LMFDB - Embedded newform 75.9.c.d.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(26,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("75.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.5533957546$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			26.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	75.26
Dual form			75.9.c.d.26.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.0000i q^{2} +81.0000i q^{3} -33.0000 q^{4} +1377.00 q^{6} -3791.00i q^{8} -6561.00 q^{9} +O(q^{10})$	$q-17.0000i q^{2} +81.0000i q^{3} -33.0000 q^{4} +1377.00 q^{6} -3791.00i q^{8} -6561.00 q^{9} -2673.00i q^{12} -72895.0 q^{16} +21118.0i q^{17} +111537. i q^{18} +203998. q^{19} -550078. i q^{23} +307071. q^{24} -531441. i q^{27} +1.83168e6 q^{31} +268719. i q^{32} +359006. q^{34} +216513. q^{36} -3.46797e6i q^{38} -9.35133e6 q^{46} -8.06592e6i q^{47} -5.90450e6i q^{48} -5.76480e6 q^{49} -1.71056e6 q^{51} -1.26197e7i q^{53} -9.03450e6 q^{54} +1.65238e7i q^{57} +1.43246e7 q^{61} -3.11386e7i q^{62} -1.40929e7 q^{64} -696894. i q^{68} +4.45563e7 q^{69} +2.48728e7i q^{72} -6.73193e6 q^{76} +6.96173e7 q^{79} +4.30467e7 q^{81} +3.84720e6i q^{83} +1.81526e7i q^{92} +1.48366e8i q^{93} -1.37121e8 q^{94} -2.17662e7 q^{96} +9.80016e7i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 66 q^{4} + 2754 q^{6} - 13122 q^{9}+O(q^{10}) $ 2 * q - 66 * q^4 + 2754 * q^6 - 13122 * q^9	$ 2 q - 66 q^{4} + 2754 q^{6} - 13122 q^{9} - 145790 q^{16} + 407996 q^{19} + 614142 q^{24} + 3663364 q^{31} + 718012 q^{34} + 433026 q^{36} - 18702652 q^{46} - 11529602 q^{49} - 3421116 q^{51} - 18068994 q^{54} + 28649284 q^{61} - 28185794 q^{64} + 89112636 q^{69} - 13463868 q^{76} + 139234556 q^{79} + 86093442 q^{81} - 274241348 q^{94} - 43532478 q^{96}+O(q^{100}) $ 2 * q - 66 * q^4 + 2754 * q^6 - 13122 * q^9 - 145790 * q^16 + 407996 * q^19 + 614142 * q^24 + 3663364 * q^31 + 718012 * q^34 + 433026 * q^36 - 18702652 * q^46 - 11529602 * q^49 - 3421116 * q^51 - 18068994 * q^54 + 28649284 * q^61 - 28185794 * q^64 + 89112636 * q^69 - 13463868 * q^76 + 139234556 * q^79 + 86093442 * q^81 - 274241348 * q^94 - 43532478 * q^96

Display 100 coefficients 10 coefficients

Character values

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

$n$	$26$	$52$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 17.0000i	− 1.06250i	−0.847215	−	0.531250i	$-0.821722\pi$
$2$	− 17.0000i	− 1.06250i	0.847215	−	0.531250i	$-0.178278\pi$
$3$	81.0000i	1.00000i
$3$	81.0000i	1.00000i
$4$	−33.0000	−0.128906
$5$	0	0
$5$	0	0
$6$	1377.00	1.06250
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	− 3791.00i	− 0.925537i
$9$	−6561.00	−1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	− 2673.00i	− 0.128906i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	−72895.0	−1.11229
$17$	21118.0i	0.252847i	0.991976	+	0.126423i	$0.0403498\pi$
$17$	21118.0i	0.252847i	−0.991976	+	0.126423i	$0.959650\pi$
$18$	111537.i	1.06250i
$19$	203998.	1.56535	0.782675	−	0.622430i	$-0.213855\pi$
$19$	203998.	1.56535	0.782675	+	0.622430i	$0.213855\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 550078.i	− 1.96568i	−0.184459	−	0.982840i	$-0.559053\pi$
$23$	− 550078.i	− 1.96568i	0.184459	−	0.982840i	$-0.440947\pi$
$24$	307071.	0.925537
$25$	0	0
$26$	0	0
$27$	− 531441.i	− 1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	1.83168e6	1.98337	0.991684	−	0.128697i	$-0.0410794\pi$
$31$	1.83168e6	1.98337	0.991684	+	0.128697i	$0.0410794\pi$
$32$	268719.i	0.256270i
$33$	0	0
$34$	359006.	0.268650
$35$	0	0
$36$	216513.	0.128906
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	− 3.46797e6i	− 1.66318i
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	−9.35133e6	−2.08854
$47$	− 8.06592e6i	− 1.65296i	−0.562965	−	0.826480i	$-0.690340\pi$
$47$	− 8.06592e6i	− 1.65296i	0.562965	−	0.826480i	$-0.309660\pi$
$48$	− 5.90450e6i	− 1.11229i
$49$	−5.76480e6	−1.00000
$50$	0	0
$51$	−1.71056e6	−0.252847
$52$	0	0
$53$	− 1.26197e7i	− 1.59935i	−0.600430	−	0.799677i	$-0.705004\pi$
$53$	− 1.26197e7i	− 1.59935i	0.600430	−	0.799677i	$-0.294996\pi$
$54$	−9.03450e6	−1.06250
$55$	0	0
$56$	0	0
$57$	1.65238e7i	1.56535i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	1.43246e7	1.03458	0.517290	−	0.855810i	$-0.326941\pi$
$61$	1.43246e7	1.03458	0.517290	+	0.855810i	$0.326941\pi$
$62$	− 3.11386e7i	− 2.10733i
$63$	0	0
$64$	−1.40929e7	−0.840002
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	− 696894.i	− 0.0325935i
$69$	4.45563e7	1.96568
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	2.48728e7i	0.925537i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	−6.73193e6	−0.201783
$77$	0	0
$78$	0	0
$79$	6.96173e7	1.78735	0.893673	−	0.448719i	$-0.148119\pi$
$79$	6.96173e7	1.78735	0.893673	+	0.448719i	$0.148119\pi$
$80$	0	0
$81$	4.30467e7	1.00000
$82$	0	0
$83$	3.84720e6i	0.0810649i	0.999178	+	0.0405324i	$0.0129054\pi$
$83$	3.84720e6i	0.0810649i	−0.999178	+	0.0405324i	$0.987095\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	1.81526e7i	0.253389i
$93$	1.48366e8i	1.98337i
$94$	−1.37121e8	−1.75627
$95$	0	0
$96$	−2.17662e7	−0.256270
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	9.80016e7i	1.06250i
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	2.90795e7i	0.268650i
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	−2.14535e8	−1.69931
$107$	1.26157e8i	0.962445i	0.876598	+	0.481223i	$0.159807\pi$
$107$	1.26157e8i	0.962445i	−0.876598	+	0.481223i	$0.840193\pi$
$108$	1.75376e7i	0.128906i
$109$	−2.59549e8	−1.83871	−0.919355	−	0.393429i	$-0.871289\pi$
$109$	−2.59549e8	−1.83871	−0.919355	+	0.393429i	$0.871289\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4.05523e7i	− 0.248715i	−0.992237	−	0.124357i	$-0.960313\pi$
$113$	− 4.05523e7i	− 0.248715i	0.992237	−	0.124357i	$-0.0396870\pi$
$114$	2.80905e8	1.66318
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.14359e8	1.00000
$122$	− 2.43519e8i	− 1.09924i
$123$	0	0
$124$	−6.04455e7	−0.255669
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	3.08371e8i	1.14877i
$129$	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
$136$	8.00583e7	0.234019
$137$	5.21273e8i	1.47973i	0.672754	+	0.739866i	$0.265111\pi$
$137$	5.21273e8i	1.47973i	−0.672754	+	0.739866i	$0.734889\pi$
$138$	− 7.57457e8i	− 2.08854i
$139$	−1.53518e8	−0.411244	−0.205622	−	0.978631i	$-0.565922\pi$
$139$	−1.53518e8	−0.411244	−0.205622	+	0.978631i	$0.565922\pi$
$140$	0	0
$141$	6.53340e8	1.65296
$142$	0	0
$143$	0	0
$144$	4.78264e8	1.11229
$145$	0	0
$146$	0	0
$147$	− 4.66949e8i	− 1.00000i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−9.17845e8	−1.76548	−0.882738	−	0.469866i	$-0.844302\pi$
$151$	−9.17845e8	−1.76548	−0.882738	+	0.469866i	$0.844302\pi$
$152$	− 7.73356e8i	− 1.44879i
$153$	− 1.38555e8i	− 0.252847i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	− 1.18349e9i	− 1.89906i
$159$	1.02219e9	1.59935
$160$	0	0
$161$	0	0
$162$	− 7.31794e8i	− 1.06250i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	6.54024e7	0.0861314
$167$	1.47924e9i	1.90183i	0.309445	+	0.950917i	$0.399857\pi$
$167$	1.47924e9i	1.90183i	−0.309445	+	0.950917i	$0.600143\pi$
$168$	0	0
$169$	−8.15731e8	−1.00000
$170$	0	0
$171$	−1.33843e9	−1.56535
$172$	0	0
$173$	− 4.85246e8i	− 0.541723i	−0.962618	−	0.270862i	$-0.912691\pi$
$173$	− 4.85246e8i	− 0.541723i	0.962618	−	0.270862i	$-0.0873086\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	4.17641e8	0.389125	0.194562	−	0.980890i	$-0.437671\pi$
$181$	4.17641e8	0.389125	0.194562	+	0.980890i	$0.437671\pi$
$182$	0	0
$183$	1.16030e9i	1.03458i
$184$	−2.08535e9	−1.81931
$185$	0	0
$186$	2.52223e9	2.10733
$187$	0	0
$188$	2.66175e8i	0.213077i
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	− 1.14152e9i	− 0.840002i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	1.90238e8	0.128906
$197$	− 8.63782e8i	− 0.573507i	−0.958004	−	0.286754i	$-0.907424\pi$
$197$	− 8.63782e8i	− 0.573507i	0.958004	−	0.286754i	$-0.0925761\pi$
$198$	0	0
$199$	−3.49007e8	−0.222547	−0.111274	−	0.993790i	$-0.535493\pi$
$199$	−3.49007e8	−0.222547	−0.111274	+	0.993790i	$0.535493\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	5.64484e7	0.0325935
$205$	0	0
$206$	0	0
$207$	3.60906e9i	1.96568i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−2.20007e9	−1.10996	−0.554979	−	0.831864i	$-0.687274\pi$
$211$	−2.20007e9	−1.10996	−0.554979	+	0.831864i	$0.687274\pi$
$212$	4.16449e8i	0.206167i
$213$	0	0
$214$	2.14467e9	1.02260
$215$	0	0
$216$	−2.01469e9	−0.925537
$217$	0	0
$218$	4.41233e9i	1.95363i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	−6.89389e8	−0.264260
$227$	− 1.93187e9i	− 0.727571i	−0.931483	−	0.363786i	$-0.881484\pi$
$227$	− 1.93187e9i	− 0.727571i	0.931483	−	0.363786i	$-0.118516\pi$
$228$	− 5.45287e8i	− 0.201783i
$229$	2.21018e9	0.803684	0.401842	−	0.915709i	$-0.368370\pi$
$229$	2.21018e9	0.803684	0.401842	+	0.915709i	$0.368370\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.64490e9i	1.91528i	0.287969	+	0.957640i	$0.407020\pi$
$233$	5.64490e9i	1.91528i	−0.287969	+	0.957640i	$0.592980\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.63900e9i	1.78735i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	5.86943e9	1.73991	0.869956	−	0.493129i	$-0.164147\pi$
$241$	5.86943e9	1.73991	0.869956	+	0.493129i	$0.164147\pi$
$242$	− 3.64410e9i	− 1.06250i
$243$	3.48678e9i	1.00000i
$244$	−4.72713e8	−0.133364
$245$	0	0
$246$	0	0
$247$	0	0
$248$	− 6.94391e9i	− 1.83568i
$249$	−3.11623e8	−0.0810649
$250$	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
$256$	1.63453e9	0.380569
$257$	1.49008e9i	0.341567i	0.985309	+	0.170784i	$0.0546299\pi$
$257$	1.49008e9i	0.341567i	−0.985309	+	0.170784i	$0.945370\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 5.89901e9i	− 1.23298i	−0.787363	−	0.616490i	$-0.788554\pi$
$263$	− 5.89901e9i	− 1.23298i	0.787363	−	0.616490i	$-0.211446\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−3.48683e9	−0.646477	−0.323238	−	0.946318i	$-0.604772\pi$
$271$	−3.48683e9	−0.646477	−0.323238	+	0.946318i	$0.604772\pi$
$272$	− 1.53940e9i	− 0.281239i
$273$	0	0
$274$	8.86165e9	1.57222
$275$	0	0
$276$	−1.47036e9	−0.253389
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	2.60980e9i	0.436947i
$279$	−1.20177e10	−1.98337
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	− 1.11068e10i	− 1.75627i
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 1.76307e9i	− 0.256270i
$289$	6.52979e9	0.936069
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.44691e10i	− 1.96323i	−0.190872	−	0.981615i	$-0.561132\pi$
$293$	− 1.44691e10i	− 1.96323i	0.190872	−	0.981615i	$-0.438868\pi$
$294$	−7.93813e9	−1.06250
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	1.56034e10i	1.87582i
$303$	0	0
$304$	−1.48704e10	−1.74112
$305$	0	0
$306$	−2.35544e9	−0.268650
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	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		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	−2.29737e9	−0.230400
$317$	− 1.33010e10i	− 1.31718i	−0.752500	−	0.658592i	$-0.771152\pi$
$317$	− 1.33010e10i	− 1.31718i	0.752500	−	0.658592i	$-0.228848\pi$
$318$	− 1.73773e10i	− 1.69931i
$319$	0	0
$320$	0	0
$321$	−1.02187e10	−0.962445
$322$	0	0
$323$	4.30803e9i	0.395793i
$324$	−1.42054e9	−0.128906
$325$	0	0
$326$	0	0
$327$	− 2.10235e10i	− 1.83871i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.77059e10	1.47505	0.737525	−	0.675320i	$-0.235994\pi$
$331$	1.77059e10	1.47505	0.737525	+	0.675320i	$0.235994\pi$
$332$	− 1.26958e8i	− 0.0104498i
$333$	0	0
$334$	2.51471e10	2.02070
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	1.38674e10i	1.06250i
$339$	3.28474e9	0.248715
$340$	0	0
$341$	0	0
$342$	2.27533e10i	1.66318i
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−8.24918e9	−0.575581
$347$	1.84744e10i	1.27424i	0.770763	+	0.637122i	$0.219875\pi$
$347$	1.84744e10i	1.27424i	−0.770763	+	0.637122i	$0.780125\pi$
$348$	0	0
$349$	2.85260e10	1.92282	0.961410	−	0.275121i	$-0.0887178\pi$
$349$	2.85260e10	1.92282	0.961410	+	0.275121i	$0.0887178\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 7.29370e8i	− 0.0469731i	−0.999724	−	0.0234865i	$-0.992523\pi$
$353$	− 7.29370e8i	− 0.0469731i	0.999724	−	0.0234865i	$-0.00747668\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	2.46316e10	1.45032
$362$	− 7.09989e9i	− 0.413445i
$363$	1.73631e10i	1.00000i
$364$	0	0
$365$	0	0
$366$	1.97250e10	1.09924
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	4.00979e10i	2.18641i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	− 4.89609e9i	− 0.255669i
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	−3.05779e10	−1.52988
$377$	0	0
$378$	0	0
$379$	−2.80519e10	−1.35958	−0.679792	−	0.733405i	$-0.737930\pi$
$379$	−2.80519e10	−1.35958	−0.679792	+	0.733405i	$0.737930\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 1.16291e10i	− 0.540445i	−0.962798	−	0.270222i	$-0.912903\pi$
$383$	− 1.16291e10i	− 0.540445i	0.962798	−	0.270222i	$-0.0870972\pi$
$384$	−2.49781e10	−1.14877
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	1.16165e10	0.497016
$392$	2.18544e10i	0.925537i
$393$	0	0
$394$	−1.46843e10	−0.609352
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	5.93312e9i	0.236456i
$399$	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
$408$	6.48473e9i	0.234019i
$409$	−4.28802e10	−1.53237	−0.766186	−	0.642619i	$-0.777848\pi$
$409$	−4.28802e10	−1.53237	−0.766186	+	0.642619i	$0.777848\pi$
$410$	0	0
$411$	−4.22231e10	−1.47973
$412$	0	0
$413$	0	0
$414$	6.13540e10	2.08854
$415$	0	0
$416$	0	0
$417$	− 1.24350e10i	− 0.411244i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−6.27660e10	−1.99800	−0.999001	−	0.0446850i	$-0.985772\pi$
$421$	−6.27660e10	−1.99800	−0.999001	+	0.0446850i	$0.985772\pi$
$422$	3.74012e10i	1.17933i
$423$	5.29205e10i	1.65296i
$424$	−4.78412e10	−1.48026
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 4.16318e9i	− 0.124065i
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	3.87394e10i	1.11229i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	8.56512e9	0.237021
$437$	− 1.12215e11i	− 3.07698i
$438$	0	0
$439$	7.42807e10	1.99994	0.999972	−	0.00748227i	$-0.00238170\pi$
$439$	7.42807e10	1.99994	0.999972	+	0.00748227i	$0.00238170\pi$
$440$	0	0
$441$	3.78229e10	1.00000
$442$	0	0
$443$	− 7.18229e10i	− 1.86487i	−0.361342	−	0.932433i	$-0.617681\pi$
$443$	− 7.18229e10i	− 1.86487i	0.361342	−	0.932433i	$-0.382319\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	1.33823e9i	0.0320609i
$453$	− 7.43455e10i	− 1.76548i
$454$	−3.28419e10	−0.773044
$455$	0	0
$456$	6.26419e10	1.44879
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	− 3.75730e10i	− 0.853914i
$459$	1.12230e10	0.252847
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	9.59632e10	2.03498
$467$	− 5.02257e9i	− 0.105599i	−0.998605	−	0.0527994i	$-0.983186\pi$
$467$	− 5.02257e9i	− 0.105599i	0.998605	−	0.0527994i	$-0.0168144\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	9.58630e10	1.89906
$475$	0	0
$476$	0	0
$477$	8.27977e10i	1.59935i
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	− 9.97803e10i	− 1.84866i
$483$	0	0
$484$	−7.07384e9	−0.128906
$485$	0	0
$486$	5.92753e10	1.06250
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	− 5.43047e10i	− 0.957543i
$489$	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
$496$	−1.33520e11	−2.20608
$497$	0	0
$498$	5.29760e9i	0.0861314i
$499$	−2.18003e10	−0.351610	−0.175805	−	0.984425i	$-0.556253\pi$
$499$	−2.18003e10	−0.351610	−0.175805	+	0.984425i	$0.556253\pi$
$500$	0	0
$501$	−1.19818e11	−1.90183
$502$	0	0
$503$	1.04314e11i	1.62956i	0.579767	+	0.814782i	$0.303144\pi$
$503$	1.04314e11i	1.62956i	−0.579767	+	0.814782i	$0.696856\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6.60742e10i	− 1.00000i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	5.11560e10i	0.744418i
$513$	− 1.08413e11i	− 1.56535i
$514$	2.53313e10	0.362915
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	3.93049e10	0.541723
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	−1.00283e11	−1.31004
$527$	3.86815e10i	0.501488i
$528$	0	0
$529$	−2.24275e11	−2.86390
$530$	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
$541$	1.61284e11	1.88279	0.941395	−	0.337305i	$-0.109515\pi$
$541$	1.61284e11	1.88279	0.941395	+	0.337305i	$0.109515\pi$
$542$	5.92760e10i	0.686882i
$543$	3.38289e10i	0.389125i
$544$	−5.67481e9	−0.0647971
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	− 1.72020e10i	− 0.190747i
$549$	−9.39840e10	−1.03458
$550$	0	0
$551$	0	0
$552$	− 1.68913e11i	− 1.81931i
$553$	0	0
$554$	0	0
$555$	0	0
$556$	5.06609e9	0.0530120
$557$	1.33216e11i	1.38399i	0.721900	+	0.691997i	$0.243269\pi$
$557$	1.33216e11i	1.38399i	−0.721900	+	0.691997i	$0.756731\pi$
$558$	2.04300e11i	2.10733i
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.71432e11i	1.70631i	0.521655	+	0.853157i	$0.325315\pi$
$563$	1.71432e11i	1.70631i	−0.521655	+	0.853157i	$0.674685\pi$
$564$	−2.15602e10	−0.213077
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	6.18846e10	0.582154	0.291077	−	0.956700i	$-0.405986\pi$
$571$	6.18846e10	0.582154	0.291077	+	0.956700i	$0.405986\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	9.24635e10	0.840002
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	− 1.11006e11i	− 0.994573i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−2.45975e11	−2.08593
$587$	2.35841e11i	1.98640i	0.116405	+	0.993202i	$0.462863\pi$
$587$	2.35841e11i	1.98640i	−0.116405	+	0.993202i	$0.537137\pi$
$588$	1.54093e10i	0.128906i
$589$	3.73659e11	3.10467
$590$	0	0
$591$	6.99663e10	0.573507
$592$	0	0
$593$	1.83361e11i	1.48282i	0.671055	+	0.741408i	$0.265841\pi$
$593$	1.83361e11i	1.48282i	−0.671055	+	0.741408i	$0.734159\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 2.82696e10i	− 0.222547i
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	1.79749e11	1.37774	0.688870	−	0.724884i	$-0.258107\pi$
$601$	1.79749e11	1.37774	0.688870	+	0.724884i	$0.258107\pi$
$602$	0	0
$603$	0	0
$604$	3.02889e10	0.227581
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	5.48181e10i	0.401153i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	4.57232e9i	0.0325935i
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.26458e11i	− 0.872581i	−0.899806	−	0.436291i	$-0.856292\pi$
$617$	− 1.26458e11i	− 0.872581i	0.899806	−	0.436291i	$-0.143708\pi$
$618$	0	0
$619$	2.15718e11	1.46934	0.734672	−	0.678422i	$-0.237336\pi$
$619$	2.15718e11	1.46934	0.734672	+	0.678422i	$0.237336\pi$
$620$	0	0
$621$	−2.92334e11	−1.96568
$622$	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
$631$	2.30059e11	1.45118	0.725591	−	0.688126i	$-0.241566\pi$
$631$	2.30059e11	1.45118	0.725591	+	0.688126i	$0.241566\pi$
$632$	− 2.63919e11i	− 1.65426i
$633$	− 1.78206e11i	− 1.10996i
$634$	−2.26117e11	−1.39951
$635$	0	0
$636$	−3.37324e10	−0.206167
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	1.73718e11i	1.02260i
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	7.32365e10	0.420531
$647$	2.35694e11i	1.34503i	0.740085	+	0.672513i	$0.234785\pi$
$647$	2.35694e11i	1.34503i	−0.740085	+	0.672513i	$0.765215\pi$
$648$	− 1.63190e11i	− 0.925537i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.12967e11i	− 1.17128i	−0.810572	−	0.585639i	$-0.800844\pi$
$653$	− 2.12967e11i	− 1.17128i	0.810572	−	0.585639i	$-0.199156\pi$
$654$	−3.57399e11	−1.95363
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−3.52354e11	−1.84575	−0.922876	−	0.385097i	$-0.874168\pi$
$661$	−3.52354e11	−1.84575	−0.922876	+	0.385097i	$0.874168\pi$
$662$	− 3.01001e11i	− 1.56724i
$663$	0	0
$664$	1.45847e10	0.0750285
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 4.88149e10i	− 0.245158i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	2.69191e10	0.128906
$677$	− 1.83259e11i	− 0.872391i	−0.899852	−	0.436196i	$-0.856326\pi$
$677$	− 1.83259e11i	− 0.872391i	0.899852	−	0.436196i	$-0.143674\pi$
$678$	− 5.58405e10i	− 0.264260i
$679$	0	0
$680$	0	0
$681$	1.56482e11	0.727571
$682$	0	0
$683$	4.21478e11i	1.93683i	0.249338	+	0.968416i	$0.419787\pi$
$683$	4.21478e11i	1.93683i	−0.249338	+	0.968416i	$0.580213\pi$
$684$	4.41682e10	0.201783
$685$	0	0
$686$	0	0
$687$	1.79024e11i	0.803684i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1.72432e11	0.756320	0.378160	−	0.925740i	$-0.376557\pi$
$691$	1.72432e11	0.756320	0.378160	+	0.925740i	$0.376557\pi$
$692$	1.60131e10i	0.0698315i
$693$	0	0
$694$	3.14065e11	1.35388
$695$	0	0
$696$	0	0
$697$	0	0
$698$	− 4.84941e11i	− 2.04300i
$699$	−4.57236e11	−1.91528
$700$	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
$706$	−1.23993e10	−0.0499089
$707$	0	0
$708$	0	0
$709$	2.98535e11	1.18144	0.590718	−	0.806878i	$-0.298844\pi$
$709$	2.98535e11	1.18144	0.590718	+	0.806878i	$0.298844\pi$
$710$	0	0
$711$	−4.56759e11	−1.78735
$712$	0	0
$713$	− 1.00757e12i	− 3.89867i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	− 4.18738e11i	− 1.54097i
$723$	4.75424e11i	1.73991i
$724$	−1.37821e10	−0.0501606
$725$	0	0
$726$	2.95172e11	1.06250
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−2.82430e11	−1.00000
$730$	0	0
$731$	0	0
$732$	− 3.82898e10i	− 0.133364i
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	1.47816e11	0.503746
$737$	0	0
$738$	0	0
$739$	−4.95951e11	−1.66288	−0.831440	−	0.555614i	$-0.812483\pi$
$739$	−4.95951e11	−1.66288	−0.831440	+	0.555614i	$0.812483\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.59192e10i	0.314739i	0.987540	+	0.157369i	$0.0503013\pi$
$743$	9.59192e10i	0.314739i	−0.987540	+	0.157369i	$0.949699\pi$
$744$	5.62456e11	1.83568
$745$	0	0
$746$	0	0
$747$	− 2.52415e10i	− 0.0810649i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.47109e11	0.776834	0.388417	−	0.921484i	$-0.373022\pi$
$751$	2.47109e11	0.776834	0.388417	+	0.921484i	$0.373022\pi$
$752$	5.87965e11i	1.83857i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	4.76883e11i	1.44456i
$759$	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
$766$	−1.97695e11	−0.574222
$767$	0	0
$768$	1.32397e11i	0.380569i
$769$	−2.07708e10	−0.0593947	−0.0296973	−	0.999559i	$-0.509454\pi$
$769$	−2.07708e10	−0.0593947	−0.0296973	+	0.999559i	$0.509454\pi$
$770$	0	0
$771$	−1.20696e11	−0.341567
$772$	0	0
$773$	5.71008e11i	1.59928i	0.600480	+	0.799639i	$0.294976\pi$
$773$	5.71008e11i	1.59928i	−0.600480	+	0.799639i	$0.705024\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	− 1.97481e11i	− 0.528079i
$783$	0	0
$784$	4.20225e11	1.11229
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	2.85048e10i	0.0739287i
$789$	4.77820e11	1.23298
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	1.15172e10	0.0286877
$797$	4.61070e11i	1.14270i	0.820705	+	0.571352i	$0.193581\pi$
$797$	4.61070e11i	1.14270i	−0.820705	+	0.571352i	$0.806419\pi$
$798$	0	0
$799$	1.70336e11	0.417946
$800$	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	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−8.44251e11	−1.95159	−0.975794	−	0.218694i	$-0.929820\pi$
$811$	−8.44251e11	−1.95159	−0.975794	+	0.218694i	$0.929820\pi$
$812$	0	0
$813$	− 2.82433e11i	− 0.646477i
$814$	0	0
$815$	0	0
$816$	1.24691e11	0.281239
$817$	0	0
$818$	7.28964e11i	1.62814i
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	7.17793e11i	1.57222i
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5.72422e11i	− 1.22375i	−0.790953	−	0.611877i	$-0.790415\pi$
$827$	− 5.72422e11i	− 1.22375i	0.790953	−	0.611877i	$-0.209585\pi$
$828$	− 1.19099e11i	− 0.253389i
$829$	−3.15356e11	−0.667703	−0.333852	−	0.942626i	$-0.608348\pi$
$829$	−3.15356e11	−0.667703	−0.333852	+	0.942626i	$0.608348\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1.21741e11i	− 0.252847i
$834$	−2.11394e11	−0.436947
$835$	0	0
$836$	0	0
$837$	− 9.73431e11i	− 1.98337i
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	5.00246e11	1.00000
$842$	1.06702e12i	2.12288i
$843$	0	0
$844$	7.26023e10	0.143081
$845$	0	0
$846$	8.99649e11	1.75627
$847$	0	0
$848$	9.19911e11i	1.77895i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	4.78261e11	0.890779
$857$	− 6.28524e11i	− 1.16520i	−0.812761	−	0.582598i	$-0.802036\pi$
$857$	− 6.28524e11i	− 1.16520i	0.812761	−	0.582598i	$-0.197964\pi$
$858$	0	0
$859$	−9.50927e11	−1.74652	−0.873262	−	0.487251i	$-0.838000\pi$
$859$	−9.50927e11	−1.74652	−0.873262	+	0.487251i	$0.838000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.91341e11i	1.60694i	0.595343	+	0.803472i	$0.297016\pi$
$863$	8.91341e11i	1.60694i	−0.595343	+	0.803472i	$0.702984\pi$
$864$	1.42808e11	0.256270
$865$	0	0
$866$	0	0
$867$	5.28913e11i	0.936069i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	9.83950e11i	1.70179i
$873$	0	0
$874$	−1.90765e12	−3.26929
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	− 1.26277e12i	− 2.12494i
$879$	1.17200e12	1.96323
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	− 6.42989e11i	− 1.06250i
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	−1.22099e12	−1.98142
$887$	− 4.41859e11i	− 0.713821i	−0.934139	−	0.356911i	$-0.883830\pi$
$887$	− 4.41859e11i	− 0.713821i	0.934139	−	0.356911i	$-0.116170\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.64543e12i	− 2.58746i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	2.66502e11	0.404391
$902$	0	0
$903$	0	0
$904$	−1.53734e11	−0.230195
$905$	0	0
$906$	−1.26387e12	−1.87582
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	6.37519e10i	0.0937885i
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	− 1.20451e12i	− 1.74112i
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−7.29358e10	−0.103600
$917$	0	0
$918$	− 1.90791e11i	− 0.268650i
$919$	1.42655e12	1.99997	0.999987	−	0.00511270i	$-0.00162743\pi$
$919$	1.42655e12	1.99997	0.999987	+	0.00511270i	$0.00162743\pi$
$920$	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	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−1.17601e12	−1.56535
$932$	− 1.86282e11i	− 0.246892i
$933$	0	0
$934$	−8.53838e10	−0.112199
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	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
$947$	1.14087e12i	1.41852i	0.704945	+	0.709262i	$0.250972\pi$
$947$	1.14087e12i	1.41852i	−0.704945	+	0.709262i	$0.749028\pi$
$948$	− 1.86087e11i	− 0.230400i
$949$	0	0
$950$	0	0
$951$	1.07738e12	1.31718
$952$	0	0
$953$	− 1.52277e12i	− 1.84613i	−0.384646	−	0.923064i	$-0.625676\pi$
$953$	− 1.52277e12i	− 1.84613i	0.384646	−	0.923064i	$-0.374324\pi$
$954$	1.40756e12	1.69931
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.50217e12	2.93375
$962$	0	0
$963$	− 8.27716e11i	− 0.962445i
$964$	−1.93691e11	−0.224286
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	− 8.12635e11i	− 0.925537i
$969$	−3.48950e11	−0.395793
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	− 1.15064e11i	− 0.128906i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−1.04419e12	−1.15075
$977$	− 1.53140e12i	− 1.68078i	−0.541983	−	0.840390i	$-0.682326\pi$
$977$	− 1.53140e12i	− 1.68078i	0.541983	−	0.840390i	$-0.317674\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1.70290e12	1.83871
$982$	0	0
$983$	1.68782e12i	1.80765i	0.427907	+	0.903823i	$0.359251\pi$
$983$	1.68782e12i	1.80765i	−0.427907	+	0.903823i	$0.640749\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−8.34017e11	−0.864730	−0.432365	−	0.901699i	$-0.642321\pi$
$991$	−8.34017e11	−0.864730	−0.432365	+	0.901699i	$0.642321\pi$
$992$	4.92208e11i	0.508279i
$993$	1.43418e12i	1.47505i
$994$	0	0
$995$	0	0
$996$	1.02836e10	0.0104498
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	3.70605e11i	0.373585i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.c.d.26.1		2
3.2	odd	2	inner	75.9.c.d.26.2		2
5.2	odd	4		15.9.d.b.14.1	yes	1
5.3	odd	4		15.9.d.a.14.1	✓	1
5.4	even	2	inner	75.9.c.d.26.2		2
15.2	even	4		15.9.d.a.14.1	✓	1
15.8	even	4		15.9.d.b.14.1	yes	1
15.14	odd	2	CM	75.9.c.d.26.1		2
20.3	even	4		240.9.c.b.209.1		1
20.7	even	4		240.9.c.a.209.1		1
60.23	odd	4		240.9.c.a.209.1		1
60.47	odd	4		240.9.c.b.209.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.9.d.a.14.1	✓	1	5.3	odd	4
15.9.d.a.14.1	✓	1	15.2	even	4
15.9.d.b.14.1	yes	1	5.2	odd	4
15.9.d.b.14.1	yes	1	15.8	even	4
75.9.c.d.26.1		2	1.1	even	1	trivial
75.9.c.d.26.1		2	15.14	odd	2	CM
75.9.c.d.26.2		2	3.2	odd	2	inner
75.9.c.d.26.2		2	5.4	even	2	inner
240.9.c.a.209.1		1	20.7	even	4
240.9.c.a.209.1		1	60.23	odd	4
240.9.c.b.209.1		1	20.3	even	4
240.9.c.b.209.1		1	60.47	odd	4

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-17.0000i q^{2} +81.0000i q^{3} -33.0000 q^{4} +1377.00 q^{6} -3791.00i q^{8} -6561.00 q^{9} +O(q^{10})\)	\(q-17.0000i q^{2} +81.0000i q^{3} -33.0000 q^{4} +1377.00 q^{6} -3791.00i q^{8} -6561.00 q^{9} -2673.00i q^{12} -72895.0 q^{16} +21118.0i q^{17} +111537. i q^{18} +203998. q^{19} -550078. i q^{23} +307071. q^{24} -531441. i q^{27} +1.83168e6 q^{31} +268719. i q^{32} +359006. q^{34} +216513. q^{36} -3.46797e6i q^{38} -9.35133e6 q^{46} -8.06592e6i q^{47} -5.90450e6i q^{48} -5.76480e6 q^{49} -1.71056e6 q^{51} -1.26197e7i q^{53} -9.03450e6 q^{54} +1.65238e7i q^{57} +1.43246e7 q^{61} -3.11386e7i q^{62} -1.40929e7 q^{64} -696894. i q^{68} +4.45563e7 q^{69} +2.48728e7i q^{72} -6.73193e6 q^{76} +6.96173e7 q^{79} +4.30467e7 q^{81} +3.84720e6i q^{83} +1.81526e7i q^{92} +1.48366e8i q^{93} -1.37121e8 q^{94} -2.17662e7 q^{96} +9.80016e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 66 q^{4} + 2754 q^{6} - 13122 q^{9}+O(q^{10}) \) 2 * q - 66 * q^4 + 2754 * q^6 - 13122 * q^9	\( 2 q - 66 q^{4} + 2754 q^{6} - 13122 q^{9} - 145790 q^{16} + 407996 q^{19} + 614142 q^{24} + 3663364 q^{31} + 718012 q^{34} + 433026 q^{36} - 18702652 q^{46} - 11529602 q^{49} - 3421116 q^{51} - 18068994 q^{54} + 28649284 q^{61} - 28185794 q^{64} + 89112636 q^{69} - 13463868 q^{76} + 139234556 q^{79} + 86093442 q^{81} - 274241348 q^{94} - 43532478 q^{96}+O(q^{100}) \) 2 * q - 66 * q^4 + 2754 * q^6 - 13122 * q^9 - 145790 * q^16 + 407996 * q^19 + 614142 * q^24 + 3663364 * q^31 + 718012 * q^34 + 433026 * q^36 - 18702652 * q^46 - 11529602 * q^49 - 3421116 * q^51 - 18068994 * q^54 + 28649284 * q^61 - 28185794 * q^64 + 89112636 * q^69 - 13463868 * q^76 + 139234556 * q^79 + 86093442 * q^81 - 274241348 * q^94 - 43532478 * q^96

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