LMFDB - Embedded newform 1682.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,2,Mod(1,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, 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("1682.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1682 = 2 \cdot 29^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1682.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:	$13.4308376200$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 2x + 1$ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	1682.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.554958 q^{3} +1.00000 q^{4} +0.198062 q^{5} +0.554958 q^{6} +0.109916 q^{7} -1.00000 q^{8} -2.69202 q^{9} -0.198062 q^{10} -1.33513 q^{11} -0.554958 q^{12} +3.93900 q^{13} -0.109916 q^{14} -0.109916 q^{15} +1.00000 q^{16} -2.91185 q^{17} +2.69202 q^{18} +1.29590 q^{19} +0.198062 q^{20} -0.0609989 q^{21} +1.33513 q^{22} +7.78986 q^{23} +0.554958 q^{24} -4.96077 q^{25} -3.93900 q^{26} +3.15883 q^{27} +0.109916 q^{28} +0.109916 q^{30} -9.34481 q^{31} -1.00000 q^{32} +0.740939 q^{33} +2.91185 q^{34} +0.0217703 q^{35} -2.69202 q^{36} -3.02715 q^{37} -1.29590 q^{38} -2.18598 q^{39} -0.198062 q^{40} +3.76271 q^{41} +0.0609989 q^{42} -6.66487 q^{43} -1.33513 q^{44} -0.533188 q^{45} -7.78986 q^{46} +0.801938 q^{47} -0.554958 q^{48} -6.98792 q^{49} +4.96077 q^{50} +1.61596 q^{51} +3.93900 q^{52} +8.33513 q^{53} -3.15883 q^{54} -0.264438 q^{55} -0.109916 q^{56} -0.719169 q^{57} -5.08815 q^{59} -0.109916 q^{60} -11.0586 q^{61} +9.34481 q^{62} -0.295897 q^{63} +1.00000 q^{64} +0.780167 q^{65} -0.740939 q^{66} -11.0000 q^{67} -2.91185 q^{68} -4.32304 q^{69} -0.0217703 q^{70} +10.9487 q^{71} +2.69202 q^{72} +7.94869 q^{73} +3.02715 q^{74} +2.75302 q^{75} +1.29590 q^{76} -0.146752 q^{77} +2.18598 q^{78} +4.89008 q^{79} +0.198062 q^{80} +6.32304 q^{81} -3.76271 q^{82} -11.6746 q^{83} -0.0609989 q^{84} -0.576728 q^{85} +6.66487 q^{86} +1.33513 q^{88} -11.4940 q^{89} +0.533188 q^{90} +0.432960 q^{91} +7.78986 q^{92} +5.18598 q^{93} -0.801938 q^{94} +0.256668 q^{95} +0.554958 q^{96} -10.5918 q^{97} +6.98792 q^{98} +3.59419 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 5 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} - 3 q^{9} - 5 q^{10} - 3 q^{11} - 2 q^{12} + 2 q^{13} - q^{14} - q^{15} + 3 q^{16} - 5 q^{17} + 3 q^{18} - 10 q^{19} + 5 q^{20}+ \cdots + 24 q^{99}+O(q^{100})$ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 5 * q^5 + 2 * q^6 + q^7 - 3 * q^8 - 3 * q^9 - 5 * q^10 - 3 * q^11 - 2 * q^12 + 2 * q^13 - q^14 - q^15 + 3 * q^16 - 5 * q^17 + 3 * q^18 - 10 * q^19 + 5 * q^20 - 10 * q^21 + 3 * q^22 + 2 * q^24 - 2 * q^25 - 2 * q^26 + q^27 + q^28 + q^30 - 5 * q^31 - 3 * q^32 - 12 * q^33 + 5 * q^34 - 3 * q^35 - 3 * q^36 - 3 * q^37 + 10 * q^38 + 8 * q^39 - 5 * q^40 - 6 * q^41 + 10 * q^42 - 21 * q^43 - 3 * q^44 - 5 * q^45 - 2 * q^47 - 2 * q^48 - 2 * q^49 + 2 * q^50 + 15 * q^51 + 2 * q^52 + 24 * q^53 - q^54 - 12 * q^55 - q^56 + 9 * q^57 - 19 * q^59 - q^60 - 2 * q^61 + 5 * q^62 + 13 * q^63 + 3 * q^64 + q^65 + 12 * q^66 - 33 * q^67 - 5 * q^68 + 7 * q^69 + 3 * q^70 + q^71 + 3 * q^72 - 8 * q^73 + 3 * q^74 + 13 * q^75 - 10 * q^76 + 27 * q^77 - 8 * q^78 + 14 * q^79 + 5 * q^80 - q^81 + 6 * q^82 - 14 * q^83 - 10 * q^84 + q^85 + 21 * q^86 + 3 * q^88 - 25 * q^89 + 5 * q^90 - 18 * q^91 + q^93 + 2 * q^94 - 26 * q^95 + 2 * q^96 - 4 * q^97 + 2 * q^98 + 24 * 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
$3$	−0.554958	−0.320405	−0.160203	−	0.987084i	$-0.551215\pi$
$3$	−0.554958	−0.320405	−0.160203	+	0.987084i	$0.551215\pi$
$4$	1.00000	0.500000
$5$	0.198062	0.0885761	0.0442881	−	0.999019i	$-0.485898\pi$
$5$	0.198062	0.0885761	0.0442881	+	0.999019i	$0.485898\pi$
$6$	0.554958	0.226561
$7$	0.109916	0.0415444	0.0207722	−	0.999784i	$-0.493388\pi$
$7$	0.109916	0.0415444	0.0207722	+	0.999784i	$0.493388\pi$
$8$	−1.00000	−0.353553
$9$	−2.69202	−0.897340
$10$	−0.198062	−0.0626328
$11$	−1.33513	−0.402556	−0.201278	−	0.979534i	$-0.564509\pi$
$11$	−1.33513	−0.402556	−0.201278	+	0.979534i	$0.564509\pi$
$12$	−0.554958	−0.160203
$13$	3.93900	1.09248	0.546241	−	0.837628i	$-0.316058\pi$
$13$	3.93900	1.09248	0.546241	+	0.837628i	$0.316058\pi$
$14$	−0.109916	−0.0293764
$15$	−0.109916	−0.0283803
$16$	1.00000	0.250000
$17$	−2.91185	−0.706228	−0.353114	−	0.935580i	$-0.614877\pi$
$17$	−2.91185	−0.706228	−0.353114	+	0.935580i	$0.614877\pi$
$18$	2.69202	0.634516
$19$	1.29590	0.297299	0.148650	−	0.988890i	$-0.452507\pi$
$19$	1.29590	0.297299	0.148650	+	0.988890i	$0.452507\pi$
$20$	0.198062	0.0442881
$21$	−0.0609989	−0.0133111
$22$	1.33513	0.284650
$23$	7.78986	1.62430	0.812149	−	0.583451i	$-0.198298\pi$
$23$	7.78986	1.62430	0.812149	+	0.583451i	$0.198298\pi$
$24$	0.554958	0.113280
$25$	−4.96077	−0.992154
$26$	−3.93900	−0.772502
$27$	3.15883	0.607918
$28$	0.109916	0.0207722
$29$	0	0
$29$	0	0
$30$	0.109916	0.0200679
$31$	−9.34481	−1.67838	−0.839189	−	0.543840i	$-0.816970\pi$
$31$	−9.34481	−1.67838	−0.839189	+	0.543840i	$0.816970\pi$
$32$	−1.00000	−0.176777
$33$	0.740939	0.128981
$34$	2.91185	0.499379
$35$	0.0217703	0.00367985
$36$	−2.69202	−0.448670
$37$	−3.02715	−0.497660	−0.248830	−	0.968547i	$-0.580046\pi$
$37$	−3.02715	−0.497660	−0.248830	+	0.968547i	$0.580046\pi$
$38$	−1.29590	−0.210222
$39$	−2.18598	−0.350037
$40$	−0.198062	−0.0313164
$41$	3.76271	0.587636	0.293818	−	0.955861i	$-0.405074\pi$
$41$	3.76271	0.587636	0.293818	+	0.955861i	$0.405074\pi$
$42$	0.0609989	0.00941234
$43$	−6.66487	−1.01638	−0.508192	−	0.861244i	$-0.669686\pi$
$43$	−6.66487	−1.01638	−0.508192	+	0.861244i	$0.669686\pi$
$44$	−1.33513	−0.201278
$45$	−0.533188	−0.0794830
$46$	−7.78986	−1.14855
$47$	0.801938	0.116975	0.0584873	−	0.998288i	$-0.481372\pi$
$47$	0.801938	0.116975	0.0584873	+	0.998288i	$0.481372\pi$
$48$	−0.554958	−0.0801013
$49$	−6.98792	−0.998274
$50$	4.96077	0.701559
$51$	1.61596	0.226279
$52$	3.93900	0.546241
$53$	8.33513	1.14492	0.572459	−	0.819934i	$-0.305990\pi$
$53$	8.33513	1.14492	0.572459	+	0.819934i	$0.305990\pi$
$54$	−3.15883	−0.429863
$55$	−0.264438	−0.0356568
$56$	−0.109916	−0.0146882
$57$	−0.719169	−0.0952562
$58$	0	0
$59$	−5.08815	−0.662420	−0.331210	−	0.943557i	$-0.607457\pi$
$59$	−5.08815	−0.662420	−0.331210	+	0.943557i	$0.607457\pi$
$60$	−0.109916	−0.0141901
$61$	−11.0586	−1.41591	−0.707955	−	0.706258i	$-0.750382\pi$
$61$	−11.0586	−1.41591	−0.707955	+	0.706258i	$0.750382\pi$
$62$	9.34481	1.18679
$63$	−0.295897	−0.0372795
$64$	1.00000	0.125000
$65$	0.780167	0.0967679
$66$	−0.740939	−0.0912033
$67$	−11.0000	−1.34386	−0.671932	−	0.740613i	$-0.734535\pi$
$67$	−11.0000	−1.34386	−0.671932	+	0.740613i	$0.734535\pi$
$68$	−2.91185	−0.353114
$69$	−4.32304	−0.520433
$70$	−0.0217703	−0.00260204
$71$	10.9487	1.29937	0.649685	−	0.760203i	$-0.274901\pi$
$71$	10.9487	1.29937	0.649685	+	0.760203i	$0.274901\pi$
$72$	2.69202	0.317258
$73$	7.94869	0.930324	0.465162	−	0.885226i	$-0.345996\pi$
$73$	7.94869	0.930324	0.465162	+	0.885226i	$0.345996\pi$
$74$	3.02715	0.351899
$75$	2.75302	0.317891
$76$	1.29590	0.148650
$77$	−0.146752	−0.0167239
$78$	2.18598	0.247514
$79$	4.89008	0.550177	0.275089	−	0.961419i	$-0.411293\pi$
$79$	4.89008	0.550177	0.275089	+	0.961419i	$0.411293\pi$
$80$	0.198062	0.0221440
$81$	6.32304	0.702560
$82$	−3.76271	−0.415522
$83$	−11.6746	−1.28145	−0.640725	−	0.767771i	$-0.721366\pi$
$83$	−11.6746	−1.28145	−0.640725	+	0.767771i	$0.721366\pi$
$84$	−0.0609989	−0.00665553
$85$	−0.576728	−0.0625550
$86$	6.66487	0.718692
$87$	0	0
$88$	1.33513	0.142325
$89$	−11.4940	−1.21836	−0.609179	−	0.793033i	$-0.708501\pi$
$89$	−11.4940	−1.21836	−0.609179	+	0.793033i	$0.708501\pi$
$90$	0.533188	0.0562029
$91$	0.432960	0.0453866
$92$	7.78986	0.812149
$93$	5.18598	0.537761
$94$	−0.801938	−0.0827136
$95$	0.256668	0.0263336
$96$	0.554958	0.0566402
$97$	−10.5918	−1.07543	−0.537717	−	0.843125i	$-0.680713\pi$
$97$	−10.5918	−1.07543	−0.537717	+	0.843125i	$0.680713\pi$
$98$	6.98792	0.705886
$99$	3.59419	0.361229
$100$	−4.96077	−0.496077
$101$	5.97285	0.594321	0.297161	−	0.954828i	$-0.403960\pi$
$101$	5.97285	0.594321	0.297161	+	0.954828i	$0.403960\pi$
$102$	−1.61596	−0.160004
$103$	−14.3937	−1.41826	−0.709128	−	0.705080i	$-0.750911\pi$
$103$	−14.3937	−1.41826	−0.709128	+	0.705080i	$0.750911\pi$
$104$	−3.93900	−0.386251
$105$	−0.0120816	−0.00117904
$106$	−8.33513	−0.809579
$107$	−16.4916	−1.59430	−0.797150	−	0.603781i	$-0.793660\pi$
$107$	−16.4916	−1.59430	−0.797150	+	0.603781i	$0.793660\pi$
$108$	3.15883	0.303959
$109$	2.84117	0.272134	0.136067	−	0.990700i	$-0.456554\pi$
$109$	2.84117	0.272134	0.136067	+	0.990700i	$0.456554\pi$
$110$	0.264438	0.0252132
$111$	1.67994	0.159453
$112$	0.109916	0.0103861
$113$	15.6746	1.47454	0.737269	−	0.675599i	$-0.236115\pi$
$113$	15.6746	1.47454	0.737269	+	0.675599i	$0.236115\pi$
$114$	0.719169	0.0673563
$115$	1.54288	0.143874
$116$	0	0
$117$	−10.6039	−0.980329
$118$	5.08815	0.468402
$119$	−0.320060	−0.0293399
$120$	0.109916	0.0100339
$121$	−9.21744	−0.837949
$122$	11.0586	1.00120
$123$	−2.08815	−0.188282
$124$	−9.34481	−0.839189
$125$	−1.97285	−0.176457
$126$	0.295897	0.0263606
$127$	−2.70841	−0.240333	−0.120166	−	0.992754i	$-0.538343\pi$
$127$	−2.70841	−0.240333	−0.120166	+	0.992754i	$0.538343\pi$
$128$	−1.00000	−0.0883883
$129$	3.69873	0.325655
$130$	−0.780167	−0.0684252
$131$	−14.6136	−1.27679	−0.638397	−	0.769708i	$-0.720402\pi$
$131$	−14.6136	−1.27679	−0.638397	+	0.769708i	$0.720402\pi$
$132$	0.740939	0.0644904
$133$	0.142440	0.0123511
$134$	11.0000	0.950255
$135$	0.625646	0.0538470
$136$	2.91185	0.249689
$137$	−1.59419	−0.136201	−0.0681003	−	0.997678i	$-0.521694\pi$
$137$	−1.59419	−0.136201	−0.0681003	+	0.997678i	$0.521694\pi$
$138$	4.32304	0.368002
$139$	1.29052	0.109460	0.0547302	−	0.998501i	$-0.482570\pi$
$139$	1.29052	0.109460	0.0547302	+	0.998501i	$0.482570\pi$
$140$	0.0217703	0.00183992
$141$	−0.445042	−0.0374793
$142$	−10.9487	−0.918794
$143$	−5.25906	−0.439785
$144$	−2.69202	−0.224335
$145$	0	0
$146$	−7.94869	−0.657838
$147$	3.87800	0.319852
$148$	−3.02715	−0.248830
$149$	−6.69740	−0.548672	−0.274336	−	0.961634i	$-0.588458\pi$
$149$	−6.69740	−0.548672	−0.274336	+	0.961634i	$0.588458\pi$
$150$	−2.75302	−0.224783
$151$	−9.49396	−0.772607	−0.386304	−	0.922372i	$-0.626248\pi$
$151$	−9.49396	−0.772607	−0.386304	+	0.922372i	$0.626248\pi$
$152$	−1.29590	−0.105111
$153$	7.83877	0.633727
$154$	0.146752	0.0118256
$155$	−1.85086	−0.148664
$156$	−2.18598	−0.175019
$157$	9.39373	0.749701	0.374851	−	0.927085i	$-0.377694\pi$
$157$	9.39373	0.749701	0.374851	+	0.927085i	$0.377694\pi$
$158$	−4.89008	−0.389034
$159$	−4.62565	−0.366838
$160$	−0.198062	−0.0156582
$161$	0.856232	0.0674805
$162$	−6.32304	−0.496785
$163$	−16.2687	−1.27427	−0.637133	−	0.770754i	$-0.719880\pi$
$163$	−16.2687	−1.27427	−0.637133	+	0.770754i	$0.719880\pi$
$164$	3.76271	0.293818
$165$	0.146752	0.0114246
$166$	11.6746	0.906122
$167$	−13.7627	−1.06499	−0.532495	−	0.846433i	$-0.678746\pi$
$167$	−13.7627	−1.06499	−0.532495	+	0.846433i	$0.678746\pi$
$168$	0.0609989	0.00470617
$169$	2.51573	0.193518
$170$	0.576728	0.0442330
$171$	−3.48858	−0.266779
$172$	−6.66487	−0.508192
$173$	−0.119605	−0.00909340	−0.00454670	−	0.999990i	$-0.501447\pi$
$173$	−0.119605	−0.00909340	−0.00454670	+	0.999990i	$0.501447\pi$
$174$	0	0
$175$	−0.545269	−0.0412185
$176$	−1.33513	−0.100639
$177$	2.82371	0.212243
$178$	11.4940	0.861509
$179$	−12.6799	−0.947743	−0.473872	−	0.880594i	$-0.657144\pi$
$179$	−12.6799	−0.947743	−0.473872	+	0.880594i	$0.657144\pi$
$180$	−0.533188	−0.0397415
$181$	7.13467	0.530316	0.265158	−	0.964205i	$-0.414576\pi$
$181$	7.13467	0.530316	0.265158	+	0.964205i	$0.414576\pi$
$182$	−0.432960	−0.0320932
$183$	6.13706	0.453665
$184$	−7.78986	−0.574276
$185$	−0.599564	−0.0440808
$186$	−5.18598	−0.380255
$187$	3.88769	0.284296
$188$	0.801938	0.0584873
$189$	0.347207	0.0252556
$190$	−0.256668	−0.0186207
$191$	3.19806	0.231404	0.115702	−	0.993284i	$-0.463088\pi$
$191$	3.19806	0.231404	0.115702	+	0.993284i	$0.463088\pi$
$192$	−0.554958	−0.0400507
$193$	−15.0707	−1.08481	−0.542406	−	0.840117i	$-0.682487\pi$
$193$	−15.0707	−1.08481	−0.542406	+	0.840117i	$0.682487\pi$
$194$	10.5918	0.760446
$195$	−0.432960	−0.0310049
$196$	−6.98792	−0.499137
$197$	−14.5187	−1.03442	−0.517208	−	0.855860i	$-0.673029\pi$
$197$	−14.5187	−1.03442	−0.517208	+	0.855860i	$0.673029\pi$
$198$	−3.59419	−0.255428
$199$	26.5284	1.88055	0.940274	−	0.340418i	$-0.110569\pi$
$199$	26.5284	1.88055	0.940274	+	0.340418i	$0.110569\pi$
$200$	4.96077	0.350780
$201$	6.10454	0.430581
$202$	−5.97285	−0.420248
$203$	0	0
$204$	1.61596	0.113140
$205$	0.745251	0.0520506
$206$	14.3937	1.00286
$207$	−20.9705	−1.45755
$208$	3.93900	0.273121
$209$	−1.73019	−0.119679
$210$	0.0120816	0.000833709 0
$211$	−16.7995	−1.15653	−0.578264	−	0.815850i	$-0.696270\pi$
$211$	−16.7995	−1.15653	−0.578264	+	0.815850i	$0.696270\pi$
$212$	8.33513	0.572459
$213$	−6.07606	−0.416325
$214$	16.4916	1.12734
$215$	−1.32006	−0.0900274
$216$	−3.15883	−0.214931
$217$	−1.02715	−0.0697273
$218$	−2.84117	−0.192428
$219$	−4.41119	−0.298081
$220$	−0.264438	−0.0178284
$221$	−11.4698	−0.771542
$222$	−1.67994	−0.112750
$223$	−1.43296	−0.0959581	−0.0479791	−	0.998848i	$-0.515278\pi$
$223$	−1.43296	−0.0959581	−0.0479791	+	0.998848i	$0.515278\pi$
$224$	−0.109916	−0.00734409
$225$	13.3545	0.890300
$226$	−15.6746	−1.04266
$227$	−13.4155	−0.890418	−0.445209	−	0.895427i	$-0.646871\pi$
$227$	−13.4155	−0.890418	−0.445209	+	0.895427i	$0.646871\pi$
$228$	−0.719169	−0.0476281
$229$	11.5090	0.760538	0.380269	−	0.924876i	$-0.375831\pi$
$229$	11.5090	0.760538	0.380269	+	0.924876i	$0.375831\pi$
$230$	−1.54288	−0.101734
$231$	0.0814412	0.00535844
$232$	0	0
$233$	−0.735562	−0.0481883	−0.0240941	−	0.999710i	$-0.507670\pi$
$233$	−0.735562	−0.0481883	−0.0240941	+	0.999710i	$0.507670\pi$
$234$	10.6039	0.693197
$235$	0.158834	0.0103612
$236$	−5.08815	−0.331210
$237$	−2.71379	−0.176280
$238$	0.320060	0.0207464
$239$	0.0881460	0.00570169	0.00285085	−	0.999996i	$-0.499093\pi$
$239$	0.0881460	0.00570169	0.00285085	+	0.999996i	$0.499093\pi$
$240$	−0.109916	−0.00709506
$241$	−6.21313	−0.400223	−0.200111	−	0.979773i	$-0.564130\pi$
$241$	−6.21313	−0.400223	−0.200111	+	0.979773i	$0.564130\pi$
$242$	9.21744	0.592519
$243$	−12.9855	−0.833022
$244$	−11.0586	−0.707955
$245$	−1.38404	−0.0884233
$246$	2.08815	0.133135
$247$	5.10454	0.324794
$248$	9.34481	0.593396
$249$	6.47889	0.410583
$250$	1.97285	0.124774
$251$	15.1914	0.958870	0.479435	−	0.877577i	$-0.340842\pi$
$251$	15.1914	0.958870	0.479435	+	0.877577i	$0.340842\pi$
$252$	−0.295897	−0.0186398
$253$	−10.4004	−0.653870
$254$	2.70841	0.169941
$255$	0.320060	0.0200429
$256$	1.00000	0.0625000
$257$	5.53079	0.345002	0.172501	−	0.985009i	$-0.444815\pi$
$257$	5.53079	0.345002	0.172501	+	0.985009i	$0.444815\pi$
$258$	−3.69873	−0.230273
$259$	−0.332733	−0.0206750
$260$	0.780167	0.0483839
$261$	0	0
$262$	14.6136	0.902829
$263$	2.18837	0.134941	0.0674704	−	0.997721i	$-0.478507\pi$
$263$	2.18837	0.134941	0.0674704	+	0.997721i	$0.478507\pi$
$264$	−0.740939	−0.0456016
$265$	1.65087	0.101412
$266$	−0.142440	−0.00873357
$267$	6.37867	0.390368
$268$	−11.0000	−0.671932
$269$	30.4577	1.85704	0.928520	−	0.371283i	$-0.121082\pi$
$269$	30.4577	1.85704	0.928520	+	0.371283i	$0.121082\pi$
$270$	−0.625646	−0.0380756
$271$	31.5459	1.91627	0.958137	−	0.286309i	$-0.0924285\pi$
$271$	31.5459	1.91627	0.958137	+	0.286309i	$0.0924285\pi$
$272$	−2.91185	−0.176557
$273$	−0.240275	−0.0145421
$274$	1.59419	0.0963083
$275$	6.62325	0.399397
$276$	−4.32304	−0.260217
$277$	14.2024	0.853338	0.426669	−	0.904408i	$-0.359687\pi$
$277$	14.2024	0.853338	0.426669	+	0.904408i	$0.359687\pi$
$278$	−1.29052	−0.0774003
$279$	25.1564	1.50608
$280$	−0.0217703	−0.00130102
$281$	20.7627	1.23860	0.619300	−	0.785155i	$-0.287417\pi$
$281$	20.7627	1.23860	0.619300	+	0.785155i	$0.287417\pi$
$282$	0.445042	0.0265019
$283$	−17.3472	−1.03118	−0.515592	−	0.856834i	$-0.672428\pi$
$283$	−17.3472	−1.03118	−0.515592	+	0.856834i	$0.672428\pi$
$284$	10.9487	0.649685
$285$	−0.142440	−0.00843743
$286$	5.25906	0.310975
$287$	0.413583	0.0244130
$288$	2.69202	0.158629
$289$	−8.52111	−0.501242
$290$	0	0
$291$	5.87800	0.344575
$292$	7.94869	0.465162
$293$	17.5278	1.02399	0.511993	−	0.858990i	$-0.328907\pi$
$293$	17.5278	1.02399	0.511993	+	0.858990i	$0.328907\pi$
$294$	−3.87800	−0.226170
$295$	−1.00777	−0.0586746
$296$	3.02715	0.175949
$297$	−4.21744	−0.244721
$298$	6.69740	0.387970
$299$	30.6843	1.77452
$300$	2.75302	0.158946
$301$	−0.732578	−0.0422251
$302$	9.49396	0.546316
$303$	−3.31468	−0.190424
$304$	1.29590	0.0743248
$305$	−2.19029	−0.125416
$306$	−7.83877	−0.448113
$307$	30.3618	1.73284	0.866420	−	0.499316i	$-0.166415\pi$
$307$	30.3618	1.73284	0.866420	+	0.499316i	$0.166415\pi$
$308$	−0.146752	−0.00836197
$309$	7.98792	0.454417
$310$	1.85086	0.105122
$311$	−23.7875	−1.34886	−0.674432	−	0.738337i	$-0.735611\pi$
$311$	−23.7875	−1.34886	−0.674432	+	0.738337i	$0.735611\pi$
$312$	2.18598	0.123757
$313$	−28.6746	−1.62078	−0.810391	−	0.585889i	$-0.800745\pi$
$313$	−28.6746	−1.62078	−0.810391	+	0.585889i	$0.800745\pi$
$314$	−9.39373	−0.530119
$315$	−0.0586060	−0.00330208
$316$	4.89008	0.275089
$317$	−7.55794	−0.424496	−0.212248	−	0.977216i	$-0.568079\pi$
$317$	−7.55794	−0.424496	−0.212248	+	0.977216i	$0.568079\pi$
$318$	4.62565	0.259393
$319$	0	0
$320$	0.198062	0.0110720
$321$	9.15213	0.510822
$322$	−0.856232	−0.0477159
$323$	−3.77346	−0.209961
$324$	6.32304	0.351280
$325$	−19.5405	−1.08391
$326$	16.2687	0.901043
$327$	−1.57673	−0.0871933
$328$	−3.76271	−0.207761
$329$	0.0881460	0.00485965
$330$	−0.146752	−0.00807843
$331$	−11.1860	−0.614837	−0.307419	−	0.951574i	$-0.599465\pi$
$331$	−11.1860	−0.614837	−0.307419	+	0.951574i	$0.599465\pi$
$332$	−11.6746	−0.640725
$333$	8.14914	0.446570
$334$	13.7627	0.753062
$335$	−2.17868	−0.119034
$336$	−0.0609989	−0.00332776
$337$	−20.7928	−1.13266	−0.566329	−	0.824179i	$-0.691637\pi$
$337$	−20.7928	−1.13266	−0.566329	+	0.824179i	$0.691637\pi$
$338$	−2.51573	−0.136838
$339$	−8.69873	−0.472450
$340$	−0.576728	−0.0312775
$341$	12.4765	0.675640
$342$	3.48858	0.188641
$343$	−1.53750	−0.0830172
$344$	6.66487	0.359346
$345$	−0.856232	−0.0460980
$346$	0.119605	0.00643000
$347$	0.0193774	0.00104023	0.000520116	−	1.00000i	$-0.499834\pi$
$347$	0.0193774	0.00104023	0.000520116		1.00000i	$0.499834\pi$
$348$	0	0
$349$	29.0315	1.55402	0.777009	−	0.629489i	$-0.216736\pi$
$349$	29.0315	1.55402	0.777009	+	0.629489i	$0.216736\pi$
$350$	0.545269	0.0291459
$351$	12.4426	0.664139
$352$	1.33513	0.0711624
$353$	20.6165	1.09731	0.548654	−	0.836049i	$-0.315140\pi$
$353$	20.6165	1.09731	0.548654	+	0.836049i	$0.315140\pi$
$354$	−2.82371	−0.150078
$355$	2.16852	0.115093
$356$	−11.4940	−0.609179
$357$	0.177620	0.00940065
$358$	12.6799	0.670156
$359$	20.8901	1.10254	0.551268	−	0.834328i	$-0.314144\pi$
$359$	20.8901	1.10254	0.551268	+	0.834328i	$0.314144\pi$
$360$	0.533188	0.0281015
$361$	−17.3207	−0.911613
$362$	−7.13467	−0.374990
$363$	5.11529	0.268483
$364$	0.432960	0.0226933
$365$	1.57434	0.0824045
$366$	−6.13706	−0.320789
$367$	3.76271	0.196412	0.0982059	−	0.995166i	$-0.468690\pi$
$367$	3.76271	0.196412	0.0982059	+	0.995166i	$0.468690\pi$
$368$	7.78986	0.406074
$369$	−10.1293	−0.527310
$370$	0.599564	0.0311698
$371$	0.916166	0.0475650
$372$	5.18598	0.268881
$373$	15.0140	0.777395	0.388698	−	0.921365i	$-0.372925\pi$
$373$	15.0140	0.777395	0.388698	+	0.921365i	$0.372925\pi$
$374$	−3.88769	−0.201028
$375$	1.09485	0.0565379
$376$	−0.801938	−0.0413568
$377$	0	0
$378$	−0.347207	−0.0178584
$379$	23.5646	1.21043	0.605217	−	0.796060i	$-0.293086\pi$
$379$	23.5646	1.21043	0.605217	+	0.796060i	$0.293086\pi$
$380$	0.256668	0.0131668
$381$	1.50306	0.0770039
$382$	−3.19806	−0.163627
$383$	−26.8213	−1.37051	−0.685253	−	0.728305i	$-0.740308\pi$
$383$	−26.8213	−1.37051	−0.685253	+	0.728305i	$0.740308\pi$
$384$	0.554958	0.0283201
$385$	−0.0290660	−0.00148134
$386$	15.0707	0.767078
$387$	17.9420	0.912042
$388$	−10.5918	−0.537717
$389$	−18.4034	−0.933090	−0.466545	−	0.884497i	$-0.654502\pi$
$389$	−18.4034	−0.933090	−0.466545	+	0.884497i	$0.654502\pi$
$390$	0.432960	0.0219238
$391$	−22.6829	−1.14712
$392$	6.98792	0.352943
$393$	8.10992	0.409091
$394$	14.5187	0.731442
$395$	0.968541	0.0487326
$396$	3.59419	0.180615
$397$	28.9071	1.45080	0.725402	−	0.688325i	$-0.241654\pi$
$397$	28.9071	1.45080	0.725402	+	0.688325i	$0.241654\pi$
$398$	−26.5284	−1.32975
$399$	−0.0790483	−0.00395737
$400$	−4.96077	−0.248039
$401$	15.3370	0.765895	0.382948	−	0.923770i	$-0.374909\pi$
$401$	15.3370	0.765895	0.382948	+	0.923770i	$0.374909\pi$
$402$	−6.10454	−0.304467
$403$	−36.8092	−1.83360
$404$	5.97285	0.297161
$405$	1.25236	0.0622301
$406$	0	0
$407$	4.04162	0.200336
$408$	−1.61596	−0.0800018
$409$	−34.7308	−1.71733	−0.858663	−	0.512540i	$-0.828705\pi$
$409$	−34.7308	−1.71733	−0.858663	+	0.512540i	$0.828705\pi$
$410$	−0.745251	−0.0368053
$411$	0.884707	0.0436394
$412$	−14.3937	−0.709128
$413$	−0.559270	−0.0275199
$414$	20.9705	1.03064
$415$	−2.31229	−0.113506
$416$	−3.93900	−0.193125
$417$	−0.716185	−0.0350717
$418$	1.73019	0.0846261
$419$	−21.9627	−1.07295	−0.536474	−	0.843917i	$-0.680244\pi$
$419$	−21.9627	−1.07295	−0.536474	+	0.843917i	$0.680244\pi$
$420$	−0.0120816	−0.000589521 0
$421$	40.4349	1.97068	0.985338	−	0.170615i	$-0.0545755\pi$
$421$	40.4349	1.97068	0.985338	+	0.170615i	$0.0545755\pi$
$422$	16.7995	0.817789
$423$	−2.15883	−0.104966
$424$	−8.33513	−0.404789
$425$	14.4450	0.700687
$426$	6.07606	0.294386
$427$	−1.21552	−0.0588232
$428$	−16.4916	−0.797150
$429$	2.91856	0.140909
$430$	1.32006	0.0636590
$431$	−22.1008	−1.06456	−0.532279	−	0.846569i	$-0.678664\pi$
$431$	−22.1008	−1.06456	−0.532279	+	0.846569i	$0.678664\pi$
$432$	3.15883	0.151979
$433$	−3.88710	−0.186802	−0.0934010	−	0.995629i	$-0.529774\pi$
$433$	−3.88710	−0.186802	−0.0934010	+	0.995629i	$0.529774\pi$
$434$	1.02715	0.0493046
$435$	0	0
$436$	2.84117	0.136067
$437$	10.0949	0.482902
$438$	4.41119	0.210775
$439$	10.1575	0.484791	0.242396	−	0.970177i	$-0.422067\pi$
$439$	10.1575	0.484791	0.242396	+	0.970177i	$0.422067\pi$
$440$	0.264438	0.0126066
$441$	18.8116	0.895792
$442$	11.4698	0.545563
$443$	19.2965	0.916804	0.458402	−	0.888745i	$-0.348422\pi$
$443$	19.2965	0.916804	0.458402	+	0.888745i	$0.348422\pi$
$444$	1.67994	0.0797264
$445$	−2.27652	−0.107917
$446$	1.43296	0.0678526
$447$	3.71678	0.175797
$448$	0.109916	0.00519306
$449$	21.6394	1.02123	0.510613	−	0.859811i	$-0.329419\pi$
$449$	21.6394	1.02123	0.510613	+	0.859811i	$0.329419\pi$
$450$	−13.3545	−0.629537
$451$	−5.02369	−0.236556
$452$	15.6746	0.737269
$453$	5.26875	0.247547
$454$	13.4155	0.629621
$455$	0.0857531	0.00402017
$456$	0.719169	0.0336782
$457$	2.53079	0.118386	0.0591928	−	0.998247i	$-0.481147\pi$
$457$	2.53079	0.118386	0.0591928	+	0.998247i	$0.481147\pi$
$458$	−11.5090	−0.537781
$459$	−9.19806	−0.429329
$460$	1.54288	0.0719370
$461$	10.7627	0.501269	0.250635	−	0.968082i	$-0.419361\pi$
$461$	10.7627	0.501269	0.250635	+	0.968082i	$0.419361\pi$
$462$	−0.0814412	−0.00378899
$463$	−6.12392	−0.284603	−0.142301	−	0.989823i	$-0.545450\pi$
$463$	−6.12392	−0.284603	−0.142301	+	0.989823i	$0.545450\pi$
$464$	0	0
$465$	1.02715	0.0476328
$466$	0.735562	0.0340743
$467$	34.9638	1.61793	0.808965	−	0.587857i	$-0.200028\pi$
$467$	34.9638	1.61793	0.808965	+	0.587857i	$0.200028\pi$
$468$	−10.6039	−0.490164
$469$	−1.20908	−0.0558301
$470$	−0.158834	−0.00732645
$471$	−5.21313	−0.240208
$472$	5.08815	0.234201
$473$	8.89844	0.409151
$474$	2.71379	0.124649
$475$	−6.42865	−0.294967
$476$	−0.320060	−0.0146699
$477$	−22.4383	−1.02738
$478$	−0.0881460	−0.00403170
$479$	−30.8732	−1.41063	−0.705317	−	0.708892i	$-0.749195\pi$
$479$	−30.8732	−1.41063	−0.705317	+	0.708892i	$0.749195\pi$
$480$	0.109916	0.00501697
$481$	−11.9239	−0.543685
$482$	6.21313	0.283000
$483$	−0.475173	−0.0216211
$484$	−9.21744	−0.418975
$485$	−2.09783	−0.0952578
$486$	12.9855	0.589035
$487$	−0.408206	−0.0184976	−0.00924879	−	0.999957i	$-0.502944\pi$
$487$	−0.408206	−0.0184976	−0.00924879	+	0.999957i	$0.502944\pi$
$488$	11.0586	0.500600
$489$	9.02848	0.408282
$490$	1.38404	0.0625247
$491$	20.1618	0.909890	0.454945	−	0.890520i	$-0.349659\pi$
$491$	20.1618	0.909890	0.454945	+	0.890520i	$0.349659\pi$
$492$	−2.08815	−0.0941409
$493$	0	0
$494$	−5.10454	−0.229664
$495$	0.711873	0.0319963
$496$	−9.34481	−0.419595
$497$	1.20344	0.0539816
$498$	−6.47889	−0.290326
$499$	−10.3787	−0.464613	−0.232306	−	0.972643i	$-0.574627\pi$
$499$	−10.3787	−0.464613	−0.232306	+	0.972643i	$0.574627\pi$
$500$	−1.97285	−0.0882287
$501$	7.63773	0.341228
$502$	−15.1914	−0.678023
$503$	22.9909	1.02511	0.512557	−	0.858653i	$-0.328698\pi$
$503$	22.9909	1.02511	0.512557	+	0.858653i	$0.328698\pi$
$504$	0.295897	0.0131803
$505$	1.18300	0.0526427
$506$	10.4004	0.462356
$507$	−1.39612	−0.0620041
$508$	−2.70841	−0.120166
$509$	25.7041	1.13931	0.569657	−	0.821882i	$-0.307076\pi$
$509$	25.7041	1.13931	0.569657	+	0.821882i	$0.307076\pi$
$510$	−0.320060	−0.0141725
$511$	0.873690	0.0386498
$512$	−1.00000	−0.0441942
$513$	4.09352	0.180733
$514$	−5.53079	−0.243953
$515$	−2.85086	−0.125624
$516$	3.69873	0.162827
$517$	−1.07069	−0.0470888
$518$	0.332733	0.0146194
$519$	0.0663757	0.00291357
$520$	−0.780167	−0.0342126
$521$	17.1535	0.751507	0.375753	−	0.926720i	$-0.377384\pi$
$521$	17.1535	0.751507	0.375753	+	0.926720i	$0.377384\pi$
$522$	0	0
$523$	22.4494	0.981642	0.490821	−	0.871261i	$-0.336697\pi$
$523$	22.4494	0.981642	0.490821	+	0.871261i	$0.336697\pi$
$524$	−14.6136	−0.638397
$525$	0.302602	0.0132066
$526$	−2.18837	−0.0954176
$527$	27.2107	1.18532
$528$	0.740939	0.0322452
$529$	37.6819	1.63834
$530$	−1.65087	−0.0717094
$531$	13.6974	0.594416
$532$	0.142440	0.00617556
$533$	14.8213	0.641982
$534$	−6.37867	−0.276032
$535$	−3.26636	−0.141217
$536$	11.0000	0.475128
$537$	7.03684	0.303662
$538$	−30.4577	−1.31313
$539$	9.32975	0.401861
$540$	0.625646	0.0269235
$541$	−12.4426	−0.534951	−0.267476	−	0.963565i	$-0.586189\pi$
$541$	−12.4426	−0.534951	−0.267476	+	0.963565i	$0.586189\pi$
$542$	−31.5459	−1.35501
$543$	−3.95944	−0.169916
$544$	2.91185	0.124845
$545$	0.562728	0.0241046
$546$	0.240275	0.0102828
$547$	−11.7638	−0.502983	−0.251491	−	0.967860i	$-0.580921\pi$
$547$	−11.7638	−0.502983	−0.251491	+	0.967860i	$0.580921\pi$
$548$	−1.59419	−0.0681003
$549$	29.7700	1.27055
$550$	−6.62325	−0.282416
$551$	0	0
$552$	4.32304	0.184001
$553$	0.537500	0.0228568
$554$	−14.2024	−0.603401
$555$	0.332733	0.0141237
$556$	1.29052	0.0547302
$557$	32.7362	1.38708	0.693538	−	0.720420i	$-0.256051\pi$
$557$	32.7362	1.38708	0.693538	+	0.720420i	$0.256051\pi$
$558$	−25.1564	−1.06496
$559$	−26.2529	−1.11038
$560$	0.0217703	0.000919962 0
$561$	−2.15751	−0.0910900
$562$	−20.7627	−0.875822
$563$	−30.1105	−1.26901	−0.634503	−	0.772920i	$-0.718795\pi$
$563$	−30.1105	−1.26901	−0.634503	+	0.772920i	$0.718795\pi$
$564$	−0.445042	−0.0187396
$565$	3.10454	0.130609
$566$	17.3472	0.729158
$567$	0.695005	0.0291875
$568$	−10.9487	−0.459397
$569$	15.5773	0.653035	0.326518	−	0.945191i	$-0.394125\pi$
$569$	15.5773	0.653035	0.326518	+	0.945191i	$0.394125\pi$
$570$	0.142440	0.00596616
$571$	6.74572	0.282300	0.141150	−	0.989988i	$-0.454920\pi$
$571$	6.74572	0.282300	0.141150	+	0.989988i	$0.454920\pi$
$572$	−5.25906	−0.219892
$573$	−1.77479	−0.0741429
$574$	−0.413583	−0.0172626
$575$	−38.6437	−1.61155
$576$	−2.69202	−0.112168
$577$	−18.6974	−0.778383	−0.389191	−	0.921157i	$-0.627246\pi$
$577$	−18.6974	−0.778383	−0.389191	+	0.921157i	$0.627246\pi$
$578$	8.52111	0.354431
$579$	8.36360	0.347579
$580$	0	0
$581$	−1.28322	−0.0532371
$582$	−5.87800	−0.243651
$583$	−11.1284	−0.460893
$584$	−7.94869	−0.328919
$585$	−2.10023	−0.0868337
$586$	−17.5278	−0.724067
$587$	−16.2500	−0.670708	−0.335354	−	0.942092i	$-0.608856\pi$
$587$	−16.2500	−0.670708	−0.335354	+	0.942092i	$0.608856\pi$
$588$	3.87800	0.159926
$589$	−12.1099	−0.498980
$590$	1.00777	0.0414892
$591$	8.05728	0.331432
$592$	−3.02715	−0.124415
$593$	31.1715	1.28006	0.640030	−	0.768350i	$-0.278922\pi$
$593$	31.1715	1.28006	0.640030	+	0.768350i	$0.278922\pi$
$594$	4.21744	0.173044
$595$	−0.0633918	−0.00259881
$596$	−6.69740	−0.274336
$597$	−14.7222	−0.602538
$598$	−30.6843	−1.25477
$599$	7.82802	0.319844	0.159922	−	0.987130i	$-0.448876\pi$
$599$	7.82802	0.319844	0.159922	+	0.987130i	$0.448876\pi$
$600$	−2.75302	−0.112392
$601$	17.5730	0.716818	0.358409	−	0.933565i	$-0.383319\pi$
$601$	17.5730	0.716818	0.358409	+	0.933565i	$0.383319\pi$
$602$	0.732578	0.0298577
$603$	29.6122	1.20590
$604$	−9.49396	−0.386304
$605$	−1.82563	−0.0742223
$606$	3.31468	0.134650
$607$	−39.3448	−1.59696	−0.798478	−	0.602023i	$-0.794361\pi$
$607$	−39.3448	−1.59696	−0.798478	+	0.602023i	$0.794361\pi$
$608$	−1.29590	−0.0525556
$609$	0	0
$610$	2.19029	0.0886824
$611$	3.15883	0.127793
$612$	7.83877	0.316864
$613$	25.1317	1.01506	0.507530	−	0.861634i	$-0.330559\pi$
$613$	25.1317	1.01506	0.507530	+	0.861634i	$0.330559\pi$
$614$	−30.3618	−1.22530
$615$	−0.413583	−0.0166773
$616$	0.146752	0.00591281
$617$	−12.6233	−0.508193	−0.254097	−	0.967179i	$-0.581778\pi$
$617$	−12.6233	−0.508193	−0.254097	+	0.967179i	$0.581778\pi$
$618$	−7.98792	−0.321321
$619$	−21.7192	−0.872967	−0.436484	−	0.899712i	$-0.643776\pi$
$619$	−21.7192	−0.872967	−0.436484	+	0.899712i	$0.643776\pi$
$620$	−1.85086	−0.0743321
$621$	24.6069	0.987439
$622$	23.7875	0.953790
$623$	−1.26337	−0.0506160
$624$	−2.18598	−0.0875093
$625$	24.4131	0.976524
$626$	28.6746	1.14607
$627$	0.960180	0.0383459
$628$	9.39373	0.374851
$629$	8.81461	0.351462
$630$	0.0586060	0.00233492
$631$	−39.3594	−1.56687	−0.783437	−	0.621472i	$-0.786535\pi$
$631$	−39.3594	−1.56687	−0.783437	+	0.621472i	$0.786535\pi$
$632$	−4.89008	−0.194517
$633$	9.32304	0.370558
$634$	7.55794	0.300164
$635$	−0.536435	−0.0212878
$636$	−4.62565	−0.183419
$637$	−27.5254	−1.09060
$638$	0	0
$639$	−29.4741	−1.16598
$640$	−0.198062	−0.00782910
$641$	−4.92825	−0.194654	−0.0973270	−	0.995252i	$-0.531029\pi$
$641$	−4.92825	−0.194654	−0.0973270	+	0.995252i	$0.531029\pi$
$642$	−9.15213	−0.361206
$643$	−7.01075	−0.276477	−0.138239	−	0.990399i	$-0.544144\pi$
$643$	−7.01075	−0.276477	−0.138239	+	0.990399i	$0.544144\pi$
$644$	0.856232	0.0337403
$645$	0.732578	0.0288452
$646$	3.77346	0.148465
$647$	−14.7289	−0.579051	−0.289526	−	0.957170i	$-0.593498\pi$
$647$	−14.7289	−0.579051	−0.289526	+	0.957170i	$0.593498\pi$
$648$	−6.32304	−0.248393
$649$	6.79331	0.266661
$650$	19.5405	0.766441
$651$	0.570024	0.0223410
$652$	−16.2687	−0.637133
$653$	−25.0175	−0.979009	−0.489504	−	0.872001i	$-0.662822\pi$
$653$	−25.0175	−0.979009	−0.489504	+	0.872001i	$0.662822\pi$
$654$	1.57673	0.0616550
$655$	−2.89440	−0.113093
$656$	3.76271	0.146909
$657$	−21.3980	−0.834817
$658$	−0.0881460	−0.00343629
$659$	48.6590	1.89549	0.947743	−	0.319034i	$-0.103358\pi$
$659$	48.6590	1.89549	0.947743	+	0.319034i	$0.103358\pi$
$660$	0.146752	0.00571231
$661$	−28.7851	−1.11961	−0.559805	−	0.828625i	$-0.689124\pi$
$661$	−28.7851	−1.11961	−0.559805	+	0.828625i	$0.689124\pi$
$662$	11.1860	0.434755
$663$	6.36526	0.247206
$664$	11.6746	0.453061
$665$	0.0282120	0.00109402
$666$	−8.14914	−0.315773
$667$	0	0
$668$	−13.7627	−0.532495
$669$	0.795233	0.0307455
$670$	2.17868	0.0841699
$671$	14.7646	0.569982
$672$	0.0609989	0.00235308
$673$	23.1341	0.891753	0.445877	−	0.895094i	$-0.352892\pi$
$673$	23.1341	0.891753	0.445877	+	0.895094i	$0.352892\pi$
$674$	20.7928	0.800910
$675$	−15.6703	−0.603148
$676$	2.51573	0.0967588
$677$	−18.0573	−0.693998	−0.346999	−	0.937866i	$-0.612799\pi$
$677$	−18.0573	−0.693998	−0.346999	+	0.937866i	$0.612799\pi$
$678$	8.69873	0.334073
$679$	−1.16421	−0.0446783
$680$	0.576728	0.0221165
$681$	7.44504	0.285295
$682$	−12.4765	−0.477750
$683$	−44.1269	−1.68847	−0.844234	−	0.535974i	$-0.819944\pi$
$683$	−44.1269	−1.68847	−0.844234	+	0.535974i	$0.819944\pi$
$684$	−3.48858	−0.133389
$685$	−0.315748	−0.0120641
$686$	1.53750	0.0587020
$687$	−6.38703	−0.243680
$688$	−6.66487	−0.254096
$689$	32.8321	1.25080
$690$	0.856232	0.0325962
$691$	23.8495	0.907279	0.453639	−	0.891185i	$-0.350125\pi$
$691$	23.8495	0.907279	0.453639	+	0.891185i	$0.350125\pi$
$692$	−0.119605	−0.00454670
$693$	0.395060	0.0150071
$694$	−0.0193774	−0.000735554 0
$695$	0.255603	0.00969559
$696$	0	0
$697$	−10.9565	−0.415005
$698$	−29.0315	−1.09886
$699$	0.408206	0.0154398
$700$	−0.545269	−0.0206092
$701$	1.12200	0.0423773	0.0211886	−	0.999775i	$-0.493255\pi$
$701$	1.12200	0.0423773	0.0211886	+	0.999775i	$0.493255\pi$
$702$	−12.4426	−0.469618
$703$	−3.92287	−0.147954
$704$	−1.33513	−0.0503194
$705$	−0.0881460	−0.00331977
$706$	−20.6165	−0.775914
$707$	0.656514	0.0246907
$708$	2.82371	0.106121
$709$	4.08383	0.153372	0.0766858	−	0.997055i	$-0.475566\pi$
$709$	4.08383	0.153372	0.0766858	+	0.997055i	$0.475566\pi$
$710$	−2.16852	−0.0813832
$711$	−13.1642	−0.493696
$712$	11.4940	0.430754
$713$	−72.7948	−2.72619
$714$	−0.177620	−0.00664726
$715$	−1.04162	−0.0389544
$716$	−12.6799	−0.473872
$717$	−0.0489173	−0.00182685
$718$	−20.8901	−0.779611
$719$	29.1237	1.08613	0.543065	−	0.839691i	$-0.317264\pi$
$719$	29.1237	1.08613	0.543065	+	0.839691i	$0.317264\pi$
$720$	−0.533188	−0.0198707
$721$	−1.58211	−0.0589207
$722$	17.3207	0.644608
$723$	3.44803	0.128233
$724$	7.13467	0.265158
$725$	0	0
$726$	−5.11529	−0.189846
$727$	−15.8586	−0.588164	−0.294082	−	0.955780i	$-0.595014\pi$
$727$	−15.8586	−0.588164	−0.294082	+	0.955780i	$0.595014\pi$
$728$	−0.432960	−0.0160466
$729$	−11.7627	−0.435656
$730$	−1.57434	−0.0582688
$731$	19.4071	0.717799
$732$	6.13706	0.226832
$733$	−21.4101	−0.790801	−0.395401	−	0.918509i	$-0.629394\pi$
$733$	−21.4101	−0.790801	−0.395401	+	0.918509i	$0.629394\pi$
$734$	−3.76271	−0.138884
$735$	0.768086	0.0283313
$736$	−7.78986	−0.287138
$737$	14.6864	0.540980
$738$	10.1293	0.372864
$739$	−33.6394	−1.23744	−0.618722	−	0.785610i	$-0.712349\pi$
$739$	−33.6394	−1.23744	−0.618722	+	0.785610i	$0.712349\pi$
$740$	−0.599564	−0.0220404
$741$	−2.83281	−0.104066
$742$	−0.916166	−0.0336335
$743$	5.43668	0.199452	0.0997262	−	0.995015i	$-0.468203\pi$
$743$	5.43668	0.199452	0.0997262	+	0.995015i	$0.468203\pi$
$744$	−5.18598	−0.190127
$745$	−1.32650	−0.0485993
$746$	−15.0140	−0.549702
$747$	31.4282	1.14990
$748$	3.88769	0.142148
$749$	−1.81269	−0.0662343
$750$	−1.09485	−0.0399783
$751$	−16.7976	−0.612954	−0.306477	−	0.951878i	$-0.599150\pi$
$751$	−16.7976	−0.612954	−0.306477	+	0.951878i	$0.599150\pi$
$752$	0.801938	0.0292437
$753$	−8.43057	−0.307227
$754$	0	0
$755$	−1.88040	−0.0684346
$756$	0.347207	0.0126278
$757$	2.51142	0.0912790	0.0456395	−	0.998958i	$-0.485467\pi$
$757$	2.51142	0.0912790	0.0456395	+	0.998958i	$0.485467\pi$
$758$	−23.5646	−0.855907
$759$	5.77181	0.209503
$760$	−0.256668	−0.00931034
$761$	−53.5387	−1.94078	−0.970388	−	0.241552i	$-0.922344\pi$
$761$	−53.5387	−1.94078	−0.970388	+	0.241552i	$0.922344\pi$
$762$	−1.50306	−0.0544500
$763$	0.312290	0.0113057
$764$	3.19806	0.115702
$765$	1.55257	0.0561331
$766$	26.8213	0.969094
$767$	−20.0422	−0.723682
$768$	−0.554958	−0.0200253
$769$	9.24219	0.333282	0.166641	−	0.986018i	$-0.446708\pi$
$769$	9.24219	0.333282	0.166641	+	0.986018i	$0.446708\pi$
$770$	0.0290660	0.00104747
$771$	−3.06936	−0.110540
$772$	−15.0707	−0.542406
$773$	−37.4373	−1.34653	−0.673263	−	0.739404i	$-0.735108\pi$
$773$	−37.4373	−1.34653	−0.673263	+	0.739404i	$0.735108\pi$
$774$	−17.9420	−0.644911
$775$	46.3575	1.66521
$776$	10.5918	0.380223
$777$	0.184653	0.00662438
$778$	18.4034	0.659795
$779$	4.87608	0.174704
$780$	−0.432960	−0.0155025
$781$	−14.6179	−0.523069
$782$	22.6829	0.811140
$783$	0	0
$784$	−6.98792	−0.249569
$785$	1.86054	0.0664057
$786$	−8.10992	−0.289271
$787$	25.9372	0.924561	0.462281	−	0.886734i	$-0.347031\pi$
$787$	25.9372	0.924561	0.462281	+	0.886734i	$0.347031\pi$
$788$	−14.5187	−0.517208
$789$	−1.21446	−0.0432358
$790$	−0.968541	−0.0344591
$791$	1.72289	0.0612589
$792$	−3.59419	−0.127714
$793$	−43.5599	−1.54686
$794$	−28.9071	−1.02587
$795$	−0.916166	−0.0324931
$796$	26.5284	0.940274
$797$	20.1914	0.715215	0.357607	−	0.933872i	$-0.383593\pi$
$797$	20.1914	0.715215	0.357607	+	0.933872i	$0.383593\pi$
$798$	0.0790483	0.00279828
$799$	−2.33513	−0.0826108
$800$	4.96077	0.175390
$801$	30.9420	1.09328
$802$	−15.3370	−0.541570
$803$	−10.6125	−0.374507
$804$	6.10454	0.215291
$805$	0.169587	0.00597716
$806$	36.8092	1.29655
$807$	−16.9028	−0.595005
$808$	−5.97285	−0.210124
$809$	34.4413	1.21089	0.605446	−	0.795886i	$-0.292995\pi$
$809$	34.4413	1.21089	0.605446	+	0.795886i	$0.292995\pi$
$810$	−1.25236	−0.0440033
$811$	32.7536	1.15013	0.575067	−	0.818106i	$-0.304976\pi$
$811$	32.7536	1.15013	0.575067	+	0.818106i	$0.304976\pi$
$812$	0	0
$813$	−17.5066	−0.613984
$814$	−4.04162	−0.141659
$815$	−3.22223	−0.112870
$816$	1.61596	0.0565698
$817$	−8.63699	−0.302170
$818$	34.7308	1.21433
$819$	−1.16554	−0.0407272
$820$	0.745251	0.0260253
$821$	4.34614	0.151681	0.0758407	−	0.997120i	$-0.475836\pi$
$821$	4.34614	0.151681	0.0758407	+	0.997120i	$0.475836\pi$
$822$	−0.884707	−0.0308577
$823$	13.4101	0.467448	0.233724	−	0.972303i	$-0.424909\pi$
$823$	13.4101	0.467448	0.233724	+	0.972303i	$0.424909\pi$
$824$	14.3937	0.501429
$825$	−3.67563	−0.127969
$826$	0.559270	0.0194595
$827$	−26.3424	−0.916016	−0.458008	−	0.888948i	$-0.651437\pi$
$827$	−26.3424	−0.916016	−0.458008	+	0.888948i	$0.651437\pi$
$828$	−20.9705	−0.728774
$829$	−10.3618	−0.359880	−0.179940	−	0.983678i	$-0.557590\pi$
$829$	−10.3618	−0.359880	−0.179940	+	0.983678i	$0.557590\pi$
$830$	2.31229	0.0802608
$831$	−7.88172	−0.273414
$832$	3.93900	0.136560
$833$	20.3478	0.705009
$834$	0.716185	0.0247994
$835$	−2.72587	−0.0943327
$836$	−1.73019	−0.0598397
$837$	−29.5187	−1.02032
$838$	21.9627	0.758689
$839$	−26.6752	−0.920929	−0.460464	−	0.887678i	$-0.652317\pi$
$839$	−26.6752	−0.920929	−0.460464	+	0.887678i	$0.652317\pi$
$840$	0.0120816	0.000416854 0
$841$	0	0
$842$	−40.4349	−1.39348
$843$	−11.5224	−0.396854
$844$	−16.7995	−0.578264
$845$	0.498271	0.0171410
$846$	2.15883	0.0742222
$847$	−1.01315	−0.0348121
$848$	8.33513	0.286229
$849$	9.62697	0.330397
$850$	−14.4450	−0.495461
$851$	−23.5810	−0.808348
$852$	−6.07606	−0.208163
$853$	−19.7071	−0.674758	−0.337379	−	0.941369i	$-0.609540\pi$
$853$	−19.7071	−0.674758	−0.337379	+	0.941369i	$0.609540\pi$
$854$	1.21552	0.0415943
$855$	−0.690957	−0.0236302
$856$	16.4916	0.563670
$857$	2.15691	0.0736788	0.0368394	−	0.999321i	$-0.488271\pi$
$857$	2.15691	0.0736788	0.0368394	+	0.999321i	$0.488271\pi$
$858$	−2.91856	−0.0996380
$859$	34.4510	1.17545	0.587727	−	0.809060i	$-0.300023\pi$
$859$	34.4510	1.17545	0.587727	+	0.809060i	$0.300023\pi$
$860$	−1.32006	−0.0450137
$861$	−0.229521	−0.00782206
$862$	22.1008	0.752757
$863$	−9.58881	−0.326407	−0.163203	−	0.986592i	$-0.552183\pi$
$863$	−9.58881	−0.326407	−0.163203	+	0.986592i	$0.552183\pi$
$864$	−3.15883	−0.107466
$865$	−0.0236892	−0.000805458 0
$866$	3.88710	0.132089
$867$	4.72886	0.160600
$868$	−1.02715	−0.0348636
$869$	−6.52888	−0.221477
$870$	0	0
$871$	−43.3290	−1.46815
$872$	−2.84117	−0.0962140
$873$	28.5133	0.965030
$874$	−10.0949	−0.341463
$875$	−0.216849	−0.00733082
$876$	−4.41119	−0.149040
$877$	−0.842231	−0.0284401	−0.0142201	−	0.999899i	$-0.504527\pi$
$877$	−0.842231	−0.0284401	−0.0142201	+	0.999899i	$0.504527\pi$
$878$	−10.1575	−0.342799
$879$	−9.72720	−0.328090
$880$	−0.264438	−0.00891420
$881$	−8.22819	−0.277215	−0.138607	−	0.990347i	$-0.544263\pi$
$881$	−8.22819	−0.277215	−0.138607	+	0.990347i	$0.544263\pi$
$882$	−18.8116	−0.633420
$883$	54.3021	1.82741	0.913706	−	0.406376i	$-0.133208\pi$
$883$	54.3021	1.82741	0.913706	+	0.406376i	$0.133208\pi$
$884$	−11.4698	−0.385771
$885$	0.559270	0.0187997
$886$	−19.2965	−0.648278
$887$	−13.5013	−0.453328	−0.226664	−	0.973973i	$-0.572782\pi$
$887$	−13.5013	−0.453328	−0.226664	+	0.973973i	$0.572782\pi$
$888$	−1.67994	−0.0563751
$889$	−0.297699	−0.00998450
$890$	2.27652	0.0763091
$891$	−8.44206	−0.282820
$892$	−1.43296	−0.0479791
$893$	1.03923	0.0347765
$894$	−3.71678	−0.124308
$895$	−2.51142	−0.0839474
$896$	−0.109916	−0.00367204
$897$	−17.0285	−0.568564
$898$	−21.6394	−0.722116
$899$	0	0
$900$	13.3545	0.445150
$901$	−24.2707	−0.808573
$902$	5.02369	0.167271
$903$	0.406550	0.0135291
$904$	−15.6746	−0.521328
$905$	1.41311	0.0469733
$906$	−5.26875	−0.175042
$907$	−46.4795	−1.54333	−0.771663	−	0.636032i	$-0.780575\pi$
$907$	−46.4795	−1.54333	−0.771663	+	0.636032i	$0.780575\pi$
$908$	−13.4155	−0.445209
$909$	−16.0790	−0.533308
$910$	−0.0857531	−0.00284269
$911$	21.1685	0.701344	0.350672	−	0.936498i	$-0.385953\pi$
$911$	21.1685	0.701344	0.350672	+	0.936498i	$0.385953\pi$
$912$	−0.719169	−0.0238141
$913$	15.5870	0.515855
$914$	−2.53079	−0.0837113
$915$	1.21552	0.0401839
$916$	11.5090	0.380269
$917$	−1.60627	−0.0530437
$918$	9.19806	0.303581
$919$	−20.1605	−0.665033	−0.332517	−	0.943097i	$-0.607898\pi$
$919$	−20.1605	−0.665033	−0.332517	+	0.943097i	$0.607898\pi$
$920$	−1.54288	−0.0508671
$921$	−16.8495	−0.555211
$922$	−10.7627	−0.354451
$923$	43.1269	1.41954
$924$	0.0814412	0.00267922
$925$	15.0170	0.493755
$926$	6.12392	0.201244
$927$	38.7482	1.27266
$928$	0	0
$929$	19.7952	0.649461	0.324730	−	0.945807i	$-0.394726\pi$
$929$	19.7952	0.649461	0.324730	+	0.945807i	$0.394726\pi$
$930$	−1.02715	−0.0336815
$931$	−9.05562	−0.296786
$932$	−0.735562	−0.0240941
$933$	13.2010	0.432183
$934$	−34.9638	−1.14405
$935$	0.770005	0.0251819
$936$	10.6039	0.346599
$937$	−56.0883	−1.83232	−0.916162	−	0.400809i	$-0.868729\pi$
$937$	−56.0883	−1.83232	−0.916162	+	0.400809i	$0.868729\pi$
$938$	1.20908	0.0394778
$939$	15.9132	0.519307
$940$	0.158834	0.00518058
$941$	−2.40688	−0.0784620	−0.0392310	−	0.999230i	$-0.512491\pi$
$941$	−2.40688	−0.0784620	−0.0392310	+	0.999230i	$0.512491\pi$
$942$	5.21313	0.169853
$943$	29.3110	0.954496
$944$	−5.08815	−0.165605
$945$	0.0687686	0.00223704
$946$	−8.89844	−0.289313
$947$	2.39745	0.0779067	0.0389534	−	0.999241i	$-0.487598\pi$
$947$	2.39745	0.0779067	0.0389534	+	0.999241i	$0.487598\pi$
$948$	−2.71379	−0.0881399
$949$	31.3099	1.01636
$950$	6.42865	0.208573
$951$	4.19434	0.136011
$952$	0.320060	0.0103732
$953$	−33.4789	−1.08449	−0.542244	−	0.840221i	$-0.682425\pi$
$953$	−33.4789	−1.08449	−0.542244	+	0.840221i	$0.682425\pi$
$954$	22.4383	0.726468
$955$	0.633415	0.0204968
$956$	0.0881460	0.00285085
$957$	0	0
$958$	30.8732	0.997468
$959$	−0.175227	−0.00565838
$960$	−0.109916	−0.00354753
$961$	56.3256	1.81695
$962$	11.9239	0.384443
$963$	44.3957	1.43063
$964$	−6.21313	−0.200111
$965$	−2.98493	−0.0960884
$966$	0.475173	0.0152884
$967$	35.1739	1.13112	0.565558	−	0.824708i	$-0.308661\pi$
$967$	35.1739	1.13112	0.565558	+	0.824708i	$0.308661\pi$
$968$	9.21744	0.296260
$969$	2.09411	0.0672726
$970$	2.09783	0.0673574
$971$	−27.5332	−0.883582	−0.441791	−	0.897118i	$-0.645657\pi$
$971$	−27.5332	−0.883582	−0.441791	+	0.897118i	$0.645657\pi$
$972$	−12.9855	−0.416511
$973$	0.141849	0.00454748
$974$	0.408206	0.0130798
$975$	10.8442	0.347291
$976$	−11.0586	−0.353977
$977$	−11.3593	−0.363416	−0.181708	−	0.983353i	$-0.558163\pi$
$977$	−11.3593	−0.363416	−0.181708	+	0.983353i	$0.558163\pi$
$978$	−9.02848	−0.288699
$979$	15.3459	0.490456
$980$	−1.38404	−0.0442116
$981$	−7.64848	−0.244197
$982$	−20.1618	−0.643389
$983$	−32.9189	−1.04995	−0.524975	−	0.851118i	$-0.675925\pi$
$983$	−32.9189	−1.04995	−0.524975	+	0.851118i	$0.675925\pi$
$984$	2.08815	0.0665677
$985$	−2.87561	−0.0916245
$986$	0	0
$987$	−0.0489173	−0.00155706
$988$	5.10454	0.162397
$989$	−51.9184	−1.65091
$990$	−0.711873	−0.0226248
$991$	22.4814	0.714145	0.357073	−	0.934077i	$-0.383775\pi$
$991$	22.4814	0.714145	0.357073	+	0.934077i	$0.383775\pi$
$992$	9.34481	0.296698
$993$	6.20775	0.196997
$994$	−1.20344	−0.0381708
$995$	5.25428	0.166572
$996$	6.47889	0.205292
$997$	−22.1244	−0.700686	−0.350343	−	0.936621i	$-0.613935\pi$
$997$	−22.1244	−0.700686	−0.350343	+	0.936621i	$0.613935\pi$
$998$	10.3787	0.328531
$999$	−9.56225	−0.302536

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	1682.2.a.m.1.2		3
29.12	odd	4		1682.2.b.g.1681.2		6
29.17	odd	4		1682.2.b.g.1681.5		6
29.23	even	7		58.2.d.a.7.1	✓	6
29.24	even	7		58.2.d.a.25.1	yes	6
29.28	even	2		1682.2.a.n.1.2		3
87.23	odd	14		522.2.k.c.181.1		6
87.53	odd	14		522.2.k.c.199.1		6
116.23	odd	14		464.2.u.b.65.1		6
116.111	odd	14		464.2.u.b.257.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.d.a.7.1	✓	6	29.23	even	7
58.2.d.a.25.1	yes	6	29.24	even	7
464.2.u.b.65.1		6	116.23	odd	14
464.2.u.b.257.1		6	116.111	odd	14
522.2.k.c.181.1		6	87.23	odd	14
522.2.k.c.199.1		6	87.53	odd	14
1682.2.a.m.1.2		3	1.1	even	1	trivial
1682.2.a.n.1.2		3	29.28	even	2
1682.2.b.g.1681.2		6	29.12	odd	4
1682.2.b.g.1681.5		6	29.17	odd	4

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

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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	1682.2.a.m.1.2

Level	$1682$
Weight	$2$
Character	1682.1
Self dual	yes
Analytic conductor	$13.431$
Analytic rank	$1$
Dimension	$3$
CM	no
Inner twists	$1$