LMFDB - Embedded newform 1421.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1421,2,Mod(1,1421)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1421, 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("1421.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1421 = 7^{2} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1421.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:	$11.3467421272$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 2$ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1421.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} -2.41421 q^{3} +3.82843 q^{4} +1.00000 q^{5} +5.82843 q^{6} -4.41421 q^{8} +2.82843 q^{9} -2.41421 q^{10} -0.414214 q^{11} -9.24264 q^{12} +3.82843 q^{13} -2.41421 q^{15} +3.00000 q^{16} -0.828427 q^{17} -6.82843 q^{18} -6.00000 q^{19} +3.82843 q^{20} +1.00000 q^{22} +3.65685 q^{23} +10.6569 q^{24} -4.00000 q^{25} -9.24264 q^{26} +0.414214 q^{27} +1.00000 q^{29} +5.82843 q^{30} -10.0711 q^{31} +1.58579 q^{32} +1.00000 q^{33} +2.00000 q^{34} +10.8284 q^{36} -4.00000 q^{37} +14.4853 q^{38} -9.24264 q^{39} -4.41421 q^{40} +4.48528 q^{41} +3.58579 q^{43} -1.58579 q^{44} +2.82843 q^{45} -8.82843 q^{46} +3.24264 q^{47} -7.24264 q^{48} +9.65685 q^{50} +2.00000 q^{51} +14.6569 q^{52} +9.48528 q^{53} -1.00000 q^{54} -0.414214 q^{55} +14.4853 q^{57} -2.41421 q^{58} +3.65685 q^{59} -9.24264 q^{60} +4.82843 q^{61} +24.3137 q^{62} -9.82843 q^{64} +3.82843 q^{65} -2.41421 q^{66} +5.65685 q^{67} -3.17157 q^{68} -8.82843 q^{69} -8.82843 q^{71} -12.4853 q^{72} -4.00000 q^{73} +9.65685 q^{74} +9.65685 q^{75} -22.9706 q^{76} +22.3137 q^{78} -2.41421 q^{79} +3.00000 q^{80} -9.48528 q^{81} -10.8284 q^{82} -7.65685 q^{83} -0.828427 q^{85} -8.65685 q^{86} -2.41421 q^{87} +1.82843 q^{88} +12.4853 q^{89} -6.82843 q^{90} +14.0000 q^{92} +24.3137 q^{93} -7.82843 q^{94} -6.00000 q^{95} -3.82843 q^{96} -4.48528 q^{97} -1.17157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 6 q^{6} - 6 q^{8} - 2 q^{10} + 2 q^{11} - 10 q^{12} + 2 q^{13} - 2 q^{15} + 6 q^{16} + 4 q^{17} - 8 q^{18} - 12 q^{19} + 2 q^{20} + 2 q^{22} - 4 q^{23} + 10 q^{24}+ \cdots - 8 q^{99}+O(q^{100})$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 6 * q^6 - 6 * q^8 - 2 * q^10 + 2 * q^11 - 10 * q^12 + 2 * q^13 - 2 * q^15 + 6 * q^16 + 4 * q^17 - 8 * q^18 - 12 * q^19 + 2 * q^20 + 2 * q^22 - 4 * q^23 + 10 * q^24 - 8 * q^25 - 10 * q^26 - 2 * q^27 + 2 * q^29 + 6 * q^30 - 6 * q^31 + 6 * q^32 + 2 * q^33 + 4 * q^34 + 16 * q^36 - 8 * q^37 + 12 * q^38 - 10 * q^39 - 6 * q^40 - 8 * q^41 + 10 * q^43 - 6 * q^44 - 12 * q^46 - 2 * q^47 - 6 * q^48 + 8 * q^50 + 4 * q^51 + 18 * q^52 + 2 * q^53 - 2 * q^54 + 2 * q^55 + 12 * q^57 - 2 * q^58 - 4 * q^59 - 10 * q^60 + 4 * q^61 + 26 * q^62 - 14 * q^64 + 2 * q^65 - 2 * q^66 - 12 * q^68 - 12 * q^69 - 12 * q^71 - 8 * q^72 - 8 * q^73 + 8 * q^74 + 8 * q^75 - 12 * q^76 + 22 * q^78 - 2 * q^79 + 6 * q^80 - 2 * q^81 - 16 * q^82 - 4 * q^83 + 4 * q^85 - 6 * q^86 - 2 * q^87 - 2 * q^88 + 8 * q^89 - 8 * q^90 + 28 * q^92 + 26 * q^93 - 10 * q^94 - 12 * q^95 - 2 * q^96 + 8 * q^97 - 8 * 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

)

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

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	3.82843	1.91421
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	5.82843	2.37945
$7$	0	0
$7$	0	0
$8$	−4.41421	−1.56066
$9$	2.82843	0.942809
$10$	−2.41421	−0.763441
$11$	−0.414214	−0.124890	−0.0624450	−	0.998048i	$-0.519890\pi$
$11$	−0.414214	−0.124890	−0.0624450	+	0.998048i	$0.519890\pi$
$12$	−9.24264	−2.66812
$13$	3.82843	1.06181	0.530907	−	0.847430i	$-0.321851\pi$
$13$	3.82843	1.06181	0.530907	+	0.847430i	$0.321851\pi$
$14$	0	0
$15$	−2.41421	−0.623347
$16$	3.00000	0.750000
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	−6.82843	−1.60948
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	3.82843	0.856062
$21$	0	0
$22$	1.00000	0.213201
$23$	3.65685	0.762507	0.381253	−	0.924471i	$-0.375493\pi$
$23$	3.65685	0.762507	0.381253	+	0.924471i	$0.375493\pi$
$24$	10.6569	2.17532
$25$	−4.00000	−0.800000
$26$	−9.24264	−1.81263
$27$	0.414214	0.0797154
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	5.82843	1.06412
$31$	−10.0711	−1.80882	−0.904409	−	0.426667i	$-0.859687\pi$
$31$	−10.0711	−1.80882	−0.904409	+	0.426667i	$0.859687\pi$
$32$	1.58579	0.280330
$33$	1.00000	0.174078
$34$	2.00000	0.342997
$35$	0	0
$36$	10.8284	1.80474
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	14.4853	2.34982
$39$	−9.24264	−1.48001
$40$	−4.41421	−0.697948
$41$	4.48528	0.700483	0.350242	−	0.936659i	$-0.386099\pi$
$41$	4.48528	0.700483	0.350242	+	0.936659i	$0.386099\pi$
$42$	0	0
$43$	3.58579	0.546827	0.273414	−	0.961897i	$-0.411847\pi$
$43$	3.58579	0.546827	0.273414	+	0.961897i	$0.411847\pi$
$44$	−1.58579	−0.239066
$45$	2.82843	0.421637
$46$	−8.82843	−1.30168
$47$	3.24264	0.472988	0.236494	−	0.971633i	$-0.424002\pi$
$47$	3.24264	0.472988	0.236494	+	0.971633i	$0.424002\pi$
$48$	−7.24264	−1.04539
$49$	0	0
$50$	9.65685	1.36569
$51$	2.00000	0.280056
$52$	14.6569	2.03254
$53$	9.48528	1.30290	0.651452	−	0.758690i	$-0.274160\pi$
$53$	9.48528	1.30290	0.651452	+	0.758690i	$0.274160\pi$
$54$	−1.00000	−0.136083
$55$	−0.414214	−0.0558525
$56$	0	0
$57$	14.4853	1.91862
$58$	−2.41421	−0.317002
$59$	3.65685	0.476082	0.238041	−	0.971255i	$-0.423495\pi$
$59$	3.65685	0.476082	0.238041	+	0.971255i	$0.423495\pi$
$60$	−9.24264	−1.19322
$61$	4.82843	0.618217	0.309108	−	0.951027i	$-0.399969\pi$
$61$	4.82843	0.618217	0.309108	+	0.951027i	$0.399969\pi$
$62$	24.3137	3.08784
$63$	0	0
$64$	−9.82843	−1.22855
$65$	3.82843	0.474858
$66$	−2.41421	−0.297169
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	−3.17157	−0.384610
$69$	−8.82843	−1.06282
$70$	0	0
$71$	−8.82843	−1.04774	−0.523871	−	0.851798i	$-0.675513\pi$
$71$	−8.82843	−1.04774	−0.523871	+	0.851798i	$0.675513\pi$
$72$	−12.4853	−1.47140
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	9.65685	1.12259
$75$	9.65685	1.11508
$76$	−22.9706	−2.63490
$77$	0	0
$78$	22.3137	2.52653
$79$	−2.41421	−0.271620	−0.135810	−	0.990735i	$-0.543364\pi$
$79$	−2.41421	−0.271620	−0.135810	+	0.990735i	$0.543364\pi$
$80$	3.00000	0.335410
$81$	−9.48528	−1.05392
$82$	−10.8284	−1.19580
$83$	−7.65685	−0.840449	−0.420224	−	0.907420i	$-0.638049\pi$
$83$	−7.65685	−0.840449	−0.420224	+	0.907420i	$0.638049\pi$
$84$	0	0
$85$	−0.828427	−0.0898555
$86$	−8.65685	−0.933493
$87$	−2.41421	−0.258831
$88$	1.82843	0.194911
$89$	12.4853	1.32344	0.661719	−	0.749752i	$-0.269827\pi$
$89$	12.4853	1.32344	0.661719	+	0.749752i	$0.269827\pi$
$90$	−6.82843	−0.719779
$91$	0	0
$92$	14.0000	1.45960
$93$	24.3137	2.52121
$94$	−7.82843	−0.807441
$95$	−6.00000	−0.615587
$96$	−3.82843	−0.390737
$97$	−4.48528	−0.455411	−0.227706	−	0.973730i	$-0.573122\pi$
$97$	−4.48528	−0.455411	−0.227706	+	0.973730i	$0.573122\pi$
$98$	0	0
$99$	−1.17157	−0.117748
$100$	−15.3137	−1.53137
$101$	2.34315	0.233152	0.116576	−	0.993182i	$-0.462808\pi$
$101$	2.34315	0.233152	0.116576	+	0.993182i	$0.462808\pi$
$102$	−4.82843	−0.478086
$103$	4.82843	0.475759	0.237880	−	0.971295i	$-0.423548\pi$
$103$	4.82843	0.475759	0.237880	+	0.971295i	$0.423548\pi$
$104$	−16.8995	−1.65713
$105$	0	0
$106$	−22.8995	−2.22420
$107$	−14.8284	−1.43352	−0.716759	−	0.697321i	$-0.754375\pi$
$107$	−14.8284	−1.43352	−0.716759	+	0.697321i	$0.754375\pi$
$108$	1.58579	0.152592
$109$	12.6569	1.21231	0.606153	−	0.795348i	$-0.292712\pi$
$109$	12.6569	1.21231	0.606153	+	0.795348i	$0.292712\pi$
$110$	1.00000	0.0953463
$111$	9.65685	0.916588
$112$	0	0
$113$	−13.3137	−1.25245	−0.626224	−	0.779643i	$-0.715401\pi$
$113$	−13.3137	−1.25245	−0.626224	+	0.779643i	$0.715401\pi$
$114$	−34.9706	−3.27529
$115$	3.65685	0.341003
$116$	3.82843	0.355461
$117$	10.8284	1.00109
$118$	−8.82843	−0.812723
$119$	0	0
$120$	10.6569	0.972833
$121$	−10.8284	−0.984402
$122$	−11.6569	−1.05536
$123$	−10.8284	−0.976366
$124$	−38.5563	−3.46246
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−4.34315	−0.385392	−0.192696	−	0.981259i	$-0.561723\pi$
$127$	−4.34315	−0.385392	−0.192696	+	0.981259i	$0.561723\pi$
$128$	20.5563	1.81694
$129$	−8.65685	−0.762194
$130$	−9.24264	−0.810633
$131$	−21.3137	−1.86219	−0.931094	−	0.364780i	$-0.881144\pi$
$131$	−21.3137	−1.86219	−0.931094	+	0.364780i	$0.881144\pi$
$132$	3.82843	0.333222
$133$	0	0
$134$	−13.6569	−1.17977
$135$	0.414214	0.0356498
$136$	3.65685	0.313573
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	21.3137	1.81434
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−7.82843	−0.659272
$142$	21.3137	1.78861
$143$	−1.58579	−0.132610
$144$	8.48528	0.707107
$145$	1.00000	0.0830455
$146$	9.65685	0.799207
$147$	0	0
$148$	−15.3137	−1.25878
$149$	−7.82843	−0.641330	−0.320665	−	0.947193i	$-0.603906\pi$
$149$	−7.82843	−0.641330	−0.320665	+	0.947193i	$0.603906\pi$
$150$	−23.3137	−1.90356
$151$	−14.1421	−1.15087	−0.575435	−	0.817847i	$-0.695167\pi$
$151$	−14.1421	−1.15087	−0.575435	+	0.817847i	$0.695167\pi$
$152$	26.4853	2.14824
$153$	−2.34315	−0.189432
$154$	0	0
$155$	−10.0711	−0.808928
$156$	−35.3848	−2.83305
$157$	−8.48528	−0.677199	−0.338600	−	0.940931i	$-0.609953\pi$
$157$	−8.48528	−0.677199	−0.338600	+	0.940931i	$0.609953\pi$
$158$	5.82843	0.463685
$159$	−22.8995	−1.81605
$160$	1.58579	0.125367
$161$	0	0
$162$	22.8995	1.79915
$163$	3.92893	0.307738	0.153869	−	0.988091i	$-0.450827\pi$
$163$	3.92893	0.307738	0.153869	+	0.988091i	$0.450827\pi$
$164$	17.1716	1.34087
$165$	1.00000	0.0778499
$166$	18.4853	1.43474
$167$	3.17157	0.245424	0.122712	−	0.992442i	$-0.460841\pi$
$167$	3.17157	0.245424	0.122712	+	0.992442i	$0.460841\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	2.00000	0.153393
$171$	−16.9706	−1.29777
$172$	13.7279	1.04674
$173$	−12.3431	−0.938432	−0.469216	−	0.883083i	$-0.655463\pi$
$173$	−12.3431	−0.938432	−0.469216	+	0.883083i	$0.655463\pi$
$174$	5.82843	0.441852
$175$	0	0
$176$	−1.24264	−0.0936676
$177$	−8.82843	−0.663585
$178$	−30.1421	−2.25925
$179$	−6.48528	−0.484733	−0.242366	−	0.970185i	$-0.577924\pi$
$179$	−6.48528	−0.484733	−0.242366	+	0.970185i	$0.577924\pi$
$180$	10.8284	0.807103
$181$	−8.31371	−0.617953	−0.308977	−	0.951070i	$-0.599986\pi$
$181$	−8.31371	−0.617953	−0.308977	+	0.951070i	$0.599986\pi$
$182$	0	0
$183$	−11.6569	−0.861699
$184$	−16.1421	−1.19001
$185$	−4.00000	−0.294086
$186$	−58.6985	−4.30398
$187$	0.343146	0.0250933
$188$	12.4142	0.905400
$189$	0	0
$190$	14.4853	1.05087
$191$	25.3137	1.83164	0.915818	−	0.401594i	$-0.131544\pi$
$191$	25.3137	1.83164	0.915818	+	0.401594i	$0.131544\pi$
$192$	23.7279	1.71242
$193$	−5.17157	−0.372258	−0.186129	−	0.982525i	$-0.559594\pi$
$193$	−5.17157	−0.372258	−0.186129	+	0.982525i	$0.559594\pi$
$194$	10.8284	0.777436
$195$	−9.24264	−0.661879
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	2.82843	0.201008
$199$	0.485281	0.0344007	0.0172003	−	0.999852i	$-0.494525\pi$
$199$	0.485281	0.0344007	0.0172003	+	0.999852i	$0.494525\pi$
$200$	17.6569	1.24853
$201$	−13.6569	−0.963280
$202$	−5.65685	−0.398015
$203$	0	0
$204$	7.65685	0.536087
$205$	4.48528	0.313266
$206$	−11.6569	−0.812172
$207$	10.3431	0.718898
$208$	11.4853	0.796361
$209$	2.48528	0.171911
$210$	0	0
$211$	−19.3848	−1.33450	−0.667252	−	0.744832i	$-0.732529\pi$
$211$	−19.3848	−1.33450	−0.667252	+	0.744832i	$0.732529\pi$
$212$	36.3137	2.49404
$213$	21.3137	1.46039
$214$	35.7990	2.44717
$215$	3.58579	0.244549
$216$	−1.82843	−0.124409
$217$	0	0
$218$	−30.5563	−2.06954
$219$	9.65685	0.652550
$220$	−1.58579	−0.106914
$221$	−3.17157	−0.213343
$222$	−23.3137	−1.56471
$223$	3.17157	0.212384	0.106192	−	0.994346i	$-0.466134\pi$
$223$	3.17157	0.212384	0.106192	+	0.994346i	$0.466134\pi$
$224$	0	0
$225$	−11.3137	−0.754247
$226$	32.1421	2.13806
$227$	8.14214	0.540413	0.270206	−	0.962802i	$-0.412908\pi$
$227$	8.14214	0.540413	0.270206	+	0.962802i	$0.412908\pi$
$228$	55.4558	3.67265
$229$	3.51472	0.232259	0.116130	−	0.993234i	$-0.462951\pi$
$229$	3.51472	0.232259	0.116130	+	0.993234i	$0.462951\pi$
$230$	−8.82843	−0.582129
$231$	0	0
$232$	−4.41421	−0.289807
$233$	18.3137	1.19977	0.599885	−	0.800086i	$-0.295213\pi$
$233$	18.3137	1.19977	0.599885	+	0.800086i	$0.295213\pi$
$234$	−26.1421	−1.70896
$235$	3.24264	0.211527
$236$	14.0000	0.911322
$237$	5.82843	0.378597
$238$	0	0
$239$	−19.6569	−1.27150	−0.635748	−	0.771897i	$-0.719308\pi$
$239$	−19.6569	−1.27150	−0.635748	+	0.771897i	$0.719308\pi$
$240$	−7.24264	−0.467510
$241$	18.3137	1.17969	0.589845	−	0.807517i	$-0.299189\pi$
$241$	18.3137	1.17969	0.589845	+	0.807517i	$0.299189\pi$
$242$	26.1421	1.68048
$243$	21.6569	1.38929
$244$	18.4853	1.18340
$245$	0	0
$246$	26.1421	1.66676
$247$	−22.9706	−1.46158
$248$	44.4558	2.82295
$249$	18.4853	1.17146
$250$	21.7279	1.37419
$251$	−20.0711	−1.26687	−0.633437	−	0.773794i	$-0.718357\pi$
$251$	−20.0711	−1.26687	−0.633437	+	0.773794i	$0.718357\pi$
$252$	0	0
$253$	−1.51472	−0.0952295
$254$	10.4853	0.657905
$255$	2.00000	0.125245
$256$	−29.9706	−1.87316
$257$	18.1716	1.13351	0.566756	−	0.823886i	$-0.308198\pi$
$257$	18.1716	1.13351	0.566756	+	0.823886i	$0.308198\pi$
$258$	20.8995	1.30115
$259$	0	0
$260$	14.6569	0.908980
$261$	2.82843	0.175075
$262$	51.4558	3.17895
$263$	2.75736	0.170026	0.0850130	−	0.996380i	$-0.472907\pi$
$263$	2.75736	0.170026	0.0850130	+	0.996380i	$0.472907\pi$
$264$	−4.41421	−0.271676
$265$	9.48528	0.582676
$266$	0	0
$267$	−30.1421	−1.84467
$268$	21.6569	1.32290
$269$	−31.4558	−1.91790	−0.958948	−	0.283581i	$-0.908478\pi$
$269$	−31.4558	−1.91790	−0.958948	+	0.283581i	$0.908478\pi$
$270$	−1.00000	−0.0608581
$271$	−16.5563	−1.00573	−0.502863	−	0.864366i	$-0.667720\pi$
$271$	−16.5563	−1.00573	−0.502863	+	0.864366i	$0.667720\pi$
$272$	−2.48528	−0.150692
$273$	0	0
$274$	−28.9706	−1.75018
$275$	1.65685	0.0999121
$276$	−33.7990	−2.03446
$277$	−17.3137	−1.04028	−0.520140	−	0.854081i	$-0.674120\pi$
$277$	−17.3137	−1.04028	−0.520140	+	0.854081i	$0.674120\pi$
$278$	33.7990	2.02713
$279$	−28.4853	−1.70537
$280$	0	0
$281$	31.9706	1.90720	0.953602	−	0.301070i	$-0.0973439\pi$
$281$	31.9706	1.90720	0.953602	+	0.301070i	$0.0973439\pi$
$282$	18.8995	1.12545
$283$	−11.6569	−0.692928	−0.346464	−	0.938063i	$-0.612618\pi$
$283$	−11.6569	−0.692928	−0.346464	+	0.938063i	$0.612618\pi$
$284$	−33.7990	−2.00560
$285$	14.4853	0.858034
$286$	3.82843	0.226380
$287$	0	0
$288$	4.48528	0.264298
$289$	−16.3137	−0.959630
$290$	−2.41421	−0.141768
$291$	10.8284	0.634774
$292$	−15.3137	−0.896167
$293$	−7.65685	−0.447318	−0.223659	−	0.974667i	$-0.571800\pi$
$293$	−7.65685	−0.447318	−0.223659	+	0.974667i	$0.571800\pi$
$294$	0	0
$295$	3.65685	0.212910
$296$	17.6569	1.02628
$297$	−0.171573	−0.00995567
$298$	18.8995	1.09482
$299$	14.0000	0.809641
$300$	36.9706	2.13450
$301$	0	0
$302$	34.1421	1.96466
$303$	−5.65685	−0.324978
$304$	−18.0000	−1.03237
$305$	4.82843	0.276475
$306$	5.65685	0.323381
$307$	−2.89949	−0.165483	−0.0827415	−	0.996571i	$-0.526368\pi$
$307$	−2.89949	−0.165483	−0.0827415	+	0.996571i	$0.526368\pi$
$308$	0	0
$309$	−11.6569	−0.663135
$310$	24.3137	1.38093
$311$	−2.68629	−0.152326	−0.0761628	−	0.997095i	$-0.524267\pi$
$311$	−2.68629	−0.152326	−0.0761628	+	0.997095i	$0.524267\pi$
$312$	40.7990	2.30979
$313$	−9.82843	−0.555536	−0.277768	−	0.960648i	$-0.589595\pi$
$313$	−9.82843	−0.555536	−0.277768	+	0.960648i	$0.589595\pi$
$314$	20.4853	1.15605
$315$	0	0
$316$	−9.24264	−0.519939
$317$	−31.4558	−1.76674	−0.883368	−	0.468680i	$-0.844730\pi$
$317$	−31.4558	−1.76674	−0.883368	+	0.468680i	$0.844730\pi$
$318$	55.2843	3.10019
$319$	−0.414214	−0.0231915
$320$	−9.82843	−0.549426
$321$	35.7990	1.99810
$322$	0	0
$323$	4.97056	0.276570
$324$	−36.3137	−2.01743
$325$	−15.3137	−0.849452
$326$	−9.48528	−0.525341
$327$	−30.5563	−1.68977
$328$	−19.7990	−1.09322
$329$	0	0
$330$	−2.41421	−0.132898
$331$	−2.41421	−0.132697	−0.0663486	−	0.997797i	$-0.521135\pi$
$331$	−2.41421	−0.132697	−0.0663486	+	0.997797i	$0.521135\pi$
$332$	−29.3137	−1.60880
$333$	−11.3137	−0.619987
$334$	−7.65685	−0.418964
$335$	5.65685	0.309067
$336$	0	0
$337$	21.7990	1.18747	0.593733	−	0.804662i	$-0.297653\pi$
$337$	21.7990	1.18747	0.593733	+	0.804662i	$0.297653\pi$
$338$	−4.00000	−0.217571
$339$	32.1421	1.74572
$340$	−3.17157	−0.172003
$341$	4.17157	0.225903
$342$	40.9706	2.21543
$343$	0	0
$344$	−15.8284	−0.853412
$345$	−8.82843	−0.475307
$346$	29.7990	1.60200
$347$	2.48528	0.133417	0.0667084	−	0.997773i	$-0.478750\pi$
$347$	2.48528	0.133417	0.0667084	+	0.997773i	$0.478750\pi$
$348$	−9.24264	−0.495458
$349$	5.14214	0.275252	0.137626	−	0.990484i	$-0.456053\pi$
$349$	5.14214	0.275252	0.137626	+	0.990484i	$0.456053\pi$
$350$	0	0
$351$	1.58579	0.0846430
$352$	−0.656854	−0.0350104
$353$	−26.9706	−1.43550	−0.717749	−	0.696302i	$-0.754828\pi$
$353$	−26.9706	−1.43550	−0.717749	+	0.696302i	$0.754828\pi$
$354$	21.3137	1.13281
$355$	−8.82843	−0.468564
$356$	47.7990	2.53334
$357$	0	0
$358$	15.6569	0.827490
$359$	3.92893	0.207361	0.103681	−	0.994611i	$-0.466938\pi$
$359$	3.92893	0.207361	0.103681	+	0.994611i	$0.466938\pi$
$360$	−12.4853	−0.658032
$361$	17.0000	0.894737
$362$	20.0711	1.05491
$363$	26.1421	1.37211
$364$	0	0
$365$	−4.00000	−0.209370
$366$	28.1421	1.47101
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	10.9706	0.571880
$369$	12.6863	0.660422
$370$	9.65685	0.502036
$371$	0	0
$372$	93.0833	4.82614
$373$	−26.3137	−1.36247	−0.681236	−	0.732064i	$-0.738557\pi$
$373$	−26.3137	−1.36247	−0.681236	+	0.732064i	$0.738557\pi$
$374$	−0.828427	−0.0428369
$375$	21.7279	1.12203
$376$	−14.3137	−0.738173
$377$	3.82843	0.197174
$378$	0	0
$379$	−6.97056	−0.358054	−0.179027	−	0.983844i	$-0.557295\pi$
$379$	−6.97056	−0.358054	−0.179027	+	0.983844i	$0.557295\pi$
$380$	−22.9706	−1.17837
$381$	10.4853	0.537177
$382$	−61.1127	−3.12680
$383$	3.51472	0.179594	0.0897969	−	0.995960i	$-0.471378\pi$
$383$	3.51472	0.179594	0.0897969	+	0.995960i	$0.471378\pi$
$384$	−49.6274	−2.53254
$385$	0	0
$386$	12.4853	0.635484
$387$	10.1421	0.515554
$388$	−17.1716	−0.871755
$389$	3.02944	0.153599	0.0767993	−	0.997047i	$-0.475530\pi$
$389$	3.02944	0.153599	0.0767993	+	0.997047i	$0.475530\pi$
$390$	22.3137	1.12990
$391$	−3.02944	−0.153205
$392$	0	0
$393$	51.4558	2.59560
$394$	−4.82843	−0.243253
$395$	−2.41421	−0.121472
$396$	−4.48528	−0.225394
$397$	−19.3431	−0.970805	−0.485402	−	0.874291i	$-0.661327\pi$
$397$	−19.3431	−0.970805	−0.485402	+	0.874291i	$0.661327\pi$
$398$	−1.17157	−0.0587256
$399$	0	0
$400$	−12.0000	−0.600000
$401$	−18.6569	−0.931679	−0.465839	−	0.884869i	$-0.654248\pi$
$401$	−18.6569	−0.931679	−0.465839	+	0.884869i	$0.654248\pi$
$402$	32.9706	1.64442
$403$	−38.5563	−1.92063
$404$	8.97056	0.446302
$405$	−9.48528	−0.471327
$406$	0	0
$407$	1.65685	0.0821272
$408$	−8.82843	−0.437072
$409$	18.9706	0.938034	0.469017	−	0.883189i	$-0.344608\pi$
$409$	18.9706	0.938034	0.469017	+	0.883189i	$0.344608\pi$
$410$	−10.8284	−0.534778
$411$	−28.9706	−1.42901
$412$	18.4853	0.910704
$413$	0	0
$414$	−24.9706	−1.22724
$415$	−7.65685	−0.375860
$416$	6.07107	0.297659
$417$	33.7990	1.65514
$418$	−6.00000	−0.293470
$419$	9.51472	0.464824	0.232412	−	0.972617i	$-0.425338\pi$
$419$	9.51472	0.464824	0.232412	+	0.972617i	$0.425338\pi$
$420$	0	0
$421$	37.1127	1.80876	0.904381	−	0.426726i	$-0.140333\pi$
$421$	37.1127	1.80876	0.904381	+	0.426726i	$0.140333\pi$
$422$	46.7990	2.27814
$423$	9.17157	0.445937
$424$	−41.8701	−2.03339
$425$	3.31371	0.160738
$426$	−51.4558	−2.49304
$427$	0	0
$428$	−56.7696	−2.74406
$429$	3.82843	0.184838
$430$	−8.65685	−0.417471
$431$	19.6569	0.946837	0.473419	−	0.880838i	$-0.343020\pi$
$431$	19.6569	0.946837	0.473419	+	0.880838i	$0.343020\pi$
$432$	1.24264	0.0597866
$433$	−30.6274	−1.47186	−0.735930	−	0.677058i	$-0.763255\pi$
$433$	−30.6274	−1.47186	−0.735930	+	0.677058i	$0.763255\pi$
$434$	0	0
$435$	−2.41421	−0.115753
$436$	48.4558	2.32061
$437$	−21.9411	−1.04959
$438$	−23.3137	−1.11397
$439$	0.343146	0.0163775	0.00818873	−	0.999966i	$-0.497393\pi$
$439$	0.343146	0.0163775	0.00818873	+	0.999966i	$0.497393\pi$
$440$	1.82843	0.0871668
$441$	0	0
$442$	7.65685	0.364199
$443$	−24.3431	−1.15658	−0.578289	−	0.815832i	$-0.696279\pi$
$443$	−24.3431	−1.15658	−0.578289	+	0.815832i	$0.696279\pi$
$444$	36.9706	1.75455
$445$	12.4853	0.591859
$446$	−7.65685	−0.362563
$447$	18.8995	0.893915
$448$	0	0
$449$	−34.9706	−1.65036	−0.825181	−	0.564868i	$-0.808927\pi$
$449$	−34.9706	−1.65036	−0.825181	+	0.564868i	$0.808927\pi$
$450$	27.3137	1.28758
$451$	−1.85786	−0.0874834
$452$	−50.9706	−2.39745
$453$	34.1421	1.60414
$454$	−19.6569	−0.922542
$455$	0	0
$456$	−63.9411	−2.99432
$457$	1.02944	0.0481550	0.0240775	−	0.999710i	$-0.492335\pi$
$457$	1.02944	0.0481550	0.0240775	+	0.999710i	$0.492335\pi$
$458$	−8.48528	−0.396491
$459$	−0.343146	−0.0160167
$460$	14.0000	0.652753
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	3.00000	0.139272
$465$	24.3137	1.12752
$466$	−44.2132	−2.04814
$467$	38.3553	1.77487	0.887437	−	0.460930i	$-0.152484\pi$
$467$	38.3553	1.77487	0.887437	+	0.460930i	$0.152484\pi$
$468$	41.4558	1.91630
$469$	0	0
$470$	−7.82843	−0.361098
$471$	20.4853	0.943912
$472$	−16.1421	−0.743002
$473$	−1.48528	−0.0682933
$474$	−14.0711	−0.646306
$475$	24.0000	1.10120
$476$	0	0
$477$	26.8284	1.22839
$478$	47.4558	2.17058
$479$	−6.89949	−0.315246	−0.157623	−	0.987499i	$-0.550383\pi$
$479$	−6.89949	−0.315246	−0.157623	+	0.987499i	$0.550383\pi$
$480$	−3.82843	−0.174743
$481$	−15.3137	−0.698245
$482$	−44.2132	−2.01386
$483$	0	0
$484$	−41.4558	−1.88436
$485$	−4.48528	−0.203666
$486$	−52.2843	−2.37166
$487$	−11.5147	−0.521782	−0.260891	−	0.965368i	$-0.584016\pi$
$487$	−11.5147	−0.521782	−0.260891	+	0.965368i	$0.584016\pi$
$488$	−21.3137	−0.964826
$489$	−9.48528	−0.428939
$490$	0	0
$491$	−21.2426	−0.958667	−0.479333	−	0.877633i	$-0.659122\pi$
$491$	−21.2426	−0.958667	−0.479333	+	0.877633i	$0.659122\pi$
$492$	−41.4558	−1.86897
$493$	−0.828427	−0.0373105
$494$	55.4558	2.49508
$495$	−1.17157	−0.0526583
$496$	−30.2132	−1.35661
$497$	0	0
$498$	−44.6274	−1.99980
$499$	18.9706	0.849239	0.424620	−	0.905372i	$-0.360408\pi$
$499$	18.9706	0.849239	0.424620	+	0.905372i	$0.360408\pi$
$500$	−34.4558	−1.54091
$501$	−7.65685	−0.342083
$502$	48.4558	2.16269
$503$	−0.272078	−0.0121314	−0.00606568	−	0.999982i	$-0.501931\pi$
$503$	−0.272078	−0.0121314	−0.00606568	+	0.999982i	$0.501931\pi$
$504$	0	0
$505$	2.34315	0.104269
$506$	3.65685	0.162567
$507$	−4.00000	−0.177646
$508$	−16.6274	−0.737722
$509$	10.5147	0.466057	0.233028	−	0.972470i	$-0.425137\pi$
$509$	10.5147	0.466057	0.233028	+	0.972470i	$0.425137\pi$
$510$	−4.82843	−0.213806
$511$	0	0
$512$	31.2426	1.38074
$513$	−2.48528	−0.109728
$514$	−43.8701	−1.93503
$515$	4.82843	0.212766
$516$	−33.1421	−1.45900
$517$	−1.34315	−0.0590715
$518$	0	0
$519$	29.7990	1.30803
$520$	−16.8995	−0.741092
$521$	29.1421	1.27674	0.638370	−	0.769730i	$-0.279609\pi$
$521$	29.1421	1.27674	0.638370	+	0.769730i	$0.279609\pi$
$522$	−6.82843	−0.298872
$523$	−4.68629	−0.204917	−0.102459	−	0.994737i	$-0.532671\pi$
$523$	−4.68629	−0.204917	−0.102459	+	0.994737i	$0.532671\pi$
$524$	−81.5980	−3.56462
$525$	0	0
$526$	−6.65685	−0.290253
$527$	8.34315	0.363433
$528$	3.00000	0.130558
$529$	−9.62742	−0.418583
$530$	−22.8995	−0.994690
$531$	10.3431	0.448854
$532$	0	0
$533$	17.1716	0.743783
$534$	72.7696	3.14905
$535$	−14.8284	−0.641089
$536$	−24.9706	−1.07856
$537$	15.6569	0.675643
$538$	75.9411	3.27405
$539$	0	0
$540$	1.58579	0.0682414
$541$	−10.3431	−0.444687	−0.222343	−	0.974968i	$-0.571371\pi$
$541$	−10.3431	−0.444687	−0.222343	+	0.974968i	$0.571371\pi$
$542$	39.9706	1.71688
$543$	20.0711	0.861332
$544$	−1.31371	−0.0563248
$545$	12.6569	0.542160
$546$	0	0
$547$	35.7990	1.53065	0.765327	−	0.643641i	$-0.222577\pi$
$547$	35.7990	1.53065	0.765327	+	0.643641i	$0.222577\pi$
$548$	45.9411	1.96251
$549$	13.6569	0.582860
$550$	−4.00000	−0.170561
$551$	−6.00000	−0.255609
$552$	38.9706	1.65870
$553$	0	0
$554$	41.7990	1.77587
$555$	9.65685	0.409911
$556$	−53.5980	−2.27306
$557$	−17.3137	−0.733605	−0.366803	−	0.930299i	$-0.619548\pi$
$557$	−17.3137	−0.733605	−0.366803	+	0.930299i	$0.619548\pi$
$558$	68.7696	2.91125
$559$	13.7279	0.580629
$560$	0	0
$561$	−0.828427	−0.0349762
$562$	−77.1838	−3.25580
$563$	0.757359	0.0319189	0.0159594	−	0.999873i	$-0.494920\pi$
$563$	0.757359	0.0319189	0.0159594	+	0.999873i	$0.494920\pi$
$564$	−29.9706	−1.26199
$565$	−13.3137	−0.560112
$566$	28.1421	1.18290
$567$	0	0
$568$	38.9706	1.63517
$569$	−39.6569	−1.66250	−0.831251	−	0.555897i	$-0.812375\pi$
$569$	−39.6569	−1.66250	−0.831251	+	0.555897i	$0.812375\pi$
$570$	−34.9706	−1.46476
$571$	14.6274	0.612138	0.306069	−	0.952009i	$-0.400986\pi$
$571$	14.6274	0.612138	0.306069	+	0.952009i	$0.400986\pi$
$572$	−6.07107	−0.253844
$573$	−61.1127	−2.55302
$574$	0	0
$575$	−14.6274	−0.610005
$576$	−27.7990	−1.15829
$577$	29.7990	1.24055	0.620274	−	0.784385i	$-0.287021\pi$
$577$	29.7990	1.24055	0.620274	+	0.784385i	$0.287021\pi$
$578$	39.3848	1.63819
$579$	12.4853	0.518871
$580$	3.82843	0.158967
$581$	0	0
$582$	−26.1421	−1.08363
$583$	−3.92893	−0.162720
$584$	17.6569	0.730646
$585$	10.8284	0.447700
$586$	18.4853	0.763620
$587$	−7.65685	−0.316032	−0.158016	−	0.987437i	$-0.550510\pi$
$587$	−7.65685	−0.316032	−0.158016	+	0.987437i	$0.550510\pi$
$588$	0	0
$589$	60.4264	2.48983
$590$	−8.82843	−0.363461
$591$	−4.82843	−0.198615
$592$	−12.0000	−0.493197
$593$	19.4853	0.800165	0.400082	−	0.916479i	$-0.368982\pi$
$593$	19.4853	0.800165	0.400082	+	0.916479i	$0.368982\pi$
$594$	0.414214	0.0169954
$595$	0	0
$596$	−29.9706	−1.22764
$597$	−1.17157	−0.0479493
$598$	−33.7990	−1.38214
$599$	9.87006	0.403280	0.201640	−	0.979460i	$-0.435373\pi$
$599$	9.87006	0.403280	0.201640	+	0.979460i	$0.435373\pi$
$600$	−42.6274	−1.74026
$601$	17.1716	0.700443	0.350222	−	0.936667i	$-0.386106\pi$
$601$	17.1716	0.700443	0.350222	+	0.936667i	$0.386106\pi$
$602$	0	0
$603$	16.0000	0.651570
$604$	−54.1421	−2.20301
$605$	−10.8284	−0.440238
$606$	13.6569	0.554772
$607$	7.72792	0.313667	0.156833	−	0.987625i	$-0.449871\pi$
$607$	7.72792	0.313667	0.156833	+	0.987625i	$0.449871\pi$
$608$	−9.51472	−0.385873
$609$	0	0
$610$	−11.6569	−0.471972
$611$	12.4142	0.502225
$612$	−8.97056	−0.362614
$613$	−9.00000	−0.363507	−0.181753	−	0.983344i	$-0.558177\pi$
$613$	−9.00000	−0.363507	−0.181753	+	0.983344i	$0.558177\pi$
$614$	7.00000	0.282497
$615$	−10.8284	−0.436644
$616$	0	0
$617$	0.686292	0.0276291	0.0138145	−	0.999905i	$-0.495603\pi$
$617$	0.686292	0.0276291	0.0138145	+	0.999905i	$0.495603\pi$
$618$	28.1421	1.13204
$619$	−33.5858	−1.34993	−0.674963	−	0.737851i	$-0.735841\pi$
$619$	−33.5858	−1.34993	−0.674963	+	0.737851i	$0.735841\pi$
$620$	−38.5563	−1.54846
$621$	1.51472	0.0607836
$622$	6.48528	0.260036
$623$	0	0
$624$	−27.7279	−1.11001
$625$	11.0000	0.440000
$626$	23.7279	0.948358
$627$	−6.00000	−0.239617
$628$	−32.4853	−1.29630
$629$	3.31371	0.132126
$630$	0	0
$631$	−36.8284	−1.46612	−0.733058	−	0.680166i	$-0.761908\pi$
$631$	−36.8284	−1.46612	−0.733058	+	0.680166i	$0.761908\pi$
$632$	10.6569	0.423907
$633$	46.7990	1.86009
$634$	75.9411	3.01601
$635$	−4.34315	−0.172352
$636$	−87.6690	−3.47630
$637$	0	0
$638$	1.00000	0.0395904
$639$	−24.9706	−0.987820
$640$	20.5563	0.812561
$641$	17.7990	0.703018	0.351509	−	0.936185i	$-0.385669\pi$
$641$	17.7990	0.703018	0.351509	+	0.936185i	$0.385669\pi$
$642$	−86.4264	−3.41098
$643$	−32.4853	−1.28109	−0.640547	−	0.767919i	$-0.721292\pi$
$643$	−32.4853	−1.28109	−0.640547	+	0.767919i	$0.721292\pi$
$644$	0	0
$645$	−8.65685	−0.340863
$646$	−12.0000	−0.472134
$647$	−39.6569	−1.55907	−0.779536	−	0.626358i	$-0.784545\pi$
$647$	−39.6569	−1.55907	−0.779536	+	0.626358i	$0.784545\pi$
$648$	41.8701	1.64481
$649$	−1.51472	−0.0594579
$650$	36.9706	1.45010
$651$	0	0
$652$	15.0416	0.589076
$653$	−30.1421	−1.17955	−0.589776	−	0.807567i	$-0.700784\pi$
$653$	−30.1421	−1.17955	−0.589776	+	0.807567i	$0.700784\pi$
$654$	73.7696	2.88462
$655$	−21.3137	−0.832796
$656$	13.4558	0.525362
$657$	−11.3137	−0.441390
$658$	0	0
$659$	14.4142	0.561498	0.280749	−	0.959781i	$-0.409417\pi$
$659$	14.4142	0.561498	0.280749	+	0.959781i	$0.409417\pi$
$660$	3.82843	0.149021
$661$	−33.3137	−1.29575	−0.647877	−	0.761745i	$-0.724343\pi$
$661$	−33.3137	−1.29575	−0.647877	+	0.761745i	$0.724343\pi$
$662$	5.82843	0.226528
$663$	7.65685	0.297368
$664$	33.7990	1.31166
$665$	0	0
$666$	27.3137	1.05838
$667$	3.65685	0.141594
$668$	12.1421	0.469793
$669$	−7.65685	−0.296031
$670$	−13.6569	−0.527610
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−21.6274	−0.833676	−0.416838	−	0.908981i	$-0.636862\pi$
$673$	−21.6274	−0.833676	−0.416838	+	0.908981i	$0.636862\pi$
$674$	−52.6274	−2.02713
$675$	−1.65685	−0.0637723
$676$	6.34315	0.243967
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	−77.5980	−2.98013
$679$	0	0
$680$	3.65685	0.140234
$681$	−19.6569	−0.753252
$682$	−10.0711	−0.385641
$683$	20.9706	0.802416	0.401208	−	0.915987i	$-0.368590\pi$
$683$	20.9706	0.802416	0.401208	+	0.915987i	$0.368590\pi$
$684$	−64.9706	−2.48421
$685$	12.0000	0.458496
$686$	0	0
$687$	−8.48528	−0.323734
$688$	10.7574	0.410120
$689$	36.3137	1.38344
$690$	21.3137	0.811399
$691$	−48.0000	−1.82601	−0.913003	−	0.407953i	$-0.866243\pi$
$691$	−48.0000	−1.82601	−0.913003	+	0.407953i	$0.866243\pi$
$692$	−47.2548	−1.79636
$693$	0	0
$694$	−6.00000	−0.227757
$695$	−14.0000	−0.531050
$696$	10.6569	0.403947
$697$	−3.71573	−0.140743
$698$	−12.4142	−0.469885
$699$	−44.2132	−1.67230
$700$	0	0
$701$	−40.1127	−1.51504	−0.757518	−	0.652814i	$-0.773588\pi$
$701$	−40.1127	−1.51504	−0.757518	+	0.652814i	$0.773588\pi$
$702$	−3.82843	−0.144495
$703$	24.0000	0.905177
$704$	4.07107	0.153434
$705$	−7.82843	−0.294836
$706$	65.1127	2.45055
$707$	0	0
$708$	−33.7990	−1.27024
$709$	29.1421	1.09446	0.547228	−	0.836984i	$-0.315683\pi$
$709$	29.1421	1.09446	0.547228	+	0.836984i	$0.315683\pi$
$710$	21.3137	0.799889
$711$	−6.82843	−0.256086
$712$	−55.1127	−2.06544
$713$	−36.8284	−1.37924
$714$	0	0
$715$	−1.58579	−0.0593051
$716$	−24.8284	−0.927882
$717$	47.4558	1.77227
$718$	−9.48528	−0.353988
$719$	20.1421	0.751175	0.375587	−	0.926787i	$-0.377441\pi$
$719$	20.1421	0.751175	0.375587	+	0.926787i	$0.377441\pi$
$720$	8.48528	0.316228
$721$	0	0
$722$	−41.0416	−1.52741
$723$	−44.2132	−1.64431
$724$	−31.8284	−1.18289
$725$	−4.00000	−0.148556
$726$	−63.1127	−2.34233
$727$	−1.31371	−0.0487228	−0.0243614	−	0.999703i	$-0.507755\pi$
$727$	−1.31371	−0.0487228	−0.0243614	+	0.999703i	$0.507755\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	9.65685	0.357416
$731$	−2.97056	−0.109870
$732$	−44.6274	−1.64948
$733$	41.2548	1.52378	0.761891	−	0.647705i	$-0.224271\pi$
$733$	41.2548	1.52378	0.761891	+	0.647705i	$0.224271\pi$
$734$	43.4558	1.60398
$735$	0	0
$736$	5.79899	0.213754
$737$	−2.34315	−0.0863109
$738$	−30.6274	−1.12741
$739$	4.07107	0.149757	0.0748783	−	0.997193i	$-0.476143\pi$
$739$	4.07107	0.149757	0.0748783	+	0.997193i	$0.476143\pi$
$740$	−15.3137	−0.562943
$741$	55.4558	2.03722
$742$	0	0
$743$	23.6569	0.867886	0.433943	−	0.900940i	$-0.357122\pi$
$743$	23.6569	0.867886	0.433943	+	0.900940i	$0.357122\pi$
$744$	−107.326	−3.93476
$745$	−7.82843	−0.286811
$746$	63.5269	2.32589
$747$	−21.6569	−0.792383
$748$	1.31371	0.0480339
$749$	0	0
$750$	−52.4558	−1.91542
$751$	25.3137	0.923710	0.461855	−	0.886955i	$-0.347184\pi$
$751$	25.3137	0.923710	0.461855	+	0.886955i	$0.347184\pi$
$752$	9.72792	0.354741
$753$	48.4558	1.76583
$754$	−9.24264	−0.336597
$755$	−14.1421	−0.514685
$756$	0	0
$757$	25.5147	0.927348	0.463674	−	0.886006i	$-0.346531\pi$
$757$	25.5147	0.927348	0.463674	+	0.886006i	$0.346531\pi$
$758$	16.8284	0.611236
$759$	3.65685	0.132735
$760$	26.4853	0.960722
$761$	−45.5980	−1.65293	−0.826463	−	0.562991i	$-0.809650\pi$
$761$	−45.5980	−1.65293	−0.826463	+	0.562991i	$0.809650\pi$
$762$	−25.3137	−0.917019
$763$	0	0
$764$	96.9117	3.50614
$765$	−2.34315	−0.0847166
$766$	−8.48528	−0.306586
$767$	14.0000	0.505511
$768$	72.3553	2.61090
$769$	49.1127	1.77105	0.885525	−	0.464592i	$-0.153799\pi$
$769$	49.1127	1.77105	0.885525	+	0.464592i	$0.153799\pi$
$770$	0	0
$771$	−43.8701	−1.57994
$772$	−19.7990	−0.712581
$773$	19.5147	0.701896	0.350948	−	0.936395i	$-0.385859\pi$
$773$	19.5147	0.701896	0.350948	+	0.936395i	$0.385859\pi$
$774$	−24.4853	−0.880105
$775$	40.2843	1.44705
$776$	19.7990	0.710742
$777$	0	0
$778$	−7.31371	−0.262209
$779$	−26.9117	−0.964211
$780$	−35.3848	−1.26698
$781$	3.65685	0.130853
$782$	7.31371	0.261538
$783$	0.414214	0.0148028
$784$	0	0
$785$	−8.48528	−0.302853
$786$	−124.225	−4.43097
$787$	54.0833	1.92786	0.963930	−	0.266156i	$-0.0857536\pi$
$787$	54.0833	1.92786	0.963930	+	0.266156i	$0.0857536\pi$
$788$	7.65685	0.272764
$789$	−6.65685	−0.236990
$790$	5.82843	0.207366
$791$	0	0
$792$	5.17157	0.183764
$793$	18.4853	0.656432
$794$	46.6985	1.65727
$795$	−22.8995	−0.812161
$796$	1.85786	0.0658503
$797$	−51.7401	−1.83273	−0.916364	−	0.400345i	$-0.868890\pi$
$797$	−51.7401	−1.83273	−0.916364	+	0.400345i	$0.868890\pi$
$798$	0	0
$799$	−2.68629	−0.0950342
$800$	−6.34315	−0.224264
$801$	35.3137	1.24775
$802$	45.0416	1.59048
$803$	1.65685	0.0584691
$804$	−52.2843	−1.84392
$805$	0	0
$806$	93.0833	3.27872
$807$	75.9411	2.67325
$808$	−10.3431	−0.363871
$809$	36.2843	1.27569	0.637844	−	0.770166i	$-0.279827\pi$
$809$	36.2843	1.27569	0.637844	+	0.770166i	$0.279827\pi$
$810$	22.8995	0.804606
$811$	−10.8284	−0.380238	−0.190119	−	0.981761i	$-0.560887\pi$
$811$	−10.8284	−0.380238	−0.190119	+	0.981761i	$0.560887\pi$
$812$	0	0
$813$	39.9706	1.40183
$814$	−4.00000	−0.140200
$815$	3.92893	0.137624
$816$	6.00000	0.210042
$817$	−21.5147	−0.752705
$818$	−45.7990	−1.60132
$819$	0	0
$820$	17.1716	0.599657
$821$	−1.48528	−0.0518367	−0.0259183	−	0.999664i	$-0.508251\pi$
$821$	−1.48528	−0.0518367	−0.0259183	+	0.999664i	$0.508251\pi$
$822$	69.9411	2.43948
$823$	−54.2843	−1.89223	−0.946115	−	0.323830i	$-0.895029\pi$
$823$	−54.2843	−1.89223	−0.946115	+	0.323830i	$0.895029\pi$
$824$	−21.3137	−0.742498
$825$	−4.00000	−0.139262
$826$	0	0
$827$	32.8995	1.14403	0.572014	−	0.820244i	$-0.306162\pi$
$827$	32.8995	1.14403	0.572014	+	0.820244i	$0.306162\pi$
$828$	39.5980	1.37612
$829$	29.7990	1.03496	0.517481	−	0.855695i	$-0.326870\pi$
$829$	29.7990	1.03496	0.517481	+	0.855695i	$0.326870\pi$
$830$	18.4853	0.641633
$831$	41.7990	1.44999
$832$	−37.6274	−1.30450
$833$	0	0
$834$	−81.5980	−2.82551
$835$	3.17157	0.109757
$836$	9.51472	0.329073
$837$	−4.17157	−0.144191
$838$	−22.9706	−0.793505
$839$	7.92893	0.273737	0.136869	−	0.990589i	$-0.456296\pi$
$839$	7.92893	0.273737	0.136869	+	0.990589i	$0.456296\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−89.5980	−3.08775
$843$	−77.1838	−2.65835
$844$	−74.2132	−2.55452
$845$	1.65685	0.0569975
$846$	−22.1421	−0.761262
$847$	0	0
$848$	28.4558	0.977178
$849$	28.1421	0.965836
$850$	−8.00000	−0.274398
$851$	−14.6274	−0.501421
$852$	81.5980	2.79550
$853$	22.9706	0.786497	0.393249	−	0.919432i	$-0.371351\pi$
$853$	22.9706	0.786497	0.393249	+	0.919432i	$0.371351\pi$
$854$	0	0
$855$	−16.9706	−0.580381
$856$	65.4558	2.23723
$857$	6.17157	0.210817	0.105408	−	0.994429i	$-0.466385\pi$
$857$	6.17157	0.210817	0.105408	+	0.994429i	$0.466385\pi$
$858$	−9.24264	−0.315539
$859$	−19.7279	−0.673108	−0.336554	−	0.941664i	$-0.609261\pi$
$859$	−19.7279	−0.673108	−0.336554	+	0.941664i	$0.609261\pi$
$860$	13.7279	0.468118
$861$	0	0
$862$	−47.4558	−1.61635
$863$	17.1127	0.582523	0.291262	−	0.956643i	$-0.405925\pi$
$863$	17.1127	0.582523	0.291262	+	0.956643i	$0.405925\pi$
$864$	0.656854	0.0223466
$865$	−12.3431	−0.419680
$866$	73.9411	2.51262
$867$	39.3848	1.33758
$868$	0	0
$869$	1.00000	0.0339227
$870$	5.82843	0.197602
$871$	21.6569	0.733815
$872$	−55.8701	−1.89200
$873$	−12.6863	−0.429366
$874$	52.9706	1.79176
$875$	0	0
$876$	36.9706	1.24912
$877$	−37.1421	−1.25420	−0.627100	−	0.778938i	$-0.715758\pi$
$877$	−37.1421	−1.25420	−0.627100	+	0.778938i	$0.715758\pi$
$878$	−0.828427	−0.0279581
$879$	18.4853	0.623493
$880$	−1.24264	−0.0418894
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	38.4264	1.29315	0.646576	−	0.762850i	$-0.276200\pi$
$883$	38.4264	1.29315	0.646576	+	0.762850i	$0.276200\pi$
$884$	−12.1421	−0.408384
$885$	−8.82843	−0.296764
$886$	58.7696	1.97440
$887$	−17.1005	−0.574179	−0.287089	−	0.957904i	$-0.592688\pi$
$887$	−17.1005	−0.574179	−0.287089	+	0.957904i	$0.592688\pi$
$888$	−42.6274	−1.43048
$889$	0	0
$890$	−30.1421	−1.01037
$891$	3.92893	0.131624
$892$	12.1421	0.406549
$893$	−19.4558	−0.651065
$894$	−45.6274	−1.52601
$895$	−6.48528	−0.216779
$896$	0	0
$897$	−33.7990	−1.12852
$898$	84.4264	2.81735
$899$	−10.0711	−0.335889
$900$	−43.3137	−1.44379
$901$	−7.85786	−0.261783
$902$	4.48528	0.149344
$903$	0	0
$904$	58.7696	1.95465
$905$	−8.31371	−0.276357
$906$	−82.4264	−2.73843
$907$	22.2843	0.739937	0.369969	−	0.929044i	$-0.379368\pi$
$907$	22.2843	0.739937	0.369969	+	0.929044i	$0.379368\pi$
$908$	31.1716	1.03446
$909$	6.62742	0.219818
$910$	0	0
$911$	−15.4437	−0.511671	−0.255835	−	0.966720i	$-0.582351\pi$
$911$	−15.4437	−0.511671	−0.255835	+	0.966720i	$0.582351\pi$
$912$	43.4558	1.43897
$913$	3.17157	0.104964
$914$	−2.48528	−0.0822058
$915$	−11.6569	−0.385364
$916$	13.4558	0.444594
$917$	0	0
$918$	0.828427	0.0273422
$919$	8.14214	0.268584	0.134292	−	0.990942i	$-0.457124\pi$
$919$	8.14214	0.268584	0.134292	+	0.990942i	$0.457124\pi$
$920$	−16.1421	−0.532190
$921$	7.00000	0.230658
$922$	33.7990	1.11311
$923$	−33.7990	−1.11251
$924$	0	0
$925$	16.0000	0.526077
$926$	62.7696	2.06274
$927$	13.6569	0.448550
$928$	1.58579	0.0520560
$929$	−18.6863	−0.613077	−0.306539	−	0.951858i	$-0.599171\pi$
$929$	−18.6863	−0.613077	−0.306539	+	0.951858i	$0.599171\pi$
$930$	−58.6985	−1.92480
$931$	0	0
$932$	70.1127	2.29662
$933$	6.48528	0.212319
$934$	−92.5980	−3.02990
$935$	0.343146	0.0112221
$936$	−47.7990	−1.56236
$937$	16.6274	0.543194	0.271597	−	0.962411i	$-0.412448\pi$
$937$	16.6274	0.543194	0.271597	+	0.962411i	$0.412448\pi$
$938$	0	0
$939$	23.7279	0.774331
$940$	12.4142	0.404907
$941$	56.5980	1.84504	0.922521	−	0.385948i	$-0.126125\pi$
$941$	56.5980	1.84504	0.922521	+	0.385948i	$0.126125\pi$
$942$	−49.4558	−1.61136
$943$	16.4020	0.534123
$944$	10.9706	0.357061
$945$	0	0
$946$	3.58579	0.116584
$947$	2.61522	0.0849834	0.0424917	−	0.999097i	$-0.486470\pi$
$947$	2.61522	0.0849834	0.0424917	+	0.999097i	$0.486470\pi$
$948$	22.3137	0.724716
$949$	−15.3137	−0.497104
$950$	−57.9411	−1.87986
$951$	75.9411	2.46256
$952$	0	0
$953$	−35.6274	−1.15409	−0.577043	−	0.816714i	$-0.695793\pi$
$953$	−35.6274	−1.15409	−0.577043	+	0.816714i	$0.695793\pi$
$954$	−64.7696	−2.09699
$955$	25.3137	0.819132
$956$	−75.2548	−2.43392
$957$	1.00000	0.0323254
$958$	16.6569	0.538159
$959$	0	0
$960$	23.7279	0.765815
$961$	70.4264	2.27182
$962$	36.9706	1.19198
$963$	−41.9411	−1.35153
$964$	70.1127	2.25818
$965$	−5.17157	−0.166479
$966$	0	0
$967$	−35.2426	−1.13333	−0.566663	−	0.823949i	$-0.691766\pi$
$967$	−35.2426	−1.13333	−0.566663	+	0.823949i	$0.691766\pi$
$968$	47.7990	1.53632
$969$	−12.0000	−0.385496
$970$	10.8284	0.347680
$971$	15.6569	0.502452	0.251226	−	0.967928i	$-0.419166\pi$
$971$	15.6569	0.502452	0.251226	+	0.967928i	$0.419166\pi$
$972$	82.9117	2.65939
$973$	0	0
$974$	27.7990	0.890737
$975$	36.9706	1.18401
$976$	14.4853	0.463663
$977$	36.1716	1.15723	0.578616	−	0.815600i	$-0.303593\pi$
$977$	36.1716	1.15723	0.578616	+	0.815600i	$0.303593\pi$
$978$	22.8995	0.732245
$979$	−5.17157	−0.165284
$980$	0	0
$981$	35.7990	1.14297
$982$	51.2843	1.63655
$983$	21.8701	0.697547	0.348773	−	0.937207i	$-0.386598\pi$
$983$	21.8701	0.697547	0.348773	+	0.937207i	$0.386598\pi$
$984$	47.7990	1.52378
$985$	2.00000	0.0637253
$986$	2.00000	0.0636930
$987$	0	0
$988$	−87.9411	−2.79778
$989$	13.1127	0.416960
$990$	2.82843	0.0898933
$991$	−12.8284	−0.407508	−0.203754	−	0.979022i	$-0.565314\pi$
$991$	−12.8284	−0.407508	−0.203754	+	0.979022i	$0.565314\pi$
$992$	−15.9706	−0.507066
$993$	5.82843	0.184960
$994$	0	0
$995$	0.485281	0.0153845
$996$	70.7696	2.24242
$997$	−28.2843	−0.895772	−0.447886	−	0.894091i	$-0.647823\pi$
$997$	−28.2843	−0.895772	−0.447886	+	0.894091i	$0.647823\pi$
$998$	−45.7990	−1.44974
$999$	−1.65685	−0.0524205

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1421.2.a.j.1.1		2
7.6	odd	2		29.2.a.a.1.1	✓	2
21.20	even	2		261.2.a.d.1.2		2
28.27	even	2		464.2.a.h.1.1		2
35.13	even	4		725.2.b.b.349.4		4
35.27	even	4		725.2.b.b.349.1		4
35.34	odd	2		725.2.a.b.1.2		2
56.13	odd	2		1856.2.a.r.1.1		2
56.27	even	2		1856.2.a.w.1.2		2
77.76	even	2		3509.2.a.j.1.2		2
84.83	odd	2		4176.2.a.bq.1.2		2
91.90	odd	2		4901.2.a.g.1.2		2
105.104	even	2		6525.2.a.o.1.1		2
119.118	odd	2		8381.2.a.e.1.1		2
203.6	odd	14		841.2.d.f.645.1		12
203.13	odd	14		841.2.d.f.778.2		12
203.20	odd	14		841.2.d.j.574.1		12
203.27	even	28		841.2.e.k.236.4		24
203.34	odd	14		841.2.d.f.605.1		12
203.41	even	4		841.2.b.a.840.1		4
203.48	even	28		841.2.e.k.651.4		24
203.55	even	28		841.2.e.k.270.4		24
203.62	odd	14		841.2.d.f.190.1		12
203.69	even	28		841.2.e.k.63.1		24
203.76	even	28		841.2.e.k.63.4		24
203.83	odd	14		841.2.d.j.190.2		12
203.90	even	28		841.2.e.k.270.1		24
203.97	even	28		841.2.e.k.651.1		24
203.104	even	4		841.2.b.a.840.4		4
203.111	odd	14		841.2.d.j.605.2		12
203.118	even	28		841.2.e.k.236.1		24
203.125	odd	14		841.2.d.f.574.2		12
203.132	odd	14		841.2.d.j.778.1		12
203.139	odd	14		841.2.d.j.645.2		12
203.153	even	28		841.2.e.k.267.1		24
203.160	even	28		841.2.e.k.196.1		24
203.167	odd	14		841.2.d.f.571.1		12
203.181	odd	14		841.2.d.j.571.2		12
203.188	even	28		841.2.e.k.196.4		24
203.195	even	28		841.2.e.k.267.4		24
203.202	odd	2		841.2.a.d.1.2		2
609.608	even	2		7569.2.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.a.a.1.1	✓	2	7.6	odd	2
261.2.a.d.1.2		2	21.20	even	2
464.2.a.h.1.1		2	28.27	even	2
725.2.a.b.1.2		2	35.34	odd	2
725.2.b.b.349.1		4	35.27	even	4
725.2.b.b.349.4		4	35.13	even	4
841.2.a.d.1.2		2	203.202	odd	2
841.2.b.a.840.1		4	203.41	even	4
841.2.b.a.840.4		4	203.104	even	4
841.2.d.f.190.1		12	203.62	odd	14
841.2.d.f.571.1		12	203.167	odd	14
841.2.d.f.574.2		12	203.125	odd	14
841.2.d.f.605.1		12	203.34	odd	14
841.2.d.f.645.1		12	203.6	odd	14
841.2.d.f.778.2		12	203.13	odd	14
841.2.d.j.190.2		12	203.83	odd	14
841.2.d.j.571.2		12	203.181	odd	14
841.2.d.j.574.1		12	203.20	odd	14
841.2.d.j.605.2		12	203.111	odd	14
841.2.d.j.645.2		12	203.139	odd	14
841.2.d.j.778.1		12	203.132	odd	14
841.2.e.k.63.1		24	203.69	even	28
841.2.e.k.63.4		24	203.76	even	28
841.2.e.k.196.1		24	203.160	even	28
841.2.e.k.196.4		24	203.188	even	28
841.2.e.k.236.1		24	203.118	even	28
841.2.e.k.236.4		24	203.27	even	28
841.2.e.k.267.1		24	203.153	even	28
841.2.e.k.267.4		24	203.195	even	28
841.2.e.k.270.1		24	203.90	even	28
841.2.e.k.270.4		24	203.55	even	28
841.2.e.k.651.1		24	203.97	even	28
841.2.e.k.651.4		24	203.48	even	28
1421.2.a.j.1.1		2	1.1	even	1	trivial
1856.2.a.r.1.1		2	56.13	odd	2
1856.2.a.w.1.2		2	56.27	even	2
3509.2.a.j.1.2		2	77.76	even	2
4176.2.a.bq.1.2		2	84.83	odd	2
4901.2.a.g.1.2		2	91.90	odd	2
6525.2.a.o.1.1		2	105.104	even	2
7569.2.a.c.1.1		2	609.608	even	2
8381.2.a.e.1.1		2	119.118	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1421.2.a.j.1.1

Level	$1421$
Weight	$2$
Character	1421.1
Self dual	yes
Analytic conductor	$11.347$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$