LMFDB - Embedded newform 2156.2.i.f.177.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2156,2,Mod(177,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2156.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2156 = 2^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2156.i (of order $3$, degree $2$, not minimal)

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:	no
Analytic conductor:	$17.2157466758$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			177.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2156.177
Dual form			2156.2.i.f.1145.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 2.44949i) q^{3} +(-0.707107 + 1.22474i) q^{5} +(-2.50000 + 4.33013i) q^{9} +O(q^{10})$	$q+(-1.41421 - 2.44949i) q^{3} +(-0.707107 + 1.22474i) q^{5} +(-2.50000 + 4.33013i) q^{9} +(0.500000 + 0.866025i) q^{11} -1.41421 q^{13} +4.00000 q^{15} +(-3.53553 - 6.12372i) q^{17} +(-1.41421 + 2.44949i) q^{19} +(-2.00000 + 3.46410i) q^{23} +(1.50000 + 2.59808i) q^{25} +5.65685 q^{27} +(2.82843 + 4.89898i) q^{31} +(1.41421 - 2.44949i) q^{33} +(4.00000 - 6.92820i) q^{37} +(2.00000 + 3.46410i) q^{39} +9.89949 q^{41} +4.00000 q^{43} +(-3.53553 - 6.12372i) q^{45} +(-10.0000 + 17.3205i) q^{51} +(3.00000 + 5.19615i) q^{53} -1.41421 q^{55} +8.00000 q^{57} +(-4.24264 - 7.34847i) q^{59} +(0.707107 - 1.22474i) q^{61} +(1.00000 - 1.73205i) q^{65} +(4.00000 + 6.92820i) q^{67} +11.3137 q^{69} +8.00000 q^{71} +(0.707107 + 1.22474i) q^{73} +(4.24264 - 7.34847i) q^{75} +(-8.00000 + 13.8564i) q^{79} +(-0.500000 - 0.866025i) q^{81} -2.82843 q^{83} +10.0000 q^{85} +(7.77817 - 13.4722i) q^{89} +(8.00000 - 13.8564i) q^{93} +(-2.00000 - 3.46410i) q^{95} +9.89949 q^{97} -5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{9}+O(q^{10}) $ 4 * q - 10 * q^9	$ 4 q - 10 q^{9} + 2 q^{11} + 16 q^{15} - 8 q^{23} + 6 q^{25} + 16 q^{37} + 8 q^{39} + 16 q^{43} - 40 q^{51} + 12 q^{53} + 32 q^{57} + 4 q^{65} + 16 q^{67} + 32 q^{71} - 32 q^{79} - 2 q^{81} + 40 q^{85} + 32 q^{93} - 8 q^{95} - 20 q^{99}+O(q^{100}) $ 4 * q - 10 * q^9 + 2 * q^11 + 16 * q^15 - 8 * q^23 + 6 * q^25 + 16 * q^37 + 8 * q^39 + 16 * q^43 - 40 * q^51 + 12 * q^53 + 32 * q^57 + 4 * q^65 + 16 * q^67 + 32 * q^71 - 32 * q^79 - 2 * q^81 + 40 * q^85 + 32 * q^93 - 8 * q^95 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times$.

$n$	$981$	$1079$	$1277$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	−1.41421	−	2.44949i	−0.816497	−	1.41421i	−0.908248	−	0.418432i	$-0.862580\pi$
$3$	−1.41421	−	2.44949i	−0.816497	−	1.41421i	0.0917517	−	0.995782i	$-0.470753\pi$
$4$		0			0
$5$	−0.707107	+	1.22474i	−0.316228	+	0.547723i	−0.979698	−	0.200480i	$-0.935750\pi$
$5$	−0.707107	+	1.22474i	−0.316228	+	0.547723i	0.663470	+	0.748203i	$0.269083\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−2.50000	+	4.33013i	−0.833333	+	1.44338i
$10$		0			0
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i
$12$		0			0
$13$	−1.41421			−0.392232			−0.196116	−	0.980581i	$-0.562833\pi$
$13$	−1.41421			−0.392232			−0.196116	+	0.980581i	$0.562833\pi$
$14$		0			0
$15$	4.00000			1.03280
$16$		0			0
$17$	−3.53553	−	6.12372i	−0.857493	−	1.48522i	−0.874313	−	0.485363i	$-0.838688\pi$
$17$	−3.53553	−	6.12372i	−0.857493	−	1.48522i	0.0168199	−	0.999859i	$-0.494646\pi$
$18$		0			0
$19$	−1.41421	+	2.44949i	−0.324443	+	0.561951i	−0.981399	−	0.191977i	$-0.938510\pi$
$19$	−1.41421	+	2.44949i	−0.324443	+	0.561951i	0.656957	+	0.753928i	$0.271843\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	−0.995639	−	0.0932891i	$-0.970262\pi$
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	0.578610	+	0.815604i	$0.303595\pi$
$24$		0			0
$25$	1.50000	+	2.59808i	0.300000	+	0.519615i
$26$		0			0
$27$	5.65685			1.08866
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	2.82843	+	4.89898i	0.508001	+	0.879883i	0.999957	+	0.00926296i	$0.00294853\pi$
$31$	2.82843	+	4.89898i	0.508001	+	0.879883i	−0.491957	+	0.870620i	$0.663718\pi$
$32$		0			0
$33$	1.41421	−	2.44949i	0.246183	−	0.426401i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.00000	−	6.92820i	0.657596	−	1.13899i	−0.323640	−	0.946180i	$-0.604907\pi$
$37$	4.00000	−	6.92820i	0.657596	−	1.13899i	0.981236	−	0.192809i	$-0.0617599\pi$
$38$		0			0
$39$	2.00000	+	3.46410i	0.320256	+	0.554700i
$40$		0			0
$41$	9.89949			1.54604			0.773021	−	0.634381i	$-0.218745\pi$
$41$	9.89949			1.54604			0.773021	+	0.634381i	$0.218745\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$	−3.53553	−	6.12372i	−0.527046	−	0.912871i
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−10.0000	+	17.3205i	−1.40028	+	2.42536i
$52$		0			0
$53$	3.00000	+	5.19615i	0.412082	+	0.713746i	0.995117	−	0.0987002i	$-0.0314685\pi$
$53$	3.00000	+	5.19615i	0.412082	+	0.713746i	−0.583036	+	0.812447i	$0.698135\pi$
$54$		0			0
$55$	−1.41421			−0.190693
$56$		0			0
$57$	8.00000			1.05963
$58$		0			0
$59$	−4.24264	−	7.34847i	−0.552345	−	0.956689i	−0.998105	−	0.0615367i	$-0.980400\pi$
$59$	−4.24264	−	7.34847i	−0.552345	−	0.956689i	0.445760	−	0.895152i	$-0.352933\pi$
$60$		0			0
$61$	0.707107	−	1.22474i	0.0905357	−	0.156813i	−0.817201	−	0.576353i	$-0.804475\pi$
$61$	0.707107	−	1.22474i	0.0905357	−	0.156813i	0.907737	+	0.419540i	$0.137809\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00000	−	1.73205i	0.124035	−	0.214834i
$66$		0			0
$67$	4.00000	+	6.92820i	0.488678	+	0.846415i	0.999915	−	0.0130248i	$-0.00414604\pi$
$67$	4.00000	+	6.92820i	0.488678	+	0.846415i	−0.511237	+	0.859440i	$0.670813\pi$
$68$		0			0
$69$	11.3137			1.36201
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$	0.707107	+	1.22474i	0.0827606	+	0.143346i	0.904435	−	0.426612i	$-0.140293\pi$
$73$	0.707107	+	1.22474i	0.0827606	+	0.143346i	−0.821674	+	0.569958i	$0.806960\pi$
$74$		0			0
$75$	4.24264	−	7.34847i	0.489898	−	0.848528i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.00000	+	13.8564i	−0.900070	+	1.55897i	−0.0726692	+	0.997356i	$0.523152\pi$
$79$	−8.00000	+	13.8564i	−0.900070	+	1.55897i	−0.827401	+	0.561611i	$0.810182\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−2.82843			−0.310460			−0.155230	−	0.987878i	$-0.549612\pi$
$83$	−2.82843			−0.310460			−0.155230	+	0.987878i	$0.549612\pi$
$84$		0			0
$85$	10.0000			1.08465
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.77817	−	13.4722i	0.824485	−	1.42805i	−0.0778275	−	0.996967i	$-0.524798\pi$
$89$	7.77817	−	13.4722i	0.824485	−	1.42805i	0.902312	−	0.431083i	$-0.141868\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	8.00000	−	13.8564i	0.829561	−	1.43684i
$94$		0			0
$95$	−2.00000	−	3.46410i	−0.205196	−	0.355409i
$96$		0			0
$97$	9.89949			1.00514			0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949			1.00514			0.502571	+	0.864536i	$0.332388\pi$
$98$		0			0
$99$	−5.00000			−0.502519
$100$		0			0
$101$	9.19239	+	15.9217i	0.914677	+	1.58427i	0.807374	+	0.590040i	$0.200888\pi$
$101$	9.19239	+	15.9217i	0.914677	+	1.58427i	0.107303	+	0.994226i	$0.465778\pi$
$102$		0			0
$103$	8.48528	−	14.6969i	0.836080	−	1.44813i	−0.0570688	−	0.998370i	$-0.518175\pi$
$103$	8.48528	−	14.6969i	0.836080	−	1.44813i	0.893148	−	0.449762i	$-0.148491\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.00000	−	6.92820i	0.386695	−	0.669775i	−0.605308	−	0.795991i	$-0.706950\pi$
$107$	4.00000	−	6.92820i	0.386695	−	0.669775i	0.992003	+	0.126217i	$0.0402834\pi$
$108$		0			0
$109$	−8.00000	−	13.8564i	−0.766261	−	1.32720i	−0.939577	−	0.342337i	$-0.888782\pi$
$109$	−8.00000	−	13.8564i	−0.766261	−	1.32720i	0.173316	−	0.984866i	$-0.444552\pi$
$110$		0			0
$111$	−22.6274			−2.14770
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	−2.82843	−	4.89898i	−0.263752	−	0.456832i
$116$		0			0
$117$	3.53553	−	6.12372i	0.326860	−	0.566139i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.0454545	+	0.0787296i
$122$		0			0
$123$	−14.0000	−	24.2487i	−1.26234	−	2.18643i
$124$		0			0
$125$	−11.3137			−1.01193
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$	−5.65685	−	9.79796i	−0.498058	−	0.862662i
$130$		0			0
$131$	−1.41421	+	2.44949i	−0.123560	+	0.214013i	−0.921169	−	0.389162i	$-0.872765\pi$
$131$	−1.41421	+	2.44949i	−0.123560	+	0.214013i	0.797609	+	0.603175i	$0.206098\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−4.00000	+	6.92820i	−0.344265	+	0.596285i
$136$		0			0
$137$	−8.00000	−	13.8564i	−0.683486	−	1.18383i	−0.973910	−	0.226935i	$-0.927130\pi$
$137$	−8.00000	−	13.8564i	−0.683486	−	1.18383i	0.290424	−	0.956898i	$-0.406204\pi$
$138$		0			0
$139$	−19.7990			−1.67933			−0.839664	−	0.543106i	$-0.817248\pi$
$139$	−19.7990			−1.67933			−0.839664	+	0.543106i	$0.817248\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.707107	−	1.22474i	−0.0591312	−	0.102418i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	11.0000	−	19.0526i	0.901155	−	1.56085i	0.0751583	−	0.997172i	$-0.476054\pi$
$149$	11.0000	−	19.0526i	0.901155	−	1.56085i	0.825997	−	0.563675i	$-0.190613\pi$
$150$		0			0
$151$	−2.00000	−	3.46410i	−0.162758	−	0.281905i	0.773099	−	0.634285i	$-0.218706\pi$
$151$	−2.00000	−	3.46410i	−0.162758	−	0.281905i	−0.935857	+	0.352381i	$0.885372\pi$
$152$		0			0
$153$	35.3553			2.85831
$154$		0			0
$155$	−8.00000			−0.642575
$156$		0			0
$157$	7.77817	+	13.4722i	0.620766	+	1.07520i	0.989343	+	0.145601i	$0.0465116\pi$
$157$	7.77817	+	13.4722i	0.620766	+	1.07520i	−0.368577	+	0.929597i	$0.620155\pi$
$158$		0			0
$159$	8.48528	−	14.6969i	0.672927	−	1.16554i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.00000	−	3.46410i	0.156652	−	0.271329i	−0.777007	−	0.629492i	$-0.783263\pi$
$163$	2.00000	−	3.46410i	0.156652	−	0.271329i	0.933659	+	0.358162i	$0.116597\pi$
$164$		0			0
$165$	2.00000	+	3.46410i	0.155700	+	0.269680i
$166$		0			0
$167$	5.65685			0.437741			0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685			0.437741			0.218870	+	0.975754i	$0.429763\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$	−7.07107	−	12.2474i	−0.540738	−	0.936586i
$172$		0			0
$173$	−0.707107	+	1.22474i	−0.0537603	+	0.0931156i	−0.891653	−	0.452719i	$-0.850454\pi$
$173$	−0.707107	+	1.22474i	−0.0537603	+	0.0931156i	0.837893	+	0.545835i	$0.183787\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−12.0000	+	20.7846i	−0.901975	+	1.56227i
$178$		0			0
$179$	8.00000	+	13.8564i	0.597948	+	1.03568i	0.993124	+	0.117071i	$0.0373504\pi$
$179$	8.00000	+	13.8564i	0.597948	+	1.03568i	−0.395175	+	0.918606i	$0.629316\pi$
$180$		0			0
$181$	12.7279			0.946059			0.473029	−	0.881047i	$-0.343160\pi$
$181$	12.7279			0.946059			0.473029	+	0.881047i	$0.343160\pi$
$182$		0			0
$183$	−4.00000			−0.295689
$184$		0			0
$185$	5.65685	+	9.79796i	0.415900	+	0.720360i
$186$		0			0
$187$	3.53553	−	6.12372i	0.258544	−	0.447811i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	−3.00000	−	5.19615i	−0.215945	−	0.374027i	0.737620	−	0.675216i	$-0.235950\pi$
$193$	−3.00000	−	5.19615i	−0.215945	−	0.374027i	−0.953564	+	0.301189i	$0.902616\pi$
$194$		0			0
$195$	−5.65685			−0.405096
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	8.48528	+	14.6969i	0.601506	+	1.04184i	0.992593	+	0.121485i	$0.0387656\pi$
$199$	8.48528	+	14.6969i	0.601506	+	1.04184i	−0.391088	+	0.920353i	$0.627901\pi$
$200$		0			0
$201$	11.3137	−	19.5959i	0.798007	−	1.38219i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−7.00000	+	12.1244i	−0.488901	+	0.846802i
$206$		0			0
$207$	−10.0000	−	17.3205i	−0.695048	−	1.20386i
$208$		0			0
$209$	−2.82843			−0.195646
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$	−11.3137	−	19.5959i	−0.775203	−	1.34269i
$214$		0			0
$215$	−2.82843	+	4.89898i	−0.192897	+	0.334108i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	2.00000	−	3.46410i	0.135147	−	0.234082i
$220$		0			0
$221$	5.00000	+	8.66025i	0.336336	+	0.582552i
$222$		0			0
$223$	−11.3137			−0.757622			−0.378811	−	0.925474i	$-0.623667\pi$
$223$	−11.3137			−0.757622			−0.378811	+	0.925474i	$0.623667\pi$
$224$		0			0
$225$	−15.0000			−1.00000
$226$		0			0
$227$	1.41421	+	2.44949i	0.0938647	+	0.162578i	0.909134	−	0.416503i	$-0.136745\pi$
$227$	1.41421	+	2.44949i	0.0938647	+	0.162578i	−0.815270	+	0.579082i	$0.803411\pi$
$228$		0			0
$229$	−6.36396	+	11.0227i	−0.420542	+	0.728401i	−0.995993	−	0.0894361i	$-0.971494\pi$
$229$	−6.36396	+	11.0227i	−0.420542	+	0.728401i	0.575450	+	0.817837i	$0.304827\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.0000	−	20.7846i	0.786146	−	1.36165i	−0.142166	−	0.989843i	$-0.545407\pi$
$233$	12.0000	−	20.7846i	0.786146	−	1.36165i	0.928312	−	0.371802i	$-0.121260\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	45.2548			2.93962
$238$		0			0
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$	13.4350	+	23.2702i	0.865426	+	1.49896i	0.866623	+	0.498963i	$0.166286\pi$
$241$	13.4350	+	23.2702i	0.865426	+	1.49896i	−0.00119700	+	0.999999i	$0.500381\pi$
$242$		0			0
$243$	7.07107	−	12.2474i	0.453609	−	0.785674i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.00000	−	3.46410i	0.127257	−	0.220416i
$248$		0			0
$249$	4.00000	+	6.92820i	0.253490	+	0.439057i
$250$		0			0
$251$	14.1421			0.892644			0.446322	−	0.894873i	$-0.352734\pi$
$251$	14.1421			0.892644			0.446322	+	0.894873i	$0.352734\pi$
$252$		0			0
$253$	−4.00000			−0.251478
$254$		0			0
$255$	−14.1421	−	24.4949i	−0.885615	−	1.53393i
$256$		0			0
$257$	0.707107	−	1.22474i	0.0441081	−	0.0763975i	−0.843129	−	0.537712i	$-0.819289\pi$
$257$	0.707107	−	1.22474i	0.0441081	−	0.0763975i	0.887237	+	0.461315i	$0.152622\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	0.952517	+	0.304487i	$0.0984850\pi$
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	−0.212565	+	0.977147i	$0.568182\pi$
$264$		0			0
$265$	−8.48528			−0.521247
$266$		0			0
$267$	−44.0000			−2.69276
$268$		0			0
$269$	3.53553	+	6.12372i	0.215565	+	0.373370i	0.953447	−	0.301560i	$-0.0975073\pi$
$269$	3.53553	+	6.12372i	0.215565	+	0.373370i	−0.737882	+	0.674930i	$0.764174\pi$
$270$		0			0
$271$	−8.48528	+	14.6969i	−0.515444	+	0.892775i	0.484395	+	0.874849i	$0.339040\pi$
$271$	−8.48528	+	14.6969i	−0.515444	+	0.892775i	−0.999839	+	0.0179261i	$0.994294\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.50000	+	2.59808i	−0.0904534	+	0.156670i
$276$		0			0
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$		0			0
$279$	−28.2843			−1.69334
$280$		0			0
$281$	24.0000			1.43172			0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000			1.43172			0.715860	+	0.698244i	$0.246035\pi$
$282$		0			0
$283$	−4.24264	−	7.34847i	−0.252199	−	0.436821i	0.711932	−	0.702248i	$-0.247820\pi$
$283$	−4.24264	−	7.34847i	−0.252199	−	0.436821i	−0.964131	+	0.265427i	$0.914487\pi$
$284$		0			0
$285$	−5.65685	+	9.79796i	−0.335083	+	0.580381i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.5000	+	28.5788i	−0.970588	+	1.68111i
$290$		0			0
$291$	−14.0000	−	24.2487i	−0.820695	−	1.42148i
$292$		0			0
$293$	−29.6985			−1.73500			−0.867502	−	0.497434i	$-0.834276\pi$
$293$	−29.6985			−1.73500			−0.867502	+	0.497434i	$0.834276\pi$
$294$		0			0
$295$	12.0000			0.698667
$296$		0			0
$297$	2.82843	+	4.89898i	0.164122	+	0.284268i
$298$		0			0
$299$	2.82843	−	4.89898i	0.163572	−	0.283315i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	26.0000	−	45.0333i	1.49366	−	2.58710i
$304$		0			0
$305$	1.00000	+	1.73205i	0.0572598	+	0.0991769i
$306$		0			0
$307$	14.1421			0.807134			0.403567	−	0.914950i	$-0.367770\pi$
$307$	14.1421			0.807134			0.403567	+	0.914950i	$0.367770\pi$
$308$		0			0
$309$	−48.0000			−2.73062
$310$		0			0
$311$	2.82843	+	4.89898i	0.160385	+	0.277796i	0.935007	−	0.354629i	$-0.115393\pi$
$311$	2.82843	+	4.89898i	0.160385	+	0.277796i	−0.774622	+	0.632425i	$0.782060\pi$
$312$		0			0
$313$	−13.4350	+	23.2702i	−0.759393	+	1.31531i	0.183768	+	0.982970i	$0.441171\pi$
$313$	−13.4350	+	23.2702i	−0.759393	+	1.31531i	−0.943161	+	0.332337i	$0.892163\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.00000	−	1.73205i	0.0561656	−	0.0972817i	−0.836576	−	0.547852i	$-0.815446\pi$
$317$	1.00000	−	1.73205i	0.0561656	−	0.0972817i	0.892741	+	0.450570i	$0.148779\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−22.6274			−1.26294
$322$		0			0
$323$	20.0000			1.11283
$324$		0			0
$325$	−2.12132	−	3.67423i	−0.117670	−	0.203810i
$326$		0			0
$327$	−22.6274	+	39.1918i	−1.25130	+	2.16731i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−14.0000	+	24.2487i	−0.769510	+	1.33283i	0.168320	+	0.985732i	$0.446166\pi$
$331$	−14.0000	+	24.2487i	−0.769510	+	1.33283i	−0.937829	+	0.347097i	$0.887167\pi$
$332$		0			0
$333$	20.0000	+	34.6410i	1.09599	+	1.89832i
$334$		0			0
$335$	−11.3137			−0.618134
$336$		0			0
$337$	16.0000			0.871576			0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000			0.871576			0.435788	+	0.900049i	$0.356470\pi$
$338$		0			0
$339$	−8.48528	−	14.6969i	−0.460857	−	0.798228i
$340$		0			0
$341$	−2.82843	+	4.89898i	−0.153168	+	0.265295i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−8.00000	+	13.8564i	−0.430706	+	0.746004i
$346$		0			0
$347$	−6.00000	−	10.3923i	−0.322097	−	0.557888i	0.658824	−	0.752297i	$-0.271054\pi$
$347$	−6.00000	−	10.3923i	−0.322097	−	0.557888i	−0.980921	+	0.194409i	$0.937721\pi$
$348$		0			0
$349$	−9.89949			−0.529908			−0.264954	−	0.964261i	$-0.585357\pi$
$349$	−9.89949			−0.529908			−0.264954	+	0.964261i	$0.585357\pi$
$350$		0			0
$351$	−8.00000			−0.427008
$352$		0			0
$353$	−4.94975	−	8.57321i	−0.263448	−	0.456306i	0.703707	−	0.710490i	$-0.251527\pi$
$353$	−4.94975	−	8.57321i	−0.263448	−	0.456306i	−0.967156	+	0.254184i	$0.918193\pi$
$354$		0			0
$355$	−5.65685	+	9.79796i	−0.300235	+	0.520022i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.0000	+	17.3205i	−0.527780	+	0.914141i	0.471696	+	0.881761i	$0.343642\pi$
$359$	−10.0000	+	17.3205i	−0.527780	+	0.914141i	−0.999476	+	0.0323801i	$0.989691\pi$
$360$		0			0
$361$	5.50000	+	9.52628i	0.289474	+	0.501383i
$362$		0			0
$363$	2.82843			0.148454
$364$		0			0
$365$	−2.00000			−0.104685
$366$		0			0
$367$	16.9706	+	29.3939i	0.885856	+	1.53435i	0.844729	+	0.535194i	$0.179761\pi$
$367$	16.9706	+	29.3939i	0.885856	+	1.53435i	0.0411270	+	0.999154i	$0.486905\pi$
$368$		0			0
$369$	−24.7487	+	42.8661i	−1.28837	+	2.23152i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	19.0000	−	32.9090i	0.983783	−	1.70396i	0.336557	−	0.941663i	$-0.390737\pi$
$373$	19.0000	−	32.9090i	0.983783	−	1.70396i	0.647225	−	0.762299i	$-0.275929\pi$
$374$		0			0
$375$	16.0000	+	27.7128i	0.826236	+	1.43108i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	36.0000			1.84920			0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000			1.84920			0.924598	+	0.380945i	$0.124401\pi$
$380$		0			0
$381$	−16.9706	−	29.3939i	−0.869428	−	1.50589i
$382$		0			0
$383$	−11.3137	+	19.5959i	−0.578103	+	1.00130i	0.417593	+	0.908634i	$0.362874\pi$
$383$	−11.3137	+	19.5959i	−0.578103	+	1.00130i	−0.995697	+	0.0926706i	$0.970460\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−10.0000	+	17.3205i	−0.508329	+	0.880451i
$388$		0			0
$389$	−4.00000	−	6.92820i	−0.202808	−	0.351274i	0.746624	−	0.665246i	$-0.231673\pi$
$389$	−4.00000	−	6.92820i	−0.202808	−	0.351274i	−0.949432	+	0.313972i	$0.898340\pi$
$390$		0			0
$391$	28.2843			1.43040
$392$		0			0
$393$	8.00000			0.403547
$394$		0			0
$395$	−11.3137	−	19.5959i	−0.569254	−	0.985978i
$396$		0			0
$397$	3.53553	−	6.12372i	0.177443	−	0.307341i	−0.763561	−	0.645736i	$-0.776551\pi$
$397$	3.53553	−	6.12372i	0.177443	−	0.307341i	0.941004	+	0.338395i	$0.109884\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0000	−	20.7846i	0.599251	−	1.03793i	−0.393680	−	0.919247i	$-0.628798\pi$
$401$	12.0000	−	20.7846i	0.599251	−	1.03793i	0.992932	−	0.118686i	$-0.0378683\pi$
$402$		0			0
$403$	−4.00000	−	6.92820i	−0.199254	−	0.345118i
$404$		0			0
$405$	1.41421			0.0702728
$406$		0			0
$407$	8.00000			0.396545
$408$		0			0
$409$	12.0208	+	20.8207i	0.594391	+	1.02952i	0.993632	+	0.112670i	$0.0359402\pi$
$409$	12.0208	+	20.8207i	0.594391	+	1.02952i	−0.399241	+	0.916846i	$0.630726\pi$
$410$		0			0
$411$	−22.6274	+	39.1918i	−1.11613	+	1.93319i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	2.00000	−	3.46410i	0.0981761	−	0.170046i
$416$		0			0
$417$	28.0000	+	48.4974i	1.37117	+	2.37493i
$418$		0			0
$419$	−25.4558			−1.24360			−0.621800	−	0.783176i	$-0.713598\pi$
$419$	−25.4558			−1.24360			−0.621800	+	0.783176i	$0.713598\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	10.6066	−	18.3712i	0.514496	−	0.891133i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−2.00000	+	3.46410i	−0.0965609	+	0.167248i
$430$		0			0
$431$	−2.00000	−	3.46410i	−0.0963366	−	0.166860i	0.813829	−	0.581104i	$-0.197379\pi$
$431$	−2.00000	−	3.46410i	−0.0963366	−	0.166860i	−0.910166	+	0.414244i	$0.864046\pi$
$432$		0			0
$433$	12.7279			0.611665			0.305832	−	0.952085i	$-0.401065\pi$
$433$	12.7279			0.611665			0.305832	+	0.952085i	$0.401065\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.65685	−	9.79796i	−0.270604	−	0.468700i
$438$		0			0
$439$	8.48528	−	14.6969i	0.404980	−	0.701447i	−0.589339	−	0.807886i	$-0.700612\pi$
$439$	8.48528	−	14.6969i	0.404980	−	0.701447i	0.994319	+	0.106439i	$0.0339450\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$	11.0000	+	19.0526i	0.521450	+	0.903178i
$446$		0			0
$447$	−62.2254			−2.94316
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	4.94975	+	8.57321i	0.233075	+	0.403697i
$452$		0			0
$453$	−5.65685	+	9.79796i	−0.265782	+	0.460348i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7.00000	+	12.1244i	−0.327446	+	0.567153i	−0.982004	−	0.188858i	$-0.939521\pi$
$457$	−7.00000	+	12.1244i	−0.327446	+	0.567153i	0.654558	+	0.756012i	$0.272855\pi$
$458$		0			0
$459$	−20.0000	−	34.6410i	−0.933520	−	1.61690i
$460$		0			0
$461$	24.0416			1.11973			0.559865	−	0.828584i	$-0.310853\pi$
$461$	24.0416			1.11973			0.559865	+	0.828584i	$0.310853\pi$
$462$		0			0
$463$	16.0000			0.743583			0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000			0.743583			0.371792	+	0.928316i	$0.378744\pi$
$464$		0			0
$465$	11.3137	+	19.5959i	0.524661	+	0.908739i
$466$		0			0
$467$	−4.24264	+	7.34847i	−0.196326	+	0.340047i	−0.947334	−	0.320246i	$-0.896234\pi$
$467$	−4.24264	+	7.34847i	−0.196326	+	0.340047i	0.751008	+	0.660293i	$0.229568\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	22.0000	−	38.1051i	1.01371	−	1.75579i
$472$		0			0
$473$	2.00000	+	3.46410i	0.0919601	+	0.159280i
$474$		0			0
$475$	−8.48528			−0.389331
$476$		0			0
$477$	−30.0000			−1.37361
$478$		0			0
$479$	16.9706	+	29.3939i	0.775405	+	1.34304i	0.934567	+	0.355788i	$0.115787\pi$
$479$	16.9706	+	29.3939i	0.775405	+	1.34304i	−0.159162	+	0.987252i	$0.550879\pi$
$480$		0			0
$481$	−5.65685	+	9.79796i	−0.257930	+	0.446748i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.00000	+	12.1244i	−0.317854	+	0.550539i
$486$		0			0
$487$	10.0000	+	17.3205i	0.453143	+	0.784867i	0.998579	−	0.0532853i	$-0.0169693\pi$
$487$	10.0000	+	17.3205i	0.453143	+	0.784867i	−0.545436	+	0.838152i	$0.683636\pi$
$488$		0			0
$489$	−11.3137			−0.511624
$490$		0			0
$491$	−32.0000			−1.44414			−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000			−1.44414			−0.722070	+	0.691820i	$0.756809\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	3.53553	−	6.12372i	0.158910	−	0.275241i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−12.0000	+	20.7846i	−0.537194	+	0.930447i	0.461860	+	0.886953i	$0.347182\pi$
$499$	−12.0000	+	20.7846i	−0.537194	+	0.930447i	−0.999054	+	0.0434940i	$0.986151\pi$
$500$		0			0
$501$	−8.00000	−	13.8564i	−0.357414	−	0.619059i
$502$		0			0
$503$	−5.65685			−0.252227			−0.126113	−	0.992016i	$-0.540250\pi$
$503$	−5.65685			−0.252227			−0.126113	+	0.992016i	$0.540250\pi$
$504$		0			0
$505$	−26.0000			−1.15698
$506$		0			0
$507$	15.5563	+	26.9444i	0.690882	+	1.19664i
$508$		0			0
$509$	14.8492	−	25.7196i	0.658181	−	1.14000i	−0.322905	−	0.946431i	$-0.604659\pi$
$509$	14.8492	−	25.7196i	0.658181	−	1.14000i	0.981086	−	0.193572i	$-0.0620072\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−8.00000	+	13.8564i	−0.353209	+	0.611775i
$514$		0			0
$515$	12.0000	+	20.7846i	0.528783	+	0.915879i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	4.00000			0.175581
$520$		0			0
$521$	3.53553	+	6.12372i	0.154895	+	0.268285i	0.933021	−	0.359823i	$-0.117163\pi$
$521$	3.53553	+	6.12372i	0.154895	+	0.268285i	−0.778126	+	0.628108i	$0.783830\pi$
$522$		0			0
$523$	−21.2132	+	36.7423i	−0.927589	+	1.60663i	−0.140244	+	0.990117i	$0.544789\pi$
$523$	−21.2132	+	36.7423i	−0.927589	+	1.60663i	−0.787344	+	0.616514i	$0.788544\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	20.0000	−	34.6410i	0.871214	−	1.50899i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$	42.4264			1.84115
$532$		0			0
$533$	−14.0000			−0.606407
$534$		0			0
$535$	5.65685	+	9.79796i	0.244567	+	0.423603i
$536$		0			0
$537$	22.6274	−	39.1918i	0.976445	−	1.69125i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	15.0000	−	25.9808i	0.644900	−	1.11700i	−0.339424	−	0.940633i	$-0.610232\pi$
$541$	15.0000	−	25.9808i	0.644900	−	1.11700i	0.984325	−	0.176367i	$-0.0564345\pi$
$542$		0			0
$543$	−18.0000	−	31.1769i	−0.772454	−	1.33793i
$544$		0			0
$545$	22.6274			0.969252
$546$		0			0
$547$	−20.0000			−0.855138			−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000			−0.855138			−0.427569	+	0.903983i	$0.640630\pi$
$548$		0			0
$549$	3.53553	+	6.12372i	0.150893	+	0.261354i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	16.0000	−	27.7128i	0.679162	−	1.17634i
$556$		0			0
$557$	9.00000	+	15.5885i	0.381342	+	0.660504i	0.991254	−	0.131965i	$-0.0421286\pi$
$557$	9.00000	+	15.5885i	0.381342	+	0.660504i	−0.609912	+	0.792469i	$0.708795\pi$
$558$		0			0
$559$	−5.65685			−0.239259
$560$		0			0
$561$	−20.0000			−0.844401
$562$		0			0
$563$	−18.3848	−	31.8434i	−0.774826	−	1.34204i	−0.934892	−	0.354932i	$-0.884504\pi$
$563$	−18.3848	−	31.8434i	−0.774826	−	1.34204i	0.160066	−	0.987106i	$-0.448829\pi$
$564$		0			0
$565$	−4.24264	+	7.34847i	−0.178489	+	0.309152i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	−22.0000	−	38.1051i	−0.920671	−	1.59465i	−0.798379	−	0.602155i	$-0.794309\pi$
$571$	−22.0000	−	38.1051i	−0.920671	−	1.59465i	−0.122292	−	0.992494i	$-0.539025\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.0000			−0.500435
$576$		0			0
$577$	10.6066	+	18.3712i	0.441559	+	0.764802i	0.997805	−	0.0662152i	$-0.0210924\pi$
$577$	10.6066	+	18.3712i	0.441559	+	0.764802i	−0.556247	+	0.831017i	$0.687759\pi$
$578$		0			0
$579$	−8.48528	+	14.6969i	−0.352636	+	0.610784i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−3.00000	+	5.19615i	−0.124247	+	0.215203i
$584$		0			0
$585$	5.00000	+	8.66025i	0.206725	+	0.358057i
$586$		0			0
$587$	−14.1421			−0.583708			−0.291854	−	0.956463i	$-0.594272\pi$
$587$	−14.1421			−0.583708			−0.291854	+	0.956463i	$0.594272\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	−8.48528	−	14.6969i	−0.349038	−	0.604551i
$592$		0			0
$593$	−3.53553	+	6.12372i	−0.145187	+	0.251471i	−0.929443	−	0.368967i	$-0.879712\pi$
$593$	−3.53553	+	6.12372i	−0.145187	+	0.251471i	0.784256	+	0.620438i	$0.213045\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	24.0000	−	41.5692i	0.982255	−	1.70131i
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−35.3553			−1.44217			−0.721087	−	0.692844i	$-0.756357\pi$
$601$	−35.3553			−1.44217			−0.721087	+	0.692844i	$0.756357\pi$
$602$		0			0
$603$	−40.0000			−1.62893
$604$		0			0
$605$	−0.707107	−	1.22474i	−0.0287480	−	0.0497930i
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−16.0000	−	27.7128i	−0.646234	−	1.11931i	−0.984015	−	0.178085i	$-0.943010\pi$
$613$	−16.0000	−	27.7128i	−0.646234	−	1.11931i	0.337781	−	0.941225i	$-0.390324\pi$
$614$		0			0
$615$	39.5980			1.59674
$616$		0			0
$617$	−24.0000			−0.966204			−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000			−0.966204			−0.483102	+	0.875564i	$0.660490\pi$
$618$		0			0
$619$	−4.24264	−	7.34847i	−0.170526	−	0.295360i	0.768078	−	0.640357i	$-0.221213\pi$
$619$	−4.24264	−	7.34847i	−0.170526	−	0.295360i	−0.938604	+	0.344997i	$0.887880\pi$
$620$		0			0
$621$	−11.3137	+	19.5959i	−0.454003	+	0.786357i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.500000	−	0.866025i	0.0200000	−	0.0346410i
$626$		0			0
$627$	4.00000	+	6.92820i	0.159745	+	0.276686i
$628$		0			0
$629$	−56.5685			−2.25554
$630$		0			0
$631$	48.0000			1.91085			0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000			1.91085			0.955425	+	0.295234i	$0.0953977\pi$
$632$		0			0
$633$	11.3137	+	19.5959i	0.449680	+	0.778868i
$634$		0			0
$635$	−8.48528	+	14.6969i	−0.336728	+	0.583230i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−20.0000	+	34.6410i	−0.791188	+	1.37038i
$640$		0			0
$641$	12.0000	+	20.7846i	0.473972	+	0.820943i	0.999556	−	0.0297987i	$-0.00948663\pi$
$641$	12.0000	+	20.7846i	0.473972	+	0.820943i	−0.525584	+	0.850741i	$0.676153\pi$
$642$		0			0
$643$	42.4264			1.67313			0.836567	−	0.547865i	$-0.184559\pi$
$643$	42.4264			1.67313			0.836567	+	0.547865i	$0.184559\pi$
$644$		0			0
$645$	16.0000			0.629999
$646$		0			0
$647$	−14.1421	−	24.4949i	−0.555985	−	0.962994i	−0.997826	−	0.0659006i	$-0.979008\pi$
$647$	−14.1421	−	24.4949i	−0.555985	−	0.962994i	0.441841	−	0.897093i	$-0.354325\pi$
$648$		0			0
$649$	4.24264	−	7.34847i	0.166538	−	0.288453i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.0000	+	20.7846i	−0.469596	+	0.813365i	−0.999396	−	0.0347583i	$-0.988934\pi$
$653$	−12.0000	+	20.7846i	−0.469596	+	0.813365i	0.529799	+	0.848123i	$0.322267\pi$
$654$		0			0
$655$	−2.00000	−	3.46410i	−0.0781465	−	0.135354i
$656$		0			0
$657$	−7.07107			−0.275869
$658$		0			0
$659$	−28.0000			−1.09073			−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000			−1.09073			−0.545363	+	0.838200i	$0.683608\pi$
$660$		0			0
$661$	10.6066	+	18.3712i	0.412549	+	0.714556i	0.995168	−	0.0981898i	$-0.0313052\pi$
$661$	10.6066	+	18.3712i	0.412549	+	0.714556i	−0.582619	+	0.812746i	$0.697972\pi$
$662$		0			0
$663$	14.1421	−	24.4949i	0.549235	−	0.951303i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	16.0000	+	27.7128i	0.618596	+	1.07144i
$670$		0			0
$671$	1.41421			0.0545951
$672$		0			0
$673$	8.00000			0.308377			0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000			0.308377			0.154189	+	0.988041i	$0.450724\pi$
$674$		0			0
$675$	8.48528	+	14.6969i	0.326599	+	0.565685i
$676$		0			0
$677$	24.7487	−	42.8661i	0.951171	−	1.64748i	0.208275	−	0.978070i	$-0.433215\pi$
$677$	24.7487	−	42.8661i	0.951171	−	1.64748i	0.742896	−	0.669407i	$-0.233452\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	4.00000	−	6.92820i	0.153280	−	0.265489i
$682$		0			0
$683$	−24.0000	−	41.5692i	−0.918334	−	1.59060i	−0.801945	−	0.597398i	$-0.796201\pi$
$683$	−24.0000	−	41.5692i	−0.918334	−	1.59060i	−0.116390	−	0.993204i	$-0.537132\pi$
$684$		0			0
$685$	22.6274			0.864549
$686$		0			0
$687$	36.0000			1.37349
$688$		0			0
$689$	−4.24264	−	7.34847i	−0.161632	−	0.279954i
$690$		0			0
$691$	1.41421	−	2.44949i	0.0537992	−	0.0931830i	−0.837872	−	0.545867i	$-0.816200\pi$
$691$	1.41421	−	2.44949i	0.0537992	−	0.0931830i	0.891671	+	0.452684i	$0.149534\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	14.0000	−	24.2487i	0.531050	−	0.919806i
$696$		0			0
$697$	−35.0000	−	60.6218i	−1.32572	−	2.29621i
$698$		0			0
$699$	−67.8823			−2.56754
$700$		0			0
$701$	32.0000			1.20862			0.604312	−	0.796748i	$-0.293448\pi$
$701$	32.0000			1.20862			0.604312	+	0.796748i	$0.293448\pi$
$702$		0			0
$703$	11.3137	+	19.5959i	0.426705	+	0.739074i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−20.0000	+	34.6410i	−0.751116	+	1.30097i	0.196167	+	0.980571i	$0.437151\pi$
$709$	−20.0000	+	34.6410i	−0.751116	+	1.30097i	−0.947282	+	0.320400i	$0.896183\pi$
$710$		0			0
$711$	−40.0000	−	69.2820i	−1.50012	−	2.59828i
$712$		0			0
$713$	−22.6274			−0.847403
$714$		0			0
$715$	2.00000			0.0747958
$716$		0			0
$717$	−28.2843	−	48.9898i	−1.05630	−	1.82956i
$718$		0			0
$719$	5.65685	−	9.79796i	0.210965	−	0.365402i	−0.741052	−	0.671448i	$-0.765673\pi$
$719$	5.65685	−	9.79796i	0.210965	−	0.365402i	0.952017	+	0.306046i	$0.0990060\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	38.0000	−	65.8179i	1.41324	−	2.44780i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−11.3137			−0.419602			−0.209801	−	0.977744i	$-0.567282\pi$
$727$	−11.3137			−0.419602			−0.209801	+	0.977744i	$0.567282\pi$
$728$		0			0
$729$	−43.0000			−1.59259
$730$		0			0
$731$	−14.1421	−	24.4949i	−0.523066	−	0.905977i
$732$		0			0
$733$	10.6066	−	18.3712i	0.391764	−	0.678555i	−0.600919	−	0.799310i	$-0.705198\pi$
$733$	10.6066	−	18.3712i	0.391764	−	0.678555i	0.992682	+	0.120756i	$0.0385317\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.00000	+	6.92820i	−0.147342	+	0.255204i
$738$		0			0
$739$	−18.0000	−	31.1769i	−0.662141	−	1.14686i	−0.980052	−	0.198741i	$-0.936315\pi$
$739$	−18.0000	−	31.1769i	−0.662141	−	1.14686i	0.317911	−	0.948120i	$-0.397019\pi$
$740$		0			0
$741$	−11.3137			−0.415619
$742$		0			0
$743$	12.0000			0.440237			0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000			0.440237			0.220119	+	0.975473i	$0.429356\pi$
$744$		0			0
$745$	15.5563	+	26.9444i	0.569941	+	0.987166i
$746$		0			0
$747$	7.07107	−	12.2474i	0.258717	−	0.448111i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	14.0000	−	24.2487i	0.510867	−	0.884848i	−0.489053	−	0.872254i	$-0.662658\pi$
$751$	14.0000	−	24.2487i	0.510867	−	0.884848i	0.999921	−	0.0125942i	$-0.00400897\pi$
$752$		0			0
$753$	−20.0000	−	34.6410i	−0.728841	−	1.26239i
$754$		0			0
$755$	5.65685			0.205874
$756$		0			0
$757$	−8.00000			−0.290765			−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000			−0.290765			−0.145382	+	0.989376i	$0.546441\pi$
$758$		0			0
$759$	5.65685	+	9.79796i	0.205331	+	0.355643i
$760$		0			0
$761$	−6.36396	+	11.0227i	−0.230693	+	0.399573i	−0.958012	−	0.286727i	$-0.907433\pi$
$761$	−6.36396	+	11.0227i	−0.230693	+	0.399573i	0.727319	+	0.686300i	$0.240766\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−25.0000	+	43.3013i	−0.903877	+	1.56556i
$766$		0			0
$767$	6.00000	+	10.3923i	0.216647	+	0.375244i
$768$		0			0
$769$	−4.24264			−0.152994			−0.0764968	−	0.997070i	$-0.524373\pi$
$769$	−4.24264			−0.152994			−0.0764968	+	0.997070i	$0.524373\pi$
$770$		0			0
$771$	−4.00000			−0.144056
$772$		0			0
$773$	4.94975	+	8.57321i	0.178030	+	0.308357i	0.941206	−	0.337834i	$-0.109694\pi$
$773$	4.94975	+	8.57321i	0.178030	+	0.308357i	−0.763176	+	0.646191i	$0.776361\pi$
$774$		0			0
$775$	−8.48528	+	14.6969i	−0.304800	+	0.527930i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−14.0000	+	24.2487i	−0.501602	+	0.868800i
$780$		0			0
$781$	4.00000	+	6.92820i	0.143131	+	0.247911i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−22.0000			−0.785214
$786$		0			0
$787$	4.24264	+	7.34847i	0.151234	+	0.261945i	0.931681	−	0.363277i	$-0.118342\pi$
$787$	4.24264	+	7.34847i	0.151234	+	0.261945i	−0.780447	+	0.625221i	$0.785009\pi$
$788$		0			0
$789$	33.9411	−	58.7878i	1.20834	−	2.09290i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.00000	+	1.73205i	−0.0355110	+	0.0615069i
$794$		0			0

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:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 - 2.44949i) q^{3} +(-0.707107 + 1.22474i) q^{5} +(-2.50000 + 4.33013i) q^{9} +O(q^{10})\)	\(q+(-1.41421 - 2.44949i) q^{3} +(-0.707107 + 1.22474i) q^{5} +(-2.50000 + 4.33013i) q^{9} +(0.500000 + 0.866025i) q^{11} -1.41421 q^{13} +4.00000 q^{15} +(-3.53553 - 6.12372i) q^{17} +(-1.41421 + 2.44949i) q^{19} +(-2.00000 + 3.46410i) q^{23} +(1.50000 + 2.59808i) q^{25} +5.65685 q^{27} +(2.82843 + 4.89898i) q^{31} +(1.41421 - 2.44949i) q^{33} +(4.00000 - 6.92820i) q^{37} +(2.00000 + 3.46410i) q^{39} +9.89949 q^{41} +4.00000 q^{43} +(-3.53553 - 6.12372i) q^{45} +(-10.0000 + 17.3205i) q^{51} +(3.00000 + 5.19615i) q^{53} -1.41421 q^{55} +8.00000 q^{57} +(-4.24264 - 7.34847i) q^{59} +(0.707107 - 1.22474i) q^{61} +(1.00000 - 1.73205i) q^{65} +(4.00000 + 6.92820i) q^{67} +11.3137 q^{69} +8.00000 q^{71} +(0.707107 + 1.22474i) q^{73} +(4.24264 - 7.34847i) q^{75} +(-8.00000 + 13.8564i) q^{79} +(-0.500000 - 0.866025i) q^{81} -2.82843 q^{83} +10.0000 q^{85} +(7.77817 - 13.4722i) q^{89} +(8.00000 - 13.8564i) q^{93} +(-2.00000 - 3.46410i) q^{95} +9.89949 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{9}+O(q^{10}) \) 4 * q - 10 * q^9	\( 4 q - 10 q^{9} + 2 q^{11} + 16 q^{15} - 8 q^{23} + 6 q^{25} + 16 q^{37} + 8 q^{39} + 16 q^{43} - 40 q^{51} + 12 q^{53} + 32 q^{57} + 4 q^{65} + 16 q^{67} + 32 q^{71} - 32 q^{79} - 2 q^{81} + 40 q^{85} + 32 q^{93} - 8 q^{95} - 20 q^{99}+O(q^{100}) \) 4 * q - 10 * q^9 + 2 * q^11 + 16 * q^15 - 8 * q^23 + 6 * q^25 + 16 * q^37 + 8 * q^39 + 16 * q^43 - 40 * q^51 + 12 * q^53 + 32 * q^57 + 4 * q^65 + 16 * q^67 + 32 * q^71 - 32 * q^79 - 2 * q^81 + 40 * q^85 + 32 * q^93 - 8 * q^95 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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