LMFDB - Embedded newform 931.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [931,2,Mod(1,931)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(931, 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("931.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 931 = 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	931.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:	$7.43407242818$
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 133)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	931.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.41421 q^{3} +2.82843 q^{5} +2.00000 q^{6} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.41421 q^{3} +2.82843 q^{5} +2.00000 q^{6} +2.82843 q^{8} -1.00000 q^{9} -4.00000 q^{10} -0.171573 q^{11} -6.24264 q^{13} -4.00000 q^{15} -4.00000 q^{16} +3.00000 q^{17} +1.41421 q^{18} -1.00000 q^{19} +0.242641 q^{22} +3.00000 q^{23} -4.00000 q^{24} +3.00000 q^{25} +8.82843 q^{26} +5.65685 q^{27} -3.17157 q^{29} +5.65685 q^{30} +2.24264 q^{31} +0.242641 q^{33} -4.24264 q^{34} -4.00000 q^{37} +1.41421 q^{38} +8.82843 q^{39} +8.00000 q^{40} +8.82843 q^{41} -10.4853 q^{43} -2.82843 q^{45} -4.24264 q^{46} +6.17157 q^{47} +5.65685 q^{48} -4.24264 q^{50} -4.24264 q^{51} -13.4142 q^{53} -8.00000 q^{54} -0.485281 q^{55} +1.41421 q^{57} +4.48528 q^{58} -10.2426 q^{59} -11.4853 q^{61} -3.17157 q^{62} +8.00000 q^{64} -17.6569 q^{65} -0.343146 q^{66} -2.00000 q^{67} -4.24264 q^{69} -13.0711 q^{71} -2.82843 q^{72} +3.00000 q^{73} +5.65685 q^{74} -4.24264 q^{75} -12.4853 q^{78} +3.75736 q^{79} -11.3137 q^{80} -5.00000 q^{81} -12.4853 q^{82} +5.48528 q^{83} +8.48528 q^{85} +14.8284 q^{86} +4.48528 q^{87} -0.485281 q^{88} -6.00000 q^{89} +4.00000 q^{90} -3.17157 q^{93} -8.72792 q^{94} -2.82843 q^{95} -6.24264 q^{97} +0.171573 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^6 - 2 * q^9	$ 2 q + 4 q^{6} - 2 q^{9} - 8 q^{10} - 6 q^{11} - 4 q^{13} - 8 q^{15} - 8 q^{16} + 6 q^{17} - 2 q^{19} - 8 q^{22} + 6 q^{23} - 8 q^{24} + 6 q^{25} + 12 q^{26} - 12 q^{29} - 4 q^{31} - 8 q^{33} - 8 q^{37} + 12 q^{39} + 16 q^{40} + 12 q^{41} - 4 q^{43} + 18 q^{47} - 24 q^{53} - 16 q^{54} + 16 q^{55} - 8 q^{58} - 12 q^{59} - 6 q^{61} - 12 q^{62} + 16 q^{64} - 24 q^{65} - 12 q^{66} - 4 q^{67} - 12 q^{71} + 6 q^{73} - 8 q^{78} + 16 q^{79} - 10 q^{81} - 8 q^{82} - 6 q^{83} + 24 q^{86} - 8 q^{87} + 16 q^{88} - 12 q^{89} + 8 q^{90} - 12 q^{93} + 8 q^{94} - 4 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q + 4 * q^6 - 2 * q^9 - 8 * q^10 - 6 * q^11 - 4 * q^13 - 8 * q^15 - 8 * q^16 + 6 * q^17 - 2 * q^19 - 8 * q^22 + 6 * q^23 - 8 * q^24 + 6 * q^25 + 12 * q^26 - 12 * q^29 - 4 * q^31 - 8 * q^33 - 8 * q^37 + 12 * q^39 + 16 * q^40 + 12 * q^41 - 4 * q^43 + 18 * q^47 - 24 * q^53 - 16 * q^54 + 16 * q^55 - 8 * q^58 - 12 * q^59 - 6 * q^61 - 12 * q^62 + 16 * q^64 - 24 * q^65 - 12 * q^66 - 4 * q^67 - 12 * q^71 + 6 * q^73 - 8 * q^78 + 16 * q^79 - 10 * q^81 - 8 * q^82 - 6 * q^83 + 24 * q^86 - 8 * q^87 + 16 * q^88 - 12 * q^89 + 8 * q^90 - 12 * q^93 + 8 * q^94 - 4 * q^97 + 6 * q^99

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
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	2.00000	0.816497
$7$	0	0
$7$	0	0
$8$	2.82843	1.00000
$9$	−1.00000	−0.333333
$10$	−4.00000	−1.26491
$11$	−0.171573	−0.0517312	−0.0258656	−	0.999665i	$-0.508234\pi$
$11$	−0.171573	−0.0517312	−0.0258656	+	0.999665i	$0.508234\pi$
$12$	0	0
$13$	−6.24264	−1.73140	−0.865699	−	0.500566i	$-0.833125\pi$
$13$	−6.24264	−1.73140	−0.865699	+	0.500566i	$0.833125\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	−4.00000	−1.00000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	1.41421	0.333333
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0.242641	0.0517312
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	−4.00000	−0.816497
$25$	3.00000	0.600000
$26$	8.82843	1.73140
$27$	5.65685	1.08866
$28$	0	0
$29$	−3.17157	−0.588946	−0.294473	−	0.955660i	$-0.595144\pi$
$29$	−3.17157	−0.588946	−0.294473	+	0.955660i	$0.595144\pi$
$30$	5.65685	1.03280
$31$	2.24264	0.402790	0.201395	−	0.979510i	$-0.435452\pi$
$31$	2.24264	0.402790	0.201395	+	0.979510i	$0.435452\pi$
$32$	0	0
$33$	0.242641	0.0422383
$34$	−4.24264	−0.727607
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	1.41421	0.229416
$39$	8.82843	1.41368
$40$	8.00000	1.26491
$41$	8.82843	1.37877	0.689384	−	0.724396i	$-0.257881\pi$
$41$	8.82843	1.37877	0.689384	+	0.724396i	$0.257881\pi$
$42$	0	0
$43$	−10.4853	−1.59899	−0.799495	−	0.600672i	$-0.794900\pi$
$43$	−10.4853	−1.59899	−0.799495	+	0.600672i	$0.794900\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	−4.24264	−0.625543
$47$	6.17157	0.900216	0.450108	−	0.892974i	$-0.351385\pi$
$47$	6.17157	0.900216	0.450108	+	0.892974i	$0.351385\pi$
$48$	5.65685	0.816497
$49$	0	0
$50$	−4.24264	−0.600000
$51$	−4.24264	−0.594089
$52$	0	0
$53$	−13.4142	−1.84258	−0.921292	−	0.388872i	$-0.872865\pi$
$53$	−13.4142	−1.84258	−0.921292	+	0.388872i	$0.872865\pi$
$54$	−8.00000	−1.08866
$55$	−0.485281	−0.0654353
$56$	0	0
$57$	1.41421	0.187317
$58$	4.48528	0.588946
$59$	−10.2426	−1.33348	−0.666739	−	0.745291i	$-0.732310\pi$
$59$	−10.2426	−1.33348	−0.666739	+	0.745291i	$0.732310\pi$
$60$	0	0
$61$	−11.4853	−1.47054	−0.735270	−	0.677775i	$-0.762945\pi$
$61$	−11.4853	−1.47054	−0.735270	+	0.677775i	$0.762945\pi$
$62$	−3.17157	−0.402790
$63$	0	0
$64$	8.00000	1.00000
$65$	−17.6569	−2.19006
$66$	−0.343146	−0.0422383
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	−4.24264	−0.510754
$70$	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
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	5.65685	0.657596
$75$	−4.24264	−0.489898
$76$	0	0
$77$	0	0
$78$	−12.4853	−1.41368
$79$	3.75736	0.422736	0.211368	−	0.977407i	$-0.432208\pi$
$79$	3.75736	0.422736	0.211368	+	0.977407i	$0.432208\pi$
$80$	−11.3137	−1.26491
$81$	−5.00000	−0.555556
$82$	−12.4853	−1.37877
$83$	5.48528	0.602088	0.301044	−	0.953610i	$-0.402665\pi$
$83$	5.48528	0.602088	0.301044	+	0.953610i	$0.402665\pi$
$84$	0	0
$85$	8.48528	0.920358
$86$	14.8284	1.59899
$87$	4.48528	0.480873
$88$	−0.485281	−0.0517312
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	4.00000	0.421637
$91$	0	0
$92$	0	0
$93$	−3.17157	−0.328877
$94$	−8.72792	−0.900216
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−6.24264	−0.633844	−0.316922	−	0.948452i	$-0.602649\pi$
$97$	−6.24264	−0.633844	−0.316922	+	0.948452i	$0.602649\pi$
$98$	0	0
$99$	0.171573	0.0172437
$100$	0	0
$101$	−5.82843	−0.579950	−0.289975	−	0.957034i	$-0.593647\pi$
$101$	−5.82843	−0.579950	−0.289975	+	0.957034i	$0.593647\pi$
$102$	6.00000	0.594089
$103$	0.242641	0.0239081	0.0119540	−	0.999929i	$-0.496195\pi$
$103$	0.242641	0.0239081	0.0119540	+	0.999929i	$0.496195\pi$
$104$	−17.6569	−1.73140
$105$	0	0
$106$	18.9706	1.84258
$107$	−0.343146	−0.0331732	−0.0165866	−	0.999862i	$-0.505280\pi$
$107$	−0.343146	−0.0331732	−0.0165866	+	0.999862i	$0.505280\pi$
$108$	0	0
$109$	−14.4853	−1.38744	−0.693719	−	0.720246i	$-0.744029\pi$
$109$	−14.4853	−1.38744	−0.693719	+	0.720246i	$0.744029\pi$
$110$	0.686292	0.0654353
$111$	5.65685	0.536925
$112$	0	0
$113$	−4.24264	−0.399114	−0.199557	−	0.979886i	$-0.563950\pi$
$113$	−4.24264	−0.399114	−0.199557	+	0.979886i	$0.563950\pi$
$114$	−2.00000	−0.187317
$115$	8.48528	0.791257
$116$	0	0
$117$	6.24264	0.577132
$118$	14.4853	1.33348
$119$	0	0
$120$	−11.3137	−1.03280
$121$	−10.9706	−0.997324
$122$	16.2426	1.47054
$123$	−12.4853	−1.12576
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	0.485281	0.0430618	0.0215309	−	0.999768i	$-0.493146\pi$
$127$	0.485281	0.0430618	0.0215309	+	0.999768i	$0.493146\pi$
$128$	−11.3137	−1.00000
$129$	14.8284	1.30557
$130$	24.9706	2.19006
$131$	8.31371	0.726372	0.363186	−	0.931717i	$-0.381689\pi$
$131$	8.31371	0.726372	0.363186	+	0.931717i	$0.381689\pi$
$132$	0	0
$133$	0	0
$134$	2.82843	0.244339
$135$	16.0000	1.37706
$136$	8.48528	0.727607
$137$	8.65685	0.739605	0.369802	−	0.929110i	$-0.379425\pi$
$137$	8.65685	0.739605	0.369802	+	0.929110i	$0.379425\pi$
$138$	6.00000	0.510754
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	0	0
$141$	−8.72792	−0.735024
$142$	18.4853	1.55125
$143$	1.07107	0.0895672
$144$	4.00000	0.333333
$145$	−8.97056	−0.744965
$146$	−4.24264	−0.351123
$147$	0	0
$148$	0	0
$149$	−9.17157	−0.751365	−0.375682	−	0.926749i	$-0.622592\pi$
$149$	−9.17157	−0.751365	−0.375682	+	0.926749i	$0.622592\pi$
$150$	6.00000	0.489898
$151$	−20.7279	−1.68681	−0.843407	−	0.537275i	$-0.819454\pi$
$151$	−20.7279	−1.68681	−0.843407	+	0.537275i	$0.819454\pi$
$152$	−2.82843	−0.229416
$153$	−3.00000	−0.242536
$154$	0	0
$155$	6.34315	0.509494
$156$	0	0
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	−5.31371	−0.422736
$159$	18.9706	1.50446
$160$	0	0
$161$	0	0
$162$	7.07107	0.555556
$163$	15.4853	1.21290	0.606450	−	0.795121i	$-0.292593\pi$
$163$	15.4853	1.21290	0.606450	+	0.795121i	$0.292593\pi$
$164$	0	0
$165$	0.686292	0.0534277
$166$	−7.75736	−0.602088
$167$	1.07107	0.0828817	0.0414409	−	0.999141i	$-0.486805\pi$
$167$	1.07107	0.0828817	0.0414409	+	0.999141i	$0.486805\pi$
$168$	0	0
$169$	25.9706	1.99774
$170$	−12.0000	−0.920358
$171$	1.00000	0.0764719
$172$	0	0
$173$	−4.92893	−0.374740	−0.187370	−	0.982289i	$-0.559996\pi$
$173$	−4.92893	−0.374740	−0.187370	+	0.982289i	$0.559996\pi$
$174$	−6.34315	−0.480873
$175$	0	0
$176$	0.686292	0.0517312
$177$	14.4853	1.08878
$178$	8.48528	0.635999
$179$	−4.58579	−0.342758	−0.171379	−	0.985205i	$-0.554822\pi$
$179$	−4.58579	−0.342758	−0.171379	+	0.985205i	$0.554822\pi$
$180$	0	0
$181$	−12.4853	−0.928024	−0.464012	−	0.885829i	$-0.653590\pi$
$181$	−12.4853	−0.928024	−0.464012	+	0.885829i	$0.653590\pi$
$182$	0	0
$183$	16.2426	1.20069
$184$	8.48528	0.625543
$185$	−11.3137	−0.831800
$186$	4.48528	0.328877
$187$	−0.514719	−0.0376400
$188$	0	0
$189$	0	0
$190$	4.00000	0.290191
$191$	−20.1421	−1.45743	−0.728717	−	0.684815i	$-0.759883\pi$
$191$	−20.1421	−1.45743	−0.728717	+	0.684815i	$0.759883\pi$
$192$	−11.3137	−0.816497
$193$	16.7279	1.20410	0.602051	−	0.798458i	$-0.294350\pi$
$193$	16.7279	1.20410	0.602051	+	0.798458i	$0.294350\pi$
$194$	8.82843	0.633844
$195$	24.9706	1.78818
$196$	0	0
$197$	−17.4853	−1.24577	−0.622887	−	0.782312i	$-0.714041\pi$
$197$	−17.4853	−1.24577	−0.622887	+	0.782312i	$0.714041\pi$
$198$	−0.242641	−0.0172437
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	8.48528	0.600000
$201$	2.82843	0.199502
$202$	8.24264	0.579950
$203$	0	0
$204$	0	0
$205$	24.9706	1.74402
$206$	−0.343146	−0.0239081
$207$	−3.00000	−0.208514
$208$	24.9706	1.73140
$209$	0.171573	0.0118679
$210$	0	0
$211$	20.7279	1.42697	0.713485	−	0.700671i	$-0.247116\pi$
$211$	20.7279	1.42697	0.713485	+	0.700671i	$0.247116\pi$
$212$	0	0
$213$	18.4853	1.26659
$214$	0.485281	0.0331732
$215$	−29.6569	−2.02258
$216$	16.0000	1.08866
$217$	0	0
$218$	20.4853	1.38744
$219$	−4.24264	−0.286691
$220$	0	0
$221$	−18.7279	−1.25978
$222$	−8.00000	−0.536925
$223$	−1.51472	−0.101433	−0.0507165	−	0.998713i	$-0.516151\pi$
$223$	−1.51472	−0.101433	−0.0507165	+	0.998713i	$0.516151\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	6.00000	0.399114
$227$	15.1716	1.00697	0.503486	−	0.864003i	$-0.332050\pi$
$227$	15.1716	1.00697	0.503486	+	0.864003i	$0.332050\pi$
$228$	0	0
$229$	−25.4853	−1.68411	−0.842057	−	0.539388i	$-0.818656\pi$
$229$	−25.4853	−1.68411	−0.842057	+	0.539388i	$0.818656\pi$
$230$	−12.0000	−0.791257
$231$	0	0
$232$	−8.97056	−0.588946
$233$	−16.7990	−1.10054	−0.550269	−	0.834987i	$-0.685475\pi$
$233$	−16.7990	−1.10054	−0.550269	+	0.834987i	$0.685475\pi$
$234$	−8.82843	−0.577132
$235$	17.4558	1.13869
$236$	0	0
$237$	−5.31371	−0.345162
$238$	0	0
$239$	2.31371	0.149661	0.0748307	−	0.997196i	$-0.476158\pi$
$239$	2.31371	0.149661	0.0748307	+	0.997196i	$0.476158\pi$
$240$	16.0000	1.03280
$241$	−16.4853	−1.06191	−0.530955	−	0.847400i	$-0.678167\pi$
$241$	−16.4853	−1.06191	−0.530955	+	0.847400i	$0.678167\pi$
$242$	15.5147	0.997324
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	17.6569	1.12576
$247$	6.24264	0.397210
$248$	6.34315	0.402790
$249$	−7.75736	−0.491603
$250$	8.00000	0.505964
$251$	−28.6274	−1.80695	−0.903473	−	0.428644i	$-0.858991\pi$
$251$	−28.6274	−1.80695	−0.903473	+	0.428644i	$0.858991\pi$
$252$	0	0
$253$	−0.514719	−0.0323601
$254$	−0.686292	−0.0430618
$255$	−12.0000	−0.751469
$256$	0	0
$257$	12.7279	0.793946	0.396973	−	0.917830i	$-0.370061\pi$
$257$	12.7279	0.793946	0.396973	+	0.917830i	$0.370061\pi$
$258$	−20.9706	−1.30557
$259$	0	0
$260$	0	0
$261$	3.17157	0.196315
$262$	−11.7574	−0.726372
$263$	28.4558	1.75466	0.877331	−	0.479885i	$-0.159322\pi$
$263$	28.4558	1.75466	0.877331	+	0.479885i	$0.159322\pi$
$264$	0.686292	0.0422383
$265$	−37.9411	−2.33070
$266$	0	0
$267$	8.48528	0.519291
$268$	0	0
$269$	8.48528	0.517357	0.258678	−	0.965964i	$-0.416713\pi$
$269$	8.48528	0.517357	0.258678	+	0.965964i	$0.416713\pi$
$270$	−22.6274	−1.37706
$271$	27.4853	1.66961	0.834806	−	0.550544i	$-0.185580\pi$
$271$	27.4853	1.66961	0.834806	+	0.550544i	$0.185580\pi$
$272$	−12.0000	−0.727607
$273$	0	0
$274$	−12.2426	−0.739605
$275$	−0.514719	−0.0310387
$276$	0	0
$277$	29.9706	1.80076	0.900378	−	0.435108i	$-0.143290\pi$
$277$	29.9706	1.80076	0.900378	+	0.435108i	$0.143290\pi$
$278$	−9.89949	−0.593732
$279$	−2.24264	−0.134263
$280$	0	0
$281$	13.4142	0.800225	0.400112	−	0.916466i	$-0.368971\pi$
$281$	13.4142	0.800225	0.400112	+	0.916466i	$0.368971\pi$
$282$	12.3431	0.735024
$283$	−2.48528	−0.147735	−0.0738673	−	0.997268i	$-0.523534\pi$
$283$	−2.48528	−0.147735	−0.0738673	+	0.997268i	$0.523534\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	−1.51472	−0.0895672
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	12.6863	0.744965
$291$	8.82843	0.517532
$292$	0	0
$293$	10.9706	0.640907	0.320454	−	0.947264i	$-0.396165\pi$
$293$	10.9706	0.640907	0.320454	+	0.947264i	$0.396165\pi$
$294$	0	0
$295$	−28.9706	−1.68673
$296$	−11.3137	−0.657596
$297$	−0.970563	−0.0563178
$298$	12.9706	0.751365
$299$	−18.7279	−1.08306
$300$	0	0
$301$	0	0
$302$	29.3137	1.68681
$303$	8.24264	0.473527
$304$	4.00000	0.229416
$305$	−32.4853	−1.86010
$306$	4.24264	0.242536
$307$	20.2426	1.15531	0.577654	−	0.816282i	$-0.303968\pi$
$307$	20.2426	1.15531	0.577654	+	0.816282i	$0.303968\pi$
$308$	0	0
$309$	−0.343146	−0.0195209
$310$	−8.97056	−0.509494
$311$	0.857864	0.0486450	0.0243225	−	0.999704i	$-0.492257\pi$
$311$	0.857864	0.0486450	0.0243225	+	0.999704i	$0.492257\pi$
$312$	24.9706	1.41368
$313$	−22.4558	−1.26928	−0.634640	−	0.772808i	$-0.718851\pi$
$313$	−22.4558	−1.26928	−0.634640	+	0.772808i	$0.718851\pi$
$314$	−4.24264	−0.239426
$315$	0	0
$316$	0	0
$317$	27.2132	1.52845	0.764223	−	0.644952i	$-0.223123\pi$
$317$	27.2132	1.52845	0.764223	+	0.644952i	$0.223123\pi$
$318$	−26.8284	−1.50446
$319$	0.544156	0.0304669
$320$	22.6274	1.26491
$321$	0.485281	0.0270858
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	−18.7279	−1.03884
$326$	−21.8995	−1.21290
$327$	20.4853	1.13284
$328$	24.9706	1.37877
$329$	0	0
$330$	−0.970563	−0.0534277
$331$	14.7279	0.809520	0.404760	−	0.914423i	$-0.367355\pi$
$331$	14.7279	0.809520	0.404760	+	0.914423i	$0.367355\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	−1.51472	−0.0828817
$335$	−5.65685	−0.309067
$336$	0	0
$337$	−16.7279	−0.911228	−0.455614	−	0.890177i	$-0.650580\pi$
$337$	−16.7279	−0.911228	−0.455614	+	0.890177i	$0.650580\pi$
$338$	−36.7279	−1.99774
$339$	6.00000	0.325875
$340$	0	0
$341$	−0.384776	−0.0208368
$342$	−1.41421	−0.0764719
$343$	0	0
$344$	−29.6569	−1.59899
$345$	−12.0000	−0.646058
$346$	6.97056	0.374740
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	16.4558	0.880861	0.440431	−	0.897787i	$-0.354826\pi$
$349$	16.4558	0.880861	0.440431	+	0.897787i	$0.354826\pi$
$350$	0	0
$351$	−35.3137	−1.88491
$352$	0	0
$353$	−11.3137	−0.602168	−0.301084	−	0.953598i	$-0.597348\pi$
$353$	−11.3137	−0.602168	−0.301084	+	0.953598i	$0.597348\pi$
$354$	−20.4853	−1.08878
$355$	−36.9706	−1.96219
$356$	0	0
$357$	0	0
$358$	6.48528	0.342758
$359$	5.14214	0.271392	0.135696	−	0.990751i	$-0.456673\pi$
$359$	5.14214	0.271392	0.135696	+	0.990751i	$0.456673\pi$
$360$	−8.00000	−0.421637
$361$	1.00000	0.0526316
$362$	17.6569	0.928024
$363$	15.5147	0.814312
$364$	0	0
$365$	8.48528	0.444140
$366$	−22.9706	−1.20069
$367$	6.97056	0.363860	0.181930	−	0.983311i	$-0.441766\pi$
$367$	6.97056	0.363860	0.181930	+	0.983311i	$0.441766\pi$
$368$	−12.0000	−0.625543
$369$	−8.82843	−0.459590
$370$	16.0000	0.831800
$371$	0	0
$372$	0	0
$373$	−22.7279	−1.17681	−0.588404	−	0.808567i	$-0.700243\pi$
$373$	−22.7279	−1.17681	−0.588404	+	0.808567i	$0.700243\pi$
$374$	0.727922	0.0376400
$375$	8.00000	0.413118
$376$	17.4558	0.900216
$377$	19.7990	1.01970
$378$	0	0
$379$	3.51472	0.180539	0.0902695	−	0.995917i	$-0.471227\pi$
$379$	3.51472	0.180539	0.0902695	+	0.995917i	$0.471227\pi$
$380$	0	0
$381$	−0.686292	−0.0351598
$382$	28.4853	1.45743
$383$	3.89949	0.199255	0.0996274	−	0.995025i	$-0.468235\pi$
$383$	3.89949	0.199255	0.0996274	+	0.995025i	$0.468235\pi$
$384$	16.0000	0.816497
$385$	0	0
$386$	−23.6569	−1.20410
$387$	10.4853	0.532997
$388$	0	0
$389$	−17.8284	−0.903937	−0.451969	−	0.892034i	$-0.649278\pi$
$389$	−17.8284	−0.903937	−0.451969	+	0.892034i	$0.649278\pi$
$390$	−35.3137	−1.78818
$391$	9.00000	0.455150
$392$	0	0
$393$	−11.7574	−0.593080
$394$	24.7279	1.24577
$395$	10.6274	0.534723
$396$	0	0
$397$	29.0000	1.45547	0.727734	−	0.685859i	$-0.240573\pi$
$397$	29.0000	1.45547	0.727734	+	0.685859i	$0.240573\pi$
$398$	−14.1421	−0.708881
$399$	0	0
$400$	−12.0000	−0.600000
$401$	−8.10051	−0.404520	−0.202260	−	0.979332i	$-0.564829\pi$
$401$	−8.10051	−0.404520	−0.202260	+	0.979332i	$0.564829\pi$
$402$	−4.00000	−0.199502
$403$	−14.0000	−0.697390
$404$	0	0
$405$	−14.1421	−0.702728
$406$	0	0
$407$	0.686292	0.0340182
$408$	−12.0000	−0.594089
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	−35.3137	−1.74402
$411$	−12.2426	−0.603885
$412$	0	0
$413$	0	0
$414$	4.24264	0.208514
$415$	15.5147	0.761588
$416$	0	0
$417$	−9.89949	−0.484780
$418$	−0.242641	−0.0118679
$419$	21.0000	1.02592	0.512959	−	0.858413i	$-0.328549\pi$
$419$	21.0000	1.02592	0.512959	+	0.858413i	$0.328549\pi$
$420$	0	0
$421$	−8.97056	−0.437198	−0.218599	−	0.975815i	$-0.570149\pi$
$421$	−8.97056	−0.437198	−0.218599	+	0.975815i	$0.570149\pi$
$422$	−29.3137	−1.42697
$423$	−6.17157	−0.300072
$424$	−37.9411	−1.84258
$425$	9.00000	0.436564
$426$	−26.1421	−1.26659
$427$	0	0
$428$	0	0
$429$	−1.51472	−0.0731313
$430$	41.9411	2.02258
$431$	−16.6274	−0.800914	−0.400457	−	0.916315i	$-0.631149\pi$
$431$	−16.6274	−0.800914	−0.400457	+	0.916315i	$0.631149\pi$
$432$	−22.6274	−1.08866
$433$	23.4558	1.12722	0.563608	−	0.826042i	$-0.309413\pi$
$433$	23.4558	1.12722	0.563608	+	0.826042i	$0.309413\pi$
$434$	0	0
$435$	12.6863	0.608261
$436$	0	0
$437$	−3.00000	−0.143509
$438$	6.00000	0.286691
$439$	−18.4853	−0.882254	−0.441127	−	0.897445i	$-0.645421\pi$
$439$	−18.4853	−0.882254	−0.441127	+	0.897445i	$0.645421\pi$
$440$	−1.37258	−0.0654353
$441$	0	0
$442$	26.4853	1.25978
$443$	11.8284	0.561986	0.280993	−	0.959710i	$-0.409336\pi$
$443$	11.8284	0.561986	0.280993	+	0.959710i	$0.409336\pi$
$444$	0	0
$445$	−16.9706	−0.804482
$446$	2.14214	0.101433
$447$	12.9706	0.613487
$448$	0	0
$449$	−10.9289	−0.515768	−0.257884	−	0.966176i	$-0.583025\pi$
$449$	−10.9289	−0.515768	−0.257884	+	0.966176i	$0.583025\pi$
$450$	4.24264	0.200000
$451$	−1.51472	−0.0713253
$452$	0	0
$453$	29.3137	1.37728
$454$	−21.4558	−1.00697
$455$	0	0
$456$	4.00000	0.187317
$457$	4.51472	0.211190	0.105595	−	0.994409i	$-0.466325\pi$
$457$	4.51472	0.211190	0.105595	+	0.994409i	$0.466325\pi$
$458$	36.0416	1.68411
$459$	16.9706	0.792118
$460$	0	0
$461$	−3.34315	−0.155706	−0.0778529	−	0.996965i	$-0.524806\pi$
$461$	−3.34315	−0.155706	−0.0778529	+	0.996965i	$0.524806\pi$
$462$	0	0
$463$	19.9706	0.928111	0.464055	−	0.885806i	$-0.346394\pi$
$463$	19.9706	0.928111	0.464055	+	0.885806i	$0.346394\pi$
$464$	12.6863	0.588946
$465$	−8.97056	−0.416000
$466$	23.7574	1.10054
$467$	−4.79899	−0.222071	−0.111035	−	0.993816i	$-0.535417\pi$
$467$	−4.79899	−0.222071	−0.111035	+	0.993816i	$0.535417\pi$
$468$	0	0
$469$	0	0
$470$	−24.6863	−1.13869
$471$	−4.24264	−0.195491
$472$	−28.9706	−1.33348
$473$	1.79899	0.0827176
$474$	7.51472	0.345162
$475$	−3.00000	−0.137649
$476$	0	0
$477$	13.4142	0.614195
$478$	−3.27208	−0.149661
$479$	11.8284	0.540455	0.270227	−	0.962797i	$-0.412901\pi$
$479$	11.8284	0.540455	0.270227	+	0.962797i	$0.412901\pi$
$480$	0	0
$481$	24.9706	1.13856
$482$	23.3137	1.06191
$483$	0	0
$484$	0	0
$485$	−17.6569	−0.801756
$486$	14.0000	0.635053
$487$	−34.7279	−1.57367	−0.786836	−	0.617162i	$-0.788282\pi$
$487$	−34.7279	−1.57367	−0.786836	+	0.617162i	$0.788282\pi$
$488$	−32.4853	−1.47054
$489$	−21.8995	−0.990329
$490$	0	0
$491$	−15.3431	−0.692426	−0.346213	−	0.938156i	$-0.612533\pi$
$491$	−15.3431	−0.692426	−0.346213	+	0.938156i	$0.612533\pi$
$492$	0	0
$493$	−9.51472	−0.428521
$494$	−8.82843	−0.397210
$495$	0.485281	0.0218118
$496$	−8.97056	−0.402790
$497$	0	0
$498$	10.9706	0.491603
$499$	6.97056	0.312045	0.156023	−	0.987753i	$-0.450133\pi$
$499$	6.97056	0.312045	0.156023	+	0.987753i	$0.450133\pi$
$500$	0	0
$501$	−1.51472	−0.0676726
$502$	40.4853	1.80695
$503$	−7.62742	−0.340090	−0.170045	−	0.985436i	$-0.554391\pi$
$503$	−7.62742	−0.340090	−0.170045	+	0.985436i	$0.554391\pi$
$504$	0	0
$505$	−16.4853	−0.733585
$506$	0.727922	0.0323601
$507$	−36.7279	−1.63114
$508$	0	0
$509$	9.51472	0.421732	0.210866	−	0.977515i	$-0.432372\pi$
$509$	9.51472	0.421732	0.210866	+	0.977515i	$0.432372\pi$
$510$	16.9706	0.751469
$511$	0	0
$512$	22.6274	1.00000
$513$	−5.65685	−0.249756
$514$	−18.0000	−0.793946
$515$	0.686292	0.0302416
$516$	0	0
$517$	−1.05887	−0.0465692
$518$	0	0
$519$	6.97056	0.305974
$520$	−49.9411	−2.19006
$521$	−1.41421	−0.0619578	−0.0309789	−	0.999520i	$-0.509862\pi$
$521$	−1.41421	−0.0619578	−0.0309789	+	0.999520i	$0.509862\pi$
$522$	−4.48528	−0.196315
$523$	−21.5147	−0.940773	−0.470386	−	0.882461i	$-0.655885\pi$
$523$	−21.5147	−0.940773	−0.470386	+	0.882461i	$0.655885\pi$
$524$	0	0
$525$	0	0
$526$	−40.2426	−1.75466
$527$	6.72792	0.293073
$528$	−0.970563	−0.0422383
$529$	−14.0000	−0.608696
$530$	53.6569	2.33070
$531$	10.2426	0.444493
$532$	0	0
$533$	−55.1127	−2.38720
$534$	−12.0000	−0.519291
$535$	−0.970563	−0.0419611
$536$	−5.65685	−0.244339
$537$	6.48528	0.279861
$538$	−12.0000	−0.517357
$539$	0	0
$540$	0	0
$541$	−28.9706	−1.24554	−0.622771	−	0.782404i	$-0.713993\pi$
$541$	−28.9706	−1.24554	−0.622771	+	0.782404i	$0.713993\pi$
$542$	−38.8701	−1.66961
$543$	17.6569	0.757728
$544$	0	0
$545$	−40.9706	−1.75499
$546$	0	0
$547$	5.02944	0.215043	0.107522	−	0.994203i	$-0.465709\pi$
$547$	5.02944	0.215043	0.107522	+	0.994203i	$0.465709\pi$
$548$	0	0
$549$	11.4853	0.490180
$550$	0.727922	0.0310387
$551$	3.17157	0.135114
$552$	−12.0000	−0.510754
$553$	0	0
$554$	−42.3848	−1.80076
$555$	16.0000	0.679162
$556$	0	0
$557$	14.1421	0.599222	0.299611	−	0.954062i	$-0.403143\pi$
$557$	14.1421	0.599222	0.299611	+	0.954062i	$0.403143\pi$
$558$	3.17157	0.134263
$559$	65.4558	2.76849
$560$	0	0
$561$	0.727922	0.0307329
$562$	−18.9706	−0.800225
$563$	−45.5563	−1.91997	−0.959986	−	0.280049i	$-0.909649\pi$
$563$	−45.5563	−1.91997	−0.959986	+	0.280049i	$0.909649\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	3.51472	0.147735
$567$	0	0
$568$	−36.9706	−1.55125
$569$	26.1421	1.09594	0.547968	−	0.836500i	$-0.315402\pi$
$569$	26.1421	1.09594	0.547968	+	0.836500i	$0.315402\pi$
$570$	−5.65685	−0.236940
$571$	45.4853	1.90350	0.951750	−	0.306875i	$-0.0992833\pi$
$571$	45.4853	1.90350	0.951750	+	0.306875i	$0.0992833\pi$
$572$	0	0
$573$	28.4853	1.18999
$574$	0	0
$575$	9.00000	0.375326
$576$	−8.00000	−0.333333
$577$	15.4853	0.644661	0.322330	−	0.946627i	$-0.395534\pi$
$577$	15.4853	0.644661	0.322330	+	0.946627i	$0.395534\pi$
$578$	11.3137	0.470588
$579$	−23.6569	−0.983145
$580$	0	0
$581$	0	0
$582$	−12.4853	−0.517532
$583$	2.30152	0.0953190
$584$	8.48528	0.351123
$585$	17.6569	0.730021
$586$	−15.5147	−0.640907
$587$	−14.3137	−0.590790	−0.295395	−	0.955375i	$-0.595451\pi$
$587$	−14.3137	−0.590790	−0.295395	+	0.955375i	$0.595451\pi$
$588$	0	0
$589$	−2.24264	−0.0924064
$590$	40.9706	1.68673
$591$	24.7279	1.01717
$592$	16.0000	0.657596
$593$	25.9706	1.06648	0.533242	−	0.845963i	$-0.320974\pi$
$593$	25.9706	1.06648	0.533242	+	0.845963i	$0.320974\pi$
$594$	1.37258	0.0563178
$595$	0	0
$596$	0	0
$597$	−14.1421	−0.578799
$598$	26.4853	1.08306
$599$	31.7990	1.29927	0.649636	−	0.760246i	$-0.274921\pi$
$599$	31.7990	1.29927	0.649636	+	0.760246i	$0.274921\pi$
$600$	−12.0000	−0.489898
$601$	−21.9411	−0.894997	−0.447499	−	0.894285i	$-0.647685\pi$
$601$	−21.9411	−0.894997	−0.447499	+	0.894285i	$0.647685\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	−31.0294	−1.26153
$606$	−11.6569	−0.473527
$607$	−23.2132	−0.942195	−0.471097	−	0.882081i	$-0.656142\pi$
$607$	−23.2132	−0.942195	−0.471097	+	0.882081i	$0.656142\pi$
$608$	0	0
$609$	0	0
$610$	45.9411	1.86010
$611$	−38.5269	−1.55863
$612$	0	0
$613$	1.97056	0.0795903	0.0397951	−	0.999208i	$-0.487329\pi$
$613$	1.97056	0.0795903	0.0397951	+	0.999208i	$0.487329\pi$
$614$	−28.6274	−1.15531
$615$	−35.3137	−1.42399
$616$	0	0
$617$	47.6569	1.91859	0.959296	−	0.282401i	$-0.0911310\pi$
$617$	47.6569	1.91859	0.959296	+	0.282401i	$0.0911310\pi$
$618$	0.485281	0.0195209
$619$	−32.9411	−1.32402	−0.662008	−	0.749497i	$-0.730295\pi$
$619$	−32.9411	−1.32402	−0.662008	+	0.749497i	$0.730295\pi$
$620$	0	0
$621$	16.9706	0.681005
$622$	−1.21320	−0.0486450
$623$	0	0
$624$	−35.3137	−1.41368
$625$	−31.0000	−1.24000
$626$	31.7574	1.26928
$627$	−0.242641	−0.00969014
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	26.9706	1.07368	0.536841	−	0.843684i	$-0.319618\pi$
$631$	26.9706	1.07368	0.536841	+	0.843684i	$0.319618\pi$
$632$	10.6274	0.422736
$633$	−29.3137	−1.16512
$634$	−38.4853	−1.52845
$635$	1.37258	0.0544693
$636$	0	0
$637$	0	0
$638$	−0.769553	−0.0304669
$639$	13.0711	0.517083
$640$	−32.0000	−1.26491
$641$	15.2132	0.600885	0.300443	−	0.953800i	$-0.402866\pi$
$641$	15.2132	0.600885	0.300443	+	0.953800i	$0.402866\pi$
$642$	−0.686292	−0.0270858
$643$	−8.51472	−0.335788	−0.167894	−	0.985805i	$-0.553697\pi$
$643$	−8.51472	−0.335788	−0.167894	+	0.985805i	$0.553697\pi$
$644$	0	0
$645$	41.9411	1.65143
$646$	4.24264	0.166924
$647$	−46.6274	−1.83311	−0.916556	−	0.399905i	$-0.869043\pi$
$647$	−46.6274	−1.83311	−0.916556	+	0.399905i	$0.869043\pi$
$648$	−14.1421	−0.555556
$649$	1.75736	0.0689824
$650$	26.4853	1.03884
$651$	0	0
$652$	0	0
$653$	5.14214	0.201227	0.100614	−	0.994926i	$-0.467919\pi$
$653$	5.14214	0.201227	0.100614	+	0.994926i	$0.467919\pi$
$654$	−28.9706	−1.13284
$655$	23.5147	0.918796
$656$	−35.3137	−1.37877
$657$	−3.00000	−0.117041
$658$	0	0
$659$	−13.0711	−0.509177	−0.254588	−	0.967050i	$-0.581940\pi$
$659$	−13.0711	−0.509177	−0.254588	+	0.967050i	$0.581940\pi$
$660$	0	0
$661$	44.2426	1.72084	0.860420	−	0.509586i	$-0.170201\pi$
$661$	44.2426	1.72084	0.860420	+	0.509586i	$0.170201\pi$
$662$	−20.8284	−0.809520
$663$	26.4853	1.02860
$664$	15.5147	0.602088
$665$	0	0
$666$	−5.65685	−0.219199
$667$	−9.51472	−0.368411
$668$	0	0
$669$	2.14214	0.0828197
$670$	8.00000	0.309067
$671$	1.97056	0.0760727
$672$	0	0
$673$	−1.02944	−0.0396819	−0.0198409	−	0.999803i	$-0.506316\pi$
$673$	−1.02944	−0.0396819	−0.0198409	+	0.999803i	$0.506316\pi$
$674$	23.6569	0.911228
$675$	16.9706	0.653197
$676$	0	0
$677$	38.1421	1.46592	0.732961	−	0.680271i	$-0.238138\pi$
$677$	38.1421	1.46592	0.732961	+	0.680271i	$0.238138\pi$
$678$	−8.48528	−0.325875
$679$	0	0
$680$	24.0000	0.920358
$681$	−21.4558	−0.822190
$682$	0.544156	0.0208368
$683$	−35.3553	−1.35283	−0.676417	−	0.736519i	$-0.736468\pi$
$683$	−35.3553	−1.35283	−0.676417	+	0.736519i	$0.736468\pi$
$684$	0	0
$685$	24.4853	0.935535
$686$	0	0
$687$	36.0416	1.37507
$688$	41.9411	1.59899
$689$	83.7401	3.19024
$690$	16.9706	0.646058
$691$	39.9706	1.52055	0.760276	−	0.649600i	$-0.225064\pi$
$691$	39.9706	1.52055	0.760276	+	0.649600i	$0.225064\pi$
$692$	0	0
$693$	0	0
$694$	−16.9706	−0.644194
$695$	19.7990	0.751018
$696$	12.6863	0.480873
$697$	26.4853	1.00320
$698$	−23.2721	−0.880861
$699$	23.7574	0.898586
$700$	0	0
$701$	17.8284	0.673370	0.336685	−	0.941617i	$-0.390694\pi$
$701$	17.8284	0.673370	0.336685	+	0.941617i	$0.390694\pi$
$702$	49.9411	1.88491
$703$	4.00000	0.150863
$704$	−1.37258	−0.0517312
$705$	−24.6863	−0.929740
$706$	16.0000	0.602168
$707$	0	0
$708$	0	0
$709$	−1.54416	−0.0579920	−0.0289960	−	0.999580i	$-0.509231\pi$
$709$	−1.54416	−0.0579920	−0.0289960	+	0.999580i	$0.509231\pi$
$710$	52.2843	1.96219
$711$	−3.75736	−0.140912
$712$	−16.9706	−0.635999
$713$	6.72792	0.251963
$714$	0	0
$715$	3.02944	0.113295
$716$	0	0
$717$	−3.27208	−0.122198
$718$	−7.27208	−0.271392
$719$	8.31371	0.310049	0.155025	−	0.987911i	$-0.450454\pi$
$719$	8.31371	0.310049	0.155025	+	0.987911i	$0.450454\pi$
$720$	11.3137	0.421637
$721$	0	0
$722$	−1.41421	−0.0526316
$723$	23.3137	0.867046
$724$	0	0
$725$	−9.51472	−0.353368
$726$	−21.9411	−0.814312
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	−12.0000	−0.444140
$731$	−31.4558	−1.16344
$732$	0	0
$733$	43.9411	1.62300	0.811501	−	0.584351i	$-0.198651\pi$
$733$	43.9411	1.62300	0.811501	+	0.584351i	$0.198651\pi$
$734$	−9.85786	−0.363860
$735$	0	0
$736$	0	0
$737$	0.343146	0.0126399
$738$	12.4853	0.459590
$739$	−46.4558	−1.70891	−0.854453	−	0.519529i	$-0.826107\pi$
$739$	−46.4558	−1.70891	−0.854453	+	0.519529i	$0.826107\pi$
$740$	0	0
$741$	−8.82843	−0.324320
$742$	0	0
$743$	0.0416306	0.00152728	0.000763639	−	1.00000i	$-0.499757\pi$
$743$	0.0416306	0.00152728	0.000763639		1.00000i	$0.499757\pi$
$744$	−8.97056	−0.328877
$745$	−25.9411	−0.950409
$746$	32.1421	1.17681
$747$	−5.48528	−0.200696
$748$	0	0
$749$	0	0
$750$	−11.3137	−0.413118
$751$	5.27208	0.192381	0.0961904	−	0.995363i	$-0.469334\pi$
$751$	5.27208	0.192381	0.0961904	+	0.995363i	$0.469334\pi$
$752$	−24.6863	−0.900216
$753$	40.4853	1.47537
$754$	−28.0000	−1.01970
$755$	−58.6274	−2.13367
$756$	0	0
$757$	−42.4558	−1.54308	−0.771542	−	0.636178i	$-0.780514\pi$
$757$	−42.4558	−1.54308	−0.771542	+	0.636178i	$0.780514\pi$
$758$	−4.97056	−0.180539
$759$	0.727922	0.0264219
$760$	−8.00000	−0.290191
$761$	29.3137	1.06262	0.531311	−	0.847177i	$-0.321700\pi$
$761$	29.3137	1.06262	0.531311	+	0.847177i	$0.321700\pi$
$762$	0.970563	0.0351598
$763$	0	0
$764$	0	0
$765$	−8.48528	−0.306786
$766$	−5.51472	−0.199255
$767$	63.9411	2.30878
$768$	0	0
$769$	9.45584	0.340986	0.170493	−	0.985359i	$-0.445464\pi$
$769$	9.45584	0.340986	0.170493	+	0.985359i	$0.445464\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	41.6569	1.49829	0.749146	−	0.662404i	$-0.230464\pi$
$773$	41.6569	1.49829	0.749146	+	0.662404i	$0.230464\pi$
$774$	−14.8284	−0.532997
$775$	6.72792	0.241674
$776$	−17.6569	−0.633844
$777$	0	0
$778$	25.2132	0.903937
$779$	−8.82843	−0.316311
$780$	0	0
$781$	2.24264	0.0802480
$782$	−12.7279	−0.455150
$783$	−17.9411	−0.641164
$784$	0	0
$785$	8.48528	0.302853
$786$	16.6274	0.593080
$787$	25.6985	0.916052	0.458026	−	0.888939i	$-0.348557\pi$
$787$	25.6985	0.916052	0.458026	+	0.888939i	$0.348557\pi$
$788$	0	0
$789$	−40.2426	−1.43268
$790$	−15.0294	−0.534723
$791$	0	0
$792$	0.485281	0.0172437
$793$	71.6985	2.54609
$794$	−41.0122	−1.45547
$795$	53.6569	1.90301
$796$	0	0
$797$	10.9706	0.388597	0.194299	−	0.980942i	$-0.437757\pi$
$797$	10.9706	0.388597	0.194299	+	0.980942i	$0.437757\pi$
$798$	0	0
$799$	18.5147	0.655004
$800$	0	0
$801$	6.00000	0.212000
$802$	11.4558	0.404520
$803$	−0.514719	−0.0181640
$804$	0	0
$805$	0	0
$806$	19.7990	0.697390
$807$	−12.0000	−0.422420
$808$	−16.4853	−0.579950
$809$	−7.79899	−0.274198	−0.137099	−	0.990557i	$-0.543778\pi$
$809$	−7.79899	−0.274198	−0.137099	+	0.990557i	$0.543778\pi$
$810$	20.0000	0.702728
$811$	−23.4558	−0.823646	−0.411823	−	0.911264i	$-0.635108\pi$
$811$	−23.4558	−0.823646	−0.411823	+	0.911264i	$0.635108\pi$
$812$	0	0
$813$	−38.8701	−1.36323
$814$	−0.970563	−0.0340182
$815$	43.7990	1.53421
$816$	16.9706	0.594089
$817$	10.4853	0.366834
$818$	5.65685	0.197787
$819$	0	0
$820$	0	0
$821$	−23.4853	−0.819642	−0.409821	−	0.912166i	$-0.634409\pi$
$821$	−23.4853	−0.819642	−0.409821	+	0.912166i	$0.634409\pi$
$822$	17.3137	0.603885
$823$	−7.48528	−0.260921	−0.130460	−	0.991454i	$-0.541645\pi$
$823$	−7.48528	−0.260921	−0.130460	+	0.991454i	$0.541645\pi$
$824$	0.686292	0.0239081
$825$	0.727922	0.0253430
$826$	0	0
$827$	−10.9289	−0.380036	−0.190018	−	0.981781i	$-0.560855\pi$
$827$	−10.9289	−0.380036	−0.190018	+	0.981781i	$0.560855\pi$
$828$	0	0
$829$	30.2426	1.05037	0.525185	−	0.850988i	$-0.323996\pi$
$829$	30.2426	1.05037	0.525185	+	0.850988i	$0.323996\pi$
$830$	−21.9411	−0.761588
$831$	−42.3848	−1.47031
$832$	−49.9411	−1.73140
$833$	0	0
$834$	14.0000	0.484780
$835$	3.02944	0.104838
$836$	0	0
$837$	12.6863	0.438502
$838$	−29.6985	−1.02592
$839$	−35.3137	−1.21916	−0.609582	−	0.792723i	$-0.708663\pi$
$839$	−35.3137	−1.21916	−0.609582	+	0.792723i	$0.708663\pi$
$840$	0	0
$841$	−18.9411	−0.653142
$842$	12.6863	0.437198
$843$	−18.9706	−0.653381
$844$	0	0
$845$	73.4558	2.52696
$846$	8.72792	0.300072
$847$	0	0
$848$	53.6569	1.84258
$849$	3.51472	0.120625
$850$	−12.7279	−0.436564
$851$	−12.0000	−0.411355
$852$	0	0
$853$	8.51472	0.291538	0.145769	−	0.989319i	$-0.453434\pi$
$853$	8.51472	0.291538	0.145769	+	0.989319i	$0.453434\pi$
$854$	0	0
$855$	2.82843	0.0967302
$856$	−0.970563	−0.0331732
$857$	−8.10051	−0.276708	−0.138354	−	0.990383i	$-0.544181\pi$
$857$	−8.10051	−0.276708	−0.138354	+	0.990383i	$0.544181\pi$
$858$	2.14214	0.0731313
$859$	9.51472	0.324638	0.162319	−	0.986738i	$-0.448103\pi$
$859$	9.51472	0.324638	0.162319	+	0.986738i	$0.448103\pi$
$860$	0	0
$861$	0	0
$862$	23.5147	0.800914
$863$	26.1421	0.889889	0.444944	−	0.895558i	$-0.353223\pi$
$863$	26.1421	0.889889	0.444944	+	0.895558i	$0.353223\pi$
$864$	0	0
$865$	−13.9411	−0.474012
$866$	−33.1716	−1.12722
$867$	11.3137	0.384234
$868$	0	0
$869$	−0.644661	−0.0218686
$870$	−17.9411	−0.608261
$871$	12.4853	0.423048
$872$	−40.9706	−1.38744
$873$	6.24264	0.211281
$874$	4.24264	0.143509
$875$	0	0
$876$	0	0
$877$	20.9706	0.708126	0.354063	−	0.935222i	$-0.384800\pi$
$877$	20.9706	0.708126	0.354063	+	0.935222i	$0.384800\pi$
$878$	26.1421	0.882254
$879$	−15.5147	−0.523298
$880$	1.94113	0.0654353
$881$	−16.4558	−0.554411	−0.277206	−	0.960811i	$-0.589408\pi$
$881$	−16.4558	−0.554411	−0.277206	+	0.960811i	$0.589408\pi$
$882$	0	0
$883$	−3.48528	−0.117289	−0.0586445	−	0.998279i	$-0.518678\pi$
$883$	−3.48528	−0.117289	−0.0586445	+	0.998279i	$0.518678\pi$
$884$	0	0
$885$	40.9706	1.37721
$886$	−16.7279	−0.561986
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	16.0000	0.536925
$889$	0	0
$890$	24.0000	0.804482
$891$	0.857864	0.0287395
$892$	0	0
$893$	−6.17157	−0.206524
$894$	−18.3431	−0.613487
$895$	−12.9706	−0.433558
$896$	0	0
$897$	26.4853	0.884318
$898$	15.4558	0.515768
$899$	−7.11270	−0.237222
$900$	0	0
$901$	−40.2426	−1.34068
$902$	2.14214	0.0713253
$903$	0	0
$904$	−12.0000	−0.399114
$905$	−35.3137	−1.17387
$906$	−41.4558	−1.37728
$907$	−6.72792	−0.223397	−0.111698	−	0.993742i	$-0.535629\pi$
$907$	−6.72792	−0.223397	−0.111698	+	0.993742i	$0.535629\pi$
$908$	0	0
$909$	5.82843	0.193317
$910$	0	0
$911$	2.44365	0.0809618	0.0404809	−	0.999180i	$-0.487111\pi$
$911$	2.44365	0.0809618	0.0404809	+	0.999180i	$0.487111\pi$
$912$	−5.65685	−0.187317
$913$	−0.941125	−0.0311467
$914$	−6.38478	−0.211190
$915$	45.9411	1.51877
$916$	0	0
$917$	0	0
$918$	−24.0000	−0.792118
$919$	−6.45584	−0.212959	−0.106479	−	0.994315i	$-0.533958\pi$
$919$	−6.45584	−0.212959	−0.106479	+	0.994315i	$0.533958\pi$
$920$	24.0000	0.791257
$921$	−28.6274	−0.943305
$922$	4.72792	0.155706
$923$	81.5980	2.68583
$924$	0	0
$925$	−12.0000	−0.394558
$926$	−28.2426	−0.928111
$927$	−0.242641	−0.00796937
$928$	0	0
$929$	−29.1421	−0.956122	−0.478061	−	0.878327i	$-0.658660\pi$
$929$	−29.1421	−0.956122	−0.478061	+	0.878327i	$0.658660\pi$
$930$	12.6863	0.416000
$931$	0	0
$932$	0	0
$933$	−1.21320	−0.0397185
$934$	6.78680	0.222071
$935$	−1.45584	−0.0476112
$936$	17.6569	0.577132
$937$	−24.9706	−0.815753	−0.407876	−	0.913037i	$-0.633731\pi$
$937$	−24.9706	−0.815753	−0.407876	+	0.913037i	$0.633731\pi$
$938$	0	0
$939$	31.7574	1.03636
$940$	0	0
$941$	9.55635	0.311528	0.155764	−	0.987794i	$-0.450216\pi$
$941$	9.55635	0.311528	0.155764	+	0.987794i	$0.450216\pi$
$942$	6.00000	0.195491
$943$	26.4853	0.862479
$944$	40.9706	1.33348
$945$	0	0
$946$	−2.54416	−0.0827176
$947$	−0.343146	−0.0111507	−0.00557537	−	0.999984i	$-0.501775\pi$
$947$	−0.343146	−0.0111507	−0.00557537	+	0.999984i	$0.501775\pi$
$948$	0	0
$949$	−18.7279	−0.607934
$950$	4.24264	0.137649
$951$	−38.4853	−1.24797
$952$	0	0
$953$	47.6569	1.54376	0.771878	−	0.635770i	$-0.219317\pi$
$953$	47.6569	1.54376	0.771878	+	0.635770i	$0.219317\pi$
$954$	−18.9706	−0.614195
$955$	−56.9706	−1.84352
$956$	0	0
$957$	−0.769553	−0.0248761
$958$	−16.7279	−0.540455
$959$	0	0
$960$	−32.0000	−1.03280
$961$	−25.9706	−0.837760
$962$	−35.3137	−1.13856
$963$	0.343146	0.0110577
$964$	0	0
$965$	47.3137	1.52308
$966$	0	0
$967$	17.5147	0.563235	0.281618	−	0.959527i	$-0.409129\pi$
$967$	17.5147	0.563235	0.281618	+	0.959527i	$0.409129\pi$
$968$	−31.0294	−0.997324
$969$	4.24264	0.136293
$970$	24.9706	0.801756
$971$	−10.2843	−0.330038	−0.165019	−	0.986290i	$-0.552769\pi$
$971$	−10.2843	−0.330038	−0.165019	+	0.986290i	$0.552769\pi$
$972$	0	0
$973$	0	0
$974$	49.1127	1.57367
$975$	26.4853	0.848208
$976$	45.9411	1.47054
$977$	24.0416	0.769160	0.384580	−	0.923092i	$-0.374346\pi$
$977$	24.0416	0.769160	0.384580	+	0.923092i	$0.374346\pi$
$978$	30.9706	0.990329
$979$	1.02944	0.0329010
$980$	0	0
$981$	14.4853	0.462479
$982$	21.6985	0.692426
$983$	28.2843	0.902128	0.451064	−	0.892492i	$-0.351045\pi$
$983$	28.2843	0.902128	0.451064	+	0.892492i	$0.351045\pi$
$984$	−35.3137	−1.12576
$985$	−49.4558	−1.57579
$986$	13.4558	0.428521
$987$	0	0
$988$	0	0
$989$	−31.4558	−1.00024
$990$	−0.686292	−0.0218118
$991$	−6.72792	−0.213719	−0.106860	−	0.994274i	$-0.534080\pi$
$991$	−6.72792	−0.213719	−0.106860	+	0.994274i	$0.534080\pi$
$992$	0	0
$993$	−20.8284	−0.660970
$994$	0	0
$995$	28.2843	0.896672
$996$	0	0
$997$	−26.5147	−0.839730	−0.419865	−	0.907587i	$-0.637923\pi$
$997$	−26.5147	−0.839730	−0.419865	+	0.907587i	$0.637923\pi$
$998$	−9.85786	−0.312045
$999$	−22.6274	−0.715900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	931.2.a.i.1.1		2
3.2	odd	2		8379.2.a.z.1.2		2
7.2	even	3		133.2.f.b.39.2	✓	4
7.3	odd	6		931.2.f.f.324.2		4
7.4	even	3		133.2.f.b.58.2	yes	4
7.5	odd	6		931.2.f.f.704.2		4
7.6	odd	2		931.2.a.h.1.1		2
21.2	odd	6		1197.2.j.g.172.1		4
21.11	odd	6		1197.2.j.g.856.1		4
21.20	even	2		8379.2.a.ba.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.f.b.39.2	✓	4	7.2	even	3
133.2.f.b.58.2	yes	4	7.4	even	3
931.2.a.h.1.1		2	7.6	odd	2
931.2.a.i.1.1		2	1.1	even	1	trivial
931.2.f.f.324.2		4	7.3	odd	6
931.2.f.f.704.2		4	7.5	odd	6
1197.2.j.g.172.1		4	21.2	odd	6
1197.2.j.g.856.1		4	21.11	odd	6
8379.2.a.z.1.2		2	3.2	odd	2
8379.2.a.ba.1.2		2	21.20	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:	\( N \)	\(=\)	\( 931 = 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	931.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.41421 q^{3} +2.82843 q^{5} +2.00000 q^{6} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.41421 q^{3} +2.82843 q^{5} +2.00000 q^{6} +2.82843 q^{8} -1.00000 q^{9} -4.00000 q^{10} -0.171573 q^{11} -6.24264 q^{13} -4.00000 q^{15} -4.00000 q^{16} +3.00000 q^{17} +1.41421 q^{18} -1.00000 q^{19} +0.242641 q^{22} +3.00000 q^{23} -4.00000 q^{24} +3.00000 q^{25} +8.82843 q^{26} +5.65685 q^{27} -3.17157 q^{29} +5.65685 q^{30} +2.24264 q^{31} +0.242641 q^{33} -4.24264 q^{34} -4.00000 q^{37} +1.41421 q^{38} +8.82843 q^{39} +8.00000 q^{40} +8.82843 q^{41} -10.4853 q^{43} -2.82843 q^{45} -4.24264 q^{46} +6.17157 q^{47} +5.65685 q^{48} -4.24264 q^{50} -4.24264 q^{51} -13.4142 q^{53} -8.00000 q^{54} -0.485281 q^{55} +1.41421 q^{57} +4.48528 q^{58} -10.2426 q^{59} -11.4853 q^{61} -3.17157 q^{62} +8.00000 q^{64} -17.6569 q^{65} -0.343146 q^{66} -2.00000 q^{67} -4.24264 q^{69} -13.0711 q^{71} -2.82843 q^{72} +3.00000 q^{73} +5.65685 q^{74} -4.24264 q^{75} -12.4853 q^{78} +3.75736 q^{79} -11.3137 q^{80} -5.00000 q^{81} -12.4853 q^{82} +5.48528 q^{83} +8.48528 q^{85} +14.8284 q^{86} +4.48528 q^{87} -0.485281 q^{88} -6.00000 q^{89} +4.00000 q^{90} -3.17157 q^{93} -8.72792 q^{94} -2.82843 q^{95} -6.24264 q^{97} +0.171573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^6 - 2 * q^9	\( 2 q + 4 q^{6} - 2 q^{9} - 8 q^{10} - 6 q^{11} - 4 q^{13} - 8 q^{15} - 8 q^{16} + 6 q^{17} - 2 q^{19} - 8 q^{22} + 6 q^{23} - 8 q^{24} + 6 q^{25} + 12 q^{26} - 12 q^{29} - 4 q^{31} - 8 q^{33} - 8 q^{37} + 12 q^{39} + 16 q^{40} + 12 q^{41} - 4 q^{43} + 18 q^{47} - 24 q^{53} - 16 q^{54} + 16 q^{55} - 8 q^{58} - 12 q^{59} - 6 q^{61} - 12 q^{62} + 16 q^{64} - 24 q^{65} - 12 q^{66} - 4 q^{67} - 12 q^{71} + 6 q^{73} - 8 q^{78} + 16 q^{79} - 10 q^{81} - 8 q^{82} - 6 q^{83} + 24 q^{86} - 8 q^{87} + 16 q^{88} - 12 q^{89} + 8 q^{90} - 12 q^{93} + 8 q^{94} - 4 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q + 4 * q^6 - 2 * q^9 - 8 * q^10 - 6 * q^11 - 4 * q^13 - 8 * q^15 - 8 * q^16 + 6 * q^17 - 2 * q^19 - 8 * q^22 + 6 * q^23 - 8 * q^24 + 6 * q^25 + 12 * q^26 - 12 * q^29 - 4 * q^31 - 8 * q^33 - 8 * q^37 + 12 * q^39 + 16 * q^40 + 12 * q^41 - 4 * q^43 + 18 * q^47 - 24 * q^53 - 16 * q^54 + 16 * q^55 - 8 * q^58 - 12 * q^59 - 6 * q^61 - 12 * q^62 + 16 * q^64 - 24 * q^65 - 12 * q^66 - 4 * q^67 - 12 * q^71 + 6 * q^73 - 8 * q^78 + 16 * q^79 - 10 * q^81 - 8 * q^82 - 6 * q^83 + 24 * q^86 - 8 * q^87 + 16 * q^88 - 12 * q^89 + 8 * q^90 - 12 * q^93 + 8 * q^94 - 4 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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