LMFDB - Embedded newform 252.12.k.e.37.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,12,Mod(37,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, 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:	$193.622481501$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.10
Character	$\chi$	$=$	252.37
Dual form			252.12.k.e.109.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2320.27 - 4018.82i) q^{5} +(9109.27 - 43524.1i) q^{7} +O(q^{10})$	$q+(2320.27 - 4018.82i) q^{5} +(9109.27 - 43524.1i) q^{7} +(322467. + 558529. i) q^{11} -127318. q^{13} +(-2.75116e6 - 4.76515e6i) q^{17} +(4.75013e6 - 8.22747e6i) q^{19} +(-8.14869e6 + 1.41139e7i) q^{23} +(1.36468e7 + 2.36369e7i) q^{25} -1.33602e8 q^{29} +(-2.72858e7 - 4.72604e7i) q^{31} +(-1.53780e8 - 1.37596e8i) q^{35} +(3.40400e8 - 5.89589e8i) q^{37} +1.24954e9 q^{41} +9.37396e8 q^{43} +(1.10271e9 - 1.90996e9i) q^{47} +(-1.81137e9 - 7.92945e8i) q^{49} +(5.16345e8 + 8.94335e8i) q^{53} +2.99284e9 q^{55} +(-2.72921e9 - 4.72712e9i) q^{59} +(-3.44994e9 + 5.97548e9i) q^{61} +(-2.95412e8 + 5.11668e8i) q^{65} +(-2.85881e9 - 4.95161e9i) q^{67} -1.68423e10 q^{71} +(1.34743e9 + 2.33381e9i) q^{73} +(2.72469e10 - 8.94729e9i) q^{77} +(-1.12173e10 + 1.94290e10i) q^{79} +1.24811e10 q^{83} -2.55337e10 q^{85} +(-2.39110e10 + 4.14150e10i) q^{89} +(-1.15977e9 + 5.54140e9i) q^{91} +(-2.20431e10 - 3.81798e10i) q^{95} +7.59317e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 85666 q^{7}+O(q^{10}) $ 28 * q + 85666 * q^7	$ 28 q + 85666 q^{7} + 2794700 q^{13} + 37414826 q^{19} - 135469250 q^{25} + 71052098 q^{31} - 395898874 q^{37} + 980170268 q^{43} - 1253333270 q^{49} - 8844657360 q^{55} + 16692685100 q^{61} + 10934009870 q^{67} + 25578690650 q^{73} - 113974174090 q^{79} + 111789733440 q^{85} - 267181531078 q^{91} + 279658007384 q^{97}+O(q^{100}) $ 28 * q + 85666 * q^7 + 2794700 * q^13 + 37414826 * q^19 - 135469250 * q^25 + 71052098 * q^31 - 395898874 * q^37 + 980170268 * q^43 - 1253333270 * q^49 - 8844657360 * q^55 + 16692685100 * q^61 + 10934009870 * q^67 + 25578690650 * q^73 - 113974174090 * q^79 + 111789733440 * q^85 - 267181531078 * q^91 + 279658007384 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	2320.27	−	4018.82i	0.332050	−	0.575127i	−0.650864	−	0.759194i	$-0.725593\pi$
$5$	2320.27	−	4018.82i	0.332050	−	0.575127i	0.982914	+	0.184068i	$0.0589265\pi$
$6$		0			0
$7$	9109.27	−	43524.1i	0.204854	−	0.978793i
$7$	9109.27	−	43524.1i	0.204854	−	0.978793i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	322467.	+	558529.i	0.603706	+	1.04565i	0.992255	+	0.124221i	$0.0396431\pi$
$11$	322467.	+	558529.i	0.603706	+	1.04565i	−0.388549	+	0.921428i	$0.627024\pi$
$12$		0			0
$13$	−127318.			−0.0951045			−0.0475523	−	0.998869i	$-0.515142\pi$
$13$	−127318.			−0.0951045			−0.0475523	+	0.998869i	$0.515142\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.75116e6	−	4.76515e6i	−0.469944	−	0.813968i	0.529465	−	0.848332i	$-0.322393\pi$
$17$	−2.75116e6	−	4.76515e6i	−0.469944	−	0.813968i	−0.999409	+	0.0343642i	$0.989059\pi$
$18$		0			0
$19$	4.75013e6	−	8.22747e6i	0.440109	−	0.762292i	−0.557588	−	0.830118i	$-0.688273\pi$
$19$	4.75013e6	−	8.22747e6i	0.440109	−	0.762292i	0.997697	+	0.0678261i	$0.0216063\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.14869e6	+	1.41139e7i	−0.263988	+	0.457241i	−0.967298	−	0.253642i	$-0.918371\pi$
$23$	−8.14869e6	+	1.41139e7i	−0.263988	+	0.457241i	0.703310	+	0.710883i	$0.251705\pi$
$24$		0			0
$25$	1.36468e7	+	2.36369e7i	0.279486	+	0.484084i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.33602e8			−1.20955			−0.604776	−	0.796395i	$-0.706738\pi$
$29$	−1.33602e8			−1.20955			−0.604776	+	0.796395i	$0.706738\pi$
$30$		0			0
$31$	−2.72858e7	−	4.72604e7i	−0.171178	−	0.296489i	0.767654	−	0.640865i	$-0.221424\pi$
$31$	−2.72858e7	−	4.72604e7i	−0.171178	−	0.296489i	−0.938832	+	0.344376i	$0.888091\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.53780e8	−	1.37596e8i	−0.494908	−	0.442825i
$36$		0			0
$37$	3.40400e8	−	5.89589e8i	0.807011	−	1.39778i	−0.107914	−	0.994160i	$-0.534417\pi$
$37$	3.40400e8	−	5.89589e8i	0.807011	−	1.39778i	0.914925	−	0.403624i	$-0.132250\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.24954e9			1.68438			0.842188	−	0.539184i	$-0.181267\pi$
$41$	1.24954e9			1.68438			0.842188	+	0.539184i	$0.181267\pi$
$42$		0			0
$43$	9.37396e8			0.972404			0.486202	−	0.873846i	$-0.338382\pi$
$43$	9.37396e8			0.972404			0.486202	+	0.873846i	$0.338382\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.10271e9	−	1.90996e9i	0.701333	−	1.21474i	−0.266665	−	0.963789i	$-0.585922\pi$
$47$	1.10271e9	−	1.90996e9i	0.701333	−	1.21474i	0.967999	−	0.250956i	$-0.0807448\pi$
$48$		0			0
$49$	−1.81137e9	−	7.92945e8i	−0.916070	−	0.401019i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.16345e8	+	8.94335e8i	0.169599	+	0.293753i	0.938279	−	0.345880i	$-0.112420\pi$
$53$	5.16345e8	+	8.94335e8i	0.169599	+	0.293753i	−0.768680	+	0.639633i	$0.779086\pi$
$54$		0			0
$55$	2.99284e9			0.801841
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.72921e9	−	4.72712e9i	−0.496993	−	0.860817i	0.503001	−	0.864286i	$-0.332229\pi$
$59$	−2.72921e9	−	4.72712e9i	−0.496993	−	0.860817i	−0.999994	+	0.00346882i	$0.998896\pi$
$60$		0			0
$61$	−3.44994e9	+	5.97548e9i	−0.522995	+	0.905855i	0.476647	+	0.879095i	$0.341852\pi$
$61$	−3.44994e9	+	5.97548e9i	−0.522995	+	0.905855i	−0.999642	+	0.0267595i	$0.991481\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.95412e8	+	5.11668e8i	−0.0315794	+	0.0546972i
$66$		0			0
$67$	−2.85881e9	−	4.95161e9i	−0.258687	−	0.448059i	0.707204	−	0.707010i	$-0.249956\pi$
$67$	−2.85881e9	−	4.95161e9i	−0.258687	−	0.448059i	−0.965890	+	0.258951i	$0.916623\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.68423e10			−1.10785			−0.553923	−	0.832568i	$-0.686870\pi$
$71$	−1.68423e10			−1.10785			−0.553923	+	0.832568i	$0.686870\pi$
$72$		0			0
$73$	1.34743e9	+	2.33381e9i	0.0760727	+	0.131762i	0.901552	−	0.432670i	$-0.142429\pi$
$73$	1.34743e9	+	2.33381e9i	0.0760727	+	0.131762i	−0.825480	+	0.564432i	$0.809095\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.72469e10	−	8.94729e9i	1.14714	−	0.376697i
$78$		0			0
$79$	−1.12173e10	+	1.94290e10i	−0.410148	+	0.710397i	−0.994906	−	0.100811i	$-0.967856\pi$
$79$	−1.12173e10	+	1.94290e10i	−0.410148	+	0.710397i	0.584758	+	0.811208i	$0.301190\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.24811e10			0.347794			0.173897	−	0.984764i	$-0.444364\pi$
$83$	1.24811e10			0.347794			0.173897	+	0.984764i	$0.444364\pi$
$84$		0			0
$85$	−2.55337e10			−0.624180
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.39110e10	+	4.14150e10i	−0.453892	+	0.786164i	−0.998624	−	0.0524464i	$-0.983298\pi$
$89$	−2.39110e10	+	4.14150e10i	−0.453892	+	0.786164i	0.544732	+	0.838610i	$0.316631\pi$
$90$		0			0
$91$	−1.15977e9	+	5.54140e9i	−0.0194825	+	0.0930876i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.20431e10	−	3.81798e10i	−0.292276	−	0.506238i
$96$		0			0
$97$	7.59317e10			0.897798			0.448899	−	0.893583i	$-0.351816\pi$
$97$	7.59317e10			0.897798			0.448899	+	0.893583i	$0.351816\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.90880e10	+	1.36984e11i	0.748760	+	1.29689i	0.948417	+	0.317025i	$0.102684\pi$
$101$	7.90880e10	+	1.36984e11i	0.748760	+	1.29689i	−0.199657	+	0.979866i	$0.563983\pi$
$102$		0			0
$103$	6.60053e10	−	1.14324e11i	0.561014	−	0.971705i	−0.436394	−	0.899756i	$-0.643745\pi$
$103$	6.60053e10	−	1.14324e11i	0.561014	−	0.971705i	0.997408	−	0.0719493i	$-0.0229220\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.11173e10	−	5.38967e10i	0.214482	−	0.371494i	−0.738630	−	0.674111i	$-0.764527\pi$
$107$	3.11173e10	−	5.38967e10i	0.214482	−	0.371494i	0.953112	+	0.302617i	$0.0978603\pi$
$108$		0			0
$109$	−9.26558e10	−	1.60485e11i	−0.576802	−	0.999051i	−0.995843	−	0.0910837i	$-0.970967\pi$
$109$	−9.26558e10	−	1.60485e11i	−0.576802	−	0.999051i	0.419041	−	0.907967i	$-0.362366\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.22692e11			−1.13703			−0.568517	−	0.822672i	$-0.692482\pi$
$113$	−2.22692e11			−1.13703			−0.568517	+	0.822672i	$0.692482\pi$
$114$		0			0
$115$	3.78143e10	+	6.54962e10i	0.175314	+	0.303653i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.32460e11	+	7.63347e10i	−0.892975	+	0.293234i
$120$		0			0
$121$	−6.53139e10	+	1.13127e11i	−0.228921	+	0.396503i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.53245e11			1.03531
$126$		0			0
$127$	2.61295e11			0.701796			0.350898	−	0.936414i	$-0.385876\pi$
$127$	2.61295e11			0.701796			0.350898	+	0.936414i	$0.385876\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.90296e10	+	1.19563e11i	−0.156330	+	0.270772i	−0.933543	−	0.358466i	$-0.883300\pi$
$131$	−6.90296e10	+	1.19563e11i	−0.156330	+	0.270772i	0.777212	+	0.629238i	$0.216633\pi$
$132$		0			0
$133$	−3.14823e11	−	2.81691e11i	−0.655968	−	0.586934i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.58431e10	−	1.48685e11i	−0.151965	−	0.263210i	0.779985	−	0.625798i	$-0.215227\pi$
$137$	−8.58431e10	−	1.48685e11i	−0.151965	−	0.263210i	−0.931950	+	0.362588i	$0.881893\pi$
$138$		0			0
$139$	−4.77688e11			−0.780842			−0.390421	−	0.920637i	$-0.627670\pi$
$139$	−4.77688e11			−0.780842			−0.390421	+	0.920637i	$0.627670\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.10558e10	−	7.11108e10i	−0.0574152	−	0.0994460i
$144$		0			0
$145$	−3.09993e11	+	5.36924e11i	−0.401632	+	0.695646i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.10014e11	−	1.40299e12i	0.903583	−	1.56505i	0.0807754	−	0.996732i	$-0.474260\pi$
$149$	8.10014e11	−	1.40299e12i	0.903583	−	1.56505i	0.822808	−	0.568320i	$-0.192406\pi$
$150$		0			0
$151$	1.93019e11	+	3.34319e11i	0.200091	+	0.346567i	0.948557	−	0.316605i	$-0.102543\pi$
$151$	1.93019e11	+	3.34319e11i	0.200091	+	0.346567i	−0.748467	+	0.663172i	$0.769210\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.53242e11			−0.227358
$156$		0			0
$157$	−6.30736e11	−	1.09247e12i	−0.527715	−	0.914029i	−0.999478	−	0.0323038i	$-0.989716\pi$
$157$	−6.30736e11	−	1.09247e12i	−0.527715	−	0.914029i	0.471763	−	0.881725i	$-0.343618\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.40068e11	+	4.83232e11i	0.393465	+	0.352057i
$162$		0			0
$163$	3.65208e11	−	6.32558e11i	0.248604	−	0.430595i	−0.714535	−	0.699600i	$-0.753362\pi$
$163$	3.65208e11	−	6.32558e11i	0.248604	−	0.430595i	0.963139	+	0.269005i	$0.0866949\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.26429e12			1.34893			0.674467	−	0.738305i	$-0.264374\pi$
$167$	2.26429e12			1.34893			0.674467	+	0.738305i	$0.264374\pi$
$168$		0			0
$169$	−1.77595e12			−0.990955
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.56105e12	+	2.70382e12i	−0.765886	+	1.32655i	0.173891	+	0.984765i	$0.444366\pi$
$173$	−1.56105e12	+	2.70382e12i	−0.765886	+	1.32655i	−0.939777	+	0.341789i	$0.888967\pi$
$174$		0			0
$175$	1.15309e12	−	3.78649e11i	0.531072	−	0.174392i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.02008e12	−	3.49888e12i	−0.821631	−	1.42311i	−0.904467	−	0.426543i	$-0.859731\pi$
$179$	−2.02008e12	−	3.49888e12i	−0.821631	−	1.42311i	0.0828362	−	0.996563i	$-0.473602\pi$
$180$		0			0
$181$	−1.15611e12			−0.442353			−0.221176	−	0.975234i	$-0.570990\pi$
$181$	−1.15611e12			−0.442353			−0.221176	+	0.975234i	$0.570990\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.57964e12	−	2.73601e12i	−0.535936	−	0.928268i
$186$		0			0
$187$	1.77431e12	−	3.07320e12i	0.567416	−	0.982794i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.06349e12	+	5.30613e12i	−0.872034	+	1.51041i	−0.0121448	+	0.999926i	$0.503866\pi$
$191$	−3.06349e12	+	5.30613e12i	−0.872034	+	1.51041i	−0.859889	+	0.510481i	$0.829467\pi$
$192$		0			0
$193$	−2.79480e12	−	4.84073e12i	−0.751251	−	1.30120i	−0.947217	−	0.320594i	$-0.896118\pi$
$193$	−2.79480e12	−	4.84073e12i	−0.751251	−	1.30120i	0.195966	−	0.980611i	$-0.437216\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.04399e11			0.217168			0.108584	−	0.994087i	$-0.465368\pi$
$197$	9.04399e11			0.217168			0.108584	+	0.994087i	$0.465368\pi$
$198$		0			0
$199$	−1.37183e12	−	2.37608e12i	−0.311609	−	0.539722i	0.667102	−	0.744966i	$-0.267534\pi$
$199$	−1.37183e12	−	2.37608e12i	−0.311609	−	0.539722i	−0.978711	+	0.205244i	$0.934201\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.21702e12	+	5.81492e12i	−0.247782	+	1.18390i
$204$		0			0
$205$	2.89927e12	−	5.02168e12i	0.559297	−	0.968730i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.12704e12			1.06279
$210$		0			0
$211$	−4.07099e12			−0.670111			−0.335056	−	0.942198i	$-0.608755\pi$
$211$	−4.07099e12			−0.670111			−0.335056	+	0.942198i	$0.608755\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.17501e12	−	3.76723e12i	0.322886	−	0.559256i
$216$		0			0
$217$	−2.30552e12	+	7.57084e11i	−0.325268	+	0.106811i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.50272e11	+	6.06689e11i	0.0446939	+	0.0774120i
$222$		0			0
$223$	−1.08157e13			−1.31334			−0.656672	−	0.754177i	$-0.728036\pi$
$223$	−1.08157e13			−1.31334			−0.656672	+	0.754177i	$0.728036\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.80714e12	+	3.13005e12i	0.198998	+	0.344675i	0.948204	−	0.317663i	$-0.102898\pi$
$227$	1.80714e12	+	3.13005e12i	0.198998	+	0.344675i	−0.749206	+	0.662337i	$0.769565\pi$
$228$		0			0
$229$	−2.52749e12	+	4.37774e12i	−0.265212	+	0.459361i	−0.967619	−	0.252414i	$-0.918775\pi$
$229$	−2.52749e12	+	4.37774e12i	−0.265212	+	0.459361i	0.702407	+	0.711776i	$0.252109\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.66684e12	−	8.08321e12i	0.445211	−	0.771127i	−0.552856	−	0.833277i	$-0.686462\pi$
$233$	4.66684e12	−	8.08321e12i	0.445211	−	0.771127i	0.998067	+	0.0621492i	$0.0197955\pi$
$234$		0			0
$235$	−5.11718e12	−	8.86321e12i	−0.465755	−	0.806711i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.14897e12			−0.261204			−0.130602	−	0.991435i	$-0.541691\pi$
$239$	−3.14897e12			−0.261204			−0.130602	+	0.991435i	$0.541691\pi$
$240$		0			0
$241$	−1.14507e13	−	1.98332e13i	−0.907275	−	1.57145i	−0.817834	−	0.575454i	$-0.804825\pi$
$241$	−1.14507e13	−	1.98332e13i	−0.907275	−	1.57145i	−0.0894407	−	0.995992i	$-0.528508\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.38957e12	+	5.43972e12i	−0.534817	+	0.393698i
$246$		0			0
$247$	−6.04777e11	+	1.04750e12i	−0.0418564	+	0.0724974i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.96124e12			−0.631114			−0.315557	−	0.948907i	$-0.602191\pi$
$251$	−9.96124e12			−0.631114			−0.315557	+	0.948907i	$0.602191\pi$
$252$		0			0
$253$	−1.05107e13			−0.637485
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.97326e12	−	5.14983e12i	0.165425	−	0.286524i	−0.771381	−	0.636373i	$-0.780434\pi$
$257$	2.97326e12	−	5.14983e12i	0.165425	−	0.286524i	0.936806	+	0.349849i	$0.113767\pi$
$258$		0			0
$259$	−2.25606e13	−	2.01863e13i	−1.20282	−	1.07624i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.88108e12	−	1.71145e13i	−0.484226	−	0.838704i	0.515610	−	0.856823i	$-0.327565\pi$
$263$	−9.88108e12	−	1.71145e13i	−0.484226	−	0.838704i	−0.999836	+	0.0181196i	$0.994232\pi$
$264$		0			0
$265$	4.79223e12			0.225261
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.57182e12	+	6.18657e12i	0.154615	+	0.267801i	0.932919	−	0.360087i	$-0.117253\pi$
$269$	3.57182e12	+	6.18657e12i	0.154615	+	0.267801i	−0.778304	+	0.627888i	$0.783920\pi$
$270$		0			0
$271$	1.44098e13	−	2.49584e13i	0.598861	−	1.03726i	−0.394129	−	0.919055i	$-0.628954\pi$
$271$	1.44098e13	−	2.49584e13i	0.598861	−	1.03726i	0.992990	−	0.118202i	$-0.0377130\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.80127e12	+	1.52442e13i	−0.337455	+	0.584489i
$276$		0			0
$277$	−1.97902e13	−	3.42777e13i	−0.729142	−	1.26291i	−0.957246	−	0.289274i	$-0.906586\pi$
$277$	−1.97902e13	−	3.42777e13i	−0.729142	−	1.26291i	0.228105	−	0.973637i	$-0.426747\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.76402e13			0.941145			0.470573	−	0.882361i	$-0.344047\pi$
$281$	2.76402e13			0.941145			0.470573	+	0.882361i	$0.344047\pi$
$282$		0			0
$283$	−1.77676e13	−	3.07743e13i	−0.581838	−	1.00777i	−0.995261	−	0.0972347i	$-0.969000\pi$
$283$	−1.77676e13	−	3.07743e13i	−0.581838	−	1.00777i	0.413423	−	0.910539i	$-0.364333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.13824e13	−	5.43851e13i	0.345051	−	1.64866i
$288$		0			0
$289$	1.99820e12	−	3.46099e12i	0.0583044	−	0.100986i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.34334e13			0.633962			0.316981	−	0.948432i	$-0.397331\pi$
$293$	2.34334e13			0.633962			0.316981	+	0.948432i	$0.397331\pi$
$294$		0			0
$295$	−2.53299e13			−0.660105
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.03747e12	−	1.79696e12i	0.0251065	−	0.0434857i
$300$		0			0
$301$	8.53899e12	−	4.07993e13i	0.199201	−	0.951782i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.60096e13	+	2.77294e13i	0.347321	+	0.601577i
$306$		0			0
$307$	−6.19514e13			−1.29655			−0.648276	−	0.761405i	$-0.724510\pi$
$307$	−6.19514e13			−1.29655			−0.648276	+	0.761405i	$0.724510\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.88766e13	+	3.26953e13i	0.367910	+	0.637239i	0.989239	−	0.146311i	$-0.0467400\pi$
$311$	1.88766e13	+	3.26953e13i	0.367910	+	0.637239i	−0.621328	+	0.783550i	$0.713407\pi$
$312$		0			0
$313$	2.66362e13	−	4.61353e13i	0.501163	−	0.868039i	−0.498836	−	0.866696i	$-0.666239\pi$
$313$	2.66362e13	−	4.61353e13i	0.501163	−	0.868039i	0.999999	−	0.00134323i	$-0.000427564\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.94358e13	+	6.83049e13i	−0.691935	+	1.19847i	0.279268	+	0.960213i	$0.409908\pi$
$317$	−3.94358e13	+	6.83049e13i	−0.691935	+	1.19847i	−0.971203	+	0.238253i	$0.923425\pi$
$318$		0			0
$319$	−4.30823e13	−	7.46207e13i	−0.730214	−	1.26477i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.22734e13			−0.827308
$324$		0			0
$325$	−1.73748e12	−	3.00940e12i	−0.0265804	−	0.0460386i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.30842e13	−	6.53929e13i	−1.04531	−	0.935305i
$330$		0			0
$331$	1.99776e13	−	3.46022e13i	0.276369	−	0.478685i	−0.694110	−	0.719869i	$-0.744202\pi$
$331$	1.99776e13	−	3.46022e13i	0.276369	−	0.478685i	0.970480	+	0.241183i	$0.0775354\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.65328e13			−0.343587
$336$		0			0
$337$	−1.05523e14			−1.32246			−0.661229	−	0.750184i	$-0.729965\pi$
$337$	−1.05523e14			−1.32246			−0.661229	+	0.750184i	$0.729965\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.75976e13	−	3.04799e13i	0.206682	−	0.357984i
$342$		0			0
$343$	−5.10125e13	+	7.16151e13i	−0.580175	+	0.814492i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.99461e13	+	1.03830e14i	0.639660	+	1.10792i	0.985507	+	0.169633i	$0.0542581\pi$
$347$	5.99461e13	+	1.03830e14i	0.639660	+	1.10792i	−0.345847	+	0.938291i	$0.612409\pi$
$348$		0			0
$349$	−5.55470e13			−0.574276			−0.287138	−	0.957889i	$-0.592704\pi$
$349$	−5.55470e13			−0.574276			−0.287138	+	0.957889i	$0.592704\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.76714e12	+	3.06077e12i	0.0171597	+	0.0297214i	0.874478	−	0.485066i	$-0.161204\pi$
$353$	1.76714e12	+	3.06077e12i	0.0171597	+	0.0297214i	−0.857318	+	0.514787i	$0.827871\pi$
$354$		0			0
$355$	−3.90785e13	+	6.76860e13i	−0.367860	+	0.637152i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.20064e13	+	3.81163e13i	−0.194774	+	0.337358i	−0.946826	−	0.321745i	$-0.895731\pi$
$359$	−2.20064e13	+	3.81163e13i	−0.194774	+	0.337358i	0.752053	+	0.659103i	$0.229064\pi$
$360$		0			0
$361$	1.31177e13	+	2.27205e13i	0.112607	+	0.195042i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.25055e13			0.101040
$366$		0			0
$367$	−1.01544e14	−	1.75880e14i	−0.796145	−	1.37896i	−0.922109	−	0.386929i	$-0.873536\pi$
$367$	−1.01544e14	−	1.75880e14i	−0.796145	−	1.37896i	0.125964	−	0.992035i	$-0.459798\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.36287e13	−	1.43267e13i	0.322267	−	0.105825i
$372$		0			0
$373$	3.26136e13	−	5.64884e13i	0.233884	−	0.405099i	−0.725064	−	0.688682i	$-0.758190\pi$
$373$	3.26136e13	−	5.64884e13i	0.233884	−	0.405099i	0.958948	+	0.283583i	$0.0915231\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.70100e13			0.115034
$378$		0			0
$379$	1.46191e14			0.960296			0.480148	−	0.877187i	$-0.340583\pi$
$379$	1.46191e14			0.960296			0.480148	+	0.877187i	$0.340583\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.53387e13	−	1.30490e14i	0.467116	−	0.809069i	−0.532178	−	0.846632i	$-0.678626\pi$
$383$	7.53387e13	−	1.30490e14i	0.467116	−	0.809069i	0.999294	+	0.0375637i	$0.0119597\pi$
$384$		0			0
$385$	2.72625e13	−	1.30261e14i	0.164260	−	0.784836i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.43466e14	−	2.48490e14i	−0.816632	−	1.41445i	−0.908150	−	0.418644i	$-0.862505\pi$
$389$	−1.43466e14	−	2.48490e14i	−0.816632	−	1.41445i	0.0915188	−	0.995803i	$-0.470828\pi$
$390$		0			0
$391$	8.96733e13			0.496239
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.20544e13	+	9.01609e13i	0.272379	+	0.471774i
$396$		0			0
$397$	−2.76686e13	+	4.79234e13i	−0.140812	+	0.243893i	−0.927803	−	0.373071i	$-0.878305\pi$
$397$	−2.76686e13	+	4.79234e13i	−0.140812	+	0.243893i	0.786991	+	0.616965i	$0.211638\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.15469e13	+	8.92819e13i	−0.248261	+	0.430001i	−0.963043	−	0.269346i	$-0.913192\pi$
$401$	−5.15469e13	+	8.92819e13i	−0.248261	+	0.430001i	0.714782	+	0.699347i	$0.246526\pi$
$402$		0			0
$403$	3.47398e12	+	6.01710e12i	0.0162798	+	0.0281974i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.39070e14			1.94879
$408$		0			0
$409$	1.40043e13	+	2.42561e13i	0.0605037	+	0.104795i	0.894691	−	0.446686i	$-0.147396\pi$
$409$	1.40043e13	+	2.42561e13i	0.0605037	+	0.104795i	−0.834187	+	0.551482i	$0.814063\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.30605e14	+	7.57256e13i	−0.944372	+	0.310111i
$414$		0			0
$415$	2.89594e13	−	5.01592e13i	0.115485	−	0.200026i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.10039e14			0.416264			0.208132	−	0.978101i	$-0.433262\pi$
$419$	1.10039e14			0.416264			0.208132	+	0.978101i	$0.433262\pi$
$420$		0			0
$421$	−3.77399e13			−0.139075			−0.0695376	−	0.997579i	$-0.522152\pi$
$421$	−3.77399e13			−0.139075			−0.0695376	+	0.997579i	$0.522152\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.50889e13	−	1.30058e14i	0.262686	−	0.454985i
$426$		0			0
$427$	2.28651e14	+	2.04588e14i	0.779506	+	0.697472i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.72060e14	−	4.71221e14i	−0.881129	−	1.52616i	−0.850088	−	0.526641i	$-0.823451\pi$
$431$	−2.72060e14	−	4.71221e14i	−0.881129	−	1.52616i	−0.0310411	−	0.999518i	$-0.509882\pi$
$432$		0			0
$433$	−6.75598e13			−0.213307			−0.106654	−	0.994296i	$-0.534014\pi$
$433$	−6.75598e13			−0.213307			−0.106654	+	0.994296i	$0.534014\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.74147e13	+	1.34086e14i	0.232367	+	0.402472i
$438$		0			0
$439$	5.53938e13	−	9.59450e13i	0.162146	−	0.280845i	−0.773492	−	0.633806i	$-0.781492\pi$
$439$	5.53938e13	−	9.59450e13i	0.162146	−	0.280845i	0.935638	+	0.352961i	$0.114825\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.13322e14	+	3.69484e14i	−0.594038	+	1.02890i	0.399644	+	0.916671i	$0.369134\pi$
$443$	−2.13322e14	+	3.69484e14i	−0.594038	+	1.02890i	−0.993682	+	0.112234i	$0.964199\pi$
$444$		0			0
$445$	1.10960e14	+	1.92188e14i	0.301429	+	0.522091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4.89336e14			−1.26547			−0.632736	−	0.774368i	$-0.718068\pi$
$449$	−4.89336e14			−1.26547			−0.632736	+	0.774368i	$0.718068\pi$
$450$		0			0
$451$	4.02935e14	+	6.97905e14i	1.01687	+	1.76127i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.95789e13	+	1.75185e13i	0.0470680	+	0.0421146i
$456$		0			0
$457$	−1.68285e14	+	2.91478e14i	−0.394918	+	0.684017i	−0.993091	−	0.117350i	$-0.962560\pi$
$457$	−1.68285e14	+	2.91478e14i	−0.394918	+	0.684017i	0.598173	+	0.801367i	$0.295894\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.72184e14			1.27991			0.639957	−	0.768411i	$-0.278952\pi$
$461$	5.72184e14			1.27991			0.639957	+	0.768411i	$0.278952\pi$
$462$		0			0
$463$	9.65176e13			0.210820			0.105410	−	0.994429i	$-0.466385\pi$
$463$	9.65176e13			0.210820			0.105410	+	0.994429i	$0.466385\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.10007e14	+	5.36947e14i	−0.645845	+	1.11864i	0.338261	+	0.941052i	$0.390161\pi$
$467$	−3.10007e14	+	5.36947e14i	−0.645845	+	1.11864i	−0.984106	+	0.177584i	$0.943172\pi$
$468$		0			0
$469$	−2.41556e14	+	7.93218e13i	−0.491550	+	0.161414i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.02279e14	+	5.23563e14i	0.587046	+	1.01679i
$474$		0			0
$475$	2.59296e14			0.492018
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.22297e14	−	7.31440e14i	−0.765196	−	1.32536i	−0.940143	−	0.340780i	$-0.889309\pi$
$479$	−4.22297e14	−	7.31440e14i	−0.765196	−	1.32536i	0.174947	−	0.984578i	$-0.444024\pi$
$480$		0			0
$481$	−4.33390e13	+	7.50653e13i	−0.0767504	+	0.132936i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.76182e14	−	3.05156e14i	0.298113	−	0.516348i
$486$		0			0
$487$	3.07884e14	+	5.33271e14i	0.509306	+	0.882143i	0.999942	+	0.0107787i	$0.00343102\pi$
$487$	3.07884e14	+	5.33271e14i	0.509306	+	0.882143i	−0.490636	+	0.871364i	$0.663236\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7.95668e13			−0.125830			−0.0629149	−	0.998019i	$-0.520040\pi$
$491$	−7.95668e13			−0.125830			−0.0629149	+	0.998019i	$0.520040\pi$
$492$		0			0
$493$	3.67561e14	+	6.36634e14i	0.568423	+	0.984537i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.53421e14	+	7.33044e14i	−0.226947	+	1.08435i
$498$		0			0
$499$	−5.37349e14	+	9.30716e14i	−0.777506	+	1.34668i	0.155870	+	0.987778i	$0.450182\pi$
$499$	−5.37349e14	+	9.30716e14i	−0.777506	+	1.34668i	−0.933375	+	0.358901i	$0.883151\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.11141e15			1.53904			0.769520	−	0.638623i	$-0.220496\pi$
$503$	1.11141e15			1.53904			0.769520	+	0.638623i	$0.220496\pi$
$504$		0			0
$505$	7.34021e14			0.994502
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.40003e13	−	1.28172e14i	0.0960033	−	0.166283i	−0.814024	−	0.580832i	$-0.802727\pi$
$509$	7.40003e13	−	1.28172e14i	0.0960033	−	0.166283i	0.910027	+	0.414549i	$0.136061\pi$
$510$		0			0
$511$	1.13851e14	−	3.73862e13i	0.144551	−	0.0474675i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.06300e14	−	5.30527e14i	−0.372569	−	0.645309i
$516$		0			0
$517$	1.42235e15			1.69360
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.24621e14	+	5.62261e14i	0.370484	+	0.641697i	0.989640	−	0.143571i	$-0.0458585\pi$
$521$	3.24621e14	+	5.62261e14i	0.370484	+	0.641697i	−0.619156	+	0.785268i	$0.712525\pi$
$522$		0			0
$523$	−2.13434e14	+	3.69679e14i	−0.238509	+	0.413110i	−0.960287	−	0.279015i	$-0.909992\pi$
$523$	−2.13434e14	+	3.69679e14i	−0.238509	+	0.413110i	0.721778	+	0.692125i	$0.243325\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.50135e14	+	2.60042e14i	−0.160888	+	0.278667i
$528$		0			0
$529$	3.43603e14	+	5.95137e14i	0.360620	+	0.624613i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.59089e14			−0.160192
$534$		0			0
$535$	−1.44401e14	−	2.50110e14i	−0.142437	−	0.246709i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.41224e14	−	1.26740e15i	−0.133712	−	1.19998i
$540$		0			0
$541$	6.27342e14	−	1.08659e15i	0.581995	−	1.00804i	−0.413248	−	0.910619i	$-0.635606\pi$
$541$	6.27342e14	−	1.08659e15i	0.581995	−	1.00804i	0.995243	−	0.0974263i	$-0.0310610\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.59945e14			−0.766108
$546$		0			0
$547$	1.21600e15			1.06170			0.530850	−	0.847466i	$-0.321873\pi$
$547$	1.21600e15			1.06170			0.530850	+	0.847466i	$0.321873\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.34628e14	+	1.09921e15i	−0.532336	+	0.922032i
$552$		0			0
$553$	7.43448e14	+	6.65208e14i	0.611311	+	0.546977i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.66709e14	−	1.15477e15i	−0.526906	−	0.912628i	−0.999508	−	0.0313521i	$-0.990019\pi$
$557$	−6.66709e14	−	1.15477e15i	−0.526906	−	0.912628i	0.472603	−	0.881276i	$-0.343315\pi$
$558$		0			0
$559$	−1.19347e14			−0.0924800
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.95512e14	+	1.03146e15i	0.443705	+	0.768519i	0.997961	−	0.0638268i	$-0.0203305\pi$
$563$	5.95512e14	+	1.03146e15i	0.443705	+	0.768519i	−0.554256	+	0.832346i	$0.686997\pi$
$564$		0			0
$565$	−5.16705e14	+	8.94959e14i	−0.377551	+	0.653938i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.01277e14	−	1.38785e15i	0.563203	−	0.975497i	−0.434011	−	0.900908i	$-0.642902\pi$
$569$	8.01277e14	−	1.38785e15i	0.563203	−	0.975497i	0.997214	−	0.0745892i	$-0.0237646\pi$
$570$		0			0
$571$	5.56886e14	+	9.64556e14i	0.383944	+	0.665011i	0.991622	−	0.129173i	$-0.0412322\pi$
$571$	5.56886e14	+	9.64556e14i	0.383944	+	0.665011i	−0.607678	+	0.794184i	$0.707899\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.44813e14			−0.295124
$576$		0			0
$577$	5.04736e14	+	8.74228e14i	0.328547	+	0.569060i	0.982224	−	0.187714i	$-0.0601078\pi$
$577$	5.04736e14	+	8.74228e14i	0.328547	+	0.569060i	−0.653677	+	0.756774i	$0.726774\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.13693e14	−	5.43228e14i	0.0712470	−	0.340418i
$582$		0			0
$583$	−3.33008e14	+	5.76787e14i	−0.204775	+	0.354681i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.21003e13			0.0545446			0.0272723	−	0.999628i	$-0.491318\pi$
$587$	9.21003e13			0.0545446			0.0272723	+	0.999628i	$0.491318\pi$
$588$		0			0
$589$	−5.18445e14			−0.301348
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.15431e15	−	1.99932e15i	0.646428	−	1.11965i	−0.337542	−	0.941310i	$-0.609596\pi$
$593$	1.15431e15	−	1.99932e15i	0.646428	−	1.11965i	0.983970	−	0.178335i	$-0.0570711\pi$
$594$		0			0
$595$	−2.32593e14	+	1.11133e15i	−0.127866	+	0.610942i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.26122e14	+	1.43089e15i	0.437720	+	0.758154i	0.997513	−	0.0704788i	$-0.0224527\pi$
$599$	8.26122e14	+	1.43089e15i	0.437720	+	0.758154i	−0.559793	+	0.828632i	$0.689119\pi$
$600$		0			0
$601$	1.20170e13			0.00625156			0.00312578	−	0.999995i	$-0.499005\pi$
$601$	1.20170e13			0.00625156			0.00312578	+	0.999995i	$0.499005\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.03091e14	+	5.24970e14i	0.152026	+	0.263317i
$606$		0			0
$607$	8.21676e13	−	1.42318e14i	0.0404728	−	0.0701009i	−0.845079	−	0.534641i	$-0.820447\pi$
$607$	8.21676e13	−	1.42318e14i	0.0404728	−	0.0701009i	0.885552	+	0.464540i	$0.153780\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.40395e14	+	2.43172e14i	−0.0667000	+	0.115528i
$612$		0			0
$613$	−1.78897e14	−	3.09858e14i	−0.0834775	−	0.144587i	0.821264	−	0.570549i	$-0.193269\pi$
$613$	−1.78897e14	−	3.09858e14i	−0.0834775	−	0.144587i	−0.904741	+	0.425961i	$0.859936\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.08164e15			0.937210			0.468605	−	0.883408i	$-0.344757\pi$
$617$	2.08164e15			0.937210			0.468605	+	0.883408i	$0.344757\pi$
$618$		0			0
$619$	8.06152e14	+	1.39630e15i	0.356548	+	0.617560i	0.987382	−	0.158359i	$-0.0506202\pi$
$619$	8.06152e14	+	1.39630e15i	0.356548	+	0.617560i	−0.630833	+	0.775918i	$0.717287\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.58474e15	+	1.41797e15i	0.676510	+	0.605315i
$624$		0			0
$625$	1.53277e14	−	2.65483e14i	0.0642890	−	0.111352i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.74597e15			−1.51700
$630$		0			0
$631$	−2.96340e15			−1.17931			−0.589657	−	0.807654i	$-0.700737\pi$
$631$	−2.96340e15			−1.17931			−0.589657	+	0.807654i	$0.700737\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.06274e14	−	1.05010e15i	0.233031	−	0.403622i
$636$		0			0
$637$	2.30620e14	+	1.00956e14i	0.0871224	+	0.0381387i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.15717e15	+	3.73633e15i	0.787346	+	1.36372i	0.927588	+	0.373606i	$0.121879\pi$
$641$	2.15717e15	+	3.73633e15i	0.787346	+	1.36372i	−0.140242	+	0.990117i	$0.544788\pi$
$642$		0			0
$643$	−2.99617e15			−1.07500			−0.537498	−	0.843265i	$-0.680630\pi$
$643$	−2.99617e15			−1.07500			−0.537498	+	0.843265i	$0.680630\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.93288e15	−	3.34785e15i	−0.670243	−	1.16089i	−0.977835	−	0.209376i	$-0.932857\pi$
$647$	−1.93288e15	−	3.34785e15i	−0.670243	−	1.16089i	0.307592	−	0.951518i	$-0.400477\pi$
$648$		0			0
$649$	1.76016e15	−	3.04868e15i	0.600075	−	1.03936i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.41629e14	+	9.38129e14i	−0.178517	+	0.309200i	−0.941373	−	0.337368i	$-0.890463\pi$
$653$	−5.41629e14	+	9.38129e14i	−0.178517	+	0.309200i	0.762856	+	0.646569i	$0.223797\pi$
$654$		0			0
$655$	3.20334e14	+	5.54835e14i	0.103819	+	0.179819i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4.08793e15			−1.28125			−0.640624	−	0.767855i	$-0.721324\pi$
$659$	−4.08793e15			−1.28125			−0.640624	+	0.767855i	$0.721324\pi$
$660$		0			0
$661$	−8.57086e14	−	1.48452e15i	−0.264190	−	0.457591i	0.703161	−	0.711031i	$-0.251771\pi$
$661$	−8.57086e14	−	1.48452e15i	−0.264190	−	0.457591i	−0.967351	+	0.253440i	$0.918438\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.86254e15	+	6.11618e14i	−0.555375	+	0.182373i
$666$		0			0
$667$	1.08868e15	−	1.88566e15i	0.319308	−	0.553057i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.44997e15			−1.26294
$672$		0			0
$673$	2.65743e15			0.741957			0.370978	−	0.928642i	$-0.379022\pi$
$673$	2.65743e15			0.741957			0.370978	+	0.928642i	$0.379022\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.98685e14	+	1.21016e15i	−0.188818	+	0.327043i	−0.944856	−	0.327485i	$-0.893799\pi$
$677$	−6.98685e14	+	1.21016e15i	−0.188818	+	0.327043i	0.756038	+	0.654527i	$0.227132\pi$
$678$		0			0
$679$	6.91682e14	−	3.30486e15i	0.183917	−	0.878758i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.56799e15	+	2.71584e15i	0.403673	+	0.699182i	0.994166	−	0.107861i	$-0.0344001\pi$
$683$	1.56799e15	+	2.71584e15i	0.403673	+	0.699182i	−0.590493	+	0.807043i	$0.701067\pi$
$684$		0			0
$685$	−7.96716e14			−0.201839
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.57399e13	−	1.13865e14i	−0.0161296	−	0.0279373i
$690$		0			0
$691$	−2.12034e15	+	3.67254e15i	−0.512008	+	0.886824i	0.487895	+	0.872902i	$0.337765\pi$
$691$	−2.12034e15	+	3.67254e15i	−0.512008	+	0.886824i	−0.999903	+	0.0139218i	$0.995568\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.10836e15	+	1.91974e15i	−0.259278	+	0.449083i
$696$		0			0
$697$	−3.43768e15	−	5.95424e15i	−0.791563	−	1.37103i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	7.37666e15			1.64593			0.822963	−	0.568094i	$-0.192319\pi$
$701$	7.37666e15			1.64593			0.822963	+	0.568094i	$0.192319\pi$
$702$		0			0
$703$	−3.23389e15	−	5.60125e15i	−0.710347	−	1.23036i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.68255e15	−	2.19441e15i	1.42277	−	0.467208i
$708$		0			0
$709$	2.08526e14	−	3.61177e14i	0.0437125	−	0.0757122i	−0.843341	−	0.537378i	$-0.819415\pi$
$709$	2.08526e14	−	3.61177e14i	0.0437125	−	0.0757122i	0.887054	+	0.461666i	$0.152748\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8.89375e14			0.180756
$714$		0			0
$715$	−3.81042e14			−0.0762587
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.90503e14	−	6.76371e14i	0.0757906	−	0.131273i	−0.825639	−	0.564198i	$-0.809185\pi$
$719$	3.90503e14	−	6.76371e14i	0.0757906	−	0.131273i	0.901430	+	0.432925i	$0.142519\pi$
$720$		0			0
$721$	−4.37461e15	−	3.91423e15i	−0.836172	−	0.748174i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.82324e15	−	3.15795e15i	−0.338053	−	0.585525i
$726$		0			0
$727$	2.88685e15			0.527212			0.263606	−	0.964630i	$-0.415088\pi$
$727$	2.88685e15			0.527212			0.263606	+	0.964630i	$0.415088\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.57893e15	−	4.46683e15i	−0.456976	−	0.791505i
$732$		0			0
$733$	2.51533e15	−	4.35667e15i	0.439059	−	0.760472i	−0.558558	−	0.829465i	$-0.688645\pi$
$733$	2.51533e15	−	4.35667e15i	0.439059	−	0.760472i	0.997617	+	0.0689931i	$0.0219787\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.84374e15	−	3.19346e15i	0.312341	−	0.540991i
$738$		0			0
$739$	−1.23835e15	−	2.14489e15i	−0.206681	−	0.357982i	0.743986	−	0.668195i	$-0.232933\pi$
$739$	−1.23835e15	−	2.14489e15i	−0.206681	−	0.357982i	−0.950667	+	0.310213i	$0.899600\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	9.91447e14			0.160632			0.0803158	−	0.996769i	$-0.474407\pi$
$743$	9.91447e14			0.160632			0.0803158	+	0.996769i	$0.474407\pi$
$744$		0			0
$745$	−3.75890e15	−	6.51060e15i	−0.600069	−	1.03935i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.06235e15	−	1.84531e15i	−0.319678	−	0.286036i
$750$		0			0
$751$	−8.60259e14	+	1.49001e15i	−0.131404	+	0.227599i	−0.924218	−	0.381865i	$-0.875282\pi$
$751$	−8.60259e14	+	1.49001e15i	−0.131404	+	0.227599i	0.792814	+	0.609464i	$0.208615\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.79142e15			0.265760
$756$		0			0
$757$	−3.57866e15			−0.523231			−0.261615	−	0.965172i	$-0.584255\pi$
$757$	−3.57866e15			−0.523231			−0.261615	+	0.965172i	$0.584255\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2.16021e15	+	3.74159e15i	−0.306818	+	0.531424i	−0.977664	−	0.210172i	$-0.932598\pi$
$761$	−2.16021e15	+	3.74159e15i	−0.306818	+	0.531424i	0.670847	+	0.741596i	$0.265931\pi$
$762$		0			0
$763$	−7.82898e15	+	2.57087e15i	−1.09602	+	0.359910i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.47477e14	+	6.01848e14i	0.0472663	+	0.0818676i
$768$		0			0
$769$	−1.09944e16			−1.47426			−0.737131	−	0.675749i	$-0.763820\pi$
$769$	−1.09944e16			−1.47426			−0.737131	+	0.675749i	$0.763820\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.36724e15	+	5.83224e15i	0.438821	+	0.760060i	0.997599	−	0.0692574i	$-0.0220630\pi$
$773$	3.36724e15	+	5.83224e15i	0.438821	+	0.760060i	−0.558778	+	0.829317i	$0.688730\pi$
$774$		0			0
$775$	7.44728e14	−	1.28991e15i	0.0956837	−	0.165729i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.93548e15	−	1.02806e16i	0.741310	−	1.28399i
$780$		0			0
$781$	−5.43107e15	−	9.40689e15i	−0.668813	−	1.15842i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.85390e15			−0.700910
$786$		0			0
$787$	7.15834e15	+	1.23986e16i	0.845184	+	1.46390i	0.885461	+	0.464713i	$0.153843\pi$
$787$	7.15834e15	+	1.23986e16i	0.845184	+	1.46390i	−0.0402771	+	0.999189i	$0.512824\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.02856e15	+	9.69247e15i	−0.232926	+	1.11292i
$792$		0			0
$793$	4.39240e14	−	7.60786e14i	0.0497392	−	0.0861509i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.20669e16			−1.32916			−0.664579	−	0.747218i	$-0.731389\pi$
$797$	−1.20669e16			−1.32916			−0.664579	+	0.747218i	$0.731389\pi$
$798$		0			0
$799$	−1.21350e16			−1.31835
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−8.69000e14	+	1.50515e15i	−0.0918511	+	0.159091i
$804$		0			0
$805$	3.19513e15	−	1.04921e15i	0.333128	−	0.109392i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.65512e15	−	2.86674e15i	−0.167924	−	0.290852i	0.769766	−	0.638326i	$-0.220373\pi$
$809$	−1.65512e15	−	2.86674e15i	−0.167924	−	0.290852i	−0.937690	+	0.347474i	$0.887040\pi$
$810$		0			0
$811$	−4.03561e13			−0.00403919			−0.00201960	−	0.999998i	$-0.500643\pi$
$811$	−4.03561e13			−0.00403919			−0.00201960	+	0.999998i	$0.500643\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.69476e15	−	2.93541e15i	−0.165098	−	0.285958i
$816$		0			0
$817$	4.45275e15	−	7.71240e15i	0.427964	−	0.741256i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.71220e15	−	1.50900e16i	0.815156	−	1.41189i	−0.0940603	−	0.995567i	$-0.529985\pi$
$821$	8.71220e15	−	1.50900e16i	0.815156	−	1.41189i	0.909216	−	0.416325i	$-0.136682\pi$
$822$		0			0
$823$	−6.49068e15	−	1.12422e16i	−0.599227	−	1.03789i	−0.992935	−	0.118656i	$-0.962141\pi$
$823$	−6.49068e15	−	1.12422e16i	−0.599227	−	1.03789i	0.393708	−	0.919235i	$-0.371192\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.05784e16			−0.950910			−0.475455	−	0.879740i	$-0.657717\pi$
$827$	−1.05784e16			−0.950910			−0.475455	+	0.879740i	$0.657717\pi$
$828$		0			0
$829$	−9.12914e15	−	1.58121e16i	−0.809804	−	1.40262i	−0.913000	−	0.407960i	$-0.866240\pi$
$829$	−9.12914e15	−	1.58121e16i	−0.809804	−	1.40262i	0.103196	−	0.994661i	$-0.467093\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.20486e15	+	1.08130e16i	0.104086	+	0.934108i
$834$		0			0
$835$	5.25375e15	−	9.09976e15i	0.447913	−	0.775808i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.25871e16			1.87573			0.937863	−	0.347005i	$-0.112801\pi$
$839$	2.25871e16			1.87573			0.937863	+	0.347005i	$0.112801\pi$
$840$		0			0
$841$	5.64906e15			0.463018
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.12068e15	+	7.13723e15i	−0.329046	+	0.569925i
$846$		0			0
$847$	4.32879e15	+	3.87323e15i	0.341199	+	0.305292i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.54762e15	+	9.60876e15i	0.426083	+	0.737997i
$852$		0			0
$853$	1.13337e15			0.0859312			0.0429656	−	0.999077i	$-0.486319\pi$
$853$	1.13337e15			0.0859312			0.0429656	+	0.999077i	$0.486319\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.59431e15	+	4.49348e15i	0.191702	+	0.332038i	0.945815	−	0.324707i	$-0.105266\pi$
$857$	2.59431e15	+	4.49348e15i	0.191702	+	0.332038i	−0.754112	+	0.656746i	$0.771932\pi$
$858$		0			0
$859$	−4.87030e15	+	8.43561e15i	−0.355298	+	0.615395i	−0.987169	−	0.159679i	$-0.948954\pi$
$859$	−4.87030e15	+	8.43561e15i	−0.355298	+	0.615395i	0.631871	+	0.775074i	$0.282287\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.00545e16	+	1.74149e16i	−0.714992	+	1.23840i	0.247971	+	0.968767i	$0.420236\pi$
$863$	−1.00545e16	+	1.74149e16i	−0.714992	+	1.23840i	−0.962963	+	0.269635i	$0.913097\pi$
$864$		0			0
$865$	7.24412e15	+	1.25472e16i	0.508624	+	0.880963i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.44689e16			−0.990435
$870$		0			0
$871$	3.63978e14	+	6.30429e14i	0.0246023	+	0.0426124i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.21781e15	−	1.53747e16i	0.212088	−	1.01336i
$876$		0			0
$877$	1.02360e16	−	1.77292e16i	0.666240	−	1.15396i	−0.312707	−	0.949850i	$-0.601236\pi$
$877$	1.02360e16	−	1.77292e16i	0.666240	−	1.15396i	0.978947	−	0.204112i	$-0.0654308\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.21251e16			0.769693			0.384846	−	0.922981i	$-0.374254\pi$
$881$	1.21251e16			0.769693			0.384846	+	0.922981i	$0.374254\pi$
$882$		0			0
$883$	−9.98434e14			−0.0625944			−0.0312972	−	0.999510i	$-0.509964\pi$
$883$	−9.98434e14			−0.0625944			−0.0312972	+	0.999510i	$0.509964\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.65790e14	−	8.06772e14i	0.0284846	−	0.0493368i	−0.851432	−	0.524466i	$-0.824265\pi$
$887$	4.65790e14	−	8.06772e14i	0.0284846	−	0.0493368i	0.879916	+	0.475129i	$0.157599\pi$
$888$		0			0
$889$	2.38021e15	−	1.13726e16i	0.143766	−	0.686912i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.04761e16	−	1.81451e16i	−0.617327	−	1.06924i
$894$		0			0
$895$	−1.87485e16			−1.09129
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.64545e15	+	6.31410e15i	0.207049	+	0.358619i
$900$		0			0
$901$	2.84109e15	−	4.92091e15i	0.159404	−	0.276096i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.68249e15	+	4.64622e15i	−0.146883	+	0.254409i
$906$		0			0
$907$	1.00462e16	+	1.74006e16i	0.543455	+	0.941291i	0.998702	+	0.0509264i	$0.0162174\pi$
$907$	1.00462e16	+	1.74006e16i	0.543455	+	0.941291i	−0.455248	+	0.890365i	$0.650449\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.43297e15			0.234069			0.117035	−	0.993128i	$-0.462661\pi$
$911$	4.43297e15			0.234069			0.117035	+	0.993128i	$0.462661\pi$
$912$		0			0
$913$	4.02473e15	+	6.97104e15i	0.209965	+	0.363671i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.57505e15	+	4.09358e15i	0.233005	+	0.208484i
$918$		0			0
$919$	−1.53713e16	+	2.66240e16i	−0.773529	+	1.33979i	0.162089	+	0.986776i	$0.448177\pi$
$919$	−1.53713e16	+	2.66240e16i	−0.773529	+	1.33979i	−0.935618	+	0.353015i	$0.885156\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.14432e15			0.105361
$924$		0			0
$925$	1.85814e16			0.902194
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.92053e15	+	8.52261e15i	−0.233306	+	0.404097i	−0.958779	−	0.284153i	$-0.908288\pi$
$929$	−4.92053e15	+	8.52261e15i	−0.233306	+	0.404097i	0.725473	+	0.688250i	$0.241621\pi$
$930$		0			0
$931$	−1.51282e16	+	1.11364e16i	−0.708864	+	0.521820i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.23377e15	−	1.42613e16i	−0.376821	−	0.652673i
$936$		0			0
$937$	2.64016e16			1.19416			0.597081	−	0.802181i	$-0.296327\pi$
$937$	2.64016e16			1.19416			0.597081	+	0.802181i	$0.296327\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.00159e16	−	1.73481e16i	−0.442536	−	0.766495i	0.555341	−	0.831623i	$-0.312588\pi$
$941$	−1.00159e16	−	1.73481e16i	−0.442536	−	0.766495i	−0.997877	+	0.0651280i	$0.979254\pi$
$942$		0			0
$943$	−1.01821e16	+	1.76359e16i	−0.444656	+	0.770166i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.88064e15	+	6.72147e15i	−0.165569	+	0.286774i	−0.936857	−	0.349712i	$-0.886279\pi$
$947$	−3.88064e15	+	6.72147e15i	−0.165569	+	0.286774i	0.771288	+	0.636486i	$0.219613\pi$
$948$		0			0
$949$	−1.71551e14	−	2.97136e14i	−0.00723486	−	0.0125311i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.36763e16			1.79984			0.899922	−	0.436052i	$-0.143624\pi$
$953$	4.36763e16			1.79984			0.899922	+	0.436052i	$0.143624\pi$
$954$		0			0
$955$	1.42162e16	+	2.46233e16i	0.579117	+	1.00306i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.25334e15	+	2.38184e15i	−0.288759	+	0.0948222i
$960$		0			0
$961$	1.12152e16	−	1.94253e16i	0.441396	−	0.764521i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.59387e16			−0.997810
$966$		0			0
$967$	1.52684e16			0.580693			0.290347	−	0.956922i	$-0.406229\pi$
$967$	1.52684e16			0.580693			0.290347	+	0.956922i	$0.406229\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6.90055e15	−	1.19521e16i	0.256553	−	0.444364i	−0.708763	−	0.705447i	$-0.750746\pi$
$971$	6.90055e15	−	1.19521e16i	0.256553	−	0.444364i	0.965316	+	0.261083i	$0.0840797\pi$
$972$		0			0
$973$	−4.35139e15	+	2.07909e16i	−0.159958	+	0.764282i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.44890e14	−	7.70571e14i	−0.0159894	−	0.0276945i	0.857920	−	0.513783i	$-0.171756\pi$
$977$	−4.44890e14	−	7.70571e14i	−0.0159894	−	0.0276945i	−0.873909	+	0.486089i	$0.838423\pi$
$978$		0			0
$979$	−3.08420e16			−1.09607
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.09153e16	−	1.89059e16i	−0.379308	−	0.656981i	0.611654	−	0.791126i	$-0.290505\pi$
$983$	−1.09153e16	−	1.89059e16i	−0.379308	−	0.656981i	−0.990962	+	0.134145i	$0.957171\pi$
$984$		0			0
$985$	2.09845e15	−	3.63462e15i	0.0721105	−	0.124899i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7.63855e15	+	1.32304e16i	−0.256703	+	0.444623i
$990$		0			0
$991$	2.08695e16	+	3.61470e16i	0.693595	+	1.20134i	0.970652	+	0.240488i	$0.0773076\pi$
$991$	2.08695e16	+	3.61470e16i	0.693595	+	1.20134i	−0.277057	+	0.960853i	$0.589359\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.27321e16			−0.413878
$996$		0			0
$997$	2.10555e16	+	3.64692e16i	0.676927	+	1.17247i	0.975902	+	0.218211i	$0.0700222\pi$
$997$	2.10555e16	+	3.64692e16i	0.676927	+	1.17247i	−0.298974	+	0.954261i	$0.596645\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.12.k.e.37.10	yes	28
3.2	odd	2	inner	252.12.k.e.37.5	✓	28
7.4	even	3	inner	252.12.k.e.109.10	yes	28
21.11	odd	6	inner	252.12.k.e.109.5	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.12.k.e.37.5	✓	28	3.2	odd	2	inner
252.12.k.e.37.10	yes	28	1.1	even	1	trivial
252.12.k.e.109.5	yes	28	21.11	odd	6	inner
252.12.k.e.109.10	yes	28	7.4	even	3	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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2320.27 - 4018.82i) q^{5} +(9109.27 - 43524.1i) q^{7} +O(q^{10})\)	\(q+(2320.27 - 4018.82i) q^{5} +(9109.27 - 43524.1i) q^{7} +(322467. + 558529. i) q^{11} -127318. q^{13} +(-2.75116e6 - 4.76515e6i) q^{17} +(4.75013e6 - 8.22747e6i) q^{19} +(-8.14869e6 + 1.41139e7i) q^{23} +(1.36468e7 + 2.36369e7i) q^{25} -1.33602e8 q^{29} +(-2.72858e7 - 4.72604e7i) q^{31} +(-1.53780e8 - 1.37596e8i) q^{35} +(3.40400e8 - 5.89589e8i) q^{37} +1.24954e9 q^{41} +9.37396e8 q^{43} +(1.10271e9 - 1.90996e9i) q^{47} +(-1.81137e9 - 7.92945e8i) q^{49} +(5.16345e8 + 8.94335e8i) q^{53} +2.99284e9 q^{55} +(-2.72921e9 - 4.72712e9i) q^{59} +(-3.44994e9 + 5.97548e9i) q^{61} +(-2.95412e8 + 5.11668e8i) q^{65} +(-2.85881e9 - 4.95161e9i) q^{67} -1.68423e10 q^{71} +(1.34743e9 + 2.33381e9i) q^{73} +(2.72469e10 - 8.94729e9i) q^{77} +(-1.12173e10 + 1.94290e10i) q^{79} +1.24811e10 q^{83} -2.55337e10 q^{85} +(-2.39110e10 + 4.14150e10i) q^{89} +(-1.15977e9 + 5.54140e9i) q^{91} +(-2.20431e10 - 3.81798e10i) q^{95} +7.59317e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 85666 q^{7}+O(q^{10}) \) 28 * q + 85666 * q^7	\( 28 q + 85666 q^{7} + 2794700 q^{13} + 37414826 q^{19} - 135469250 q^{25} + 71052098 q^{31} - 395898874 q^{37} + 980170268 q^{43} - 1253333270 q^{49} - 8844657360 q^{55} + 16692685100 q^{61} + 10934009870 q^{67} + 25578690650 q^{73} - 113974174090 q^{79} + 111789733440 q^{85} - 267181531078 q^{91} + 279658007384 q^{97}+O(q^{100}) \) 28 * q + 85666 * q^7 + 2794700 * q^13 + 37414826 * q^19 - 135469250 * q^25 + 71052098 * q^31 - 395898874 * q^37 + 980170268 * q^43 - 1253333270 * q^49 - 8844657360 * q^55 + 16692685100 * q^61 + 10934009870 * q^67 + 25578690650 * q^73 - 113974174090 * q^79 + 111789733440 * q^85 - 267181531078 * q^91 + 279658007384 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more