LMFDB - Embedded newform 75.10.e.c.32.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,10,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, 10, names="a")

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

chi := DirichletCharacter("75.32");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
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:	$38.6276877123$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{4} $
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			75.10.e.c.68.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(11.0227 - 11.0227i) q^{2} +(99.2043 + 99.2043i) q^{3} +269.000i q^{4} +2187.00 q^{6} +(8608.73 + 8608.73i) q^{8} +19683.0i q^{9} +O(q^{10})$	\(q+(11.0227 - 11.0227i) q^{2} +(99.2043 + 99.2043i) q^{3} +269.000i q^{4} +2187.00 q^{6} +(8608.73 + 8608.73i) q^{8} +19683.0i q^{9} +(-26686.0 + 26686.0i) q^{12} +52055.0 q^{16} +(210269. - 210269. i) q^{17} +(216960. + 216960. i) q^{18} +1.03632e6i q^{19} +(-347237. - 347237. i) q^{23} +1.70805e6i q^{24} +(-1.95264e6 + 1.95264e6i) q^{27} -8.24737e6 q^{31} +(-3.83388e6 + 3.83388e6i) q^{32} -4.63547e6i q^{34} -5.29473e6 q^{36} +(1.14230e7 + 1.14230e7i) q^{38} -7.65499e6 q^{46} +(-4.70102e7 + 4.70102e7i) q^{47} +(5.16408e6 + 5.16408e6i) q^{48} -4.03536e7i q^{49} +4.17192e7 q^{51} +(7.68054e7 + 7.68054e7i) q^{53} +4.30467e7i q^{54} +(-1.02807e8 + 1.02807e8i) q^{57} +1.97895e8 q^{61} +(-9.09083e7 + 9.09083e7i) q^{62} +1.11172e8i q^{64} +(5.65624e7 + 5.65624e7i) q^{68} -6.88949e7i q^{69} +(-1.69446e8 + 1.69446e8i) q^{72} -2.78769e8 q^{76} -4.21557e8i q^{79} -3.87420e8 q^{81} +(6.04524e8 + 6.04524e8i) q^{83} +(9.34068e7 - 9.34068e7i) q^{92} +(-8.18175e8 - 8.18175e8i) q^{93} +1.03636e9i q^{94} -7.60676e8 q^{96} +(-4.44806e8 - 4.44806e8i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8748 q^{6}+O(q^{10}) $ 4 * q + 8748 * q^6	$ 4 q + 8748 q^{6} + 208220 q^{16} - 32989472 q^{31} - 21178908 q^{36} - 30619944 q^{46} + 166876848 q^{51} + 791579528 q^{61} - 1115076016 q^{76} - 1549681956 q^{81} - 3042703116 q^{96}+O(q^{100}) $ 4 * q + 8748 * q^6 + 208220 * q^16 - 32989472 * q^31 - 21178908 * q^36 - 30619944 * q^46 + 166876848 * q^51 + 791579528 * q^61 - 1115076016 * q^76 - 1549681956 * q^81 - 3042703116 * q^96

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 $

$2$	11.0227	−	11.0227i	0.487139	−	0.487139i	−0.420263	−	0.907402i	$-0.638062\pi$
$2$	11.0227	−	11.0227i	0.487139	−	0.487139i	0.907402	+	0.420263i	$0.138062\pi$
$3$	99.2043	+	99.2043i	0.707107	+	0.707107i
$3$	99.2043	+	99.2043i	0.707107	+	0.707107i
$4$			269.000i			0.525391i
$5$		0			0
$5$		0			0
$6$	2187.00			0.688919
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	8608.73	+	8608.73i	0.743078	+	0.743078i
$9$			19683.0i			1.00000i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−26686.0	+	26686.0i	−0.371507	+	0.371507i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	52055.0			0.198574
$17$	210269.	−	210269.i	0.610598	−	0.610598i	−0.332504	−	0.943102i	$-0.607893\pi$
$17$	210269.	−	210269.i	0.610598	−	0.610598i	0.943102	+	0.332504i	$0.107893\pi$
$18$	216960.	+	216960.i	0.487139	+	0.487139i
$19$			1.03632e6i			1.82432i	0.409834	+	0.912160i	$0.365586\pi$
$19$			1.03632e6i			1.82432i	−0.409834	+	0.912160i	$0.634414\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−347237.	−	347237.i	−0.258733	−	0.258733i	0.565806	−	0.824538i	$-0.308565\pi$
$23$	−347237.	−	347237.i	−0.258733	−	0.258733i	−0.824538	+	0.565806i	$0.808565\pi$
$24$			1.70805e6i			1.05087i
$25$		0			0
$26$		0			0
$27$	−1.95264e6	+	1.95264e6i	−0.707107	+	0.707107i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−8.24737e6			−1.60394			−0.801969	−	0.597365i	$-0.796214\pi$
$31$	−8.24737e6			−1.60394			−0.801969	+	0.597365i	$0.796214\pi$
$32$	−3.83388e6	+	3.83388e6i	−0.646344	+	0.646344i
$33$		0			0
$34$		−	4.63547e6i		−	0.594892i
$35$		0			0
$36$	−5.29473e6			−0.525391
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	1.14230e7	+	1.14230e7i	0.888698	+	0.888698i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$	−7.65499e6			−0.252078
$47$	−4.70102e7	+	4.70102e7i	−1.40524	+	1.40524i	−0.623112	+	0.782132i	$0.714132\pi$
$47$	−4.70102e7	+	4.70102e7i	−1.40524	+	1.40524i	−0.782132	+	0.623112i	$0.785868\pi$
$48$	5.16408e6	+	5.16408e6i	0.140413	+	0.140413i
$49$		−	4.03536e7i		−	1.00000i
$50$		0			0
$51$	4.17192e7			0.863516
$52$		0			0
$53$	7.68054e7	+	7.68054e7i	1.33706	+	1.33706i	0.898896	+	0.438163i	$0.144371\pi$
$53$	7.68054e7	+	7.68054e7i	1.33706	+	1.33706i	0.438163	+	0.898896i	$0.355629\pi$
$54$			4.30467e7i			0.688919i
$55$		0			0
$56$		0			0
$57$	−1.02807e8	+	1.02807e8i	−1.28999	+	1.28999i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	1.97895e8			1.83000			0.914998	−	0.403458i	$-0.132192\pi$
$61$	1.97895e8			1.83000			0.914998	+	0.403458i	$0.132192\pi$
$62$	−9.09083e7	+	9.09083e7i	−0.781342	+	0.781342i
$63$		0			0
$64$			1.11172e8i			0.828294i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	5.65624e7	+	5.65624e7i	0.320802	+	0.320802i
$69$		−	6.88949e7i		−	0.365903i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−1.69446e8	+	1.69446e8i	−0.743078	+	0.743078i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$		0			0
$76$	−2.78769e8			−0.958481
$77$		0			0
$78$		0			0
$79$		−	4.21557e8i		−	1.21768i	−0.793292	−	0.608842i	$-0.791634\pi$
$79$		−	4.21557e8i		−	1.21768i	0.793292	−	0.608842i	$-0.208366\pi$
$80$		0			0
$81$	−3.87420e8			−1.00000
$82$		0			0
$83$	6.04524e8	+	6.04524e8i	1.39818	+	1.39818i	0.805273	+	0.592904i	$0.202019\pi$
$83$	6.04524e8	+	6.04524e8i	1.39818	+	1.39818i	0.592904	+	0.805273i	$0.297981\pi$
$84$		0			0
$85$		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
$91$		0			0
$92$	9.34068e7	−	9.34068e7i	0.135936	−	0.135936i
$93$	−8.18175e8	−	8.18175e8i	−1.13416	−	1.13416i
$94$			1.03636e9i			1.36910i
$95$		0			0
$96$	−7.60676e8			−0.914069
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	−4.44806e8	−	4.44806e8i	−0.487139	−	0.487139i
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	4.59859e8	−	4.59859e8i	0.420652	−	0.420652i
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$		0			0
$106$	1.69321e9			1.30267
$107$	1.91697e9	−	1.91697e9i	1.41380	−	1.41380i	0.689758	−	0.724040i	$-0.257717\pi$
$107$	1.91697e9	−	1.91697e9i	1.41380	−	1.41380i	0.724040	−	0.689758i	$-0.242283\pi$
$108$	−5.25260e8	−	5.25260e8i	−0.371507	−	0.371507i
$109$		−	9.56488e8i		−	0.649023i	−0.945882	−	0.324512i	$-0.894800\pi$
$109$		−	9.56488e8i		−	0.649023i	0.945882	−	0.324512i	$-0.105200\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.09596e8	−	8.09596e8i	−0.467106	−	0.467106i	0.433870	−	0.900976i	$-0.357148\pi$
$113$	−8.09596e8	−	8.09596e8i	−0.467106	−	0.467106i	−0.900976	+	0.433870i	$0.857148\pi$
$114$			2.26642e9i			1.25681i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.35795e9			1.00000
$122$	2.18134e9	−	2.18134e9i	0.891463	−	0.891463i
$123$		0			0
$124$		−	2.21854e9i		−	0.842694i
$125$		0			0
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	−7.37536e8	−	7.37536e8i	−0.242850	−	0.242850i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	3.62030e9			0.907443
$137$	2.47036e9	−	2.47036e9i	0.599126	−	0.599126i	−0.340954	−	0.940080i	$-0.610750\pi$
$137$	2.47036e9	−	2.47036e9i	0.599126	−	0.599126i	0.940080	+	0.340954i	$0.110750\pi$
$138$	−7.59408e8	−	7.59408e8i	−0.178246	−	0.178246i
$139$		−	8.79640e9i		−	1.99866i	−0.0366291	−	0.999329i	$-0.511662\pi$
$139$		−	8.79640e9i		−	1.99866i	0.0366291	−	0.999329i	$-0.488338\pi$
$140$		0			0
$141$	−9.32724e9			−1.98732
$142$		0			0
$143$		0			0
$144$			1.02460e9i			0.198574i
$145$		0			0
$146$		0			0
$147$	4.00325e9	−	4.00325e9i	0.707107	−	0.707107i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−2.00509e9			−0.313861			−0.156931	−	0.987610i	$-0.550160\pi$
$151$	−2.00509e9			−0.313861			−0.156931	+	0.987610i	$0.550160\pi$
$152$	−8.92137e9	+	8.92137e9i	−1.35561	+	1.35561i
$153$	4.13873e9	+	4.13873e9i	0.610598	+	0.610598i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$	−4.64670e9	−	4.64670e9i	−0.593182	−	0.593182i
$159$			1.52389e10i			1.89089i
$160$		0			0
$161$		0			0
$162$	−4.27042e9	+	4.27042e9i	−0.487139	+	0.487139i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$	1.33270e10			1.36221
$167$	−1.42041e10	+	1.42041e10i	−1.41315	+	1.41315i	−0.679173	+	0.733979i	$0.737661\pi$
$167$	−1.42041e10	+	1.42041e10i	−1.41315	+	1.41315i	−0.733979	+	0.679173i	$0.762339\pi$
$168$		0			0
$169$			1.06045e10i			1.00000i
$170$		0			0
$171$	−2.03978e10			−1.82432
$172$		0			0
$173$	1.60107e10	+	1.60107e10i	1.35895	+	1.35895i	0.875211	+	0.483741i	$0.160722\pi$
$173$	1.60107e10	+	1.60107e10i	1.35895	+	1.35895i	0.483741	+	0.875211i	$0.339278\pi$
$174$		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
$181$	2.09035e10			1.44766			0.723829	−	0.689979i	$-0.242380\pi$
$181$	2.09035e10			1.44766			0.723829	+	0.689979i	$0.242380\pi$
$182$		0			0
$183$	1.96320e10	+	1.96320e10i	1.29400	+	1.29400i
$184$		−	5.97854e9i		−	0.384517i
$185$		0			0
$186$	−1.80370e10			−1.10498
$187$		0			0
$188$	−1.26457e10	−	1.26457e10i	−0.738302	−	0.738302i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−1.10287e10	+	1.10287e10i	−0.585692	+	0.585692i
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	1.08551e10			0.525391
$197$	−3.90049e9	+	3.90049e9i	−0.184510	+	0.184510i	−0.793318	−	0.608808i	$-0.791648\pi$
$197$	−3.90049e9	+	3.90049e9i	−0.184510	+	0.184510i	0.608808	+	0.793318i	$0.291648\pi$
$198$		0			0
$199$		−	3.34242e10i		−	1.51085i	−0.655233	−	0.755427i	$-0.727430\pi$
$199$		−	3.34242e10i		−	1.51085i	0.655233	−	0.755427i	$-0.272570\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$			1.12225e10i			0.453683i
$205$		0			0
$206$		0			0
$207$	6.83467e9	−	6.83467e9i	0.258733	−	0.258733i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	4.35730e10			1.51337			0.756687	−	0.653778i	$-0.226817\pi$
$211$	4.35730e10			1.51337			0.756687	+	0.653778i	$0.226817\pi$
$212$	−2.06607e10	+	2.06607e10i	−0.702478	+	0.702478i
$213$		0			0
$214$		−	4.22603e10i		−	1.37743i
$215$		0			0
$216$	−3.36195e10			−1.05087
$217$		0			0
$218$	−1.05431e10	−	1.05431e10i	−0.316165	−	0.316165i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$		0			0
$226$	−1.78479e10			−0.455091
$227$	4.78475e10	−	4.78475e10i	1.19603	−	1.19603i	0.220689	−	0.975344i	$-0.429170\pi$
$227$	4.78475e10	−	4.78475e10i	1.19603	−	1.19603i	0.975344	−	0.220689i	$-0.0708304\pi$
$228$	−2.76551e10	−	2.76551e10i	−0.677748	−	0.677748i
$229$			1.02803e10i			0.247028i	0.992343	+	0.123514i	$0.0394165\pi$
$229$			1.02803e10i			0.247028i	−0.992343	+	0.123514i	$0.960584\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.80618e10	+	2.80618e10i	0.623754	+	0.623754i	0.946489	−	0.322735i	$-0.104602\pi$
$233$	2.80618e10	+	2.80618e10i	0.623754	+	0.623754i	−0.322735	+	0.946489i	$0.604602\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	4.18203e10	−	4.18203e10i	0.861032	−	0.861032i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	5.74117e10			1.09629			0.548143	−	0.836384i	$-0.315335\pi$
$241$	5.74117e10			1.09629			0.548143	+	0.836384i	$0.315335\pi$
$242$	2.59910e10	−	2.59910e10i	0.487139	−	0.487139i
$243$	−3.84338e10	−	3.84338e10i	−0.707107	−	0.707107i
$244$			5.32337e10i			0.961463i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−7.09994e10	−	7.09994e10i	−1.19185	−	1.19185i
$249$			1.19943e11i			1.97732i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−7.31792e10			−1.06490
$257$	9.14988e10	−	9.14988e10i	1.30833	−	1.30833i	0.385704	−	0.922623i	$-0.373959\pi$
$257$	9.14988e10	−	9.14988e10i	1.30833	−	1.30833i	0.922623	−	0.385704i	$-0.126041\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	8.88477e10	+	8.88477e10i	1.14511	+	1.14511i	0.987503	+	0.157603i	$0.0503766\pi$
$263$	8.88477e10	+	8.88477e10i	1.14511	+	1.14511i	0.157603	+	0.987503i	$0.449623\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−9.53249e10			−1.07361			−0.536803	−	0.843708i	$-0.680368\pi$
$271$	−9.53249e10			−1.07361			−0.536803	+	0.843708i	$0.680368\pi$
$272$	1.09456e10	−	1.09456e10i	0.121249	−	0.121249i
$273$		0			0
$274$		−	5.44601e10i		−	0.583715i
$275$		0			0
$276$	1.85327e10			0.192242
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	−9.69601e10	−	9.69601e10i	−0.973625	−	0.973625i
$279$		−	1.62333e11i		−	1.60394i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−1.02811e11	+	1.02811e11i	−0.968100	+	0.968100i
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−7.54623e10	−	7.54623e10i	−0.646344	−	0.646344i
$289$			3.01617e10i			0.254340i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.02023e11	+	1.02023e11i	0.808714	+	0.808714i	0.984439	−	0.175725i	$-0.0562271\pi$
$293$	1.02023e11	+	1.02023e11i	0.808714	+	0.808714i	−0.175725	+	0.984439i	$0.556227\pi$
$294$		−	8.82533e10i		−	0.688919i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−2.21015e10	+	2.21015e10i	−0.152894	+	0.152894i
$303$		0			0
$304$			5.39454e10i			0.362263i
$305$		0			0
$306$	9.12399e10			0.594892
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$	1.13399e11			0.639760
$317$	−2.50470e11	+	2.50470e11i	−1.39312	+	1.39312i	−0.574886	+	0.818234i	$0.694954\pi$
$317$	−2.50470e11	+	2.50470e11i	−1.39312	+	1.39312i	−0.818234	+	0.574886i	$0.805046\pi$
$318$	1.67973e11	+	1.67973e11i	0.921125	+	0.921125i
$319$		0			0
$320$		0			0
$321$	3.80343e11			1.99941
$322$		0			0
$323$	2.17905e11	+	2.17905e11i	1.11393	+	1.11393i
$324$		−	1.04216e11i		−	0.525391i
$325$		0			0
$326$		0			0
$327$	9.48877e10	−	9.48877e10i	0.458929	−	0.458929i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.36256e11			−1.99763			−0.998816	−	0.0486552i	$-0.984506\pi$
$331$	−4.36256e11			−1.99763			−0.998816	+	0.0486552i	$0.984506\pi$
$332$	−1.62617e11	+	1.62617e11i	−0.734589	+	0.734589i
$333$		0			0
$334$			3.13134e11i			1.37680i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	1.16890e11	+	1.16890e11i	0.487139	+	0.487139i
$339$		−	1.60631e11i		−	0.660588i
$340$		0			0
$341$		0			0
$342$	−2.24839e11	+	2.24839e11i	−0.888698	+	0.888698i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	3.52963e11			1.32400
$347$	−2.14823e11	+	2.14823e11i	−0.795421	+	0.795421i	−0.982370	−	0.186948i	$-0.940140\pi$
$347$	−2.14823e11	+	2.14823e11i	−0.795421	+	0.795421i	0.186948	+	0.982370i	$0.440140\pi$
$348$		0			0
$349$			4.21710e11i			1.52160i	0.648988	+	0.760799i	$0.275193\pi$
$349$			4.21710e11i			1.52160i	−0.648988	+	0.760799i	$0.724807\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.48979e11	−	3.48979e11i	−1.19622	−	1.19622i	−0.975288	−	0.220936i	$-0.929089\pi$
$353$	−3.48979e11	−	3.48979e11i	−1.19622	−	1.19622i	−0.220936	−	0.975288i	$-0.570911\pi$
$354$		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
$361$	−7.51263e11			−2.32814
$362$	2.30413e11	−	2.30413e11i	0.705211	−	0.705211i
$363$	2.33919e11	+	2.33919e11i	0.707107	+	0.707107i
$364$		0			0
$365$		0			0
$366$	4.32796e11			1.26072
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$	−1.80754e10	−	1.80754e10i	−0.0513776	−	0.0513776i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	2.20089e11	−	2.20089e11i	0.595875	−	0.595875i
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$	−8.09397e11			−2.08841
$377$		0			0
$378$		0			0
$379$		−	6.42170e11i		−	1.59872i	−0.600850	−	0.799362i	$-0.705171\pi$
$379$		−	6.42170e11i		−	1.59872i	0.600850	−	0.799362i	$-0.294829\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.87696e11	−	2.87696e11i	−0.683188	−	0.683188i	0.277530	−	0.960717i	$-0.410484\pi$
$383$	−2.87696e11	−	2.87696e11i	−0.683188	−	0.683188i	−0.960717	+	0.277530i	$0.910484\pi$
$384$		−	1.46334e11i		−	0.343442i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$	−1.46027e11			−0.315963
$392$	3.47393e11	−	3.47393e11i	0.743078	−	0.743078i
$393$		0			0
$394$			8.59878e10i			0.179765i
$395$		0			0
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$	−3.68425e11	−	3.68425e11i	−0.735996	−	0.735996i
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	3.59150e11	+	3.59150e11i	0.641659	+	0.641659i
$409$			2.79515e8i			0.000493912i	1.00000		0.000246956i	$7.86085e-5\pi$
$409$			2.79515e8i			0.000493912i	−1.00000		0.000246956i	$0.999921\pi$
$410$		0			0
$411$	4.90141e11			0.847292
$412$		0			0
$413$		0			0
$414$		−	1.50673e11i		−	0.252078i
$415$		0			0
$416$		0			0
$417$	8.72641e11	−	8.72641e11i	1.41326	−	1.41326i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	1.21430e12			1.88390			0.941948	−	0.335760i	$-0.108993\pi$
$421$	1.21430e12			1.88390			0.941948	+	0.335760i	$0.108993\pi$
$422$	4.80292e11	−	4.80292e11i	0.737224	−	0.737224i
$423$	−9.25302e11	−	9.25302e11i	−1.40524	−	1.40524i
$424$			1.32239e12i			1.98708i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	5.15664e11	+	5.15664e11i	0.742796	+	0.742796i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−1.01645e11	+	1.01645e11i	−0.140413	+	0.140413i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$	2.57295e11			0.340991
$437$	3.59847e11	−	3.59847e11i	0.472011	−	0.472011i
$438$		0			0
$439$		−	5.83482e11i		−	0.749786i	−0.927068	−	0.374893i	$-0.877679\pi$
$439$		−	5.83482e11i		−	0.749786i	0.927068	−	0.374893i	$-0.122321\pi$
$440$		0			0
$441$	7.94280e11			1.00000
$442$		0			0
$443$	−7.91579e11	−	7.91579e11i	−0.976512	−	0.976512i	0.0232185	−	0.999730i	$-0.492609\pi$
$443$	−7.91579e11	−	7.91579e11i	−0.976512	−	0.976512i	−0.999730	+	0.0232185i	$0.992609\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	2.17781e11	−	2.17781e11i	0.245413	−	0.245413i
$453$	−1.98914e11	−	1.98914e11i	−0.221933	−	0.221933i
$454$		−	1.05482e12i		−	1.16527i
$455$		0			0
$456$	−1.77008e12			−1.91712
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$	1.13317e11	+	1.13317e11i	0.120337	+	0.120337i
$459$			8.21159e11i			0.863516i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$	6.18633e11			0.607710
$467$	−1.15852e12	+	1.15852e12i	−1.12714	+	1.12714i	−0.136495	+	0.990641i	$0.543584\pi$
$467$	−1.15852e12	+	1.15852e12i	−1.12714	+	1.12714i	−0.990641	+	0.136495i	$0.956416\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		−	9.21945e11i		−	0.838885i
$475$		0			0
$476$		0			0
$477$	−1.51176e12	+	1.51176e12i	−1.33706	+	1.33706i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	6.32833e11	−	6.32833e11i	0.534044	−	0.534044i
$483$		0			0
$484$			6.34288e11i			0.525391i
$485$		0			0
$486$	−8.47289e11			−0.688919
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	1.70362e12	+	1.70362e12i	1.35983	+	1.35983i
$489$		0			0
$490$		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
$496$	−4.29317e11			−0.318501
$497$		0			0
$498$	1.32209e12	+	1.32209e12i	0.963231	+	0.963231i
$499$		−	6.82548e9i		−	0.00492812i	−0.999997	−	0.00246406i	$-0.999216\pi$
$499$		−	6.82548e9i		−	0.00492812i	0.999997	−	0.00246406i	$-0.000784335\pi$
$500$		0			0
$501$	−2.81821e12			−1.99850
$502$		0			0
$503$	1.30137e12	+	1.30137e12i	0.906454	+	0.906454i	0.995984	−	0.0895304i	$-0.0285366\pi$
$503$	1.30137e12	+	1.30137e12i	0.906454	+	0.906454i	−0.0895304	+	0.995984i	$0.528537\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.05201e12	+	1.05201e12i	−0.707107	+	0.707107i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−4.29014e11	+	4.29014e11i	−0.275903	+	0.275903i
$513$	−2.02355e12	−	2.02355e12i	−1.28999	−	1.28999i
$514$		−	2.01713e12i		−	1.27467i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			3.17667e12i			1.92185i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$	1.95868e12			1.11565
$527$	−1.73417e12	+	1.73417e12i	−0.979362	+	0.979362i
$528$		0			0
$529$		−	1.56001e12i		−	0.866115i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.50417e12			0.754934			0.377467	−	0.926023i	$-0.376795\pi$
$541$	1.50417e12			0.754934			0.377467	+	0.926023i	$0.376795\pi$
$542$	−1.05074e12	+	1.05074e12i	−0.522995	+	0.522995i
$543$	2.07372e12	+	2.07372e12i	1.02365	+	1.02365i
$544$			1.61229e12i			0.789313i
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	6.64527e11	+	6.64527e11i	0.314775	+	0.314775i
$549$			3.89516e12i			1.83000i
$550$		0			0
$551$		0			0
$552$	5.93097e11	−	5.93097e11i	0.271894	−	0.271894i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	2.36623e12			1.05008
$557$	−8.59191e11	+	8.59191e11i	−0.378217	+	0.378217i	−0.870459	−	0.492241i	$-0.836178\pi$
$557$	−8.59191e11	+	8.59191e11i	−0.378217	+	0.378217i	0.492241	+	0.870459i	$0.336178\pi$
$558$	−1.78935e12	−	1.78935e12i	−0.781342	−	0.781342i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.63759e11	−	5.63759e11i	−0.236486	−	0.236486i	0.578907	−	0.815393i	$-0.303479\pi$
$563$	−5.63759e11	−	5.63759e11i	−0.236486	−	0.236486i	−0.815393	+	0.578907i	$0.803479\pi$
$564$		−	2.50903e12i		−	1.04412i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	3.07243e12			1.20954			0.604769	−	0.796401i	$-0.293265\pi$
$571$	3.07243e12			1.20954			0.604769	+	0.796401i	$0.293265\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−2.18819e12			−0.828294
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	3.32463e11	+	3.32463e11i	0.123899	+	0.123899i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	2.24914e12			0.787913
$587$	3.92981e12	−	3.92981e12i	1.36615	−	1.36615i	0.500306	−	0.865848i	$-0.333221\pi$
$587$	3.92981e12	−	3.92981e12i	1.36615	−	1.36615i	0.865848	−	0.500306i	$-0.166779\pi$
$588$	1.07687e12	+	1.07687e12i	0.371507	+	0.371507i
$589$		−	8.54688e12i		−	2.92610i
$590$		0			0
$591$	−7.73890e11			−0.260937
$592$		0			0
$593$	−2.88157e12	−	2.88157e12i	−0.956936	−	0.956936i	0.0421739	−	0.999110i	$-0.486572\pi$
$593$	−2.88157e12	−	2.88157e12i	−0.956936	−	0.956936i	−0.999110	+	0.0421739i	$0.986572\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.31583e12	−	3.31583e12i	1.06833	−	1.06833i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−6.34546e12			−1.98394			−0.991968	−	0.126487i	$-0.959630\pi$
$601$	−6.34546e12			−1.98394			−0.991968	+	0.126487i	$0.959630\pi$
$602$		0			0
$603$		0			0
$604$		−	5.39369e11i		−	0.164900i
$605$		0			0
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$	−3.97312e12	−	3.97312e12i	−1.17914	−	1.17914i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−1.11332e12	+	1.11332e12i	−0.320802	+	0.320802i
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.84585e12	−	4.84585e12i	1.34613	−	1.34613i	0.456308	−	0.889822i	$-0.349172\pi$
$617$	4.84585e12	−	4.84585e12i	1.34613	−	1.34613i	0.889822	−	0.456308i	$-0.150828\pi$
$618$		0			0
$619$			2.43128e12i			0.665621i	0.942994	+	0.332810i	$0.107997\pi$
$619$			2.43128e12i			0.665621i	−0.942994	+	0.332810i	$0.892003\pi$
$620$		0			0
$621$	1.35606e12			0.365903
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−5.44094e11			−0.136629			−0.0683143	−	0.997664i	$-0.521762\pi$
$631$	−5.44094e11			−0.136629			−0.0683143	+	0.997664i	$0.521762\pi$
$632$	3.62907e12	−	3.62907e12i	0.904834	−	0.904834i
$633$	4.32263e12	+	4.32263e12i	1.07012	+	1.07012i
$634$			5.52170e12i			1.35729i
$635$		0			0
$636$	−4.09925e12			−0.993454
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	4.19240e12	−	4.19240e12i	0.973992	−	0.973992i
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$	4.80381e12			1.08527
$647$	−6.12174e12	+	6.12174e12i	−1.37343	+	1.37343i	−0.518116	+	0.855310i	$0.673367\pi$
$647$	−6.12174e12	+	6.12174e12i	−1.37343	+	1.37343i	−0.855310	+	0.518116i	$0.826633\pi$
$648$	−3.33520e12	−	3.33520e12i	−0.743078	−	0.743078i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.08337e12	−	4.08337e12i	−0.878839	−	0.878839i	0.114576	−	0.993415i	$-0.463449\pi$
$653$	−4.08337e12	−	4.08337e12i	−0.878839	−	0.878839i	−0.993415	+	0.114576i	$0.963449\pi$
$654$		−	2.09184e12i		−	0.447124i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−7.29256e12			−1.48585			−0.742923	−	0.669377i	$-0.766561\pi$
$661$	−7.29256e12			−1.48585			−0.742923	+	0.669377i	$0.766561\pi$
$662$	−4.80872e12	+	4.80872e12i	−0.973125	+	0.973125i
$663$		0			0
$664$			1.04084e13i			2.07791i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−3.82089e12	−	3.82089e12i	−0.742456	−	0.742456i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$		0			0
$676$	−2.85261e12			−0.525391
$677$	6.87766e12	−	6.87766e12i	1.25832	−	1.25832i	0.306428	−	0.951894i	$-0.400866\pi$
$677$	6.87766e12	−	6.87766e12i	1.25832	−	1.25832i	0.951894	−	0.306428i	$-0.0991338\pi$
$678$	−1.77059e12	−	1.77059e12i	−0.321798	−	0.321798i
$679$		0			0
$680$		0			0
$681$	9.49336e12			1.69145
$682$		0			0
$683$	7.05089e12	+	7.05089e12i	1.23980	+	1.23980i	0.960083	+	0.279714i	$0.0902397\pi$
$683$	7.05089e12	+	7.05089e12i	1.23980	+	1.23980i	0.279714	+	0.960083i	$0.409760\pi$
$684$		−	5.48701e12i		−	0.958481i
$685$		0			0
$686$		0			0
$687$	−1.01985e12	+	1.01985e12i	−0.174675	+	0.174675i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	2.84838e12			0.475278			0.237639	−	0.971354i	$-0.423627\pi$
$691$	2.84838e12			0.475278			0.237639	+	0.971354i	$0.423627\pi$
$692$	−4.30689e12	+	4.30689e12i	−0.713981	+	0.713981i
$693$		0			0
$694$			4.73585e12i			0.774962i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	4.64839e12	+	4.64839e12i	0.741230	+	0.741230i
$699$			5.56770e12i			0.882121i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−7.69338e12			−1.16546
$707$		0			0
$708$		0			0
$709$		−	1.33568e13i		−	1.98516i	−0.121592	−	0.992580i	$-0.538800\pi$
$709$		−	1.33568e13i		−	1.98516i	0.121592	−	0.992580i	$-0.461200\pi$
$710$		0			0
$711$	8.29751e12			1.21768
$712$		0			0
$713$	2.86379e12	+	2.86379e12i	0.414991	+	0.414991i
$714$		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
$721$		0			0
$722$	−8.28095e12	+	8.28095e12i	−1.13413	+	1.13413i
$723$	5.69549e12	+	5.69549e12i	0.775192	+	0.775192i
$724$			5.62305e12i			0.760586i
$725$		0			0
$726$	5.15683e12			0.688919
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	7.62560e12i		−	1.00000i
$730$		0			0
$731$		0			0
$732$	−5.28102e12	+	5.28102e12i	−0.679857	+	0.679857i
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$	2.66253e12			0.334461
$737$		0			0
$738$		0			0
$739$		−	9.97704e12i		−	1.23056i	−0.788310	−	0.615279i	$-0.789043\pi$
$739$		−	9.97704e12i		−	1.23056i	0.788310	−	0.615279i	$-0.210957\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.45993e12	+	8.45993e12i	1.01840	+	1.01840i	0.999828	+	0.0185689i	$0.00591102\pi$
$743$	8.45993e12	+	8.45993e12i	1.01840	+	1.01840i	0.0185689	+	0.999828i	$0.494089\pi$
$744$		−	1.40869e13i		−	1.68553i
$745$		0			0
$746$		0			0
$747$	−1.18988e13	+	1.18988e13i	−1.39818	+	1.39818i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.68820e13			−1.93662			−0.968309	−	0.249756i	$-0.919650\pi$
$751$	−1.68820e13			−1.93662			−0.968309	+	0.249756i	$0.919650\pi$
$752$	−2.44712e12	+	2.44712e12i	−0.279045	+	0.279045i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	−7.07845e12	−	7.07845e12i	−0.778801	−	0.778801i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−6.34238e12			−0.665615
$767$		0			0
$768$	−7.25969e12	−	7.25969e12i	−0.752996	−	0.752996i
$769$			1.13083e13i			1.16608i	0.812445	+	0.583038i	$0.198136\pi$
$769$			1.13083e13i			1.16608i	−0.812445	+	0.583038i	$0.801864\pi$
$770$		0			0
$771$	1.81541e13			1.85025
$772$		0			0
$773$	1.05114e13	+	1.05114e13i	1.05890	+	1.05890i	0.998153	+	0.0607462i	$0.0193480\pi$
$773$	1.05114e13	+	1.05114e13i	1.05890	+	1.05890i	0.0607462	+	0.998153i	$0.480652\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−1.60961e12	+	1.60961e12i	−0.153918	+	0.153918i
$783$		0			0
$784$		−	2.10061e12i		−	0.198574i
$785$		0			0
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−1.04923e12	−	1.04923e12i	−0.0969400	−	0.0969400i
$789$			1.76282e13i			1.61942i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	8.99111e12			0.793788
$797$	1.60365e13	−	1.60365e13i	1.40782	−	1.40782i	0.636750	−	0.771070i	$-0.280278\pi$
$797$	1.60365e13	−	1.60365e13i	1.40782	−	1.40782i	0.771070	−	0.636750i	$-0.219722\pi$
$798$		0			0
$799$			1.97696e13i			1.71608i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		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
$811$	−1.97521e13			−1.60331			−0.801657	−	0.597784i	$-0.796048\pi$
$811$	−1.97521e13			−1.60331			−0.801657	+	0.597784i	$0.796048\pi$
$812$		0			0
$813$	−9.45665e12	−	9.45665e12i	−0.759154	−	0.759154i
$814$		0			0
$815$		0			0
$816$	2.17169e12			0.171472
$817$		0			0
$818$	3.08101e9	+	3.08101e9i	0.000240604	+	0.000240604i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	5.40268e12	−	5.40268e12i	0.412749	−	0.412749i
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.84736e13	+	1.84736e13i	−1.37334	+	1.37334i	−0.517893	+	0.855445i	$0.673284\pi$
$827$	−1.84736e13	+	1.84736e13i	−1.37334	+	1.37334i	−0.855445	+	0.517893i	$0.826716\pi$
$828$	1.83853e12	+	1.83853e12i	0.135936	+	0.135936i
$829$			1.53301e13i			1.12732i	0.826006	+	0.563662i	$0.190608\pi$
$829$			1.53301e13i			1.12732i	−0.826006	+	0.563662i	$0.809392\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8.48512e12	−	8.48512e12i	−0.610598	−	0.610598i
$834$		−	1.92377e13i		−	1.37691i
$835$		0			0
$836$		0			0
$837$	1.61041e13	−	1.61041e13i	1.13416	−	1.13416i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−1.45071e13			−1.00000
$842$	1.33849e13	−	1.33849e13i	0.917719	−	0.917719i
$843$		0			0
$844$			1.17211e13i			0.795112i
$845$		0			0
$846$	−2.03987e13			−1.36910
$847$		0			0
$848$	3.99811e12	+	3.99811e12i	0.265505	+	0.265505i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$	3.30053e13			2.10112
$857$	1.95627e13	−	1.95627e13i	1.23884	−	1.23884i	0.278360	−	0.960477i	$-0.410209\pi$
$857$	1.95627e13	−	1.95627e13i	1.23884	−	1.23884i	0.960477	−	0.278360i	$-0.0897908\pi$
$858$		0			0
$859$		−	3.11951e13i		−	1.95487i	−0.211238	−	0.977435i	$-0.567750\pi$
$859$		−	3.11951e13i		−	1.95487i	0.211238	−	0.977435i	$-0.432250\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.56568e12	+	1.56568e12i	0.0960849	+	0.0960849i	0.753515	−	0.657430i	$-0.228357\pi$
$863$	1.56568e12	+	1.56568e12i	0.0960849	+	0.0960849i	−0.657430	+	0.753515i	$0.728357\pi$
$864$		−	1.49724e13i		−	0.914069i
$865$		0			0
$866$		0			0
$867$	−2.99217e12	+	2.99217e12i	−0.179846	+	0.179846i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	8.23415e12	−	8.23415e12i	0.482275	−	0.482275i
$873$		0			0
$874$		−	7.93298e12i		−	0.459870i
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$	−6.43155e12	−	6.43155e12i	−0.365250	−	0.365250i
$879$			2.02423e13i			1.14369i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	8.75511e12	−	8.75511e12i	0.487139	−	0.487139i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$	−1.74507e13			−0.951395
$887$	5.54271e12	−	5.54271e12i	0.300653	−	0.300653i	−0.540616	−	0.841269i	$-0.681809\pi$
$887$	5.54271e12	−	5.54271e12i	0.300653	−	0.300653i	0.841269	+	0.540616i	$0.181809\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.87174e13	−	4.87174e13i	−2.56362	−	2.56362i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	3.22996e13			1.63281
$902$		0			0
$903$		0			0
$904$		−	1.39392e13i		−	0.694192i
$905$		0			0
$906$	−4.38513e12			−0.216225
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	1.28710e13	+	1.28710e13i	0.628384	+	0.628384i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−5.35162e12	+	5.35162e12i	−0.256158	+	0.256158i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−2.76541e12			−0.129786
$917$		0			0
$918$	9.05140e12	+	9.05140e12i	0.420652	+	0.420652i
$919$		−	3.98586e13i		−	1.84333i	−0.387991	−	0.921663i	$-0.626831\pi$
$919$		−	3.98586e13i		−	1.84333i	0.387991	−	0.921663i	$-0.373169\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	4.18191e13			1.82432
$932$	−7.54861e12	+	7.54861e12i	−0.327714	+	0.327714i
$933$		0			0
$934$			2.55400e13i			1.09814i
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		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
$947$	−3.09253e13	+	3.09253e13i	−1.24951	+	1.24951i	−0.293569	+	0.955938i	$0.594843\pi$
$947$	−3.09253e13	+	3.09253e13i	−1.24951	+	1.24951i	−0.955938	+	0.293569i	$0.905157\pi$
$948$	1.12497e13	+	1.12497e13i	0.452378	+	0.452378i
$949$		0			0
$950$		0			0
$951$	−4.96953e13			−1.97017
$952$		0			0
$953$	1.85294e12	+	1.85294e12i	0.0727683	+	0.0727683i	0.742554	−	0.669786i	$-0.233614\pi$
$953$	1.85294e12	+	1.85294e12i	0.0727683	+	0.0727683i	−0.669786	+	0.742554i	$0.733614\pi$
$954$			3.33274e13i			1.30267i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	4.15795e13			1.57262
$962$		0			0
$963$	3.77316e13	+	3.77316e13i	1.41380	+	1.41380i
$964$			1.54438e13i			0.575979i
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	2.02989e13	+	2.02989e13i	0.743078	+	0.743078i
$969$			4.32343e13i			1.57533i
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	1.03387e13	−	1.03387e13i	0.371507	−	0.371507i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.03014e13			0.363390
$977$	−3.21988e13	+	3.21988e13i	−1.13061	+	1.13061i	−0.140538	+	0.990075i	$0.544883\pi$
$977$	−3.21988e13	+	3.21988e13i	−1.13061	+	1.13061i	−0.990075	+	0.140538i	$0.955117\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.88266e13			0.649023
$982$		0			0
$983$	1.97557e13	+	1.97557e13i	0.674840	+	0.674840i	0.958828	−	0.283988i	$-0.0916574\pi$
$983$	1.97557e13	+	1.97557e13i	0.674840	+	0.674840i	−0.283988	+	0.958828i	$0.591657\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−6.04995e13			−1.99260			−0.996300	−	0.0859393i	$-0.972611\pi$
$991$	−6.04995e13			−1.99260			−0.996300	+	0.0859393i	$0.972611\pi$
$992$	3.16195e13	−	3.16195e13i	1.03670	−	1.03670i
$993$	−4.32785e13	−	4.32785e13i	−1.41254	−	1.41254i
$994$		0			0
$995$		0			0
$996$	−3.22646e13			−1.03887
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$	−7.52353e10	−	7.52353e10i	−0.00240068	−	0.00240068i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.10.e.c.32.2	yes	4
3.2	odd	2	inner	75.10.e.c.32.1	✓	4
5.2	odd	4	inner	75.10.e.c.68.2	yes	4
5.3	odd	4	inner	75.10.e.c.68.1	yes	4
5.4	even	2	inner	75.10.e.c.32.1	✓	4
15.2	even	4	inner	75.10.e.c.68.1	yes	4
15.8	even	4	inner	75.10.e.c.68.2	yes	4
15.14	odd	2	CM	75.10.e.c.32.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.10.e.c.32.1	✓	4	3.2	odd	2	inner
75.10.e.c.32.1	✓	4	5.4	even	2	inner
75.10.e.c.32.2	yes	4	1.1	even	1	trivial
75.10.e.c.32.2	yes	4	15.14	odd	2	CM
75.10.e.c.68.1	yes	4	5.3	odd	4	inner
75.10.e.c.68.1	yes	4	15.2	even	4	inner
75.10.e.c.68.2	yes	4	5.2	odd	4	inner
75.10.e.c.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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(11.0227 - 11.0227i) q^{2} +(99.2043 + 99.2043i) q^{3} +269.000i q^{4} +2187.00 q^{6} +(8608.73 + 8608.73i) q^{8} +19683.0i q^{9} +O(q^{10})\)	\(q+(11.0227 - 11.0227i) q^{2} +(99.2043 + 99.2043i) q^{3} +269.000i q^{4} +2187.00 q^{6} +(8608.73 + 8608.73i) q^{8} +19683.0i q^{9} +(-26686.0 + 26686.0i) q^{12} +52055.0 q^{16} +(210269. - 210269. i) q^{17} +(216960. + 216960. i) q^{18} +1.03632e6i q^{19} +(-347237. - 347237. i) q^{23} +1.70805e6i q^{24} +(-1.95264e6 + 1.95264e6i) q^{27} -8.24737e6 q^{31} +(-3.83388e6 + 3.83388e6i) q^{32} -4.63547e6i q^{34} -5.29473e6 q^{36} +(1.14230e7 + 1.14230e7i) q^{38} -7.65499e6 q^{46} +(-4.70102e7 + 4.70102e7i) q^{47} +(5.16408e6 + 5.16408e6i) q^{48} -4.03536e7i q^{49} +4.17192e7 q^{51} +(7.68054e7 + 7.68054e7i) q^{53} +4.30467e7i q^{54} +(-1.02807e8 + 1.02807e8i) q^{57} +1.97895e8 q^{61} +(-9.09083e7 + 9.09083e7i) q^{62} +1.11172e8i q^{64} +(5.65624e7 + 5.65624e7i) q^{68} -6.88949e7i q^{69} +(-1.69446e8 + 1.69446e8i) q^{72} -2.78769e8 q^{76} -4.21557e8i q^{79} -3.87420e8 q^{81} +(6.04524e8 + 6.04524e8i) q^{83} +(9.34068e7 - 9.34068e7i) q^{92} +(-8.18175e8 - 8.18175e8i) q^{93} +1.03636e9i q^{94} -7.60676e8 q^{96} +(-4.44806e8 - 4.44806e8i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8748 q^{6}+O(q^{10}) \) 4 * q + 8748 * q^6	\( 4 q + 8748 q^{6} + 208220 q^{16} - 32989472 q^{31} - 21178908 q^{36} - 30619944 q^{46} + 166876848 q^{51} + 791579528 q^{61} - 1115076016 q^{76} - 1549681956 q^{81} - 3042703116 q^{96}+O(q^{100}) \) 4 * q + 8748 * q^6 + 208220 * q^16 - 32989472 * q^31 - 21178908 * q^36 - 30619944 * q^46 + 166876848 * q^51 + 791579528 * q^61 - 1115076016 * q^76 - 1549681956 * q^81 - 3042703116 * q^96

Introduction

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