	−0.402327	−0.201163	−	0.979558i	$-0.564472\pi$
$271$	−6.62313	−0.402327	−0.201163	+	0.979558i	$0.564472\pi$
$272$	−0.464467	−0.0281625
$273$	1.77794	0.107606
$274$	14.1885	0.857157
$275$	−0.0903585	−0.00544882
$276$	15.5332	0.934988
$277$	15.6367	0.939521	0.469760	−	0.882794i	$-0.344340\pi$
$277$	15.6367	0.939521	0.469760	+	0.882794i	$0.344340\pi$
$278$	−10.4139	−0.624582
$279$	0.0907469	0.00543288
$280$	−0.345421	−0.0206429
$281$	24.6991	1.47342	0.736711	−	0.676208i	$-0.236378\pi$
$281$	24.6991	1.47342	0.736711	+	0.676208i	$0.236378\pi$
$282$	23.2762	1.38608
$283$	6.03169	0.358547	0.179273	−	0.983799i	$-0.442625\pi$
$283$	6.03169	0.358547	0.179273	+	0.983799i	$0.442625\pi$
$284$	−0.438915	−0.0260448
$285$	−22.4581	−1.33030
$286$	4.24107	0.250780
$287$	0.148631	0.00877339
$288$	−0.0472515	−0.00278432
$289$	−16.7843	−0.987310
$290$	−0.139724	−0.00820489
$291$	3.62552	0.212532
$292$	3.32578	0.194627
$293$	−2.26491	−0.132317	−0.0661586	−	0.997809i	$-0.521074\pi$
$293$	−2.26491	−0.132317	−0.0661586	+	0.997809i	$0.521074\pi$
$294$	−12.1766	−0.710155
$295$	−25.9958	−1.51353
$296$	2.74863	0.159761
$297$	3.36247	0.195110
$298$	16.5280	0.957440
$299$	57.8501	3.34556
$300$	0.241794	0.0139600
$301$	−0.887392	−0.0511484
$302$	−1.02621	−0.0590516
$303$	15.6866	0.901174
$304$	5.83491	0.334655
$305$	−1.96296	−0.112399
$306$	0.0219468	0.00125461
$307$	−7.37287	−0.420792	−0.210396	−	0.977616i	$-0.567475\pi$
$307$	−7.37287	−0.420792	−0.210396	+	0.977616i	$0.567475\pi$
$308$	0.102198	0.00582327
$309$	−23.3767	−1.32985
$310$	−4.23449	−0.240503
$311$	8.74208	0.495718	0.247859	−	0.968796i	$-0.420273\pi$
$311$	8.74208	0.495718	0.247859	+	0.968796i	$0.420273\pi$
$312$	−11.3489	−0.642502
$313$	9.08282	0.513391	0.256696	−	0.966492i	$-0.417366\pi$
$313$	9.08282	0.513391	0.256696	+	0.966492i	$0.417366\pi$
$314$	−23.6676	−1.33564
$315$	0.0163217	0.000919621 0
$316$	−6.31109	−0.355027
$317$	13.8119	0.775753	0.387877	−	0.921711i	$-0.373209\pi$
$317$	13.8119	0.775753	0.387877	+	0.921711i	$0.373209\pi$
$318$	−21.8249	−1.22388
$319$	0.0413395	0.00231457
$320$	2.20488	0.123256
$321$	−18.9197	−1.05600
$322$	1.39403	0.0776860
$323$	−2.71013	−0.150795
$324$	−9.13952	−0.507751
$325$	0.900510	0.0499513
$326$	−24.9066	−1.37945
$327$	−25.9707	−1.43619
$328$	−0.948733	−0.0523850
$329$	2.08893	0.115166
$330$	2.51083	0.138216
$331$	−6.17996	−0.339681	−0.169841	−	0.985472i	$-0.554325\pi$
$331$	−6.17996	−0.339681	−0.169841	+	0.985472i	$0.554325\pi$
$332$	−10.4609	−0.574117
$333$	−0.129877	−0.00711721
$334$	20.9722	1.14755
$335$	−24.3919	−1.33267
$336$	−0.273476	−0.0149193
$337$	1.84183	0.100331	0.0501654	−	0.998741i	$-0.484025\pi$
$337$	1.84183	0.100331	0.0501654	+	0.998741i	$0.484025\pi$
$338$	−29.2664	−1.59188
$339$	9.38186	0.509553
$340$	−1.02409	−0.0555393
$341$	1.25284	0.0678450
$342$	−0.275708	−0.0149086
$343$	−2.18943	−0.118218
$344$	5.66437	0.305402
$345$	34.2488	1.84389
$346$	19.9984	1.07512
$347$	9.65504	0.518310	0.259155	−	0.965836i	$-0.416556\pi$
$347$	9.65504	0.518310	0.259155	+	0.965836i	$0.416556\pi$
$348$	−0.110622	−0.00592996
$349$	4.04766	0.216666	0.108333	−	0.994115i	$-0.465449\pi$
$349$	4.04766	0.216666	0.108333	+	0.994115i	$0.465449\pi$
$350$	0.0216998	0.00115990
$351$	−33.5103	−1.78865
$352$	−0.652346	−0.0347702
$353$	18.1060	0.963684	0.481842	−	0.876258i	$-0.339968\pi$
$353$	18.1060	0.963684	0.481842	+	0.876258i	$0.339968\pi$
$354$	−20.5813	−1.09388
$355$	−0.967755	−0.0513631
$356$	−15.7312	−0.833750
$357$	0.127020	0.00672263
$358$	−10.8889	−0.575494
$359$	−6.25537	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$359$	−6.25537	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$360$	−0.104184	−0.00549097
$361$	15.0462	0.791906
$362$	−0.428557	−0.0225245
$363$	18.4592	0.968854
$364$	−1.01850	−0.0533841
$365$	7.33294	0.383824
$366$	−1.55411	−0.0812345
$367$	9.55570	0.498804	0.249402	−	0.968400i	$-0.419766\pi$
$367$	9.55570	0.498804	0.249402	+	0.968400i	$0.419766\pi$
$368$	−8.89828	−0.463855
$369$	0.0448290	0.00233371
$370$	6.06040	0.315065
$371$	−1.95867	−0.101689
$372$	−3.35252	−0.173820
$373$	16.1010	0.833676	0.416838	−	0.908981i	$-0.363138\pi$
$373$	16.1010	0.833676	0.416838	+	0.908981i	$0.363138\pi$
$374$	0.302993	0.0156674
$375$	19.7777	1.02132
$376$	−13.3339	−0.687646
$377$	−0.411989	−0.0212185
$378$	−0.807505	−0.0415336
$379$	24.3896	1.25281	0.626405	−	0.779497i	$-0.284525\pi$
$379$	24.3896	1.25281	0.626405	+	0.779497i	$0.284525\pi$
$380$	12.8653	0.659975
$381$	−1.73490	−0.0888815
$382$	−18.9964	−0.971940
$383$	−28.5792	−1.46033	−0.730164	−	0.683272i	$-0.760556\pi$
$383$	−28.5792	−1.46033	−0.730164	+	0.683272i	$0.760556\pi$
$384$	1.74564	0.0890817
$385$	0.225334	0.0114841
$386$	−4.16406	−0.211945
$387$	−0.267650	−0.0136054
$388$	−2.07690	−0.105439
$389$	28.3934	1.43960	0.719801	−	0.694180i	$-0.244233\pi$
$389$	28.3934	1.43960	0.719801	+	0.694180i	$0.244233\pi$
$390$	−25.0228	−1.26708
$391$	4.13296	0.209013
$392$	6.97546	0.352314
$393$	−21.2897	−1.07392
$394$	−14.3473	−0.722807
$395$	−13.9152	−0.700149
$396$	0.0308243	0.00154898
$397$	14.9933	0.752491	0.376246	−	0.926520i	$-0.377215\pi$
$397$	14.9933	0.752491	0.376246	+	0.926520i	$0.377215\pi$
$398$	9.42430	0.472398
$399$	−1.59571	−0.0798852
$400$	−0.138513	−0.00692565
$401$	27.9414	1.39533	0.697663	−	0.716426i	$-0.254223\pi$
$401$	27.9414	1.39533	0.697663	+	0.716426i	$0.254223\pi$
$402$	−19.3115	−0.963169
$403$	−12.4857	−0.621959
$404$	−8.98620	−0.447080
$405$	−20.1515	−1.00134
$406$	−0.00992777	−0.000492707 0
$407$	−1.79306	−0.0888787
$408$	−0.810792	−0.0401402
$409$	−16.2981	−0.805888	−0.402944	−	0.915225i	$-0.632013\pi$
$409$	−16.2981	−0.805888	−0.402944	+	0.915225i	$0.632013\pi$
$410$	−2.09184	−0.103309
$411$	24.7679	1.22171
$412$	13.3915	0.659751
$413$	−1.84707	−0.0908883
$414$	0.420457	0.0206643
$415$	−23.0650	−1.13222
$416$	6.50126	0.318751
$417$	−18.1788	−0.890221
$418$	−3.80638	−0.186176
$419$	6.30919	0.308224	0.154112	−	0.988053i	$-0.450748\pi$
$419$	6.30919	0.308224	0.154112	+	0.988053i	$0.450748\pi$
$420$	−0.602980	−0.0294224
$421$	5.89283	0.287199	0.143600	−	0.989636i	$-0.454132\pi$
$421$	5.89283	0.287199	0.143600	+	0.989636i	$0.454132\pi$
$422$	0.781632	0.0380492
$423$	0.630049	0.0306340
$424$	12.5025	0.607176
$425$	0.0643348	0.00312070
$426$	−0.766187	−0.0371219
$427$	−0.139473	−0.00674959
$428$	10.8383	0.523888
$429$	7.40338	0.357439
$430$	12.4892	0.602284
$431$	−1.00000	−0.0481683
$431$	−1.00000	−0.0481683
$432$	5.15443	0.247993
$433$	−31.6288	−1.51998	−0.759990	−	0.649934i	$-0.774796\pi$
$433$	−31.6288	−1.51998	−0.759990	+	0.649934i	$0.774796\pi$
$434$	−0.300872	−0.0144423
$435$	−0.243908	−0.0116945
$436$	14.8775	0.712504
$437$	−51.9207	−2.48370
$438$	5.80561	0.277403
$439$	−18.3571	−0.876138	−0.438069	−	0.898941i	$-0.644337\pi$
$439$	−18.3571	−0.876138	−0.438069	+	0.898941i	$0.644337\pi$
$440$	−1.43834	−0.0685703
$441$	−0.329601	−0.0156953
$442$	−3.01962	−0.143629
$443$	27.1503	1.28995	0.644975	−	0.764203i	$-0.276868\pi$
$443$	27.1503	1.28995	0.644975	+	0.764203i	$0.276868\pi$
$444$	4.79812	0.227709
$445$	−34.6853	−1.64424
$446$	1.70500	0.0807340
$447$	28.8519	1.36465
$448$	0.156662	0.00740160
$449$	0.862033	0.0406818	0.0203409	−	0.999793i	$-0.493525\pi$
$449$	0.862033	0.0406818	0.0203409	+	0.999793i	$0.493525\pi$
$450$	0.00654495	0.000308532 0
$451$	0.618902	0.0291430
$452$	−5.37446	−0.252793
$453$	−1.79139	−0.0841667
$454$	20.5767	0.965714
$455$	−2.24567	−0.105279
$456$	10.1856	0.476987
$457$	−23.1644	−1.08358	−0.541792	−	0.840512i	$-0.682254\pi$
$457$	−23.1644	−1.08358	−0.541792	+	0.840512i	$0.682254\pi$
$458$	−20.5148	−0.958593
$459$	−2.39406	−0.111745
$460$	−19.6196	−0.914770
$461$	−3.81137	−0.177513	−0.0887566	−	0.996053i	$-0.528289\pi$
$461$	−3.81137	−0.177513	−0.0887566	+	0.996053i	$0.528289\pi$
$462$	0.178401	0.00829995
$463$	7.93945	0.368978	0.184489	−	0.982835i	$-0.440937\pi$
$463$	7.93945	0.368978	0.184489	+	0.982835i	$0.440937\pi$
$464$	0.0633705	0.00294190
$465$	−7.39189	−0.342791
$466$	13.9158	0.644635
$467$	−24.4002	−1.12911	−0.564554	−	0.825396i	$-0.690952\pi$
$467$	−24.4002	−1.12911	−0.564554	+	0.825396i	$0.690952\pi$
$468$	−0.307194	−0.0142001
$469$	−1.73311	−0.0800275
$470$	−29.3997	−1.35611
$471$	−41.3151	−1.90370
$472$	11.7901	0.542685
$473$	−3.69513	−0.169902
$474$	−11.0169	−0.506022
$475$	−0.808212	−0.0370833
$476$	−0.0727645	−0.00333515
$477$	−0.590763	−0.0270492
$478$	29.5089	1.34971
$479$	35.3294	1.61424	0.807120	−	0.590387i	$-0.201025\pi$
$479$	35.3294	1.61424	0.807120	+	0.590387i	$0.201025\pi$
$480$	3.84892	0.175678
$481$	17.8696	0.814783
$482$	8.99428	0.409678
$483$	2.43346	0.110726
$484$	−10.5744	−0.480657
$485$	−4.57932	−0.207936
$486$	−0.491006	−0.0222725
$487$	27.2982	1.23700	0.618500	−	0.785785i	$-0.287741\pi$
$487$	27.2982	1.23700	0.618500	+	0.785785i	$0.287741\pi$
$488$	0.890281	0.0403011
$489$	−43.4779	−1.96614
$490$	15.3800	0.694799
$491$	−14.0711	−0.635019	−0.317510	−	0.948255i	$-0.602847\pi$
$491$	−14.0711	−0.635019	−0.317510	+	0.948255i	$0.602847\pi$
$492$	−1.65614	−0.0746647
$493$	−0.0294335	−0.00132562
$494$	37.9343	1.70675
$495$	0.0679639	0.00305475
$496$	1.92051	0.0862335
$497$	−0.0687615	−0.00308437
$498$	−18.2610	−0.818293
$499$	−43.5032	−1.94747	−0.973736	−	0.227681i	$-0.926886\pi$
$499$	−43.5032	−1.94747	−0.973736	+	0.227681i	$0.926886\pi$
$500$	−11.3298	−0.506684
$501$	36.6099	1.63561
$502$	5.35945	0.239204
$503$	−25.8081	−1.15073	−0.575364	−	0.817898i	$-0.695139\pi$
$503$	−25.8081	−1.15073	−0.575364	+	0.817898i	$0.695139\pi$
$504$	−0.00740252	−0.000329735 0
$505$	−19.8135	−0.881688
$506$	5.80476	0.258053
$507$	−51.0886	−2.26892
$508$	0.993847	0.0440948
$509$	−3.13398	−0.138911	−0.0694557	−	0.997585i	$-0.522126\pi$
$509$	−3.13398	−0.138911	−0.0694557	+	0.997585i	$0.522126\pi$
$510$	−1.78770	−0.0791605
$511$	0.521024	0.0230488
$512$	−1.00000	−0.0441942
$513$	30.0757	1.32787
$514$	10.2059	0.450162
$515$	29.5266	1.30110
$516$	9.88793	0.435292
$517$	8.69835	0.382553
$518$	0.430607	0.0189198
$519$	34.9100	1.53238
$520$	14.3345	0.628609
$521$	−33.6451	−1.47402	−0.737010	−	0.675882i	$-0.763763\pi$
$521$	−33.6451	−1.47402	−0.737010	+	0.675882i	$0.763763\pi$
$522$	−0.00299435	−0.000131059 0
$523$	−4.34015	−0.189782	−0.0948908	−	0.995488i	$-0.530250\pi$
$523$	−4.34015	−0.189782	−0.0948908	+	0.995488i	$0.530250\pi$
$524$	12.1959	0.532782
$525$	0.0378799	0.00165322
$526$	−19.0954	−0.832599
$527$	−0.892014	−0.0388567
$528$	−1.13876	−0.0495582
$529$	56.1795	2.44259
$530$	27.5665	1.19741
$531$	−0.557101	−0.0241761
$532$	0.914111	0.0396317
$533$	−6.16796	−0.267164
$534$	−27.4609	−1.18835
$535$	23.8971	1.03316
$536$	11.0627	0.477836
$537$	−19.0080	−0.820255
$538$	−18.4709	−0.796336
$539$	−4.55041	−0.196000
$540$	11.3649	0.489067
$541$	−35.9399	−1.54518	−0.772589	−	0.634906i	$-0.781039\pi$
$541$	−35.9399	−1.54518	−0.772589	+	0.634906i	$0.781039\pi$
$542$	6.62313	0.284488
$543$	−0.748106	−0.0321043
$544$	0.464467	0.0199139
$545$	32.8031	1.40513
$546$	−1.77794	−0.0760887
$547$	−14.7702	−0.631528	−0.315764	−	0.948838i	$-0.602261\pi$
$547$	−14.7702	−0.631528	−0.315764	+	0.948838i	$0.602261\pi$
$548$	−14.1885	−0.606102
$549$	−0.0420671	−0.00179538
$550$	0.0903585	0.00385290
$551$	0.369762	0.0157524
$552$	−15.5332	−0.661136
$553$	−0.988710	−0.0420442
$554$	−15.6367	−0.664341
$555$	10.5793	0.449065
$556$	10.4139	0.441646
$557$	16.0324	0.679315	0.339658	−	0.940549i	$-0.389689\pi$
$557$	16.0324	0.679315	0.339658	+	0.940549i	$0.389689\pi$
$558$	−0.0907469	−0.00384162
$559$	36.8255	1.55755
$560$	0.345421	0.0145967
$561$	0.528917	0.0223309
$562$	−24.6991	−1.04187
$563$	−5.84799	−0.246463	−0.123232	−	0.992378i	$-0.539326\pi$
$563$	−5.84799	−0.246463	−0.123232	+	0.992378i	$0.539326\pi$
$564$	−23.2762	−0.980107
$565$	−11.8500	−0.498535
$566$	−6.03169	−0.253531
$567$	−1.43182	−0.0601307
$568$	0.438915	0.0184165
$569$	3.06263	0.128392	0.0641960	−	0.997937i	$-0.479552\pi$
$569$	3.06263	0.128392	0.0641960	+	0.997937i	$0.479552\pi$
$570$	22.4581	0.940667
$571$	46.9297	1.96395	0.981974	−	0.189018i	$-0.0605306\pi$
$571$	46.9297	1.96395	0.981974	+	0.189018i	$0.0605306\pi$
$572$	−4.24107	−0.177328
$573$	−33.1608	−1.38531
$574$	−0.148631	−0.00620372
$575$	1.23253	0.0514000
$576$	0.0472515	0.00196881
$577$	−14.3249	−0.596355	−0.298177	−	0.954510i	$-0.596379\pi$
$577$	−14.3249	−0.596355	−0.298177	+	0.954510i	$0.596379\pi$
$578$	16.7843	0.698134
$579$	−7.26895	−0.302087
$580$	0.139724	0.00580174
$581$	−1.63883	−0.0679901
$582$	−3.62552	−0.150283
$583$	−8.15597	−0.337786
$584$	−3.32578	−0.137622
$585$	−0.677326	−0.0280040
$586$	2.26491	0.0935624
$587$	−26.9205	−1.11113	−0.555565	−	0.831473i	$-0.687498\pi$
$587$	−26.9205	−1.11113	−0.555565	+	0.831473i	$0.687498\pi$
$588$	12.1766	0.502155
$589$	11.2060	0.461736
$590$	25.9958	1.07023
$591$	−25.0452	−1.03022
$592$	−2.74863	−0.112968
$593$	9.48794	0.389623	0.194811	−	0.980841i	$-0.437591\pi$
$593$	9.48794	0.389623	0.194811	+	0.980841i	$0.437591\pi$
$594$	−3.36247	−0.137964
$595$	−0.160437	−0.00657727
$596$	−16.5280	−0.677012
$597$	16.4514	0.673312
$598$	−57.8501	−2.36567
$599$	−25.1443	−1.02737	−0.513684	−	0.857979i	$-0.671720\pi$
$599$	−25.1443	−1.02737	−0.513684	+	0.857979i	$0.671720\pi$
$600$	−0.241794	−0.00987119
$601$	8.28149	0.337809	0.168905	−	0.985632i	$-0.445977\pi$
$601$	8.28149	0.337809	0.168905	+	0.985632i	$0.445977\pi$
$602$	0.887392	0.0361674
$603$	−0.522729	−0.0212872
$604$	1.02621	0.0417558
$605$	−23.3154	−0.947904
$606$	−15.6866	−0.637226
$607$	27.3851	1.11153	0.555763	−	0.831340i	$-0.312426\pi$
$607$	27.3851	1.11153	0.555763	+	0.831340i	$0.312426\pi$
$608$	−5.83491	−0.236637
$609$	−0.0173303	−0.000702259 0
$610$	1.96296	0.0794780
$611$	−86.6875	−3.50700
$612$	−0.0219468	−0.000887145 0
$613$	−3.30066	−0.133312	−0.0666562	−	0.997776i	$-0.521233\pi$
$613$	−3.30066	−0.133312	−0.0666562	+	0.997776i	$0.521233\pi$
$614$	7.37287	0.297545
$615$	−3.65159	−0.147247
$616$	−0.102198	−0.00411768
$617$	−20.3991	−0.821238	−0.410619	−	0.911807i	$-0.634687\pi$
$617$	−20.3991	−0.821238	−0.410619	+	0.911807i	$0.634687\pi$
$618$	23.3767	0.940348
$619$	−2.14567	−0.0862416	−0.0431208	−	0.999070i	$-0.513730\pi$
$619$	−2.14567	−0.0862416	−0.0431208	+	0.999070i	$0.513730\pi$
$620$	4.23449	0.170061
$621$	−45.8656	−1.84052
$622$	−8.74208	−0.350525
$623$	−2.46448	−0.0987373
$624$	11.3489	0.454318
$625$	−24.2882	−0.971530
$626$	−9.08282	−0.363022
$627$	−6.64457	−0.265358
$628$	23.6676	0.944441
$629$	1.27665	0.0509033
$630$	−0.0163217	−0.000650271 0
$631$	44.3696	1.76632	0.883162	−	0.469068i	$-0.155410\pi$
$631$	44.3696	1.76632	0.883162	+	0.469068i	$0.155410\pi$
$632$	6.31109	0.251042
$633$	1.36445	0.0542319
$634$	−13.8119	−0.548540
$635$	2.19131	0.0869596
$636$	21.8249	0.865413
$637$	45.3493	1.79680
$638$	−0.0413395	−0.00163665
$639$	−0.0207394	−0.000820438 0
$640$	−2.20488	−0.0871555
$641$	24.6478	0.973531	0.486766	−	0.873533i	$-0.338177\pi$
$641$	24.6478	0.973531	0.486766	+	0.873533i	$0.338177\pi$
$642$	18.9197	0.746702
$643$	25.9806	1.02457	0.512287	−	0.858814i	$-0.328798\pi$
$643$	25.9806	1.02457	0.512287	+	0.858814i	$0.328798\pi$
$644$	−1.39403	−0.0549323
$645$	21.8017	0.858440
$646$	2.71013	0.106628
$647$	0.751480	0.0295437	0.0147719	−	0.999891i	$-0.495298\pi$
$647$	0.751480	0.0295437	0.0147719	+	0.999891i	$0.495298\pi$
$648$	9.13952	0.359034
$649$	−7.69125	−0.301908
$650$	−0.900510	−0.0353209
$651$	−0.525213	−0.0205847
$652$	24.9066	0.975417
$653$	38.1197	1.49174	0.745869	−	0.666093i	$-0.232035\pi$
$653$	38.1197	1.49174	0.745869	+	0.666093i	$0.232035\pi$
$654$	25.9707	1.01554
$655$	26.8906	1.05070
$656$	0.948733	0.0370418
$657$	0.157148	0.00613093
$658$	−2.08893	−0.0814348
$659$	−12.9150	−0.503096	−0.251548	−	0.967845i	$-0.580940\pi$
$659$	−12.9150	−0.503096	−0.251548	+	0.967845i	$0.580940\pi$
$660$	−2.51083	−0.0977338
$661$	22.5406	0.876728	0.438364	−	0.898797i	$-0.355558\pi$
$661$	22.5406	0.876728	0.438364	+	0.898797i	$0.355558\pi$
$662$	6.17996	0.240191
$663$	−5.27117	−0.204715
$664$	10.4609	0.405962
$665$	2.01550	0.0781578
$666$	0.129877	0.00503263
$667$	−0.563889	−0.0218339
$668$	−20.9722	−0.811439
$669$	2.97631	0.115071
$670$	24.3919	0.942342
$671$	−0.580771	−0.0224204
$672$	0.273476	0.0105495
$673$	−41.7604	−1.60975	−0.804873	−	0.593447i	$-0.797767\pi$
$673$	−41.7604	−1.60975	−0.804873	+	0.593447i	$0.797767\pi$
$674$	−1.84183	−0.0709446
$675$	−0.713956	−0.0274802
$676$	29.2664	1.12563
$677$	−10.3065	−0.396111	−0.198055	−	0.980191i	$-0.563463\pi$
$677$	−10.3065	−0.396111	−0.198055	+	0.980191i	$0.563463\pi$
$678$	−9.38186	−0.360308
$679$	−0.325372	−0.0124866
$680$	1.02409	0.0392722
$681$	35.9195	1.37644
$682$	−1.25284	−0.0479736
$683$	34.8815	1.33470	0.667351	−	0.744743i	$-0.267428\pi$
$683$	34.8815	1.33470	0.667351	+	0.744743i	$0.267428\pi$
$684$	0.275708	0.0105420
$685$	−31.2839	−1.19530
$686$	2.18943	0.0835927
$687$	−35.8114	−1.36629
$688$	−5.66437	−0.215952
$689$	81.2822	3.09661
$690$	−34.2488	−1.30383
$691$	−10.0208	−0.381211	−0.190605	−	0.981667i	$-0.561045\pi$
$691$	−10.0208	−0.381211	−0.190605	+	0.981667i	$0.561045\pi$
$692$	−19.9984	−0.760226
$693$	0.00482901	0.000183439 0
$694$	−9.65504	−0.366500
$695$	22.9613	0.870971
$696$	0.110622	0.00419312
$697$	−0.440655	−0.0166910
$698$	−4.04766	−0.153206
$699$	24.2919	0.918804
$700$	−0.0216998	−0.000820174 0
$701$	−22.6094	−0.853946	−0.426973	−	0.904264i	$-0.640420\pi$
$701$	−22.6094	−0.853946	−0.426973	+	0.904264i	$0.640420\pi$
$702$	33.5103	1.26477
$703$	−16.0380	−0.604886
$704$	0.652346	0.0245862
$705$	−51.3213	−1.93287
$706$	−18.1060	−0.681427
$707$	−1.40780	−0.0529457
$708$	20.5813	0.773493
$709$	20.6029	0.773758	0.386879	−	0.922130i	$-0.373553\pi$
$709$	20.6029	0.773758	0.386879	+	0.922130i	$0.373553\pi$
$710$	0.967755	0.0363192
$711$	−0.298208	−0.0111837
$712$	15.7312	0.589550
$713$	−17.0892	−0.639997
$714$	−0.127020	−0.00475362
$715$	−9.35105	−0.349710
$716$	10.8889	0.406935
$717$	51.5119	1.92375
$718$	6.25537	0.233448
$719$	27.3996	1.02183	0.510917	−	0.859630i	$-0.329306\pi$
$719$	27.3996	1.02183	0.510917	+	0.859630i	$0.329306\pi$
$720$	0.104184	0.00388270
$721$	2.09794	0.0781314
$722$	−15.0462	−0.559962
$723$	15.7008	0.583917
$724$	0.428557	0.0159272
$725$	−0.00877765	−0.000325994 0
$726$	−18.4592	−0.685083
$727$	2.77657	0.102977	0.0514887	−	0.998674i	$-0.483603\pi$
$727$	2.77657	0.102977	0.0514887	+	0.998674i	$0.483603\pi$
$728$	1.01850	0.0377482
$729$	26.5614	0.983757
$730$	−7.33294	−0.271404
$731$	2.63091	0.0973078
$732$	1.55411	0.0574415
$733$	−14.6613	−0.541526	−0.270763	−	0.962646i	$-0.587276\pi$
$733$	−14.6613	−0.541526	−0.270763	+	0.962646i	$0.587276\pi$
$734$	−9.55570	−0.352707
$735$	26.8480	0.990302
$736$	8.89828	0.327995
$737$	−7.21671	−0.265831
$738$	−0.0448290	−0.00165018
$739$	−41.0890	−1.51148	−0.755742	−	0.654869i	$-0.772724\pi$
$739$	−41.0890	−1.51148	−0.755742	+	0.654869i	$0.772724\pi$
$740$	−6.06040	−0.222785
$741$	66.2196	2.43264
$742$	1.95867	0.0719052
$743$	−30.4819	−1.11827	−0.559136	−	0.829076i	$-0.688867\pi$
$743$	−30.4819	−1.11827	−0.559136	+	0.829076i	$0.688867\pi$
$744$	3.35252	0.122909
$745$	−36.4422	−1.33514
$746$	−16.1010	−0.589498
$747$	−0.494293	−0.0180852
$748$	−0.302993	−0.0110785
$749$	1.69795	0.0620418
$750$	−19.7777	−0.722180
$751$	−45.8869	−1.67444	−0.837219	−	0.546868i	$-0.815820\pi$
$751$	−45.8869	−1.67444	−0.837219	+	0.546868i	$0.815820\pi$
$752$	13.3339	0.486239
$753$	9.35565	0.340939
$754$	0.411989	0.0150037
$755$	2.26266	0.0823467
$756$	0.807505	0.0293687
$757$	2.64634	0.0961827	0.0480913	−	0.998843i	$-0.484686\pi$
$757$	2.64634	0.0961827	0.0480913	+	0.998843i	$0.484686\pi$
$758$	−24.3896	−0.885871
$759$	10.1330	0.367805
$760$	−12.8653	−0.466672
$761$	−46.3450	−1.68000	−0.840002	−	0.542583i	$-0.817446\pi$
$761$	−46.3450	−1.68000	−0.840002	+	0.542583i	$0.817446\pi$
$762$	1.73490	0.0628487
$763$	2.33074	0.0843786
$764$	18.9964	0.687266
$765$	−0.0483899	−0.00174954
$766$	28.5792	1.03261
$767$	76.6508	2.76770
$768$	−1.74564	−0.0629903
$769$	−13.0789	−0.471636	−0.235818	−	0.971797i	$-0.575777\pi$
$769$	−13.0789	−0.471636	−0.235818	+	0.971797i	$0.575777\pi$
$770$	−0.225334	−0.00812048
$771$	17.8158	0.641620
$772$	4.16406	0.149868
$773$	−19.8118	−0.712579	−0.356290	−	0.934376i	$-0.615958\pi$
$773$	−19.8118	−0.712579	−0.356290	+	0.934376i	$0.615958\pi$
$774$	0.267650	0.00962047
$775$	−0.266016	−0.00955557
$776$	2.07690	0.0745565
$777$	0.751684	0.0269665
$778$	−28.3934	−1.01795
$779$	5.53577	0.198340
$780$	25.0228	0.895961
$781$	−0.286325	−0.0102455
$782$	−4.13296	−0.147794
$783$	0.326639	0.0116731
$784$	−6.97546	−0.249123
$785$	52.1842	1.86253
$786$	21.2897	0.759378
$787$	39.7402	1.41659	0.708293	−	0.705919i	$-0.249466\pi$
$787$	39.7402	1.41659	0.708293	+	0.705919i	$0.249466\pi$
$788$	14.3473	0.511101
$789$	−33.3336	−1.18671
$790$	13.9152	0.495080
$791$	−0.841975	−0.0299372
$792$	−0.0308243	−0.00109529
$793$	5.78795	0.205536
$794$	−14.9933	−0.532091
$795$	48.1212	1.70668
$796$	−9.42430	−0.334036
$797$	2.95456	0.104656	0.0523279	−	0.998630i	$-0.483336\pi$
$797$	2.95456	0.104656	0.0523279	+	0.998630i	$0.483336\pi$
$798$	1.59571	0.0564874
$799$	−6.19318	−0.219099
$800$	0.138513	0.00489718
$801$	−0.743321	−0.0262640
$802$	−27.9414	−0.986645
$803$	2.16956	0.0765621
$804$	19.3115	0.681063
$805$	−3.07366	−0.108332
$806$	12.4857	0.439792
$807$	−32.2435	−1.13502
$808$	8.98620	0.316133
$809$	−34.8978	−1.22694	−0.613470	−	0.789718i	$-0.710227\pi$
$809$	−34.8978	−1.22694	−0.613470	+	0.789718i	$0.710227\pi$
$810$	20.1515	0.708053
$811$	16.5904	0.582567	0.291284	−	0.956637i	$-0.405918\pi$
$811$	16.5904	0.582567	0.291284	+	0.956637i	$0.405918\pi$
$812$	0.00992777	0.000348397 0
$813$	11.5616	0.405483
$814$	1.79306	0.0628467
$815$	54.9160	1.92362
$816$	0.810792	0.0283834
$817$	−33.0511	−1.15631
$818$	16.2981	0.569849
$819$	−0.0481258	−0.00168165
$820$	2.09184	0.0730502
$821$	21.2733	0.742443	0.371221	−	0.928544i	$-0.378939\pi$
$821$	21.2733	0.742443	0.371221	+	0.928544i	$0.378939\pi$
$822$	−24.7679	−0.863881
$823$	−41.2205	−1.43686	−0.718429	−	0.695601i	$-0.755138\pi$
$823$	−41.2205	−1.43686	−0.718429	+	0.695601i	$0.755138\pi$
$824$	−13.3915	−0.466515
$825$	0.157733	0.00549156
$826$	1.84707	0.0642677
$827$	−33.1469	−1.15263	−0.576316	−	0.817227i	$-0.695510\pi$
$827$	−33.1469	−1.15263	−0.576316	+	0.817227i	$0.695510\pi$
$828$	−0.420457	−0.0146119
$829$	18.4481	0.640731	0.320365	−	0.947294i	$-0.396194\pi$
$829$	18.4481	0.640731	0.320365	+	0.947294i	$0.396194\pi$
$830$	23.0650	0.800598
$831$	−27.2961	−0.946891
$832$	−6.50126	−0.225391
$833$	3.23987	0.112255
$834$	18.1788	0.629481
$835$	−46.2411	−1.60024
$836$	3.80638	0.131647
$837$	9.89914	0.342164
$838$	−6.30919	−0.217947
$839$	19.4048	0.669929	0.334965	−	0.942231i	$-0.391276\pi$
$839$	19.4048	0.669929	0.334965	+	0.942231i	$0.391276\pi$
$840$	0.602980	0.0208048
$841$	−28.9960	−0.999862
$842$	−5.89283	−0.203080
$843$	−43.1156	−1.48498
$844$	−0.781632	−0.0269049
$845$	64.5289	2.21986
$846$	−0.630049	−0.0216615
$847$	−1.65662	−0.0569220
$848$	−12.5025	−0.429338
$849$	−10.5291	−0.361359
$850$	−0.0643348	−0.00220666
$851$	24.4581	0.838413
$852$	0.766187	0.0262492
$853$	12.5100	0.428335	0.214167	−	0.976797i	$-0.431296\pi$
$853$	12.5100	0.428335	0.214167	+	0.976797i	$0.431296\pi$
$854$	0.139473	0.00477268
$855$	0.607903	0.0207898
$856$	−10.8383	−0.370445
$857$	17.0216	0.581446	0.290723	−	0.956807i	$-0.406104\pi$
$857$	17.0216	0.581446	0.290723	+	0.956807i	$0.406104\pi$
$858$	−7.40338	−0.252747
$859$	18.7774	0.640676	0.320338	−	0.947303i	$-0.396204\pi$
$859$	18.7774	0.640676	0.320338	+	0.947303i	$0.396204\pi$
$860$	−12.4892	−0.425879
$861$	−0.259455	−0.00884221
$862$	1.00000	0.0340601
$863$	18.8828	0.642778	0.321389	−	0.946947i	$-0.395850\pi$
$863$	18.8828	0.642778	0.321389	+	0.946947i	$0.395850\pi$
$864$	−5.15443	−0.175357
$865$	−44.0941	−1.49924
$866$	31.6288	1.07479
$867$	29.2993	0.995055
$868$	0.300872	0.0102122
$869$	−4.11702	−0.139660
$870$	0.243908	0.00826926
$871$	71.9216	2.43697
$872$	−14.8775	−0.503816
$873$	−0.0981367	−0.00332143
$874$	51.9207	1.75624
$875$	−1.77495	−0.0600043
$876$	−5.80561	−0.196153
$877$	−33.5075	−1.13147	−0.565734	−	0.824588i	$-0.691407\pi$
$877$	−33.5075	−1.13147	−0.565734	+	0.824588i	$0.691407\pi$
$878$	18.3571	0.619523
$879$	3.95371	0.133355
$880$	1.43834	0.0484866
$881$	56.5883	1.90651	0.953254	−	0.302169i	$-0.0977106\pi$
$881$	56.5883	1.90651	0.953254	+	0.302169i	$0.0977106\pi$
$882$	0.329601	0.0110982
$883$	28.0681	0.944565	0.472283	−	0.881447i	$-0.343430\pi$
$883$	28.0681	0.944565	0.472283	+	0.881447i	$0.343430\pi$
$884$	3.01962	0.101561
$885$	45.3792	1.52541
$886$	−27.1503	−0.912133
$887$	−8.23335	−0.276449	−0.138224	−	0.990401i	$-0.544139\pi$
$887$	−8.23335	−0.276449	−0.138224	+	0.990401i	$0.544139\pi$
$888$	−4.79812	−0.161014
$889$	0.155698	0.00522196
$890$	34.6853	1.16265
$891$	−5.96213	−0.199739
$892$	−1.70500	−0.0570876
$893$	77.8024	2.60356
$894$	−28.8519	−0.964951
$895$	24.0086	0.802519
$896$	−0.156662	−0.00523372
$897$	−100.985	−3.37180
$898$	−0.862033	−0.0287664
$899$	0.121704	0.00405905
$900$	−0.00654495	−0.000218165 0
$901$	5.80701	0.193460
$902$	−0.618902	−0.0206072
$903$	1.54907	0.0515497
$904$	5.37446	0.178752
$905$	0.944917	0.0314101
$906$	1.79139	0.0595149
$907$	−8.92366	−0.296305	−0.148153	−	0.988965i	$-0.547333\pi$
$907$	−8.92366	−0.296305	−0.148153	+	0.988965i	$0.547333\pi$
$908$	−20.5767	−0.682863
$909$	−0.424611	−0.0140835
$910$	2.24567	0.0744434
$911$	−2.76378	−0.0915682	−0.0457841	−	0.998951i	$-0.514579\pi$
$911$	−2.76378	−0.0915682	−0.0457841	+	0.998951i	$0.514579\pi$
$912$	−10.1856	−0.337280
$913$	−6.82413	−0.225846
$914$	23.1644	0.766210
$915$	3.42662	0.113281
$916$	20.5148	0.677828
$917$	1.91064	0.0630950
$918$	2.39406	0.0790159
$919$	−41.1042	−1.35590	−0.677950	−	0.735108i	$-0.737132\pi$
$919$	−41.1042	−1.35590	−0.677950	+	0.735108i	$0.737132\pi$
$920$	19.6196	0.646840
$921$	12.8704	0.424093
$922$	3.81137	0.125521
$923$	2.85351	0.0939243
$924$	−0.178401	−0.00586895
$925$	0.380722	0.0125180
$926$	−7.93945	−0.260907
$927$	0.632768	0.0207828
$928$	−0.0633705	−0.00208024
$929$	25.2162	0.827318	0.413659	−	0.910432i	$-0.364251\pi$
$929$	25.2162	0.827318	0.413659	+	0.910432i	$0.364251\pi$
$930$	7.39189	0.242390
$931$	−40.7012	−1.33393
$932$	−13.9158	−0.455826
$933$	−15.2605	−0.499606
$934$	24.4002	0.798400
$935$	−0.668063	−0.0218480
$936$	0.307194	0.0100410
$937$	−0.681892	−0.0222765	−0.0111382	−	0.999938i	$-0.503545\pi$
$937$	−0.681892	−0.0222765	−0.0111382	+	0.999938i	$0.503545\pi$
$938$	1.73311	0.0565880
$939$	−15.8553	−0.517419
$940$	29.3997	0.958913
$941$	−27.8038	−0.906379	−0.453190	−	0.891414i	$-0.649714\pi$
$941$	−27.8038	−0.906379	−0.453190	+	0.891414i	$0.649714\pi$
$942$	41.3151	1.34612
$943$	−8.44209	−0.274912
$944$	−11.7901	−0.383736
$945$	1.78045	0.0579180
$946$	3.69513	0.120139
$947$	−14.1900	−0.461112	−0.230556	−	0.973059i	$-0.574054\pi$
$947$	−14.1900	−0.461112	−0.230556	+	0.973059i	$0.574054\pi$
$948$	11.0169	0.357812
$949$	−21.6218	−0.701873
$950$	0.808212	0.0262219
$951$	−24.1106	−0.781838
$952$	0.0727645	0.00235831
$953$	−34.1073	−1.10484	−0.552422	−	0.833565i	$-0.686296\pi$
$953$	−34.1073	−1.10484	−0.552422	+	0.833565i	$0.686296\pi$
$954$	0.590763	0.0191266
$955$	41.8847	1.35536
$956$	−29.5089	−0.954387
$957$	−0.0721638	−0.00233273
$958$	−35.3294	−1.14144
$959$	−2.22280	−0.0717779
$960$	−3.84892	−0.124223
$961$	−27.3116	−0.881021
$962$	−17.8696	−0.576139
$963$	0.512125	0.0165030
$964$	−8.99428	−0.289686
$965$	9.18125	0.295555
$966$	−2.43346	−0.0782954
$967$	−52.1206	−1.67609	−0.838043	−	0.545605i	$-0.816300\pi$
$967$	−52.1206	−1.67609	−0.838043	+	0.545605i	$0.816300\pi$
$968$	10.5744	0.339876
$969$	4.73090	0.151978
$970$	4.57932	0.147033
$971$	12.6660	0.406470	0.203235	−	0.979130i	$-0.434854\pi$
$971$	12.6660	0.406470	0.203235	+	0.979130i	$0.434854\pi$
$972$	0.491006	0.0157490
$973$	1.63146	0.0523022
$974$	−27.2982	−0.874691
$975$	−1.57196	−0.0503432
$976$	−0.890281	−0.0284972
$977$	−29.5467	−0.945284	−0.472642	−	0.881255i	$-0.656700\pi$
$977$	−29.5467	−0.945284	−0.472642	+	0.881255i	$0.656700\pi$
$978$	43.4779	1.39027
$979$	−10.2622	−0.327980
$980$	−15.3800	−0.491297
$981$	0.702984	0.0224446
$982$	14.0711	0.449026
$983$	43.4813	1.38684	0.693420	−	0.720534i	$-0.256103\pi$
$983$	43.4813	1.38684	0.693420	+	0.720534i	$0.256103\pi$
$984$	1.65614	0.0527959
$985$	31.6340	1.00794
$986$	0.0294335	0.000937355 0
$987$	−3.64651	−0.116070
$988$	−37.9343	−1.20685
$989$	50.4031	1.60273
$990$	−0.0679639	−0.00216003
$991$	−31.5743	−1.00299	−0.501496	−	0.865160i	$-0.667217\pi$
$991$	−31.5743	−1.00299	−0.501496	+	0.865160i	$0.667217\pi$
$992$	−1.92051	−0.0609763
$993$	10.7880	0.342346
$994$	0.0687615	0.00218098
$995$	−20.7794	−0.658753
$996$	18.2610	0.578620
$997$	39.9641	1.26568	0.632838	−	0.774284i	$-0.281890\pi$
$997$	39.9641	1.26568	0.632838	+	0.774284i	$0.281890\pi$
$998$	43.5032	1.37707
$999$	−14.1676	−0.448244

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	862.2.a.i.1.2	✓	6
3.2	odd	2		7758.2.a.t.1.2		6
4.3	odd	2		6896.2.a.q.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
862.2.a.i.1.2	✓	6	1.1	even	1	trivial
6896.2.a.q.1.5		6	4.3	odd	2
7758.2.a.t.1.2		6	3.2	odd	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	862.2.a.i.1.2

Level	$862$
Weight	$2$
Character	862.1
Self dual	yes
Analytic conductor	$6.883$
Analytic rank	$1$
Dimension	$6$
CM	no
Inner twists	$1$