LMFDB - Embedded newform 637.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, 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("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.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:	$5.08647060876$
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 91)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	637.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.41421 q^{3} -1.58579 q^{5} +2.00000 q^{6} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.41421 q^{3} -1.58579 q^{5} +2.00000 q^{6} +2.82843 q^{8} -1.00000 q^{9} +2.24264 q^{10} +4.24264 q^{11} +1.00000 q^{13} +2.24264 q^{15} -4.00000 q^{16} -1.41421 q^{17} +1.41421 q^{18} +7.24264 q^{19} -6.00000 q^{22} -5.82843 q^{23} -4.00000 q^{24} -2.48528 q^{25} -1.41421 q^{26} +5.65685 q^{27} +0.171573 q^{29} -3.17157 q^{30} -3.24264 q^{31} -6.00000 q^{33} +2.00000 q^{34} +2.24264 q^{37} -10.2426 q^{38} -1.41421 q^{39} -4.48528 q^{40} -8.82843 q^{41} -5.00000 q^{43} +1.58579 q^{45} +8.24264 q^{46} -1.58579 q^{47} +5.65685 q^{48} +3.51472 q^{50} +2.00000 q^{51} -0.171573 q^{53} -8.00000 q^{54} -6.72792 q^{55} -10.2426 q^{57} -0.242641 q^{58} -0.343146 q^{59} -6.00000 q^{61} +4.58579 q^{62} +8.00000 q^{64} -1.58579 q^{65} +8.48528 q^{66} -14.4853 q^{67} +8.24264 q^{69} -13.0711 q^{71} -2.82843 q^{72} +9.24264 q^{73} -3.17157 q^{74} +3.51472 q^{75} +2.00000 q^{78} +15.4853 q^{79} +6.34315 q^{80} -5.00000 q^{81} +12.4853 q^{82} -13.2426 q^{83} +2.24264 q^{85} +7.07107 q^{86} -0.242641 q^{87} +12.0000 q^{88} -1.58579 q^{89} -2.24264 q^{90} +4.58579 q^{93} +2.24264 q^{94} -11.4853 q^{95} -11.7279 q^{97} -4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 6 * q^5 + 4 * q^6 - 2 * q^9	$ 2 q - 6 q^{5} + 4 q^{6} - 2 q^{9} - 4 q^{10} + 2 q^{13} - 4 q^{15} - 8 q^{16} + 6 q^{19} - 12 q^{22} - 6 q^{23} - 8 q^{24} + 12 q^{25} + 6 q^{29} - 12 q^{30} + 2 q^{31} - 12 q^{33} + 4 q^{34} - 4 q^{37} - 12 q^{38} + 8 q^{40} - 12 q^{41} - 10 q^{43} + 6 q^{45} + 8 q^{46} - 6 q^{47} + 24 q^{50} + 4 q^{51} - 6 q^{53} - 16 q^{54} + 12 q^{55} - 12 q^{57} + 8 q^{58} - 12 q^{59} - 12 q^{61} + 12 q^{62} + 16 q^{64} - 6 q^{65} - 12 q^{67} + 8 q^{69} - 12 q^{71} + 10 q^{73} - 12 q^{74} + 24 q^{75} + 4 q^{78} + 14 q^{79} + 24 q^{80} - 10 q^{81} + 8 q^{82} - 18 q^{83} - 4 q^{85} + 8 q^{87} + 24 q^{88} - 6 q^{89} + 4 q^{90} + 12 q^{93} - 4 q^{94} - 6 q^{95} + 2 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 + 4 * q^6 - 2 * q^9 - 4 * q^10 + 2 * q^13 - 4 * q^15 - 8 * q^16 + 6 * q^19 - 12 * q^22 - 6 * q^23 - 8 * q^24 + 12 * q^25 + 6 * q^29 - 12 * q^30 + 2 * q^31 - 12 * q^33 + 4 * q^34 - 4 * q^37 - 12 * q^38 + 8 * q^40 - 12 * q^41 - 10 * q^43 + 6 * q^45 + 8 * q^46 - 6 * q^47 + 24 * q^50 + 4 * q^51 - 6 * q^53 - 16 * q^54 + 12 * q^55 - 12 * q^57 + 8 * q^58 - 12 * q^59 - 12 * q^61 + 12 * q^62 + 16 * q^64 - 6 * q^65 - 12 * q^67 + 8 * q^69 - 12 * q^71 + 10 * q^73 - 12 * q^74 + 24 * q^75 + 4 * q^78 + 14 * q^79 + 24 * q^80 - 10 * q^81 + 8 * q^82 - 18 * q^83 - 4 * q^85 + 8 * q^87 + 24 * q^88 - 6 * q^89 + 4 * q^90 + 12 * q^93 - 4 * q^94 - 6 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	−1.58579	−0.709185	−0.354593	−	0.935021i	$-0.615380\pi$
$5$	−1.58579	−0.709185	−0.354593	+	0.935021i	$0.615380\pi$
$6$	2.00000	0.816497
$7$	0	0
$7$	0	0
$8$	2.82843	1.00000
$9$	−1.00000	−0.333333
$10$	2.24264	0.709185
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.24264	0.579047
$16$	−4.00000	−1.00000
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	1.41421	0.333333
$19$	7.24264	1.66158	0.830788	−	0.556589i	$-0.187890\pi$
$19$	7.24264	1.66158	0.830788	+	0.556589i	$0.187890\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−5.82843	−1.21531	−0.607656	−	0.794201i	$-0.707890\pi$
$23$	−5.82843	−1.21531	−0.607656	+	0.794201i	$0.707890\pi$
$24$	−4.00000	−0.816497
$25$	−2.48528	−0.497056
$26$	−1.41421	−0.277350
$27$	5.65685	1.08866
$28$	0	0
$29$	0.171573	0.0318603	0.0159301	−	0.999873i	$-0.494929\pi$
$29$	0.171573	0.0318603	0.0159301	+	0.999873i	$0.494929\pi$
$30$	−3.17157	−0.579047
$31$	−3.24264	−0.582395	−0.291198	−	0.956663i	$-0.594054\pi$
$31$	−3.24264	−0.582395	−0.291198	+	0.956663i	$0.594054\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	2.24264	0.368688	0.184344	−	0.982862i	$-0.440984\pi$
$37$	2.24264	0.368688	0.184344	+	0.982862i	$0.440984\pi$
$38$	−10.2426	−1.66158
$39$	−1.41421	−0.226455
$40$	−4.48528	−0.709185
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	1.58579	0.236395
$46$	8.24264	1.21531
$47$	−1.58579	−0.231311	−0.115655	−	0.993289i	$-0.536897\pi$
$47$	−1.58579	−0.231311	−0.115655	+	0.993289i	$0.536897\pi$
$48$	5.65685	0.816497
$49$	0	0
$50$	3.51472	0.497056
$51$	2.00000	0.280056
$52$	0	0
$53$	−0.171573	−0.0235673	−0.0117837	−	0.999931i	$-0.503751\pi$
$53$	−0.171573	−0.0235673	−0.0117837	+	0.999931i	$0.503751\pi$
$54$	−8.00000	−1.08866
$55$	−6.72792	−0.907193
$56$	0	0
$57$	−10.2426	−1.35667
$58$	−0.242641	−0.0318603
$59$	−0.343146	−0.0446738	−0.0223369	−	0.999751i	$-0.507111\pi$
$59$	−0.343146	−0.0446738	−0.0223369	+	0.999751i	$0.507111\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	4.58579	0.582395
$63$	0	0
$64$	8.00000	1.00000
$65$	−1.58579	−0.196693
$66$	8.48528	1.04447
$67$	−14.4853	−1.76966	−0.884829	−	0.465915i	$-0.845725\pi$
$67$	−14.4853	−1.76966	−0.884829	+	0.465915i	$0.845725\pi$
$68$	0	0
$69$	8.24264	0.992297
$70$	0	0
$71$	−13.0711	−1.55125	−0.775625	−	0.631194i	$-0.782565\pi$
$71$	−13.0711	−1.55125	−0.775625	+	0.631194i	$0.782565\pi$
$72$	−2.82843	−0.333333
$73$	9.24264	1.08177	0.540885	−	0.841097i	$-0.318090\pi$
$73$	9.24264	1.08177	0.540885	+	0.841097i	$0.318090\pi$
$74$	−3.17157	−0.368688
$75$	3.51472	0.405845
$76$	0	0
$77$	0	0
$78$	2.00000	0.226455
$79$	15.4853	1.74223	0.871115	−	0.491079i	$-0.163397\pi$
$79$	15.4853	1.74223	0.871115	+	0.491079i	$0.163397\pi$
$80$	6.34315	0.709185
$81$	−5.00000	−0.555556
$82$	12.4853	1.37877
$83$	−13.2426	−1.45357	−0.726784	−	0.686866i	$-0.758986\pi$
$83$	−13.2426	−1.45357	−0.726784	+	0.686866i	$0.758986\pi$
$84$	0	0
$85$	2.24264	0.243249
$86$	7.07107	0.762493
$87$	−0.242641	−0.0260138
$88$	12.0000	1.27920
$89$	−1.58579	−0.168093	−0.0840465	−	0.996462i	$-0.526784\pi$
$89$	−1.58579	−0.168093	−0.0840465	+	0.996462i	$0.526784\pi$
$90$	−2.24264	−0.236395
$91$	0	0
$92$	0	0
$93$	4.58579	0.475524
$94$	2.24264	0.231311
$95$	−11.4853	−1.17837
$96$	0	0
$97$	−11.7279	−1.19079	−0.595395	−	0.803433i	$-0.703004\pi$
$97$	−11.7279	−1.19079	−0.595395	+	0.803433i	$0.703004\pi$
$98$	0	0
$99$	−4.24264	−0.426401
$100$	0	0
$101$	−10.2426	−1.01918	−0.509590	−	0.860417i	$-0.670203\pi$
$101$	−10.2426	−1.01918	−0.509590	+	0.860417i	$0.670203\pi$
$102$	−2.82843	−0.280056
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	0.242641	0.0235673
$107$	−20.1421	−1.94721	−0.973607	−	0.228232i	$-0.926706\pi$
$107$	−20.1421	−1.94721	−0.973607	+	0.228232i	$0.926706\pi$
$108$	0	0
$109$	16.7279	1.60224	0.801122	−	0.598501i	$-0.204237\pi$
$109$	16.7279	1.60224	0.801122	+	0.598501i	$0.204237\pi$
$110$	9.51472	0.907193
$111$	−3.17157	−0.301032
$112$	0	0
$113$	2.31371	0.217655	0.108828	−	0.994061i	$-0.465290\pi$
$113$	2.31371	0.217655	0.108828	+	0.994061i	$0.465290\pi$
$114$	14.4853	1.35667
$115$	9.24264	0.861881
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0.485281	0.0446738
$119$	0	0
$120$	6.34315	0.579047
$121$	7.00000	0.636364
$122$	8.48528	0.768221
$123$	12.4853	1.12576
$124$	0	0
$125$	11.8701	1.06169
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−11.3137	−1.00000
$129$	7.07107	0.622573
$130$	2.24264	0.196693
$131$	2.82843	0.247121	0.123560	−	0.992337i	$-0.460569\pi$
$131$	2.82843	0.247121	0.123560	+	0.992337i	$0.460569\pi$
$132$	0	0
$133$	0	0
$134$	20.4853	1.76966
$135$	−8.97056	−0.772063
$136$	−4.00000	−0.342997
$137$	−4.58579	−0.391790	−0.195895	−	0.980625i	$-0.562761\pi$
$137$	−4.58579	−0.391790	−0.195895	+	0.980625i	$0.562761\pi$
$138$	−11.6569	−0.992297
$139$	−6.24264	−0.529494	−0.264747	−	0.964318i	$-0.585288\pi$
$139$	−6.24264	−0.529494	−0.264747	+	0.964318i	$0.585288\pi$
$140$	0	0
$141$	2.24264	0.188864
$142$	18.4853	1.55125
$143$	4.24264	0.354787
$144$	4.00000	0.333333
$145$	−0.272078	−0.0225948
$146$	−13.0711	−1.08177
$147$	0	0
$148$	0	0
$149$	16.2426	1.33065	0.665324	−	0.746554i	$-0.268293\pi$
$149$	16.2426	1.33065	0.665324	+	0.746554i	$0.268293\pi$
$150$	−4.97056	−0.405845
$151$	−9.75736	−0.794043	−0.397021	−	0.917809i	$-0.629956\pi$
$151$	−9.75736	−0.794043	−0.397021	+	0.917809i	$0.629956\pi$
$152$	20.4853	1.66158
$153$	1.41421	0.114332
$154$	0	0
$155$	5.14214	0.413026
$156$	0	0
$157$	3.75736	0.299870	0.149935	−	0.988696i	$-0.452094\pi$
$157$	3.75736	0.299870	0.149935	+	0.988696i	$0.452094\pi$
$158$	−21.8995	−1.74223
$159$	0.242641	0.0192427
$160$	0	0
$161$	0	0
$162$	7.07107	0.555556
$163$	8.48528	0.664619	0.332309	−	0.943170i	$-0.392172\pi$
$163$	8.48528	0.664619	0.332309	+	0.943170i	$0.392172\pi$
$164$	0	0
$165$	9.51472	0.740720
$166$	18.7279	1.45357
$167$	−15.3848	−1.19051	−0.595255	−	0.803537i	$-0.702949\pi$
$167$	−15.3848	−1.19051	−0.595255	+	0.803537i	$0.702949\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−3.17157	−0.243249
$171$	−7.24264	−0.553859
$172$	0	0
$173$	−24.7279	−1.88003	−0.940015	−	0.341134i	$-0.889189\pi$
$173$	−24.7279	−1.88003	−0.940015	+	0.341134i	$0.889189\pi$
$174$	0.343146	0.0260138
$175$	0	0
$176$	−16.9706	−1.27920
$177$	0.485281	0.0364760
$178$	2.24264	0.168093
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	18.7279	1.39204	0.696018	−	0.718025i	$-0.254953\pi$
$181$	18.7279	1.39204	0.696018	+	0.718025i	$0.254953\pi$
$182$	0	0
$183$	8.48528	0.627250
$184$	−16.4853	−1.21531
$185$	−3.55635	−0.261468
$186$	−6.48528	−0.475524
$187$	−6.00000	−0.438763
$188$	0	0
$189$	0	0
$190$	16.2426	1.17837
$191$	15.1716	1.09778	0.548888	−	0.835896i	$-0.315051\pi$
$191$	15.1716	1.09778	0.548888	+	0.835896i	$0.315051\pi$
$192$	−11.3137	−0.816497
$193$	−14.4853	−1.04267	−0.521337	−	0.853351i	$-0.674566\pi$
$193$	−14.4853	−1.04267	−0.521337	+	0.853351i	$0.674566\pi$
$194$	16.5858	1.19079
$195$	2.24264	0.160599
$196$	0	0
$197$	12.3431	0.879413	0.439706	−	0.898142i	$-0.355083\pi$
$197$	12.3431	0.879413	0.439706	+	0.898142i	$0.355083\pi$
$198$	6.00000	0.426401
$199$	3.75736	0.266352	0.133176	−	0.991092i	$-0.457482\pi$
$199$	3.75736	0.266352	0.133176	+	0.991092i	$0.457482\pi$
$200$	−7.02944	−0.497056
$201$	20.4853	1.44492
$202$	14.4853	1.01918
$203$	0	0
$204$	0	0
$205$	14.0000	0.977802
$206$	−11.3137	−0.788263
$207$	5.82843	0.405104
$208$	−4.00000	−0.277350
$209$	30.7279	2.12549
$210$	0	0
$211$	−15.9706	−1.09946	−0.549729	−	0.835343i	$-0.685269\pi$
$211$	−15.9706	−1.09946	−0.549729	+	0.835343i	$0.685269\pi$
$212$	0	0
$213$	18.4853	1.26659
$214$	28.4853	1.94721
$215$	7.92893	0.540749
$216$	16.0000	1.08866
$217$	0	0
$218$	−23.6569	−1.60224
$219$	−13.0711	−0.883261
$220$	0	0
$221$	−1.41421	−0.0951303
$222$	4.48528	0.301032
$223$	−0.757359	−0.0507165	−0.0253583	−	0.999678i	$-0.508073\pi$
$223$	−0.757359	−0.0507165	−0.0253583	+	0.999678i	$0.508073\pi$
$224$	0	0
$225$	2.48528	0.165685
$226$	−3.27208	−0.217655
$227$	−26.8284	−1.78067	−0.890333	−	0.455311i	$-0.849528\pi$
$227$	−26.8284	−1.78067	−0.890333	+	0.455311i	$0.849528\pi$
$228$	0	0
$229$	−29.4558	−1.94650	−0.973248	−	0.229755i	$-0.926207\pi$
$229$	−29.4558	−1.94650	−0.973248	+	0.229755i	$0.926207\pi$
$230$	−13.0711	−0.861881
$231$	0	0
$232$	0.485281	0.0318603
$233$	−14.6569	−0.960202	−0.480101	−	0.877213i	$-0.659400\pi$
$233$	−14.6569	−0.960202	−0.480101	+	0.877213i	$0.659400\pi$
$234$	1.41421	0.0924500
$235$	2.51472	0.164042
$236$	0	0
$237$	−21.8995	−1.42253
$238$	0	0
$239$	3.51472	0.227348	0.113674	−	0.993518i	$-0.463738\pi$
$239$	3.51472	0.227348	0.113674	+	0.993518i	$0.463738\pi$
$240$	−8.97056	−0.579047
$241$	20.2132	1.30205	0.651023	−	0.759058i	$-0.274340\pi$
$241$	20.2132	1.30205	0.651023	+	0.759058i	$0.274340\pi$
$242$	−9.89949	−0.636364
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	−17.6569	−1.12576
$247$	7.24264	0.460838
$248$	−9.17157	−0.582395
$249$	18.7279	1.18683
$250$	−16.7868	−1.06169
$251$	−16.5858	−1.04689	−0.523443	−	0.852061i	$-0.675353\pi$
$251$	−16.5858	−1.04689	−0.523443	+	0.852061i	$0.675353\pi$
$252$	0	0
$253$	−24.7279	−1.55463
$254$	−2.82843	−0.177471
$255$	−3.17157	−0.198612
$256$	0	0
$257$	19.4142	1.21103	0.605513	−	0.795836i	$-0.292968\pi$
$257$	19.4142	1.21103	0.605513	+	0.795836i	$0.292968\pi$
$258$	−10.0000	−0.622573
$259$	0	0
$260$	0	0
$261$	−0.171573	−0.0106201
$262$	−4.00000	−0.247121
$263$	1.97056	0.121510	0.0607551	−	0.998153i	$-0.480649\pi$
$263$	1.97056	0.121510	0.0607551	+	0.998153i	$0.480649\pi$
$264$	−16.9706	−1.04447
$265$	0.272078	0.0167136
$266$	0	0
$267$	2.24264	0.137247
$268$	0	0
$269$	−9.17157	−0.559201	−0.279600	−	0.960116i	$-0.590202\pi$
$269$	−9.17157	−0.559201	−0.279600	+	0.960116i	$0.590202\pi$
$270$	12.6863	0.772063
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	5.65685	0.342997
$273$	0	0
$274$	6.48528	0.391790
$275$	−10.5442	−0.635837
$276$	0	0
$277$	−7.48528	−0.449747	−0.224873	−	0.974388i	$-0.572197\pi$
$277$	−7.48528	−0.449747	−0.224873	+	0.974388i	$0.572197\pi$
$278$	8.82843	0.529494
$279$	3.24264	0.194132
$280$	0	0
$281$	15.5563	0.928014	0.464007	−	0.885832i	$-0.346411\pi$
$281$	15.5563	0.928014	0.464007	+	0.885832i	$0.346411\pi$
$282$	−3.17157	−0.188864
$283$	8.48528	0.504398	0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528	0.504398	0.252199	+	0.967675i	$0.418846\pi$
$284$	0	0
$285$	16.2426	0.962131
$286$	−6.00000	−0.354787
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0.384776	0.0225948
$291$	16.5858	0.972276
$292$	0	0
$293$	−15.3848	−0.898788	−0.449394	−	0.893334i	$-0.648360\pi$
$293$	−15.3848	−0.898788	−0.449394	+	0.893334i	$0.648360\pi$
$294$	0	0
$295$	0.544156	0.0316820
$296$	6.34315	0.368688
$297$	24.0000	1.39262
$298$	−22.9706	−1.33065
$299$	−5.82843	−0.337067
$300$	0	0
$301$	0	0
$302$	13.7990	0.794043
$303$	14.4853	0.832158
$304$	−28.9706	−1.66158
$305$	9.51472	0.544811
$306$	−2.00000	−0.114332
$307$	−13.2426	−0.755797	−0.377899	−	0.925847i	$-0.623353\pi$
$307$	−13.2426	−0.755797	−0.377899	+	0.925847i	$0.623353\pi$
$308$	0	0
$309$	−11.3137	−0.643614
$310$	−7.27208	−0.413026
$311$	7.41421	0.420421	0.210211	−	0.977656i	$-0.432585\pi$
$311$	7.41421	0.420421	0.210211	+	0.977656i	$0.432585\pi$
$312$	−4.00000	−0.226455
$313$	−23.2132	−1.31209	−0.656044	−	0.754723i	$-0.727771\pi$
$313$	−23.2132	−1.31209	−0.656044	+	0.754723i	$0.727771\pi$
$314$	−5.31371	−0.299870
$315$	0	0
$316$	0	0
$317$	−11.3137	−0.635441	−0.317721	−	0.948184i	$-0.602917\pi$
$317$	−11.3137	−0.635441	−0.317721	+	0.948184i	$0.602917\pi$
$318$	−0.343146	−0.0192427
$319$	0.727922	0.0407558
$320$	−12.6863	−0.709185
$321$	28.4853	1.58989
$322$	0	0
$323$	−10.2426	−0.569916
$324$	0	0
$325$	−2.48528	−0.137859
$326$	−12.0000	−0.664619
$327$	−23.6569	−1.30823
$328$	−24.9706	−1.37877
$329$	0	0
$330$	−13.4558	−0.740720
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	0	0
$333$	−2.24264	−0.122896
$334$	21.7574	1.19051
$335$	22.9706	1.25502
$336$	0	0
$337$	−33.0000	−1.79762	−0.898812	−	0.438334i	$-0.855569\pi$
$337$	−33.0000	−1.79762	−0.898812	+	0.438334i	$0.855569\pi$
$338$	−1.41421	−0.0769231
$339$	−3.27208	−0.177715
$340$	0	0
$341$	−13.7574	−0.745003
$342$	10.2426	0.553859
$343$	0	0
$344$	−14.1421	−0.762493
$345$	−13.0711	−0.703723
$346$	34.9706	1.88003
$347$	−5.65685	−0.303676	−0.151838	−	0.988405i	$-0.548519\pi$
$347$	−5.65685	−0.303676	−0.151838	+	0.988405i	$0.548519\pi$
$348$	0	0
$349$	25.7279	1.37718	0.688592	−	0.725149i	$-0.258229\pi$
$349$	25.7279	1.37718	0.688592	+	0.725149i	$0.258229\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	8.48528	0.451626	0.225813	−	0.974171i	$-0.427496\pi$
$353$	8.48528	0.451626	0.225813	+	0.974171i	$0.427496\pi$
$354$	−0.686292	−0.0364760
$355$	20.7279	1.10012
$356$	0	0
$357$	0	0
$358$	12.7279	0.672692
$359$	−27.8995	−1.47248	−0.736240	−	0.676721i	$-0.763400\pi$
$359$	−27.8995	−1.47248	−0.736240	+	0.676721i	$0.763400\pi$
$360$	4.48528	0.236395
$361$	33.4558	1.76083
$362$	−26.4853	−1.39204
$363$	−9.89949	−0.519589
$364$	0	0
$365$	−14.6569	−0.767175
$366$	−12.0000	−0.627250
$367$	−10.2426	−0.534661	−0.267331	−	0.963605i	$-0.586142\pi$
$367$	−10.2426	−0.534661	−0.267331	+	0.963605i	$0.586142\pi$
$368$	23.3137	1.21531
$369$	8.82843	0.459590
$370$	5.02944	0.261468
$371$	0	0
$372$	0	0
$373$	8.48528	0.439351	0.219676	−	0.975573i	$-0.429500\pi$
$373$	8.48528	0.439351	0.219676	+	0.975573i	$0.429500\pi$
$374$	8.48528	0.438763
$375$	−16.7868	−0.866866
$376$	−4.48528	−0.231311
$377$	0.171573	0.00883645
$378$	0	0
$379$	23.7574	1.22033	0.610167	−	0.792273i	$-0.291102\pi$
$379$	23.7574	1.22033	0.610167	+	0.792273i	$0.291102\pi$
$380$	0	0
$381$	−2.82843	−0.144905
$382$	−21.4558	−1.09778
$383$	20.4853	1.04675	0.523374	−	0.852103i	$-0.324673\pi$
$383$	20.4853	1.04675	0.523374	+	0.852103i	$0.324673\pi$
$384$	16.0000	0.816497
$385$	0	0
$386$	20.4853	1.04267
$387$	5.00000	0.254164
$388$	0	0
$389$	29.6569	1.50366	0.751831	−	0.659356i	$-0.229171\pi$
$389$	29.6569	1.50366	0.751831	+	0.659356i	$0.229171\pi$
$390$	−3.17157	−0.160599
$391$	8.24264	0.416848
$392$	0	0
$393$	−4.00000	−0.201773
$394$	−17.4558	−0.879413
$395$	−24.5563	−1.23556
$396$	0	0
$397$	18.2132	0.914094	0.457047	−	0.889442i	$-0.348907\pi$
$397$	18.2132	0.914094	0.457047	+	0.889442i	$0.348907\pi$
$398$	−5.31371	−0.266352
$399$	0	0
$400$	9.94113	0.497056
$401$	6.34315	0.316762	0.158381	−	0.987378i	$-0.449373\pi$
$401$	6.34315	0.316762	0.158381	+	0.987378i	$0.449373\pi$
$402$	−28.9706	−1.44492
$403$	−3.24264	−0.161527
$404$	0	0
$405$	7.92893	0.393992
$406$	0	0
$407$	9.51472	0.471627
$408$	5.65685	0.280056
$409$	−3.24264	−0.160338	−0.0801691	−	0.996781i	$-0.525546\pi$
$409$	−3.24264	−0.160338	−0.0801691	+	0.996781i	$0.525546\pi$
$410$	−19.7990	−0.977802
$411$	6.48528	0.319895
$412$	0	0
$413$	0	0
$414$	−8.24264	−0.405104
$415$	21.0000	1.03085
$416$	0	0
$417$	8.82843	0.432330
$418$	−43.4558	−2.12549
$419$	20.8701	1.01957	0.509785	−	0.860302i	$-0.329725\pi$
$419$	20.8701	1.01957	0.509785	+	0.860302i	$0.329725\pi$
$420$	0	0
$421$	6.72792	0.327899	0.163949	−	0.986469i	$-0.447577\pi$
$421$	6.72792	0.327899	0.163949	+	0.986469i	$0.447577\pi$
$422$	22.5858	1.09946
$423$	1.58579	0.0771036
$424$	−0.485281	−0.0235673
$425$	3.51472	0.170489
$426$	−26.1421	−1.26659
$427$	0	0
$428$	0	0
$429$	−6.00000	−0.289683
$430$	−11.2132	−0.540749
$431$	−12.3431	−0.594548	−0.297274	−	0.954792i	$-0.596078\pi$
$431$	−12.3431	−0.594548	−0.297274	+	0.954792i	$0.596078\pi$
$432$	−22.6274	−1.08866
$433$	24.9706	1.20001	0.600004	−	0.799997i	$-0.295166\pi$
$433$	24.9706	1.20001	0.600004	+	0.799997i	$0.295166\pi$
$434$	0	0
$435$	0.384776	0.0184486
$436$	0	0
$437$	−42.2132	−2.01933
$438$	18.4853	0.883261
$439$	34.4853	1.64589	0.822946	−	0.568119i	$-0.192329\pi$
$439$	34.4853	1.64589	0.822946	+	0.568119i	$0.192329\pi$
$440$	−19.0294	−0.907193
$441$	0	0
$442$	2.00000	0.0951303
$443$	−3.68629	−0.175141	−0.0875705	−	0.996158i	$-0.527910\pi$
$443$	−3.68629	−0.175141	−0.0875705	+	0.996158i	$0.527910\pi$
$444$	0	0
$445$	2.51472	0.119209
$446$	1.07107	0.0507165
$447$	−22.9706	−1.08647
$448$	0	0
$449$	21.1716	0.999148	0.499574	−	0.866271i	$-0.333490\pi$
$449$	21.1716	0.999148	0.499574	+	0.866271i	$0.333490\pi$
$450$	−3.51472	−0.165685
$451$	−37.4558	−1.76373
$452$	0	0
$453$	13.7990	0.648333
$454$	37.9411	1.78067
$455$	0	0
$456$	−28.9706	−1.35667
$457$	−7.21320	−0.337419	−0.168710	−	0.985666i	$-0.553960\pi$
$457$	−7.21320	−0.337419	−0.168710	+	0.985666i	$0.553960\pi$
$458$	41.6569	1.94650
$459$	−8.00000	−0.373408
$460$	0	0
$461$	−8.82843	−0.411181	−0.205590	−	0.978638i	$-0.565911\pi$
$461$	−8.82843	−0.411181	−0.205590	+	0.978638i	$0.565911\pi$
$462$	0	0
$463$	−4.24264	−0.197172	−0.0985861	−	0.995129i	$-0.531432\pi$
$463$	−4.24264	−0.197172	−0.0985861	+	0.995129i	$0.531432\pi$
$464$	−0.686292	−0.0318603
$465$	−7.27208	−0.337235
$466$	20.7279	0.960202
$467$	3.89949	0.180447	0.0902236	−	0.995922i	$-0.471242\pi$
$467$	3.89949	0.180447	0.0902236	+	0.995922i	$0.471242\pi$
$468$	0	0
$469$	0	0
$470$	−3.55635	−0.164042
$471$	−5.31371	−0.244843
$472$	−0.970563	−0.0446738
$473$	−21.2132	−0.975384
$474$	30.9706	1.42253
$475$	−18.0000	−0.825897
$476$	0	0
$477$	0.171573	0.00785578
$478$	−4.97056	−0.227348
$479$	6.21320	0.283889	0.141944	−	0.989875i	$-0.454665\pi$
$479$	6.21320	0.283889	0.141944	+	0.989875i	$0.454665\pi$
$480$	0	0
$481$	2.24264	0.102256
$482$	−28.5858	−1.30205
$483$	0	0
$484$	0	0
$485$	18.5980	0.844491
$486$	14.0000	0.635053
$487$	−11.4558	−0.519114	−0.259557	−	0.965728i	$-0.583577\pi$
$487$	−11.4558	−0.519114	−0.259557	+	0.965728i	$0.583577\pi$
$488$	−16.9706	−0.768221
$489$	−12.0000	−0.542659
$490$	0	0
$491$	16.6274	0.750385	0.375192	−	0.926947i	$-0.377577\pi$
$491$	16.6274	0.750385	0.375192	+	0.926947i	$0.377577\pi$
$492$	0	0
$493$	−0.242641	−0.0109280
$494$	−10.2426	−0.460838
$495$	6.72792	0.302398
$496$	12.9706	0.582395
$497$	0	0
$498$	−26.4853	−1.18683
$499$	−13.2721	−0.594140	−0.297070	−	0.954856i	$-0.596009\pi$
$499$	−13.2721	−0.594140	−0.297070	+	0.954856i	$0.596009\pi$
$500$	0	0
$501$	21.7574	0.972047
$502$	23.4558	1.04689
$503$	28.6274	1.27643	0.638217	−	0.769857i	$-0.279672\pi$
$503$	28.6274	1.27643	0.638217	+	0.769857i	$0.279672\pi$
$504$	0	0
$505$	16.2426	0.722788
$506$	34.9706	1.55463
$507$	−1.41421	−0.0628074
$508$	0	0
$509$	5.10051	0.226076	0.113038	−	0.993591i	$-0.463942\pi$
$509$	5.10051	0.226076	0.113038	+	0.993591i	$0.463942\pi$
$510$	4.48528	0.198612
$511$	0	0
$512$	22.6274	1.00000
$513$	40.9706	1.80889
$514$	−27.4558	−1.21103
$515$	−12.6863	−0.559025
$516$	0	0
$517$	−6.72792	−0.295894
$518$	0	0
$519$	34.9706	1.53504
$520$	−4.48528	−0.196693
$521$	6.34315	0.277898	0.138949	−	0.990300i	$-0.455628\pi$
$521$	6.34315	0.277898	0.138949	+	0.990300i	$0.455628\pi$
$522$	0.242641	0.0106201
$523$	−30.9706	−1.35425	−0.677124	−	0.735869i	$-0.736774\pi$
$523$	−30.9706	−1.35425	−0.677124	+	0.735869i	$0.736774\pi$
$524$	0	0
$525$	0	0
$526$	−2.78680	−0.121510
$527$	4.58579	0.199760
$528$	24.0000	1.04447
$529$	10.9706	0.476981
$530$	−0.384776	−0.0167136
$531$	0.343146	0.0148913
$532$	0	0
$533$	−8.82843	−0.382402
$534$	−3.17157	−0.137247
$535$	31.9411	1.38094
$536$	−40.9706	−1.76966
$537$	12.7279	0.549250
$538$	12.9706	0.559201
$539$	0	0
$540$	0	0
$541$	−7.21320	−0.310120	−0.155060	−	0.987905i	$-0.549557\pi$
$541$	−7.21320	−0.310120	−0.155060	+	0.987905i	$0.549557\pi$
$542$	28.2843	1.21491
$543$	−26.4853	−1.13659
$544$	0	0
$545$	−26.5269	−1.13629
$546$	0	0
$547$	35.4853	1.51724	0.758621	−	0.651533i	$-0.225874\pi$
$547$	35.4853	1.51724	0.758621	+	0.651533i	$0.225874\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	14.9117	0.635837
$551$	1.24264	0.0529383
$552$	23.3137	0.992297
$553$	0	0
$554$	10.5858	0.449747
$555$	5.02944	0.213488
$556$	0	0
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	−4.58579	−0.194132
$559$	−5.00000	−0.211477
$560$	0	0
$561$	8.48528	0.358249
$562$	−22.0000	−0.928014
$563$	6.34315	0.267332	0.133666	−	0.991026i	$-0.457325\pi$
$563$	6.34315	0.267332	0.133666	+	0.991026i	$0.457325\pi$
$564$	0	0
$565$	−3.66905	−0.154358
$566$	−12.0000	−0.504398
$567$	0	0
$568$	−36.9706	−1.55125
$569$	5.14214	0.215570	0.107785	−	0.994174i	$-0.465624\pi$
$569$	5.14214	0.215570	0.107785	+	0.994174i	$0.465624\pi$
$570$	−22.9706	−0.962131
$571$	42.4558	1.77672	0.888361	−	0.459146i	$-0.151844\pi$
$571$	42.4558	1.77672	0.888361	+	0.459146i	$0.151844\pi$
$572$	0	0
$573$	−21.4558	−0.896331
$574$	0	0
$575$	14.4853	0.604078
$576$	−8.00000	−0.333333
$577$	6.97056	0.290188	0.145094	−	0.989418i	$-0.453651\pi$
$577$	6.97056	0.290188	0.145094	+	0.989418i	$0.453651\pi$
$578$	21.2132	0.882353
$579$	20.4853	0.851339
$580$	0	0
$581$	0	0
$582$	−23.4558	−0.972276
$583$	−0.727922	−0.0301475
$584$	26.1421	1.08177
$585$	1.58579	0.0655642
$586$	21.7574	0.898788
$587$	6.55635	0.270609	0.135305	−	0.990804i	$-0.456799\pi$
$587$	6.55635	0.270609	0.135305	+	0.990804i	$0.456799\pi$
$588$	0	0
$589$	−23.4853	−0.967694
$590$	−0.769553	−0.0316820
$591$	−17.4558	−0.718037
$592$	−8.97056	−0.368688
$593$	0.556349	0.0228465	0.0114233	−	0.999935i	$-0.496364\pi$
$593$	0.556349	0.0228465	0.0114233	+	0.999935i	$0.496364\pi$
$594$	−33.9411	−1.39262
$595$	0	0
$596$	0	0
$597$	−5.31371	−0.217476
$598$	8.24264	0.337067
$599$	−28.7990	−1.17669	−0.588347	−	0.808608i	$-0.700221\pi$
$599$	−28.7990	−1.17669	−0.588347	+	0.808608i	$0.700221\pi$
$600$	9.94113	0.405845
$601$	38.9706	1.58964	0.794821	−	0.606844i	$-0.207565\pi$
$601$	38.9706	1.58964	0.794821	+	0.606844i	$0.207565\pi$
$602$	0	0
$603$	14.4853	0.589886
$604$	0	0
$605$	−11.1005	−0.451300
$606$	−20.4853	−0.832158
$607$	26.7279	1.08485	0.542426	−	0.840103i	$-0.317506\pi$
$607$	26.7279	1.08485	0.542426	+	0.840103i	$0.317506\pi$
$608$	0	0
$609$	0	0
$610$	−13.4558	−0.544811
$611$	−1.58579	−0.0641541
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	18.7279	0.755797
$615$	−19.7990	−0.798372
$616$	0	0
$617$	12.3431	0.496916	0.248458	−	0.968643i	$-0.420076\pi$
$617$	12.3431	0.496916	0.248458	+	0.968643i	$0.420076\pi$
$618$	16.0000	0.643614
$619$	−0.970563	−0.0390102	−0.0195051	−	0.999810i	$-0.506209\pi$
$619$	−0.970563	−0.0390102	−0.0195051	+	0.999810i	$0.506209\pi$
$620$	0	0
$621$	−32.9706	−1.32306
$622$	−10.4853	−0.420421
$623$	0	0
$624$	5.65685	0.226455
$625$	−6.39697	−0.255879
$626$	32.8284	1.31209
$627$	−43.4558	−1.73546
$628$	0	0
$629$	−3.17157	−0.126459
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	43.7990	1.74223
$633$	22.5858	0.897704
$634$	16.0000	0.635441
$635$	−3.17157	−0.125860
$636$	0	0
$637$	0	0
$638$	−1.02944	−0.0407558
$639$	13.0711	0.517083
$640$	17.9411	0.709185
$641$	15.3431	0.606018	0.303009	−	0.952988i	$-0.402009\pi$
$641$	15.3431	0.606018	0.303009	+	0.952988i	$0.402009\pi$
$642$	−40.2843	−1.58989
$643$	12.4853	0.492371	0.246186	−	0.969223i	$-0.420823\pi$
$643$	12.4853	0.492371	0.246186	+	0.969223i	$0.420823\pi$
$644$	0	0
$645$	−11.2132	−0.441519
$646$	14.4853	0.569916
$647$	−14.5269	−0.571112	−0.285556	−	0.958362i	$-0.592178\pi$
$647$	−14.5269	−0.571112	−0.285556	+	0.958362i	$0.592178\pi$
$648$	−14.1421	−0.555556
$649$	−1.45584	−0.0571469
$650$	3.51472	0.137859
$651$	0	0
$652$	0	0
$653$	17.3137	0.677538	0.338769	−	0.940870i	$-0.389990\pi$
$653$	17.3137	0.677538	0.338769	+	0.940870i	$0.389990\pi$
$654$	33.4558	1.30823
$655$	−4.48528	−0.175254
$656$	35.3137	1.37877
$657$	−9.24264	−0.360590
$658$	0	0
$659$	−15.3431	−0.597684	−0.298842	−	0.954303i	$-0.596600\pi$
$659$	−15.3431	−0.597684	−0.298842	+	0.954303i	$0.596600\pi$
$660$	0	0
$661$	10.7574	0.418413	0.209206	−	0.977872i	$-0.432912\pi$
$661$	10.7574	0.418413	0.209206	+	0.977872i	$0.432912\pi$
$662$	25.4558	0.989369
$663$	2.00000	0.0776736
$664$	−37.4558	−1.45357
$665$	0	0
$666$	3.17157	0.122896
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	1.07107	0.0414099
$670$	−32.4853	−1.25502
$671$	−25.4558	−0.982712
$672$	0	0
$673$	−26.9411	−1.03850	−0.519252	−	0.854621i	$-0.673789\pi$
$673$	−26.9411	−1.03850	−0.519252	+	0.854621i	$0.673789\pi$
$674$	46.6690	1.79762
$675$	−14.0589	−0.541126
$676$	0	0
$677$	41.3553	1.58941	0.794707	−	0.606993i	$-0.207624\pi$
$677$	41.3553	1.58941	0.794707	+	0.606993i	$0.207624\pi$
$678$	4.62742	0.177715
$679$	0	0
$680$	6.34315	0.243249
$681$	37.9411	1.45391
$682$	19.4558	0.745003
$683$	−21.1716	−0.810108	−0.405054	−	0.914293i	$-0.632747\pi$
$683$	−21.1716	−0.810108	−0.405054	+	0.914293i	$0.632747\pi$
$684$	0	0
$685$	7.27208	0.277852
$686$	0	0
$687$	41.6569	1.58931
$688$	20.0000	0.762493
$689$	−0.171573	−0.00653641
$690$	18.4853	0.703723
$691$	−28.6985	−1.09174	−0.545871	−	0.837869i	$-0.683801\pi$
$691$	−28.6985	−1.09174	−0.545871	+	0.837869i	$0.683801\pi$
$692$	0	0
$693$	0	0
$694$	8.00000	0.303676
$695$	9.89949	0.375509
$696$	−0.686292	−0.0260138
$697$	12.4853	0.472914
$698$	−36.3848	−1.37718
$699$	20.7279	0.784002
$700$	0	0
$701$	−10.7990	−0.407872	−0.203936	−	0.978984i	$-0.565373\pi$
$701$	−10.7990	−0.407872	−0.203936	+	0.978984i	$0.565373\pi$
$702$	−8.00000	−0.301941
$703$	16.2426	0.612603
$704$	33.9411	1.27920
$705$	−3.55635	−0.133940
$706$	−12.0000	−0.451626
$707$	0	0
$708$	0	0
$709$	0.727922	0.0273377	0.0136688	−	0.999907i	$-0.495649\pi$
$709$	0.727922	0.0273377	0.0136688	+	0.999907i	$0.495649\pi$
$710$	−29.3137	−1.10012
$711$	−15.4853	−0.580743
$712$	−4.48528	−0.168093
$713$	18.8995	0.707792
$714$	0	0
$715$	−6.72792	−0.251610
$716$	0	0
$717$	−4.97056	−0.185629
$718$	39.4558	1.47248
$719$	1.75736	0.0655384	0.0327692	−	0.999463i	$-0.489567\pi$
$719$	1.75736	0.0655384	0.0327692	+	0.999463i	$0.489567\pi$
$720$	−6.34315	−0.236395
$721$	0	0
$722$	−47.3137	−1.76083
$723$	−28.5858	−1.06312
$724$	0	0
$725$	−0.426407	−0.0158364
$726$	14.0000	0.519589
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	20.7279	0.767175
$731$	7.07107	0.261533
$732$	0	0
$733$	16.6985	0.616773	0.308386	−	0.951261i	$-0.400211\pi$
$733$	16.6985	0.616773	0.308386	+	0.951261i	$0.400211\pi$
$734$	14.4853	0.534661
$735$	0	0
$736$	0	0
$737$	−61.4558	−2.26376
$738$	−12.4853	−0.459590
$739$	−17.6985	−0.651049	−0.325525	−	0.945534i	$-0.605541\pi$
$739$	−17.6985	−0.651049	−0.325525	+	0.945534i	$0.605541\pi$
$740$	0	0
$741$	−10.2426	−0.376273
$742$	0	0
$743$	−29.6569	−1.08800	−0.544002	−	0.839084i	$-0.683092\pi$
$743$	−29.6569	−1.08800	−0.544002	+	0.839084i	$0.683092\pi$
$744$	12.9706	0.475524
$745$	−25.7574	−0.943677
$746$	−12.0000	−0.439351
$747$	13.2426	0.484523
$748$	0	0
$749$	0	0
$750$	23.7401	0.866866
$751$	15.4853	0.565066	0.282533	−	0.959258i	$-0.408825\pi$
$751$	15.4853	0.565066	0.282533	+	0.959258i	$0.408825\pi$
$752$	6.34315	0.231311
$753$	23.4558	0.854778
$754$	−0.242641	−0.00883645
$755$	15.4731	0.563123
$756$	0	0
$757$	21.4853	0.780896	0.390448	−	0.920625i	$-0.372320\pi$
$757$	21.4853	0.780896	0.390448	+	0.920625i	$0.372320\pi$
$758$	−33.5980	−1.22033
$759$	34.9706	1.26935
$760$	−32.4853	−1.17837
$761$	−34.7574	−1.25995	−0.629977	−	0.776614i	$-0.716936\pi$
$761$	−34.7574	−1.25995	−0.629977	+	0.776614i	$0.716936\pi$
$762$	4.00000	0.144905
$763$	0	0
$764$	0	0
$765$	−2.24264	−0.0810828
$766$	−28.9706	−1.04675
$767$	−0.343146	−0.0123903
$768$	0	0
$769$	−52.2132	−1.88286	−0.941428	−	0.337214i	$-0.890516\pi$
$769$	−52.2132	−1.88286	−0.941428	+	0.337214i	$0.890516\pi$
$770$	0	0
$771$	−27.4558	−0.988798
$772$	0	0
$773$	32.8284	1.18076	0.590378	−	0.807127i	$-0.298979\pi$
$773$	32.8284	1.18076	0.590378	+	0.807127i	$0.298979\pi$
$774$	−7.07107	−0.254164
$775$	8.05887	0.289483
$776$	−33.1716	−1.19079
$777$	0	0
$778$	−41.9411	−1.50366
$779$	−63.9411	−2.29093
$780$	0	0
$781$	−55.4558	−1.98437
$782$	−11.6569	−0.416848
$783$	0.970563	0.0346851
$784$	0	0
$785$	−5.95837	−0.212663
$786$	5.65685	0.201773
$787$	24.7574	0.882505	0.441252	−	0.897383i	$-0.354534\pi$
$787$	24.7574	0.882505	0.441252	+	0.897383i	$0.354534\pi$
$788$	0	0
$789$	−2.78680	−0.0992126
$790$	34.7279	1.23556
$791$	0	0
$792$	−12.0000	−0.426401
$793$	−6.00000	−0.213066
$794$	−25.7574	−0.914094
$795$	−0.384776	−0.0136466
$796$	0	0
$797$	24.3431	0.862278	0.431139	−	0.902285i	$-0.358112\pi$
$797$	24.3431	0.862278	0.431139	+	0.902285i	$0.358112\pi$
$798$	0	0
$799$	2.24264	0.0793389
$800$	0	0
$801$	1.58579	0.0560310
$802$	−8.97056	−0.316762
$803$	39.2132	1.38380
$804$	0	0
$805$	0	0
$806$	4.58579	0.161527
$807$	12.9706	0.456585
$808$	−28.9706	−1.01918
$809$	17.4853	0.614750	0.307375	−	0.951589i	$-0.400549\pi$
$809$	17.4853	0.614750	0.307375	+	0.951589i	$0.400549\pi$
$810$	−11.2132	−0.393992
$811$	21.9411	0.770457	0.385229	−	0.922821i	$-0.374123\pi$
$811$	21.9411	0.770457	0.385229	+	0.922821i	$0.374123\pi$
$812$	0	0
$813$	28.2843	0.991973
$814$	−13.4558	−0.471627
$815$	−13.4558	−0.471338
$816$	−8.00000	−0.280056
$817$	−36.2132	−1.26694
$818$	4.58579	0.160338
$819$	0	0
$820$	0	0
$821$	−20.1421	−0.702965	−0.351483	−	0.936194i	$-0.614322\pi$
$821$	−20.1421	−0.702965	−0.351483	+	0.936194i	$0.614322\pi$
$822$	−9.17157	−0.319895
$823$	10.4853	0.365494	0.182747	−	0.983160i	$-0.441501\pi$
$823$	10.4853	0.365494	0.182747	+	0.983160i	$0.441501\pi$
$824$	22.6274	0.788263
$825$	14.9117	0.519158
$826$	0	0
$827$	−49.4558	−1.71975	−0.859874	−	0.510506i	$-0.829458\pi$
$827$	−49.4558	−1.71975	−0.859874	+	0.510506i	$0.829458\pi$
$828$	0	0
$829$	−50.7279	−1.76185	−0.880927	−	0.473253i	$-0.843080\pi$
$829$	−50.7279	−1.76185	−0.880927	+	0.473253i	$0.843080\pi$
$830$	−29.6985	−1.03085
$831$	10.5858	0.367217
$832$	8.00000	0.277350
$833$	0	0
$834$	−12.4853	−0.432330
$835$	24.3970	0.844292
$836$	0	0
$837$	−18.3431	−0.634032
$838$	−29.5147	−1.01957
$839$	33.1716	1.14521	0.572605	−	0.819831i	$-0.305933\pi$
$839$	33.1716	1.14521	0.572605	+	0.819831i	$0.305933\pi$
$840$	0	0
$841$	−28.9706	−0.998985
$842$	−9.51472	−0.327899
$843$	−22.0000	−0.757720
$844$	0	0
$845$	−1.58579	−0.0545527
$846$	−2.24264	−0.0771036
$847$	0	0
$848$	0.686292	0.0235673
$849$	−12.0000	−0.411839
$850$	−4.97056	−0.170489
$851$	−13.0711	−0.448070
$852$	0	0
$853$	−39.7279	−1.36026	−0.680129	−	0.733092i	$-0.738076\pi$
$853$	−39.7279	−1.36026	−0.680129	+	0.733092i	$0.738076\pi$
$854$	0	0
$855$	11.4853	0.392788
$856$	−56.9706	−1.94721
$857$	54.7696	1.87089	0.935446	−	0.353469i	$-0.114998\pi$
$857$	54.7696	1.87089	0.935446	+	0.353469i	$0.114998\pi$
$858$	8.48528	0.289683
$859$	−30.9706	−1.05670	−0.528351	−	0.849026i	$-0.677189\pi$
$859$	−30.9706	−1.05670	−0.528351	+	0.849026i	$0.677189\pi$
$860$	0	0
$861$	0	0
$862$	17.4558	0.594548
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	0	0
$865$	39.2132	1.33329
$866$	−35.3137	−1.20001
$867$	21.2132	0.720438
$868$	0	0
$869$	65.6985	2.22867
$870$	−0.544156	−0.0184486
$871$	−14.4853	−0.490815
$872$	47.3137	1.60224
$873$	11.7279	0.396930
$874$	59.6985	2.01933
$875$	0	0
$876$	0	0
$877$	−10.2426	−0.345869	−0.172935	−	0.984933i	$-0.555325\pi$
$877$	−10.2426	−0.345869	−0.172935	+	0.984933i	$0.555325\pi$
$878$	−48.7696	−1.64589
$879$	21.7574	0.733858
$880$	26.9117	0.907193
$881$	−13.1127	−0.441778	−0.220889	−	0.975299i	$-0.570896\pi$
$881$	−13.1127	−0.441778	−0.220889	+	0.975299i	$0.570896\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	−0.769553	−0.0258682
$886$	5.21320	0.175141
$887$	−19.1127	−0.641742	−0.320871	−	0.947123i	$-0.603976\pi$
$887$	−19.1127	−0.641742	−0.320871	+	0.947123i	$0.603976\pi$
$888$	−8.97056	−0.301032
$889$	0	0
$890$	−3.55635	−0.119209
$891$	−21.2132	−0.710669
$892$	0	0
$893$	−11.4853	−0.384340
$894$	32.4853	1.08647
$895$	14.2721	0.477063
$896$	0	0
$897$	8.24264	0.275214
$898$	−29.9411	−0.999148
$899$	−0.556349	−0.0185553
$900$	0	0
$901$	0.242641	0.00808353
$902$	52.9706	1.76373
$903$	0	0
$904$	6.54416	0.217655
$905$	−29.6985	−0.987211
$906$	−19.5147	−0.648333
$907$	1.97056	0.0654315	0.0327157	−	0.999465i	$-0.489584\pi$
$907$	1.97056	0.0654315	0.0327157	+	0.999465i	$0.489584\pi$
$908$	0	0
$909$	10.2426	0.339727
$910$	0	0
$911$	−43.9706	−1.45681	−0.728405	−	0.685147i	$-0.759738\pi$
$911$	−43.9706	−1.45681	−0.728405	+	0.685147i	$0.759738\pi$
$912$	40.9706	1.35667
$913$	−56.1838	−1.85941
$914$	10.2010	0.337419
$915$	−13.4558	−0.444836
$916$	0	0
$917$	0	0
$918$	11.3137	0.373408
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	26.1421	0.861881
$921$	18.7279	0.617106
$922$	12.4853	0.411181
$923$	−13.0711	−0.430239
$924$	0	0
$925$	−5.57359	−0.183259
$926$	6.00000	0.197172
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−36.8995	−1.21063	−0.605317	−	0.795985i	$-0.706953\pi$
$929$	−36.8995	−1.21063	−0.605317	+	0.795985i	$0.706953\pi$
$930$	10.2843	0.337235
$931$	0	0
$932$	0	0
$933$	−10.4853	−0.343273
$934$	−5.51472	−0.180447
$935$	9.51472	0.311165
$936$	−2.82843	−0.0924500
$937$	−45.2132	−1.47705	−0.738525	−	0.674226i	$-0.764478\pi$
$937$	−45.2132	−1.47705	−0.738525	+	0.674226i	$0.764478\pi$
$938$	0	0
$939$	32.8284	1.07132
$940$	0	0
$941$	−40.0711	−1.30628	−0.653140	−	0.757237i	$-0.726549\pi$
$941$	−40.0711	−1.30628	−0.653140	+	0.757237i	$0.726549\pi$
$942$	7.51472	0.244843
$943$	51.4558	1.67563
$944$	1.37258	0.0446738
$945$	0	0
$946$	30.0000	0.975384
$947$	15.1716	0.493010	0.246505	−	0.969142i	$-0.420718\pi$
$947$	15.1716	0.493010	0.246505	+	0.969142i	$0.420718\pi$
$948$	0	0
$949$	9.24264	0.300029
$950$	25.4558	0.825897
$951$	16.0000	0.518836
$952$	0	0
$953$	11.1421	0.360929	0.180465	−	0.983581i	$-0.442240\pi$
$953$	11.1421	0.360929	0.180465	+	0.983581i	$0.442240\pi$
$954$	−0.242641	−0.00785578
$955$	−24.0589	−0.778527
$956$	0	0
$957$	−1.02944	−0.0332770
$958$	−8.78680	−0.283889
$959$	0	0
$960$	17.9411	0.579047
$961$	−20.4853	−0.660816
$962$	−3.17157	−0.102256
$963$	20.1421	0.649071
$964$	0	0
$965$	22.9706	0.739449
$966$	0	0
$967$	17.6985	0.569145	0.284572	−	0.958655i	$-0.408148\pi$
$967$	17.6985	0.569145	0.284572	+	0.958655i	$0.408148\pi$
$968$	19.7990	0.636364
$969$	14.4853	0.465334
$970$	−26.3015	−0.844491
$971$	−43.4558	−1.39456	−0.697282	−	0.716797i	$-0.745608\pi$
$971$	−43.4558	−1.39456	−0.697282	+	0.716797i	$0.745608\pi$
$972$	0	0
$973$	0	0
$974$	16.2010	0.519114
$975$	3.51472	0.112561
$976$	24.0000	0.768221
$977$	24.0416	0.769160	0.384580	−	0.923092i	$-0.374346\pi$
$977$	24.0416	0.769160	0.384580	+	0.923092i	$0.374346\pi$
$978$	16.9706	0.542659
$979$	−6.72792	−0.215025
$980$	0	0
$981$	−16.7279	−0.534081
$982$	−23.5147	−0.750385
$983$	15.0416	0.479754	0.239877	−	0.970803i	$-0.422893\pi$
$983$	15.0416	0.479754	0.239877	+	0.970803i	$0.422893\pi$
$984$	35.3137	1.12576
$985$	−19.5736	−0.623667
$986$	0.343146	0.0109280
$987$	0	0
$988$	0	0
$989$	29.1421	0.926666
$990$	−9.51472	−0.302398
$991$	1.02944	0.0327012	0.0163506	−	0.999866i	$-0.494795\pi$
$991$	1.02944	0.0327012	0.0163506	+	0.999866i	$0.494795\pi$
$992$	0	0
$993$	25.4558	0.807817
$994$	0	0
$995$	−5.95837	−0.188893
$996$	0	0
$997$	11.5147	0.364675	0.182337	−	0.983236i	$-0.441634\pi$
$997$	11.5147	0.364675	0.182337	+	0.983236i	$0.441634\pi$
$998$	18.7696	0.594140
$999$	12.6863	0.401377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.a.g.1.1		2
3.2	odd	2		5733.2.a.s.1.2		2
7.2	even	3		637.2.e.g.508.2		4
7.3	odd	6		637.2.e.f.79.2		4
7.4	even	3		637.2.e.g.79.2		4
7.5	odd	6		637.2.e.f.508.2		4
7.6	odd	2		91.2.a.c.1.1	✓	2
13.12	even	2		8281.2.a.v.1.2		2
21.20	even	2		819.2.a.h.1.2		2
28.27	even	2		1456.2.a.q.1.1		2
35.34	odd	2		2275.2.a.j.1.2		2
56.13	odd	2		5824.2.a.bl.1.1		2
56.27	even	2		5824.2.a.bk.1.2		2
91.34	even	4		1183.2.c.d.337.2		4
91.83	even	4		1183.2.c.d.337.4		4
91.90	odd	2		1183.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.c.1.1	✓	2	7.6	odd	2
637.2.a.g.1.1		2	1.1	even	1	trivial
637.2.e.f.79.2		4	7.3	odd	6
637.2.e.f.508.2		4	7.5	odd	6
637.2.e.g.79.2		4	7.4	even	3
637.2.e.g.508.2		4	7.2	even	3
819.2.a.h.1.2		2	21.20	even	2
1183.2.a.d.1.2		2	91.90	odd	2
1183.2.c.d.337.2		4	91.34	even	4
1183.2.c.d.337.4		4	91.83	even	4
1456.2.a.q.1.1		2	28.27	even	2
2275.2.a.j.1.2		2	35.34	odd	2
5733.2.a.s.1.2		2	3.2	odd	2
5824.2.a.bk.1.2		2	56.27	even	2
5824.2.a.bl.1.1		2	56.13	odd	2
8281.2.a.v.1.2		2	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.41421 q^{3} -1.58579 q^{5} +2.00000 q^{6} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.41421 q^{3} -1.58579 q^{5} +2.00000 q^{6} +2.82843 q^{8} -1.00000 q^{9} +2.24264 q^{10} +4.24264 q^{11} +1.00000 q^{13} +2.24264 q^{15} -4.00000 q^{16} -1.41421 q^{17} +1.41421 q^{18} +7.24264 q^{19} -6.00000 q^{22} -5.82843 q^{23} -4.00000 q^{24} -2.48528 q^{25} -1.41421 q^{26} +5.65685 q^{27} +0.171573 q^{29} -3.17157 q^{30} -3.24264 q^{31} -6.00000 q^{33} +2.00000 q^{34} +2.24264 q^{37} -10.2426 q^{38} -1.41421 q^{39} -4.48528 q^{40} -8.82843 q^{41} -5.00000 q^{43} +1.58579 q^{45} +8.24264 q^{46} -1.58579 q^{47} +5.65685 q^{48} +3.51472 q^{50} +2.00000 q^{51} -0.171573 q^{53} -8.00000 q^{54} -6.72792 q^{55} -10.2426 q^{57} -0.242641 q^{58} -0.343146 q^{59} -6.00000 q^{61} +4.58579 q^{62} +8.00000 q^{64} -1.58579 q^{65} +8.48528 q^{66} -14.4853 q^{67} +8.24264 q^{69} -13.0711 q^{71} -2.82843 q^{72} +9.24264 q^{73} -3.17157 q^{74} +3.51472 q^{75} +2.00000 q^{78} +15.4853 q^{79} +6.34315 q^{80} -5.00000 q^{81} +12.4853 q^{82} -13.2426 q^{83} +2.24264 q^{85} +7.07107 q^{86} -0.242641 q^{87} +12.0000 q^{88} -1.58579 q^{89} -2.24264 q^{90} +4.58579 q^{93} +2.24264 q^{94} -11.4853 q^{95} -11.7279 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 6 * q^5 + 4 * q^6 - 2 * q^9	\( 2 q - 6 q^{5} + 4 q^{6} - 2 q^{9} - 4 q^{10} + 2 q^{13} - 4 q^{15} - 8 q^{16} + 6 q^{19} - 12 q^{22} - 6 q^{23} - 8 q^{24} + 12 q^{25} + 6 q^{29} - 12 q^{30} + 2 q^{31} - 12 q^{33} + 4 q^{34} - 4 q^{37} - 12 q^{38} + 8 q^{40} - 12 q^{41} - 10 q^{43} + 6 q^{45} + 8 q^{46} - 6 q^{47} + 24 q^{50} + 4 q^{51} - 6 q^{53} - 16 q^{54} + 12 q^{55} - 12 q^{57} + 8 q^{58} - 12 q^{59} - 12 q^{61} + 12 q^{62} + 16 q^{64} - 6 q^{65} - 12 q^{67} + 8 q^{69} - 12 q^{71} + 10 q^{73} - 12 q^{74} + 24 q^{75} + 4 q^{78} + 14 q^{79} + 24 q^{80} - 10 q^{81} + 8 q^{82} - 18 q^{83} - 4 q^{85} + 8 q^{87} + 24 q^{88} - 6 q^{89} + 4 q^{90} + 12 q^{93} - 4 q^{94} - 6 q^{95} + 2 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 + 4 * q^6 - 2 * q^9 - 4 * q^10 + 2 * q^13 - 4 * q^15 - 8 * q^16 + 6 * q^19 - 12 * q^22 - 6 * q^23 - 8 * q^24 + 12 * q^25 + 6 * q^29 - 12 * q^30 + 2 * q^31 - 12 * q^33 + 4 * q^34 - 4 * q^37 - 12 * q^38 + 8 * q^40 - 12 * q^41 - 10 * q^43 + 6 * q^45 + 8 * q^46 - 6 * q^47 + 24 * q^50 + 4 * q^51 - 6 * q^53 - 16 * q^54 + 12 * q^55 - 12 * q^57 + 8 * q^58 - 12 * q^59 - 12 * q^61 + 12 * q^62 + 16 * q^64 - 6 * q^65 - 12 * q^67 + 8 * q^69 - 12 * q^71 + 10 * q^73 - 12 * q^74 + 24 * q^75 + 4 * q^78 + 14 * q^79 + 24 * q^80 - 10 * q^81 + 8 * q^82 - 18 * q^83 - 4 * q^85 + 8 * q^87 + 24 * q^88 - 6 * q^89 + 4 * q^90 + 12 * q^93 - 4 * q^94 - 6 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more