LMFDB - Embedded newform 252.9.d.d.181.7

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			181.7
Root			$10.9569 + 18.9779i$ of defining polynomial
Character	$\chi$	$=$	252.181
Dual form			252.9.d.d.181.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+307.159i q^{5} +(222.189 + 2390.70i) q^{7} +O(q^{10})$	$q+307.159i q^{5} +(222.189 + 2390.70i) q^{7} -5381.78 q^{11} -48978.5i q^{13} +55408.7i q^{17} +67798.4i q^{19} +344567. q^{23} +296278. q^{25} +1.17408e6 q^{29} +125625. i q^{31} +(-734325. + 68247.3i) q^{35} -734456. q^{37} +1.18960e6i q^{41} -3.84354e6 q^{43} -4.82991e6i q^{47} +(-5.66607e6 + 1.06237e6i) q^{49} +1.19800e7 q^{53} -1.65306e6i q^{55} +1.90443e7i q^{59} -1.11077e7i q^{61} +1.50442e7 q^{65} -2.92913e7 q^{67} -2.79462e6 q^{71} +3.34290e7i q^{73} +(-1.19577e6 - 1.28662e7i) q^{77} -4.96171e6 q^{79} +3.33137e7i q^{83} -1.70193e7 q^{85} -3.09244e7i q^{89} +(1.17093e8 - 1.08825e7i) q^{91} -2.08249e7 q^{95} +5.05301e7i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			307.159i			0.491455i	0.969339	+	0.245727i	$0.0790268\pi$
$5$			307.159i			0.491455i	−0.969339	+	0.245727i	$0.920973\pi$
$6$		0			0
$7$	222.189	+	2390.70i	0.0925400	+	0.995709i
$7$	222.189	+	2390.70i	0.0925400	+	0.995709i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5381.78			−0.367583			−0.183791	−	0.982965i	$-0.558837\pi$
$11$	−5381.78			−0.367583			−0.183791	+	0.982965i	$0.558837\pi$
$12$		0			0
$13$		−	48978.5i		−	1.71487i	−0.514589	−	0.857437i	$-0.672056\pi$
$13$		−	48978.5i		−	1.71487i	0.514589	−	0.857437i	$-0.327944\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			55408.7i			0.663411i	0.943383	+	0.331705i	$0.107624\pi$
$17$			55408.7i			0.663411i	−0.943383	+	0.331705i	$0.892376\pi$
$18$		0			0
$19$			67798.4i			0.520242i	0.965576	+	0.260121i	$0.0837624\pi$
$19$			67798.4i			0.520242i	−0.965576	+	0.260121i	$0.916238\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	344567.			1.23130			0.615648	−	0.788021i	$-0.288894\pi$
$23$	344567.			1.23130			0.615648	+	0.788021i	$0.288894\pi$
$24$		0			0
$25$	296278.			0.758472
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.17408e6			1.65999			0.829995	−	0.557771i	$-0.188343\pi$
$29$	1.17408e6			1.65999			0.829995	+	0.557771i	$0.188343\pi$
$30$		0			0
$31$			125625.i			0.136029i	0.997684	+	0.0680144i	$0.0216664\pi$
$31$			125625.i			0.136029i	−0.997684	+	0.0680144i	$0.978334\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−734325.	+	68247.3i	−0.489346	+	0.0454793i
$36$		0			0
$37$	−734456.			−0.391885			−0.195943	−	0.980615i	$-0.562777\pi$
$37$	−734456.			−0.391885			−0.195943	+	0.980615i	$0.562777\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.18960e6i			0.420984i	0.977596	+	0.210492i	$0.0675065\pi$
$41$			1.18960e6i			0.420984i	−0.977596	+	0.210492i	$0.932493\pi$
$42$		0			0
$43$	−3.84354e6			−1.12424			−0.562119	−	0.827057i	$-0.690014\pi$
$43$	−3.84354e6			−1.12424			−0.562119	+	0.827057i	$0.690014\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	4.82991e6i		−	0.989799i	−0.868950	−	0.494900i	$-0.835205\pi$
$47$		−	4.82991e6i		−	0.989799i	0.868950	−	0.494900i	$-0.164795\pi$
$48$		0			0
$49$	−5.66607e6	+	1.06237e6i	−0.982873	+	0.184286i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.19800e7			1.51829			0.759143	−	0.650924i	$-0.225618\pi$
$53$	1.19800e7			1.51829			0.759143	+	0.650924i	$0.225618\pi$
$54$		0			0
$55$		−	1.65306e6i		−	0.180650i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			1.90443e7i			1.57165i	0.618448	+	0.785826i	$0.287762\pi$
$59$			1.90443e7i			1.57165i	−0.618448	+	0.785826i	$0.712238\pi$
$60$		0			0
$61$		−	1.11077e7i		−	0.802242i	−0.916025	−	0.401121i	$-0.868621\pi$
$61$		−	1.11077e7i		−	0.802242i	0.916025	−	0.401121i	$-0.131379\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.50442e7			0.842783
$66$		0			0
$67$	−2.92913e7			−1.45358			−0.726792	−	0.686858i	$-0.758989\pi$
$67$	−2.92913e7			−1.45358			−0.726792	+	0.686858i	$0.758989\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.79462e6			−0.109974			−0.0549869	−	0.998487i	$-0.517512\pi$
$71$	−2.79462e6			−0.109974			−0.0549869	+	0.998487i	$0.517512\pi$
$72$		0			0
$73$			3.34290e7i			1.17715i	0.808443	+	0.588574i	$0.200311\pi$
$73$			3.34290e7i			1.17715i	−0.808443	+	0.588574i	$0.799689\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.19577e6	−	1.28662e7i	−0.0340161	−	0.366006i
$78$		0			0
$79$	−4.96171e6			−0.127386			−0.0636932	−	0.997970i	$-0.520288\pi$
$79$	−4.96171e6			−0.127386			−0.0636932	+	0.997970i	$0.520288\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.33137e7i			0.701957i	0.936384	+	0.350978i	$0.114151\pi$
$83$			3.33137e7i			0.701957i	−0.936384	+	0.350978i	$0.885849\pi$
$84$		0			0
$85$	−1.70193e7			−0.326036
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	3.09244e7i		−	0.492879i	−0.969158	−	0.246440i	$-0.920739\pi$
$89$		−	3.09244e7i		−	0.492879i	0.969158	−	0.246440i	$-0.0792607\pi$
$90$		0			0
$91$	1.17093e8	−	1.08825e7i	1.70751	−	0.158694i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.08249e7			−0.255675
$96$		0			0
$97$			5.05301e7i			0.570773i	0.958413	+	0.285386i	$0.0921220\pi$
$97$			5.05301e7i			0.570773i	−0.958413	+	0.285386i	$0.907878\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			1.90888e8i			1.83439i	0.398434	+	0.917197i	$0.369554\pi$
$101$			1.90888e8i			1.83439i	−0.398434	+	0.917197i	$0.630446\pi$
$102$		0			0
$103$		−	4.51846e6i		−	0.0401459i	−0.999799	−	0.0200730i	$-0.993610\pi$
$103$		−	4.51846e6i		−	0.0401459i	0.999799	−	0.0200730i	$-0.00638985\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.12055e8			−0.854861			−0.427431	−	0.904048i	$-0.640581\pi$
$107$	−1.12055e8			−0.854861			−0.427431	+	0.904048i	$0.640581\pi$
$108$		0			0
$109$	1.56777e8			1.11065			0.555325	−	0.831634i	$-0.312594\pi$
$109$	1.56777e8			1.11065			0.555325	+	0.831634i	$0.312594\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.97902e8			−1.21377			−0.606885	−	0.794790i	$-0.707581\pi$
$113$	−1.97902e8			−1.21377			−0.606885	+	0.794790i	$0.707581\pi$
$114$		0			0
$115$			1.05837e8i			0.605127i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.32466e8	+	1.23112e7i	−0.660564	+	0.0613921i
$120$		0			0
$121$	−1.85395e8			−0.864883
$122$		0			0
$123$		0			0
$124$		0			0
$125$			2.10989e8i			0.864210i
$126$		0			0
$127$	4.18632e8			1.60923			0.804614	−	0.593799i	$-0.202372\pi$
$127$	4.18632e8			1.60923			0.804614	+	0.593799i	$0.202372\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			2.79773e8i			0.949994i	0.879988	+	0.474997i	$0.157551\pi$
$131$			2.79773e8i			0.949994i	−0.879988	+	0.474997i	$0.842449\pi$
$132$		0			0
$133$	−1.62086e8	+	1.50640e7i	−0.518009	+	0.0481432i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.92940e7			0.196704			0.0983521	−	0.995152i	$-0.468643\pi$
$137$	6.92940e7			0.196704			0.0983521	+	0.995152i	$0.468643\pi$
$138$		0			0
$139$			4.57712e8i			1.22612i	0.790036	+	0.613060i	$0.210062\pi$
$139$			4.57712e8i			1.22612i	−0.790036	+	0.613060i	$0.789938\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2.63592e8i			0.630358i
$144$		0			0
$145$			3.60629e8i			0.815810i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.29858e8			1.48079			0.740395	−	0.672172i	$-0.234639\pi$
$149$	7.29858e8			1.48079			0.740395	+	0.672172i	$0.234639\pi$
$150$		0			0
$151$	−2.20692e8			−0.424501			−0.212250	−	0.977215i	$-0.568079\pi$
$151$	−2.20692e8			−0.424501			−0.212250	+	0.977215i	$0.568079\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.85870e7			−0.0668520
$156$		0			0
$157$		−	4.57190e8i		−	0.752485i	−0.926521	−	0.376243i	$-0.877216\pi$
$157$		−	4.57190e8i		−	0.752485i	0.926521	−	0.376243i	$-0.122784\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.65589e7	+	8.23756e8i	0.113944	+	1.22601i
$162$		0			0
$163$	−1.58414e8			−0.224411			−0.112205	−	0.993685i	$-0.535791\pi$
$163$	−1.58414e8			−0.224411			−0.112205	+	0.993685i	$0.535791\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			5.55510e8i			0.714210i	0.934064	+	0.357105i	$0.116236\pi$
$167$			5.55510e8i			0.714210i	−0.934064	+	0.357105i	$0.883764\pi$
$168$		0			0
$169$	−1.58316e9			−1.94079
$170$		0			0
$171$		0			0
$172$		0			0
$173$			1.42883e9i			1.59513i	0.603231	+	0.797566i	$0.293880\pi$
$173$			1.42883e9i			1.59513i	−0.603231	+	0.797566i	$0.706120\pi$
$174$		0			0
$175$	6.58296e7	+	7.08311e8i	0.0701890	+	0.755218i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.76429e8			−0.658886			−0.329443	−	0.944175i	$-0.606861\pi$
$179$	−6.76429e8			−0.658886			−0.329443	+	0.944175i	$0.606861\pi$
$180$		0			0
$181$			9.05179e8i			0.843374i	0.906742	+	0.421687i	$0.138562\pi$
$181$			9.05179e8i			0.843374i	−0.906742	+	0.421687i	$0.861438\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	2.25595e8i		−	0.192594i
$186$		0			0
$187$		−	2.98198e8i		−	0.243858i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.71513e8			0.204013			0.102006	−	0.994784i	$-0.467474\pi$
$191$	2.71513e8			0.204013			0.102006	+	0.994784i	$0.467474\pi$
$192$		0			0
$193$	−7.48364e8			−0.539366			−0.269683	−	0.962949i	$-0.586919\pi$
$193$	−7.48364e8			−0.539366			−0.269683	+	0.962949i	$0.586919\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.09695e9			0.728320			0.364160	−	0.931336i	$-0.381356\pi$
$197$	1.09695e9			0.728320			0.364160	+	0.931336i	$0.381356\pi$
$198$		0			0
$199$			1.37706e8i			0.0878094i	0.999036	+	0.0439047i	$0.0139798\pi$
$199$			1.37706e8i			0.0878094i	−0.999036	+	0.0439047i	$0.986020\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.60867e8	+	2.80687e9i	0.153616	+	1.65287i
$204$		0			0
$205$	−3.65396e8			−0.206894
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.64876e8i		−	0.191232i
$210$		0			0
$211$	3.35246e9			1.69135			0.845676	−	0.533696i	$-0.179197\pi$
$211$	3.35246e9			1.69135			0.845676	+	0.533696i	$0.179197\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1.18058e9i		−	0.552512i
$216$		0			0
$217$	−3.00332e8	+	2.79125e7i	−0.135445	+	0.0125881i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.71384e9			1.13767
$222$		0			0
$223$			2.34771e9i			0.949348i	0.880162	+	0.474674i	$0.157434\pi$
$223$			2.34771e9i			0.949348i	−0.880162	+	0.474674i	$0.842566\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			2.87854e9i			1.08410i	0.840347	+	0.542049i	$0.182351\pi$
$227$			2.87854e9i			1.08410i	−0.840347	+	0.542049i	$0.817649\pi$
$228$		0			0
$229$			7.05735e8i			0.256625i	0.991734	+	0.128313i	$0.0409561\pi$
$229$			7.05735e8i			0.256625i	−0.991734	+	0.128313i	$0.959044\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.01041e9			−1.70000			−0.850002	−	0.526780i	$-0.823399\pi$
$233$	−5.01041e9			−1.70000			−0.850002	+	0.526780i	$0.823399\pi$
$234$		0			0
$235$	1.48355e9			0.486442
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.19477e9			−0.979147			−0.489574	−	0.871962i	$-0.662848\pi$
$239$	−3.19477e9			−0.979147			−0.489574	+	0.871962i	$0.662848\pi$
$240$		0			0
$241$		−	1.70613e9i		−	0.505759i	−0.967498	−	0.252879i	$-0.918622\pi$
$241$		−	1.70613e9i		−	0.505759i	0.967498	−	0.252879i	$-0.0813776\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.26317e8	−	1.74038e9i	−0.0905682	−	0.483038i
$246$		0			0
$247$	3.32067e9			0.892149
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	2.60073e9i		−	0.655239i	−0.944810	−	0.327620i	$-0.893754\pi$
$251$		−	2.60073e9i		−	0.655239i	0.944810	−	0.327620i	$-0.106246\pi$
$252$		0			0
$253$	−1.85439e9			−0.452603
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	1.84643e9i		−	0.423252i	−0.977351	−	0.211626i	$-0.932124\pi$
$257$		−	1.84643e9i		−	0.423252i	0.977351	−	0.211626i	$-0.0678759\pi$
$258$		0			0
$259$	−1.63188e8	−	1.75586e9i	−0.0362651	−	0.390204i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.63329e9			−1.38646			−0.693228	−	0.720718i	$-0.743812\pi$
$263$	−6.63329e9			−1.38646			−0.693228	+	0.720718i	$0.743812\pi$
$264$		0			0
$265$			3.67977e9i			0.746169i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			5.20298e9i			0.993673i	0.867844	+	0.496836i	$0.165505\pi$
$269$			5.20298e9i			0.993673i	−0.867844	+	0.496836i	$0.834495\pi$
$270$		0			0
$271$		−	5.27608e9i		−	0.978215i	−0.872224	−	0.489107i	$-0.837323\pi$
$271$		−	5.27608e9i		−	0.978215i	0.872224	−	0.489107i	$-0.162677\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.59450e9			−0.278801
$276$		0			0
$277$	1.54281e9			0.262055			0.131027	−	0.991379i	$-0.458172\pi$
$277$	1.54281e9			0.262055			0.131027	+	0.991379i	$0.458172\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.16602e9			1.47013			0.735065	−	0.677997i	$-0.237152\pi$
$281$	9.16602e9			1.47013			0.735065	+	0.677997i	$0.237152\pi$
$282$		0			0
$283$		−	4.15888e9i		−	0.648381i	−0.945992	−	0.324191i	$-0.894908\pi$
$283$		−	4.15888e9i		−	0.648381i	0.945992	−	0.324191i	$-0.105092\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.84397e9	+	2.64315e8i	−0.419177	+	0.0389578i
$288$		0			0
$289$	3.90563e9			0.559886
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1.30089e10i		−	1.76511i	−0.470210	−	0.882554i	$-0.655822\pi$
$293$		−	1.30089e10i		−	1.76511i	0.470210	−	0.882554i	$-0.344178\pi$
$294$		0			0
$295$	−5.84963e9			−0.772396
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	1.68764e10i		−	2.11152i
$300$		0			0
$301$	−8.53992e8	−	9.18875e9i	−0.104037	−	1.11941i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.41184e9			0.394266
$306$		0			0
$307$			1.60817e10i			1.81042i	0.424966	+	0.905209i	$0.360286\pi$
$307$			1.60817e10i			1.81042i	−0.424966	+	0.905209i	$0.639714\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			3.74219e9i			0.400022i	0.979794	+	0.200011i	$0.0640978\pi$
$311$			3.74219e9i			0.400022i	−0.979794	+	0.200011i	$0.935902\pi$
$312$		0			0
$313$			2.81712e8i			0.0293514i	0.999892	+	0.0146757i	$0.00467158\pi$
$313$			2.81712e8i			0.0293514i	−0.999892	+	0.0146757i	$0.995328\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.18485e10			1.17335			0.586675	−	0.809822i	$-0.300436\pi$
$317$	1.18485e10			1.17335			0.586675	+	0.809822i	$0.300436\pi$
$318$		0			0
$319$	−6.31864e9			−0.610184
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.75662e9			−0.345134
$324$		0			0
$325$		−	1.45113e10i		−	1.30068i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.15468e10	−	1.07315e9i	0.985552	−	0.0915961i
$330$		0			0
$331$	−2.46682e9			−0.205507			−0.102753	−	0.994707i	$-0.532765\pi$
$331$	−2.46682e9			−0.205507			−0.102753	+	0.994707i	$0.532765\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	8.99711e9i		−	0.714371i
$336$		0			0
$337$	−1.20989e9			−0.0938051			−0.0469025	−	0.998899i	$-0.514935\pi$
$337$	−1.20989e9			−0.0938051			−0.0469025	+	0.998899i	$0.514935\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	6.76088e8i		−	0.0500018i
$342$		0			0
$343$	−3.79874e9	−	1.33098e10i	−0.274450	−	0.961601i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.61251e10			1.80194			0.900971	−	0.433880i	$-0.142856\pi$
$347$	2.61251e10			1.80194			0.900971	+	0.433880i	$0.142856\pi$
$348$		0			0
$349$		−	2.46730e9i		−	0.166311i	−0.996537	−	0.0831555i	$-0.973500\pi$
$349$		−	2.46730e9i		−	0.166311i	0.996537	−	0.0831555i	$-0.0264998\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	9.27283e9i		−	0.597191i	−0.954380	−	0.298596i	$-0.903482\pi$
$353$		−	9.27283e9i		−	0.597191i	0.954380	−	0.298596i	$-0.0965182\pi$
$354$		0			0
$355$		−	8.58394e8i		−	0.0540472i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.76277e9			0.106125			0.0530624	−	0.998591i	$-0.483102\pi$
$359$	1.76277e9			0.106125			0.0530624	+	0.998591i	$0.483102\pi$
$360$		0			0
$361$	1.23869e10			0.729349
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.02680e10			−0.578516
$366$		0			0
$367$		−	1.47281e10i		−	0.811862i	−0.913904	−	0.405931i	$-0.866947\pi$
$367$		−	1.47281e10i		−	0.811862i	0.913904	−	0.405931i	$-0.133053\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.66182e9	+	2.86406e10i	0.140502	+	1.51177i
$372$		0			0
$373$	−2.99301e10			−1.54623			−0.773114	−	0.634268i	$-0.781302\pi$
$373$	−2.99301e10			−1.54623			−0.773114	+	0.634268i	$0.781302\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	5.75047e10i		−	2.84667i
$378$		0			0
$379$	6.21942e9			0.301435			0.150717	−	0.988577i	$-0.451842\pi$
$379$	6.21942e9			0.301435			0.150717	+	0.988577i	$0.451842\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	1.42959e10i		−	0.664378i	−0.943213	−	0.332189i	$-0.892213\pi$
$383$		−	1.42959e10i		−	0.664378i	0.943213	−	0.332189i	$-0.107787\pi$
$384$		0			0
$385$	3.95198e9	−	3.67292e8i	0.179875	−	0.0167174i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.15618e10			−1.81508			−0.907541	−	0.419964i	$-0.862043\pi$
$389$	−4.15618e10			−1.81508			−0.907541	+	0.419964i	$0.862043\pi$
$390$		0			0
$391$			1.90920e10i			0.816855i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	1.52403e9i		−	0.0626046i
$396$		0			0
$397$			1.59996e10i			0.644091i	0.946724	+	0.322045i	$0.104370\pi$
$397$			1.59996e10i			0.644091i	−0.946724	+	0.322045i	$0.895630\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.03741e10			−0.787953			−0.393976	−	0.919121i	$-0.628901\pi$
$401$	−2.03741e10			−0.787953			−0.393976	+	0.919121i	$0.628901\pi$
$402$		0			0
$403$	6.15294e9			0.233272
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.95268e9			0.144050
$408$		0			0
$409$		−	3.77473e10i		−	1.34894i	−0.738302	−	0.674471i	$-0.764372\pi$
$409$		−	3.77473e10i		−	1.34894i	0.738302	−	0.674471i	$-0.235628\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.55291e10	+	4.23142e9i	−1.56491	+	0.145441i
$414$		0			0
$415$	−1.02326e10			−0.344980
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	5.28934e9i		−	0.171611i	−0.996312	−	0.0858056i	$-0.972654\pi$
$419$		−	5.28934e9i		−	0.171611i	0.996312	−	0.0858056i	$-0.0273464\pi$
$420$		0			0
$421$	3.20225e10			1.01936			0.509679	−	0.860364i	$-0.329764\pi$
$421$	3.20225e10			1.01936			0.509679	+	0.860364i	$0.329764\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			1.64164e10i			0.503179i
$426$		0			0
$427$	2.65552e10	−	2.46801e9i	0.798800	−	0.0742395i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.05036e10			0.883978			0.441989	−	0.897021i	$-0.354273\pi$
$431$	3.05036e10			0.883978			0.441989	+	0.897021i	$0.354273\pi$
$432$		0			0
$433$			2.62208e10i			0.745924i	0.927847	+	0.372962i	$0.121658\pi$
$433$			2.62208e10i			0.745924i	−0.927847	+	0.372962i	$0.878342\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			2.33611e10i			0.640572i
$438$		0			0
$439$			3.44122e10i			0.926519i	0.886223	+	0.463259i	$0.153320\pi$
$439$			3.44122e10i			0.926519i	−0.886223	+	0.463259i	$0.846680\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.61747e10			0.419974			0.209987	−	0.977704i	$-0.432658\pi$
$443$	1.61747e10			0.419974			0.209987	+	0.977704i	$0.432658\pi$
$444$		0			0
$445$	9.49870e9			0.242228
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.25204e9			0.178433			0.0892165	−	0.996012i	$-0.471564\pi$
$449$	7.25204e9			0.178433			0.0892165	+	0.996012i	$0.471564\pi$
$450$		0			0
$451$		−	6.40216e9i		−	0.154746i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.34265e9	+	3.59661e10i	0.0779912	+	0.839167i
$456$		0			0
$457$	7.99631e10			1.83326			0.916632	−	0.399733i	$-0.130897\pi$
$457$	7.99631e10			1.83326			0.916632	+	0.399733i	$0.130897\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	5.61132e10i		−	1.24240i	−0.783652	−	0.621200i	$-0.786646\pi$
$461$		−	5.61132e10i		−	1.24240i	0.783652	−	0.621200i	$-0.213354\pi$
$462$		0			0
$463$	1.32046e10			0.287344			0.143672	−	0.989625i	$-0.454109\pi$
$463$	1.32046e10			0.287344			0.143672	+	0.989625i	$0.454109\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			6.17464e10i			1.29821i	0.760700	+	0.649104i	$0.224856\pi$
$467$			6.17464e10i			1.29821i	−0.760700	+	0.649104i	$0.775144\pi$
$468$		0			0
$469$	−6.50820e9	−	7.00267e10i	−0.134515	−	1.44735i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.06851e10			0.413250
$474$		0			0
$475$			2.00872e10i			0.394589i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			7.70479e10i			1.46359i	0.681526	+	0.731794i	$0.261317\pi$
$479$			7.70479e10i			1.46359i	−0.681526	+	0.731794i	$0.738683\pi$
$480$		0			0
$481$			3.59725e10i			0.672033i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.55208e10			−0.280509
$486$		0			0
$487$	−8.33487e10			−1.48178			−0.740889	−	0.671627i	$-0.765596\pi$
$487$	−8.33487e10			−1.48178			−0.740889	+	0.671627i	$0.765596\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.56139e10			−0.612764			−0.306382	−	0.951909i	$-0.599119\pi$
$491$	−3.56139e10			−0.612764			−0.306382	+	0.951909i	$0.599119\pi$
$492$		0			0
$493$			6.50543e10i			1.10126i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.20933e8	−	6.68109e9i	−0.0101770	−	0.109502i
$498$		0			0
$499$	−2.49153e10			−0.401851			−0.200925	−	0.979607i	$-0.564395\pi$
$499$	−2.49153e10			−0.401851			−0.200925	+	0.979607i	$0.564395\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			6.27049e10i			0.979557i	0.871847	+	0.489778i	$0.162922\pi$
$503$			6.27049e10i			0.979557i	−0.871847	+	0.489778i	$0.837078\pi$
$504$		0			0
$505$	−5.86330e10			−0.901522
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	6.00554e10i		−	0.894707i	−0.894357	−	0.447353i	$-0.852367\pi$
$509$		−	6.00554e10i		−	0.894707i	0.894357	−	0.447353i	$-0.147633\pi$
$510$		0			0
$511$	−7.99185e10	+	7.42754e9i	−1.17210	+	0.108933i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.38789e9			0.0197299
$516$		0			0
$517$			2.59935e10i			0.363833i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			9.52594e10i			1.29288i	0.762966	+	0.646438i	$0.223742\pi$
$521$			9.52594e10i			1.29288i	−0.762966	+	0.646438i	$0.776258\pi$
$522$		0			0
$523$			2.86838e10i			0.383381i	0.981455	+	0.191690i	$0.0613969\pi$
$523$			2.86838e10i			0.383381i	−0.981455	+	0.191690i	$0.938603\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.96074e9			−0.0902429
$528$		0			0
$529$	4.04156e10			0.516091
$530$		0			0
$531$		0			0
$532$		0			0
$533$	5.82648e10			0.721934
$534$		0			0
$535$		−	3.44187e10i		−	0.420126i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.04935e10	−	5.71745e9i	0.361287	−	0.0677403i
$540$		0			0
$541$	−8.73664e10			−1.01990			−0.509948	−	0.860205i	$-0.670335\pi$
$541$	−8.73664e10			−1.01990			−0.509948	+	0.860205i	$0.670335\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			4.81556e10i			0.545834i
$546$		0			0
$547$	−4.69144e10			−0.524030			−0.262015	−	0.965064i	$-0.584387\pi$
$547$	−4.69144e10			−0.524030			−0.262015	+	0.965064i	$0.584387\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			7.96007e10i			0.863596i
$552$		0			0
$553$	−1.10244e9	−	1.18619e10i	−0.0117883	−	0.126840i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.29283e11			1.34314			0.671569	−	0.740942i	$-0.265621\pi$
$557$	1.29283e11			1.34314			0.671569	+	0.740942i	$0.265621\pi$
$558$		0			0
$559$			1.88251e11i			1.92793i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			1.05189e11i			1.04698i	0.852032	+	0.523490i	$0.175370\pi$
$563$			1.05189e11i			1.04698i	−0.852032	+	0.523490i	$0.824630\pi$
$564$		0			0
$565$		−	6.07874e10i		−	0.596513i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.55660e10			−0.434703			−0.217351	−	0.976093i	$-0.569742\pi$
$569$	−4.55660e10			−0.434703			−0.217351	+	0.976093i	$0.569742\pi$
$570$		0			0
$571$	1.10385e11			1.03840			0.519202	−	0.854652i	$-0.326229\pi$
$571$	1.10385e11			1.03840			0.519202	+	0.854652i	$0.326229\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.02088e11			0.933904
$576$		0			0
$577$		−	7.38745e9i		−	0.0666487i	−0.999445	−	0.0333243i	$-0.989391\pi$
$577$		−	7.38745e9i		−	0.0666487i	0.999445	−	0.0333243i	$-0.0106094\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.96429e10	+	7.40192e9i	−0.698945	+	0.0649591i
$582$		0			0
$583$	−6.44738e10			−0.558096
$584$		0			0
$585$		0			0
$586$		0			0
$587$			8.31559e10i			0.700391i	0.936677	+	0.350196i	$0.113885\pi$
$587$			8.31559e10i			0.700391i	−0.936677	+	0.350196i	$0.886115\pi$
$588$		0			0
$589$	−8.51720e9			−0.0707678
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6.67065e10i		−	0.539447i	−0.962938	−	0.269724i	$-0.913068\pi$
$593$		−	6.67065e10i		−	0.539447i	0.962938	−	0.269724i	$-0.0869324\pi$
$594$		0			0
$595$	−3.78150e9	−	4.06880e10i	−0.0301714	−	0.324637i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7.85940e10			−0.610495			−0.305247	−	0.952273i	$-0.598739\pi$
$599$	−7.85940e10			−0.610495			−0.305247	+	0.952273i	$0.598739\pi$
$600$		0			0
$601$		−	2.40655e11i		−	1.84458i	−0.386497	−	0.922291i	$-0.626315\pi$
$601$		−	2.40655e11i		−	1.84458i	0.386497	−	0.922291i	$-0.373685\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	5.69459e10i		−	0.425051i
$606$		0			0
$607$			7.02833e10i			0.517723i	0.965914	+	0.258862i	$0.0833473\pi$
$607$			7.02833e10i			0.517723i	−0.965914	+	0.258862i	$0.916653\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.36562e11			−1.69738
$612$		0			0
$613$	1.51121e11			1.07024			0.535121	−	0.844776i	$-0.320266\pi$
$613$	1.51121e11			1.07024			0.535121	+	0.844776i	$0.320266\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.18283e10			0.0816175			0.0408088	−	0.999167i	$-0.487007\pi$
$617$	1.18283e10			0.0816175			0.0408088	+	0.999167i	$0.487007\pi$
$618$		0			0
$619$			2.21107e11i			1.50605i	0.657992	+	0.753025i	$0.271406\pi$
$619$			2.21107e11i			1.50605i	−0.657992	+	0.753025i	$0.728594\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.39308e10	−	6.87104e9i	0.490764	−	0.0456111i
$624$		0			0
$625$	5.09265e10			0.333752
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	4.06953e10i		−	0.259981i
$630$		0			0
$631$	2.03138e11			1.28137			0.640685	−	0.767804i	$-0.278650\pi$
$631$	2.03138e11			1.28137			0.640685	+	0.767804i	$0.278650\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			1.28587e11i			0.790863i
$636$		0			0
$637$	5.20334e10	+	2.77515e11i	0.316027	+	1.68550i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.41763e8			0.00143205			0.000716023	−	1.00000i	$-0.499772\pi$
$641$	2.41763e8			0.00143205			0.000716023		1.00000i	$0.499772\pi$
$642$		0			0
$643$		−	1.17705e11i		−	0.688577i	−0.938864	−	0.344288i	$-0.888120\pi$
$643$		−	1.17705e11i		−	0.688577i	0.938864	−	0.344288i	$-0.111880\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.63521e11i		−	0.933163i	−0.884478	−	0.466581i	$-0.845485\pi$
$647$		−	1.63521e11i		−	0.933163i	0.884478	−	0.466581i	$-0.154515\pi$
$648$		0			0
$649$		−	1.02492e11i		−	0.577712i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.14460e10			0.392939			0.196470	−	0.980510i	$-0.437052\pi$
$653$	7.14460e10			0.392939			0.196470	+	0.980510i	$0.437052\pi$
$654$		0			0
$655$	−8.59349e10			−0.466879
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.87788e11			−1.52592			−0.762959	−	0.646447i	$-0.776254\pi$
$659$	−2.87788e11			−1.52592			−0.762959	+	0.646447i	$0.776254\pi$
$660$		0			0
$661$		−	1.81369e11i		−	0.950073i	−0.879966	−	0.475036i	$-0.842435\pi$
$661$		−	1.81369e11i		−	0.950073i	0.879966	−	0.475036i	$-0.157565\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.62706e9	−	4.97861e10i	−0.0236602	−	0.254578i
$666$		0			0
$667$	4.04549e11			2.04394
$668$		0			0
$669$		0			0
$670$		0			0
$671$			5.97793e10i			0.294890i
$672$		0			0
$673$	−7.10534e10			−0.346358			−0.173179	−	0.984890i	$-0.555404\pi$
$673$	−7.10534e10			−0.346358			−0.173179	+	0.984890i	$0.555404\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	2.56950e11i		−	1.22319i	−0.791171	−	0.611595i	$-0.790528\pi$
$677$		−	2.56950e11i		−	1.22319i	0.791171	−	0.611595i	$-0.209472\pi$
$678$		0			0
$679$	−1.20802e11	+	1.12272e10i	−0.568324	+	0.0528194i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8.54657e10			0.392744			0.196372	−	0.980530i	$-0.437084\pi$
$683$	8.54657e10			0.392744			0.196372	+	0.980530i	$0.437084\pi$
$684$		0			0
$685$			2.12843e10i			0.0966712i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	5.86763e11i		−	2.60367i
$690$		0			0
$691$		−	3.41062e10i		−	0.149596i	−0.997199	−	0.0747981i	$-0.976169\pi$
$691$		−	3.41062e10i		−	0.149596i	0.997199	−	0.0747981i	$-0.0238312\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.40591e11			−0.602583
$696$		0			0
$697$	−6.59142e10			−0.279285
$698$		0			0
$699$		0			0
$700$		0			0
$701$	9.13882e10			0.378458			0.189229	−	0.981933i	$-0.439401\pi$
$701$	9.13882e10			0.378458			0.189229	+	0.981933i	$0.439401\pi$
$702$		0			0
$703$		−	4.97949e10i		−	0.203875i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.56355e11	+	4.24131e10i	−1.82652	+	0.169755i
$708$		0			0
$709$	−4.31905e11			−1.70924			−0.854620	−	0.519254i	$-0.826210\pi$
$709$	−4.31905e11			−1.70924			−0.854620	+	0.519254i	$0.826210\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			4.32864e10i			0.167492i
$714$		0			0
$715$	−8.09646e10			−0.309793
$716$		0			0
$717$		0			0
$718$		0			0
$719$			4.56445e11i			1.70794i	0.520322	+	0.853970i	$0.325812\pi$
$719$			4.56445e11i			1.70794i	−0.520322	+	0.853970i	$0.674188\pi$
$720$		0			0
$721$	1.08023e10	−	1.00395e9i	0.0399737	−	0.00371511i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.47854e11			1.25906
$726$		0			0
$727$		−	5.00871e11i		−	1.79303i	−0.443012	−	0.896516i	$-0.646090\pi$
$727$		−	5.00871e11i		−	1.79303i	0.443012	−	0.896516i	$-0.353910\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	2.12966e11i		−	0.745831i
$732$		0			0
$733$		−	3.92835e11i		−	1.36080i	−0.732841	−	0.680400i	$-0.761806\pi$
$733$		−	3.92835e11i		−	1.36080i	0.732841	−	0.680400i	$-0.238194\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.57640e11			0.534312
$738$		0			0
$739$	3.54424e11			1.18835			0.594176	−	0.804335i	$-0.297478\pi$
$739$	3.54424e11			1.18835			0.594176	+	0.804335i	$0.297478\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4.28146e11			1.40487			0.702436	−	0.711747i	$-0.252096\pi$
$743$	4.28146e11			1.40487			0.702436	+	0.711747i	$0.252096\pi$
$744$		0			0
$745$			2.24183e11i			0.727741i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.48973e10	−	2.67889e11i	−0.0791089	−	0.851193i
$750$		0			0
$751$	−2.38599e11			−0.750082			−0.375041	−	0.927008i	$-0.622371\pi$
$751$	−2.38599e11			−0.750082			−0.375041	+	0.927008i	$0.622371\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	6.77876e10i		−	0.208623i
$756$		0			0
$757$	−1.86981e11			−0.569395			−0.284697	−	0.958617i	$-0.591893\pi$
$757$	−1.86981e11			−0.569395			−0.284697	+	0.958617i	$0.591893\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	4.41696e11i		−	1.31700i	−0.752582	−	0.658498i	$-0.771192\pi$
$761$		−	4.41696e11i		−	1.31700i	0.752582	−	0.658498i	$-0.228808\pi$
$762$		0			0
$763$	3.48341e10	+	3.74807e11i	0.102780	+	1.10588i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9.32760e11			2.69518
$768$		0			0
$769$		−	3.57622e11i		−	1.02263i	−0.859393	−	0.511315i	$-0.829158\pi$
$769$		−	3.57622e11i		−	1.02263i	0.859393	−	0.511315i	$-0.170842\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	2.35131e11i		−	0.658555i	−0.944233	−	0.329277i	$-0.893195\pi$
$773$		−	2.35131e11i		−	0.658555i	0.944233	−	0.329277i	$-0.106805\pi$
$774$		0			0
$775$			3.72201e10i			0.103174i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.06529e10			−0.219013
$780$		0			0
$781$	1.50400e10			0.0404245
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.40430e11			0.369813
$786$		0			0
$787$			1.15901e11i			0.302125i	0.988524	+	0.151063i	$0.0482695\pi$
$787$			1.15901e11i			0.302125i	−0.988524	+	0.151063i	$0.951730\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.39716e10	−	4.73124e11i	−0.112322	−	1.20856i
$792$		0			0
$793$	−5.44039e11			−1.37574
$794$		0			0
$795$		0			0
$796$		0			0
$797$			3.96381e11i			0.982381i	0.871052	+	0.491190i	$0.163438\pi$
$797$			3.96381e11i			0.982381i	−0.871052	+	0.491190i	$0.836562\pi$
$798$		0			0
$799$	2.67619e11			0.656644
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	1.79907e11i		−	0.432700i
$804$		0			0
$805$	−2.53024e11	+	2.35158e10i	−0.602530	+	0.0559984i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.29402e11			0.302098			0.151049	−	0.988526i	$-0.451735\pi$
$809$	1.29402e11			0.302098			0.151049	+	0.988526i	$0.451735\pi$
$810$		0			0
$811$			4.80667e11i			1.11112i	0.831476	+	0.555560i	$0.187496\pi$
$811$			4.80667e11i			1.11112i	−0.831476	+	0.555560i	$0.812504\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	4.86584e10i		−	0.110288i
$816$		0			0
$817$		−	2.60586e11i		−	0.584875i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6.26912e11			−1.37986			−0.689929	−	0.723877i	$-0.742358\pi$
$821$	−6.26912e11			−1.37986			−0.689929	+	0.723877i	$0.742358\pi$
$822$		0			0
$823$	−9.14176e10			−0.199265			−0.0996324	−	0.995024i	$-0.531767\pi$
$823$	−9.14176e10			−0.199265			−0.0996324	+	0.995024i	$0.531767\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.29762e11			−0.918769			−0.459384	−	0.888238i	$-0.651930\pi$
$827$	−4.29762e11			−0.918769			−0.459384	+	0.888238i	$0.651930\pi$
$828$		0			0
$829$		−	3.66115e11i		−	0.775174i	−0.921833	−	0.387587i	$-0.873309\pi$
$829$		−	3.66115e11i		−	0.775174i	0.921833	−	0.387587i	$-0.126691\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.88647e10	−	3.13950e11i	−0.122257	−	0.652048i
$834$		0			0
$835$	−1.70630e11			−0.351002
$836$		0			0
$837$		0			0
$838$		0			0
$839$			8.61714e11i			1.73906i	0.493878	+	0.869531i	$0.335579\pi$
$839$			8.61714e11i			1.73906i	−0.493878	+	0.869531i	$0.664421\pi$
$840$		0			0
$841$	8.78216e11			1.75557
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	4.86283e11i		−	0.953811i
$846$		0			0
$847$	−4.11927e10	−	4.43224e11i	−0.0800363	−	0.861172i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.53069e11			−0.482527
$852$		0			0
$853$		−	7.23327e11i		−	1.36628i	−0.730289	−	0.683138i	$-0.760615\pi$
$853$		−	7.23327e11i		−	1.36628i	0.730289	−	0.683138i	$-0.239385\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.28445e11i			0.238119i	0.992887	+	0.119059i	$0.0379879\pi$
$857$			1.28445e11i			0.238119i	−0.992887	+	0.119059i	$0.962012\pi$
$858$		0			0
$859$		−	6.59898e11i		−	1.21200i	−0.795463	−	0.606002i	$-0.792772\pi$
$859$		−	6.59898e11i		−	1.21200i	0.795463	−	0.606002i	$-0.207228\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.41184e11			0.254532			0.127266	−	0.991869i	$-0.459380\pi$
$863$	1.41184e11			0.254532			0.127266	+	0.991869i	$0.459380\pi$
$864$		0			0
$865$	−4.38879e11			−0.783936
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.67028e10			0.0468250
$870$		0			0
$871$			1.43465e12i			2.49271i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.04410e11	+	4.68793e10i	−0.860501	+	0.0799740i
$876$		0			0
$877$	1.05924e12			1.79059			0.895294	−	0.445476i	$-0.146965\pi$
$877$	1.05924e12			1.79059			0.895294	+	0.445476i	$0.146965\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1.11273e11i			0.184709i	0.995726	+	0.0923543i	$0.0294392\pi$
$881$			1.11273e11i			0.184709i	−0.995726	+	0.0923543i	$0.970561\pi$
$882$		0			0
$883$	5.74127e10			0.0944420			0.0472210	−	0.998884i	$-0.484963\pi$
$883$	5.74127e10			0.0944420			0.0472210	+	0.998884i	$0.484963\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			2.33977e11i			0.377989i	0.981978	+	0.188995i	$0.0605228\pi$
$887$			2.33977e11i			0.377989i	−0.981978	+	0.188995i	$0.939477\pi$
$888$		0			0
$889$	9.30153e10	+	1.00082e12i	0.148918	+	1.60232i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.27460e11			0.514935
$894$		0			0
$895$		−	2.07771e11i		−	0.323813i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1.47494e11i			0.225806i
$900$		0			0
$901$			6.63797e11i			1.00725i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.78034e11			−0.414480
$906$		0			0
$907$	1.28107e12			1.89297			0.946485	−	0.322747i	$-0.104606\pi$
$907$	1.28107e12			1.89297			0.946485	+	0.322747i	$0.104606\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7.60592e11			1.10428			0.552139	−	0.833752i	$-0.313812\pi$
$911$	7.60592e11			1.10428			0.552139	+	0.833752i	$0.313812\pi$
$912$		0			0
$913$		−	1.79287e11i		−	0.258027i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.68853e11	+	6.21624e10i	−0.945917	+	0.0879124i
$918$		0			0
$919$	−1.16164e12			−1.62859			−0.814293	−	0.580454i	$-0.802875\pi$
$919$	−1.16164e12			−1.62859			−0.814293	+	0.580454i	$0.802875\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1.36876e11i			0.188591i
$924$		0			0
$925$	−2.17603e11			−0.297234
$926$		0			0
$927$		0			0
$928$		0			0
$929$			4.81419e11i			0.646339i	0.946341	+	0.323169i	$0.104748\pi$
$929$			4.81419e11i			0.646339i	−0.946341	+	0.323169i	$0.895252\pi$
$930$		0			0
$931$	−7.20271e10	−	3.84150e11i	−0.0958732	−	0.511331i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	9.15942e10			0.119845
$936$		0			0
$937$			3.25575e10i			0.0422370i	0.999777	+	0.0211185i	$0.00672272\pi$
$937$			3.25575e10i			0.0422370i	−0.999777	+	0.0211185i	$0.993277\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			9.07253e11i			1.15710i	0.815648	+	0.578549i	$0.196381\pi$
$941$			9.07253e11i			1.15710i	−0.815648	+	0.578549i	$0.803619\pi$
$942$		0			0
$943$			4.09897e11i			0.518356i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.39273e12			−1.73168			−0.865838	−	0.500325i	$-0.833214\pi$
$947$	−1.39273e12			−1.73168			−0.865838	+	0.500325i	$0.833214\pi$
$948$		0			0
$949$	1.63730e12			2.01866
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.43413e11			−0.295102			−0.147551	−	0.989054i	$-0.547139\pi$
$953$	−2.43413e11			−0.295102			−0.147551	+	0.989054i	$0.547139\pi$
$954$		0			0
$955$			8.33978e10i			0.100263i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.53964e10	+	1.65661e11i	0.0182030	+	0.195860i
$960$		0			0
$961$	8.37109e11			0.981496
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	2.29867e11i		−	0.265074i
$966$		0			0
$967$	−1.61333e12			−1.84509			−0.922545	−	0.385890i	$-0.873894\pi$
$967$	−1.61333e12			−1.84509			−0.922545	+	0.385890i	$0.873894\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			9.23466e11i			1.03883i	0.854523	+	0.519414i	$0.173850\pi$
$971$			9.23466e11i			1.03883i	−0.854523	+	0.519414i	$0.826150\pi$
$972$		0			0
$973$	−1.09425e12	+	1.01698e11i	−1.22086	+	0.113465i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.47197e12			1.61555			0.807775	−	0.589491i	$-0.200672\pi$
$977$	1.47197e12			1.61555			0.807775	+	0.589491i	$0.200672\pi$
$978$		0			0
$979$			1.66428e11i			0.181174i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	6.45672e11i		−	0.691509i	−0.938325	−	0.345754i	$-0.887623\pi$
$983$		−	6.45672e11i		−	0.691509i	0.938325	−	0.345754i	$-0.112377\pi$
$984$		0			0
$985$			3.36939e11i			0.357937i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.32436e12			−1.38427
$990$		0			0
$991$	4.11657e11			0.426816			0.213408	−	0.976963i	$-0.431544\pi$
$991$	4.11657e11			0.426816			0.213408	+	0.976963i	$0.431544\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4.22977e10			−0.0431543
$996$		0			0
$997$			1.26600e12i			1.28131i	0.767831	+	0.640653i	$0.221336\pi$
$997$			1.26600e12i			1.28131i	−0.767831	+	0.640653i	$0.778664\pi$
$998$		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.181.7		10
3.2	odd	2		84.9.d.a.13.2	✓	10
7.6	odd	2	inner	252.9.d.d.181.4		10
12.11	even	2		336.9.f.a.97.7		10
21.20	even	2		84.9.d.a.13.9	yes	10
84.83	odd	2		336.9.f.a.97.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.d.a.13.2	✓	10	3.2	odd	2
84.9.d.a.13.9	yes	10	21.20	even	2
252.9.d.d.181.4		10	7.6	odd	2	inner
252.9.d.d.181.7		10	1.1	even	1	trivial
336.9.f.a.97.4		10	84.83	odd	2
336.9.f.a.97.7		10	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+307.159i q^{5} +(222.189 + 2390.70i) q^{7} +O(q^{10})\)	\(q+307.159i q^{5} +(222.189 + 2390.70i) q^{7} -5381.78 q^{11} -48978.5i q^{13} +55408.7i q^{17} +67798.4i q^{19} +344567. q^{23} +296278. q^{25} +1.17408e6 q^{29} +125625. i q^{31} +(-734325. + 68247.3i) q^{35} -734456. q^{37} +1.18960e6i q^{41} -3.84354e6 q^{43} -4.82991e6i q^{47} +(-5.66607e6 + 1.06237e6i) q^{49} +1.19800e7 q^{53} -1.65306e6i q^{55} +1.90443e7i q^{59} -1.11077e7i q^{61} +1.50442e7 q^{65} -2.92913e7 q^{67} -2.79462e6 q^{71} +3.34290e7i q^{73} +(-1.19577e6 - 1.28662e7i) q^{77} -4.96171e6 q^{79} +3.33137e7i q^{83} -1.70193e7 q^{85} -3.09244e7i q^{89} +(1.17093e8 - 1.08825e7i) q^{91} -2.08249e7 q^{95} +5.05301e7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2338 q^{7}+O(q^{10}) \) 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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