LMFDB - Embedded newform 1911.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1911,2,Mod(1,1911)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1911, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1911.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1911 = 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1911.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:	$15.2594118263$
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:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1911.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.00000 q^{3} +1.00000 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.00000 q^{3} +1.00000 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} -1.41421 q^{10} -2.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} -4.00000 q^{16} -1.41421 q^{17} -1.41421 q^{18} +3.58579 q^{19} +2.82843 q^{22} +0.414214 q^{23} -2.82843 q^{24} -4.00000 q^{25} +1.41421 q^{26} -1.00000 q^{27} +1.24264 q^{29} +1.41421 q^{30} +0.414214 q^{31} +2.00000 q^{33} +2.00000 q^{34} -4.00000 q^{37} -5.07107 q^{38} +1.00000 q^{39} +2.82843 q^{40} -5.17157 q^{41} +3.82843 q^{43} +1.00000 q^{45} -0.585786 q^{46} +4.65685 q^{47} +4.00000 q^{48} +5.65685 q^{50} +1.41421 q^{51} +2.41421 q^{53} +1.41421 q^{54} -2.00000 q^{55} -3.58579 q^{57} -1.75736 q^{58} -7.65685 q^{59} +11.6569 q^{61} -0.585786 q^{62} +8.00000 q^{64} -1.00000 q^{65} -2.82843 q^{66} -4.58579 q^{67} -0.414214 q^{69} -6.82843 q^{71} +2.82843 q^{72} +5.58579 q^{73} +5.65685 q^{74} +4.00000 q^{75} -1.41421 q^{78} +1.48528 q^{79} -4.00000 q^{80} +1.00000 q^{81} +7.31371 q^{82} +5.48528 q^{83} -1.41421 q^{85} -5.41421 q^{86} -1.24264 q^{87} -5.65685 q^{88} -2.65685 q^{89} -1.41421 q^{90} -0.414214 q^{93} -6.58579 q^{94} +3.58579 q^{95} +16.8995 q^{97} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} - 8 q^{16} + 10 q^{19} - 2 q^{23} - 8 q^{25} - 2 q^{27} - 6 q^{29} - 2 q^{31} + 4 q^{33} + 4 q^{34} - 8 q^{37} + 4 q^{38} + 2 q^{39} - 16 q^{41} + 2 q^{43} + 2 q^{45} - 4 q^{46} - 2 q^{47} + 8 q^{48} + 2 q^{53} - 4 q^{55} - 10 q^{57} - 12 q^{58} - 4 q^{59} + 12 q^{61} - 4 q^{62} + 16 q^{64} - 2 q^{65} - 12 q^{67} + 2 q^{69} - 8 q^{71} + 14 q^{73} + 8 q^{75} - 14 q^{79} - 8 q^{80} + 2 q^{81} - 8 q^{82} - 6 q^{83} - 8 q^{86} + 6 q^{87} + 6 q^{89} + 2 q^{93} - 16 q^{94} + 10 q^{95} + 14 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 - 8 * q^16 + 10 * q^19 - 2 * q^23 - 8 * q^25 - 2 * q^27 - 6 * q^29 - 2 * q^31 + 4 * q^33 + 4 * q^34 - 8 * q^37 + 4 * q^38 + 2 * q^39 - 16 * q^41 + 2 * q^43 + 2 * q^45 - 4 * q^46 - 2 * q^47 + 8 * q^48 + 2 * q^53 - 4 * q^55 - 10 * q^57 - 12 * q^58 - 4 * q^59 + 12 * q^61 - 4 * q^62 + 16 * q^64 - 2 * q^65 - 12 * q^67 + 2 * q^69 - 8 * q^71 + 14 * q^73 + 8 * q^75 - 14 * q^79 - 8 * q^80 + 2 * q^81 - 8 * q^82 - 6 * q^83 - 8 * q^86 + 6 * q^87 + 6 * q^89 + 2 * q^93 - 16 * q^94 + 10 * q^95 + 14 * q^97 - 4 * 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	$0.666667\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	1.41421	0.577350
$7$	0	0
$7$	0	0
$8$	2.82843	1.00000
$9$	1.00000	0.333333
$10$	−1.41421	−0.447214
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	−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
$19$	3.58579	0.822636	0.411318	−	0.911492i	$-0.365069\pi$
$19$	3.58579	0.822636	0.411318	+	0.911492i	$0.365069\pi$
$20$	0	0
$21$	0	0
$22$	2.82843	0.603023
$23$	0.414214	0.0863695	0.0431847	−	0.999067i	$-0.486250\pi$
$23$	0.414214	0.0863695	0.0431847	+	0.999067i	$0.486250\pi$
$24$	−2.82843	−0.577350
$25$	−4.00000	−0.800000
$26$	1.41421	0.277350
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.24264	0.230753	0.115376	−	0.993322i	$-0.463193\pi$
$29$	1.24264	0.230753	0.115376	+	0.993322i	$0.463193\pi$
$30$	1.41421	0.258199
$31$	0.414214	0.0743950	0.0371975	−	0.999308i	$-0.488157\pi$
$31$	0.414214	0.0743950	0.0371975	+	0.999308i	$0.488157\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	2.00000	0.342997
$35$	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	$0.606644\pi$
$38$	−5.07107	−0.822636
$39$	1.00000	0.160128
$40$	2.82843	0.447214
$41$	−5.17157	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$41$	−5.17157	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$42$	0	0
$43$	3.82843	0.583830	0.291915	−	0.956444i	$-0.405708\pi$
$43$	3.82843	0.583830	0.291915	+	0.956444i	$0.405708\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	−0.585786	−0.0863695
$47$	4.65685	0.679272	0.339636	−	0.940557i	$-0.389696\pi$
$47$	4.65685	0.679272	0.339636	+	0.940557i	$0.389696\pi$
$48$	4.00000	0.577350
$49$	0	0
$50$	5.65685	0.800000
$51$	1.41421	0.198030
$52$	0	0
$53$	2.41421	0.331618	0.165809	−	0.986158i	$-0.446977\pi$
$53$	2.41421	0.331618	0.165809	+	0.986158i	$0.446977\pi$
$54$	1.41421	0.192450
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−3.58579	−0.474949
$58$	−1.75736	−0.230753
$59$	−7.65685	−0.996838	−0.498419	−	0.866936i	$-0.666086\pi$
$59$	−7.65685	−0.996838	−0.498419	+	0.866936i	$0.666086\pi$
$60$	0	0
$61$	11.6569	1.49251	0.746254	−	0.665662i	$-0.231851\pi$
$61$	11.6569	1.49251	0.746254	+	0.665662i	$0.231851\pi$
$62$	−0.585786	−0.0743950
$63$	0	0
$64$	8.00000	1.00000
$65$	−1.00000	−0.124035
$66$	−2.82843	−0.348155
$67$	−4.58579	−0.560243	−0.280121	−	0.959965i	$-0.590375\pi$
$67$	−4.58579	−0.560243	−0.280121	+	0.959965i	$0.590375\pi$
$68$	0	0
$69$	−0.414214	−0.0498655
$70$	0	0
$71$	−6.82843	−0.810385	−0.405193	−	0.914231i	$-0.632796\pi$
$71$	−6.82843	−0.810385	−0.405193	+	0.914231i	$0.632796\pi$
$72$	2.82843	0.333333
$73$	5.58579	0.653767	0.326883	−	0.945065i	$-0.394002\pi$
$73$	5.58579	0.653767	0.326883	+	0.945065i	$0.394002\pi$
$74$	5.65685	0.657596
$75$	4.00000	0.461880
$76$	0	0
$77$	0	0
$78$	−1.41421	−0.160128
$79$	1.48528	0.167107	0.0835536	−	0.996503i	$-0.473373\pi$
$79$	1.48528	0.167107	0.0835536	+	0.996503i	$0.473373\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	7.31371	0.807664
$83$	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
$85$	−1.41421	−0.153393
$86$	−5.41421	−0.583830
$87$	−1.24264	−0.133225
$88$	−5.65685	−0.603023
$89$	−2.65685	−0.281626	−0.140813	−	0.990036i	$-0.544972\pi$
$89$	−2.65685	−0.281626	−0.140813	+	0.990036i	$0.544972\pi$
$90$	−1.41421	−0.149071
$91$	0	0
$92$	0	0
$93$	−0.414214	−0.0429519
$94$	−6.58579	−0.679272
$95$	3.58579	0.367894
$96$	0	0
$97$	16.8995	1.71588	0.857942	−	0.513747i	$-0.171743\pi$
$97$	16.8995	1.71588	0.857942	+	0.513747i	$0.171743\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−7.65685	−0.761885	−0.380943	−	0.924599i	$-0.624401\pi$
$101$	−7.65685	−0.761885	−0.380943	+	0.924599i	$0.624401\pi$
$102$	−2.00000	−0.198030
$103$	−4.48528	−0.441948	−0.220974	−	0.975280i	$-0.570924\pi$
$103$	−4.48528	−0.441948	−0.220974	+	0.975280i	$0.570924\pi$
$104$	−2.82843	−0.277350
$105$	0	0
$106$	−3.41421	−0.331618
$107$	−16.4853	−1.59369	−0.796846	−	0.604182i	$-0.793500\pi$
$107$	−16.4853	−1.59369	−0.796846	+	0.604182i	$0.793500\pi$
$108$	0	0
$109$	−17.0711	−1.63511	−0.817556	−	0.575849i	$-0.804672\pi$
$109$	−17.0711	−1.63511	−0.817556	+	0.575849i	$0.804672\pi$
$110$	2.82843	0.269680
$111$	4.00000	0.379663
$112$	0	0
$113$	−14.8995	−1.40163	−0.700813	−	0.713345i	$-0.747179\pi$
$113$	−14.8995	−1.40163	−0.700813	+	0.713345i	$0.747179\pi$
$114$	5.07107	0.474949
$115$	0.414214	0.0386256
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	10.8284	0.996838
$119$	0	0
$120$	−2.82843	−0.258199
$121$	−7.00000	−0.636364
$122$	−16.4853	−1.49251
$123$	5.17157	0.466305
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−5.31371	−0.471515	−0.235758	−	0.971812i	$-0.575757\pi$
$127$	−5.31371	−0.471515	−0.235758	+	0.971812i	$0.575757\pi$
$128$	−11.3137	−1.00000
$129$	−3.82843	−0.337074
$130$	1.41421	0.124035
$131$	−12.2426	−1.06964	−0.534822	−	0.844965i	$-0.679621\pi$
$131$	−12.2426	−1.06964	−0.534822	+	0.844965i	$0.679621\pi$
$132$	0	0
$133$	0	0
$134$	6.48528	0.560243
$135$	−1.00000	−0.0860663
$136$	−4.00000	−0.342997
$137$	−3.51472	−0.300283	−0.150141	−	0.988665i	$-0.547973\pi$
$137$	−3.51472	−0.300283	−0.150141	+	0.988665i	$0.547973\pi$
$138$	0.585786	0.0498655
$139$	10.9706	0.930511	0.465255	−	0.885176i	$-0.345962\pi$
$139$	10.9706	0.930511	0.465255	+	0.885176i	$0.345962\pi$
$140$	0	0
$141$	−4.65685	−0.392178
$142$	9.65685	0.810385
$143$	2.00000	0.167248
$144$	−4.00000	−0.333333
$145$	1.24264	0.103196
$146$	−7.89949	−0.653767
$147$	0	0
$148$	0	0
$149$	−20.1421	−1.65011	−0.825054	−	0.565054i	$-0.808855\pi$
$149$	−20.1421	−1.65011	−0.825054	+	0.565054i	$0.808855\pi$
$150$	−5.65685	−0.461880
$151$	−10.8284	−0.881205	−0.440602	−	0.897702i	$-0.645235\pi$
$151$	−10.8284	−0.881205	−0.440602	+	0.897702i	$0.645235\pi$
$152$	10.1421	0.822636
$153$	−1.41421	−0.114332
$154$	0	0
$155$	0.414214	0.0332704
$156$	0	0
$157$	2.24264	0.178982	0.0894911	−	0.995988i	$-0.471476\pi$
$157$	2.24264	0.178982	0.0894911	+	0.995988i	$0.471476\pi$
$158$	−2.10051	−0.167107
$159$	−2.41421	−0.191460
$160$	0	0
$161$	0	0
$162$	−1.41421	−0.111111
$163$	3.75736	0.294299	0.147150	−	0.989114i	$-0.452990\pi$
$163$	3.75736	0.294299	0.147150	+	0.989114i	$0.452990\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	−7.75736	−0.602088
$167$	−17.9706	−1.39060	−0.695302	−	0.718718i	$-0.744729\pi$
$167$	−17.9706	−1.39060	−0.695302	+	0.718718i	$0.744729\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.00000	0.153393
$171$	3.58579	0.274212
$172$	0	0
$173$	−14.8284	−1.12738	−0.563692	−	0.825985i	$-0.690620\pi$
$173$	−14.8284	−1.12738	−0.563692	+	0.825985i	$0.690620\pi$
$174$	1.75736	0.133225
$175$	0	0
$176$	8.00000	0.603023
$177$	7.65685	0.575524
$178$	3.75736	0.281626
$179$	12.7574	0.953530	0.476765	−	0.879031i	$-0.341809\pi$
$179$	12.7574	0.953530	0.476765	+	0.879031i	$0.341809\pi$
$180$	0	0
$181$	−8.82843	−0.656212	−0.328106	−	0.944641i	$-0.606410\pi$
$181$	−8.82843	−0.656212	−0.328106	+	0.944641i	$0.606410\pi$
$182$	0	0
$183$	−11.6569	−0.861699
$184$	1.17157	0.0863695
$185$	−4.00000	−0.294086
$186$	0.585786	0.0429519
$187$	2.82843	0.206835
$188$	0	0
$189$	0	0
$190$	−5.07107	−0.367894
$191$	6.34315	0.458974	0.229487	−	0.973312i	$-0.426295\pi$
$191$	6.34315	0.458974	0.229487	+	0.973312i	$0.426295\pi$
$192$	−8.00000	−0.577350
$193$	−4.58579	−0.330092	−0.165046	−	0.986286i	$-0.552777\pi$
$193$	−4.58579	−0.330092	−0.165046	+	0.986286i	$0.552777\pi$
$194$	−23.8995	−1.71588
$195$	1.00000	0.0716115
$196$	0	0
$197$	−20.8284	−1.48396	−0.741982	−	0.670420i	$-0.766114\pi$
$197$	−20.8284	−1.48396	−0.741982	+	0.670420i	$0.766114\pi$
$198$	2.82843	0.201008
$199$	7.41421	0.525580	0.262790	−	0.964853i	$-0.415357\pi$
$199$	7.41421	0.525580	0.262790	+	0.964853i	$0.415357\pi$
$200$	−11.3137	−0.800000
$201$	4.58579	0.323456
$202$	10.8284	0.761885
$203$	0	0
$204$	0	0
$205$	−5.17157	−0.361198
$206$	6.34315	0.441948
$207$	0.414214	0.0287898
$208$	4.00000	0.277350
$209$	−7.17157	−0.496068
$210$	0	0
$211$	−7.14214	−0.491685	−0.245842	−	0.969310i	$-0.579065\pi$
$211$	−7.14214	−0.491685	−0.245842	+	0.969310i	$0.579065\pi$
$212$	0	0
$213$	6.82843	0.467876
$214$	23.3137	1.59369
$215$	3.82843	0.261097
$216$	−2.82843	−0.192450
$217$	0	0
$218$	24.1421	1.63511
$219$	−5.58579	−0.377452
$220$	0	0
$221$	1.41421	0.0951303
$222$	−5.65685	−0.379663
$223$	22.0711	1.47799	0.738994	−	0.673712i	$-0.235301\pi$
$223$	22.0711	1.47799	0.738994	+	0.673712i	$0.235301\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	21.0711	1.40163
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	0	0
$229$	−18.4853	−1.22154	−0.610771	−	0.791807i	$-0.709140\pi$
$229$	−18.4853	−1.22154	−0.610771	+	0.791807i	$0.709140\pi$
$230$	−0.585786	−0.0386256
$231$	0	0
$232$	3.51472	0.230753
$233$	−29.7279	−1.94754	−0.973770	−	0.227533i	$-0.926934\pi$
$233$	−29.7279	−1.94754	−0.973770	+	0.227533i	$0.926934\pi$
$234$	1.41421	0.0924500
$235$	4.65685	0.303780
$236$	0	0
$237$	−1.48528	−0.0964794
$238$	0	0
$239$	4.58579	0.296630	0.148315	−	0.988940i	$-0.452615\pi$
$239$	4.58579	0.296630	0.148315	+	0.988940i	$0.452615\pi$
$240$	4.00000	0.258199
$241$	18.0711	1.16406	0.582030	−	0.813167i	$-0.302259\pi$
$241$	18.0711	1.16406	0.582030	+	0.813167i	$0.302259\pi$
$242$	9.89949	0.636364
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	−7.31371	−0.466305
$247$	−3.58579	−0.228158
$248$	1.17157	0.0743950
$249$	−5.48528	−0.347616
$250$	12.7279	0.804984
$251$	−1.51472	−0.0956082	−0.0478041	−	0.998857i	$-0.515222\pi$
$251$	−1.51472	−0.0956082	−0.0478041	+	0.998857i	$0.515222\pi$
$252$	0	0
$253$	−0.828427	−0.0520828
$254$	7.51472	0.471515
$255$	1.41421	0.0885615
$256$	0	0
$257$	−18.4853	−1.15308	−0.576540	−	0.817069i	$-0.695598\pi$
$257$	−18.4853	−1.15308	−0.576540	+	0.817069i	$0.695598\pi$
$258$	5.41421	0.337074
$259$	0	0
$260$	0	0
$261$	1.24264	0.0769175
$262$	17.3137	1.06964
$263$	7.58579	0.467760	0.233880	−	0.972266i	$-0.424858\pi$
$263$	7.58579	0.467760	0.233880	+	0.972266i	$0.424858\pi$
$264$	5.65685	0.348155
$265$	2.41421	0.148304
$266$	0	0
$267$	2.65685	0.162597
$268$	0	0
$269$	13.2132	0.805623	0.402812	−	0.915283i	$-0.368033\pi$
$269$	13.2132	0.805623	0.402812	+	0.915283i	$0.368033\pi$
$270$	1.41421	0.0860663
$271$	−2.97056	−0.180449	−0.0902244	−	0.995921i	$-0.528758\pi$
$271$	−2.97056	−0.180449	−0.0902244	+	0.995921i	$0.528758\pi$
$272$	5.65685	0.342997
$273$	0	0
$274$	4.97056	0.300283
$275$	8.00000	0.482418
$276$	0	0
$277$	−23.6274	−1.41963	−0.709817	−	0.704386i	$-0.751222\pi$
$277$	−23.6274	−1.41963	−0.709817	+	0.704386i	$0.751222\pi$
$278$	−15.5147	−0.930511
$279$	0.414214	0.0247983
$280$	0	0
$281$	−14.5858	−0.870115	−0.435058	−	0.900403i	$-0.643272\pi$
$281$	−14.5858	−0.870115	−0.435058	+	0.900403i	$0.643272\pi$
$282$	6.58579	0.392178
$283$	1.17157	0.0696428	0.0348214	−	0.999394i	$-0.488914\pi$
$283$	1.17157	0.0696428	0.0348214	+	0.999394i	$0.488914\pi$
$284$	0	0
$285$	−3.58579	−0.212404
$286$	−2.82843	−0.167248
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	−1.75736	−0.103196
$291$	−16.8995	−0.990666
$292$	0	0
$293$	−10.6569	−0.622580	−0.311290	−	0.950315i	$-0.600761\pi$
$293$	−10.6569	−0.622580	−0.311290	+	0.950315i	$0.600761\pi$
$294$	0	0
$295$	−7.65685	−0.445799
$296$	−11.3137	−0.657596
$297$	2.00000	0.116052
$298$	28.4853	1.65011
$299$	−0.414214	−0.0239546
$300$	0	0
$301$	0	0
$302$	15.3137	0.881205
$303$	7.65685	0.439875
$304$	−14.3431	−0.822636
$305$	11.6569	0.667470
$306$	2.00000	0.114332
$307$	18.4142	1.05095	0.525477	−	0.850808i	$-0.323887\pi$
$307$	18.4142	1.05095	0.525477	+	0.850808i	$0.323887\pi$
$308$	0	0
$309$	4.48528	0.255159
$310$	−0.585786	−0.0332704
$311$	7.41421	0.420421	0.210211	−	0.977656i	$-0.432585\pi$
$311$	7.41421	0.420421	0.210211	+	0.977656i	$0.432585\pi$
$312$	2.82843	0.160128
$313$	−25.7990	−1.45825	−0.729123	−	0.684383i	$-0.760072\pi$
$313$	−25.7990	−1.45825	−0.729123	+	0.684383i	$0.760072\pi$
$314$	−3.17157	−0.178982
$315$	0	0
$316$	0	0
$317$	17.7574	0.997353	0.498676	−	0.866788i	$-0.333820\pi$
$317$	17.7574	0.997353	0.498676	+	0.866788i	$0.333820\pi$
$318$	3.41421	0.191460
$319$	−2.48528	−0.139149
$320$	8.00000	0.447214
$321$	16.4853	0.920119
$322$	0	0
$323$	−5.07107	−0.282162
$324$	0	0
$325$	4.00000	0.221880
$326$	−5.31371	−0.294299
$327$	17.0711	0.944032
$328$	−14.6274	−0.807664
$329$	0	0
$330$	−2.82843	−0.155700
$331$	−14.9706	−0.822857	−0.411428	−	0.911442i	$-0.634970\pi$
$331$	−14.9706	−0.822857	−0.411428	+	0.911442i	$0.634970\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	25.4142	1.39060
$335$	−4.58579	−0.250548
$336$	0	0
$337$	28.7990	1.56878	0.784390	−	0.620267i	$-0.212976\pi$
$337$	28.7990	1.56878	0.784390	+	0.620267i	$0.212976\pi$
$338$	−1.41421	−0.0769231
$339$	14.8995	0.809229
$340$	0	0
$341$	−0.828427	−0.0448618
$342$	−5.07107	−0.274212
$343$	0	0
$344$	10.8284	0.583830
$345$	−0.414214	−0.0223005
$346$	20.9706	1.12738
$347$	21.4558	1.15181	0.575905	−	0.817517i	$-0.304650\pi$
$347$	21.4558	1.15181	0.575905	+	0.817517i	$0.304650\pi$
$348$	0	0
$349$	−19.0416	−1.01928	−0.509638	−	0.860389i	$-0.670221\pi$
$349$	−19.0416	−1.01928	−0.509638	+	0.860389i	$0.670221\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	15.7990	0.840895	0.420448	−	0.907317i	$-0.361873\pi$
$353$	15.7990	0.840895	0.420448	+	0.907317i	$0.361873\pi$
$354$	−10.8284	−0.575524
$355$	−6.82843	−0.362415
$356$	0	0
$357$	0	0
$358$	−18.0416	−0.953530
$359$	14.7279	0.777310	0.388655	−	0.921383i	$-0.372940\pi$
$359$	14.7279	0.777310	0.388655	+	0.921383i	$0.372940\pi$
$360$	2.82843	0.149071
$361$	−6.14214	−0.323270
$362$	12.4853	0.656212
$363$	7.00000	0.367405
$364$	0	0
$365$	5.58579	0.292373
$366$	16.4853	0.861699
$367$	−1.41421	−0.0738213	−0.0369107	−	0.999319i	$-0.511752\pi$
$367$	−1.41421	−0.0738213	−0.0369107	+	0.999319i	$0.511752\pi$
$368$	−1.65685	−0.0863695
$369$	−5.17157	−0.269221
$370$	5.65685	0.294086
$371$	0	0
$372$	0	0
$373$	29.1716	1.51045	0.755223	−	0.655467i	$-0.227528\pi$
$373$	29.1716	1.51045	0.755223	+	0.655467i	$0.227528\pi$
$374$	−4.00000	−0.206835
$375$	9.00000	0.464758
$376$	13.1716	0.679272
$377$	−1.24264	−0.0639993
$378$	0	0
$379$	−8.34315	−0.428559	−0.214279	−	0.976772i	$-0.568740\pi$
$379$	−8.34315	−0.428559	−0.214279	+	0.976772i	$0.568740\pi$
$380$	0	0
$381$	5.31371	0.272230
$382$	−8.97056	−0.458974
$383$	−2.97056	−0.151789	−0.0758943	−	0.997116i	$-0.524181\pi$
$383$	−2.97056	−0.151789	−0.0758943	+	0.997116i	$0.524181\pi$
$384$	11.3137	0.577350
$385$	0	0
$386$	6.48528	0.330092
$387$	3.82843	0.194610
$388$	0	0
$389$	10.4853	0.531625	0.265812	−	0.964025i	$-0.414360\pi$
$389$	10.4853	0.531625	0.265812	+	0.964025i	$0.414360\pi$
$390$	−1.41421	−0.0716115
$391$	−0.585786	−0.0296245
$392$	0	0
$393$	12.2426	0.617560
$394$	29.4558	1.48396
$395$	1.48528	0.0747326
$396$	0	0
$397$	12.4142	0.623052	0.311526	−	0.950238i	$-0.399160\pi$
$397$	12.4142	0.623052	0.311526	+	0.950238i	$0.399160\pi$
$398$	−10.4853	−0.525580
$399$	0	0
$400$	16.0000	0.800000
$401$	−20.5858	−1.02801	−0.514003	−	0.857789i	$-0.671838\pi$
$401$	−20.5858	−1.02801	−0.514003	+	0.857789i	$0.671838\pi$
$402$	−6.48528	−0.323456
$403$	−0.414214	−0.0206334
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	8.00000	0.396545
$408$	4.00000	0.198030
$409$	−15.7279	−0.777696	−0.388848	−	0.921302i	$-0.627127\pi$
$409$	−15.7279	−0.777696	−0.388848	+	0.921302i	$0.627127\pi$
$410$	7.31371	0.361198
$411$	3.51472	0.173368
$412$	0	0
$413$	0	0
$414$	−0.585786	−0.0287898
$415$	5.48528	0.269262
$416$	0	0
$417$	−10.9706	−0.537231
$418$	10.1421	0.496068
$419$	−16.1421	−0.788595	−0.394297	−	0.918983i	$-0.629012\pi$
$419$	−16.1421	−0.788595	−0.394297	+	0.918983i	$0.629012\pi$
$420$	0	0
$421$	5.21320	0.254076	0.127038	−	0.991898i	$-0.459453\pi$
$421$	5.21320	0.254076	0.127038	+	0.991898i	$0.459453\pi$
$422$	10.1005	0.491685
$423$	4.65685	0.226424
$424$	6.82843	0.331618
$425$	5.65685	0.274398
$426$	−9.65685	−0.467876
$427$	0	0
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	−5.41421	−0.261097
$431$	−26.9706	−1.29913	−0.649563	−	0.760308i	$-0.725048\pi$
$431$	−26.9706	−1.29913	−0.649563	+	0.760308i	$0.725048\pi$
$432$	4.00000	0.192450
$433$	−18.7279	−0.900006	−0.450003	−	0.893027i	$-0.648577\pi$
$433$	−18.7279	−0.900006	−0.450003	+	0.893027i	$0.648577\pi$
$434$	0	0
$435$	−1.24264	−0.0595801
$436$	0	0
$437$	1.48528	0.0710506
$438$	7.89949	0.377452
$439$	−38.2843	−1.82721	−0.913604	−	0.406604i	$-0.866713\pi$
$439$	−38.2843	−1.82721	−0.913604	+	0.406604i	$0.866713\pi$
$440$	−5.65685	−0.269680
$441$	0	0
$442$	−2.00000	−0.0951303
$443$	−3.24264	−0.154063	−0.0770313	−	0.997029i	$-0.524544\pi$
$443$	−3.24264	−0.154063	−0.0770313	+	0.997029i	$0.524544\pi$
$444$	0	0
$445$	−2.65685	−0.125947
$446$	−31.2132	−1.47799
$447$	20.1421	0.952690
$448$	0	0
$449$	1.55635	0.0734487	0.0367243	−	0.999325i	$-0.488308\pi$
$449$	1.55635	0.0734487	0.0367243	+	0.999325i	$0.488308\pi$
$450$	5.65685	0.266667
$451$	10.3431	0.487040
$452$	0	0
$453$	10.8284	0.508764
$454$	−14.1421	−0.663723
$455$	0	0
$456$	−10.1421	−0.474949
$457$	39.0711	1.82767	0.913834	−	0.406089i	$-0.133108\pi$
$457$	39.0711	1.82767	0.913834	+	0.406089i	$0.133108\pi$
$458$	26.1421	1.22154
$459$	1.41421	0.0660098
$460$	0	0
$461$	−10.3431	−0.481728	−0.240864	−	0.970559i	$-0.577431\pi$
$461$	−10.3431	−0.481728	−0.240864	+	0.970559i	$0.577431\pi$
$462$	0	0
$463$	−10.9289	−0.507911	−0.253955	−	0.967216i	$-0.581732\pi$
$463$	−10.9289	−0.507911	−0.253955	+	0.967216i	$0.581732\pi$
$464$	−4.97056	−0.230753
$465$	−0.414214	−0.0192087
$466$	42.0416	1.94754
$467$	−5.55635	−0.257117	−0.128559	−	0.991702i	$-0.541035\pi$
$467$	−5.55635	−0.257117	−0.128559	+	0.991702i	$0.541035\pi$
$468$	0	0
$469$	0	0
$470$	−6.58579	−0.303780
$471$	−2.24264	−0.103335
$472$	−21.6569	−0.996838
$473$	−7.65685	−0.352063
$474$	2.10051	0.0964794
$475$	−14.3431	−0.658109
$476$	0	0
$477$	2.41421	0.110539
$478$	−6.48528	−0.296630
$479$	8.79899	0.402036	0.201018	−	0.979588i	$-0.435575\pi$
$479$	8.79899	0.402036	0.201018	+	0.979588i	$0.435575\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	−25.5563	−1.16406
$483$	0	0
$484$	0	0
$485$	16.8995	0.767367
$486$	1.41421	0.0641500
$487$	37.5980	1.70373	0.851864	−	0.523764i	$-0.175473\pi$
$487$	37.5980	1.70373	0.851864	+	0.523764i	$0.175473\pi$
$488$	32.9706	1.49251
$489$	−3.75736	−0.169914
$490$	0	0
$491$	26.9706	1.21716	0.608582	−	0.793491i	$-0.291739\pi$
$491$	26.9706	1.21716	0.608582	+	0.793491i	$0.291739\pi$
$492$	0	0
$493$	−1.75736	−0.0791475
$494$	5.07107	0.228158
$495$	−2.00000	−0.0898933
$496$	−1.65685	−0.0743950
$497$	0	0
$498$	7.75736	0.347616
$499$	38.1838	1.70934	0.854670	−	0.519172i	$-0.173759\pi$
$499$	38.1838	1.70934	0.854670	+	0.519172i	$0.173759\pi$
$500$	0	0
$501$	17.9706	0.802866
$502$	2.14214	0.0956082
$503$	35.9411	1.60254	0.801268	−	0.598306i	$-0.204159\pi$
$503$	35.9411	1.60254	0.801268	+	0.598306i	$0.204159\pi$
$504$	0	0
$505$	−7.65685	−0.340726
$506$	1.17157	0.0520828
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−1.14214	−0.0506243	−0.0253121	−	0.999680i	$-0.508058\pi$
$509$	−1.14214	−0.0506243	−0.0253121	+	0.999680i	$0.508058\pi$
$510$	−2.00000	−0.0885615
$511$	0	0
$512$	22.6274	1.00000
$513$	−3.58579	−0.158316
$514$	26.1421	1.15308
$515$	−4.48528	−0.197645
$516$	0	0
$517$	−9.31371	−0.409616
$518$	0	0
$519$	14.8284	0.650896
$520$	−2.82843	−0.124035
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	−1.75736	−0.0769175
$523$	−27.9411	−1.22178	−0.610890	−	0.791715i	$-0.709188\pi$
$523$	−27.9411	−1.22178	−0.610890	+	0.791715i	$0.709188\pi$
$524$	0	0
$525$	0	0
$526$	−10.7279	−0.467760
$527$	−0.585786	−0.0255173
$528$	−8.00000	−0.348155
$529$	−22.8284	−0.992540
$530$	−3.41421	−0.148304
$531$	−7.65685	−0.332279
$532$	0	0
$533$	5.17157	0.224006
$534$	−3.75736	−0.162597
$535$	−16.4853	−0.712721
$536$	−12.9706	−0.560243
$537$	−12.7574	−0.550521
$538$	−18.6863	−0.805623
$539$	0	0
$540$	0	0
$541$	−11.3137	−0.486414	−0.243207	−	0.969974i	$-0.578199\pi$
$541$	−11.3137	−0.486414	−0.243207	+	0.969974i	$0.578199\pi$
$542$	4.20101	0.180449
$543$	8.82843	0.378864
$544$	0	0
$545$	−17.0711	−0.731244
$546$	0	0
$547$	36.1127	1.54407	0.772034	−	0.635582i	$-0.219240\pi$
$547$	36.1127	1.54407	0.772034	+	0.635582i	$0.219240\pi$
$548$	0	0
$549$	11.6569	0.497502
$550$	−11.3137	−0.482418
$551$	4.45584	0.189825
$552$	−1.17157	−0.0498655
$553$	0	0
$554$	33.4142	1.41963
$555$	4.00000	0.169791
$556$	0	0
$557$	39.1127	1.65726	0.828629	−	0.559798i	$-0.189121\pi$
$557$	39.1127	1.65726	0.828629	+	0.559798i	$0.189121\pi$
$558$	−0.585786	−0.0247983
$559$	−3.82843	−0.161925
$560$	0	0
$561$	−2.82843	−0.119416
$562$	20.6274	0.870115
$563$	−29.4142	−1.23966	−0.619831	−	0.784736i	$-0.712799\pi$
$563$	−29.4142	−1.23966	−0.619831	+	0.784736i	$0.712799\pi$
$564$	0	0
$565$	−14.8995	−0.626826
$566$	−1.65685	−0.0696428
$567$	0	0
$568$	−19.3137	−0.810385
$569$	27.5269	1.15399	0.576994	−	0.816748i	$-0.304226\pi$
$569$	27.5269	1.15399	0.576994	+	0.816748i	$0.304226\pi$
$570$	5.07107	0.212404
$571$	−0.798990	−0.0334367	−0.0167183	−	0.999860i	$-0.505322\pi$
$571$	−0.798990	−0.0334367	−0.0167183	+	0.999860i	$0.505322\pi$
$572$	0	0
$573$	−6.34315	−0.264989
$574$	0	0
$575$	−1.65685	−0.0690956
$576$	8.00000	0.333333
$577$	−10.6863	−0.444876	−0.222438	−	0.974947i	$-0.571402\pi$
$577$	−10.6863	−0.444876	−0.222438	+	0.974947i	$0.571402\pi$
$578$	21.2132	0.882353
$579$	4.58579	0.190579
$580$	0	0
$581$	0	0
$582$	23.8995	0.990666
$583$	−4.82843	−0.199973
$584$	15.7990	0.653767
$585$	−1.00000	−0.0413449
$586$	15.0711	0.622580
$587$	42.9411	1.77237	0.886185	−	0.463332i	$-0.153346\pi$
$587$	42.9411	1.77237	0.886185	+	0.463332i	$0.153346\pi$
$588$	0	0
$589$	1.48528	0.0612000
$590$	10.8284	0.445799
$591$	20.8284	0.856767
$592$	16.0000	0.657596
$593$	11.9706	0.491572	0.245786	−	0.969324i	$-0.420954\pi$
$593$	11.9706	0.491572	0.245786	+	0.969324i	$0.420954\pi$
$594$	−2.82843	−0.116052
$595$	0	0
$596$	0	0
$597$	−7.41421	−0.303444
$598$	0.585786	0.0239546
$599$	−4.89949	−0.200188	−0.100094	−	0.994978i	$-0.531914\pi$
$599$	−4.89949	−0.200188	−0.100094	+	0.994978i	$0.531914\pi$
$600$	11.3137	0.461880
$601$	−45.2132	−1.84429	−0.922143	−	0.386850i	$-0.873563\pi$
$601$	−45.2132	−1.84429	−0.922143	+	0.386850i	$0.873563\pi$
$602$	0	0
$603$	−4.58579	−0.186748
$604$	0	0
$605$	−7.00000	−0.284590
$606$	−10.8284	−0.439875
$607$	29.3137	1.18981	0.594903	−	0.803797i	$-0.297190\pi$
$607$	29.3137	1.18981	0.594903	+	0.803797i	$0.297190\pi$
$608$	0	0
$609$	0	0
$610$	−16.4853	−0.667470
$611$	−4.65685	−0.188396
$612$	0	0
$613$	13.6985	0.553277	0.276638	−	0.960974i	$-0.410780\pi$
$613$	13.6985	0.553277	0.276638	+	0.960974i	$0.410780\pi$
$614$	−26.0416	−1.05095
$615$	5.17157	0.208538
$616$	0	0
$617$	−0.142136	−0.00572216	−0.00286108	−	0.999996i	$-0.500911\pi$
$617$	−0.142136	−0.00572216	−0.00286108	+	0.999996i	$0.500911\pi$
$618$	−6.34315	−0.255159
$619$	−23.7990	−0.956562	−0.478281	−	0.878207i	$-0.658740\pi$
$619$	−23.7990	−0.956562	−0.478281	+	0.878207i	$0.658740\pi$
$620$	0	0
$621$	−0.414214	−0.0166218
$622$	−10.4853	−0.420421
$623$	0	0
$624$	−4.00000	−0.160128
$625$	11.0000	0.440000
$626$	36.4853	1.45825
$627$	7.17157	0.286405
$628$	0	0
$629$	5.65685	0.225554
$630$	0	0
$631$	−45.7990	−1.82323	−0.911614	−	0.411046i	$-0.865163\pi$
$631$	−45.7990	−1.82323	−0.911614	+	0.411046i	$0.865163\pi$
$632$	4.20101	0.167107
$633$	7.14214	0.283874
$634$	−25.1127	−0.997353
$635$	−5.31371	−0.210868
$636$	0	0
$637$	0	0
$638$	3.51472	0.139149
$639$	−6.82843	−0.270128
$640$	−11.3137	−0.447214
$641$	−38.0711	−1.50372	−0.751858	−	0.659325i	$-0.770842\pi$
$641$	−38.0711	−1.50372	−0.751858	+	0.659325i	$0.770842\pi$
$642$	−23.3137	−0.920119
$643$	33.1716	1.30816	0.654080	−	0.756426i	$-0.273056\pi$
$643$	33.1716	1.30816	0.654080	+	0.756426i	$0.273056\pi$
$644$	0	0
$645$	−3.82843	−0.150744
$646$	7.17157	0.282162
$647$	−38.2426	−1.50347	−0.751737	−	0.659463i	$-0.770784\pi$
$647$	−38.2426	−1.50347	−0.751737	+	0.659463i	$0.770784\pi$
$648$	2.82843	0.111111
$649$	15.3137	0.601116
$650$	−5.65685	−0.221880
$651$	0	0
$652$	0	0
$653$	−40.8284	−1.59774	−0.798870	−	0.601504i	$-0.794568\pi$
$653$	−40.8284	−1.59774	−0.798870	+	0.601504i	$0.794568\pi$
$654$	−24.1421	−0.944032
$655$	−12.2426	−0.478360
$656$	20.6863	0.807664
$657$	5.58579	0.217922
$658$	0	0
$659$	−25.2426	−0.983314	−0.491657	−	0.870789i	$-0.663609\pi$
$659$	−25.2426	−0.983314	−0.491657	+	0.870789i	$0.663609\pi$
$660$	0	0
$661$	12.8995	0.501732	0.250866	−	0.968022i	$-0.419285\pi$
$661$	12.8995	0.501732	0.250866	+	0.968022i	$0.419285\pi$
$662$	21.1716	0.822857
$663$	−1.41421	−0.0549235
$664$	15.5147	0.602088
$665$	0	0
$666$	5.65685	0.219199
$667$	0.514719	0.0199300
$668$	0	0
$669$	−22.0711	−0.853317
$670$	6.48528	0.250548
$671$	−23.3137	−0.900016
$672$	0	0
$673$	−27.8284	−1.07271	−0.536354	−	0.843993i	$-0.680199\pi$
$673$	−27.8284	−1.07271	−0.536354	+	0.843993i	$0.680199\pi$
$674$	−40.7279	−1.56878
$675$	4.00000	0.153960
$676$	0	0
$677$	23.6985	0.910807	0.455403	−	0.890285i	$-0.349495\pi$
$677$	23.6985	0.910807	0.455403	+	0.890285i	$0.349495\pi$
$678$	−21.0711	−0.809229
$679$	0	0
$680$	−4.00000	−0.153393
$681$	−10.0000	−0.383201
$682$	1.17157	0.0448618
$683$	36.9706	1.41464	0.707320	−	0.706894i	$-0.249904\pi$
$683$	36.9706	1.41464	0.707320	+	0.706894i	$0.249904\pi$
$684$	0	0
$685$	−3.51472	−0.134290
$686$	0	0
$687$	18.4853	0.705257
$688$	−15.3137	−0.583830
$689$	−2.41421	−0.0919742
$690$	0.585786	0.0223005
$691$	21.8701	0.831976	0.415988	−	0.909370i	$-0.363436\pi$
$691$	21.8701	0.831976	0.415988	+	0.909370i	$0.363436\pi$
$692$	0	0
$693$	0	0
$694$	−30.3431	−1.15181
$695$	10.9706	0.416137
$696$	−3.51472	−0.133225
$697$	7.31371	0.277026
$698$	26.9289	1.01928
$699$	29.7279	1.12441
$700$	0	0
$701$	−17.0416	−0.643654	−0.321827	−	0.946799i	$-0.604297\pi$
$701$	−17.0416	−0.643654	−0.321827	+	0.946799i	$0.604297\pi$
$702$	−1.41421	−0.0533761
$703$	−14.3431	−0.540962
$704$	−16.0000	−0.603023
$705$	−4.65685	−0.175387
$706$	−22.3431	−0.840895
$707$	0	0
$708$	0	0
$709$	−17.1127	−0.642681	−0.321340	−	0.946964i	$-0.604133\pi$
$709$	−17.1127	−0.642681	−0.321340	+	0.946964i	$0.604133\pi$
$710$	9.65685	0.362415
$711$	1.48528	0.0557024
$712$	−7.51472	−0.281626
$713$	0.171573	0.00642545
$714$	0	0
$715$	2.00000	0.0747958
$716$	0	0
$717$	−4.58579	−0.171259
$718$	−20.8284	−0.777310
$719$	−43.8995	−1.63717	−0.818587	−	0.574382i	$-0.805242\pi$
$719$	−43.8995	−1.63717	−0.818587	+	0.574382i	$0.805242\pi$
$720$	−4.00000	−0.149071
$721$	0	0
$722$	8.68629	0.323270
$723$	−18.0711	−0.672070
$724$	0	0
$725$	−4.97056	−0.184602
$726$	−9.89949	−0.367405
$727$	20.2426	0.750758	0.375379	−	0.926871i	$-0.377513\pi$
$727$	20.2426	0.750758	0.375379	+	0.926871i	$0.377513\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−7.89949	−0.292373
$731$	−5.41421	−0.200252
$732$	0	0
$733$	−28.0711	−1.03683	−0.518414	−	0.855130i	$-0.673477\pi$
$733$	−28.0711	−1.03683	−0.518414	+	0.855130i	$0.673477\pi$
$734$	2.00000	0.0738213
$735$	0	0
$736$	0	0
$737$	9.17157	0.337839
$738$	7.31371	0.269221
$739$	28.1421	1.03523	0.517613	−	0.855615i	$-0.326821\pi$
$739$	28.1421	1.03523	0.517613	+	0.855615i	$0.326821\pi$
$740$	0	0
$741$	3.58579	0.131727
$742$	0	0
$743$	−23.4142	−0.858984	−0.429492	−	0.903071i	$-0.641307\pi$
$743$	−23.4142	−0.858984	−0.429492	+	0.903071i	$0.641307\pi$
$744$	−1.17157	−0.0429519
$745$	−20.1421	−0.737951
$746$	−41.2548	−1.51045
$747$	5.48528	0.200696
$748$	0	0
$749$	0	0
$750$	−12.7279	−0.464758
$751$	10.3137	0.376353	0.188176	−	0.982135i	$-0.439742\pi$
$751$	10.3137	0.376353	0.188176	+	0.982135i	$0.439742\pi$
$752$	−18.6274	−0.679272
$753$	1.51472	0.0551994
$754$	1.75736	0.0639993
$755$	−10.8284	−0.394087
$756$	0	0
$757$	7.48528	0.272057	0.136029	−	0.990705i	$-0.456566\pi$
$757$	7.48528	0.272057	0.136029	+	0.990705i	$0.456566\pi$
$758$	11.7990	0.428559
$759$	0.828427	0.0300700
$760$	10.1421	0.367894
$761$	17.1421	0.621402	0.310701	−	0.950508i	$-0.399436\pi$
$761$	17.1421	0.621402	0.310701	+	0.950508i	$0.399436\pi$
$762$	−7.51472	−0.272230
$763$	0	0
$764$	0	0
$765$	−1.41421	−0.0511310
$766$	4.20101	0.151789
$767$	7.65685	0.276473
$768$	0	0
$769$	27.2426	0.982395	0.491197	−	0.871048i	$-0.336559\pi$
$769$	27.2426	0.982395	0.491197	+	0.871048i	$0.336559\pi$
$770$	0	0
$771$	18.4853	0.665731
$772$	0	0
$773$	11.5147	0.414156	0.207078	−	0.978324i	$-0.433605\pi$
$773$	11.5147	0.414156	0.207078	+	0.978324i	$0.433605\pi$
$774$	−5.41421	−0.194610
$775$	−1.65685	−0.0595160
$776$	47.7990	1.71588
$777$	0	0
$778$	−14.8284	−0.531625
$779$	−18.5442	−0.664413
$780$	0	0
$781$	13.6569	0.488681
$782$	0.828427	0.0296245
$783$	−1.24264	−0.0444084
$784$	0	0
$785$	2.24264	0.0800433
$786$	−17.3137	−0.617560
$787$	8.61522	0.307100	0.153550	−	0.988141i	$-0.450929\pi$
$787$	8.61522	0.307100	0.153550	+	0.988141i	$0.450929\pi$
$788$	0	0
$789$	−7.58579	−0.270061
$790$	−2.10051	−0.0747326
$791$	0	0
$792$	−5.65685	−0.201008
$793$	−11.6569	−0.413947
$794$	−17.5563	−0.623052
$795$	−2.41421	−0.0856233
$796$	0	0
$797$	9.27208	0.328434	0.164217	−	0.986424i	$-0.447490\pi$
$797$	9.27208	0.328434	0.164217	+	0.986424i	$0.447490\pi$
$798$	0	0
$799$	−6.58579	−0.232988
$800$	0	0
$801$	−2.65685	−0.0938753
$802$	29.1127	1.02801
$803$	−11.1716	−0.394236
$804$	0	0
$805$	0	0
$806$	0.585786	0.0206334
$807$	−13.2132	−0.465127
$808$	−21.6569	−0.761885
$809$	−2.12994	−0.0748848	−0.0374424	−	0.999299i	$-0.511921\pi$
$809$	−2.12994	−0.0748848	−0.0374424	+	0.999299i	$0.511921\pi$
$810$	−1.41421	−0.0496904
$811$	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
$813$	2.97056	0.104182
$814$	−11.3137	−0.396545
$815$	3.75736	0.131615
$816$	−5.65685	−0.198030
$817$	13.7279	0.480279
$818$	22.2426	0.777696
$819$	0	0
$820$	0	0
$821$	1.61522	0.0563717	0.0281858	−	0.999603i	$-0.491027\pi$
$821$	1.61522	0.0563717	0.0281858	+	0.999603i	$0.491027\pi$
$822$	−4.97056	−0.173368
$823$	−54.0833	−1.88522	−0.942612	−	0.333890i	$-0.891639\pi$
$823$	−54.0833	−1.88522	−0.942612	+	0.333890i	$0.891639\pi$
$824$	−12.6863	−0.441948
$825$	−8.00000	−0.278524
$826$	0	0
$827$	−15.2132	−0.529015	−0.264507	−	0.964384i	$-0.585209\pi$
$827$	−15.2132	−0.529015	−0.264507	+	0.964384i	$0.585209\pi$
$828$	0	0
$829$	−10.8701	−0.377533	−0.188766	−	0.982022i	$-0.560449\pi$
$829$	−10.8701	−0.377533	−0.188766	+	0.982022i	$0.560449\pi$
$830$	−7.75736	−0.269262
$831$	23.6274	0.819626
$832$	−8.00000	−0.277350
$833$	0	0
$834$	15.5147	0.537231
$835$	−17.9706	−0.621897
$836$	0	0
$837$	−0.414214	−0.0143173
$838$	22.8284	0.788595
$839$	53.5980	1.85041	0.925204	−	0.379470i	$-0.123894\pi$
$839$	53.5980	1.85041	0.925204	+	0.379470i	$0.123894\pi$
$840$	0	0
$841$	−27.4558	−0.946753
$842$	−7.37258	−0.254076
$843$	14.5858	0.502361
$844$	0	0
$845$	1.00000	0.0344010
$846$	−6.58579	−0.226424
$847$	0	0
$848$	−9.65685	−0.331618
$849$	−1.17157	−0.0402083
$850$	−8.00000	−0.274398
$851$	−1.65685	−0.0567962
$852$	0	0
$853$	0.129942	0.00444914	0.00222457	−	0.999998i	$-0.499292\pi$
$853$	0.129942	0.00444914	0.00222457	+	0.999998i	$0.499292\pi$
$854$	0	0
$855$	3.58579	0.122631
$856$	−46.6274	−1.59369
$857$	−23.1716	−0.791526	−0.395763	−	0.918353i	$-0.629520\pi$
$857$	−23.1716	−0.791526	−0.395763	+	0.918353i	$0.629520\pi$
$858$	2.82843	0.0965609
$859$	−52.7279	−1.79905	−0.899527	−	0.436866i	$-0.856088\pi$
$859$	−52.7279	−1.79905	−0.899527	+	0.436866i	$0.856088\pi$
$860$	0	0
$861$	0	0
$862$	38.1421	1.29913
$863$	45.9411	1.56385	0.781927	−	0.623370i	$-0.214237\pi$
$863$	45.9411	1.56385	0.781927	+	0.623370i	$0.214237\pi$
$864$	0	0
$865$	−14.8284	−0.504182
$866$	26.4853	0.900006
$867$	15.0000	0.509427
$868$	0	0
$869$	−2.97056	−0.100769
$870$	1.75736	0.0595801
$871$	4.58579	0.155383
$872$	−48.2843	−1.63511
$873$	16.8995	0.571961
$874$	−2.10051	−0.0710506
$875$	0	0
$876$	0	0
$877$	18.3848	0.620810	0.310405	−	0.950604i	$-0.399535\pi$
$877$	18.3848	0.620810	0.310405	+	0.950604i	$0.399535\pi$
$878$	54.1421	1.82721
$879$	10.6569	0.359447
$880$	8.00000	0.269680
$881$	−10.5269	−0.354661	−0.177330	−	0.984151i	$-0.556746\pi$
$881$	−10.5269	−0.354661	−0.177330	+	0.984151i	$0.556746\pi$
$882$	0	0
$883$	33.6569	1.13264	0.566322	−	0.824184i	$-0.308366\pi$
$883$	33.6569	1.13264	0.566322	+	0.824184i	$0.308366\pi$
$884$	0	0
$885$	7.65685	0.257382
$886$	4.58579	0.154063
$887$	−40.2426	−1.35122	−0.675608	−	0.737261i	$-0.736119\pi$
$887$	−40.2426	−1.35122	−0.675608	+	0.737261i	$0.736119\pi$
$888$	11.3137	0.379663
$889$	0	0
$890$	3.75736	0.125947
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	16.6985	0.558793
$894$	−28.4853	−0.952690
$895$	12.7574	0.426431
$896$	0	0
$897$	0.414214	0.0138302
$898$	−2.20101	−0.0734487
$899$	0.514719	0.0171668
$900$	0	0
$901$	−3.41421	−0.113744
$902$	−14.6274	−0.487040
$903$	0	0
$904$	−42.1421	−1.40163
$905$	−8.82843	−0.293467
$906$	−15.3137	−0.508764
$907$	−13.2843	−0.441097	−0.220548	−	0.975376i	$-0.570785\pi$
$907$	−13.2843	−0.441097	−0.220548	+	0.975376i	$0.570785\pi$
$908$	0	0
$909$	−7.65685	−0.253962
$910$	0	0
$911$	14.6152	0.484224	0.242112	−	0.970248i	$-0.422160\pi$
$911$	14.6152	0.484224	0.242112	+	0.970248i	$0.422160\pi$
$912$	14.3431	0.474949
$913$	−10.9706	−0.363073
$914$	−55.2548	−1.82767
$915$	−11.6569	−0.385364
$916$	0	0
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	−17.1127	−0.564496	−0.282248	−	0.959342i	$-0.591080\pi$
$919$	−17.1127	−0.564496	−0.282248	+	0.959342i	$0.591080\pi$
$920$	1.17157	0.0386256
$921$	−18.4142	−0.606769
$922$	14.6274	0.481728
$923$	6.82843	0.224760
$924$	0	0
$925$	16.0000	0.526077
$926$	15.4558	0.507911
$927$	−4.48528	−0.147316
$928$	0	0
$929$	25.9706	0.852067	0.426033	−	0.904707i	$-0.359911\pi$
$929$	25.9706	0.852067	0.426033	+	0.904707i	$0.359911\pi$
$930$	0.585786	0.0192087
$931$	0	0
$932$	0	0
$933$	−7.41421	−0.242730
$934$	7.85786	0.257117
$935$	2.82843	0.0924995
$936$	−2.82843	−0.0924500
$937$	50.8284	1.66049	0.830246	−	0.557397i	$-0.188200\pi$
$937$	50.8284	1.66049	0.830246	+	0.557397i	$0.188200\pi$
$938$	0	0
$939$	25.7990	0.841918
$940$	0	0
$941$	17.0000	0.554184	0.277092	−	0.960843i	$-0.410629\pi$
$941$	17.0000	0.554184	0.277092	+	0.960843i	$0.410629\pi$
$942$	3.17157	0.103335
$943$	−2.14214	−0.0697575
$944$	30.6274	0.996838
$945$	0	0
$946$	10.8284	0.352063
$947$	28.7279	0.933532	0.466766	−	0.884381i	$-0.345419\pi$
$947$	28.7279	0.933532	0.466766	+	0.884381i	$0.345419\pi$
$948$	0	0
$949$	−5.58579	−0.181322
$950$	20.2843	0.658109
$951$	−17.7574	−0.575822
$952$	0	0
$953$	−31.9289	−1.03428	−0.517140	−	0.855901i	$-0.673003\pi$
$953$	−31.9289	−1.03428	−0.517140	+	0.855901i	$0.673003\pi$
$954$	−3.41421	−0.110539
$955$	6.34315	0.205259
$956$	0	0
$957$	2.48528	0.0803377
$958$	−12.4437	−0.402036
$959$	0	0
$960$	−8.00000	−0.258199
$961$	−30.8284	−0.994465
$962$	−5.65685	−0.182384
$963$	−16.4853	−0.531231
$964$	0	0
$965$	−4.58579	−0.147622
$966$	0	0
$967$	17.5147	0.563235	0.281618	−	0.959527i	$-0.409129\pi$
$967$	17.5147	0.563235	0.281618	+	0.959527i	$0.409129\pi$
$968$	−19.7990	−0.636364
$969$	5.07107	0.162906
$970$	−23.8995	−0.767367
$971$	4.34315	0.139378	0.0696891	−	0.997569i	$-0.477799\pi$
$971$	4.34315	0.139378	0.0696891	+	0.997569i	$0.477799\pi$
$972$	0	0
$973$	0	0
$974$	−53.1716	−1.70373
$975$	−4.00000	−0.128103
$976$	−46.6274	−1.49251
$977$	15.6569	0.500907	0.250454	−	0.968129i	$-0.419420\pi$
$977$	15.6569	0.500907	0.250454	+	0.968129i	$0.419420\pi$
$978$	5.31371	0.169914
$979$	5.31371	0.169827
$980$	0	0
$981$	−17.0711	−0.545037
$982$	−38.1421	−1.21716
$983$	−34.4558	−1.09897	−0.549485	−	0.835503i	$-0.685176\pi$
$983$	−34.4558	−1.09897	−0.549485	+	0.835503i	$0.685176\pi$
$984$	14.6274	0.466305
$985$	−20.8284	−0.663649
$986$	2.48528	0.0791475
$987$	0	0
$988$	0	0
$989$	1.58579	0.0504251
$990$	2.82843	0.0898933
$991$	47.9411	1.52290	0.761450	−	0.648224i	$-0.224488\pi$
$991$	47.9411	1.52290	0.761450	+	0.648224i	$0.224488\pi$
$992$	0	0
$993$	14.9706	0.475076
$994$	0	0
$995$	7.41421	0.235046
$996$	0	0
$997$	51.1127	1.61876	0.809378	−	0.587288i	$-0.199805\pi$
$997$	51.1127	1.61876	0.809378	+	0.587288i	$0.199805\pi$
$998$	−54.0000	−1.70934
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1911.2.a.i.1.1	✓	2
3.2	odd	2		5733.2.a.q.1.2		2
7.6	odd	2		1911.2.a.j.1.1	yes	2
21.20	even	2		5733.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1911.2.a.i.1.1	✓	2	1.1	even	1	trivial
1911.2.a.j.1.1	yes	2	7.6	odd	2
5733.2.a.q.1.2		2	3.2	odd	2
5733.2.a.r.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:	\( N \)	\(=\)	\( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1911.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.00000 q^{3} +1.00000 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.00000 q^{3} +1.00000 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} -1.41421 q^{10} -2.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} -4.00000 q^{16} -1.41421 q^{17} -1.41421 q^{18} +3.58579 q^{19} +2.82843 q^{22} +0.414214 q^{23} -2.82843 q^{24} -4.00000 q^{25} +1.41421 q^{26} -1.00000 q^{27} +1.24264 q^{29} +1.41421 q^{30} +0.414214 q^{31} +2.00000 q^{33} +2.00000 q^{34} -4.00000 q^{37} -5.07107 q^{38} +1.00000 q^{39} +2.82843 q^{40} -5.17157 q^{41} +3.82843 q^{43} +1.00000 q^{45} -0.585786 q^{46} +4.65685 q^{47} +4.00000 q^{48} +5.65685 q^{50} +1.41421 q^{51} +2.41421 q^{53} +1.41421 q^{54} -2.00000 q^{55} -3.58579 q^{57} -1.75736 q^{58} -7.65685 q^{59} +11.6569 q^{61} -0.585786 q^{62} +8.00000 q^{64} -1.00000 q^{65} -2.82843 q^{66} -4.58579 q^{67} -0.414214 q^{69} -6.82843 q^{71} +2.82843 q^{72} +5.58579 q^{73} +5.65685 q^{74} +4.00000 q^{75} -1.41421 q^{78} +1.48528 q^{79} -4.00000 q^{80} +1.00000 q^{81} +7.31371 q^{82} +5.48528 q^{83} -1.41421 q^{85} -5.41421 q^{86} -1.24264 q^{87} -5.65685 q^{88} -2.65685 q^{89} -1.41421 q^{90} -0.414214 q^{93} -6.58579 q^{94} +3.58579 q^{95} +16.8995 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} - 8 q^{16} + 10 q^{19} - 2 q^{23} - 8 q^{25} - 2 q^{27} - 6 q^{29} - 2 q^{31} + 4 q^{33} + 4 q^{34} - 8 q^{37} + 4 q^{38} + 2 q^{39} - 16 q^{41} + 2 q^{43} + 2 q^{45} - 4 q^{46} - 2 q^{47} + 8 q^{48} + 2 q^{53} - 4 q^{55} - 10 q^{57} - 12 q^{58} - 4 q^{59} + 12 q^{61} - 4 q^{62} + 16 q^{64} - 2 q^{65} - 12 q^{67} + 2 q^{69} - 8 q^{71} + 14 q^{73} + 8 q^{75} - 14 q^{79} - 8 q^{80} + 2 q^{81} - 8 q^{82} - 6 q^{83} - 8 q^{86} + 6 q^{87} + 6 q^{89} + 2 q^{93} - 16 q^{94} + 10 q^{95} + 14 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 - 8 * q^16 + 10 * q^19 - 2 * q^23 - 8 * q^25 - 2 * q^27 - 6 * q^29 - 2 * q^31 + 4 * q^33 + 4 * q^34 - 8 * q^37 + 4 * q^38 + 2 * q^39 - 16 * q^41 + 2 * q^43 + 2 * q^45 - 4 * q^46 - 2 * q^47 + 8 * q^48 + 2 * q^53 - 4 * q^55 - 10 * q^57 - 12 * q^58 - 4 * q^59 + 12 * q^61 - 4 * q^62 + 16 * q^64 - 2 * q^65 - 12 * q^67 + 2 * q^69 - 8 * q^71 + 14 * q^73 + 8 * q^75 - 14 * q^79 - 8 * q^80 + 2 * q^81 - 8 * q^82 - 6 * q^83 - 8 * q^86 + 6 * q^87 + 6 * q^89 + 2 * q^93 - 16 * q^94 + 10 * q^95 + 14 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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