LMFDB - Embedded newform 75.12.e.a.32.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,12,Mod(32,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("75.32");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	75.e (of order $4$, 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:	$57.6257385420$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			32.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	75.32
Dual form			75.12.e.a.68.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(297.613 - 297.613i) q^{3} +2048.00i q^{4} +(-31607.0 - 31607.0i) q^{7} -177147. i q^{9} +O(q^{10})$	$q+(297.613 - 297.613i) q^{3} +2048.00i q^{4} +(-31607.0 - 31607.0i) q^{7} -177147. i q^{9} +(609511. + 609511. i) q^{12} +(1.02771e6 - 1.02771e6i) q^{13} -4.19430e6 q^{16} +2.05819e7i q^{19} -1.88133e7 q^{21} +(-5.27213e7 - 5.27213e7i) q^{27} +(6.47311e7 - 6.47311e7i) q^{28} -2.96477e8 q^{31} +3.62797e8 q^{36} +(-2.22157e8 - 2.22157e8i) q^{37} -6.11719e8i q^{39} +(-5.43178e8 + 5.43178e8i) q^{43} +(-1.24828e9 + 1.24828e9i) q^{48} +2.06770e7i q^{49} +(2.10475e9 + 2.10475e9i) q^{52} +(6.12544e9 + 6.12544e9i) q^{57} -1.29773e10 q^{61} +(-5.59908e9 + 5.59908e9i) q^{63} -8.58993e9i q^{64} +(-1.50516e10 - 1.50516e10i) q^{67} +(-2.07683e10 + 2.07683e10i) q^{73} -4.21517e10 q^{76} -3.28858e10i q^{79} -3.13811e10 q^{81} -3.85296e10i q^{84} -6.49656e10 q^{91} +(-8.82354e10 + 8.82354e10i) q^{93} +(1.13592e11 + 1.13592e11i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 16777216 q^{16} - 75253212 q^{21} - 1185907772 q^{31} + 1451188224 q^{36} - 51909171652 q^{61} - 168606932992 q^{76} - 125524238436 q^{81} - 259862347764 q^{91}+O(q^{100}) $ 4 * q - 16777216 * q^16 - 75253212 * q^21 - 1185907772 * q^31 + 1451188224 * q^36 - 51909171652 * q^61 - 168606932992 * q^76 - 125524238436 * q^81 - 259862347764 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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		−0.707107	−	0.707107i	$-0.750000\pi$
$2$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$3$	297.613	−	297.613i	0.707107	−	0.707107i
$3$	297.613	−	297.613i	0.707107	−	0.707107i
$4$			2048.00i			1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−31607.0	−	31607.0i	−0.710794	−	0.710794i	0.255907	−	0.966701i	$-0.417626\pi$
$7$	−31607.0	−	31607.0i	−0.710794	−	0.710794i	−0.966701	+	0.255907i	$0.917626\pi$
$8$		0			0
$9$		−	177147.i		−	1.00000i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	609511.	+	609511.i	0.707107	+	0.707107i
$13$	1.02771e6	−	1.02771e6i	0.767683	−	0.767683i	−0.210015	−	0.977698i	$-0.567351\pi$
$13$	1.02771e6	−	1.02771e6i	0.767683	−	0.767683i	0.977698	+	0.210015i	$0.0673514\pi$
$14$		0			0
$15$		0			0
$16$	−4.19430e6			−1.00000
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$		0			0
$19$			2.05819e7i			1.90696i	0.301463	+	0.953478i	$0.402525\pi$
$19$			2.05819e7i			1.90696i	−0.301463	+	0.953478i	$0.597475\pi$
$20$		0			0
$21$	−1.88133e7			−1.00521
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−5.27213e7	−	5.27213e7i	−0.707107	−	0.707107i
$28$	6.47311e7	−	6.47311e7i	0.710794	−	0.710794i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−2.96477e8			−1.85995			−0.929976	−	0.367621i	$-0.880172\pi$
$31$	−2.96477e8			−1.85995			−0.929976	+	0.367621i	$0.880172\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	3.62797e8			1.00000
$37$	−2.22157e8	−	2.22157e8i	−0.526684	−	0.526684i	0.392898	−	0.919582i	$-0.371472\pi$
$37$	−2.22157e8	−	2.22157e8i	−0.526684	−	0.526684i	−0.919582	+	0.392898i	$0.871472\pi$
$38$		0			0
$39$		−	6.11719e8i		−	1.08567i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−5.43178e8	+	5.43178e8i	−0.563463	+	0.563463i	−0.930289	−	0.366826i	$-0.880444\pi$
$43$	−5.43178e8	+	5.43178e8i	−0.563463	+	0.563463i	0.366826	+	0.930289i	$0.380444\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	−1.24828e9	+	1.24828e9i	−0.707107	+	0.707107i
$49$			2.06770e7i			0.0104570i
$50$		0			0
$51$		0			0
$52$	2.10475e9	+	2.10475e9i	0.767683	+	0.767683i
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.12544e9	+	6.12544e9i	1.34842	+	1.34842i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−1.29773e10			−1.96730			−0.983649	−	0.180098i	$-0.942359\pi$
$61$	−1.29773e10			−1.96730			−0.983649	+	0.180098i	$0.942359\pi$
$62$		0			0
$63$	−5.59908e9	+	5.59908e9i	−0.710794	+	0.710794i
$64$		−	8.58993e9i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	−1.50516e10	−	1.50516e10i	−1.36198	−	1.36198i	−0.871387	−	0.490597i	$-0.836779\pi$
$67$	−1.50516e10	−	1.50516e10i	−1.36198	−	1.36198i	−0.490597	−	0.871387i	$-0.663221\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−2.07683e10	+	2.07683e10i	−1.17254	+	1.17254i	−0.190933	+	0.981603i	$0.561151\pi$
$73$	−2.07683e10	+	2.07683e10i	−1.17254	+	1.17254i	−0.981603	+	0.190933i	$0.938849\pi$
$74$		0			0
$75$		0			0
$76$	−4.21517e10			−1.90696
$77$		0			0
$78$		0			0
$79$		−	3.28858e10i		−	1.20243i	−0.799087	−	0.601215i	$-0.794684\pi$
$79$		−	3.28858e10i		−	1.20243i	0.799087	−	0.601215i	$-0.205316\pi$
$80$		0			0
$81$	−3.13811e10			−1.00000
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		−	3.85296e10i		−	1.00521i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−6.49656e10			−1.09133
$92$		0			0
$93$	−8.82354e10	+	8.82354e10i	−1.31518	+	1.31518i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.13592e11	+	1.13592e11i	1.34309	+	1.34309i	0.892969	+	0.450117i	$0.148618\pi$
$97$	1.13592e11	+	1.13592e11i	1.34309	+	1.34309i	0.450117	+	0.892969i	$0.351382\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−1.58162e11	+	1.58162e11i	−1.34430	+	1.34430i	−0.452579	+	0.891724i	$0.649496\pi$
$103$	−1.58162e11	+	1.58162e11i	−1.34430	+	1.34430i	−0.891724	+	0.452579i	$0.850504\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$	1.07973e11	−	1.07973e11i	0.707107	−	0.707107i
$109$		−	2.57951e9i		−	0.0160580i	−0.999968	−	0.00802899i	$-0.997444\pi$
$109$		−	2.57951e9i		−	0.0160580i	0.999968	−	0.00802899i	$-0.00255573\pi$
$110$		0			0
$111$	−1.32233e11			−0.744843
$112$	1.32569e11	+	1.32569e11i	0.710794	+	0.710794i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.82056e11	−	1.82056e11i	−0.767683	−	0.767683i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.85312e11			1.00000
$122$		0			0
$123$		0			0
$124$		−	6.07185e11i		−	1.85995i
$125$		0			0
$126$		0			0
$127$	−4.46400e11	−	4.46400e11i	−1.19896	−	1.19896i	−0.974479	−	0.224478i	$-0.927932\pi$
$127$	−4.46400e11	−	4.46400e11i	−1.19896	−	1.19896i	−0.224478	−	0.974479i	$-0.572068\pi$
$128$		0			0
$129$			3.23314e11i			0.796857i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	6.50532e11	−	6.50532e11i	1.35545	−	1.35545i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$			1.14947e12i			1.87896i	0.342607	+	0.939479i	$0.388690\pi$
$139$			1.14947e12i			1.87896i	−0.342607	+	0.939479i	$0.611310\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			7.43008e11i			1.00000i
$145$		0			0
$146$		0			0
$147$	6.15375e9	+	6.15375e9i	0.00739425	+	0.00739425i
$148$	4.54977e11	−	4.54977e11i	0.526684	−	0.526684i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	1.31670e12			1.36494			0.682470	−	0.730913i	$-0.260906\pi$
$151$	1.31670e12			1.36494			0.682470	+	0.730913i	$0.260906\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.25280e12			1.08567
$157$	1.16557e12	+	1.16557e12i	0.975190	+	0.975190i	0.999700	−	0.0245092i	$-0.00780231\pi$
$157$	1.16557e12	+	1.16557e12i	0.975190	+	0.975190i	−0.0245092	+	0.999700i	$0.507802\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.61096e12	−	1.61096e12i	1.09661	−	1.09661i	0.101806	−	0.994804i	$-0.467538\pi$
$163$	1.61096e12	−	1.61096e12i	1.09661	−	1.09661i	0.994804	−	0.101806i	$-0.0324620\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	3.20212e11i		−	0.178674i
$170$		0			0
$171$	3.64602e12			1.90696
$172$	−1.11243e12	−	1.11243e12i	−0.563463	−	0.563463i
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−1.27601e12			−0.488228			−0.244114	−	0.969747i	$-0.578497\pi$
$181$	−1.27601e12			−0.488228			−0.244114	+	0.969747i	$0.578497\pi$
$182$		0			0
$183$	−3.86221e12	+	3.86221e12i	−1.39109	+	1.39109i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$			3.33272e12i			1.00521i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−2.55648e12	−	2.55648e12i	−0.707107	−	0.707107i
$193$	5.09139e12	−	5.09139e12i	1.36858	−	1.36858i	0.506123	−	0.862461i	$-0.331078\pi$
$193$	5.09139e12	−	5.09139e12i	1.36858	−	1.36858i	0.862461	−	0.506123i	$-0.168922\pi$
$194$		0			0
$195$		0			0
$196$	−4.23465e10			−0.0104570
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		−	8.39365e12i		−	1.90660i	−0.302028	−	0.953299i	$-0.597664\pi$
$199$		−	8.39365e12i		−	1.90660i	0.302028	−	0.953299i	$-0.402336\pi$
$200$		0			0
$201$	−8.95912e12			−1.92614
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−4.31052e12	+	4.31052e12i	−0.767683	+	0.767683i
$209$		0			0
$210$		0			0
$211$	−1.12332e13			−1.84906			−0.924531	−	0.381106i	$-0.875543\pi$
$211$	−1.12332e13			−1.84906			−0.924531	+	0.381106i	$0.875543\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	9.37074e12	+	9.37074e12i	1.32204	+	1.32204i
$218$		0			0
$219$			1.23619e13i			1.65822i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−3.21242e12	+	3.21242e12i	−0.390081	+	0.390081i	−0.874716	−	0.484635i	$-0.838952\pi$
$223$	−3.21242e12	+	3.21242e12i	−0.390081	+	0.390081i	0.484635	+	0.874716i	$0.338952\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$	−1.25449e13	+	1.25449e13i	−1.34842	+	1.34842i
$229$		−	1.90357e13i		−	1.99744i	−0.0505740	−	0.998720i	$-0.516105\pi$
$229$		−	1.90357e13i		−	1.99744i	0.0505740	−	0.998720i	$-0.483895\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−9.78725e12	−	9.78725e12i	−0.850246	−	0.850246i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−2.27780e12			−0.180477			−0.0902386	−	0.995920i	$-0.528763\pi$
$241$	−2.27780e12			−0.180477			−0.0902386	+	0.995920i	$0.528763\pi$
$242$		0			0
$243$	−9.33941e12	+	9.33941e12i	−0.707107	+	0.707107i
$244$		−	2.65775e13i		−	1.96730i
$245$		0			0
$246$		0			0
$247$	2.11522e13	+	2.11522e13i	1.46394	+	1.46394i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−1.14669e13	−	1.14669e13i	−0.710794	−	0.710794i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.75922e13			1.00000
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$			1.40434e13i			0.748728i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	3.08257e13	−	3.08257e13i	1.36198	−	1.36198i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−4.72629e13			−1.96422			−0.982109	−	0.188315i	$-0.939697\pi$
$271$	−4.72629e13			−1.96422			−0.982109	+	0.188315i	$0.939697\pi$
$272$		0			0
$273$	−1.93346e13	+	1.93346e13i	−0.771686	+	0.771686i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.36566e12	−	3.36566e12i	−0.124003	−	0.124003i	0.642382	−	0.766385i	$-0.277946\pi$
$277$	−3.36566e12	−	3.36566e12i	−0.124003	−	0.124003i	−0.766385	+	0.642382i	$0.777946\pi$
$278$		0			0
$279$			5.25200e13i			1.85995i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	4.22277e13	−	4.22277e13i	1.38284	−	1.38284i	0.543302	−	0.839537i	$-0.317174\pi$
$283$	4.22277e13	−	4.22277e13i	1.38284	−	1.38284i	0.839537	−	0.543302i	$-0.182826\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			3.42719e13i			1.00000i
$290$		0			0
$291$	6.76130e13			1.89941
$292$	−4.25336e13	−	4.25336e13i	−1.17254	−	1.17254i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	3.43364e13			0.801013
$302$		0			0
$303$		0			0
$304$		−	8.63267e13i		−	1.90696i
$305$		0			0
$306$		0			0
$307$	3.49686e13	+	3.49686e13i	0.731841	+	0.731841i	0.970984	−	0.239143i	$-0.0768664\pi$
$307$	3.49686e13	+	3.49686e13i	0.731841	+	0.731841i	−0.239143	+	0.970984i	$0.576866\pi$
$308$		0			0
$309$			9.41421e13i			1.90113i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	7.50682e13	−	7.50682e13i	1.41242	−	1.41242i	0.670557	−	0.741858i	$-0.266055\pi$
$313$	7.50682e13	−	7.50682e13i	1.41242	−	1.41242i	0.741858	−	0.670557i	$-0.233945\pi$
$314$		0			0
$315$		0			0
$316$	6.73502e13			1.20243
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	6.42684e13i		−	1.00000i
$325$		0			0
$326$		0			0
$327$	−7.67694e11	−	7.67694e11i	−0.0113547	−	0.0113547i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	9.80000e13			1.35573			0.677863	−	0.735188i	$-0.262906\pi$
$331$	9.80000e13			1.35573			0.677863	+	0.735188i	$0.262906\pi$
$332$		0			0
$333$	−3.93544e13	+	3.93544e13i	−0.526684	+	0.526684i
$334$		0			0
$335$		0			0
$336$	7.89087e13			1.00521
$337$	−7.67948e12	−	7.67948e12i	−0.0962426	−	0.0962426i	0.657346	−	0.753589i	$-0.271679\pi$
$337$	−7.67948e12	−	7.67948e12i	−0.0962426	−	0.0962426i	−0.753589	+	0.657346i	$0.771679\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6.18438e13	+	6.18438e13i	−0.703361	+	0.703361i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$			1.70909e14i			1.76695i	0.468478	+	0.883475i	$0.344803\pi$
$349$			1.70909e14i			1.76695i	−0.468478	+	0.883475i	$0.655197\pi$
$350$		0			0
$351$	−1.08364e14			−1.08567
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−3.07124e14			−2.63648
$362$		0			0
$363$	8.49125e13	−	8.49125e13i	0.707107	−	0.707107i
$364$		−	1.33050e14i		−	1.09133i
$365$		0			0
$366$		0			0
$367$	1.79414e14	+	1.79414e14i	1.40668	+	1.40668i	0.776245	+	0.630431i	$0.217122\pi$
$367$	1.79414e14	+	1.79414e14i	1.40668	+	1.40668i	0.630431	+	0.776245i	$0.282878\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.80706e14	−	1.80706e14i	−1.31518	−	1.31518i
$373$	2.17763e13	−	2.17763e13i	0.156166	−	0.156166i	−0.624699	−	0.780865i	$-0.714779\pi$
$373$	2.17763e13	−	2.17763e13i	0.156166	−	0.156166i	0.780865	+	0.624699i	$0.214779\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	5.50919e13i		−	0.361886i	−0.983494	−	0.180943i	$-0.942085\pi$
$379$		−	5.50919e13i		−	0.361886i	0.983494	−	0.180943i	$-0.0579150\pi$
$380$		0			0
$381$	−2.65709e14			−1.69558
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	9.62223e13	+	9.62223e13i	0.563463	+	0.563463i
$388$	−2.32637e14	+	2.32637e14i	−1.34309	+	1.34309i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.99277e14	−	1.99277e14i	−1.01416	−	1.01416i	−0.999898	−	0.0142662i	$-0.995459\pi$
$397$	−1.99277e14	−	1.99277e14i	−1.01416	−	1.01416i	−0.0142662	−	0.999898i	$-0.504541\pi$
$398$		0			0
$399$		−	3.87214e14i		−	1.91690i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−3.04692e14	+	3.04692e14i	−1.42785	+	1.42785i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	8.33592e13i		−	0.360143i	−0.983654	−	0.180072i	$-0.942367\pi$
$409$		−	8.33592e13i		−	0.360143i	0.983654	−	0.180072i	$-0.0576329\pi$
$410$		0			0
$411$		0			0
$412$	−3.23916e14	−	3.23916e14i	−1.34430	−	1.34430i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.42098e14	+	3.42098e14i	1.32862	+	1.32862i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	3.20990e13			0.118288			0.0591438	−	0.998249i	$-0.481163\pi$
$421$	3.20990e13			0.118288			0.0591438	+	0.998249i	$0.481163\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.10173e14	+	4.10173e14i	1.39834	+	1.39834i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	2.21129e14	+	2.21129e14i	0.707107	+	0.707107i
$433$	2.34313e12	−	2.34313e12i	0.00739796	−	0.00739796i	−0.703398	−	0.710796i	$-0.748335\pi$
$433$	2.34313e12	−	2.34313e12i	0.00739796	−	0.00739796i	0.710796	+	0.703398i	$0.248335\pi$
$434$		0			0
$435$		0			0
$436$	5.28283e12			0.0160580
$437$		0			0
$438$		0			0
$439$			4.50030e14i			1.31731i	0.752446	+	0.658654i	$0.228874\pi$
$439$			4.50030e14i			1.31731i	−0.752446	+	0.658654i	$0.771126\pi$
$440$		0			0
$441$	3.66287e12			0.0104570
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		−	2.70814e14i		−	0.744843i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−2.71502e14	+	2.71502e14i	−0.710794	+	0.710794i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	3.91867e14	−	3.91867e14i	0.965159	−	0.965159i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.15408e14	−	5.15408e14i	−1.20952	−	1.20952i	−0.971183	−	0.238335i	$-0.923398\pi$
$457$	−5.15408e14	−	5.15408e14i	−1.20952	−	1.20952i	−0.238335	−	0.971183i	$-0.576602\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	3.67905e14	−	3.67905e14i	0.803600	−	0.803600i	−0.180057	−	0.983656i	$-0.557628\pi$
$463$	3.67905e14	−	3.67905e14i	0.803600	−	0.803600i	0.983656	+	0.180057i	$0.0576281\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$	3.72850e14	−	3.72850e14i	0.767683	−	0.767683i
$469$			9.51473e14i			1.93618i
$470$		0			0
$471$	6.93776e14			1.37913
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−4.56625e14			−0.808652
$482$		0			0
$483$		0			0
$484$			5.84318e14i			1.00000i
$485$		0			0
$486$		0			0
$487$	−8.05794e14	−	8.05794e14i	−1.33295	−	1.33295i	−0.902716	−	0.430236i	$-0.858430\pi$
$487$	−8.05794e14	−	8.05794e14i	−1.33295	−	1.33295i	−0.430236	−	0.902716i	$-0.641570\pi$
$488$		0			0
$489$		−	9.58884e14i		−	1.55084i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.24351e15			1.85995
$497$		0			0
$498$		0			0
$499$			5.44758e14i			0.788225i	0.919062	+	0.394113i	$0.128948\pi$
$499$			5.44758e14i			0.788225i	−0.919062	+	0.394113i	$0.871052\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−9.52992e13	−	9.52992e13i	−0.126341	−	0.126341i
$508$	9.14227e14	−	9.14227e14i	1.19896	−	1.19896i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	1.31285e15			1.66686
$512$		0			0
$513$	1.08510e15	−	1.08510e15i	1.34842	−	1.34842i
$514$		0			0
$515$		0			0
$516$	−6.62146e14			−0.796857
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−1.24928e15	+	1.24928e15i	−1.39605	+	1.39605i	−0.585046	+	0.811000i	$0.698923\pi$
$523$	−1.24928e15	+	1.24928e15i	−1.39605	+	1.39605i	−0.811000	+	0.585046i	$0.801077\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	9.52810e14i		−	1.00000i
$530$		0			0
$531$		0			0
$532$	1.33229e15	+	1.33229e15i	1.35545	+	1.35545i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.78270e15			1.65384			0.826921	−	0.562319i	$-0.190091\pi$
$541$	1.78270e15			1.65384			0.826921	+	0.562319i	$0.190091\pi$
$542$		0			0
$543$	−3.79758e14	+	3.79758e14i	−0.345229	+	0.345229i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	4.60712e14	+	4.60712e14i	0.402253	+	0.402253i	0.879026	−	0.476773i	$-0.158194\pi$
$547$	4.60712e14	+	4.60712e14i	0.402253	+	0.402253i	−0.476773	+	0.879026i	$0.658194\pi$
$548$		0			0
$549$			2.29889e15i			1.96730i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.03942e15	+	1.03942e15i	−0.854680	+	0.854680i
$554$		0			0
$555$		0			0
$556$	−2.35412e15			−1.87896
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$			1.11646e15i			0.865122i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	9.91861e14	+	9.91861e14i	0.710794	+	0.710794i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−1.21874e15			−0.840261			−0.420131	−	0.907464i	$-0.638016\pi$
$571$	−1.21874e15			−0.840261			−0.420131	+	0.907464i	$0.638016\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1.52168e15			−1.00000
$577$	−1.95934e15	−	1.95934e15i	−1.27539	−	1.27539i	−0.943220	−	0.332169i	$-0.892220\pi$
$577$	−1.95934e15	−	1.95934e15i	−1.27539	−	1.27539i	−0.332169	−	0.943220i	$-0.607780\pi$
$578$		0			0
$579$		−	3.03053e15i		−	1.93547i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$	−1.26029e13	+	1.26029e13i	−0.00739425	+	0.00739425i
$589$		−	6.10206e15i		−	3.54685i
$590$		0			0
$591$		0			0
$592$	9.31793e14	+	9.31793e14i	0.526684	+	0.526684i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−2.49806e15	−	2.49806e15i	−1.34817	−	1.34817i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−3.04058e15			−1.58178			−0.790892	−	0.611956i	$-0.790383\pi$
$601$	−3.04058e15			−1.58178			−0.790892	+	0.611956i	$0.790383\pi$
$602$		0			0
$603$	−2.66635e15	+	2.66635e15i	−1.36198	+	1.36198i
$604$			2.69660e15i			1.36494i
$605$		0			0
$606$		0			0
$607$	3.29288e14	+	3.29288e14i	0.162195	+	0.162195i	0.783539	−	0.621343i	$-0.213413\pi$
$607$	3.29288e14	+	3.29288e14i	0.162195	+	0.162195i	−0.621343	+	0.783539i	$0.713413\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.87256e15	−	1.87256e15i	0.873784	−	0.873784i	−0.119099	−	0.992882i	$-0.538000\pi$
$613$	1.87256e15	−	1.87256e15i	0.873784	−	0.873784i	0.992882	+	0.119099i	$0.0380005\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$		−	1.56531e15i		−	0.692312i	−0.938177	−	0.346156i	$-0.887487\pi$
$619$		−	1.56531e15i		−	0.692312i	0.938177	−	0.346156i	$-0.112513\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$			2.56574e15i			1.08567i
$625$		0			0
$626$		0			0
$627$		0			0
$628$	−2.38708e15	+	2.38708e15i	−0.975190	+	0.975190i
$629$		0			0
$630$		0			0
$631$	−4.79425e15			−1.90791			−0.953957	−	0.299943i	$-0.903032\pi$
$631$	−4.79425e15			−1.90791			−0.953957	+	0.299943i	$0.903032\pi$
$632$		0			0
$633$	−3.34316e15	+	3.34316e15i	−1.30748	+	1.30748i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.12499e13	+	2.12499e13i	0.00802770	+	0.00802770i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	3.31582e15	−	3.31582e15i	1.18968	−	1.18968i	0.212524	−	0.977156i	$-0.431832\pi$
$643$	3.31582e15	−	3.31582e15i	1.18968	−	1.18968i	0.977156	−	0.212524i	$-0.0681685\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	5.57771e15			1.86965
$652$	3.29924e15	+	3.29924e15i	1.09661	+	1.09661i
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	3.67905e15	+	3.67905e15i	1.17254	+	1.17254i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	1.77508e15			0.547155			0.273578	−	0.961850i	$-0.411793\pi$
$661$	1.77508e15			0.547155			0.273578	+	0.961850i	$0.411793\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			1.91211e15i			0.551658i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	8.46023e14	−	8.46023e14i	0.236211	−	0.236211i	−0.579068	−	0.815279i	$-0.696584\pi$
$673$	8.46023e14	−	8.46023e14i	0.236211	−	0.236211i	0.815279	+	0.579068i	$0.196584\pi$
$674$		0			0
$675$		0			0
$676$	6.55794e14			0.178674
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		−	7.18062e15i		−	1.90932i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$			7.46705e15i			1.90696i
$685$		0			0
$686$		0			0
$687$	−5.66527e15	−	5.66527e15i	−1.41240	−	1.41240i
$688$	2.27825e15	−	2.27825e15i	0.563463	−	0.563463i
$689$		0			0
$690$		0			0
$691$	8.23279e15			1.98801			0.994003	−	0.109351i	$-0.0348773\pi$
$691$	8.23279e15			1.98801			0.994003	+	0.109351i	$0.0348773\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	4.57241e15	−	4.57241e15i	1.00436	−	1.00436i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			4.99833e15i			1.04778i	0.851785	+	0.523891i	$0.175520\pi$
$709$			4.99833e15i			1.04778i	−0.851785	+	0.523891i	$0.824480\pi$
$710$		0			0
$711$	−5.82563e15			−1.20243
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	9.99805e15			1.91105
$722$		0			0
$723$	−6.77904e14	+	6.77904e14i	−0.127617	+	0.127617i
$724$		−	2.61327e15i		−	0.488228i
$725$		0			0
$726$		0			0
$727$	−4.04370e15	−	4.04370e15i	−0.738481	−	0.738481i	0.233803	−	0.972284i	$-0.424883\pi$
$727$	−4.04370e15	−	4.04370e15i	−0.738481	−	0.738481i	−0.972284	+	0.233803i	$0.924883\pi$
$728$		0			0
$729$			5.55906e15i			1.00000i
$730$		0			0
$731$		0			0
$732$	−7.90981e15	−	7.90981e15i	−1.39109	−	1.39109i
$733$	7.52748e15	−	7.52748e15i	1.31395	−	1.31395i	0.395466	−	0.918481i	$-0.370583\pi$
$733$	7.52748e15	−	7.52748e15i	1.31395	−	1.31395i	0.918481	−	0.395466i	$-0.129417\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	1.72182e15i		−	0.287372i	−0.989623	−	0.143686i	$-0.954105\pi$
$739$		−	1.72182e15i		−	0.287372i	0.989623	−	0.143686i	$-0.0458955\pi$
$740$		0			0
$741$	1.25903e16			2.07032
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.20840e16			−1.84582			−0.922911	−	0.385013i	$-0.874197\pi$
$751$	−1.20840e16			−1.84582			−0.922911	+	0.385013i	$0.874197\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−6.82541e15			−1.00521
$757$	3.28303e15	+	3.28303e15i	0.480006	+	0.480006i	0.905134	−	0.425127i	$-0.139771\pi$
$757$	3.28303e15	+	3.28303e15i	0.480006	+	0.480006i	−0.425127	+	0.905134i	$0.639771\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−8.15304e13	+	8.15304e13i	−0.0114139	+	0.0114139i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	5.23566e15	−	5.23566e15i	0.707107	−	0.707107i
$769$			9.03163e15i			1.21108i	0.795817	+	0.605538i	$0.207042\pi$
$769$			9.03163e15i			1.21108i	−0.795817	+	0.605538i	$0.792958\pi$
$770$		0			0
$771$		0			0
$772$	1.04272e16	+	1.04272e16i	1.36858	+	1.36858i
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	4.17950e15	+	4.17950e15i	0.529430	+	0.529430i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	8.67256e13i		−	0.0104570i
$785$		0			0
$786$		0			0
$787$	4.30470e15	+	4.30470e15i	0.508255	+	0.508255i	0.913991	−	0.405735i	$-0.132985\pi$
$787$	4.30470e15	+	4.30470e15i	0.508255	+	0.508255i	−0.405735	+	0.913991i	$0.632985\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.33369e16	+	1.33369e16i	−1.51026	+	1.51026i
$794$		0			0
$795$		0			0
$796$	1.71902e16			1.90660
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−	1.83483e16i		−	1.92614i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−1.99246e16			−1.99423			−0.997113	−	0.0759375i	$-0.975805\pi$
$811$	−1.99246e16			−1.99423			−0.997113	+	0.0759375i	$0.975805\pi$
$812$		0			0
$813$	−1.40661e16	+	1.40661e16i	−1.38891	+	1.38891i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.11796e16	−	1.11796e16i	−1.07450	−	1.07450i
$818$		0			0
$819$			1.15085e16i			1.09133i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−8.78784e15	+	8.78784e15i	−0.811303	+	0.811303i	−0.984829	−	0.173526i	$-0.944484\pi$
$823$	−8.78784e15	+	8.78784e15i	−0.811303	+	0.811303i	0.173526	+	0.984829i	$0.444484\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$		−	1.54331e16i		−	1.36900i	−0.729012	−	0.684501i	$-0.760020\pi$
$829$		−	1.54331e16i		−	1.36900i	0.729012	−	0.684501i	$-0.239980\pi$
$830$		0			0
$831$	−2.00333e15			−0.175367
$832$	−8.82795e15	−	8.82795e15i	−0.767683	−	0.767683i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.56306e16	+	1.56306e16i	1.31518	+	1.31518i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−1.22005e16			−1.00000
$842$		0			0
$843$		0			0
$844$		−	2.30057e16i		−	1.84906i
$845$		0			0
$846$		0			0
$847$	−9.01784e15	−	9.01784e15i	−0.710794	−	0.710794i
$848$		0			0
$849$		−	2.51350e16i		−	1.95563i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.32054e16	−	1.32054e16i	1.00122	−	1.00122i	0.00122395	−	0.999999i	$-0.499610\pi$
$853$	1.32054e16	−	1.32054e16i	1.00122	−	1.00122i	0.999999	−	0.00122395i	$-0.000389597\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$			2.70738e16i			1.97509i	0.157335	+	0.987545i	$0.449710\pi$
$859$			2.70738e16i			1.97509i	−0.157335	+	0.987545i	$0.550290\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.01998e16	+	1.01998e16i	0.707107	+	0.707107i
$868$	−1.91913e16	+	1.91913e16i	−1.32204	+	1.32204i
$869$		0			0
$870$		0			0
$871$	−3.09374e16			−2.09114
$872$		0			0
$873$	2.01225e16	−	2.01225e16i	1.34309	−	1.34309i
$874$		0			0
$875$		0			0
$876$	−2.53171e16			−1.65822
$877$	−1.75621e15	−	1.75621e15i	−0.114308	−	0.114308i	0.647639	−	0.761947i	$-0.275757\pi$
$877$	−1.75621e15	−	1.75621e15i	−0.114308	−	0.114308i	−0.761947	+	0.647639i	$0.775757\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−2.04282e16	+	2.04282e16i	−1.28070	+	1.28070i	−0.340425	+	0.940272i	$0.610571\pi$
$883$	−2.04282e16	+	2.04282e16i	−1.28070	+	1.28070i	−0.940272	+	0.340425i	$0.889429\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$			2.82187e16i			1.70442i
$890$		0			0
$891$		0			0
$892$	−6.57903e15	−	6.57903e15i	−0.390081	−	0.390081i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	1.02190e16	−	1.02190e16i	0.566402	−	0.566402i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.51966e15	+	3.51966e15i	0.190397	+	0.190397i	0.795868	−	0.605470i	$-0.207015\pi$
$907$	3.51966e15	+	3.51966e15i	0.190397	+	0.190397i	−0.605470	+	0.795868i	$0.707015\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−2.56920e16	−	2.56920e16i	−1.34842	−	1.34842i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	3.89851e16			1.99744
$917$		0			0
$918$		0			0
$919$			7.17141e15i			0.360885i	0.983586	+	0.180443i	$0.0577530\pi$
$919$			7.17141e15i			0.360885i	−0.983586	+	0.180443i	$0.942247\pi$
$920$		0			0
$921$	2.08142e16			1.03498
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	2.80179e16	+	2.80179e16i	1.34430	+	1.34430i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−4.25572e14			−0.0199411
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	2.86610e16	+	2.86610e16i	1.29635	+	1.29635i	0.930785	+	0.365567i	$0.119125\pi$
$937$	2.86610e16	+	2.86610e16i	1.29635	+	1.29635i	0.365567	+	0.930785i	$0.380875\pi$
$938$		0			0
$939$		−	4.46826e16i		−	1.99746i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$	2.00443e16	−	2.00443e16i	0.850246	−	0.850246i
$949$			4.26876e16i			1.80027i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	6.24901e16			2.45942
$962$		0			0
$963$		0			0
$964$		−	4.66494e15i		−	0.180477i
$965$		0			0
$966$		0			0
$967$	−3.33231e16	−	3.33231e16i	−1.26736	−	1.26736i	−0.947447	−	0.319912i	$-0.896347\pi$
$967$	−3.33231e16	−	3.33231e16i	−1.26736	−	1.26736i	−0.319912	−	0.947447i	$-0.603653\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−1.91271e16	−	1.91271e16i	−0.707107	−	0.707107i
$973$	3.63313e16	−	3.63313e16i	1.33555	−	1.33555i
$974$		0			0
$975$		0			0
$976$	5.44307e16			1.96730
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−4.56952e14			−0.0160580
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−4.33197e16	+	4.33197e16i	−1.46394	+	1.46394i
$989$		0			0
$990$		0			0
$991$	−8.04799e15			−0.267474			−0.133737	−	0.991017i	$-0.542698\pi$
$991$	−8.04799e15			−0.267474			−0.133737	+	0.991017i	$0.542698\pi$
$992$		0			0
$993$	2.91661e16	−	2.91661e16i	0.958643	−	0.958643i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	7.80745e15	+	7.80745e15i	0.251007	+	0.251007i	0.821383	−	0.570376i	$-0.193203\pi$
$997$	7.80745e15	+	7.80745e15i	0.251007	+	0.251007i	−0.570376	+	0.821383i	$0.693203\pi$
$998$		0			0
$999$			2.34248e16i			0.744843i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.12.e.a.32.2	yes	4
3.2	odd	2	CM	75.12.e.a.32.2	yes	4
5.2	odd	4	inner	75.12.e.a.68.1	yes	4
5.3	odd	4	inner	75.12.e.a.68.2	yes	4
5.4	even	2	inner	75.12.e.a.32.1	✓	4
15.2	even	4	inner	75.12.e.a.68.1	yes	4
15.8	even	4	inner	75.12.e.a.68.2	yes	4
15.14	odd	2	inner	75.12.e.a.32.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.12.e.a.32.1	✓	4	5.4	even	2	inner
75.12.e.a.32.1	✓	4	15.14	odd	2	inner
75.12.e.a.32.2	yes	4	1.1	even	1	trivial
75.12.e.a.32.2	yes	4	3.2	odd	2	CM
75.12.e.a.68.1	yes	4	5.2	odd	4	inner
75.12.e.a.68.1	yes	4	15.2	even	4	inner
75.12.e.a.68.2	yes	4	5.3	odd	4	inner
75.12.e.a.68.2	yes	4	15.8	even	4	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(297.613 - 297.613i) q^{3} +2048.00i q^{4} +(-31607.0 - 31607.0i) q^{7} -177147. i q^{9} +O(q^{10})\)	\(q+(297.613 - 297.613i) q^{3} +2048.00i q^{4} +(-31607.0 - 31607.0i) q^{7} -177147. i q^{9} +(609511. + 609511. i) q^{12} +(1.02771e6 - 1.02771e6i) q^{13} -4.19430e6 q^{16} +2.05819e7i q^{19} -1.88133e7 q^{21} +(-5.27213e7 - 5.27213e7i) q^{27} +(6.47311e7 - 6.47311e7i) q^{28} -2.96477e8 q^{31} +3.62797e8 q^{36} +(-2.22157e8 - 2.22157e8i) q^{37} -6.11719e8i q^{39} +(-5.43178e8 + 5.43178e8i) q^{43} +(-1.24828e9 + 1.24828e9i) q^{48} +2.06770e7i q^{49} +(2.10475e9 + 2.10475e9i) q^{52} +(6.12544e9 + 6.12544e9i) q^{57} -1.29773e10 q^{61} +(-5.59908e9 + 5.59908e9i) q^{63} -8.58993e9i q^{64} +(-1.50516e10 - 1.50516e10i) q^{67} +(-2.07683e10 + 2.07683e10i) q^{73} -4.21517e10 q^{76} -3.28858e10i q^{79} -3.13811e10 q^{81} -3.85296e10i q^{84} -6.49656e10 q^{91} +(-8.82354e10 + 8.82354e10i) q^{93} +(1.13592e11 + 1.13592e11i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 16777216 q^{16} - 75253212 q^{21} - 1185907772 q^{31} + 1451188224 q^{36} - 51909171652 q^{61} - 168606932992 q^{76} - 125524238436 q^{81} - 259862347764 q^{91}+O(q^{100}) \) 4 * q - 16777216 * q^16 - 75253212 * q^21 - 1185907772 * q^31 + 1451188224 * q^36 - 51909171652 * q^61 - 168606932992 * q^76 - 125524238436 * q^81 - 259862347764 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more