LMFDB - Embedded newform 1716.2.p.a.857.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1716,2,Mod(857,1716)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1716, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1716.857");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1716.p (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:	$13.7023289869$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			857.5
Character	$\chi$	$=$	1716.857
Dual form			1716.2.p.a.857.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.65331 + 0.516316i) q^{3} -0.870582 q^{5} -3.25162 q^{7} +(2.46684 - 1.70725i) q^{9} +O(q^{10})$	\(q+(-1.65331 + 0.516316i) q^{3} -0.870582 q^{5} -3.25162 q^{7} +(2.46684 - 1.70725i) q^{9} +(1.49742 + 2.95935i) q^{11} +(0.579838 - 3.55862i) q^{13} +(1.43934 - 0.449495i) q^{15} -6.27509 q^{17} +2.40706 q^{19} +(5.37592 - 1.67886i) q^{21} -5.74231i q^{23} -4.24209 q^{25} +(-3.19695 + 4.09628i) q^{27} +2.64055 q^{29} +5.11377i q^{31} +(-4.00365 - 4.11956i) q^{33} +2.83080 q^{35} -11.0111i q^{37} +(0.878723 + 6.18287i) q^{39} +2.88932i q^{41} +4.95928i q^{43} +(-2.14758 + 1.48630i) q^{45} +7.73345 q^{47} +3.57302 q^{49} +(10.3746 - 3.23992i) q^{51} +7.26000i q^{53} +(-1.30363 - 2.57635i) q^{55} +(-3.97961 + 1.24280i) q^{57} +10.7142 q^{59} +5.38232i q^{61} +(-8.02121 + 5.55134i) q^{63} +(-0.504796 + 3.09807i) q^{65} +6.09988i q^{67} +(2.96485 + 9.49380i) q^{69} +14.4650 q^{71} +2.81137 q^{73} +(7.01346 - 2.19026i) q^{75} +(-4.86904 - 9.62266i) q^{77} -9.34981i q^{79} +(3.17056 - 8.42304i) q^{81} +16.3716i q^{83} +5.46298 q^{85} +(-4.36563 + 1.36336i) q^{87} +8.93558 q^{89} +(-1.88541 + 11.5713i) q^{91} +(-2.64032 - 8.45462i) q^{93} -2.09555 q^{95} +12.1653i q^{97} +(8.74625 + 4.74374i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 8 q^{9}+O(q^{10}) $ 56 * q - 8 * q^9	$ 56 q - 8 q^{9} + 48 q^{25} - 12 q^{27} + 48 q^{49} - 16 q^{55} - 12 q^{69} + 44 q^{75} - 40 q^{81} - 24 q^{91}+O(q^{100}) $ 56 * q - 8 * q^9 + 48 * q^25 - 12 * q^27 + 48 * q^49 - 16 * q^55 - 12 * q^69 + 44 * q^75 - 40 * q^81 - 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1716\mathbb{Z}\right)^\times$.

$n$	$859$	$925$	$937$	$1145$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	−1.65331	+	0.516316i	−0.954536	+	0.298095i
$3$	−1.65331	+	0.516316i	−0.954536	+	0.298095i
$4$		0			0
$5$	−0.870582			−0.389336			−0.194668	−	0.980869i	$-0.562363\pi$
$5$	−0.870582			−0.389336			−0.194668	+	0.980869i	$0.562363\pi$
$6$		0			0
$7$	−3.25162			−1.22900			−0.614498	−	0.788918i	$-0.710641\pi$
$7$	−3.25162			−1.22900			−0.614498	+	0.788918i	$0.710641\pi$
$8$		0			0
$9$	2.46684	−	1.70725i	0.822279	−	0.569085i
$10$		0			0
$11$	1.49742	+	2.95935i	0.451490	+	0.892276i
$11$	1.49742	+	2.95935i	0.451490	+	0.892276i
$12$		0			0
$13$	0.579838	−	3.55862i	0.160818	−	0.986984i
$13$	0.579838	−	3.55862i	0.160818	−	0.986984i
$14$		0			0
$15$	1.43934	−	0.449495i	0.371635	−	0.116059i
$16$		0			0
$17$	−6.27509			−1.52193			−0.760966	−	0.648792i	$-0.775275\pi$
$17$	−6.27509			−1.52193			−0.760966	+	0.648792i	$0.775275\pi$
$18$		0			0
$19$	2.40706			0.552218			0.276109	−	0.961126i	$-0.410955\pi$
$19$	2.40706			0.552218			0.276109	+	0.961126i	$0.410955\pi$
$20$		0			0
$21$	5.37592	−	1.67886i	1.17312	−	0.366357i
$22$		0			0
$23$		−	5.74231i		−	1.19736i	−0.800990	−	0.598678i	$-0.795693\pi$
$23$		−	5.74231i		−	1.19736i	0.800990	−	0.598678i	$-0.204307\pi$
$24$		0			0
$25$	−4.24209			−0.848417
$26$		0			0
$27$	−3.19695	+	4.09628i	−0.615254	+	0.788329i
$28$		0			0
$29$	2.64055			0.490337			0.245169	−	0.969480i	$-0.421157\pi$
$29$	2.64055			0.490337			0.245169	+	0.969480i	$0.421157\pi$
$30$		0			0
$31$			5.11377i			0.918460i	0.888317	+	0.459230i	$0.151875\pi$
$31$			5.11377i			0.918460i	−0.888317	+	0.459230i	$0.848125\pi$
$32$		0			0
$33$	−4.00365	−	4.11956i	−0.696946	−	0.717123i
$34$		0			0
$35$	2.83080			0.478492
$36$		0			0
$37$		−	11.0111i		−	1.81021i	−0.425183	−	0.905107i	$-0.639790\pi$
$37$		−	11.0111i		−	1.81021i	0.425183	−	0.905107i	$-0.360210\pi$
$38$		0			0
$39$	0.878723	+	6.18287i	0.140708	+	0.990051i
$40$		0			0
$41$			2.88932i			0.451236i	0.974216	+	0.225618i	$0.0724402\pi$
$41$			2.88932i			0.451236i	−0.974216	+	0.225618i	$0.927560\pi$
$42$		0			0
$43$			4.95928i			0.756283i	0.925748	+	0.378142i	$0.123437\pi$
$43$			4.95928i			0.756283i	−0.925748	+	0.378142i	$0.876563\pi$
$44$		0			0
$45$	−2.14758	+	1.48630i	−0.320143	+	0.221565i
$46$		0			0
$47$	7.73345			1.12804			0.564020	−	0.825761i	$-0.309254\pi$
$47$	7.73345			1.12804			0.564020	+	0.825761i	$0.309254\pi$
$48$		0			0
$49$	3.57302			0.510431
$50$		0			0
$51$	10.3746	−	3.23992i	1.45274	−	0.453680i
$52$		0			0
$53$			7.26000i			0.997238i	0.866821	+	0.498619i	$0.166159\pi$
$53$			7.26000i			0.997238i	−0.866821	+	0.498619i	$0.833841\pi$
$54$		0			0
$55$	−1.30363	−	2.57635i	−0.175781	−	0.347395i
$56$		0			0
$57$	−3.97961	+	1.24280i	−0.527112	+	0.164614i
$58$		0			0
$59$	10.7142			1.39487			0.697434	−	0.716649i	$-0.254325\pi$
$59$	10.7142			1.39487			0.697434	+	0.716649i	$0.254325\pi$
$60$		0			0
$61$			5.38232i			0.689135i	0.938761	+	0.344568i	$0.111975\pi$
$61$			5.38232i			0.689135i	−0.938761	+	0.344568i	$0.888025\pi$
$62$		0			0
$63$	−8.02121	+	5.55134i	−1.01058	+	0.699403i
$64$		0			0
$65$	−0.504796	+	3.09807i	−0.0626123	+	0.384268i
$66$		0			0
$67$			6.09988i			0.745220i	0.927988	+	0.372610i	$0.121537\pi$
$67$			6.09988i			0.745220i	−0.927988	+	0.372610i	$0.878463\pi$
$68$		0			0
$69$	2.96485	+	9.49380i	0.356926	+	1.14292i
$70$		0			0
$71$	14.4650			1.71668			0.858341	−	0.513080i	$-0.171496\pi$
$71$	14.4650			1.71668			0.858341	+	0.513080i	$0.171496\pi$
$72$		0			0
$73$	2.81137			0.329046			0.164523	−	0.986373i	$-0.447392\pi$
$73$	2.81137			0.329046			0.164523	+	0.986373i	$0.447392\pi$
$74$		0			0
$75$	7.01346	−	2.19026i	0.809845	−	0.252909i
$76$		0			0
$77$	−4.86904	−	9.62266i	−0.554879	−	1.09660i
$78$		0			0
$79$		−	9.34981i		−	1.05194i	−0.850504	−	0.525968i	$-0.823703\pi$
$79$		−	9.34981i		−	1.05194i	0.850504	−	0.525968i	$-0.176297\pi$
$80$		0			0
$81$	3.17056	−	8.42304i	0.352285	−	0.935893i
$82$		0			0
$83$			16.3716i			1.79701i	0.438961	+	0.898506i	$0.355346\pi$
$83$			16.3716i			1.79701i	−0.438961	+	0.898506i	$0.644654\pi$
$84$		0			0
$85$	5.46298			0.592543
$86$		0			0
$87$	−4.36563	+	1.36336i	−0.468045	+	0.146167i
$88$		0			0
$89$	8.93558			0.947170			0.473585	−	0.880748i	$-0.342960\pi$
$89$	8.93558			0.947170			0.473585	+	0.880748i	$0.342960\pi$
$90$		0			0
$91$	−1.88541	+	11.5713i	−0.197645	+	1.21300i
$92$		0			0
$93$	−2.64032	−	8.45462i	−0.273788	−	0.876704i
$94$		0			0
$95$	−2.09555			−0.214999
$96$		0			0
$97$			12.1653i			1.23520i	0.786491	+	0.617602i	$0.211896\pi$
$97$			12.1653i			1.23520i	−0.786491	+	0.617602i	$0.788104\pi$
$98$		0			0
$99$	8.74625	+	4.74374i	0.879031	+	0.476764i
$100$		0			0
$101$	18.0240			1.79346			0.896729	−	0.442579i	$-0.145937\pi$
$101$	18.0240			1.79346			0.896729	+	0.442579i	$0.145937\pi$
$102$		0			0
$103$	10.2060			1.00563			0.502816	−	0.864394i	$-0.332297\pi$
$103$	10.2060			1.00563			0.502816	+	0.864394i	$0.332297\pi$
$104$		0			0
$105$	−4.68018	+	1.46159i	−0.456738	+	0.142636i
$106$		0			0
$107$	−4.39418			−0.424801			−0.212401	−	0.977183i	$-0.568128\pi$
$107$	−4.39418			−0.424801			−0.212401	+	0.977183i	$0.568128\pi$
$108$		0			0
$109$	−17.5719			−1.68308			−0.841542	−	0.540191i	$-0.818352\pi$
$109$	−17.5719			−1.68308			−0.841542	+	0.540191i	$0.818352\pi$
$110$		0			0
$111$	5.68520	+	18.2047i	0.539616	+	1.72792i
$112$		0			0
$113$		−	5.91383i		−	0.556327i	−0.960534	−	0.278163i	$-0.910274\pi$
$113$		−	5.91383i		−	0.556327i	0.960534	−	0.278163i	$-0.0897257\pi$
$114$		0			0
$115$			4.99915i			0.466174i
$116$		0			0
$117$	−4.64511	−	9.76847i	−0.429440	−	0.903095i
$118$		0			0
$119$	20.4042			1.87045
$120$		0			0
$121$	−6.51546	+	8.86278i	−0.592314	+	0.805707i
$122$		0			0
$123$	−1.49180	−	4.77693i	−0.134511	−	0.430721i
$124$		0			0
$125$	8.04599			0.719656
$126$		0			0
$127$			12.5026i			1.10943i	0.832042	+	0.554713i	$0.187172\pi$
$127$			12.5026i			1.10943i	−0.832042	+	0.554713i	$0.812828\pi$
$128$		0			0
$129$	−2.56055	−	8.19921i	−0.225444	−	0.721900i
$130$		0			0
$131$	−4.07926			−0.356406			−0.178203	−	0.983994i	$-0.557028\pi$
$131$	−4.07926			−0.356406			−0.178203	+	0.983994i	$0.557028\pi$
$132$		0			0
$133$	−7.82685			−0.678674
$134$		0			0
$135$	2.78321	−	3.56615i	0.239540	−	0.306925i
$136$		0			0
$137$	7.02023			0.599779			0.299889	−	0.953974i	$-0.403050\pi$
$137$	7.02023			0.599779			0.299889	+	0.953974i	$0.403050\pi$
$138$		0			0
$139$		−	8.94203i		−	0.758453i	−0.925304	−	0.379227i	$-0.876190\pi$
$139$		−	8.94203i		−	0.758453i	0.925304	−	0.379227i	$-0.123810\pi$
$140$		0			0
$141$	−12.7857	+	3.99290i	−1.07675	+	0.336263i
$142$		0			0
$143$	11.3995	−	3.61282i	0.953270	−	0.302119i
$144$		0			0
$145$	−2.29881			−0.190906
$146$		0			0
$147$	−5.90729	+	1.84480i	−0.487225	+	0.152157i
$148$		0			0
$149$		−	1.11821i		−	0.0916073i	−0.998950	−	0.0458037i	$-0.985415\pi$
$149$		−	1.11821i		−	0.0916073i	0.998950	−	0.0458037i	$-0.0145849\pi$
$150$		0			0
$151$	−13.6354			−1.10963			−0.554815	−	0.831974i	$-0.687211\pi$
$151$	−13.6354			−1.10963			−0.554815	+	0.831974i	$0.687211\pi$
$152$		0			0
$153$	−15.4796	+	10.7132i	−1.25145	+	0.866108i
$154$		0			0
$155$		−	4.45196i		−	0.357590i
$156$		0			0
$157$	2.61901			0.209019			0.104510	−	0.994524i	$-0.466673\pi$
$157$	2.61901			0.209019			0.104510	+	0.994524i	$0.466673\pi$
$158$		0			0
$159$	−3.74845	−	12.0030i	−0.297272	−	0.951900i
$160$		0			0
$161$			18.6718i			1.47154i
$162$		0			0
$163$		−	13.0651i		−	1.02334i	−0.859182	−	0.511670i	$-0.829027\pi$
$163$		−	13.0651i		−	1.02334i	0.859182	−	0.511670i	$-0.170973\pi$
$164$		0			0
$165$	3.48551	+	3.58641i	0.271346	+	0.279202i
$166$		0			0
$167$			17.9582i			1.38964i	0.719181	+	0.694822i	$0.244517\pi$
$167$			17.9582i			1.38964i	−0.719181	+	0.694822i	$0.755483\pi$
$168$		0			0
$169$	−12.3276	−	4.12685i	−0.948275	−	0.317450i
$170$		0			0
$171$	5.93783	−	4.10947i	0.454077	−	0.314259i
$172$		0			0
$173$	16.7401			1.27273			0.636363	−	0.771390i	$-0.280438\pi$
$173$	16.7401			1.27273			0.636363	+	0.771390i	$0.280438\pi$
$174$		0			0
$175$	13.7936			1.04270
$176$		0			0
$177$	−17.7138	+	5.53190i	−1.33145	+	0.415803i
$178$		0			0
$179$		−	18.8161i		−	1.40638i	−0.711002	−	0.703190i	$-0.751758\pi$
$179$		−	18.8161i		−	1.40638i	0.711002	−	0.703190i	$-0.248242\pi$
$180$		0			0
$181$	7.28059			0.541162			0.270581	−	0.962697i	$-0.412784\pi$
$181$	7.28059			0.541162			0.270581	+	0.962697i	$0.412784\pi$
$182$		0			0
$183$	−2.77898	−	8.89862i	−0.205428	−	0.657805i
$184$		0			0
$185$			9.58607i			0.704782i
$186$		0			0
$187$	−9.39645	−	18.5702i	−0.687137	−	1.35798i
$188$		0			0
$189$	10.3953	−	13.3195i	0.756144	−	0.968853i
$190$		0			0
$191$			18.8671i			1.36517i	0.730805	+	0.682587i	$0.239145\pi$
$191$			18.8671i			1.36517i	−0.730805	+	0.682587i	$0.760855\pi$
$192$		0			0
$193$	3.12633			0.225039			0.112519	−	0.993650i	$-0.464108\pi$
$193$	3.12633			0.225039			0.112519	+	0.993650i	$0.464108\pi$
$194$		0			0
$195$	−0.765000	−	5.38269i	−0.0547828	−	0.385463i
$196$		0			0
$197$		−	10.6990i		−	0.762270i	−0.924519	−	0.381135i	$-0.875533\pi$
$197$		−	10.6990i		−	0.762270i	0.924519	−	0.381135i	$-0.124467\pi$
$198$		0			0
$199$	20.3173			1.44025			0.720127	−	0.693842i	$-0.244084\pi$
$199$	20.3173			1.44025			0.720127	+	0.693842i	$0.244084\pi$
$200$		0			0
$201$	−3.14947	−	10.0850i	−0.222146	−	0.711339i
$202$		0			0
$203$	−8.58605			−0.602623
$204$		0			0
$205$		−	2.51539i		−	0.175683i
$206$		0			0
$207$	−9.80359	−	14.1653i	−0.681397	−	0.984560i
$208$		0			0
$209$	3.60439	+	7.12334i	0.249321	+	0.492731i
$210$		0			0
$211$		−	2.92014i		−	0.201030i	−0.994936	−	0.100515i	$-0.967951\pi$
$211$		−	2.92014i		−	0.201030i	0.994936	−	0.100515i	$-0.0320491\pi$
$212$		0			0
$213$	−23.9151	+	7.46851i	−1.63863	+	0.511734i
$214$		0			0
$215$		−	4.31746i		−	0.294448i
$216$		0			0
$217$		−	16.6280i		−	1.12878i
$218$		0			0
$219$	−4.64805	+	1.45155i	−0.314086	+	0.0980869i
$220$		0			0
$221$	−3.63853	+	22.3307i	−0.244754	+	1.50212i
$222$		0			0
$223$			13.2322i			0.886092i	0.896499	+	0.443046i	$0.146102\pi$
$223$			13.2322i			0.886092i	−0.896499	+	0.443046i	$0.853898\pi$
$224$		0			0
$225$	−10.4645	+	7.24232i	−0.697636	+	0.482821i
$226$		0			0
$227$		−	5.67345i		−	0.376560i	−0.982115	−	0.188280i	$-0.939709\pi$
$227$		−	5.67345i		−	0.376560i	0.982115	−	0.188280i	$-0.0602912\pi$
$228$		0			0
$229$			13.8966i			0.918313i	0.888355	+	0.459156i	$0.151848\pi$
$229$			13.8966i			0.918313i	−0.888355	+	0.459156i	$0.848152\pi$
$230$		0			0
$231$	13.0183	+	13.3952i	0.856544	+	0.881342i
$232$		0			0
$233$	11.1167			0.728282			0.364141	−	0.931344i	$-0.381363\pi$
$233$	11.1167			0.728282			0.364141	+	0.931344i	$0.381363\pi$
$234$		0			0
$235$	−6.73260			−0.439186
$236$		0			0
$237$	4.82745	+	15.4581i	0.313577	+	1.00411i
$238$		0			0
$239$			23.1099i			1.49486i	0.664342	+	0.747429i	$0.268712\pi$
$239$			23.1099i			1.49486i	−0.664342	+	0.747429i	$0.731288\pi$
$240$		0			0
$241$	24.5078			1.57868			0.789342	−	0.613954i	$-0.210422\pi$
$241$	24.5078			1.57868			0.789342	+	0.613954i	$0.210422\pi$
$242$		0			0
$243$	−0.892966	+	15.5629i	−0.0572838	+	0.998358i
$244$		0			0
$245$	−3.11060			−0.198729
$246$		0			0
$247$	1.39571	−	8.56583i	0.0888067	−	0.545031i
$248$		0			0
$249$	−8.45289	−	27.0672i	−0.535680	−	1.71531i
$250$		0			0
$251$		−	10.9496i		−	0.691133i	−0.938394	−	0.345567i	$-0.887687\pi$
$251$		−	10.9496i		−	0.691133i	0.938394	−	0.345567i	$-0.112313\pi$
$252$		0			0
$253$	16.9935	−	8.59867i	1.06837	−	0.540594i
$254$		0			0
$255$	−9.03197	+	2.82062i	−0.565604	+	0.176634i
$256$		0			0
$257$		−	11.0720i		−	0.690655i	−0.938482	−	0.345328i	$-0.887768\pi$
$257$		−	11.0720i		−	0.690655i	0.938482	−	0.345328i	$-0.112232\pi$
$258$		0			0
$259$			35.8039i			2.22475i
$260$		0			0
$261$	6.51380	−	4.50809i	0.403194	−	0.279044i
$262$		0			0
$263$	12.7449			0.785886			0.392943	−	0.919563i	$-0.371457\pi$
$263$	12.7449			0.785886			0.392943	+	0.919563i	$0.371457\pi$
$264$		0			0
$265$		−	6.32043i		−	0.388261i
$266$		0			0
$267$	−14.7732	+	4.61358i	−0.904108	+	0.282346i
$268$		0			0
$269$		−	11.6244i		−	0.708755i	−0.935103	−	0.354377i	$-0.884693\pi$
$269$		−	11.6244i		−	0.708755i	0.935103	−	0.354377i	$-0.115307\pi$
$270$		0			0
$271$	−7.89508			−0.479592			−0.239796	−	0.970823i	$-0.577081\pi$
$271$	−7.89508			−0.479592			−0.239796	+	0.970823i	$0.577081\pi$
$272$		0			0
$273$	−2.85727	−	20.1043i	−0.172930	−	1.21677i
$274$		0			0
$275$	−6.35219	−	12.5538i	−0.383052	−	0.757023i
$276$		0			0
$277$		−	2.03696i		−	0.122389i	−0.998126	−	0.0611946i	$-0.980509\pi$
$277$		−	2.03696i		−	0.122389i	0.998126	−	0.0611946i	$-0.0194910\pi$
$278$		0			0
$279$	8.73051	+	12.6148i	0.522682	+	0.755231i
$280$		0			0
$281$		−	3.53913i		−	0.211127i	−0.994413	−	0.105564i	$-0.966335\pi$
$281$		−	3.53913i		−	0.211127i	0.994413	−	0.105564i	$-0.0336646\pi$
$282$		0			0
$283$		−	28.9010i		−	1.71799i	−0.511987	−	0.858993i	$-0.671090\pi$
$283$		−	28.9010i		−	1.71799i	0.511987	−	0.858993i	$-0.328910\pi$
$284$		0			0
$285$	3.46458	−	1.08196i	0.205224	−	0.0640900i
$286$		0			0
$287$		−	9.39497i		−	0.554568i
$288$		0			0
$289$	22.3767			1.31628
$290$		0			0
$291$	−6.28116	−	20.1130i	−0.368208	−	1.17905i
$292$		0			0
$293$		−	26.1433i		−	1.52731i	−0.645627	−	0.763653i	$-0.723404\pi$
$293$		−	26.1433i		−	1.52731i	0.645627	−	0.763653i	$-0.276596\pi$
$294$		0			0
$295$	−9.32758			−0.543072
$296$		0			0
$297$	−16.9095	−	3.32703i	−0.981188	−	0.193054i
$298$		0			0
$299$	−20.4347	−	3.32961i	−1.18177	−	0.192556i
$300$		0			0
$301$		−	16.1257i		−	0.929469i
$302$		0			0
$303$	−29.7992	+	9.30609i	−1.71192	+	0.534621i
$304$		0			0
$305$		−	4.68575i		−	0.268305i
$306$		0			0
$307$	−6.74794			−0.385125			−0.192562	−	0.981285i	$-0.561680\pi$
$307$	−6.74794			−0.385125			−0.192562	+	0.981285i	$0.561680\pi$
$308$		0			0
$309$	−16.8737	+	5.26954i	−0.959911	+	0.299774i
$310$		0			0
$311$			11.1800i			0.633961i	0.948432	+	0.316980i	$0.102669\pi$
$311$			11.1800i			0.633961i	−0.948432	+	0.316980i	$0.897331\pi$
$312$		0			0
$313$	10.1785			0.575321			0.287661	−	0.957732i	$-0.407123\pi$
$313$	10.1785			0.575321			0.287661	+	0.957732i	$0.407123\pi$
$314$		0			0
$315$	6.98312	−	4.83289i	0.393454	−	0.272303i
$316$		0			0
$317$	−24.6301			−1.38337			−0.691683	−	0.722201i	$-0.743131\pi$
$317$	−24.6301			−1.38337			−0.691683	+	0.722201i	$0.743131\pi$
$318$		0			0
$319$	3.95401	+	7.81429i	0.221382	+	0.437516i
$320$		0			0
$321$	7.26492	−	2.26878i	0.405488	−	0.126631i
$322$		0			0
$323$	−15.1045			−0.840439
$324$		0			0
$325$	−2.45972	+	15.0960i	−0.136441	+	0.837374i
$326$		0			0
$327$	29.0517	−	9.07266i	1.60657	−	0.501719i
$328$		0			0
$329$	−25.1462			−1.38636
$330$		0			0
$331$		−	14.1045i		−	0.775253i	−0.921817	−	0.387626i	$-0.873295\pi$
$331$		−	14.1045i		−	0.775253i	0.921817	−	0.387626i	$-0.126705\pi$
$332$		0			0
$333$	−18.7988	−	27.1626i	−1.03017	−	1.48850i
$334$		0			0
$335$		−	5.31045i		−	0.290141i
$336$		0			0
$337$		−	9.93686i		−	0.541295i	−0.962678	−	0.270648i	$-0.912762\pi$
$337$		−	9.93686i		−	0.541295i	0.962678	−	0.270648i	$-0.0872378\pi$
$338$		0			0
$339$	3.05340	+	9.77737i	0.165838	+	0.531034i
$340$		0			0
$341$	−15.1334	+	7.65747i	−0.819521	+	0.414675i
$342$		0			0
$343$	11.1432			0.601678
$344$		0			0
$345$	−2.58114	−	8.26513i	−0.138964	−	0.444980i
$346$		0			0
$347$	−32.0908			−1.72272			−0.861362	−	0.507991i	$-0.830388\pi$
$347$	−32.0908			−1.72272			−0.861362	+	0.507991i	$0.830388\pi$
$348$		0			0
$349$	−7.29161			−0.390311			−0.195156	−	0.980772i	$-0.562521\pi$
$349$	−7.29161			−0.390311			−0.195156	+	0.980772i	$0.562521\pi$
$350$		0			0
$351$	12.7234	+	13.7519i	0.679124	+	0.734023i
$352$		0			0
$353$	0.144832			0.00770861			0.00385431	−	0.999993i	$-0.498773\pi$
$353$	0.144832			0.00770861			0.00385431	+	0.999993i	$0.498773\pi$
$354$		0			0
$355$	−12.5930			−0.668366
$356$		0			0
$357$	−33.7343	+	10.5350i	−1.78541	+	0.557571i
$358$		0			0
$359$			20.8264i			1.09918i	0.835436	+	0.549588i	$0.185215\pi$
$359$			20.8264i			1.09918i	−0.835436	+	0.549588i	$0.814785\pi$
$360$		0			0
$361$	−13.2060			−0.695055
$362$		0			0
$363$	6.19605	−	18.0169i	0.325208	−	0.945642i
$364$		0			0
$365$	−2.44753			−0.128109
$366$		0			0
$367$	2.02728			0.105823			0.0529116	−	0.998599i	$-0.483150\pi$
$367$	2.02728			0.105823			0.0529116	+	0.998599i	$0.483150\pi$
$368$		0			0
$369$	4.93281	+	7.12749i	0.256792	+	0.371042i
$370$		0			0
$371$		−	23.6067i		−	1.22560i
$372$		0			0
$373$		−	4.05386i		−	0.209901i	−0.994477	−	0.104950i	$-0.966532\pi$
$373$		−	4.05386i		−	0.209901i	0.994477	−	0.104950i	$-0.0334684\pi$
$374$		0			0
$375$	−13.3025	+	4.15427i	−0.686937	+	0.214526i
$376$		0			0
$377$	1.53109	−	9.39671i	0.0788551	−	0.483955i
$378$		0			0
$379$			29.1270i			1.49615i	0.663612	+	0.748077i	$0.269023\pi$
$379$			29.1270i			1.49615i	−0.663612	+	0.748077i	$0.730977\pi$
$380$		0			0
$381$	−6.45529	−	20.6706i	−0.330714	−	1.05899i
$382$		0			0
$383$	−14.5295			−0.742423			−0.371211	−	0.928548i	$-0.621057\pi$
$383$	−14.5295			−0.742423			−0.371211	+	0.928548i	$0.621057\pi$
$384$		0			0
$385$	4.23890	+	8.37731i	0.216034	+	0.426947i
$386$		0			0
$387$	8.46676	+	12.2337i	0.430389	+	0.621876i
$388$		0			0
$389$			9.08538i			0.460647i	0.973114	+	0.230323i	$0.0739784\pi$
$389$			9.08538i			0.460647i	−0.973114	+	0.230323i	$0.926022\pi$
$390$		0			0
$391$			36.0335i			1.82229i
$392$		0			0
$393$	6.74426	−	2.10618i	0.340203	−	0.106243i
$394$		0			0
$395$			8.13978i			0.409557i
$396$		0			0
$397$		−	17.2471i		−	0.865607i	−0.901488	−	0.432803i	$-0.857524\pi$
$397$		−	17.2471i		−	0.865607i	0.901488	−	0.432803i	$-0.142476\pi$
$398$		0			0
$399$	12.9402	−	4.04113i	0.647819	−	0.202309i
$400$		0			0
$401$	−19.5813			−0.977842			−0.488921	−	0.872328i	$-0.662609\pi$
$401$	−19.5813			−0.977842			−0.488921	+	0.872328i	$0.662609\pi$
$402$		0			0
$403$	18.1980	+	2.96516i	0.906506	+	0.147705i
$404$		0			0
$405$	−2.76024	+	7.33294i	−0.137157	+	0.364377i
$406$		0			0
$407$	32.5857	−	16.4883i	1.61521	−	0.817293i
$408$		0			0
$409$	−24.5707			−1.21495			−0.607473	−	0.794341i	$-0.707817\pi$
$409$	−24.5707			−1.21495			−0.607473	+	0.794341i	$0.707817\pi$
$410$		0			0
$411$	−11.6066	+	3.62466i	−0.572511	+	0.178791i
$412$		0			0
$413$	−34.8384			−1.71429
$414$		0			0
$415$		−	14.2528i		−	0.699642i
$416$		0			0
$417$	4.61691	+	14.7839i	0.226091	+	0.723971i
$418$		0			0
$419$		−	5.83931i		−	0.285269i	−0.989775	−	0.142634i	$-0.954443\pi$
$419$		−	5.83931i		−	0.285269i	0.989775	−	0.142634i	$-0.0455573\pi$
$420$		0			0
$421$		−	20.3182i		−	0.990247i	−0.868823	−	0.495124i	$-0.835123\pi$
$421$		−	20.3182i		−	0.990247i	0.868823	−	0.495124i	$-0.164877\pi$
$422$		0			0
$423$	19.0772	−	13.2030i	0.927563	−	0.641950i
$424$		0			0
$425$	26.6195			1.29123
$426$		0			0
$427$		−	17.5012i		−	0.846945i
$428$		0			0
$429$	−16.9814	+	11.8588i	−0.819871	+	0.572549i
$430$		0			0
$431$		−	22.8532i		−	1.10080i	−0.834902	−	0.550399i	$-0.814476\pi$
$431$		−	22.8532i		−	1.10080i	0.834902	−	0.550399i	$-0.185524\pi$
$432$		0			0
$433$	−4.68978			−0.225377			−0.112688	−	0.993630i	$-0.535946\pi$
$433$	−4.68978			−0.225377			−0.112688	+	0.993630i	$0.535946\pi$
$434$		0			0
$435$	3.80064	−	1.18691i	0.182227	−	0.0569081i
$436$		0			0
$437$		−	13.8221i		−	0.661202i
$438$		0			0
$439$		−	22.9469i		−	1.09520i	−0.836742	−	0.547598i	$-0.815542\pi$
$439$		−	22.9469i		−	1.09520i	0.836742	−	0.547598i	$-0.184458\pi$
$440$		0			0
$441$	8.81405	−	6.10005i	0.419716	−	0.290478i
$442$		0			0
$443$			18.5312i			0.880442i	0.897889	+	0.440221i	$0.145100\pi$
$443$			18.5312i			0.880442i	−0.897889	+	0.440221i	$0.854900\pi$
$444$		0			0
$445$	−7.77916			−0.368767
$446$		0			0
$447$	0.577349	+	1.84874i	0.0273077	+	0.0874425i
$448$		0			0
$449$	27.8554			1.31458			0.657290	−	0.753638i	$-0.271703\pi$
$449$	27.8554			1.31458			0.657290	+	0.753638i	$0.271703\pi$
$450$		0			0
$451$	−8.55051	+	4.32654i	−0.402628	+	0.203729i
$452$		0			0
$453$	22.5434	−	7.04014i	1.05918	−	0.330775i
$454$		0			0
$455$	1.64140	−	10.0737i	0.0769502	−	0.472264i
$456$		0			0
$457$	36.9551			1.72869			0.864344	−	0.502901i	$-0.167734\pi$
$457$	36.9551			1.72869			0.864344	+	0.502901i	$0.167734\pi$
$458$		0			0
$459$	20.0611	−	25.7045i	0.936374	−	1.19978i
$460$		0			0
$461$			13.9805i			0.651135i	0.945519	+	0.325568i	$0.105555\pi$
$461$			13.9805i			0.651135i	−0.945519	+	0.325568i	$0.894445\pi$
$462$		0			0
$463$		−	35.9508i		−	1.67078i	−0.549660	−	0.835388i	$-0.685243\pi$
$463$		−	35.9508i		−	1.67078i	0.549660	−	0.835388i	$-0.314757\pi$
$464$		0			0
$465$	2.29861	+	7.36044i	0.106596	+	0.341332i
$466$		0			0
$467$			39.5107i			1.82834i	0.405333	+	0.914169i	$0.367155\pi$
$467$			39.5107i			1.82834i	−0.405333	+	0.914169i	$0.632845\pi$
$468$		0			0
$469$		−	19.8345i		−	0.915872i
$470$		0			0
$471$	−4.33002	+	1.35223i	−0.199517	+	0.0623076i
$472$		0			0
$473$	−14.6762	+	7.42614i	−0.674814	+	0.341454i
$474$		0			0
$475$	−10.2110			−0.468512
$476$		0			0
$477$	12.3947	+	17.9092i	0.567513	+	0.820008i
$478$		0			0
$479$			26.7946i			1.22428i	0.790751	+	0.612138i	$0.209690\pi$
$479$			26.7946i			1.22428i	−0.790751	+	0.612138i	$0.790310\pi$
$480$		0			0
$481$	−39.1844	−	6.38466i	−1.78665	−	0.291115i
$482$		0			0
$483$	−9.64054	−	30.8702i	−0.438660	−	1.40464i
$484$		0			0
$485$		−	10.5909i		−	0.480909i
$486$		0			0
$487$			7.85099i			0.355763i	0.984052	+	0.177881i	$0.0569243\pi$
$487$			7.85099i			0.355763i	−0.984052	+	0.177881i	$0.943076\pi$
$488$		0			0
$489$	6.74574	+	21.6007i	0.305053	+	0.976816i
$490$		0			0
$491$	8.44376			0.381062			0.190531	−	0.981681i	$-0.438979\pi$
$491$	8.44376			0.381062			0.190531	+	0.981681i	$0.438979\pi$
$492$		0			0
$493$	−16.5697			−0.746260
$494$		0			0
$495$	−7.61433	−	4.12982i	−0.342239	−	0.185621i
$496$		0			0
$497$	−47.0347			−2.10979
$498$		0			0
$499$			24.9424i			1.11658i	0.829647	+	0.558288i	$0.188541\pi$
$499$			24.9424i			1.11658i	−0.829647	+	0.558288i	$0.811459\pi$
$500$		0			0
$501$	−9.27208	−	29.6903i	−0.414246	−	1.32647i
$502$		0			0
$503$	33.6542			1.50057			0.750283	−	0.661117i	$-0.229917\pi$
$503$	33.6542			1.50057			0.750283	+	0.661117i	$0.229917\pi$
$504$		0			0
$505$	−15.6914			−0.698258
$506$		0			0
$507$	22.5120	+	0.458018i	0.999793	+	0.0203413i
$508$		0			0
$509$	17.9484			0.795547			0.397774	−	0.917484i	$-0.369783\pi$
$509$	17.9484			0.795547			0.397774	+	0.917484i	$0.369783\pi$
$510$		0			0
$511$	−9.14150			−0.404396
$512$		0			0
$513$	−7.69527	+	9.86001i	−0.339754	+	0.435330i
$514$		0			0
$515$	−8.88520			−0.391529
$516$		0			0
$517$	11.5802	+	22.8859i	0.509298	+	1.00652i
$518$		0			0
$519$	−27.6765	+	8.64318i	−1.21486	+	0.379393i
$520$		0			0
$521$		−	36.2132i		−	1.58653i	−0.608877	−	0.793264i	$-0.708380\pi$
$521$		−	36.2132i		−	1.58653i	0.608877	−	0.793264i	$-0.291620\pi$
$522$		0			0
$523$			43.4175i			1.89851i	0.314502	+	0.949257i	$0.398162\pi$
$523$			43.4175i			1.89851i	−0.314502	+	0.949257i	$0.601838\pi$
$524$		0			0
$525$	−22.8051	+	7.12187i	−0.995296	+	0.310824i
$526$		0			0
$527$		−	32.0894i		−	1.39783i
$528$		0			0
$529$	−9.97417			−0.433659
$530$		0			0
$531$	26.4301	−	18.2918i	1.14697	−	0.793798i
$532$		0			0
$533$	10.2820	+	1.67534i	0.445363	+	0.0725670i
$534$		0			0
$535$	3.82549			0.165390
$536$		0			0
$537$	9.71503	+	31.1087i	0.419235	+	1.34244i
$538$		0			0
$539$	5.35031	+	10.5738i	0.230454	+	0.455445i
$540$		0			0
$541$	6.03306			0.259381			0.129691	−	0.991554i	$-0.458602\pi$
$541$	6.03306			0.259381			0.129691	+	0.991554i	$0.458602\pi$
$542$		0			0
$543$	−12.0370	+	3.75908i	−0.516558	+	0.161318i
$544$		0			0
$545$	15.2978			0.655286
$546$		0			0
$547$		−	31.8364i		−	1.36123i	−0.732643	−	0.680613i	$-0.761713\pi$
$547$		−	31.8364i		−	1.36123i	0.732643	−	0.680613i	$-0.238287\pi$
$548$		0			0
$549$	9.18899	+	13.2773i	0.392177	+	0.566662i
$550$		0			0
$551$	6.35597			0.270773
$552$		0			0
$553$			30.4020i			1.29283i
$554$		0			0
$555$	−4.94944	−	15.8487i	−0.210092	−	0.672740i
$556$		0			0
$557$			1.66492i			0.0705448i	0.999378	+	0.0352724i	$0.0112299\pi$
$557$			1.66492i			0.0705448i	−0.999378	+	0.0352724i	$0.988770\pi$
$558$		0			0
$559$	17.6482	+	2.87558i	0.746440	+	0.121624i
$560$		0			0
$561$	25.1233	+	25.8506i	1.06070	+	1.09141i
$562$		0			0
$563$	33.0416			1.39254			0.696269	−	0.717781i	$-0.254842\pi$
$563$	33.0416			1.39254			0.696269	+	0.717781i	$0.254842\pi$
$564$		0			0
$565$			5.14848i			0.216598i
$566$		0			0
$567$	−10.3095	+	27.3885i	−0.432957	+	1.15021i
$568$		0			0
$569$	27.2794			1.14361			0.571806	−	0.820389i	$-0.306243\pi$
$569$	27.2794			1.14361			0.571806	+	0.820389i	$0.306243\pi$
$570$		0			0
$571$			2.35735i			0.0986521i	0.998783	+	0.0493260i	$0.0157073\pi$
$571$			2.35735i			0.0986521i	−0.998783	+	0.0493260i	$0.984293\pi$
$572$		0			0
$573$	−9.74136	−	31.1930i	−0.406951	−	1.30311i
$574$		0			0
$575$			24.3594i			1.01586i
$576$		0			0
$577$			20.6405i			0.859275i	0.903001	+	0.429638i	$0.141359\pi$
$577$			20.6405i			0.859275i	−0.903001	+	0.429638i	$0.858641\pi$
$578$		0			0
$579$	−5.16879	+	1.61418i	−0.214807	+	0.0670828i
$580$		0			0
$581$		−	53.2340i		−	2.20852i
$582$		0			0
$583$	−21.4849	+	10.8713i	−0.889812	+	0.450243i
$584$		0			0
$585$	4.04395	+	8.50425i	0.167197	+	0.351608i
$586$		0			0
$587$	4.40058			0.181631			0.0908157	−	0.995868i	$-0.471053\pi$
$587$	4.40058			0.181631			0.0908157	+	0.995868i	$0.471053\pi$
$588$		0			0
$589$			12.3092i			0.507191i
$590$		0			0
$591$	5.52404	+	17.6887i	0.227229	+	0.727614i
$592$		0			0
$593$			21.6607i			0.889500i	0.895655	+	0.444750i	$0.146707\pi$
$593$			21.6607i			0.889500i	−0.895655	+	0.444750i	$0.853293\pi$
$594$		0			0
$595$	−17.7635			−0.728233
$596$		0			0
$597$	−33.5907	+	10.4901i	−1.37477	+	0.429332i
$598$		0			0
$599$			9.96665i			0.407226i	0.979051	+	0.203613i	$0.0652685\pi$
$599$			9.96665i			0.407226i	−0.979051	+	0.203613i	$0.934732\pi$
$600$		0			0
$601$		−	30.0884i		−	1.22733i	−0.789566	−	0.613666i	$-0.789694\pi$
$601$		−	30.0884i		−	1.22733i	0.789566	−	0.613666i	$-0.210306\pi$
$602$		0			0
$603$	10.4141	+	15.0474i	0.424093	+	0.612778i
$604$		0			0
$605$	5.67224	−	7.71577i	0.230609	−	0.313691i
$606$		0			0
$607$			19.6228i			0.796464i	0.917285	+	0.398232i	$0.130376\pi$
$607$			19.6228i			0.796464i	−0.917285	+	0.398232i	$0.869624\pi$
$608$		0			0
$609$	14.1954	−	4.43311i	0.575225	−	0.179639i
$610$		0			0
$611$	4.48415	−	27.5204i	0.181409	−	1.11336i
$612$		0			0
$613$	40.4393			1.63333			0.816664	−	0.577114i	$-0.195821\pi$
$613$	40.4393			1.63333			0.816664	+	0.577114i	$0.195821\pi$
$614$		0			0
$615$	1.29874	+	4.15871i	0.0523701	+	0.167695i
$616$		0			0
$617$	2.14657			0.0864177			0.0432088	−	0.999066i	$-0.486242\pi$
$617$	2.14657			0.0864177			0.0432088	+	0.999066i	$0.486242\pi$
$618$		0			0
$619$			11.0786i			0.445288i	0.974900	+	0.222644i	$0.0714688\pi$
$619$			11.0786i			0.445288i	−0.974900	+	0.222644i	$0.928531\pi$
$620$		0			0
$621$	23.5221	+	18.3579i	0.943910	+	0.736677i
$622$		0			0
$623$	−29.0551			−1.16407
$624$		0			0
$625$	14.2057			0.568230
$626$		0			0
$627$	−9.63705	−	9.91604i	−0.384867	−	0.396009i
$628$		0			0
$629$			69.0956i			2.75502i
$630$		0			0
$631$			33.4037i			1.32978i	0.746941	+	0.664890i	$0.231522\pi$
$631$			33.4037i			1.32978i	−0.746941	+	0.664890i	$0.768478\pi$
$632$		0			0
$633$	1.50771	+	4.82788i	0.0599262	+	0.191891i
$634$		0			0
$635$		−	10.8845i		−	0.431940i
$636$		0			0
$637$	2.07177	−	12.7150i	0.0820865	−	0.503787i
$638$		0			0
$639$	35.6828	−	24.6955i	1.41159	−	0.976937i
$640$		0			0
$641$			21.8013i			0.861101i	0.902566	+	0.430551i	$0.141681\pi$
$641$			21.8013i			0.861101i	−0.902566	+	0.430551i	$0.858319\pi$
$642$		0			0
$643$		−	13.9891i		−	0.551675i	−0.961204	−	0.275837i	$-0.911045\pi$
$643$		−	13.9891i		−	0.551675i	0.961204	−	0.275837i	$-0.0889551\pi$
$644$		0			0
$645$	2.22917	+	7.13808i	0.0877736	+	0.281062i
$646$		0			0
$647$			39.8346i			1.56606i	0.621984	+	0.783030i	$0.286327\pi$
$647$			39.8346i			1.56606i	−0.621984	+	0.783030i	$0.713673\pi$
$648$		0			0
$649$	16.0437	+	31.7070i	0.629769	+	1.24461i
$650$		0			0
$651$	8.58531	+	27.4912i	0.336485	+	1.07747i
$652$		0			0
$653$			43.1547i			1.68877i	0.535733	+	0.844387i	$0.320035\pi$
$653$			43.1547i			1.68877i	−0.535733	+	0.844387i	$0.679965\pi$
$654$		0			0
$655$	3.55133			0.138762
$656$		0			0
$657$	6.93519	−	4.79972i	0.270567	−	0.187255i
$658$		0			0
$659$	−10.9237			−0.425528			−0.212764	−	0.977104i	$-0.568247\pi$
$659$	−10.9237			−0.425528			−0.212764	+	0.977104i	$0.568247\pi$
$660$		0			0
$661$		−	11.9384i		−	0.464350i	−0.972674	−	0.232175i	$-0.925416\pi$
$661$		−	11.9384i		−	0.464350i	0.972674	−	0.232175i	$-0.0745842\pi$
$662$		0			0
$663$	−5.51406	−	38.7980i	−0.214148	−	1.50679i
$664$		0			0
$665$	6.81392			0.264232
$666$		0			0
$667$		−	15.1629i		−	0.587108i
$668$		0			0
$669$	−6.83198	−	21.8768i	−0.264140	−	0.845807i
$670$		0			0
$671$	−15.9281	+	8.05960i	−0.614899	+	0.311138i
$672$		0			0
$673$			6.00646i			0.231532i	0.993277	+	0.115766i	$0.0369323\pi$
$673$			6.00646i			0.231532i	−0.993277	+	0.115766i	$0.963068\pi$
$674$		0			0
$675$	13.5617	−	17.3768i	0.521992	−	0.668832i
$676$		0			0
$677$	10.7362			0.412627			0.206314	−	0.978486i	$-0.433853\pi$
$677$	10.7362			0.412627			0.206314	+	0.978486i	$0.433853\pi$
$678$		0			0
$679$		−	39.5570i		−	1.51806i
$680$		0			0
$681$	2.92929	+	9.37994i	0.112251	+	0.359440i
$682$		0			0
$683$	−4.08782			−0.156416			−0.0782081	−	0.996937i	$-0.524920\pi$
$683$	−4.08782			−0.156416			−0.0782081	+	0.996937i	$0.524920\pi$
$684$		0			0
$685$	−6.11169			−0.233516
$686$		0			0
$687$	−7.17503	−	22.9753i	−0.273744	−	0.876563i
$688$		0			0
$689$	25.8356	+	4.20962i	0.984258	+	0.160374i
$690$		0			0
$691$		−	23.9280i		−	0.910263i	−0.890424	−	0.455132i	$-0.849592\pi$
$691$		−	23.9280i		−	0.910263i	0.890424	−	0.455132i	$-0.150408\pi$
$692$		0			0
$693$	−28.4395	−	15.4248i	−1.08033	−	0.585941i
$694$		0			0
$695$			7.78477i			0.295293i
$696$		0			0
$697$		−	18.1307i		−	0.686751i
$698$		0			0
$699$	−18.3794	+	5.73975i	−0.695172	+	0.217097i
$700$		0			0
$701$	−31.7585			−1.19950			−0.599750	−	0.800187i	$-0.704733\pi$
$701$	−31.7585			−1.19950			−0.599750	+	0.800187i	$0.704733\pi$
$702$		0			0
$703$		−	26.5044i		−	0.999634i
$704$		0			0
$705$	11.1310	−	3.47615i	0.419219	−	0.130919i
$706$		0			0
$707$	−58.6073			−2.20415
$708$		0			0
$709$			13.2115i			0.496169i	0.968738	+	0.248084i	$0.0798010\pi$
$709$			13.2115i			0.496169i	−0.968738	+	0.248084i	$0.920199\pi$
$710$		0			0
$711$	−15.9625	−	23.0645i	−0.598641	−	0.864985i
$712$		0			0
$713$	29.3649			1.09972
$714$		0			0
$715$	−9.92416	+	3.14525i	−0.371142	+	0.117626i
$716$		0			0
$717$	−11.9320	−	38.2078i	−0.445610	−	1.42690i
$718$		0			0
$719$		−	41.6116i		−	1.55185i	−0.630824	−	0.775926i	$-0.717283\pi$
$719$		−	41.6116i		−	1.55185i	0.630824	−	0.775926i	$-0.282717\pi$
$720$		0			0
$721$	−33.1861			−1.23592
$722$		0			0
$723$	−40.5188	+	12.6537i	−1.50691	+	0.470598i
$724$		0			0
$725$	−11.2014			−0.416011
$726$		0			0
$727$	−9.95266			−0.369124			−0.184562	−	0.982821i	$-0.559087\pi$
$727$	−9.95266			−0.369124			−0.184562	+	0.982821i	$0.559087\pi$
$728$		0			0
$729$	−6.55900	−	26.1912i	−0.242926	−	0.970045i
$730$		0			0
$731$		−	31.1199i		−	1.15101i
$732$		0			0
$733$	−22.0742			−0.815330			−0.407665	−	0.913132i	$-0.633657\pi$
$733$	−22.0742			−0.815330			−0.407665	+	0.913132i	$0.633657\pi$
$734$		0			0
$735$	5.14278	−	1.60605i	0.189694	−	0.0592401i
$736$		0			0
$737$	−18.0517	+	9.13410i	−0.664942	+	0.336459i
$738$		0			0
$739$	44.7482			1.64609			0.823045	−	0.567977i	$-0.192274\pi$
$739$	44.7482			1.64609			0.823045	+	0.567977i	$0.192274\pi$
$740$		0			0
$741$	2.11514	+	14.8826i	0.0777017	+	0.546724i
$742$		0			0
$743$		−	13.9650i		−	0.512326i	−0.966634	−	0.256163i	$-0.917542\pi$
$743$		−	13.9650i		−	0.512326i	0.966634	−	0.256163i	$-0.0824584\pi$
$744$		0			0
$745$			0.973494i			0.0356660i
$746$		0			0
$747$	27.9504	+	40.3860i	1.02265	+	1.47764i
$748$		0			0
$749$	14.2882			0.522079
$750$		0			0
$751$	−7.66003			−0.279519			−0.139759	−	0.990186i	$-0.544633\pi$
$751$	−7.66003			−0.279519			−0.139759	+	0.990186i	$0.544633\pi$
$752$		0			0
$753$	5.65345	+	18.1030i	0.206023	+	0.659712i
$754$		0			0
$755$	11.8707			0.432019
$756$		0			0
$757$	23.5103			0.854496			0.427248	−	0.904134i	$-0.359483\pi$
$757$	23.5103			0.854496			0.427248	+	0.904134i	$0.359483\pi$
$758$		0			0
$759$	−23.6558	+	22.9902i	−0.858651	+	0.834492i
$760$		0			0
$761$		−	30.6029i		−	1.10935i	−0.832066	−	0.554676i	$-0.812842\pi$
$761$		−	30.6029i		−	1.10935i	0.832066	−	0.554676i	$-0.187158\pi$
$762$		0			0
$763$	57.1372			2.06850
$764$		0			0
$765$	13.4763	−	9.32669i	0.487236	−	0.337207i
$766$		0			0
$767$	6.21249	−	38.1277i	0.224320	−	1.37671i
$768$		0			0
$769$	−11.0935			−0.400041			−0.200020	−	0.979792i	$-0.564101\pi$
$769$	−11.0935			−0.400041			−0.200020	+	0.979792i	$0.564101\pi$
$770$		0			0
$771$	5.71667	+	18.3055i	0.205881	+	0.659256i
$772$		0			0
$773$	17.2806			0.621540			0.310770	−	0.950485i	$-0.399413\pi$
$773$	17.2806			0.621540			0.310770	+	0.950485i	$0.399413\pi$
$774$		0			0
$775$		−	21.6931i		−	0.779238i
$776$		0			0
$777$	−18.4861	−	59.1948i	−0.663186	−	2.12360i
$778$		0			0
$779$			6.95479i			0.249181i
$780$		0			0
$781$	21.6602	+	42.8070i	0.775064	+	1.53175i
$782$		0			0
$783$	−8.44170	+	10.8164i	−0.301682	+	0.386547i
$784$		0			0
$785$	−2.28006			−0.0813788
$786$		0			0
$787$	37.1726			1.32506			0.662530	−	0.749036i	$-0.269483\pi$
$787$	37.1726			1.32506			0.662530	+	0.749036i	$0.269483\pi$
$788$		0			0
$789$	−21.0713	+	6.58041i	−0.750157	+	0.234269i
$790$		0			0
$791$			19.2295i			0.683723i
$792$		0			0
$793$	19.1536	+	3.12087i	0.680166	+	0.110825i
$794$		0			0
$795$	3.26333	+	10.4496i	0.115739	+	0.370609i
$796$		0			0
$797$		−	25.0395i		−	0.886945i	−0.896288	−	0.443472i	$-0.853746\pi$
$797$		−	25.0395i		−	0.886945i	0.896288	−	0.443472i	$-0.146254\pi$
$798$		0			0
$799$	−48.5281			−1.71680
$800$		0			0
$801$	22.0426	−	15.2553i	0.778838	−	0.539020i
$802$		0			0
$803$	4.20981	+	8.31981i	0.148561	+	0.293600i
$804$		0			0
$805$		−	16.2553i		−	0.572925i
$806$		0			0
$807$	6.00188	+	19.2187i	0.211276	+	0.676532i
$808$		0			0
$809$	31.0841			1.09286			0.546430	−	0.837505i	$-0.315986\pi$
$809$	31.0841			1.09286			0.546430	+	0.837505i	$0.315986\pi$
$810$		0			0
$811$	0.164157			0.00576432			0.00288216	−	0.999996i	$-0.499083\pi$
$811$	0.164157			0.00576432			0.00288216	+	0.999996i	$0.499083\pi$
$812$		0			0
$813$	13.0530	−	4.07635i	0.457788	−	0.142964i
$814$		0			0
$815$			11.3743i			0.398424i
$816$		0			0
$817$			11.9373i			0.417634i
$818$		0			0
$819$	15.1041	+	31.7633i	0.527780	+	1.10990i
$820$		0			0
$821$		−	3.63912i		−	0.127006i	−0.997982	−	0.0635031i	$-0.979773\pi$
$821$		−	3.63912i		−	0.127006i	0.997982	−	0.0635031i	$-0.0202273\pi$
$822$		0			0
$823$	20.1842			0.703577			0.351788	−	0.936080i	$-0.385574\pi$
$823$	20.1842			0.703577			0.351788	+	0.936080i	$0.385574\pi$
$824$		0			0
$825$	16.9838	+	17.4755i	0.591301	+	0.608420i
$826$		0			0
$827$		−	9.18037i		−	0.319233i	−0.987179	−	0.159616i	$-0.948974\pi$
$827$		−	9.18037i		−	0.319233i	0.987179	−	0.159616i	$-0.0510257\pi$
$828$		0			0
$829$	−2.47296			−0.0858894			−0.0429447	−	0.999077i	$-0.513674\pi$
$829$	−2.47296			−0.0858894			−0.0429447	+	0.999077i	$0.513674\pi$
$830$		0			0
$831$	1.05171	+	3.36772i	0.0364836	+	0.116825i
$832$		0			0
$833$	−22.4210			−0.776841
$834$		0			0
$835$		−	15.6341i		−	0.541039i
$836$		0			0
$837$	−20.9474	−	16.3485i	−0.724049	−	0.565086i
$838$		0			0
$839$	43.5297			1.50281			0.751406	−	0.659841i	$-0.229376\pi$
$839$	43.5297			1.50281			0.751406	+	0.659841i	$0.229376\pi$
$840$		0			0
$841$	−22.0275			−0.759569
$842$		0			0
$843$	1.82731	+	5.85127i	0.0629359	+	0.201528i
$844$		0			0
$845$	10.7322	+	3.59276i	0.369198	+	0.123595i
$846$		0			0
$847$	21.1858	−	28.8184i	0.727952	−	0.990211i
$848$		0			0
$849$	14.9220	+	47.7822i	0.512123	+	1.63988i
$850$		0			0
$851$	−63.2292			−2.16747
$852$		0			0
$853$	−14.0349			−0.480547			−0.240274	−	0.970705i	$-0.577237\pi$
$853$	−14.0349			−0.480547			−0.240274	+	0.970705i	$0.577237\pi$
$854$		0			0
$855$	−5.16937	+	3.57763i	−0.176789	+	0.122352i
$856$		0			0
$857$	15.8288			0.540702			0.270351	−	0.962762i	$-0.412860\pi$
$857$	15.8288			0.540702			0.270351	+	0.962762i	$0.412860\pi$
$858$		0			0
$859$	−40.7599			−1.39071			−0.695355	−	0.718667i	$-0.744753\pi$
$859$	−40.7599			−1.39071			−0.695355	+	0.718667i	$0.744753\pi$
$860$		0			0
$861$	4.85077	+	15.5328i	0.165314	+	0.529355i
$862$		0			0
$863$	0.496465			0.0168999			0.00844993	−	0.999964i	$-0.497310\pi$
$863$	0.496465			0.0168999			0.00844993	+	0.999964i	$0.497310\pi$
$864$		0			0
$865$	−14.5736			−0.495518
$866$		0			0
$867$	−36.9955	+	11.5534i	−1.25643	+	0.392376i
$868$		0			0
$869$	27.6693	−	14.0006i	0.938618	−	0.474938i
$870$		0			0
$871$	21.7072	+	3.53694i	0.735520	+	0.119845i
$872$		0			0
$873$	20.7693	+	30.0099i	0.702935	+	1.01568i
$874$		0			0
$875$	−26.1625			−0.884454
$876$		0			0
$877$	19.5161			0.659010			0.329505	−	0.944154i	$-0.393118\pi$
$877$	19.5161			0.659010			0.329505	+	0.944154i	$0.393118\pi$
$878$		0			0
$879$	13.4982	+	43.2228i	0.455282	+	1.45787i
$880$		0			0
$881$			29.4437i			0.991984i	0.868327	+	0.495992i	$0.165195\pi$
$881$			29.4437i			0.991984i	−0.868327	+	0.495992i	$0.834805\pi$
$882$		0			0
$883$	−10.8940			−0.366612			−0.183306	−	0.983056i	$-0.558680\pi$
$883$	−10.8940			−0.366612			−0.183306	+	0.983056i	$0.558680\pi$
$884$		0			0
$885$	15.4213	−	4.81597i	0.518382	−	0.161887i
$886$		0			0
$887$	21.7292			0.729595			0.364797	−	0.931087i	$-0.381138\pi$
$887$	21.7292			0.729595			0.364797	+	0.931087i	$0.381138\pi$
$888$		0			0
$889$		−	40.6537i		−	1.36348i
$890$		0			0
$891$	29.6743	−	3.23004i	0.994128	−	0.108210i
$892$		0			0
$893$	18.6149			0.622924
$894$		0			0
$895$			16.3809i			0.547554i
$896$		0			0
$897$	35.5040	−	5.04590i	1.18544	−	0.168478i
$898$		0			0
$899$			13.5032i			0.450355i
$900$		0			0
$901$		−	45.5571i		−	1.51773i
$902$		0			0
$903$	8.32594	+	26.6607i	0.277070	+	0.887212i
$904$		0			0
$905$	−6.33835			−0.210694
$906$		0			0
$907$	15.0799			0.500719			0.250359	−	0.968153i	$-0.419451\pi$
$907$	15.0799			0.500719			0.250359	+	0.968153i	$0.419451\pi$
$908$		0			0
$909$	44.4623	−	30.7716i	1.47472	−	1.02063i
$910$		0			0
$911$		−	46.2692i		−	1.53297i	−0.642265	−	0.766483i	$-0.722005\pi$
$911$		−	46.2692i		−	1.53297i	0.642265	−	0.766483i	$-0.277995\pi$
$912$		0			0
$913$	−48.4491	+	24.5151i	−1.60343	+	0.811332i
$914$		0			0
$915$	2.41933	+	7.74698i	0.0799805	+	0.256107i
$916$		0			0
$917$	13.2642			0.438022
$918$		0			0
$919$			54.3995i			1.79447i	0.441549	+	0.897237i	$0.354429\pi$
$919$			54.3995i			1.79447i	−0.441549	+	0.897237i	$0.645571\pi$
$920$		0			0
$921$	11.1564	−	3.48406i	0.367616	−	0.114804i
$922$		0			0
$923$	8.38736	−	51.4755i	0.276073	−	1.69434i
$924$		0			0
$925$			46.7101i			1.53582i
$926$		0			0
$927$	25.1766	−	17.4243i	0.826909	−	0.572289i
$928$		0			0
$929$	12.8045			0.420101			0.210051	−	0.977691i	$-0.432637\pi$
$929$	12.8045			0.420101			0.210051	+	0.977691i	$0.432637\pi$
$930$		0			0
$931$	8.60048			0.281869
$932$		0			0
$933$	−5.77242	−	18.4840i	−0.188980	−	0.605138i
$934$		0			0
$935$	8.18038	+	16.1668i	0.267527	+	0.528712i
$936$		0			0
$937$			23.0760i			0.753860i	0.926242	+	0.376930i	$0.123020\pi$
$937$			23.0760i			0.753860i	−0.926242	+	0.376930i	$0.876980\pi$
$938$		0			0
$939$	−16.8281	+	5.25530i	−0.549165	+	0.171500i
$940$		0			0
$941$			47.7484i			1.55655i	0.627922	+	0.778277i	$0.283906\pi$
$941$			47.7484i			1.55655i	−0.627922	+	0.778277i	$0.716094\pi$
$942$		0			0
$943$	16.5914			0.540290
$944$		0			0
$945$	−9.04993	+	11.5957i	−0.294394	+	0.377210i
$946$		0			0
$947$	−35.6861			−1.15964			−0.579822	−	0.814744i	$-0.696878\pi$
$947$	−35.6861			−1.15964			−0.579822	+	0.814744i	$0.696878\pi$
$948$		0			0
$949$	1.63014	−	10.0046i	0.0529165	−	0.324763i
$950$		0			0
$951$	40.7212	−	12.7169i	1.32047	−	0.412375i
$952$		0			0
$953$	6.22531			0.201658			0.100829	−	0.994904i	$-0.467851\pi$
$953$	6.22531			0.201658			0.100829	+	0.994904i	$0.467851\pi$
$954$		0			0
$955$		−	16.4253i		−	0.531511i
$956$		0			0
$957$	−10.5718	−	10.8779i	−0.341739	−	0.351632i
$958$		0			0
$959$	−22.8271			−0.737126
$960$		0			0
$961$	4.84934			0.156430
$962$		0			0
$963$	−10.8397	+	7.50198i	−0.349305	+	0.241748i
$964$		0			0
$965$	−2.72173			−0.0876156
$966$		0			0
$967$	−53.3892			−1.71688			−0.858440	−	0.512914i	$-0.828566\pi$
$967$	−53.3892			−1.71688			−0.858440	+	0.512914i	$0.828566\pi$
$968$		0			0
$969$	24.9724	−	7.79871i	0.802229	−	0.250531i
$970$		0			0
$971$		−	2.26367i		−	0.0726448i	−0.999340	−	0.0363224i	$-0.988436\pi$
$971$		−	2.26367i		−	0.0726448i	0.999340	−	0.0363224i	$-0.0115643\pi$
$972$		0			0
$973$			29.0761i			0.932136i
$974$		0			0
$975$	−3.72762	−	26.2283i	−0.119379	−	0.839977i
$976$		0			0
$977$	−39.1439			−1.25232			−0.626162	−	0.779693i	$-0.715375\pi$
$977$	−39.1439			−1.25232			−0.626162	+	0.779693i	$0.715375\pi$
$978$		0			0
$979$	13.3803	+	26.4435i	0.427637	+	0.845137i
$980$		0			0
$981$	−43.3470	+	29.9997i	−1.38396	+	0.957818i
$982$		0			0
$983$	21.3407			0.680662			0.340331	−	0.940306i	$-0.389461\pi$
$983$	21.3407			0.680662			0.340331	+	0.940306i	$0.389461\pi$
$984$		0			0
$985$			9.31432i			0.296779i
$986$		0			0
$987$	41.5744	−	12.9834i	1.32333	−	0.413266i
$988$		0			0
$989$	28.4778			0.905540
$990$		0			0
$991$	−20.0693			−0.637524			−0.318762	−	0.947835i	$-0.603267\pi$
$991$	−20.0693			−0.637524			−0.318762	+	0.947835i	$0.603267\pi$
$992$		0			0
$993$	7.28237	+	23.3190i	0.231099	+	0.740007i
$994$		0			0
$995$	−17.6879			−0.560743
$996$		0			0
$997$		−	0.798764i		−	0.0252971i	−0.999920	−	0.0126486i	$-0.995974\pi$
$997$		−	0.798764i		−	0.0252971i	0.999920	−	0.0126486i	$-0.00402627\pi$
$998$		0			0
$999$	45.1046	+	35.2020i	1.42705	+	1.11374i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1716.2.p.a.857.5	yes	56
3.2	odd	2	inner	1716.2.p.a.857.4	yes	56
11.10	odd	2	inner	1716.2.p.a.857.6	yes	56
13.12	even	2	inner	1716.2.p.a.857.8	yes	56
33.32	even	2	inner	1716.2.p.a.857.3	yes	56
39.38	odd	2	inner	1716.2.p.a.857.1	✓	56
143.142	odd	2	inner	1716.2.p.a.857.7	yes	56
429.428	even	2	inner	1716.2.p.a.857.2	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.p.a.857.1	✓	56	39.38	odd	2	inner
1716.2.p.a.857.2	yes	56	429.428	even	2	inner
1716.2.p.a.857.3	yes	56	33.32	even	2	inner
1716.2.p.a.857.4	yes	56	3.2	odd	2	inner
1716.2.p.a.857.5	yes	56	1.1	even	1	trivial
1716.2.p.a.857.6	yes	56	11.10	odd	2	inner
1716.2.p.a.857.7	yes	56	143.142	odd	2	inner
1716.2.p.a.857.8	yes	56	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1716.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.65331 + 0.516316i) q^{3} -0.870582 q^{5} -3.25162 q^{7} +(2.46684 - 1.70725i) q^{9} +O(q^{10})\)	\(q+(-1.65331 + 0.516316i) q^{3} -0.870582 q^{5} -3.25162 q^{7} +(2.46684 - 1.70725i) q^{9} +(1.49742 + 2.95935i) q^{11} +(0.579838 - 3.55862i) q^{13} +(1.43934 - 0.449495i) q^{15} -6.27509 q^{17} +2.40706 q^{19} +(5.37592 - 1.67886i) q^{21} -5.74231i q^{23} -4.24209 q^{25} +(-3.19695 + 4.09628i) q^{27} +2.64055 q^{29} +5.11377i q^{31} +(-4.00365 - 4.11956i) q^{33} +2.83080 q^{35} -11.0111i q^{37} +(0.878723 + 6.18287i) q^{39} +2.88932i q^{41} +4.95928i q^{43} +(-2.14758 + 1.48630i) q^{45} +7.73345 q^{47} +3.57302 q^{49} +(10.3746 - 3.23992i) q^{51} +7.26000i q^{53} +(-1.30363 - 2.57635i) q^{55} +(-3.97961 + 1.24280i) q^{57} +10.7142 q^{59} +5.38232i q^{61} +(-8.02121 + 5.55134i) q^{63} +(-0.504796 + 3.09807i) q^{65} +6.09988i q^{67} +(2.96485 + 9.49380i) q^{69} +14.4650 q^{71} +2.81137 q^{73} +(7.01346 - 2.19026i) q^{75} +(-4.86904 - 9.62266i) q^{77} -9.34981i q^{79} +(3.17056 - 8.42304i) q^{81} +16.3716i q^{83} +5.46298 q^{85} +(-4.36563 + 1.36336i) q^{87} +8.93558 q^{89} +(-1.88541 + 11.5713i) q^{91} +(-2.64032 - 8.45462i) q^{93} -2.09555 q^{95} +12.1653i q^{97} +(8.74625 + 4.74374i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 8 q^{9}+O(q^{10}) \) 56 * q - 8 * q^9	\( 56 q - 8 q^{9} + 48 q^{25} - 12 q^{27} + 48 q^{49} - 16 q^{55} - 12 q^{69} + 44 q^{75} - 40 q^{81} - 24 q^{91}+O(q^{100}) \) 56 * q - 8 * q^9 + 48 * q^25 - 12 * q^27 + 48 * q^49 - 16 * q^55 - 12 * q^69 + 44 * q^75 - 40 * q^81 - 24 * q^91

Introduction

L-functions

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