LMFDB - Embedded newform 712.2.e.c.177.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [712,2,Mod(177,712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(712, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("712.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 712 = 2^{3} \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	712.e (of order $2$, degree $1$, 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:	$5.68534862392$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 19x^{8} + 113x^{6} + 217x^{4} + 136x^{2} + 16 $ x^10 + 19x^8 + 113x^6 + 217x^4 + 136x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			177.6
Root			$2.80350i$ of defining polynomial
Character	$\chi$	$=$	712.177
Dual form			712.2.e.c.177.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.255270i q^{3} +2.64044 q^{5} -1.18098i q^{7} +2.93484 q^{9} +O(q^{10})$	$q+0.255270i q^{3} +2.64044 q^{5} -1.18098i q^{7} +2.93484 q^{9} -1.17398 q^{11} -4.59897i q^{13} +0.674025i q^{15} +1.67752 q^{17} +4.09202i q^{19} +0.301469 q^{21} -7.04326i q^{23} +1.97192 q^{25} +1.51499i q^{27} +1.00803i q^{29} +6.87031i q^{31} -0.299683i q^{33} -3.11831i q^{35} -3.66968i q^{37} +1.17398 q^{39} +5.30732i q^{41} -3.67326i q^{43} +7.74926 q^{45} +4.01649 q^{47} +5.60528 q^{49} +0.428220i q^{51} +0.503535 q^{53} -3.09982 q^{55} -1.04457 q^{57} -5.13437i q^{59} +9.70261i q^{61} -3.46599i q^{63} -12.1433i q^{65} +3.09982 q^{67} +1.79793 q^{69} -0.916661 q^{71} +2.14590 q^{73} +0.503371i q^{75} +1.38645i q^{77} -5.19047 q^{79} +8.41778 q^{81} +15.9072i q^{83} +4.42938 q^{85} -0.257321 q^{87} +(-9.05073 - 2.66165i) q^{89} -5.43130 q^{91} -1.75379 q^{93} +10.8047i q^{95} -0.911281 q^{97} -3.44544 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^9	$ 10 q - 10 q^{9} + 8 q^{11} + 8 q^{17} + 12 q^{21} - 2 q^{25} - 8 q^{39} - 8 q^{45} - 4 q^{47} - 14 q^{49} + 16 q^{53} + 24 q^{55} + 12 q^{57} - 24 q^{67} + 16 q^{69} - 20 q^{71} - 20 q^{73} + 12 q^{79} - 6 q^{81} + 20 q^{85} + 28 q^{87} - 14 q^{89} - 4 q^{91} + 24 q^{93} + 8 q^{97} - 28 q^{99}+O(q^{100}) $ 10 * q - 10 * q^9 + 8 * q^11 + 8 * q^17 + 12 * q^21 - 2 * q^25 - 8 * q^39 - 8 * q^45 - 4 * q^47 - 14 * q^49 + 16 * q^53 + 24 * q^55 + 12 * q^57 - 24 * q^67 + 16 * q^69 - 20 * q^71 - 20 * q^73 + 12 * q^79 - 6 * q^81 + 20 * q^85 + 28 * q^87 - 14 * q^89 - 4 * q^91 + 24 * q^93 + 8 * q^97 - 28 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/712\mathbb{Z}\right)^\times$.

$n$	$357$	$535$	$537$
$\chi(n)$	$1$	$1$	$-1$

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
$3$			0.255270i			0.147380i	0.997281	+	0.0736902i	$0.0234776\pi$
$3$			0.255270i			0.147380i	−0.997281	+	0.0736902i	$0.976522\pi$
$4$		0			0
$5$	2.64044			1.18084			0.590420	−	0.807096i	$-0.298962\pi$
$5$	2.64044			1.18084			0.590420	+	0.807096i	$0.298962\pi$
$6$		0			0
$7$		−	1.18098i		−	0.446369i	−0.974776	−	0.223185i	$-0.928355\pi$
$7$		−	1.18098i		−	0.446369i	0.974776	−	0.223185i	$-0.0716452\pi$
$8$		0			0
$9$	2.93484			0.978279
$10$		0			0
$11$	−1.17398			−0.353969			−0.176984	−	0.984214i	$-0.556634\pi$
$11$	−1.17398			−0.353969			−0.176984	+	0.984214i	$0.556634\pi$
$12$		0			0
$13$		−	4.59897i		−	1.27553i	−0.770233	−	0.637763i	$-0.779860\pi$
$13$		−	4.59897i		−	1.27553i	0.770233	−	0.637763i	$-0.220140\pi$
$14$		0			0
$15$			0.674025i			0.174033i
$16$		0			0
$17$	1.67752			0.406858			0.203429	−	0.979090i	$-0.434791\pi$
$17$	1.67752			0.406858			0.203429	+	0.979090i	$0.434791\pi$
$18$		0			0
$19$			4.09202i			0.938773i	0.882993	+	0.469387i	$0.155525\pi$
$19$			4.09202i			0.938773i	−0.882993	+	0.469387i	$0.844475\pi$
$20$		0			0
$21$	0.301469			0.0657860
$22$		0			0
$23$		−	7.04326i		−	1.46862i	−0.678814	−	0.734310i	$-0.737506\pi$
$23$		−	7.04326i		−	1.46862i	0.678814	−	0.734310i	$-0.262494\pi$
$24$		0			0
$25$	1.97192			0.394383
$26$		0			0
$27$			1.51499i			0.291559i
$28$		0			0
$29$			1.00803i			0.187187i	0.995611	+	0.0935934i	$0.0298354\pi$
$29$			1.00803i			0.187187i	−0.995611	+	0.0935934i	$0.970165\pi$
$30$		0			0
$31$			6.87031i			1.23394i	0.786985	+	0.616972i	$0.211641\pi$
$31$			6.87031i			1.23394i	−0.786985	+	0.616972i	$0.788359\pi$
$32$		0			0
$33$		−	0.299683i		−	0.0521680i
$34$		0			0
$35$		−	3.11831i		−	0.527090i
$36$		0			0
$37$		−	3.66968i		−	0.603291i	−0.953420	−	0.301646i	$-0.902464\pi$
$37$		−	3.66968i		−	0.603291i	0.953420	−	0.301646i	$-0.0975359\pi$
$38$		0			0
$39$	1.17398			0.187987
$40$		0			0
$41$			5.30732i			0.828864i	0.910080	+	0.414432i	$0.136020\pi$
$41$			5.30732i			0.828864i	−0.910080	+	0.414432i	$0.863980\pi$
$42$		0			0
$43$		−	3.67326i		−	0.560167i	−0.959976	−	0.280084i	$-0.909638\pi$
$43$		−	3.67326i		−	0.560167i	0.959976	−	0.280084i	$-0.0903622\pi$
$44$		0			0
$45$	7.74926			1.15519
$46$		0			0
$47$	4.01649			0.585865			0.292932	−	0.956133i	$-0.405369\pi$
$47$	4.01649			0.585865			0.292932	+	0.956133i	$0.405369\pi$
$48$		0			0
$49$	5.60528			0.800755
$50$		0			0
$51$			0.428220i			0.0599628i
$52$		0			0
$53$	0.503535			0.0691659			0.0345830	−	0.999402i	$-0.488990\pi$
$53$	0.503535			0.0691659			0.0345830	+	0.999402i	$0.488990\pi$
$54$		0			0
$55$	−3.09982			−0.417980
$56$		0			0
$57$	−1.04457			−0.138357
$58$		0			0
$59$		−	5.13437i		−	0.668438i	−0.942495	−	0.334219i	$-0.891527\pi$
$59$		−	5.13437i		−	0.668438i	0.942495	−	0.334219i	$-0.108473\pi$
$60$		0			0
$61$			9.70261i			1.24229i	0.783695	+	0.621146i	$0.213332\pi$
$61$			9.70261i			1.24229i	−0.783695	+	0.621146i	$0.786668\pi$
$62$		0			0
$63$		−	3.46599i		−	0.436673i
$64$		0			0
$65$		−	12.1433i		−	1.50619i
$66$		0			0
$67$	3.09982			0.378704			0.189352	−	0.981909i	$-0.439361\pi$
$67$	3.09982			0.378704			0.189352	+	0.981909i	$0.439361\pi$
$68$		0			0
$69$	1.79793			0.216446
$70$		0			0
$71$	−0.916661			−0.108788			−0.0543938	−	0.998520i	$-0.517323\pi$
$71$	−0.916661			−0.108788			−0.0543938	+	0.998520i	$0.517323\pi$
$72$		0			0
$73$	2.14590			0.251158			0.125579	−	0.992084i	$-0.459921\pi$
$73$	2.14590			0.251158			0.125579	+	0.992084i	$0.459921\pi$
$74$		0			0
$75$			0.503371i			0.0581243i
$76$		0			0
$77$			1.38645i			0.158001i
$78$		0			0
$79$	−5.19047			−0.583973			−0.291987	−	0.956422i	$-0.594316\pi$
$79$	−5.19047			−0.583973			−0.291987	+	0.956422i	$0.594316\pi$
$80$		0			0
$81$	8.41778			0.935309
$82$		0			0
$83$			15.9072i			1.74604i	0.487683	+	0.873021i	$0.337842\pi$
$83$			15.9072i			1.74604i	−0.487683	+	0.873021i	$0.662158\pi$
$84$		0			0
$85$	4.42938			0.480434
$86$		0			0
$87$	−0.257321			−0.0275877
$88$		0			0
$89$	−9.05073	−	2.66165i	−0.959375	−	0.282134i
$89$	−9.05073	−	2.66165i	−0.959375	−	0.282134i
$90$		0			0
$91$	−5.43130			−0.569355
$92$		0			0
$93$	−1.75379			−0.181859
$94$		0			0
$95$			10.8047i			1.10854i
$96$		0			0
$97$	−0.911281			−0.0925265			−0.0462633	−	0.998929i	$-0.514731\pi$
$97$	−0.911281			−0.0925265			−0.0462633	+	0.998929i	$0.514731\pi$
$98$		0			0
$99$	−3.44544			−0.346280
$100$		0			0
$101$		−	12.5175i		−	1.24554i	−0.782405	−	0.622770i	$-0.786007\pi$
$101$		−	12.5175i		−	1.24554i	0.782405	−	0.622770i	$-0.213993\pi$
$102$		0			0
$103$			5.10005i			0.502523i	0.967919	+	0.251261i	$0.0808454\pi$
$103$			5.10005i			0.502523i	−0.967919	+	0.251261i	$0.919155\pi$
$104$		0			0
$105$	0.796012			0.0776828
$106$		0			0
$107$	−8.38220			−0.810338			−0.405169	−	0.914242i	$-0.632787\pi$
$107$	−8.38220			−0.810338			−0.405169	+	0.914242i	$0.632787\pi$
$108$		0			0
$109$	−2.44737			−0.234415			−0.117208	−	0.993107i	$-0.537394\pi$
$109$	−2.44737			−0.234415			−0.117208	+	0.993107i	$0.537394\pi$
$110$		0			0
$111$	0.936760			0.0889133
$112$		0			0
$113$			3.84020i			0.361255i	0.983552	+	0.180628i	$0.0578129\pi$
$113$			3.84020i			0.361255i	−0.983552	+	0.180628i	$0.942187\pi$
$114$		0			0
$115$		−	18.5973i		−	1.73421i
$116$		0			0
$117$		−	13.4972i		−	1.24782i
$118$		0			0
$119$		−	1.98112i		−	0.181609i
$120$		0			0
$121$	−9.62177			−0.874706
$122$		0			0
$123$	−1.35480			−0.122158
$124$		0			0
$125$	−7.99547			−0.715137
$126$		0			0
$127$		−	1.34792i		−	0.119608i	−0.998210	−	0.0598042i	$-0.980952\pi$
$127$		−	1.34792i		−	0.119608i	0.998210	−	0.0598042i	$-0.0190476\pi$
$128$		0			0
$129$	0.937675			0.0825576
$130$		0			0
$131$	−16.1179			−1.40823			−0.704115	−	0.710086i	$-0.748656\pi$
$131$	−16.1179			−1.40823			−0.704115	+	0.710086i	$0.748656\pi$
$132$		0			0
$133$	4.83260			0.419039
$134$		0			0
$135$			4.00023i			0.344285i
$136$		0			0
$137$			18.0044i			1.53822i	0.639115	+	0.769111i	$0.279301\pi$
$137$			18.0044i			1.53822i	−0.639115	+	0.769111i	$0.720699\pi$
$138$		0			0
$139$	−13.1506			−1.11542			−0.557708	−	0.830037i	$-0.688319\pi$
$139$	−13.1506			−1.11542			−0.557708	+	0.830037i	$0.688319\pi$
$140$		0			0
$141$			1.02529i			0.0863449i
$142$		0			0
$143$			5.39911i			0.451496i
$144$		0			0
$145$			2.66165i			0.221038i
$146$		0			0
$147$			1.43086i			0.118016i
$148$		0			0
$149$			21.3543i			1.74941i	0.484654	+	0.874706i	$0.338946\pi$
$149$			21.3543i			1.74941i	−0.484654	+	0.874706i	$0.661054\pi$
$150$		0			0
$151$		−	5.60943i		−	0.456489i	−0.973604	−	0.228245i	$-0.926701\pi$
$151$		−	5.60943i		−	0.456489i	0.973604	−	0.228245i	$-0.0732986\pi$
$152$		0			0
$153$	4.92324			0.398020
$154$		0			0
$155$			18.1406i			1.45709i
$156$		0			0
$157$	−6.89323			−0.550140			−0.275070	−	0.961424i	$-0.588701\pi$
$157$	−6.89323			−0.550140			−0.275070	+	0.961424i	$0.588701\pi$
$158$		0			0
$159$			0.128538i			0.0101937i
$160$		0			0
$161$	−8.31796			−0.655547
$162$		0			0
$163$		−	6.78440i		−	0.531395i	−0.964056	−	0.265698i	$-0.914398\pi$
$163$		−	6.78440i		−	0.531395i	0.964056	−	0.265698i	$-0.0856023\pi$
$164$		0			0
$165$		−	0.791293i		−	0.0616021i
$166$		0			0
$167$	0.494542			0.0382688			0.0191344	−	0.999817i	$-0.493909\pi$
$167$	0.494542			0.0382688			0.0191344	+	0.999817i	$0.493909\pi$
$168$		0			0
$169$	−8.15055			−0.626965
$170$		0			0
$171$			12.0094i			0.918382i
$172$		0			0
$173$	−13.9720			−1.06227			−0.531137	−	0.847286i	$-0.678235\pi$
$173$	−13.9720			−1.06227			−0.531137	+	0.847286i	$0.678235\pi$
$174$		0			0
$175$		−	2.32880i		−	0.176040i
$176$		0			0
$177$	1.31065			0.0985147
$178$		0			0
$179$	−17.5328			−1.31046			−0.655230	−	0.755430i	$-0.727428\pi$
$179$	−17.5328			−1.31046			−0.655230	+	0.755430i	$0.727428\pi$
$180$		0			0
$181$		−	16.7009i		−	1.24137i	−0.784060	−	0.620685i	$-0.786855\pi$
$181$		−	16.7009i		−	1.24137i	0.784060	−	0.620685i	$-0.213145\pi$
$182$		0			0
$183$	−2.47679			−0.183089
$184$		0			0
$185$		−	9.68956i		−	0.712390i
$186$		0			0
$187$	−1.96937			−0.144015
$188$		0			0
$189$	1.78917			0.130143
$190$		0			0
$191$		−	9.55924i		−	0.691682i	−0.938293	−	0.345841i	$-0.887594\pi$
$191$		−	9.55924i		−	0.691682i	0.938293	−	0.345841i	$-0.112406\pi$
$192$		0			0
$193$			6.36449i			0.458126i	0.973412	+	0.229063i	$0.0735662\pi$
$193$			6.36449i			0.458126i	−0.973412	+	0.229063i	$0.926434\pi$
$194$		0			0
$195$	3.09982			0.221983
$196$		0			0
$197$			2.39934i			0.170946i	0.996340	+	0.0854729i	$0.0272401\pi$
$197$			2.39934i			0.170946i	−0.996340	+	0.0854729i	$0.972760\pi$
$198$		0			0
$199$	19.9696			1.41561			0.707805	−	0.706408i	$-0.249686\pi$
$199$	19.9696			1.41561			0.707805	+	0.706408i	$0.249686\pi$
$200$		0			0
$201$			0.791293i			0.0558135i
$202$		0			0
$203$	1.19047			0.0835544
$204$		0			0
$205$			14.0137i			0.978756i
$206$		0			0
$207$		−	20.6708i		−	1.43672i
$208$		0			0
$209$		−	4.80395i		−	0.332296i
$210$		0			0
$211$			11.8129i			0.813232i	0.913599	+	0.406616i	$0.133291\pi$
$211$			11.8129i			0.813232i	−0.913599	+	0.406616i	$0.866709\pi$
$212$		0			0
$213$		−	0.233996i		−	0.0160332i
$214$		0			0
$215$		−	9.69902i		−	0.661468i
$216$		0			0
$217$	8.11370			0.550794
$218$		0			0
$219$			0.547784i			0.0370158i
$220$		0			0
$221$		−	7.71485i		−	0.518957i
$222$		0			0
$223$	19.3496			1.29574			0.647872	−	0.761749i	$-0.275659\pi$
$223$	19.3496			1.29574			0.647872	+	0.761749i	$0.275659\pi$
$224$		0			0
$225$	5.78725			0.385817
$226$		0			0
$227$	−4.37859			−0.290617			−0.145309	−	0.989386i	$-0.546417\pi$
$227$	−4.37859			−0.290617			−0.145309	+	0.989386i	$0.546417\pi$
$228$		0			0
$229$			25.1027i			1.65883i	0.558630	+	0.829417i	$0.311327\pi$
$229$			25.1027i			1.65883i	−0.558630	+	0.829417i	$0.688673\pi$
$230$		0			0
$231$	−0.353919			−0.0232862
$232$		0			0
$233$	−10.7284			−0.702843			−0.351421	−	0.936217i	$-0.614302\pi$
$233$	−10.7284			−0.702843			−0.351421	+	0.936217i	$0.614302\pi$
$234$		0			0
$235$	10.6053			0.691812
$236$		0			0
$237$		−	1.32497i		−	0.0860662i
$238$		0			0
$239$		−	10.8800i		−	0.703769i	−0.936043	−	0.351884i	$-0.885541\pi$
$239$		−	10.8800i		−	0.703769i	0.936043	−	0.351884i	$-0.114459\pi$
$240$		0			0
$241$		−	22.9919i		−	1.48104i	−0.672034	−	0.740521i	$-0.734579\pi$
$241$		−	22.9919i		−	1.48104i	0.672034	−	0.740521i	$-0.265421\pi$
$242$		0			0
$243$			6.69377i			0.429406i
$244$		0			0
$245$	14.8004			0.945563
$246$		0			0
$247$	18.8191			1.19743
$248$		0			0
$249$	−4.06063			−0.257332
$250$		0			0
$251$	−19.0728			−1.20386			−0.601932	−	0.798548i	$-0.705602\pi$
$251$	−19.0728			−1.20386			−0.601932	+	0.798548i	$0.705602\pi$
$252$		0			0
$253$			8.26865i			0.519846i
$254$		0			0
$255$			1.13069i			0.0708065i
$256$		0			0
$257$	−16.8145			−1.04886			−0.524431	−	0.851453i	$-0.675722\pi$
$257$	−16.8145			−1.04886			−0.524431	+	0.851453i	$0.675722\pi$
$258$		0			0
$259$	−4.33382			−0.269291
$260$		0			0
$261$			2.95841i			0.183121i
$262$		0			0
$263$	20.5077			1.26456			0.632279	−	0.774740i	$-0.282119\pi$
$263$	20.5077			1.26456			0.632279	+	0.774740i	$0.282119\pi$
$264$		0			0
$265$	1.32955			0.0816739
$266$		0			0
$267$	0.679439	−	2.31038i	0.0415810	−	0.141393i
$268$		0			0
$269$	−17.0852			−1.04170			−0.520850	−	0.853648i	$-0.674385\pi$
$269$	−17.0852			−1.04170			−0.520850	+	0.853648i	$0.674385\pi$
$270$		0			0
$271$	23.0205			1.39839			0.699197	−	0.714929i	$-0.253541\pi$
$271$	23.0205			1.39839			0.699197	+	0.714929i	$0.253541\pi$
$272$		0			0
$273$		−	1.38645i		−	0.0839118i
$274$		0			0
$275$	−2.31499			−0.139599
$276$		0			0
$277$	−0.933830			−0.0561084			−0.0280542	−	0.999606i	$-0.508931\pi$
$277$	−0.933830			−0.0561084			−0.0280542	+	0.999606i	$0.508931\pi$
$278$		0			0
$279$			20.1632i			1.20714i
$280$		0			0
$281$		−	8.49844i		−	0.506974i	−0.967339	−	0.253487i	$-0.918422\pi$
$281$		−	8.49844i		−	0.506974i	0.967339	−	0.253487i	$-0.0815776\pi$
$282$		0			0
$283$	19.9802			1.18770			0.593849	−	0.804576i	$-0.297608\pi$
$283$	19.9802			1.18770			0.593849	+	0.804576i	$0.297608\pi$
$284$		0			0
$285$	−2.75812			−0.163377
$286$		0			0
$287$	6.26785			0.369979
$288$		0			0
$289$	−14.1859			−0.834467
$290$		0			0
$291$		−	0.232623i		−	0.0136366i
$292$		0			0
$293$			16.6463i			0.972486i	0.873824	+	0.486243i	$0.161633\pi$
$293$			16.6463i			0.972486i	−0.873824	+	0.486243i	$0.838367\pi$
$294$		0			0
$295$		−	13.5570i		−	0.789319i
$296$		0			0
$297$		−	1.77857i		−	0.103203i
$298$		0			0
$299$	−32.3917			−1.87326
$300$		0			0
$301$	−4.33805			−0.250041
$302$		0			0
$303$	3.19535			0.183568
$304$		0			0
$305$			25.6191i			1.46695i
$306$		0			0
$307$	−3.37002			−0.192337			−0.0961685	−	0.995365i	$-0.530659\pi$
$307$	−3.37002			−0.192337			−0.0961685	+	0.995365i	$0.530659\pi$
$308$		0			0
$309$	−1.30189			−0.0740620
$310$		0			0
$311$	26.9934			1.53066			0.765328	−	0.643640i	$-0.222577\pi$
$311$	26.9934			1.53066			0.765328	+	0.643640i	$0.222577\pi$
$312$		0			0
$313$		−	10.5187i		−	0.594553i	−0.954791	−	0.297276i	$-0.903922\pi$
$313$		−	10.5187i		−	0.594553i	0.954791	−	0.297276i	$-0.0960782\pi$
$314$		0			0
$315$		−	9.15173i		−	0.515641i
$316$		0			0
$317$	−23.7171			−1.33208			−0.666042	−	0.745915i	$-0.732013\pi$
$317$	−23.7171			−1.33208			−0.666042	+	0.745915i	$0.732013\pi$
$318$		0			0
$319$		−	1.18341i		−	0.0662582i
$320$		0			0
$321$		−	2.13973i		−	0.119428i
$322$		0			0
$323$			6.86443i			0.381947i
$324$		0			0
$325$		−	9.06878i		−	0.503046i
$326$		0			0
$327$		−	0.624740i		−	0.0345482i
$328$		0			0
$329$		−	4.74339i		−	0.261512i
$330$		0			0
$331$	28.2434			1.55240			0.776200	−	0.630487i	$-0.217145\pi$
$331$	28.2434			1.55240			0.776200	+	0.630487i	$0.217145\pi$
$332$		0			0
$333$		−	10.7699i		−	0.590187i
$334$		0			0
$335$	8.18490			0.447189
$336$		0			0
$337$			17.1930i			0.936563i	0.883579	+	0.468282i	$0.155127\pi$
$337$			17.1930i			0.936563i	−0.883579	+	0.468282i	$0.844873\pi$
$338$		0			0
$339$	−0.980289			−0.0532420
$340$		0			0
$341$		−	8.06561i		−	0.436777i
$342$		0			0
$343$		−	14.8866i		−	0.803801i
$344$		0			0
$345$	4.74733			0.255588
$346$		0			0
$347$	−15.7558			−0.845817			−0.422909	−	0.906172i	$-0.638991\pi$
$347$	−15.7558			−0.845817			−0.422909	+	0.906172i	$0.638991\pi$
$348$		0			0
$349$		−	6.59492i		−	0.353018i	−0.984299	−	0.176509i	$-0.943520\pi$
$349$		−	6.59492i		−	0.353018i	0.984299	−	0.176509i	$-0.0564804\pi$
$350$		0			0
$351$	6.96739			0.371892
$352$		0			0
$353$			14.2056i			0.756087i	0.925788	+	0.378044i	$0.123403\pi$
$353$			14.2056i			0.756087i	−0.925788	+	0.378044i	$0.876597\pi$
$354$		0			0
$355$	−2.42039			−0.128461
$356$		0			0
$357$	0.505720			0.0267655
$358$		0			0
$359$			3.29304i			0.173800i	0.996217	+	0.0869000i	$0.0276960\pi$
$359$			3.29304i			0.173800i	−0.996217	+	0.0869000i	$0.972304\pi$
$360$		0			0
$361$	2.25540			0.118705
$362$		0			0
$363$		−	2.45615i		−	0.128915i
$364$		0			0
$365$	5.66611			0.296578
$366$		0			0
$367$	37.1436			1.93888			0.969440	−	0.245328i	$-0.0788955\pi$
$367$	37.1436			1.93888			0.969440	+	0.245328i	$0.0788955\pi$
$368$		0			0
$369$			15.5761i			0.810861i
$370$		0			0
$371$		−	0.594666i		−	0.0308735i
$372$		0			0
$373$	16.4467			0.851579			0.425789	−	0.904822i	$-0.359996\pi$
$373$	16.4467			0.851579			0.425789	+	0.904822i	$0.359996\pi$
$374$		0			0
$375$		−	2.04101i		−	0.105397i
$376$		0			0
$377$	4.63591			0.238762
$378$		0			0
$379$		−	21.0742i		−	1.08251i	−0.840859	−	0.541254i	$-0.817950\pi$
$379$		−	21.0742i		−	1.08251i	0.840859	−	0.541254i	$-0.182050\pi$
$380$		0			0
$381$	0.344083			0.0176279
$382$		0			0
$383$		−	30.2431i		−	1.54535i	−0.634802	−	0.772675i	$-0.718918\pi$
$383$		−	30.2431i		−	1.54535i	0.634802	−	0.772675i	$-0.281082\pi$
$384$		0			0
$385$			3.66084i			0.186573i
$386$		0			0
$387$		−	10.7804i		−	0.548000i
$388$		0			0
$389$		−	33.1862i		−	1.68261i	−0.540564	−	0.841303i	$-0.681789\pi$
$389$		−	33.1862i		−	1.68261i	0.540564	−	0.841303i	$-0.318211\pi$
$390$		0			0
$391$		−	11.8152i		−	0.597519i
$392$		0			0
$393$		−	4.11443i		−	0.207546i
$394$		0			0
$395$	−13.7051			−0.689579
$396$		0			0
$397$		−	4.70602i		−	0.236188i	−0.993002	−	0.118094i	$-0.962322\pi$
$397$		−	4.70602i		−	0.236188i	0.993002	−	0.118094i	$-0.0376785\pi$
$398$		0			0
$399$			1.23362i			0.0617582i
$400$		0			0
$401$	36.1693			1.80621			0.903104	−	0.429423i	$-0.141283\pi$
$401$	36.1693			1.80621			0.903104	+	0.429423i	$0.141283\pi$
$402$		0			0
$403$	31.5964			1.57393
$404$		0			0
$405$	22.2266			1.10445
$406$		0			0
$407$			4.30813i			0.213546i
$408$		0			0
$409$	12.2161			0.604049			0.302025	−	0.953300i	$-0.402338\pi$
$409$	12.2161			0.604049			0.302025	+	0.953300i	$0.402338\pi$
$410$		0			0
$411$	−4.59599			−0.226704
$412$		0			0
$413$	−6.06360			−0.298370
$414$		0			0
$415$			42.0020i			2.06180i
$416$		0			0
$417$		−	3.35694i		−	0.164390i
$418$		0			0
$419$			26.2825i			1.28399i	0.766710	+	0.641993i	$0.221892\pi$
$419$			26.2825i			1.28399i	−0.766710	+	0.641993i	$0.778108\pi$
$420$		0			0
$421$			8.42741i			0.410727i	0.978686	+	0.205363i	$0.0658376\pi$
$421$			8.42741i			0.410727i	−0.978686	+	0.205363i	$0.934162\pi$
$422$		0			0
$423$	11.7877			0.573139
$424$		0			0
$425$	3.30792			0.160458
$426$		0			0
$427$	11.4586			0.554520
$428$		0			0
$429$	−1.37823			−0.0665417
$430$		0			0
$431$			14.0184i			0.675244i	0.941282	+	0.337622i	$0.109623\pi$
$431$			14.0184i			0.675244i	−0.941282	+	0.337622i	$0.890377\pi$
$432$		0			0
$433$		−	27.9189i		−	1.34170i	−0.741594	−	0.670849i	$-0.765930\pi$
$433$		−	27.9189i		−	1.34170i	0.741594	−	0.670849i	$-0.234070\pi$
$434$		0			0
$435$	−0.679439			−0.0325766
$436$		0			0
$437$	28.8211			1.37870
$438$		0			0
$439$			5.40383i			0.257911i	0.991650	+	0.128955i	$0.0411624\pi$
$439$			5.40383i			0.257911i	−0.991650	+	0.128955i	$0.958838\pi$
$440$		0			0
$441$	16.4506			0.783362
$442$		0			0
$443$	5.21390			0.247720			0.123860	−	0.992300i	$-0.460473\pi$
$443$	5.21390			0.247720			0.123860	+	0.992300i	$0.460473\pi$
$444$		0			0
$445$	−23.8979	−	7.02791i	−1.13287	−	0.333155i
$446$		0			0
$447$	−5.45112			−0.257829
$448$		0			0
$449$	−11.8525			−0.559354			−0.279677	−	0.960094i	$-0.590227\pi$
$449$	−11.8525			−0.559354			−0.279677	+	0.960094i	$0.590227\pi$
$450$		0			0
$451$		−	6.23070i		−	0.293392i
$452$		0			0
$453$	1.43192			0.0672775
$454$		0			0
$455$	−14.3410			−0.672317
$456$		0			0
$457$			9.33524i			0.436684i	0.975872	+	0.218342i	$0.0700648\pi$
$457$			9.33524i			0.436684i	−0.975872	+	0.218342i	$0.929935\pi$
$458$		0			0
$459$			2.54142i			0.118623i
$460$		0			0
$461$	20.3374			0.947207			0.473604	−	0.880738i	$-0.342953\pi$
$461$	20.3374			0.947207			0.473604	+	0.880738i	$0.342953\pi$
$462$		0			0
$463$	2.19904			0.102198			0.0510990	−	0.998694i	$-0.483728\pi$
$463$	2.19904			0.102198			0.0510990	+	0.998694i	$0.483728\pi$
$464$		0			0
$465$	−4.63076			−0.214746
$466$		0			0
$467$	−19.2971			−0.892965			−0.446482	−	0.894792i	$-0.647323\pi$
$467$	−19.2971			−0.892965			−0.446482	+	0.894792i	$0.647323\pi$
$468$		0			0
$469$		−	3.66084i		−	0.169042i
$470$		0			0
$471$		−	1.75964i		−	0.0810798i
$472$		0			0
$473$			4.31234i			0.198282i
$474$		0			0
$475$			8.06911i			0.370236i
$476$		0			0
$477$	1.47779			0.0676636
$478$		0			0
$479$	−6.85899			−0.313395			−0.156698	−	0.987647i	$-0.550085\pi$
$479$	−6.85899			−0.313395			−0.156698	+	0.987647i	$0.550085\pi$
$480$		0			0
$481$	−16.8767			−0.769513
$482$		0			0
$483$		−	2.12333i		−	0.0966147i
$484$		0			0
$485$	−2.40618			−0.109259
$486$		0			0
$487$	−19.4316			−0.880528			−0.440264	−	0.897868i	$-0.645115\pi$
$487$	−19.4316			−0.880528			−0.440264	+	0.897868i	$0.645115\pi$
$488$		0			0
$489$	1.73186			0.0783172
$490$		0			0
$491$		−	13.5186i		−	0.610086i	−0.952339	−	0.305043i	$-0.901329\pi$
$491$		−	13.5186i		−	0.610086i	0.952339	−	0.305043i	$-0.0986709\pi$
$492$		0			0
$493$			1.69099i			0.0761583i
$494$		0			0
$495$	−9.09748			−0.408901
$496$		0			0
$497$			1.08256i			0.0485594i
$498$		0			0
$499$		−	25.9924i		−	1.16358i	−0.813340	−	0.581789i	$-0.802353\pi$
$499$		−	25.9924i		−	1.16358i	0.813340	−	0.581789i	$-0.197647\pi$
$500$		0			0
$501$			0.126242i			0.00564007i
$502$		0			0
$503$			2.62201i			0.116909i	0.998290	+	0.0584547i	$0.0186173\pi$
$503$			2.62201i			0.116909i	−0.998290	+	0.0584547i	$0.981383\pi$
$504$		0			0
$505$		−	33.0517i		−	1.47078i
$506$		0			0
$507$		−	2.08059i		−	0.0924024i
$508$		0			0
$509$	−24.0441			−1.06573			−0.532867	−	0.846199i	$-0.678886\pi$
$509$	−24.0441			−1.06573			−0.532867	+	0.846199i	$0.678886\pi$
$510$		0			0
$511$		−	2.53426i		−	0.112109i
$512$		0			0
$513$	−6.19935			−0.273708
$514$		0			0
$515$			13.4664i			0.593399i
$516$		0			0
$517$	−4.71528			−0.207378
$518$		0			0
$519$		−	3.56665i		−	0.156558i
$520$		0			0
$521$			41.8290i			1.83256i	0.400535	+	0.916281i	$0.368824\pi$
$521$			41.8290i			1.83256i	−0.400535	+	0.916281i	$0.631176\pi$
$522$		0			0
$523$	5.52321			0.241513			0.120757	−	0.992682i	$-0.461468\pi$
$523$	5.52321			0.241513			0.120757	+	0.992682i	$0.461468\pi$
$524$		0			0
$525$	0.594472			0.0259449
$526$		0			0
$527$			11.5251i			0.502039i
$528$		0			0
$529$	−26.6075			−1.15685
$530$		0			0
$531$		−	15.0685i		−	0.653919i
$532$		0			0
$533$	24.4082			1.05724
$534$		0			0
$535$	−22.1327			−0.956880
$536$		0			0
$537$		−	4.47559i		−	0.193136i
$538$		0			0
$539$	−6.58050			−0.283442
$540$		0			0
$541$			26.7990i			1.15218i	0.817387	+	0.576089i	$0.195422\pi$
$541$			26.7990i			1.15218i	−0.817387	+	0.576089i	$0.804578\pi$
$542$		0			0
$543$	4.26325			0.182953
$544$		0			0
$545$	−6.46212			−0.276807
$546$		0			0
$547$			18.1826i			0.777434i	0.921357	+	0.388717i	$0.127082\pi$
$547$			18.1826i			0.777434i	−0.921357	+	0.388717i	$0.872918\pi$
$548$		0			0
$549$			28.4756i			1.21531i
$550$		0			0
$551$	−4.12488			−0.175726
$552$		0			0
$553$			6.12984i			0.260668i
$554$		0			0
$555$	2.47346			0.104992
$556$		0			0
$557$			11.4206i			0.483907i	0.970288	+	0.241954i	$0.0777882\pi$
$557$			11.4206i			0.483907i	−0.970288	+	0.241954i	$0.922212\pi$
$558$		0			0
$559$	−16.8932			−0.714508
$560$		0			0
$561$		−	0.502722i		−	0.0212250i
$562$		0			0
$563$			4.74941i			0.200164i	0.994979	+	0.100082i	$0.0319105\pi$
$563$			4.74941i			0.200164i	−0.994979	+	0.100082i	$0.968090\pi$
$564$		0			0
$565$			10.1398i			0.426585i
$566$		0			0
$567$		−	9.94124i		−	0.417493i
$568$		0			0
$569$		−	16.2891i		−	0.682875i	−0.939904	−	0.341437i	$-0.889086\pi$
$569$		−	16.2891i		−	0.682875i	0.939904	−	0.341437i	$-0.110914\pi$
$570$		0			0
$571$			36.1135i			1.51130i	0.654973	+	0.755652i	$0.272680\pi$
$571$			36.1135i			1.51130i	−0.654973	+	0.755652i	$0.727320\pi$
$572$		0			0
$573$	2.44019			0.101940
$574$		0			0
$575$		−	13.8887i		−	0.579199i
$576$		0			0
$577$			32.6167i			1.35785i	0.734206	+	0.678926i	$0.237554\pi$
$577$			32.6167i			1.35785i	−0.734206	+	0.678926i	$0.762446\pi$
$578$		0			0
$579$	−1.62467			−0.0675188
$580$		0			0
$581$	18.7861			0.779379
$582$		0			0
$583$	−0.591141			−0.0244826
$584$		0			0
$585$		−	35.6386i		−	1.47348i
$586$		0			0
$587$	4.18884			0.172892			0.0864460	−	0.996257i	$-0.472449\pi$
$587$	4.18884			0.172892			0.0864460	+	0.996257i	$0.472449\pi$
$588$		0			0
$589$	−28.1134			−1.15839
$590$		0			0
$591$	−0.612480			−0.0251941
$592$		0			0
$593$		−	0.441185i		−	0.0181173i	−0.999959	−	0.00905865i	$-0.997117\pi$
$593$		−	0.441185i		−	0.0181173i	0.999959	−	0.00905865i	$-0.00288350\pi$
$594$		0			0
$595$		−	5.23101i		−	0.214451i
$596$		0			0
$597$			5.09765i			0.208633i
$598$		0			0
$599$			8.91486i			0.364251i	0.983275	+	0.182126i	$0.0582977\pi$
$599$			8.91486i			0.364251i	−0.983275	+	0.182126i	$0.941702\pi$
$600$		0			0
$601$	25.7422			1.05005			0.525024	−	0.851087i	$-0.324056\pi$
$601$	25.7422			1.05005			0.525024	+	0.851087i	$0.324056\pi$
$602$		0			0
$603$	9.09748			0.370478
$604$		0			0
$605$	−25.4057			−1.03289
$606$		0			0
$607$	−33.6141			−1.36435			−0.682177	−	0.731187i	$-0.738967\pi$
$607$	−33.6141			−1.36435			−0.682177	+	0.731187i	$0.738967\pi$
$608$		0			0
$609$			0.303891i			0.0123143i
$610$		0			0
$611$		−	18.4717i		−	0.747285i
$612$		0			0
$613$	−21.4802			−0.867578			−0.433789	−	0.901014i	$-0.642824\pi$
$613$	−21.4802			−0.867578			−0.433789	+	0.901014i	$0.642824\pi$
$614$		0			0
$615$	−3.57727			−0.144249
$616$		0			0
$617$		−	38.9584i		−	1.56841i	−0.620504	−	0.784203i	$-0.713072\pi$
$617$		−	38.9584i		−	1.56841i	0.620504	−	0.784203i	$-0.286928\pi$
$618$		0			0
$619$	24.3716			0.979579			0.489790	−	0.871841i	$-0.337074\pi$
$619$	24.3716			0.979579			0.489790	+	0.871841i	$0.337074\pi$
$620$		0			0
$621$	10.6704			0.428190
$622$		0			0
$623$	−3.14335	+	10.6887i	−0.125936	+	0.428235i
$624$		0			0
$625$	−30.9711			−1.23885
$626$		0			0
$627$	1.22631			0.0489739
$628$		0			0
$629$		−	6.15594i		−	0.245454i
$630$		0			0
$631$	−18.6053			−0.740665			−0.370332	−	0.928899i	$-0.620756\pi$
$631$	−18.6053			−0.740665			−0.370332	+	0.928899i	$0.620756\pi$
$632$		0			0
$633$	−3.01548			−0.119854
$634$		0			0
$635$		−	3.55909i		−	0.141238i
$636$		0			0
$637$		−	25.7785i		−	1.02138i
$638$		0			0
$639$	−2.69025			−0.106425
$640$		0			0
$641$	−10.2150			−0.403467			−0.201733	−	0.979440i	$-0.564657\pi$
$641$	−10.2150			−0.403467			−0.201733	+	0.979440i	$0.564657\pi$
$642$		0			0
$643$	38.5878			1.52175			0.760877	−	0.648896i	$-0.224769\pi$
$643$	38.5878			1.52175			0.760877	+	0.648896i	$0.224769\pi$
$644$		0			0
$645$	2.47587			0.0974874
$646$		0			0
$647$			35.0831i			1.37926i	0.724162	+	0.689630i	$0.242227\pi$
$647$			35.0831i			1.37926i	−0.724162	+	0.689630i	$0.757773\pi$
$648$		0			0
$649$			6.02766i			0.236606i
$650$		0			0
$651$			2.07119i			0.0811762i
$652$		0			0
$653$		−	15.8592i		−	0.620618i	−0.950636	−	0.310309i	$-0.899567\pi$
$653$		−	15.8592i		−	0.620618i	0.950636	−	0.310309i	$-0.100433\pi$
$654$		0			0
$655$	−42.5584			−1.66290
$656$		0			0
$657$	6.29786			0.245703
$658$		0			0
$659$	−37.7795			−1.47168			−0.735841	−	0.677155i	$-0.763213\pi$
$659$	−37.7795			−1.47168			−0.735841	+	0.677155i	$0.763213\pi$
$660$		0			0
$661$		−	26.2600i		−	1.02140i	−0.859760	−	0.510698i	$-0.829387\pi$
$661$		−	26.2600i		−	1.02140i	0.859760	−	0.510698i	$-0.170613\pi$
$662$		0			0
$663$	1.96937			0.0764841
$664$		0			0
$665$	12.7602			0.494818
$666$		0			0
$667$	7.09982			0.274906
$668$		0			0
$669$			4.93938i			0.190967i
$670$		0			0
$671$		−	11.3907i		−	0.439732i
$672$		0			0
$673$	−44.1584			−1.70218			−0.851091	−	0.525019i	$-0.824058\pi$
$673$	−44.1584			−1.70218			−0.851091	+	0.525019i	$0.824058\pi$
$674$		0			0
$675$			2.98743i			0.114986i
$676$		0			0
$677$		−	9.23111i		−	0.354781i	−0.984141	−	0.177390i	$-0.943235\pi$
$677$		−	9.23111i		−	0.354781i	0.984141	−	0.177390i	$-0.0567655\pi$
$678$		0			0
$679$			1.07621i			0.0413010i
$680$		0			0
$681$		−	1.11772i		−	0.0428313i
$682$		0			0
$683$		−	7.66813i		−	0.293413i	−0.989180	−	0.146706i	$-0.953133\pi$
$683$		−	7.66813i		−	0.293413i	0.989180	−	0.146706i	$-0.0468673\pi$
$684$		0			0
$685$			47.5396i			1.81639i
$686$		0			0
$687$	−6.40798			−0.244480
$688$		0			0
$689$		−	2.31575i		−	0.0882229i
$690$		0			0
$691$	3.68989			0.140370			0.0701850	−	0.997534i	$-0.477641\pi$
$691$	3.68989			0.140370			0.0701850	+	0.997534i	$0.477641\pi$
$692$		0			0
$693$			4.06900i			0.154569i
$694$		0			0
$695$	−34.7232			−1.31713
$696$		0			0
$697$			8.90312i			0.337230i
$698$		0			0
$699$		−	2.73865i		−	0.103585i
$700$		0			0
$701$	−0.667713			−0.0252192			−0.0126096	−	0.999920i	$-0.504014\pi$
$701$	−0.667713			−0.0252192			−0.0126096	+	0.999920i	$0.504014\pi$
$702$		0			0
$703$	15.0164			0.566354
$704$		0			0
$705$			2.70721i			0.101960i
$706$		0			0
$707$	−14.7830			−0.555970
$708$		0			0
$709$		−	31.9891i		−	1.20137i	−0.799484	−	0.600687i	$-0.794894\pi$
$709$		−	31.9891i		−	1.20137i	0.799484	−	0.600687i	$-0.205106\pi$
$710$		0			0
$711$	−15.2332			−0.571289
$712$		0			0
$713$	48.3893			1.81219
$714$		0			0
$715$			14.2560i			0.533145i
$716$		0			0
$717$	2.77734			0.103722
$718$		0			0
$719$			26.6415i			0.993560i	0.867877	+	0.496780i	$0.165484\pi$
$719$			26.6415i			0.993560i	−0.867877	+	0.496780i	$0.834516\pi$
$720$		0			0
$721$	6.02306			0.224311
$722$		0			0
$723$	5.86916			0.218276
$724$		0			0
$725$			1.98775i			0.0738233i
$726$		0			0
$727$		−	32.2895i		−	1.19755i	−0.800917	−	0.598775i	$-0.795654\pi$
$727$		−	32.2895i		−	1.19755i	0.800917	−	0.598775i	$-0.204346\pi$
$728$		0			0
$729$	23.5446			0.872023
$730$		0			0
$731$		−	6.16196i		−	0.227908i
$732$		0			0
$733$	−26.6418			−0.984037			−0.492019	−	0.870585i	$-0.663741\pi$
$733$	−26.6418			−0.984037			−0.492019	+	0.870585i	$0.663741\pi$
$734$		0			0
$735$			3.77810i			0.139357i
$736$		0			0
$737$	−3.63914			−0.134049
$738$		0			0
$739$			33.0150i			1.21448i	0.794520	+	0.607238i	$0.207722\pi$
$739$			33.0150i			1.21448i	−0.794520	+	0.607238i	$0.792278\pi$
$740$		0			0
$741$			4.80395i			0.176478i
$742$		0			0
$743$		−	27.6210i		−	1.01332i	−0.862147	−	0.506658i	$-0.830881\pi$
$743$		−	27.6210i		−	1.01332i	0.862147	−	0.506658i	$-0.169119\pi$
$744$		0			0
$745$			56.3847i			2.06578i
$746$		0			0
$747$			46.6850i			1.70812i
$748$		0			0
$749$			9.89923i			0.361710i
$750$		0			0
$751$	−23.9690			−0.874642			−0.437321	−	0.899306i	$-0.644073\pi$
$751$	−23.9690			−0.874642			−0.437321	+	0.899306i	$0.644073\pi$
$752$		0			0
$753$		−	4.86871i		−	0.177426i
$754$		0			0
$755$		−	14.8114i		−	0.539041i
$756$		0			0
$757$	38.8211			1.41098			0.705489	−	0.708721i	$-0.250727\pi$
$757$	38.8211			1.41098			0.705489	+	0.708721i	$0.250727\pi$
$758$		0			0
$759$	−2.11074			−0.0766150
$760$		0			0
$761$	25.1101			0.910240			0.455120	−	0.890430i	$-0.349596\pi$
$761$	25.1101			0.910240			0.455120	+	0.890430i	$0.349596\pi$
$762$		0			0
$763$			2.89029i			0.104636i
$764$		0			0
$765$	12.9995			0.469998
$766$		0			0
$767$	−23.6128			−0.852610
$768$		0			0
$769$	−20.9422			−0.755194			−0.377597	−	0.925970i	$-0.623249\pi$
$769$	−20.9422			−0.755194			−0.377597	+	0.925970i	$0.623249\pi$
$770$		0			0
$771$		−	4.29225i		−	0.154582i
$772$		0			0
$773$		−	24.2440i		−	0.871995i	−0.899948	−	0.435997i	$-0.856396\pi$
$773$		−	24.2440i		−	0.871995i	0.899948	−	0.435997i	$-0.143604\pi$
$774$		0			0
$775$			13.5477i			0.486646i
$776$		0			0
$777$		−	1.10630i		−	0.0396881i
$778$		0			0
$779$	−21.7177			−0.778116
$780$		0			0
$781$	1.07614			0.0385074
$782$		0			0
$783$	−1.52716			−0.0545761
$784$		0			0
$785$	−18.2012			−0.649627
$786$		0			0
$787$		−	52.2795i		−	1.86356i	−0.363021	−	0.931781i	$-0.618255\pi$
$787$		−	52.2795i		−	1.86356i	0.363021	−	0.931781i	$-0.381745\pi$
$788$		0			0
$789$			5.23501i			0.186371i
$790$		0			0
$791$	4.53520			0.161253
$792$		0			0
$793$	44.6220			1.58457
$794$		0			0
$795$			0.339396i			0.0120371i
$796$		0			0
$797$	48.2416			1.70880			0.854402	−	0.519613i	$-0.173924\pi$
$797$	48.2416			1.70880			0.854402	+	0.519613i	$0.173924\pi$
$798$		0			0
$799$	6.73772			0.238363
$800$		0			0
$801$	−26.5624	−	7.81150i	−0.938536	−	0.276006i
$802$		0			0
$803$	−2.51924			−0.0889021
$804$		0			0
$805$	−21.9630			−0.774096
$806$		0			0
$807$		−	4.36133i		−	0.153526i
$808$		0			0
$809$	10.6665			0.375013			0.187506	−	0.982263i	$-0.439959\pi$
$809$	10.6665			0.375013			0.187506	+	0.982263i	$0.439959\pi$
$810$		0			0
$811$	−18.9845			−0.666635			−0.333318	−	0.942815i	$-0.608168\pi$
$811$	−18.9845			−0.666635			−0.333318	+	0.942815i	$0.608168\pi$
$812$		0			0
$813$			5.87644i			0.206096i
$814$		0			0
$815$		−	17.9138i		−	0.627493i
$816$		0			0
$817$	15.0310			0.525870
$818$		0			0
$819$	−15.9400			−0.556988
$820$		0			0
$821$	25.7318			0.898046			0.449023	−	0.893520i	$-0.351772\pi$
$821$	25.7318			0.898046			0.449023	+	0.893520i	$0.351772\pi$
$822$		0			0
$823$	−27.9601			−0.974627			−0.487314	−	0.873227i	$-0.662023\pi$
$823$	−27.9601			−0.974627			−0.487314	+	0.873227i	$0.662023\pi$
$824$		0			0
$825$		−	0.590948i		−	0.0205742i
$826$		0			0
$827$			28.6346i			0.995721i	0.867257	+	0.497861i	$0.165881\pi$
$827$			28.6346i			0.995721i	−0.867257	+	0.497861i	$0.834119\pi$
$828$		0			0
$829$			52.6461i			1.82848i	0.405178	+	0.914238i	$0.367209\pi$
$829$			52.6461i			1.82848i	−0.405178	+	0.914238i	$0.632791\pi$
$830$		0			0
$831$		−	0.238379i		−	0.00826927i
$832$		0			0
$833$	9.40295			0.325793
$834$		0			0
$835$	1.30581			0.0451894
$836$		0			0
$837$	−10.4084			−0.359768
$838$		0			0
$839$		−	9.25196i		−	0.319413i	−0.987165	−	0.159706i	$-0.948945\pi$
$839$		−	9.25196i		−	0.319413i	0.987165	−	0.159706i	$-0.0510548\pi$
$840$		0			0
$841$	27.9839			0.964961
$842$		0			0
$843$	2.16940			0.0747181
$844$		0			0
$845$	−21.5210			−0.740346
$846$		0			0
$847$			11.3631i			0.390442i
$848$		0			0
$849$			5.10035i			0.175043i
$850$		0			0
$851$	−25.8465			−0.886006
$852$		0			0
$853$		−	17.7194i		−	0.606701i	−0.952879	−	0.303351i	$-0.901895\pi$
$853$		−	17.7194i		−	0.606701i	0.952879	−	0.303351i	$-0.0981054\pi$
$854$		0			0
$855$			31.7101i			1.08446i
$856$		0			0
$857$		−	18.6686i		−	0.637706i	−0.947804	−	0.318853i	$-0.896702\pi$
$857$		−	18.6686i		−	0.637706i	0.947804	−	0.318853i	$-0.103298\pi$
$858$		0			0
$859$		−	28.0675i		−	0.957652i	−0.877910	−	0.478826i	$-0.841062\pi$
$859$		−	28.0675i		−	0.957652i	0.877910	−	0.478826i	$-0.158938\pi$
$860$		0			0
$861$			1.60000i			0.0545277i
$862$		0			0
$863$		−	7.80840i		−	0.265801i	−0.991129	−	0.132901i	$-0.957571\pi$
$863$		−	7.80840i		−	0.265801i	0.991129	−	0.132901i	$-0.0424291\pi$
$864$		0			0
$865$	−36.8923			−1.25438
$866$		0			0
$867$		−	3.62125i		−	0.122984i
$868$		0			0
$869$	6.09351			0.206708
$870$		0			0
$871$		−	14.2560i		−	0.483046i
$872$		0			0
$873$	−2.67446			−0.0905168
$874$		0			0
$875$			9.44250i			0.319215i
$876$		0			0
$877$		−	54.1974i		−	1.83012i	−0.403321	−	0.915058i	$-0.632144\pi$
$877$		−	54.1974i		−	1.83012i	0.403321	−	0.915058i	$-0.367856\pi$
$878$		0			0
$879$	−4.24930			−0.143325
$880$		0			0
$881$	13.8382			0.466221			0.233110	−	0.972450i	$-0.425110\pi$
$881$	13.8382			0.466221			0.233110	+	0.972450i	$0.425110\pi$
$882$		0			0
$883$		−	38.1831i		−	1.28496i	−0.766302	−	0.642481i	$-0.777905\pi$
$883$		−	38.1831i		−	1.28496i	0.766302	−	0.642481i	$-0.222095\pi$
$884$		0			0
$885$	3.46070			0.116330
$886$		0			0
$887$			45.6562i			1.53298i	0.642254	+	0.766492i	$0.277999\pi$
$887$			45.6562i			1.53298i	−0.642254	+	0.766492i	$0.722001\pi$
$888$		0			0
$889$	−1.59187			−0.0533895
$890$		0			0
$891$	−9.88231			−0.331070
$892$		0			0
$893$			16.4355i			0.549994i
$894$		0			0
$895$	−46.2942			−1.54744
$896$		0			0
$897$		−	8.26865i		−	0.276082i
$898$		0			0
$899$	−6.92549			−0.230978
$900$		0			0
$901$	0.844689			0.0281407
$902$		0			0
$903$		−	1.10738i		−	0.0368512i
$904$		0			0
$905$		−	44.0977i		−	1.46586i
$906$		0			0
$907$	20.3189			0.674677			0.337338	−	0.941383i	$-0.390473\pi$
$907$	20.3189			0.674677			0.337338	+	0.941383i	$0.390473\pi$
$908$		0			0
$909$		−	36.7369i		−	1.21848i
$910$		0			0
$911$	31.3087			1.03730			0.518651	−	0.854986i	$-0.326434\pi$
$911$	31.3087			1.03730			0.518651	+	0.854986i	$0.326434\pi$
$912$		0			0
$913$		−	18.6747i		−	0.618044i
$914$		0			0
$915$	−6.53980			−0.216199
$916$		0			0
$917$			19.0350i			0.628591i
$918$		0			0
$919$			53.8052i			1.77487i	0.460932	+	0.887436i	$0.347515\pi$
$919$			53.8052i			1.77487i	−0.460932	+	0.887436i	$0.652485\pi$
$920$		0			0
$921$		−	0.860265i		−	0.0283467i
$922$		0			0
$923$			4.21570i			0.138761i
$924$		0			0
$925$		−	7.23629i		−	0.237928i
$926$		0			0
$927$			14.9678i			0.491607i
$928$		0			0
$929$	−15.5102			−0.508874			−0.254437	−	0.967089i	$-0.581890\pi$
$929$	−15.5102			−0.508874			−0.254437	+	0.967089i	$0.581890\pi$
$930$		0			0
$931$			22.9369i			0.751727i
$932$		0			0
$933$			6.89062i			0.225589i
$934$		0			0
$935$	−5.20001			−0.170058
$936$		0			0
$937$	22.0565			0.720555			0.360278	−	0.932845i	$-0.382682\pi$
$937$	22.0565			0.720555			0.360278	+	0.932845i	$0.382682\pi$
$938$		0			0
$939$	2.68511			0.0876254
$940$		0			0
$941$			7.95429i			0.259302i	0.991560	+	0.129651i	$0.0413858\pi$
$941$			7.95429i			0.259302i	−0.991560	+	0.129651i	$0.958614\pi$
$942$		0			0
$943$	37.3808			1.21729
$944$		0			0
$945$	4.72420			0.153678
$946$		0			0
$947$	46.4953			1.51089			0.755446	−	0.655210i	$-0.227420\pi$
$947$	46.4953			1.51089			0.755446	+	0.655210i	$0.227420\pi$
$948$		0			0
$949$		−	9.86892i		−	0.320359i
$950$		0			0
$951$		−	6.05426i		−	0.196323i
$952$		0			0
$953$			25.8656i			0.837870i	0.908016	+	0.418935i	$0.137597\pi$
$953$			25.8656i			0.837870i	−0.908016	+	0.418935i	$0.862403\pi$
$954$		0			0
$955$		−	25.2406i		−	0.816766i
$956$		0			0
$957$	0.302089			0.00976516
$958$		0			0
$959$	21.2629			0.686615
$960$		0			0
$961$	−16.2011			−0.522616
$962$		0			0
$963$	−24.6004			−0.792737
$964$		0			0
$965$			16.8050i			0.540974i
$966$		0			0
$967$			22.3983i			0.720280i	0.932898	+	0.360140i	$0.117271\pi$
$967$			22.3983i			0.720280i	−0.932898	+	0.360140i	$0.882729\pi$
$968$		0			0
$969$	−1.75228			−0.0562915
$970$		0			0
$971$	−27.9937			−0.898360			−0.449180	−	0.893441i	$-0.648284\pi$
$971$	−27.9937			−0.898360			−0.449180	+	0.893441i	$0.648284\pi$
$972$		0			0
$973$			15.5306i			0.497887i
$974$		0			0
$975$	2.31499			0.0741391
$976$		0			0
$977$	24.8602			0.795348			0.397674	−	0.917527i	$-0.369818\pi$
$977$	24.8602			0.795348			0.397674	+	0.917527i	$0.369818\pi$
$978$		0			0
$979$	10.6254	+	3.12472i	0.339589	+	0.0998665i
$980$		0			0
$981$	−7.18262			−0.229323
$982$		0			0
$983$	47.9933			1.53075			0.765375	−	0.643585i	$-0.222554\pi$
$983$	47.9933			1.53075			0.765375	+	0.643585i	$0.222554\pi$
$984$		0			0
$985$			6.33531i			0.201860i
$986$		0			0
$987$	1.21085			0.0385417
$988$		0			0
$989$	−25.8717			−0.822673
$990$		0			0
$991$		−	41.7300i		−	1.32560i	−0.748798	−	0.662798i	$-0.769369\pi$
$991$		−	41.7300i		−	1.32560i	0.748798	−	0.662798i	$-0.230631\pi$
$992$		0			0
$993$			7.20971i			0.228793i
$994$		0			0
$995$	52.7286			1.67161
$996$		0			0
$997$	7.81943			0.247644			0.123822	−	0.992304i	$-0.460485\pi$
$997$	7.81943			0.247644			0.123822	+	0.992304i	$0.460485\pi$
$998$		0			0
$999$	5.55952			0.175895

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	712.2.e.c.177.6	yes	10
4.3	odd	2		1424.2.e.h.177.5		10
89.88	even	2	inner	712.2.e.c.177.5	✓	10
356.355	odd	2		1424.2.e.h.177.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.2.e.c.177.5	✓	10	89.88	even	2	inner
712.2.e.c.177.6	yes	10	1.1	even	1	trivial
1424.2.e.h.177.5		10	4.3	odd	2
1424.2.e.h.177.6		10	356.355	odd	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:	\( N \)	\(=\)	\( 712 = 2^{3} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	712.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.255270i q^{3} +2.64044 q^{5} -1.18098i q^{7} +2.93484 q^{9} +O(q^{10})\)	\(q+0.255270i q^{3} +2.64044 q^{5} -1.18098i q^{7} +2.93484 q^{9} -1.17398 q^{11} -4.59897i q^{13} +0.674025i q^{15} +1.67752 q^{17} +4.09202i q^{19} +0.301469 q^{21} -7.04326i q^{23} +1.97192 q^{25} +1.51499i q^{27} +1.00803i q^{29} +6.87031i q^{31} -0.299683i q^{33} -3.11831i q^{35} -3.66968i q^{37} +1.17398 q^{39} +5.30732i q^{41} -3.67326i q^{43} +7.74926 q^{45} +4.01649 q^{47} +5.60528 q^{49} +0.428220i q^{51} +0.503535 q^{53} -3.09982 q^{55} -1.04457 q^{57} -5.13437i q^{59} +9.70261i q^{61} -3.46599i q^{63} -12.1433i q^{65} +3.09982 q^{67} +1.79793 q^{69} -0.916661 q^{71} +2.14590 q^{73} +0.503371i q^{75} +1.38645i q^{77} -5.19047 q^{79} +8.41778 q^{81} +15.9072i q^{83} +4.42938 q^{85} -0.257321 q^{87} +(-9.05073 - 2.66165i) q^{89} -5.43130 q^{91} -1.75379 q^{93} +10.8047i q^{95} -0.911281 q^{97} -3.44544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^9	\( 10 q - 10 q^{9} + 8 q^{11} + 8 q^{17} + 12 q^{21} - 2 q^{25} - 8 q^{39} - 8 q^{45} - 4 q^{47} - 14 q^{49} + 16 q^{53} + 24 q^{55} + 12 q^{57} - 24 q^{67} + 16 q^{69} - 20 q^{71} - 20 q^{73} + 12 q^{79} - 6 q^{81} + 20 q^{85} + 28 q^{87} - 14 q^{89} - 4 q^{91} + 24 q^{93} + 8 q^{97} - 28 q^{99}+O(q^{100}) \) 10 * q - 10 * q^9 + 8 * q^11 + 8 * q^17 + 12 * q^21 - 2 * q^25 - 8 * q^39 - 8 * q^45 - 4 * q^47 - 14 * q^49 + 16 * q^53 + 24 * q^55 + 12 * q^57 - 24 * q^67 + 16 * q^69 - 20 * q^71 - 20 * q^73 + 12 * q^79 - 6 * q^81 + 20 * q^85 + 28 * q^87 - 14 * q^89 - 4 * q^91 + 24 * q^93 + 8 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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