LMFDB - Embedded newform 252.11.d.c.181.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,11,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, 11, names="a")

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

chi := DirichletCharacter("252.181");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 11 $
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:	$160.110027674$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 8700283 x^{10} + 1743363790 x^{9} + 25853580960505 x^{8} + \cdots + 27\!\cdots\!64 $ x^12 - 2x^11 + 8700283x^10 + 1743363790x^9 + 25853580960505x^8 + 7667110891620628x^7 + 33402856492133203204x^6 + 11376460542571911048304x^5 + 18926138237170427869601620x^4 + 5259537754347160642438909440x^3 + 3707356408577721628213927744128x^2 + 604475470219454050864216450904064*x + 27069230748460125474237471709860864
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{17}\cdot 7^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.4
Root			$211.922 - 1215.76i$ of defining polynomial
Character	$\chi$	$=$	252.181
Dual form			252.11.d.c.181.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4316.86i q^{5} +(16495.1 + 3223.01i) q^{7} +O(q^{10})$	$q-4316.86i q^{5} +(16495.1 + 3223.01i) q^{7} -79802.8 q^{11} +692956. i q^{13} -1.16694e6i q^{17} +2.57322e6i q^{19} +1.05327e7 q^{23} -8.86968e6 q^{25} -3.47392e7 q^{29} -1.06645e7i q^{31} +(1.39133e7 - 7.12070e7i) q^{35} -1.04883e8 q^{37} -1.13353e8i q^{41} -2.97789e7 q^{43} -2.53443e8i q^{47} +(2.61700e8 + 1.06328e8i) q^{49} -5.73403e8 q^{53} +3.44498e8i q^{55} -6.88838e8i q^{59} -3.99770e8i q^{61} +2.99139e9 q^{65} +1.04557e9 q^{67} -2.84547e9 q^{71} -3.05273e9i q^{73} +(-1.31635e9 - 2.57205e8i) q^{77} +8.17922e8 q^{79} +6.05909e9i q^{83} -5.03753e9 q^{85} +7.06701e9i q^{89} +(-2.23340e9 + 1.14304e10i) q^{91} +1.11082e10 q^{95} +1.35607e9i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 25788 q^{7}+O(q^{10}) $ 12 * q + 25788 * q^7	$ 12 q + 25788 q^{7} - 57218340 q^{25} + 179925528 q^{37} - 1040727816 q^{43} + 845565756 q^{49} + 8004513864 q^{67} - 10108183224 q^{79} - 11317946640 q^{85} + 6460110048 q^{91}+O(q^{100}) $ 12 * q + 25788 * q^7 - 57218340 * q^25 + 179925528 * q^37 - 1040727816 * q^43 + 845565756 * q^49 + 8004513864 * q^67 - 10108183224 * q^79 - 11317946640 * q^85 + 6460110048 * q^91

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
$5$		−	4316.86i		−	1.38140i	−0.723143	−	0.690698i	$-0.757303\pi$
$5$		−	4316.86i		−	1.38140i	0.723143	−	0.690698i	$-0.242697\pi$
$6$		0			0
$7$	16495.1	+	3223.01i	0.981441	+	0.191766i
$7$	16495.1	+	3223.01i	0.981441	+	0.191766i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−79802.8			−0.495512			−0.247756	−	0.968822i	$-0.579693\pi$
$11$	−79802.8			−0.495512			−0.247756	+	0.968822i	$0.579693\pi$
$12$		0			0
$13$			692956.i			1.86633i	0.359447	+	0.933166i	$0.382965\pi$
$13$			692956.i			1.86633i	−0.359447	+	0.933166i	$0.617035\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.16694e6i		−	0.821873i	−0.911664	−	0.410936i	$-0.865202\pi$
$17$		−	1.16694e6i		−	0.821873i	0.911664	−	0.410936i	$-0.134798\pi$
$18$		0			0
$19$			2.57322e6i			1.03922i	0.854403	+	0.519611i	$0.173923\pi$
$19$			2.57322e6i			1.03922i	−0.854403	+	0.519611i	$0.826077\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.05327e7			1.63644			0.818218	−	0.574908i	$-0.194962\pi$
$23$	1.05327e7			1.63644			0.818218	+	0.574908i	$0.194962\pi$
$24$		0			0
$25$	−8.86968e6			−0.908255
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.47392e7			−1.69367			−0.846837	−	0.531852i	$-0.821496\pi$
$29$	−3.47392e7			−1.69367			−0.846837	+	0.531852i	$0.821496\pi$
$30$		0			0
$31$		−	1.06645e7i		−	0.372504i	−0.982502	−	0.186252i	$-0.940366\pi$
$31$		−	1.06645e7i		−	0.372504i	0.982502	−	0.186252i	$-0.0596341\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.39133e7	−	7.12070e7i	0.264905	−	1.35576i
$36$		0			0
$37$	−1.04883e8			−1.51250			−0.756251	−	0.654282i	$-0.772971\pi$
$37$	−1.04883e8			−1.51250			−0.756251	+	0.654282i	$0.772971\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.13353e8i		−	0.978391i	−0.872174	−	0.489195i	$-0.837290\pi$
$41$		−	1.13353e8i		−	0.978391i	0.872174	−	0.489195i	$-0.162710\pi$
$42$		0			0
$43$	−2.97789e7			−0.202566			−0.101283	−	0.994858i	$-0.532295\pi$
$43$	−2.97789e7			−0.202566			−0.101283	+	0.994858i	$0.532295\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.53443e8i		−	1.10507i	−0.833489	−	0.552536i	$-0.813660\pi$
$47$		−	2.53443e8i		−	1.10507i	0.833489	−	0.552536i	$-0.186340\pi$
$48$		0			0
$49$	2.61700e8	+	1.06328e8i	0.926452	+	0.376414i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.73403e8			−1.37114			−0.685569	−	0.728008i	$-0.740446\pi$
$53$	−5.73403e8			−1.37114			−0.685569	+	0.728008i	$0.740446\pi$
$54$		0			0
$55$			3.44498e8i			0.684499i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	6.88838e8i		−	0.963512i	−0.876305	−	0.481756i	$-0.839999\pi$
$59$		−	6.88838e8i		−	0.963512i	0.876305	−	0.481756i	$-0.160001\pi$
$60$		0			0
$61$		−	3.99770e8i		−	0.473327i	−0.971592	−	0.236664i	$-0.923946\pi$
$61$		−	3.99770e8i		−	0.473327i	0.971592	−	0.236664i	$-0.0760539\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.99139e9			2.57814
$66$		0			0
$67$	1.04557e9			0.774425			0.387213	−	0.921990i	$-0.373438\pi$
$67$	1.04557e9			0.774425			0.387213	+	0.921990i	$0.373438\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.84547e9			−1.57711			−0.788555	−	0.614964i	$-0.789171\pi$
$71$	−2.84547e9			−1.57711			−0.788555	+	0.614964i	$0.789171\pi$
$72$		0			0
$73$		−	3.05273e9i		−	1.47256i	−0.676675	−	0.736281i	$-0.736580\pi$
$73$		−	3.05273e9i		−	1.47256i	0.676675	−	0.736281i	$-0.263420\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.31635e9	−	2.57205e8i	−0.486316	−	0.0950224i
$78$		0			0
$79$	8.17922e8			0.265813			0.132907	−	0.991129i	$-0.457569\pi$
$79$	8.17922e8			0.265813			0.132907	+	0.991129i	$0.457569\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			6.05909e9i			1.53821i	0.639120	+	0.769107i	$0.279299\pi$
$83$			6.05909e9i			1.53821i	−0.639120	+	0.769107i	$0.720701\pi$
$84$		0			0
$85$	−5.03753e9			−1.13533
$86$		0			0
$87$		0			0
$88$		0			0
$89$			7.06701e9i			1.26557i	0.774328	+	0.632785i	$0.218088\pi$
$89$			7.06701e9i			1.26557i	−0.774328	+	0.632785i	$0.781912\pi$
$90$		0			0
$91$	−2.23340e9	+	1.14304e10i	−0.357899	+	1.83169i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.11082e10			1.43558
$96$		0			0
$97$			1.35607e9i			0.157915i	0.996878	+	0.0789576i	$0.0251592\pi$
$97$			1.35607e9i			0.157915i	−0.996878	+	0.0789576i	$0.974841\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			1.05048e9i			0.0999499i	0.998750	+	0.0499750i	$0.0159142\pi$
$101$			1.05048e9i			0.0999499i	−0.998750	+	0.0499750i	$0.984086\pi$
$102$		0			0
$103$			6.96490e9i			0.600798i	0.953814	+	0.300399i	$0.0971199\pi$
$103$			6.96490e9i			0.600798i	−0.953814	+	0.300399i	$0.902880\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.53044e10			−1.09119			−0.545593	−	0.838050i	$-0.683696\pi$
$107$	−1.53044e10			−1.09119			−0.545593	+	0.838050i	$0.683696\pi$
$108$		0			0
$109$	−1.31406e10			−0.854048			−0.427024	−	0.904240i	$-0.640438\pi$
$109$	−1.31406e10			−0.854048			−0.427024	+	0.904240i	$0.640438\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.58501e9			0.303132			0.151566	−	0.988447i	$-0.451568\pi$
$113$	5.58501e9			0.303132			0.151566	+	0.988447i	$0.451568\pi$
$114$		0			0
$115$		−	4.54681e10i		−	2.26057i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.76106e9	−	1.92488e10i	0.157607	−	0.806619i
$120$		0			0
$121$	−1.95689e10			−0.754467
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	3.86767e9i		−	0.126736i
$126$		0			0
$127$	3.12610e10			0.946204			0.473102	−	0.881008i	$-0.343134\pi$
$127$	3.12610e10			0.946204			0.473102	+	0.881008i	$0.343134\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	3.06060e10i		−	0.793322i	−0.917965	−	0.396661i	$-0.870169\pi$
$131$		−	3.06060e10i		−	0.793322i	0.917965	−	0.396661i	$-0.129831\pi$
$132$		0			0
$133$	−8.29350e9	+	4.24454e10i	−0.199287	+	1.01993i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.09419e10			−0.226720			−0.113360	−	0.993554i	$-0.536161\pi$
$137$	−1.09419e10			−0.226720			−0.113360	+	0.993554i	$0.536161\pi$
$138$		0			0
$139$		−	4.14478e10i		−	0.798780i	−0.916781	−	0.399390i	$-0.869222\pi$
$139$		−	4.14478e10i		−	0.798780i	0.916781	−	0.399390i	$-0.130778\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	5.52998e10i		−	0.924790i
$144$		0			0
$145$			1.49964e11i			2.33964i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.00376e10			−0.953674			−0.476837	−	0.878992i	$-0.658217\pi$
$149$	−7.00376e10			−0.953674			−0.476837	+	0.878992i	$0.658217\pi$
$150$		0			0
$151$	−6.51341e10			−0.829704			−0.414852	−	0.909889i	$-0.636167\pi$
$151$	−6.51341e10			−0.829704			−0.414852	+	0.909889i	$0.636167\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.60370e10			−0.514575
$156$		0			0
$157$		−	1.14511e11i		−	1.20046i	−0.799826	−	0.600231i	$-0.795075\pi$
$157$		−	1.14511e11i		−	1.20046i	0.799826	−	0.600231i	$-0.204925\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.73737e11	+	3.39469e10i	1.60606	+	0.313813i
$162$		0			0
$163$	−9.04758e10			−0.786311			−0.393156	−	0.919472i	$-0.628617\pi$
$163$	−9.04758e10			−0.786311			−0.393156	+	0.919472i	$0.628617\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1.37407e11i		−	1.05786i	−0.848666	−	0.528928i	$-0.822594\pi$
$167$		−	1.37407e11i		−	1.05786i	0.848666	−	0.528928i	$-0.177406\pi$
$168$		0			0
$169$	−3.42329e11			−2.48319
$170$		0			0
$171$		0			0
$172$		0			0
$173$			2.55661e11i			1.64981i	0.565271	+	0.824905i	$0.308771\pi$
$173$			2.55661e11i			1.64981i	−0.565271	+	0.824905i	$0.691229\pi$
$174$		0			0
$175$	−1.46306e11	−	2.85871e10i	−0.891399	−	0.174172i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.92333e10			−0.213496			−0.106748	−	0.994286i	$-0.534044\pi$
$179$	−3.92333e10			−0.213496			−0.106748	+	0.994286i	$0.534044\pi$
$180$		0			0
$181$			1.31921e10i			0.0679081i	0.999423	+	0.0339540i	$0.0108100\pi$
$181$			1.31921e10i			0.0679081i	−0.999423	+	0.0339540i	$0.989190\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			4.52765e11i			2.08936i
$186$		0			0
$187$			9.31252e10i			0.407248i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.53113e11			1.78254			0.891270	−	0.453472i	$-0.149815\pi$
$191$	4.53113e11			1.78254			0.891270	+	0.453472i	$0.149815\pi$
$192$		0			0
$193$	−2.51466e11			−0.939059			−0.469529	−	0.882917i	$-0.655576\pi$
$193$	−2.51466e11			−0.939059			−0.469529	+	0.882917i	$0.655576\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.41684e11			−0.814548			−0.407274	−	0.913306i	$-0.633521\pi$
$197$	−2.41684e11			−0.814548			−0.407274	+	0.913306i	$0.633521\pi$
$198$		0			0
$199$			5.31941e11i			1.70451i	0.523130	+	0.852253i	$0.324764\pi$
$199$			5.31941e11i			1.70451i	−0.523130	+	0.852253i	$0.675236\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.73026e11	−	1.11965e11i	−1.66224	−	0.324789i
$204$		0			0
$205$	−4.89328e11			−1.35155
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	2.05350e11i		−	0.514947i
$210$		0			0
$211$	−3.64978e11			−0.872678			−0.436339	−	0.899782i	$-0.643725\pi$
$211$	−3.64978e11			−0.872678			−0.436339	+	0.899782i	$0.643725\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			1.28551e11i			0.279824i
$216$		0			0
$217$	3.43717e10	−	1.75911e11i	0.0714335	−	0.365590i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.08639e11			1.53389
$222$		0			0
$223$			2.67615e11i			0.485272i	0.970117	+	0.242636i	$0.0780121\pi$
$223$			2.67615e11i			0.485272i	−0.970117	+	0.242636i	$0.921988\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	4.99538e11i		−	0.828779i	−0.910100	−	0.414390i	$-0.863995\pi$
$227$		−	4.99538e11i		−	0.828779i	0.910100	−	0.414390i	$-0.136005\pi$
$228$		0			0
$229$		−	1.15560e12i		−	1.83497i	−0.397769	−	0.917486i	$-0.630215\pi$
$229$		−	1.15560e12i		−	1.83497i	0.397769	−	0.917486i	$-0.369785\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−8.90005e11			−1.29602			−0.648012	−	0.761630i	$-0.724399\pi$
$233$	−8.90005e11			−1.29602			−0.648012	+	0.761630i	$0.724399\pi$
$234$		0			0
$235$	−1.09408e12			−1.52654
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.24631e11			0.416294			0.208147	−	0.978098i	$-0.433257\pi$
$239$	3.24631e11			0.416294			0.208147	+	0.978098i	$0.433257\pi$
$240$		0			0
$241$			9.98953e11i			1.22874i	0.789018	+	0.614370i	$0.210590\pi$
$241$			9.98953e11i			1.22874i	−0.789018	+	0.614370i	$0.789410\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.59001e11	−	1.12972e12i	0.519976	−	1.27980i
$246$		0			0
$247$	−1.78312e12			−1.93953
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	1.62879e12i		−	1.63492i	−0.575989	−	0.817458i	$-0.695383\pi$
$251$		−	1.62879e12i		−	1.63492i	0.575989	−	0.817458i	$-0.304617\pi$
$252$		0			0
$253$	−8.40536e11			−0.810874
$254$		0			0
$255$		0			0
$256$		0			0
$257$			1.74327e12i			1.55489i	0.628951	+	0.777445i	$0.283485\pi$
$257$			1.74327e12i			1.55489i	−0.628951	+	0.777445i	$0.716515\pi$
$258$		0			0
$259$	−1.73005e12	−	3.38038e11i	−1.48443	−	0.290046i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.92643e12			−1.53100			−0.765499	−	0.643437i	$-0.777508\pi$
$263$	−1.92643e12			−1.53100			−0.765499	+	0.643437i	$0.777508\pi$
$264$		0			0
$265$			2.47530e12i			1.89408i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			3.21960e10i			0.0228581i	0.999935	+	0.0114291i	$0.00363806\pi$
$269$			3.21960e10i			0.0228581i	−0.999935	+	0.0114291i	$0.996362\pi$
$270$		0			0
$271$		−	7.16277e11i		−	0.490044i	−0.969518	−	0.245022i	$-0.921205\pi$
$271$		−	7.16277e11i		−	0.490044i	0.969518	−	0.245022i	$-0.0787951\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.07825e11			0.450052
$276$		0			0
$277$	−2.70064e12			−1.65603			−0.828016	−	0.560705i	$-0.810530\pi$
$277$	−2.70064e12			−1.65603			−0.828016	+	0.560705i	$0.810530\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.39685e12			0.797296			0.398648	−	0.917104i	$-0.369480\pi$
$281$	1.39685e12			0.797296			0.398648	+	0.917104i	$0.369480\pi$
$282$		0			0
$283$			2.08337e12i			1.14772i	0.818955	+	0.573858i	$0.194554\pi$
$283$			2.08337e12i			1.14772i	−0.818955	+	0.573858i	$0.805446\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.65337e11	−	1.86976e12i	0.187622	−	0.960233i
$288$		0			0
$289$	6.54241e11			0.324525
$290$		0			0
$291$		0			0
$292$		0			0
$293$			3.57961e11i			0.165767i	0.996559	+	0.0828833i	$0.0264129\pi$
$293$			3.57961e11i			0.165767i	−0.996559	+	0.0828833i	$0.973587\pi$
$294$		0			0
$295$	−2.97362e12			−1.33099
$296$		0			0
$297$		0			0
$298$		0			0
$299$			7.29867e12i			3.05413i
$300$		0			0
$301$	−4.91205e11	−	9.59776e10i	−0.198806	−	0.0388452i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.72575e12			−0.653852
$306$		0			0
$307$		−	4.04689e12i		−	1.48399i	−0.670407	−	0.741993i	$-0.733881\pi$
$307$		−	4.04689e12i		−	1.48399i	0.670407	−	0.741993i	$-0.266119\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			3.52111e12i			1.21026i	0.796127	+	0.605129i	$0.206878\pi$
$311$			3.52111e12i			1.21026i	−0.796127	+	0.605129i	$0.793122\pi$
$312$		0			0
$313$			2.37203e12i			0.789584i	0.918770	+	0.394792i	$0.129183\pi$
$313$			2.37203e12i			0.789584i	−0.918770	+	0.394792i	$0.870817\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.99854e11			−0.249870			−0.124935	−	0.992165i	$-0.539872\pi$
$317$	−7.99854e11			−0.249870			−0.124935	+	0.992165i	$0.539872\pi$
$318$		0			0
$319$	2.77229e12			0.839237
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.00279e12			0.854108
$324$		0			0
$325$		−	6.14630e12i		−	1.69511i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.16848e11	−	4.18056e12i	0.211915	−	1.08456i
$330$		0			0
$331$	−3.00327e12			−0.755883			−0.377942	−	0.925829i	$-0.623368\pi$
$331$	−3.00327e12			−0.755883			−0.377942	+	0.925829i	$0.623368\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	4.51359e12i		−	1.06979i
$336$		0			0
$337$	1.33583e12			0.307327			0.153663	−	0.988123i	$-0.450893\pi$
$337$	1.33583e12			0.307327			0.153663	+	0.988123i	$0.450893\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			8.51054e11i			0.184580i
$342$		0			0
$343$	3.97406e12	+	2.59734e12i	0.837074	+	0.547089i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.79373e12			0.754083			0.377041	−	0.926196i	$-0.376941\pi$
$347$	3.79373e12			0.754083			0.377041	+	0.926196i	$0.376941\pi$
$348$		0			0
$349$			4.89891e11i			0.0946177i	0.998880	+	0.0473089i	$0.0150645\pi$
$349$			4.89891e11i			0.0946177i	−0.998880	+	0.0473089i	$0.984935\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			1.09124e12i			0.199089i	0.995033	+	0.0995444i	$0.0317385\pi$
$353$			1.09124e12i			0.199089i	−0.995033	+	0.0995444i	$0.968261\pi$
$354$		0			0
$355$			1.22835e13i			2.17861i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.43386e12			0.408154			0.204077	−	0.978955i	$-0.434581\pi$
$359$	2.43386e12			0.408154			0.204077	+	0.978955i	$0.434581\pi$
$360$		0			0
$361$	−4.90375e11			−0.0799820
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.31782e13			−2.03419
$366$		0			0
$367$		−	4.30950e12i		−	0.647287i	−0.946179	−	0.323644i	$-0.895092\pi$
$367$		−	4.30950e12i		−	0.647287i	0.946179	−	0.323644i	$-0.104908\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.45833e12	−	1.84808e12i	−1.34569	−	0.262937i
$372$		0			0
$373$	−5.74050e11			−0.0795071			−0.0397535	−	0.999210i	$-0.512657\pi$
$373$	−5.74050e11			−0.0795071			−0.0397535	+	0.999210i	$0.512657\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	2.40727e13i		−	3.16096i
$378$		0			0
$379$	3.37288e12			0.431326			0.215663	−	0.976468i	$-0.430809\pi$
$379$	3.37288e12			0.431326			0.215663	+	0.976468i	$0.430809\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			1.56450e13i			1.89838i	0.314709	+	0.949188i	$0.398093\pi$
$383$			1.56450e13i			1.89838i	−0.314709	+	0.949188i	$0.601907\pi$
$384$		0			0
$385$	−1.11032e12	+	5.68251e12i	−0.131264	+	0.671795i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.12447e13			1.26240			0.631202	−	0.775619i	$-0.282562\pi$
$389$	1.12447e13			1.26240			0.631202	+	0.775619i	$0.282562\pi$
$390$		0			0
$391$		−	1.22910e13i		−	1.34494i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	3.53086e12i		−	0.367193i
$396$		0			0
$397$		−	8.71420e12i		−	0.883639i	−0.897104	−	0.441820i	$-0.854333\pi$
$397$		−	8.71420e12i		−	0.883639i	0.897104	−	0.441820i	$-0.145667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.23427e12			0.408373			0.204186	−	0.978932i	$-0.434545\pi$
$401$	4.23427e12			0.408373			0.204186	+	0.978932i	$0.434545\pi$
$402$		0			0
$403$	7.39000e12			0.695215
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.36994e12			0.749463
$408$		0			0
$409$		−	1.20285e13i		−	1.05098i	−0.850800	−	0.525490i	$-0.823882\pi$
$409$		−	1.20285e13i		−	1.05098i	0.850800	−	0.525490i	$-0.176118\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.22013e12	−	1.13624e13i	0.184769	−	0.945630i
$414$		0			0
$415$	2.61563e13			2.12488
$416$		0			0
$417$		0			0
$418$		0			0
$419$			3.30064e12i			0.255581i	0.991801	+	0.127790i	$0.0407885\pi$
$419$			3.30064e12i			0.255581i	−0.991801	+	0.127790i	$0.959212\pi$
$420$		0			0
$421$	1.66504e13			1.25896			0.629482	−	0.777015i	$-0.283267\pi$
$421$	1.66504e13			1.25896			0.629482	+	0.777015i	$0.283267\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			1.03504e13i			0.746470i
$426$		0			0
$427$	1.28846e12	−	6.59424e12i	0.0907680	−	0.464543i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.20733e13			−1.48416			−0.742080	−	0.670311i	$-0.766161\pi$
$431$	−2.20733e13			−1.48416			−0.742080	+	0.670311i	$0.766161\pi$
$432$		0			0
$433$			2.76169e13i			1.81441i	0.420689	+	0.907205i	$0.361788\pi$
$433$			2.76169e13i			1.81441i	−0.420689	+	0.907205i	$0.638212\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			2.71028e13i			1.70062i
$438$		0			0
$439$		−	1.98594e12i		−	0.121799i	−0.998144	−	0.0608994i	$-0.980603\pi$
$439$		−	1.98594e12i		−	0.121799i	0.998144	−	0.0608994i	$-0.0193969\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.72296e13			−1.00985			−0.504923	−	0.863164i	$-0.668479\pi$
$443$	−1.72296e13			−1.00985			−0.504923	+	0.863164i	$0.668479\pi$
$444$		0			0
$445$	3.05073e13			1.74825
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.20485e12			0.0660241			0.0330121	−	0.999455i	$-0.489490\pi$
$449$	1.20485e12			0.0660241			0.0330121	+	0.999455i	$0.489490\pi$
$450$		0			0
$451$			9.04586e12i			0.484805i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.93433e13	+	9.64129e12i	2.53029	+	0.494400i
$456$		0			0
$457$	2.52799e13			1.26822			0.634110	−	0.773243i	$-0.281367\pi$
$457$	2.52799e13			1.26822			0.634110	+	0.773243i	$0.281367\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			2.91512e13i			1.40008i	0.714105	+	0.700038i	$0.246834\pi$
$461$			2.91512e13i			1.40008i	−0.714105	+	0.700038i	$0.753166\pi$
$462$		0			0
$463$	2.88971e13			1.35815			0.679077	−	0.734067i	$-0.262380\pi$
$463$	2.88971e13			1.35815			0.679077	+	0.734067i	$0.262380\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	2.12368e12i		−	0.0956101i	−0.998857	−	0.0478051i	$-0.984777\pi$
$467$		−	2.12368e12i		−	0.0956101i	0.998857	−	0.0478051i	$-0.0152226\pi$
$468$		0			0
$469$	1.72468e13	+	3.36988e12i	0.760053	+	0.148508i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.37644e12			0.100374
$474$		0			0
$475$		−	2.28236e13i		−	0.943879i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	1.84876e13i		−	0.733166i	−0.930385	−	0.366583i	$-0.880528\pi$
$479$		−	1.84876e13i		−	0.733166i	0.930385	−	0.366583i	$-0.119472\pi$
$480$		0			0
$481$		−	7.26792e13i		−	2.82283i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.85397e12			0.218143
$486$		0			0
$487$	−1.18488e13			−0.432542			−0.216271	−	0.976333i	$-0.569389\pi$
$487$	−1.18488e13			−0.432542			−0.216271	+	0.976333i	$0.569389\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.66976e13			−0.935544			−0.467772	−	0.883849i	$-0.654943\pi$
$491$	−2.66976e13			−0.935544			−0.467772	+	0.883849i	$0.654943\pi$
$492$		0			0
$493$			4.05386e13i			1.39198i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.69362e13	−	9.17097e12i	−1.54784	−	0.302436i
$498$		0			0
$499$	−3.36588e13			−1.08792			−0.543958	−	0.839112i	$-0.683075\pi$
$499$	−3.36588e13			−1.08792			−0.543958	+	0.839112i	$0.683075\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	2.61832e13i		−	0.813171i	−0.913613	−	0.406586i	$-0.866719\pi$
$503$		−	2.61832e13i		−	0.813171i	0.913613	−	0.406586i	$-0.133281\pi$
$504$		0			0
$505$	4.53480e12			0.138070
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	5.20495e12i		−	0.152345i	−0.997095	−	0.0761724i	$-0.975730\pi$
$509$		−	5.20495e12i		−	0.152345i	0.997095	−	0.0761724i	$-0.0242699\pi$
$510$		0			0
$511$	9.83897e12	−	5.03550e13i	0.282387	−	1.44523i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.00665e13			0.829941
$516$		0			0
$517$			2.02254e13i			0.547577i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	5.13135e13i		−	1.33673i	−0.743835	−	0.668363i	$-0.766995\pi$
$521$		−	5.13135e13i		−	1.33673i	0.743835	−	0.668363i	$-0.233005\pi$
$522$		0			0
$523$			5.38369e13i			1.37585i	0.725781	+	0.687926i	$0.241478\pi$
$523$			5.38369e13i			1.37585i	−0.725781	+	0.687926i	$0.758522\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.24448e13			−0.306151
$528$		0			0
$529$	6.95105e13			1.67792
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7.85484e13			1.82600
$534$		0			0
$535$			6.60672e13i			1.50736i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.08844e13	−	8.48523e12i	−0.459068	−	0.186518i
$540$		0			0
$541$	1.34240e13			0.289666			0.144833	−	0.989456i	$-0.453736\pi$
$541$	1.34240e13			0.289666			0.144833	+	0.989456i	$0.453736\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			5.67261e13i			1.17978i
$546$		0			0
$547$	−3.99623e13			−0.816044			−0.408022	−	0.912972i	$-0.633781\pi$
$547$	−3.99623e13			−0.816044			−0.408022	+	0.912972i	$0.633781\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	8.93915e13i		−	1.76010i
$552$		0			0
$553$	1.34917e13	+	2.63617e12i	0.260880	+	0.0509739i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.15693e13			−0.402309			−0.201155	−	0.979560i	$-0.564469\pi$
$557$	−2.15693e13			−0.402309			−0.201155	+	0.979560i	$0.564469\pi$
$558$		0			0
$559$		−	2.06354e13i		−	0.378055i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1.69542e13i		−	0.299734i	−0.988706	−	0.149867i	$-0.952115\pi$
$563$		−	1.69542e13i		−	0.299734i	0.988706	−	0.149867i	$-0.0478845\pi$
$564$		0			0
$565$		−	2.41097e13i		−	0.418745i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.68043e12			0.0784738			0.0392369	−	0.999230i	$-0.487507\pi$
$569$	4.68043e12			0.0784738			0.0392369	+	0.999230i	$0.487507\pi$
$570$		0			0
$571$	−5.06247e13			−0.834031			−0.417015	−	0.908899i	$-0.636924\pi$
$571$	−5.06247e13			−0.834031			−0.417015	+	0.908899i	$0.636924\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−9.34214e13			−1.48630
$576$		0			0
$577$		−	9.95580e13i		−	1.55667i	−0.627848	−	0.778336i	$-0.716064\pi$
$577$		−	9.95580e13i		−	1.55667i	0.627848	−	0.778336i	$-0.283936\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.95285e13	+	9.99451e13i	−0.294977	+	1.50967i
$582$		0			0
$583$	4.57592e13			0.679416
$584$		0			0
$585$		0			0
$586$		0			0
$587$			3.58217e13i			0.513992i	0.966413	+	0.256996i	$0.0827327\pi$
$587$			3.58217e13i			0.513992i	−0.966413	+	0.256996i	$0.917267\pi$
$588$		0			0
$589$	2.74420e13			0.387114
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6.72199e13i		−	0.916693i	−0.888773	−	0.458347i	$-0.848442\pi$
$593$		−	6.72199e13i		−	0.916693i	0.888773	−	0.458347i	$-0.151558\pi$
$594$		0			0
$595$	−8.30944e13	−	1.62360e13i	−1.11426	−	0.217718i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.09418e13			0.271569			0.135785	−	0.990738i	$-0.456645\pi$
$599$	2.09418e13			0.271569			0.135785	+	0.990738i	$0.456645\pi$
$600$		0			0
$601$		−	5.12511e13i		−	0.653628i	−0.945089	−	0.326814i	$-0.894025\pi$
$601$		−	5.12511e13i		−	0.653628i	0.945089	−	0.326814i	$-0.105975\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			8.44764e13i			1.04222i
$606$		0			0
$607$			1.21503e14i			1.47450i	0.675622	+	0.737248i	$0.263875\pi$
$607$			1.21503e14i			1.47450i	−0.675622	+	0.737248i	$0.736125\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.75625e14			2.06243
$612$		0			0
$613$	−4.77950e13			−0.552179			−0.276089	−	0.961132i	$-0.589039\pi$
$613$	−4.77950e13			−0.552179			−0.276089	+	0.961132i	$0.589039\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.87316e13			0.209483			0.104741	−	0.994499i	$-0.466599\pi$
$617$	1.87316e13			0.209483			0.104741	+	0.994499i	$0.466599\pi$
$618$		0			0
$619$		−	5.60492e13i		−	0.616760i	−0.951263	−	0.308380i	$-0.900213\pi$
$619$		−	5.60492e13i		−	0.616760i	0.951263	−	0.308380i	$-0.0997868\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.27770e13	+	1.16571e14i	−0.242693	+	1.24208i
$624$		0			0
$625$	−1.03314e14			−1.08333
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1.22392e14i			1.24308i
$630$		0			0
$631$	−1.09387e14			−1.09350			−0.546748	−	0.837297i	$-0.684135\pi$
$631$	−1.09387e14			−1.09350			−0.546748	+	0.837297i	$0.684135\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	1.34950e14i		−	1.30708i
$636$		0			0
$637$	−7.36803e13	+	1.81346e14i	−0.702512	+	1.72907i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.16112e13			−0.569337			−0.284669	−	0.958626i	$-0.591884\pi$
$641$	−6.16112e13			−0.569337			−0.284669	+	0.958626i	$0.591884\pi$
$642$		0			0
$643$			8.59309e13i			0.781797i	0.920434	+	0.390899i	$0.127836\pi$
$643$			8.59309e13i			0.781797i	−0.920434	+	0.390899i	$0.872164\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			9.41299e13i			0.830245i	0.909766	+	0.415122i	$0.136261\pi$
$647$			9.41299e13i			0.830245i	−0.909766	+	0.415122i	$0.863739\pi$
$648$		0			0
$649$			5.49712e13i			0.477432i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.50710e14			−1.26933			−0.634666	−	0.772787i	$-0.718862\pi$
$653$	−1.50710e14			−1.26933			−0.634666	+	0.772787i	$0.718862\pi$
$654$		0			0
$655$	−1.32122e14			−1.09589
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.28228e14			1.03170			0.515852	−	0.856677i	$-0.327475\pi$
$659$	1.28228e14			1.03170			0.515852	+	0.856677i	$0.327475\pi$
$660$		0			0
$661$		−	2.55398e13i		−	0.202400i	−0.994866	−	0.101200i	$-0.967732\pi$
$661$		−	2.55398e13i		−	0.202400i	0.994866	−	0.101200i	$-0.0322681\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.83231e14	+	3.58019e13i	1.40893	+	0.275295i
$666$		0			0
$667$	−3.65896e14			−2.77159
$668$		0			0
$669$		0			0
$670$		0			0
$671$			3.19028e13i			0.234540i
$672$		0			0
$673$	−8.60021e12			−0.0622922			−0.0311461	−	0.999515i	$-0.509916\pi$
$673$	−8.60021e12			−0.0622922			−0.0311461	+	0.999515i	$0.509916\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1.41454e14i		−	0.994654i	−0.867563	−	0.497327i	$-0.834315\pi$
$677$		−	1.41454e14i		−	0.994654i	0.867563	−	0.497327i	$-0.165685\pi$
$678$		0			0
$679$	−4.37063e12	+	2.23685e13i	−0.0302827	+	0.154984i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.33407e14			−1.57040			−0.785200	−	0.619242i	$-0.787440\pi$
$683$	−2.33407e14			−1.57040			−0.785200	+	0.619242i	$0.787440\pi$
$684$		0			0
$685$			4.72347e13i			0.313190i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	3.97343e14i		−	2.55900i
$690$		0			0
$691$			1.47505e13i			0.0936306i	0.998904	+	0.0468153i	$0.0149072\pi$
$691$			1.47505e13i			0.0936306i	−0.998904	+	0.0468153i	$0.985093\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.78924e14			−1.10343
$696$		0			0
$697$	−1.32276e14			−0.804113
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.15452e13			0.363583			0.181792	−	0.983337i	$-0.441810\pi$
$701$	6.15452e13			0.363583			0.181792	+	0.983337i	$0.441810\pi$
$702$		0			0
$703$		−	2.69886e14i		−	1.57182i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.38572e12	+	1.73278e13i	−0.0191670	+	0.0980949i
$708$		0			0
$709$	7.78708e13			0.434654			0.217327	−	0.976099i	$-0.430266\pi$
$709$	7.78708e13			0.434654			0.217327	+	0.976099i	$0.430266\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.12325e14i		−	0.609579i
$714$		0			0
$715$	−2.38722e14			−1.27750
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	3.45279e12i		−	0.0179691i	−0.999960	−	0.00898454i	$-0.997140\pi$
$719$		−	3.45279e12i		−	0.0179691i	0.999960	−	0.00898454i	$-0.00285991\pi$
$720$		0			0
$721$	−2.24479e13	+	1.14887e14i	−0.115213	+	0.589648i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.08126e14			1.53829
$726$		0			0
$727$		−	7.37074e13i		−	0.362943i	−0.983396	−	0.181472i	$-0.941914\pi$
$727$		−	7.37074e13i		−	0.362943i	0.983396	−	0.181472i	$-0.0580861\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			3.47502e13i			0.166483i
$732$		0			0
$733$		−	1.87586e14i		−	0.886502i	−0.896398	−	0.443251i	$-0.853825\pi$
$733$		−	1.87586e14i		−	0.886502i	0.896398	−	0.443251i	$-0.146175\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.34395e13			−0.383737
$738$		0			0
$739$	2.29593e14			1.04168			0.520842	−	0.853653i	$-0.325618\pi$
$739$	2.29593e14			1.04168			0.520842	+	0.853653i	$0.325618\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.57320e14			−0.694769			−0.347385	−	0.937723i	$-0.612930\pi$
$743$	−1.57320e14			−0.694769			−0.347385	+	0.937723i	$0.612930\pi$
$744$		0			0
$745$			3.02343e14i			1.31740i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.52448e14	−	4.93264e13i	−1.07093	−	0.209252i
$750$		0			0
$751$	−1.43017e14			−0.598670			−0.299335	−	0.954148i	$-0.596765\pi$
$751$	−1.43017e14			−0.598670			−0.299335	+	0.954148i	$0.596765\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			2.81175e14i			1.14615i
$756$		0			0
$757$	6.09091e12			0.0245021			0.0122510	−	0.999925i	$-0.496100\pi$
$757$	6.09091e12			0.0245021			0.0122510	+	0.999925i	$0.496100\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	3.03750e14i		−	1.19012i	−0.803679	−	0.595062i	$-0.797127\pi$
$761$		−	3.03750e14i		−	1.19012i	0.803679	−	0.595062i	$-0.202873\pi$
$762$		0			0
$763$	−2.16755e14	−	4.23522e13i	−0.838197	−	0.163777i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.77334e14			1.79823
$768$		0			0
$769$			1.49968e13i			0.0557657i	0.999611	+	0.0278829i	$0.00887654\pi$
$769$			1.49968e13i			0.0557657i	−0.999611	+	0.0278829i	$0.991123\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			7.54196e13i			0.273267i	0.990622	+	0.136633i	$0.0436282\pi$
$773$			7.54196e13i			0.273267i	−0.990622	+	0.136633i	$0.956372\pi$
$774$		0			0
$775$			9.45904e13i			0.338329i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.91681e14			1.01677
$780$		0			0
$781$	2.27076e14			0.781478
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.94328e14			−1.65831
$786$		0			0
$787$			4.78456e14i			1.58478i	0.610017	+	0.792388i	$0.291162\pi$
$787$			4.78456e14i			1.58478i	−0.610017	+	0.792388i	$0.708838\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.21251e13	+	1.80005e13i	0.297506	+	0.0581303i
$792$		0			0
$793$	2.77023e14			0.883385
$794$		0			0
$795$		0			0
$796$		0			0
$797$			3.28323e14i			1.02096i	0.859889	+	0.510481i	$0.170533\pi$
$797$			3.28323e14i			1.02096i	−0.859889	+	0.510481i	$0.829467\pi$
$798$		0			0
$799$	−2.95753e14			−0.908228
$800$		0			0
$801$		0			0
$802$		0			0
$803$			2.43616e14i			0.729673i
$804$		0			0
$805$	1.46544e14	−	7.49999e14i	0.433499	−	2.21861i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.19006e14			−0.343422			−0.171711	−	0.985147i	$-0.554929\pi$
$809$	−1.19006e14			−0.343422			−0.171711	+	0.985147i	$0.554929\pi$
$810$		0			0
$811$			4.31775e14i			1.23070i	0.788253	+	0.615351i	$0.210986\pi$
$811$			4.31775e14i			1.23070i	−0.788253	+	0.615351i	$0.789014\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			3.90572e14i			1.08621i
$816$		0			0
$817$		−	7.66275e13i		−	0.210511i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.47773e14			0.664260			0.332130	−	0.943234i	$-0.392233\pi$
$821$	2.47773e14			0.664260			0.332130	+	0.943234i	$0.392233\pi$
$822$		0			0
$823$	−3.29528e14			−0.872756			−0.436378	−	0.899763i	$-0.643739\pi$
$823$	−3.29528e14			−0.872756			−0.436378	+	0.899763i	$0.643739\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3.62897e14			0.938115			0.469057	−	0.883168i	$-0.344594\pi$
$827$	3.62897e14			0.938115			0.469057	+	0.883168i	$0.344594\pi$
$828$		0			0
$829$		−	2.25744e13i		−	0.0576558i	−0.999584	−	0.0288279i	$-0.990823\pi$
$829$		−	2.25744e13i		−	0.0576558i	0.999584	−	0.0288279i	$-0.00917747\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.24078e14	−	3.05388e14i	0.309364	−	0.761425i
$834$		0			0
$835$	−5.93168e14			−1.46132
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	4.03224e14i		−	0.969922i	−0.874536	−	0.484961i	$-0.838834\pi$
$839$		−	4.03224e14i		−	0.969922i	0.874536	−	0.484961i	$-0.161166\pi$
$840$		0			0
$841$	7.86105e14			1.86853
$842$		0			0
$843$		0			0
$844$		0			0
$845$			1.47779e15i			3.43027i
$846$		0			0
$847$	−3.22791e14	−	6.30709e13i	−0.740465	−	0.144681i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.10470e15			−2.47511
$852$		0			0
$853$		−	7.60728e14i		−	1.68455i	−0.539047	−	0.842276i	$-0.681216\pi$
$853$		−	7.60728e14i		−	1.68455i	0.539047	−	0.842276i	$-0.318784\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	3.98792e14i		−	0.862666i	−0.902193	−	0.431333i	$-0.858043\pi$
$857$		−	3.98792e14i		−	0.862666i	0.902193	−	0.431333i	$-0.141957\pi$
$858$		0			0
$859$		−	5.31473e14i		−	1.13636i	−0.822905	−	0.568179i	$-0.807648\pi$
$859$		−	5.31473e14i		−	1.13636i	0.822905	−	0.568179i	$-0.192352\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.16264e14			0.242880			0.121440	−	0.992599i	$-0.461249\pi$
$863$	1.16264e14			0.242880			0.121440	+	0.992599i	$0.461249\pi$
$864$		0			0
$865$	1.10365e15			2.27904
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.52725e13			−0.131714
$870$		0			0
$871$			7.24534e14i			1.44533i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.24655e13	−	6.37974e13i	0.0243036	−	0.124384i
$876$		0			0
$877$	4.91454e14			0.947294			0.473647	−	0.880715i	$-0.342937\pi$
$877$	4.91454e14			0.947294			0.473647	+	0.880715i	$0.342937\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	6.97609e14i		−	1.31441i	−0.753710	−	0.657207i	$-0.771738\pi$
$881$		−	6.97609e14i		−	1.31441i	0.753710	−	0.657207i	$-0.228262\pi$
$882$		0			0
$883$	9.51155e14			1.77194			0.885968	−	0.463747i	$-0.153495\pi$
$883$	9.51155e14			1.77194			0.885968	+	0.463747i	$0.153495\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	5.44972e14i		−	0.992559i	−0.868163	−	0.496280i	$-0.834699\pi$
$887$		−	5.44972e14i		−	0.992559i	0.868163	−	0.496280i	$-0.165301\pi$
$888$		0			0
$889$	5.15653e14	+	1.00755e14i	0.928643	+	0.181450i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.52163e14			1.14841
$894$		0			0
$895$			1.69365e14i			0.294922i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			3.70475e14i			0.630900i
$900$		0			0
$901$			6.69128e14i			1.12690i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.69485e13			0.0938079
$906$		0			0
$907$	1.72946e14			0.281757			0.140878	−	0.990027i	$-0.455007\pi$
$907$	1.72946e14			0.281757			0.140878	+	0.990027i	$0.455007\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6.50732e12			−0.0103708			−0.00518538	−	0.999987i	$-0.501651\pi$
$911$	−6.50732e12			−0.0103708			−0.00518538	+	0.999987i	$0.501651\pi$
$912$		0			0
$913$		−	4.83532e14i		−	0.762204i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.86433e13	−	5.04848e14i	0.152132	−	0.778599i
$918$		0			0
$919$	3.86017e14			0.588883			0.294442	−	0.955669i	$-0.404866\pi$
$919$	3.86017e14			0.588883			0.294442	+	0.955669i	$0.404866\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	1.97178e15i		−	2.94341i
$924$		0			0
$925$	9.30277e14			1.37374
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1.44933e14i			0.209454i	0.994501	+	0.104727i	$0.0333970\pi$
$929$			1.44933e14i			0.209454i	−0.994501	+	0.104727i	$0.966603\pi$
$930$		0			0
$931$	−2.73604e14	+	6.73410e14i	−0.391177	+	0.962789i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.02009e14			0.562571
$936$		0			0
$937$			2.11436e14i			0.292739i	0.989230	+	0.146369i	$0.0467588\pi$
$937$			2.11436e14i			0.292739i	−0.989230	+	0.146369i	$0.953241\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1.47440e14i			0.199833i	0.994996	+	0.0999167i	$0.0318576\pi$
$941$			1.47440e14i			0.199833i	−0.994996	+	0.0999167i	$0.968142\pi$
$942$		0			0
$943$		−	1.19391e15i		−	1.60107i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.47781e15			−1.94029			−0.970147	−	0.242518i	$-0.922027\pi$
$947$	−1.47781e15			−1.94029			−0.970147	+	0.242518i	$0.922027\pi$
$948$		0			0
$949$	2.11541e15			2.74829
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.57059e14			0.199801			0.0999006	−	0.994997i	$-0.468148\pi$
$953$	1.57059e14			0.199801			0.0999006	+	0.994997i	$0.468148\pi$
$954$		0			0
$955$		−	1.95603e15i		−	2.46240i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.80487e14	−	3.52658e13i	−0.222512	−	0.0434771i
$960$		0			0
$961$	7.05897e14			0.861241
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1.08554e15i			1.29721i
$966$		0			0
$967$	−1.29025e15			−1.52595			−0.762976	−	0.646427i	$-0.776262\pi$
$967$	−1.29025e15			−1.52595			−0.762976	+	0.646427i	$0.776262\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1.13702e15i			1.31726i	0.752467	+	0.658629i	$0.228864\pi$
$971$			1.13702e15i			1.31726i	−0.752467	+	0.658629i	$0.771136\pi$
$972$		0			0
$973$	1.33586e14	−	6.83684e14i	0.153179	−	0.783955i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.73267e14			−1.09335			−0.546675	−	0.837345i	$-0.684107\pi$
$977$	−9.73267e14			−1.09335			−0.546675	+	0.837345i	$0.684107\pi$
$978$		0			0
$979$		−	5.63967e14i		−	0.627105i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			3.79308e14i			0.413261i	0.978419	+	0.206630i	$0.0662498\pi$
$983$			3.79308e14i			0.413261i	−0.978419	+	0.206630i	$0.933750\pi$
$984$		0			0
$985$			1.04332e15i			1.12521i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.13651e14			−0.331486
$990$		0			0
$991$	8.28868e13			0.0867196			0.0433598	−	0.999060i	$-0.486194\pi$
$991$	8.28868e13			0.0867196			0.0433598	+	0.999060i	$0.486194\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.29632e15			2.35460
$996$		0			0
$997$			9.60604e13i			0.0975144i	0.998811	+	0.0487572i	$0.0155261\pi$
$997$			9.60604e13i			0.0975144i	−0.998811	+	0.0487572i	$0.984474\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.11.d.c.181.4	yes	12
3.2	odd	2	inner	252.11.d.c.181.10	yes	12
7.6	odd	2	inner	252.11.d.c.181.9	yes	12
21.20	even	2	inner	252.11.d.c.181.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.11.d.c.181.3	✓	12	21.20	even	2	inner
252.11.d.c.181.4	yes	12	1.1	even	1	trivial
252.11.d.c.181.9	yes	12	7.6	odd	2	inner
252.11.d.c.181.10	yes	12	3.2	odd	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	252.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4316.86i q^{5} +(16495.1 + 3223.01i) q^{7} +O(q^{10})\)	\(q-4316.86i q^{5} +(16495.1 + 3223.01i) q^{7} -79802.8 q^{11} +692956. i q^{13} -1.16694e6i q^{17} +2.57322e6i q^{19} +1.05327e7 q^{23} -8.86968e6 q^{25} -3.47392e7 q^{29} -1.06645e7i q^{31} +(1.39133e7 - 7.12070e7i) q^{35} -1.04883e8 q^{37} -1.13353e8i q^{41} -2.97789e7 q^{43} -2.53443e8i q^{47} +(2.61700e8 + 1.06328e8i) q^{49} -5.73403e8 q^{53} +3.44498e8i q^{55} -6.88838e8i q^{59} -3.99770e8i q^{61} +2.99139e9 q^{65} +1.04557e9 q^{67} -2.84547e9 q^{71} -3.05273e9i q^{73} +(-1.31635e9 - 2.57205e8i) q^{77} +8.17922e8 q^{79} +6.05909e9i q^{83} -5.03753e9 q^{85} +7.06701e9i q^{89} +(-2.23340e9 + 1.14304e10i) q^{91} +1.11082e10 q^{95} +1.35607e9i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 25788 q^{7}+O(q^{10}) \) 12 * q + 25788 * q^7	\( 12 q + 25788 q^{7} - 57218340 q^{25} + 179925528 q^{37} - 1040727816 q^{43} + 845565756 q^{49} + 8004513864 q^{67} - 10108183224 q^{79} - 11317946640 q^{85} + 6460110048 q^{91}+O(q^{100}) \) 12 * q + 25788 * q^7 - 57218340 * q^25 + 179925528 * q^37 - 1040727816 * q^43 + 845565756 * q^49 + 8004513864 * q^67 - 10108183224 * q^79 - 11317946640 * q^85 + 6460110048 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more