LMFDB - Embedded newform 252.9.d.d.181.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(181,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("252.181");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.d (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:	$102.659409735$
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} - x^{9} + 14312 x^{8} - 30343 x^{7} + 170123918 x^{6} - 875537263 x^{5} + 496509566533 x^{4} + \cdots + 39\!\cdots\!76 $ x^10 - x^9 + 14312x^8 - 30343x^7 + 170123918x^6 - 875537263x^5 + 496509566533x^4 - 17126006720262x^3 + 1218271024906002x^2 - 21704094932515380x + 391152460437174276
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{15}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.10
Root			$19.5272 - 33.8221i$ of defining polynomial
Character	$\chi$	$=$	252.181
Dual form			252.9.d.d.181.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1053.92i q^{5} +(582.142 - 2329.36i) q^{7} +O(q^{10})$	$q+1053.92i q^{5} +(582.142 - 2329.36i) q^{7} +16535.9 q^{11} -41240.1i q^{13} -118608. i q^{17} +199074. i q^{19} +58484.3 q^{23} -720119. q^{25} -785872. q^{29} +250038. i q^{31} +(2.45495e6 + 613530. i) q^{35} -2.76760e6 q^{37} +3.01147e6i q^{41} -3.93429e6 q^{43} -9.05265e6i q^{47} +(-5.08702e6 - 2.71203e6i) q^{49} -721013. q^{53} +1.74275e7i q^{55} -1.39786e6i q^{59} -1.42791e7i q^{61} +4.34637e7 q^{65} +2.37544e7 q^{67} +9.77501e6 q^{71} -1.55346e7i q^{73} +(9.62623e6 - 3.85180e7i) q^{77} -5.74978e7 q^{79} +3.30636e7i q^{83} +1.25003e8 q^{85} -9.46109e7i q^{89} +(-9.60630e7 - 2.40076e7i) q^{91} -2.09807e8 q^{95} +1.81246e6i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2338 q^{7}+O(q^{10}) $ 10 * q - 2338 * q^7	$ 10 q - 2338 q^{7} + 37596 q^{11} + 22380 q^{23} - 303998 q^{25} + 308892 q^{29} + 1480584 q^{35} - 5471108 q^{37} + 1177324 q^{43} - 6064142 q^{49} + 129132 q^{53} + 106801008 q^{65} - 5722372 q^{67} - 26985540 q^{71} - 48770148 q^{77} - 181197556 q^{79} + 337759224 q^{85} + 67638816 q^{91} - 103302096 q^{95}+O(q^{100}) $ 10 * q - 2338 * q^7 + 37596 * q^11 + 22380 * q^23 - 303998 * q^25 + 308892 * q^29 + 1480584 * q^35 - 5471108 * q^37 + 1177324 * q^43 - 6064142 * q^49 + 129132 * q^53 + 106801008 * q^65 - 5722372 * q^67 - 26985540 * q^71 - 48770148 * q^77 - 181197556 * q^79 + 337759224 * q^85 + 67638816 * q^91 - 103302096 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$-1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$			1053.92i			1.68627i	0.537702	+	0.843135i	$0.319292\pi$
$5$			1053.92i			1.68627i	−0.537702	+	0.843135i	$0.680708\pi$
$6$		0			0
$7$	582.142	−	2329.36i	0.242458	−	0.970162i
$7$	582.142	−	2329.36i	0.242458	−	0.970162i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	16535.9			1.12942			0.564711	−	0.825288i	$-0.308987\pi$
$11$	16535.9			1.12942			0.564711	+	0.825288i	$0.308987\pi$
$12$		0			0
$13$		−	41240.1i		−	1.44393i	−0.691929	−	0.721965i	$-0.743239\pi$
$13$		−	41240.1i		−	1.44393i	0.691929	−	0.721965i	$-0.256761\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	118608.i		−	1.42010i	−0.704151	−	0.710050i	$-0.748672\pi$
$17$		−	118608.i		−	1.42010i	0.704151	−	0.710050i	$-0.251328\pi$
$18$		0			0
$19$			199074.i			1.52756i	0.645474	+	0.763782i	$0.276660\pi$
$19$			199074.i			1.52756i	−0.645474	+	0.763782i	$0.723340\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	58484.3			0.208991			0.104496	−	0.994525i	$-0.466677\pi$
$23$	58484.3			0.208991			0.104496	+	0.994525i	$0.466677\pi$
$24$		0			0
$25$	−720119.			−1.84350
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−785872.			−1.11112			−0.555559	−	0.831477i	$-0.687496\pi$
$29$	−785872.			−1.11112			−0.555559	+	0.831477i	$0.687496\pi$
$30$		0			0
$31$			250038.i			0.270744i	0.990795	+	0.135372i	$0.0432230\pi$
$31$			250038.i			0.270744i	−0.990795	+	0.135372i	$0.956777\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.45495e6	+	613530.i	1.63595	+	0.408850i
$36$		0			0
$37$	−2.76760e6			−1.47671			−0.738356	−	0.674411i	$-0.764398\pi$
$37$	−2.76760e6			−1.47671			−0.738356	+	0.674411i	$0.764398\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.01147e6i			1.06572i	0.846203	+	0.532860i	$0.178883\pi$
$41$			3.01147e6i			1.06572i	−0.846203	+	0.532860i	$0.821117\pi$
$42$		0			0
$43$	−3.93429e6			−1.15078			−0.575390	−	0.817879i	$-0.695150\pi$
$43$	−3.93429e6			−1.15078			−0.575390	+	0.817879i	$0.695150\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	9.05265e6i		−	1.85517i	−0.373610	−	0.927586i	$-0.621880\pi$
$47$		−	9.05265e6i		−	1.85517i	0.373610	−	0.927586i	$-0.378120\pi$
$48$		0			0
$49$	−5.08702e6	−	2.71203e6i	−0.882428	−	0.470447i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−721013.			−0.0913775			−0.0456888	−	0.998956i	$-0.514548\pi$
$53$	−721013.			−0.0913775			−0.0456888	+	0.998956i	$0.514548\pi$
$54$		0			0
$55$			1.74275e7i			1.90451i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	1.39786e6i		−	0.115360i	−0.998335	−	0.0576801i	$-0.981630\pi$
$59$		−	1.39786e6i		−	0.115360i	0.998335	−	0.0576801i	$-0.0183703\pi$
$60$		0			0
$61$		−	1.42791e7i		−	1.03129i	−0.856801	−	0.515646i	$-0.827552\pi$
$61$		−	1.42791e7i		−	1.03129i	0.856801	−	0.515646i	$-0.172448\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.34637e7			2.43486
$66$		0			0
$67$	2.37544e7			1.17881			0.589405	−	0.807837i	$-0.299362\pi$
$67$	2.37544e7			1.17881			0.589405	+	0.807837i	$0.299362\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.77501e6			0.384666			0.192333	−	0.981330i	$-0.438395\pi$
$71$	9.77501e6			0.384666			0.192333	+	0.981330i	$0.438395\pi$
$72$		0			0
$73$		−	1.55346e7i		−	0.547028i	−0.961868	−	0.273514i	$-0.911814\pi$
$73$		−	1.55346e7i		−	0.547028i	0.961868	−	0.273514i	$-0.0881860\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.62623e6	−	3.85180e7i	0.273838	−	1.09572i
$78$		0			0
$79$	−5.74978e7			−1.47619			−0.738095	−	0.674696i	$-0.764275\pi$
$79$	−5.74978e7			−1.47619			−0.738095	+	0.674696i	$0.764275\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.30636e7i			0.696688i	0.937367	+	0.348344i	$0.113256\pi$
$83$			3.30636e7i			0.696688i	−0.937367	+	0.348344i	$0.886744\pi$
$84$		0			0
$85$	1.25003e8			2.39467
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	9.46109e7i		−	1.50793i	−0.656914	−	0.753965i	$-0.728139\pi$
$89$		−	9.46109e7i		−	1.50793i	0.656914	−	0.753965i	$-0.271861\pi$
$90$		0			0
$91$	−9.60630e7	−	2.40076e7i	−1.40085	−	0.350093i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.09807e8			−2.57588
$96$		0			0
$97$			1.81246e6i			0.0204730i	0.999948	+	0.0102365i	$0.00325843\pi$
$97$			1.81246e6i			0.0204730i	−0.999948	+	0.0102365i	$0.996742\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			7.11933e7i			0.684153i	0.939672	+	0.342077i	$0.111130\pi$
$101$			7.11933e7i			0.684153i	−0.939672	+	0.342077i	$0.888870\pi$
$102$		0			0
$103$		−	8.35896e7i		−	0.742683i	−0.928496	−	0.371342i	$-0.878898\pi$
$103$		−	8.35896e7i		−	0.742683i	0.928496	−	0.371342i	$-0.121102\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.72921e7			−0.513368			−0.256684	−	0.966495i	$-0.582630\pi$
$107$	−6.72921e7			−0.513368			−0.256684	+	0.966495i	$0.582630\pi$
$108$		0			0
$109$	−1.20671e8			−0.854864			−0.427432	−	0.904048i	$-0.640582\pi$
$109$	−1.20671e8			−0.854864			−0.427432	+	0.904048i	$0.640582\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.06668e8			−0.654215			−0.327107	−	0.944987i	$-0.606074\pi$
$113$	−1.06668e8			−0.654215			−0.327107	+	0.944987i	$0.606074\pi$
$114$		0			0
$115$			6.16377e7i			0.352416i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.76281e8	−	6.90468e7i	−1.37773	−	0.344315i
$120$		0			0
$121$	5.90764e7			0.275596
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	3.47260e8i		−	1.42238i
$126$		0			0
$127$	1.76348e8			0.677885			0.338943	−	0.940807i	$-0.389931\pi$
$127$	1.76348e8			0.677885			0.338943	+	0.940807i	$0.389931\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	1.67867e8i		−	0.570006i	−0.958527	−	0.285003i	$-0.908005\pi$
$131$		−	1.67867e8i		−	0.570006i	0.958527	−	0.285003i	$-0.0919946\pi$
$132$		0			0
$133$	4.63714e8	+	1.15889e8i	1.48198	+	0.370370i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.87725e8			1.38450			0.692250	−	0.721658i	$-0.256619\pi$
$137$	4.87725e8			1.38450			0.692250	+	0.721658i	$0.256619\pi$
$138$		0			0
$139$		−	5.32457e8i		−	1.42635i	−0.700988	−	0.713173i	$-0.747257\pi$
$139$		−	5.32457e8i		−	1.42635i	0.700988	−	0.713173i	$-0.252743\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	6.81941e8i		−	1.63081i
$144$		0			0
$145$		−	8.28245e8i		−	1.87364i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.07143e8			0.623155			0.311578	−	0.950221i	$-0.399143\pi$
$149$	3.07143e8			0.623155			0.311578	+	0.950221i	$0.399143\pi$
$150$		0			0
$151$	−4.09049e8			−0.786806			−0.393403	−	0.919366i	$-0.628702\pi$
$151$	−4.09049e8			−0.786806			−0.393403	+	0.919366i	$0.628702\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.63520e8			−0.456548
$156$		0			0
$157$		−	3.78279e8i		−	0.622607i	−0.950311	−	0.311303i	$-0.899235\pi$
$157$		−	3.78279e8i		−	0.622607i	0.950311	−	0.311303i	$-0.100765\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.40462e7	−	1.36231e8i	0.0506716	−	0.202755i
$162$		0			0
$163$	1.88913e8			0.267616			0.133808	−	0.991007i	$-0.457279\pi$
$163$	1.88913e8			0.267616			0.133808	+	0.991007i	$0.457279\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2.66334e8i		−	0.342421i	−0.985234	−	0.171211i	$-0.945232\pi$
$167$		−	2.66334e8i		−	0.342421i	0.985234	−	0.171211i	$-0.0547679\pi$
$168$		0			0
$169$	−8.85016e8			−1.08494
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	1.18976e9i		−	1.32824i	−0.747627	−	0.664119i	$-0.768807\pi$
$173$		−	1.18976e9i		−	1.32824i	0.747627	−	0.664119i	$-0.231193\pi$
$174$		0			0
$175$	−4.19211e8	+	1.67742e9i	−0.446973	+	1.78850i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.28341e9			1.25013			0.625063	−	0.780575i	$-0.285073\pi$
$179$	1.28341e9			1.25013			0.625063	+	0.780575i	$0.285073\pi$
$180$		0			0
$181$			1.54218e9i			1.43688i	0.695589	+	0.718440i	$0.255144\pi$
$181$			1.54218e9i			1.43688i	−0.695589	+	0.718440i	$0.744856\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	2.91682e9i		−	2.49013i
$186$		0			0
$187$		−	1.96129e9i		−	1.60389i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.13999e9			−1.60797			−0.803984	−	0.594651i	$-0.797290\pi$
$191$	−2.13999e9			−1.60797			−0.803984	+	0.594651i	$0.797290\pi$
$192$		0			0
$193$	−1.49034e9			−1.07413			−0.537063	−	0.843542i	$-0.680466\pi$
$193$	−1.49034e9			−1.07413			−0.537063	+	0.843542i	$0.680466\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.72554e8			0.446542			0.223271	−	0.974756i	$-0.428327\pi$
$197$	6.72554e8			0.446542			0.223271	+	0.974756i	$0.428327\pi$
$198$		0			0
$199$		−	2.42375e9i		−	1.54552i	−0.634698	−	0.772760i	$-0.718875\pi$
$199$		−	2.42375e9i		−	1.54552i	0.634698	−	0.772760i	$-0.281125\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.57489e8	+	1.83058e9i	−0.269399	+	1.07796i
$204$		0			0
$205$	−3.17385e9			−1.79709
$206$		0			0
$207$		0			0
$208$		0			0
$209$			3.29186e9i			1.72527i
$210$		0			0
$211$	−1.07705e9			−0.543382			−0.271691	−	0.962384i	$-0.587583\pi$
$211$	−1.07705e9			−0.543382			−0.271691	+	0.962384i	$0.587583\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	4.14642e9i		−	1.94053i
$216$		0			0
$217$	5.82428e8	+	1.45558e8i	0.262666	+	0.0656442i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.89141e9			−2.05053
$222$		0			0
$223$			6.13565e8i			0.248108i	0.992275	+	0.124054i	$0.0395896\pi$
$223$			6.13565e8i			0.248108i	−0.992275	+	0.124054i	$0.960410\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	1.96217e9i		−	0.738981i	−0.929234	−	0.369491i	$-0.879532\pi$
$227$		−	1.96217e9i		−	0.738981i	0.929234	−	0.369491i	$-0.120468\pi$
$228$		0			0
$229$		−	4.32029e9i		−	1.57098i	−0.618874	−	0.785490i	$-0.712411\pi$
$229$		−	4.32029e9i		−	1.57098i	0.618874	−	0.785490i	$-0.287589\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.32959e9			1.80830			0.904150	−	0.427215i	$-0.140505\pi$
$233$	5.32959e9			1.80830			0.904150	+	0.427215i	$0.140505\pi$
$234$		0			0
$235$	9.54075e9			3.12832
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.71304e9			−0.525020			−0.262510	−	0.964929i	$-0.584550\pi$
$239$	−1.71304e9			−0.525020			−0.262510	+	0.964929i	$0.584550\pi$
$240$		0			0
$241$			3.29117e9i			0.975624i	0.872949	+	0.487812i	$0.162205\pi$
$241$			3.29117e9i			0.975624i	−0.872949	+	0.487812i	$0.837795\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.85826e9	−	5.36131e9i	0.793301	−	1.48801i
$246$		0			0
$247$	8.20982e9			2.20570
$248$		0			0
$249$		0			0
$250$		0			0
$251$			3.46086e9i			0.871944i	0.899960	+	0.435972i	$0.143595\pi$
$251$			3.46086e9i			0.871944i	−0.899960	+	0.435972i	$0.856405\pi$
$252$		0			0
$253$	9.67090e8			0.236039
$254$		0			0
$255$		0			0
$256$		0			0
$257$			5.51574e9i			1.26436i	0.774821	+	0.632180i	$0.217840\pi$
$257$			5.51574e9i			1.26436i	−0.774821	+	0.632180i	$0.782160\pi$
$258$		0			0
$259$	−1.61113e9	+	6.44672e9i	−0.358041	+	1.43265i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.40848e9			−0.503407			−0.251704	−	0.967804i	$-0.580991\pi$
$263$	−2.40848e9			−0.503407			−0.251704	+	0.967804i	$0.580991\pi$
$264$		0			0
$265$		−	7.59888e8i		−	0.154087i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			3.85853e9i			0.736907i	0.929646	+	0.368453i	$0.120113\pi$
$269$			3.85853e9i			0.736907i	−0.929646	+	0.368453i	$0.879887\pi$
$270$		0			0
$271$		−	9.79803e9i		−	1.81661i	−0.418310	−	0.908304i	$-0.637377\pi$
$271$		−	9.79803e9i		−	1.81661i	0.418310	−	0.908304i	$-0.362623\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.19078e10			−2.08210
$276$		0			0
$277$	−1.71032e9			−0.290508			−0.145254	−	0.989394i	$-0.546400\pi$
$277$	−1.71032e9			−0.290508			−0.145254	+	0.989394i	$0.546400\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.01383e9			−0.804163			−0.402081	−	0.915604i	$-0.631713\pi$
$281$	−5.01383e9			−0.804163			−0.402081	+	0.915604i	$0.631713\pi$
$282$		0			0
$283$			2.66685e9i			0.415769i	0.978153	+	0.207885i	$0.0666578\pi$
$283$			2.66685e9i			0.415769i	−0.978153	+	0.207885i	$0.933342\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.01480e9	+	1.75310e9i	1.03392	+	0.258393i
$288$		0			0
$289$	−7.09214e9			−1.01668
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.09531e10i			1.48616i	0.669204	+	0.743079i	$0.266635\pi$
$293$			1.09531e10i			1.48616i	−0.669204	+	0.743079i	$0.733365\pi$
$294$		0			0
$295$	1.47323e9			0.194528
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	2.41190e9i		−	0.301769i
$300$		0			0
$301$	−2.29031e9	+	9.16437e9i	−0.279016	+	1.11644i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.50490e10			1.73904
$306$		0			0
$307$		−	3.89411e9i		−	0.438384i	−0.975682	−	0.219192i	$-0.929658\pi$
$307$		−	3.89411e9i		−	0.438384i	0.975682	−	0.219192i	$-0.0703421\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			3.76673e9i			0.402646i	0.979525	+	0.201323i	$0.0645241\pi$
$311$			3.76673e9i			0.402646i	−0.979525	+	0.201323i	$0.935476\pi$
$312$		0			0
$313$			5.65992e8i			0.0589702i	0.999565	+	0.0294851i	$0.00938677\pi$
$313$			5.65992e8i			0.0589702i	−0.999565	+	0.0294851i	$0.990613\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.45236e9			0.936059			0.468030	−	0.883713i	$-0.344964\pi$
$317$	9.45236e9			0.936059			0.468030	+	0.883713i	$0.344964\pi$
$318$		0			0
$319$	−1.29951e10			−1.25492
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.36118e10			2.16929
$324$		0			0
$325$			2.96978e10i			2.66189i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.10869e10	−	5.26992e9i	−1.79982	−	0.449801i
$330$		0			0
$331$	−1.56242e9			−0.130162			−0.0650812	−	0.997880i	$-0.520731\pi$
$331$	−1.56242e9			−0.130162			−0.0650812	+	0.997880i	$0.520731\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			2.50352e10i			1.98779i
$336$		0			0
$337$	−4.30099e9			−0.333464			−0.166732	−	0.986002i	$-0.553321\pi$
$337$	−4.30099e9			−0.333464			−0.166732	+	0.986002i	$0.553321\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			4.13460e9i			0.305785i
$342$		0			0
$343$	−9.27867e9	+	1.02707e10i	−0.670362	+	0.742034i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.28621e9			0.0887143			0.0443571	−	0.999016i	$-0.485876\pi$
$347$	1.28621e9			0.0887143			0.0443571	+	0.999016i	$0.485876\pi$
$348$		0			0
$349$		−	2.41126e10i		−	1.62533i	−0.582728	−	0.812667i	$-0.698015\pi$
$349$		−	2.41126e10i		−	1.62533i	0.582728	−	0.812667i	$-0.301985\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	4.23185e9i		−	0.272541i	−0.990672	−	0.136270i	$-0.956488\pi$
$353$		−	4.23185e9i		−	0.272541i	0.990672	−	0.136270i	$-0.0435116\pi$
$354$		0			0
$355$			1.03021e10i			0.648650i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.95354e9			0.298221			0.149110	−	0.988821i	$-0.452359\pi$
$359$	4.95354e9			0.298221			0.149110	+	0.988821i	$0.452359\pi$
$360$		0			0
$361$	−2.26467e10			−1.33345
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.63722e10			0.922436
$366$		0			0
$367$		−	9.69184e8i		−	0.0534247i	−0.999643	−	0.0267123i	$-0.991496\pi$
$367$		−	9.69184e8i		−	0.0534247i	0.999643	−	0.0267123i	$-0.00850381\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.19732e8	+	1.67950e9i	−0.0221552	+	0.0886510i
$372$		0			0
$373$	3.48007e10			1.79785			0.898923	−	0.438106i	$-0.144350\pi$
$373$	3.48007e10			1.79785			0.898923	+	0.438106i	$0.144350\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3.24095e10i			1.60438i
$378$		0			0
$379$	1.73116e10			0.839036			0.419518	−	0.907747i	$-0.362199\pi$
$379$	1.73116e10			0.839036			0.419518	+	0.907747i	$0.362199\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			2.23270e10i			1.03761i	0.854892	+	0.518806i	$0.173623\pi$
$383$			2.23270e10i			1.03761i	−0.854892	+	0.518806i	$0.826377\pi$
$384$		0			0
$385$	4.05948e10	+	1.01453e10i	1.84768	+	0.461764i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.49315e9			0.283568			0.141784	−	0.989898i	$-0.454716\pi$
$389$	6.49315e9			0.283568			0.141784	+	0.989898i	$0.454716\pi$
$390$		0			0
$391$		−	6.93672e9i		−	0.296788i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	6.05979e10i		−	2.48926i
$396$		0			0
$397$			5.68971e9i			0.229049i	0.993420	+	0.114525i	$0.0365344\pi$
$397$			5.68971e9i			0.229049i	−0.993420	+	0.114525i	$0.963466\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.75346e9			−0.183837			−0.0919185	−	0.995767i	$-0.529300\pi$
$401$	−4.75346e9			−0.183837			−0.0919185	+	0.995767i	$0.529300\pi$
$402$		0			0
$403$	1.03116e10			0.390936
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.57646e10			−1.66783
$408$		0			0
$409$		−	1.30619e10i		−	0.466782i	−0.972383	−	0.233391i	$-0.925018\pi$
$409$		−	1.30619e10i		−	0.466782i	0.972383	−	0.233391i	$-0.0749822\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.25612e9	−	8.13753e8i	−0.111918	−	0.0279700i
$414$		0			0
$415$	−3.48464e10			−1.17480
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1.40616e10i			0.456225i	0.973635	+	0.228112i	$0.0732554\pi$
$419$			1.40616e10i			0.456225i	−0.973635	+	0.228112i	$0.926745\pi$
$420$		0			0
$421$	4.45706e10			1.41880			0.709398	−	0.704808i	$-0.248967\pi$
$421$	4.45706e10			1.41880			0.709398	+	0.704808i	$0.248967\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			8.54120e10i			2.61796i
$426$		0			0
$427$	−3.32612e10	−	8.31247e9i	−1.00052	−	0.250045i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.88711e9			0.228565			0.114282	−	0.993448i	$-0.463543\pi$
$431$	7.88711e9			0.228565			0.114282	+	0.993448i	$0.463543\pi$
$432$		0			0
$433$		−	4.02386e10i		−	1.14470i	−0.820010	−	0.572350i	$-0.806032\pi$
$433$		−	4.02386e10i		−	1.14470i	0.820010	−	0.572350i	$-0.193968\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			1.16427e10i			0.319247i
$438$		0			0
$439$			2.74017e10i			0.737767i	0.929476	+	0.368884i	$0.120260\pi$
$439$			2.74017e10i			0.737767i	−0.929476	+	0.368884i	$0.879740\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.40907e9			−0.166410			−0.0832051	−	0.996532i	$-0.526516\pi$
$443$	−6.40907e9			−0.166410			−0.0832051	+	0.996532i	$0.526516\pi$
$444$		0			0
$445$	9.97122e10			2.54278
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.73314e10			−1.41061			−0.705305	−	0.708904i	$-0.749190\pi$
$449$	−5.73314e10			−1.41061			−0.705305	+	0.708904i	$0.749190\pi$
$450$		0			0
$451$			4.97974e10i			1.20365i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.53020e10	−	1.01243e11i	0.590351	−	2.36220i
$456$		0			0
$457$	−7.30294e10			−1.67430			−0.837149	−	0.546974i	$-0.815780\pi$
$457$	−7.30294e10			−1.67430			−0.837149	+	0.546974i	$0.815780\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	4.96488e10i		−	1.09927i	−0.835404	−	0.549636i	$-0.814766\pi$
$461$		−	4.96488e10i		−	1.09927i	0.835404	−	0.549636i	$-0.185234\pi$
$462$		0			0
$463$	−8.49730e10			−1.84908			−0.924542	−	0.381080i	$-0.875552\pi$
$463$	−8.49730e10			−1.84908			−0.924542	+	0.381080i	$0.875552\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	5.06419e10i		−	1.06474i	−0.846512	−	0.532369i	$-0.821302\pi$
$467$		−	5.06419e10i		−	1.06474i	0.846512	−	0.532369i	$-0.178698\pi$
$468$		0			0
$469$	1.38284e10	−	5.53324e10i	0.285812	−	1.14364i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.50569e10			−1.29972
$474$		0			0
$475$		−	1.43357e11i		−	2.81607i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	4.46222e10i		−	0.847635i	−0.905748	−	0.423817i	$-0.860690\pi$
$479$		−	4.46222e10i		−	0.847635i	0.905748	−	0.423817i	$-0.139310\pi$
$480$		0			0
$481$			1.14136e11i			2.13227i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.91018e9			−0.0345230
$486$		0			0
$487$	2.33975e10			0.415962			0.207981	−	0.978133i	$-0.433311\pi$
$487$	2.33975e10			0.415962			0.207981	+	0.978133i	$0.433311\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.29678e10			0.395178			0.197589	−	0.980285i	$-0.436689\pi$
$491$	2.29678e10			0.395178			0.197589	+	0.980285i	$0.436689\pi$
$492$		0			0
$493$			9.32109e10i			1.57790i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.69044e9	−	2.27695e10i	0.0932654	−	0.373188i
$498$		0			0
$499$	−2.43213e10			−0.392270			−0.196135	−	0.980577i	$-0.562839\pi$
$499$	−2.43213e10			−0.392270			−0.196135	+	0.980577i	$0.562839\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			3.25660e10i			0.508737i	0.967107	+	0.254368i	$0.0818675\pi$
$503$			3.25660e10i			0.508737i	−0.967107	+	0.254368i	$0.918132\pi$
$504$		0			0
$505$	−7.50319e10			−1.15367
$506$		0			0
$507$		0			0
$508$		0			0
$509$			2.52761e10i			0.376564i	0.982115	+	0.188282i	$0.0602919\pi$
$509$			2.52761e10i			0.376564i	−0.982115	+	0.188282i	$0.939708\pi$
$510$		0			0
$511$	−3.61857e10	−	9.04336e9i	−0.530706	−	0.132631i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.80967e10			1.25236
$516$		0			0
$517$		−	1.49693e11i		−	2.09527i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			2.49267e10i			0.338309i	0.985590	+	0.169155i	$0.0541037\pi$
$521$			2.49267e10i			0.338309i	−0.985590	+	0.169155i	$0.945896\pi$
$522$		0			0
$523$		−	6.64887e10i		−	0.888672i	−0.895860	−	0.444336i	$-0.853440\pi$
$523$		−	6.64887e10i		−	0.888672i	0.895860	−	0.444336i	$-0.146560\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.96566e10			0.384484
$528$		0			0
$529$	−7.48906e10			−0.956323
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.24193e11			1.53883
$534$		0			0
$535$		−	7.09204e10i		−	0.865677i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.41184e10	−	4.48459e10i	−0.996634	−	0.531334i
$540$		0			0
$541$	1.25330e11			1.46307			0.731534	−	0.681805i	$-0.238805\pi$
$541$	1.25330e11			1.46307			0.731534	+	0.681805i	$0.238805\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1.27177e11i		−	1.44153i
$546$		0			0
$547$	1.24241e11			1.38776			0.693880	−	0.720091i	$-0.255900\pi$
$547$	1.24241e11			1.38776			0.693880	+	0.720091i	$0.255900\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.56446e11i		−	1.69730i
$552$		0			0
$553$	−3.34718e10	+	1.33933e11i	−0.357914	+	1.43214i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.71985e9			−0.0802025			−0.0401013	−	0.999196i	$-0.512768\pi$
$557$	−7.71985e9			−0.0802025			−0.0401013	+	0.999196i	$0.512768\pi$
$558$		0			0
$559$			1.62250e11i			1.66165i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	4.32531e10i		−	0.430511i	−0.976558	−	0.215255i	$-0.930942\pi$
$563$		−	4.32531e10i		−	0.430511i	0.976558	−	0.215255i	$-0.0690584\pi$
$564$		0			0
$565$		−	1.12419e11i		−	1.10318i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.95976e10			−0.568564			−0.284282	−	0.958741i	$-0.591755\pi$
$569$	−5.95976e10			−0.568564			−0.284282	+	0.958741i	$0.591755\pi$
$570$		0			0
$571$	−6.88676e10			−0.647844			−0.323922	−	0.946084i	$-0.605002\pi$
$571$	−6.88676e10			−0.647844			−0.323922	+	0.946084i	$0.605002\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.21157e10			−0.385276
$576$		0			0
$577$			6.70033e10i			0.604495i	0.953229	+	0.302248i	$0.0977369\pi$
$577$			6.70033e10i			0.604495i	−0.953229	+	0.302248i	$0.902263\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.70171e10	+	1.92477e10i	0.675900	+	0.168918i
$582$		0			0
$583$	−1.19226e10			−0.103204
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	7.10252e10i		−	0.598219i	−0.954219	−	0.299109i	$-0.903310\pi$
$587$		−	7.10252e10i		−	0.598219i	0.954219	−	0.299109i	$-0.0966895\pi$
$588$		0			0
$589$	−4.97760e10			−0.413579
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	1.05215e11i		−	0.850861i	−0.904991	−	0.425431i	$-0.860123\pi$
$593$		−	1.05215e11i		−	0.850861i	0.904991	−	0.425431i	$-0.139877\pi$
$594$		0			0
$595$	7.27697e10	−	2.91178e11i	0.580607	−	2.32322i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.41734e11			1.87772			0.938859	−	0.344302i	$-0.111884\pi$
$599$	2.41734e11			1.87772			0.938859	+	0.344302i	$0.111884\pi$
$600$		0			0
$601$		−	6.73897e10i		−	0.516530i	−0.966074	−	0.258265i	$-0.916849\pi$
$601$		−	6.73897e10i		−	0.516530i	0.966074	−	0.258265i	$-0.0831508\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			6.22617e10i			0.464729i
$606$		0			0
$607$			3.52201e10i			0.259440i	0.991551	+	0.129720i	$0.0414078\pi$
$607$			3.52201e10i			0.259440i	−0.991551	+	0.129720i	$0.958592\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.73332e11			−2.67874
$612$		0			0
$613$	−9.75749e10			−0.691029			−0.345515	−	0.938413i	$-0.612296\pi$
$613$	−9.75749e10			−0.691029			−0.345515	+	0.938413i	$0.612296\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.93555e11			1.33556			0.667781	−	0.744357i	$-0.267244\pi$
$617$	1.93555e11			1.33556			0.667781	+	0.744357i	$0.267244\pi$
$618$		0			0
$619$		−	1.11013e11i		−	0.756154i	−0.925774	−	0.378077i	$-0.876585\pi$
$619$		−	1.11013e11i		−	0.756154i	0.925774	−	0.378077i	$-0.123415\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.20383e11	−	5.50770e10i	−1.46294	−	0.365610i
$624$		0			0
$625$	8.46871e10			0.555005
$626$		0			0
$627$		0			0
$628$		0			0
$629$			3.28260e11i			2.09708i
$630$		0			0
$631$	1.09537e11			0.690947			0.345473	−	0.938429i	$-0.387718\pi$
$631$	1.09537e11			0.690947			0.345473	+	0.938429i	$0.387718\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			1.85857e11i			1.14310i
$636$		0			0
$637$	−1.11845e11	+	2.09789e11i	−0.679293	+	1.27417i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.13846e10			−0.0674351			−0.0337176	−	0.999431i	$-0.510735\pi$
$641$	−1.13846e10			−0.0674351			−0.0337176	+	0.999431i	$0.510735\pi$
$642$		0			0
$643$			2.77622e11i			1.62409i	0.583594	+	0.812046i	$0.301646\pi$
$643$			2.77622e11i			1.62409i	−0.583594	+	0.812046i	$0.698354\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.06586e11i		−	0.608253i	−0.952632	−	0.304127i	$-0.901635\pi$
$647$		−	1.06586e11i		−	0.608253i	0.952632	−	0.304127i	$-0.0983646\pi$
$648$		0			0
$649$		−	2.31149e10i		−	0.130290i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.53718e11			−1.39540			−0.697699	−	0.716391i	$-0.745793\pi$
$653$	−2.53718e11			−1.39540			−0.697699	+	0.716391i	$0.745793\pi$
$654$		0			0
$655$	1.76918e11			0.961184
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.73636e11			−1.45088			−0.725441	−	0.688284i	$-0.758364\pi$
$659$	−2.73636e11			−1.45088			−0.725441	+	0.688284i	$0.758364\pi$
$660$		0			0
$661$		−	1.48224e11i		−	0.776449i	−0.921565	−	0.388224i	$-0.873089\pi$
$661$		−	1.48224e11i		−	0.776449i	0.921565	−	0.388224i	$-0.126911\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.22138e11	+	4.88717e11i	−0.624544	+	2.49902i
$666$		0			0
$667$	−4.59612e10			−0.232214
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	2.36118e11i		−	1.16477i
$672$		0			0
$673$	−2.80387e11			−1.36678			−0.683388	−	0.730056i	$-0.739494\pi$
$673$	−2.80387e11			−1.36678			−0.683388	+	0.730056i	$0.739494\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			5.24105e10i			0.249496i	0.992188	+	0.124748i	$0.0398123\pi$
$677$			5.24105e10i			0.249496i	−0.992188	+	0.124748i	$0.960188\pi$
$678$		0			0
$679$	4.22187e9	+	1.05511e9i	0.0198621	+	0.00496384i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.79293e11			−0.823913			−0.411957	−	0.911203i	$-0.635154\pi$
$683$	−1.79293e11			−0.823913			−0.411957	+	0.911203i	$0.635154\pi$
$684$		0			0
$685$			5.14023e11i			2.33464i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			2.97346e10i			0.131943i
$690$		0			0
$691$			5.64066e9i			0.0247410i	0.999923	+	0.0123705i	$0.00393776\pi$
$691$			5.64066e9i			0.0247410i	−0.999923	+	0.0123705i	$0.996062\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.61166e11			2.40520
$696$		0			0
$697$	3.57185e11			1.51343
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.18463e10			0.131882			0.0659412	−	0.997824i	$-0.478995\pi$
$701$	3.18463e10			0.131882			0.0659412	+	0.997824i	$0.478995\pi$
$702$		0			0
$703$		−	5.50955e11i		−	2.25577i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.65835e11	+	4.14446e10i	0.663740	+	0.165879i
$708$		0			0
$709$	−2.64843e10			−0.104810			−0.0524051	−	0.998626i	$-0.516689\pi$
$709$	−2.64843e10			−0.104810			−0.0524051	+	0.998626i	$0.516689\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1.46233e10i			0.0565832i
$714$		0			0
$715$	7.18711e11			2.74998
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1.19899e11i		−	0.448644i	−0.974515	−	0.224322i	$-0.927983\pi$
$719$		−	1.19899e11i		−	0.448644i	0.974515	−	0.224322i	$-0.0720167\pi$
$720$		0			0
$721$	−1.94710e11	−	4.86610e10i	−0.720523	−	0.180070i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.65922e11			2.04835
$726$		0			0
$727$			3.78528e10i			0.135507i	0.997702	+	0.0677533i	$0.0215831\pi$
$727$			3.78528e10i			0.135507i	−0.997702	+	0.0677533i	$0.978417\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			4.66639e11i			1.63422i
$732$		0			0
$733$		−	4.53456e10i		−	0.157079i	−0.996911	−	0.0785397i	$-0.974974\pi$
$733$		−	4.53456e10i		−	0.157079i	0.996911	−	0.0785397i	$-0.0250257\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.92799e11			1.33138
$738$		0			0
$739$	9.48224e10			0.317931			0.158966	−	0.987284i	$-0.449184\pi$
$739$	9.48224e10			0.317931			0.158966	+	0.987284i	$0.449184\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.41144e11			0.791265			0.395632	−	0.918409i	$-0.370525\pi$
$743$	2.41144e11			0.791265			0.395632	+	0.918409i	$0.370525\pi$
$744$		0			0
$745$			3.23704e11i			1.05081i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.91736e10	+	1.56747e11i	−0.124470	+	0.498050i
$750$		0			0
$751$	−5.07696e11			−1.59604			−0.798020	−	0.602630i	$-0.794119\pi$
$751$	−5.07696e11			−1.59604			−0.798020	+	0.602630i	$0.794119\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	4.31104e11i		−	1.32677i
$756$		0			0
$757$	1.93568e11			0.589453			0.294727	−	0.955582i	$-0.404771\pi$
$757$	1.93568e11			0.589453			0.294727	+	0.955582i	$0.404771\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			3.29897e11i			0.983648i	0.870695	+	0.491824i	$0.163670\pi$
$761$			3.29897e11i			0.983648i	−0.870695	+	0.491824i	$0.836330\pi$
$762$		0			0
$763$	−7.02477e10	+	2.81086e11i	−0.207269	+	0.829356i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.76479e10			−0.166572
$768$		0			0
$769$			6.73612e11i			1.92621i	0.269117	+	0.963107i	$0.413268\pi$
$769$			6.73612e11i			1.92621i	−0.269117	+	0.963107i	$0.586732\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			3.95540e11i			1.10783i	0.832574	+	0.553914i	$0.186866\pi$
$773$			3.95540e11i			1.10783i	−0.832574	+	0.553914i	$0.813134\pi$
$774$		0			0
$775$		−	1.80057e11i		−	0.499119i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.99505e11			−1.62796
$780$		0			0
$781$	1.61638e11			0.434451
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.98675e11			1.04988
$786$		0			0
$787$		−	8.45468e10i		−	0.220393i	−0.993910	−	0.110197i	$-0.964852\pi$
$787$		−	8.45468e10i		−	0.220393i	0.993910	−	0.110197i	$-0.0351481\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.20959e10	+	2.48468e11i	−0.158620	+	0.634694i
$792$		0			0
$793$	−5.88872e11			−1.48912
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.06722e10i			0.0264497i	0.999913	+	0.0132248i	$0.00420972\pi$
$797$			1.06722e10i			0.0264497i	−0.999913	+	0.0132248i	$0.995790\pi$
$798$		0			0
$799$	−1.07372e12			−2.63453
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	2.56879e11i		−	0.617826i
$804$		0			0
$805$	1.43576e11	+	3.58819e10i	0.341900	+	0.0854460i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.59950e11			1.54070			0.770348	−	0.637623i	$-0.220082\pi$
$809$	6.59950e11			1.54070			0.770348	+	0.637623i	$0.220082\pi$
$810$		0			0
$811$			6.50590e11i			1.50392i	0.659211	+	0.751958i	$0.270891\pi$
$811$			6.50590e11i			1.50392i	−0.659211	+	0.751958i	$0.729109\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			1.99099e11i			0.451273i
$816$		0			0
$817$		−	7.83213e11i		−	1.75789i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2.16646e11			−0.476846			−0.238423	−	0.971161i	$-0.576631\pi$
$821$	−2.16646e11			−0.476846			−0.238423	+	0.971161i	$0.576631\pi$
$822$		0			0
$823$	−4.40900e11			−0.961039			−0.480520	−	0.876984i	$-0.659552\pi$
$823$	−4.40900e11			−0.961039			−0.480520	+	0.876984i	$0.659552\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9.73507e10			−0.208122			−0.104061	−	0.994571i	$-0.533184\pi$
$827$	−9.73507e10			−0.208122			−0.104061	+	0.994571i	$0.533184\pi$
$828$		0			0
$829$			4.03736e11i			0.854830i	0.904056	+	0.427415i	$0.140576\pi$
$829$			4.03736e11i			0.854830i	−0.904056	+	0.427415i	$0.859424\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.21669e11	+	6.03363e11i	−0.668082	+	1.25314i
$834$		0			0
$835$	2.80694e11			0.577415
$836$		0			0
$837$		0			0
$838$		0			0
$839$			5.47033e11i			1.10399i	0.833847	+	0.551996i	$0.186134\pi$
$839$			5.47033e11i			1.10399i	−0.833847	+	0.551996i	$0.813866\pi$
$840$		0			0
$841$	1.17349e11			0.234582
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	9.32734e11i		−	1.82949i
$846$		0			0
$847$	3.43909e10	−	1.37610e11i	0.0668205	−	0.267373i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.61861e11			−0.308620
$852$		0			0
$853$		−	6.43629e11i		−	1.21574i	−0.794038	−	0.607869i	$-0.792025\pi$
$853$		−	6.43629e11i		−	1.21574i	0.794038	−	0.607869i	$-0.207975\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			5.42427e11i			1.00558i	0.864408	+	0.502791i	$0.167694\pi$
$857$			5.42427e11i			1.00558i	−0.864408	+	0.502791i	$0.832306\pi$
$858$		0			0
$859$			6.00152e11i			1.10227i	0.834416	+	0.551136i	$0.185805\pi$
$859$			6.00152e11i			1.10227i	−0.834416	+	0.551136i	$0.814195\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.93494e9			−0.00889690			−0.00444845	−	0.999990i	$-0.501416\pi$
$863$	−4.93494e9			−0.00889690			−0.00444845	+	0.999990i	$0.501416\pi$
$864$		0			0
$865$	1.25391e12			2.23977
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.50776e11			−1.66724
$870$		0			0
$871$		−	9.79632e11i		−	1.70212i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.08893e11	−	2.02155e11i	−1.37994	−	0.344867i
$876$		0			0
$877$	−5.08947e11			−0.860348			−0.430174	−	0.902746i	$-0.641548\pi$
$877$	−5.08947e11			−0.860348			−0.430174	+	0.902746i	$0.641548\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	5.26964e11i		−	0.874737i	−0.899282	−	0.437369i	$-0.855911\pi$
$881$		−	5.26964e11i		−	0.874737i	0.899282	−	0.437369i	$-0.144089\pi$
$882$		0			0
$883$	−4.47485e11			−0.736098			−0.368049	−	0.929806i	$-0.619974\pi$
$883$	−4.47485e11			−0.736098			−0.368049	+	0.929806i	$0.619974\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	7.61748e11i		−	1.23060i	−0.788293	−	0.615300i	$-0.789035\pi$
$887$		−	7.61748e11i		−	1.23060i	0.788293	−	0.615300i	$-0.210965\pi$
$888$		0			0
$889$	1.02660e11	−	4.10778e11i	0.164359	−	0.657658i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.80214e12			2.83389
$894$		0			0
$895$			1.35261e12i			2.10805i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1.96498e11i		−	0.300829i
$900$		0			0
$901$			8.55180e10i			0.129765i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.62533e12			−2.42297
$906$		0			0
$907$	2.89495e11			0.427772			0.213886	−	0.976859i	$-0.431388\pi$
$907$	2.89495e11			0.427772			0.213886	+	0.976859i	$0.431388\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.60521e11			−0.233055			−0.116528	−	0.993187i	$-0.537176\pi$
$911$	−1.60521e11			−0.233055			−0.116528	+	0.993187i	$0.537176\pi$
$912$		0			0
$913$			5.46736e11i			0.786855i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.91022e11	−	9.77223e10i	−0.552998	−	0.138203i
$918$		0			0
$919$	1.18079e12			1.65543			0.827717	−	0.561145i	$-0.189639\pi$
$919$	1.18079e12			1.65543			0.827717	+	0.561145i	$0.189639\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	4.03122e11i		−	0.555431i
$924$		0			0
$925$	1.99300e12			2.72233
$926$		0			0
$927$		0			0
$928$		0			0
$929$			3.28571e11i			0.441129i	0.975372	+	0.220565i	$0.0707900\pi$
$929$			3.28571e11i			0.441129i	−0.975372	+	0.220565i	$0.929210\pi$
$930$		0			0
$931$	5.39895e11	−	1.01269e12i	0.718638	−	1.34797i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.06704e12			2.70460
$936$		0			0
$937$		−	6.69384e11i		−	0.868394i	−0.900818	−	0.434197i	$-0.857032\pi$
$937$		−	6.69384e11i		−	0.868394i	0.900818	−	0.434197i	$-0.142968\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			4.99128e11i			0.636581i	0.947993	+	0.318290i	$0.103109\pi$
$941$			4.99128e11i			0.636581i	−0.947993	+	0.318290i	$0.896891\pi$
$942$		0			0
$943$			1.76124e11i			0.222726i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.35005e12			−1.67861			−0.839304	−	0.543663i	$-0.817037\pi$
$947$	−1.35005e12			−1.67861			−0.839304	+	0.543663i	$0.817037\pi$
$948$		0			0
$949$	−6.40650e11			−0.789870
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.15828e12			1.40424			0.702120	−	0.712058i	$-0.252237\pi$
$953$	1.15828e12			1.40424			0.702120	+	0.712058i	$0.252237\pi$
$954$		0			0
$955$		−	2.25537e12i		−	2.71147i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.83925e11	−	1.13609e12i	0.335683	−	1.34319i
$960$		0			0
$961$	7.90372e11			0.926697
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.57069e12i		−	1.81127i
$966$		0			0
$967$	−3.64075e11			−0.416376			−0.208188	−	0.978089i	$-0.566757\pi$
$967$	−3.64075e11			−0.416376			−0.208188	+	0.978089i	$0.566757\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1.48578e12i			1.67139i	0.549190	+	0.835697i	$0.314936\pi$
$971$			1.48578e12i			1.67139i	−0.549190	+	0.835697i	$0.685064\pi$
$972$		0			0
$973$	−1.24028e12	−	3.09965e11i	−1.38379	−	0.345829i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.63000e12			−1.78900			−0.894498	−	0.447073i	$-0.852467\pi$
$977$	−1.63000e12			−1.78900			−0.894498	+	0.447073i	$0.852467\pi$
$978$		0			0
$979$		−	1.56447e12i		−	1.70309i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1.37874e12i		−	1.47662i	−0.674464	−	0.738308i	$-0.735625\pi$
$983$		−	1.37874e12i		−	1.47662i	0.674464	−	0.738308i	$-0.264375\pi$
$984$		0			0
$985$			7.08817e11i			0.752990i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.30094e11			−0.240503
$990$		0			0
$991$	−2.74164e11			−0.284260			−0.142130	−	0.989848i	$-0.545395\pi$
$991$	−2.74164e11			−0.284260			−0.142130	+	0.989848i	$0.545395\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.55443e12			2.60616
$996$		0			0
$997$			1.45964e12i			1.47729i	0.674095	+	0.738645i	$0.264534\pi$
$997$			1.45964e12i			1.47729i	−0.674095	+	0.738645i	$0.735466\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.d.d.181.10		10
3.2	odd	2		84.9.d.a.13.6	yes	10
7.6	odd	2	inner	252.9.d.d.181.1		10
12.11	even	2		336.9.f.a.97.1		10
21.20	even	2		84.9.d.a.13.5	✓	10
84.83	odd	2		336.9.f.a.97.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.d.a.13.5	✓	10	21.20	even	2
84.9.d.a.13.6	yes	10	3.2	odd	2
252.9.d.d.181.1		10	7.6	odd	2	inner
252.9.d.d.181.10		10	1.1	even	1	trivial
336.9.f.a.97.1		10	12.11	even	2
336.9.f.a.97.10		10	84.83	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1053.92i q^{5} +(582.142 - 2329.36i) q^{7} +O(q^{10})\)	\(q+1053.92i q^{5} +(582.142 - 2329.36i) q^{7} +16535.9 q^{11} -41240.1i q^{13} -118608. i q^{17} +199074. i q^{19} +58484.3 q^{23} -720119. q^{25} -785872. q^{29} +250038. i q^{31} +(2.45495e6 + 613530. i) q^{35} -2.76760e6 q^{37} +3.01147e6i q^{41} -3.93429e6 q^{43} -9.05265e6i q^{47} +(-5.08702e6 - 2.71203e6i) q^{49} -721013. q^{53} +1.74275e7i q^{55} -1.39786e6i q^{59} -1.42791e7i q^{61} +4.34637e7 q^{65} +2.37544e7 q^{67} +9.77501e6 q^{71} -1.55346e7i q^{73} +(9.62623e6 - 3.85180e7i) q^{77} -5.74978e7 q^{79} +3.30636e7i q^{83} +1.25003e8 q^{85} -9.46109e7i q^{89} +(-9.60630e7 - 2.40076e7i) q^{91} -2.09807e8 q^{95} +1.81246e6i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2338 q^{7}+O(q^{10}) \) 10 * q - 2338 * q^7	\( 10 q - 2338 q^{7} + 37596 q^{11} + 22380 q^{23} - 303998 q^{25} + 308892 q^{29} + 1480584 q^{35} - 5471108 q^{37} + 1177324 q^{43} - 6064142 q^{49} + 129132 q^{53} + 106801008 q^{65} - 5722372 q^{67} - 26985540 q^{71} - 48770148 q^{77} - 181197556 q^{79} + 337759224 q^{85} + 67638816 q^{91} - 103302096 q^{95}+O(q^{100}) \) 10 * q - 2338 * q^7 + 37596 * q^11 + 22380 * q^23 - 303998 * q^25 + 308892 * q^29 + 1480584 * q^35 - 5471108 * q^37 + 1177324 * q^43 - 6064142 * q^49 + 129132 * q^53 + 106801008 * q^65 - 5722372 * q^67 - 26985540 * q^71 - 48770148 * q^77 - 181197556 * q^79 + 337759224 * q^85 + 67638816 * q^91 - 103302096 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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