LMFDB - Embedded newform 180.9.l.a.73.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [180,9,Mod(37,180)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("180.37"); S:= CuspForms(chi, 9); N := Newforms(S);

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	180.l (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-894] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$73.3281498110$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4 x^{7} + 8 x^{6} + 22254 x^{5} + 4820745 x^{4} + 50131374 x^{3} + 307615702 x^{2} + \cdots + 2405464244 $ x^8 - 4x^7 + 8x^6 + 22254x^5 + 4820745x^4 + 50131374x^3 + 307615702x^2 - 1770757924*x + 2405464244
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{2}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			73.4
Root			$-29.1758 - 30.1758i$ of defining polynomial
Character	$\chi$	$=$	180.73
Dual form			180.9.l.a.37.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(577.254 + 239.589i) q^{5} +(-190.873 + 190.873i) q^{7} +19026.8 q^{11} +(-27902.1 - 27902.1i) q^{13} +(22518.5 - 22518.5i) q^{17} -63405.3i q^{19} +(-196829. - 196829. i) q^{23} +(275820. + 276607. i) q^{25} -595585. i q^{29} +249340. q^{31} +(-155913. + 64451.1i) q^{35} +(229540. - 229540. i) q^{37} +4.30046e6 q^{41} +(-3.59483e6 - 3.59483e6i) q^{43} +(-6.19225e6 + 6.19225e6i) q^{47} +5.69194e6i q^{49} +(253926. + 253926. i) q^{53} +(1.09833e7 + 4.55859e6i) q^{55} -7.27642e6i q^{59} -3.24983e6 q^{61} +(-9.42158e6 - 2.27916e7i) q^{65} +(2.68654e7 - 2.68654e7i) q^{67} +3.70958e7 q^{71} +(-2.76665e7 - 2.76665e7i) q^{73} +(-3.63169e6 + 3.63169e6i) q^{77} -2.03584e7i q^{79} +(2.23434e7 + 2.23434e7i) q^{83} +(1.83941e7 - 7.60373e6i) q^{85} -2.64461e7i q^{89} +1.06515e7 q^{91} +(1.51912e7 - 3.66010e7i) q^{95} +(9.78714e7 - 9.78714e7i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 894 q^{5} - 2030 q^{7} + 420 q^{11} + 33180 q^{13} - 43620 q^{17} + 663270 q^{23} + 163396 q^{25} - 3178492 q^{31} - 2571618 q^{35} + 5344080 q^{37} + 10185252 q^{41} - 10342710 q^{43} - 19232250 q^{47}+ \cdots - 179570760 q^{97}+O(q^{100}) $ 8 * q - 894 * q^5 - 2030 * q^7 + 420 * q^11 + 33180 * q^13 - 43620 * q^17 + 663270 * q^23 + 163396 * q^25 - 3178492 * q^31 - 2571618 * q^35 + 5344080 * q^37 + 10185252 * q^41 - 10342710 * q^43 - 19232250 * q^47 + 24320640 * q^53 + 21483180 * q^55 - 82515684 * q^61 - 72045768 * q^65 + 100675930 * q^67 + 99290076 * q^71 - 93528520 * q^73 - 134199660 * q^77 + 10450350 * q^83 + 51676156 * q^85 - 130681068 * q^91 - 84367944 * q^95 - 179570760 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$.

$n$	$37$	$91$	$101$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	577.254	+	239.589i	0.923607	+	0.383342i
$5$	577.254	+	239.589i	0.923607	+	0.383342i
$6$		0			0
$7$	−190.873	+	190.873i	−0.0794971	+	0.0794971i	−0.745737	−	0.666240i	$-0.767903\pi$
$7$	−190.873	+	190.873i	−0.0794971	+	0.0794971i	0.666240	+	0.745737i	$0.267903\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	19026.8			1.29955			0.649776	−	0.760125i	$-0.274863\pi$
$11$	19026.8			1.29955			0.649776	+	0.760125i	$0.274863\pi$
$12$		0			0
$13$	−27902.1	−	27902.1i	−0.976930	−	0.976930i	0.0228094	−	0.999740i	$-0.492739\pi$
$13$	−27902.1	−	27902.1i	−0.976930	−	0.976930i	−0.999740	+	0.0228094i	$0.992739\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	22518.5	−	22518.5i	0.269615	−	0.269615i	−0.559330	−	0.828945i	$-0.688942\pi$
$17$	22518.5	−	22518.5i	0.269615	−	0.269615i	0.828945	+	0.559330i	$0.188942\pi$
$18$		0			0
$19$		−	63405.3i		−	0.486532i	−0.969960	−	0.243266i	$-0.921781\pi$
$19$		−	63405.3i		−	0.486532i	0.969960	−	0.243266i	$-0.0782188\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−196829.	−	196829.i	−0.703360	−	0.703360i	0.261770	−	0.965130i	$-0.415694\pi$
$23$	−196829.	−	196829.i	−0.703360	−	0.703360i	−0.965130	+	0.261770i	$0.915694\pi$
$24$		0			0
$25$	275820.	+	276607.i	0.706098	+	0.708114i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	595585.i		−	0.842077i	−0.907043	−	0.421039i	$-0.861666\pi$
$29$		−	595585.i		−	0.842077i	0.907043	−	0.421039i	$-0.138334\pi$
$30$		0			0
$31$	249340.			0.269988			0.134994	−	0.990846i	$-0.456898\pi$
$31$	249340.			0.269988			0.134994	+	0.990846i	$0.456898\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−155913.	+	64451.1i	−0.103899	+	0.0429495i
$36$		0			0
$37$	229540.	−	229540.i	0.122476	−	0.122476i	−0.643212	−	0.765688i	$-0.722399\pi$
$37$	229540.	−	229540.i	0.122476	−	0.122476i	0.765688	+	0.643212i	$0.222399\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.30046e6			1.52188			0.760938	−	0.648824i	$-0.224739\pi$
$41$	4.30046e6			1.52188			0.760938	+	0.648824i	$0.224739\pi$
$42$		0			0
$43$	−3.59483e6	−	3.59483e6i	−1.05149	−	1.05149i	−0.998600	−	0.0528880i	$-0.983157\pi$
$43$	−3.59483e6	−	3.59483e6i	−1.05149	−	1.05149i	−0.0528880	−	0.998600i	$-0.516843\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.19225e6	+	6.19225e6i	−1.26899	+	1.26899i	−0.322375	+	0.946612i	$0.604481\pi$
$47$	−6.19225e6	+	6.19225e6i	−1.26899	+	1.26899i	−0.946612	+	0.322375i	$0.895519\pi$
$48$		0			0
$49$			5.69194e6i			0.987360i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	253926.	+	253926.i	0.0321813	+	0.0321813i	0.723014	−	0.690833i	$-0.242756\pi$
$53$	253926.	+	253926.i	0.0321813	+	0.0321813i	−0.690833	+	0.723014i	$0.742756\pi$
$54$		0			0
$55$	1.09833e7	+	4.55859e6i	1.20028	+	0.498173i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	7.27642e6i		−	0.600496i	−0.953861	−	0.300248i	$-0.902931\pi$
$59$		−	7.27642e6i		−	0.600496i	0.953861	−	0.300248i	$-0.0970694\pi$
$60$		0			0
$61$	−3.24983e6			−0.234715			−0.117358	−	0.993090i	$-0.537442\pi$
$61$	−3.24983e6			−0.234715			−0.117358	+	0.993090i	$0.537442\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−9.42158e6	−	2.27916e7i	−0.527801	−	1.27680i
$66$		0			0
$67$	2.68654e7	−	2.68654e7i	1.33320	−	1.33320i	0.430705	−	0.902493i	$-0.358265\pi$
$67$	2.68654e7	−	2.68654e7i	1.33320	−	1.33320i	0.902493	−	0.430705i	$-0.141735\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.70958e7			1.45979			0.729896	−	0.683559i	$-0.239569\pi$
$71$	3.70958e7			1.45979			0.729896	+	0.683559i	$0.239569\pi$
$72$		0			0
$73$	−2.76665e7	−	2.76665e7i	−0.974234	−	0.974234i	0.0254420	−	0.999676i	$-0.491901\pi$
$73$	−2.76665e7	−	2.76665e7i	−0.974234	−	0.974234i	−0.999676	+	0.0254420i	$0.991901\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.63169e6	+	3.63169e6i	−0.103311	+	0.103311i
$78$		0			0
$79$		−	2.03584e7i		−	0.522681i	−0.965247	−	0.261340i	$-0.915836\pi$
$79$		−	2.03584e7i		−	0.522681i	0.965247	−	0.261340i	$-0.0841645\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.23434e7	+	2.23434e7i	0.470800	+	0.470800i	0.902173	−	0.431374i	$-0.141971\pi$
$83$	2.23434e7	+	2.23434e7i	0.470800	+	0.470800i	−0.431374	+	0.902173i	$0.641971\pi$
$84$		0			0
$85$	1.83941e7	−	7.60373e6i	0.352373	−	0.145663i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	2.64461e7i		−	0.421504i	−0.977540	−	0.210752i	$-0.932409\pi$
$89$		−	2.64461e7i		−	0.421504i	0.977540	−	0.210752i	$-0.0675912\pi$
$90$		0			0
$91$	1.06515e7			0.155326
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.51912e7	−	3.66010e7i	0.186508	−	0.449364i
$96$		0			0
$97$	9.78714e7	−	9.78714e7i	1.10553	−	1.10553i	0.111795	−	0.993731i	$-0.464340\pi$
$97$	9.78714e7	−	9.78714e7i	1.10553	−	1.10553i	0.993731	−	0.111795i	$-0.0356600\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.04946e8			−1.00851			−0.504257	−	0.863553i	$-0.668234\pi$
$101$	−1.04946e8			−1.00851			−0.504257	+	0.863553i	$0.668234\pi$
$102$		0			0
$103$	1.04986e8	+	1.04986e8i	0.932789	+	0.932789i	0.997879	−	0.0650901i	$-0.0207335\pi$
$103$	1.04986e8	+	1.04986e8i	0.932789	+	0.932789i	−0.0650901	+	0.997879i	$0.520733\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.19346e8	−	1.19346e8i	0.910485	−	0.910485i	−0.0858251	−	0.996310i	$-0.527353\pi$
$107$	1.19346e8	−	1.19346e8i	0.910485	−	0.910485i	0.996310	+	0.0858251i	$0.0273526\pi$
$108$		0			0
$109$		−	1.74582e8i		−	1.23678i	−0.785871	−	0.618390i	$-0.787785\pi$
$109$		−	1.74582e8i		−	1.23678i	0.785871	−	0.618390i	$-0.212215\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.45939e6	−	5.45939e6i	−0.0334835	−	0.0334835i	0.690167	−	0.723650i	$-0.257537\pi$
$113$	−5.45939e6	−	5.45939e6i	−0.0334835	−	0.0334835i	−0.723650	+	0.690167i	$0.757537\pi$
$114$		0			0
$115$	−6.64623e7	−	1.60778e8i	−0.380001	−	0.919255i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			8.59633e6i			0.0428672i
$120$		0			0
$121$	1.47658e8			0.688838
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.29461e7	+	2.25756e8i	0.380707	+	0.924696i
$126$		0			0
$127$	6.35128e6	−	6.35128e6i	0.0244144	−	0.0244144i	−0.694794	−	0.719209i	$-0.744505\pi$
$127$	6.35128e6	−	6.35128e6i	0.0244144	−	0.0244144i	0.719209	+	0.694794i	$0.244505\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.24615e8			0.762699			0.381350	−	0.924431i	$-0.375459\pi$
$131$	2.24615e8			0.762699			0.381350	+	0.924431i	$0.375459\pi$
$132$		0			0
$133$	1.21023e7	+	1.21023e7i	0.0386779	+	0.0386779i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.38143e8	−	3.38143e8i	0.959882	−	0.959882i	−0.0393439	−	0.999226i	$-0.512527\pi$
$137$	3.38143e8	−	3.38143e8i	0.959882	−	0.959882i	0.999226	+	0.0393439i	$0.0125268\pi$
$138$		0			0
$139$		−	5.48983e8i		−	1.47062i	−0.677732	−	0.735309i	$-0.737037\pi$
$139$		−	5.48983e8i		−	1.47062i	0.677732	−	0.735309i	$-0.262963\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.30887e8	−	5.30887e8i	−1.26957	−	1.26957i
$144$		0			0
$145$	1.42695e8	−	3.43804e8i	0.322803	−	0.777748i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	5.66486e8i		−	1.14933i	−0.818389	−	0.574665i	$-0.805133\pi$
$149$		−	5.66486e8i		−	1.14933i	0.818389	−	0.574665i	$-0.194867\pi$
$150$		0			0
$151$	9.43633e8			1.81508			0.907539	−	0.419968i	$-0.137959\pi$
$151$	9.43633e8			1.81508			0.907539	+	0.419968i	$0.137959\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.43932e8	+	5.97389e7i	0.249363	+	0.103498i
$156$		0			0
$157$	−1.37850e8	+	1.37850e8i	−0.226886	+	0.226886i	−0.811390	−	0.584505i	$-0.801289\pi$
$157$	−1.37850e8	+	1.37850e8i	−0.226886	+	0.226886i	0.584505	+	0.811390i	$0.301289\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.51385e7			0.111830
$162$		0			0
$163$	−1.86924e8	−	1.86924e8i	−0.264798	−	0.264798i	0.562202	−	0.827000i	$-0.309954\pi$
$163$	−1.86924e8	−	1.86924e8i	−0.264798	−	0.264798i	−0.827000	+	0.562202i	$0.809954\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.37971e8	+	5.37971e8i	−0.691661	+	0.691661i	−0.962597	−	0.270936i	$-0.912667\pi$
$167$	−5.37971e8	+	5.37971e8i	−0.691661	+	0.691661i	0.270936	+	0.962597i	$0.412667\pi$
$168$		0			0
$169$			7.41325e8i			0.908786i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.79607e8	+	4.79607e8i	0.535428	+	0.535428i	0.922183	−	0.386754i	$-0.126404\pi$
$173$	4.79607e8	+	4.79607e8i	0.535428	+	0.535428i	−0.386754	+	0.922183i	$0.626404\pi$
$174$		0			0
$175$	−1.05443e8	−	150288.i	−0.112426	−	0.000160240i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			1.39076e9i			1.35469i	0.735665	+	0.677346i	$0.236870\pi$
$179$			1.39076e9i			1.35469i	−0.735665	+	0.677346i	$0.763130\pi$
$180$		0			0
$181$	−1.45715e9			−1.35766			−0.678830	−	0.734296i	$-0.737512\pi$
$181$	−1.45715e9			−1.35766			−0.678830	+	0.734296i	$0.737512\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.87498e8	−	7.75078e7i	0.160070	−	0.0661696i
$186$		0			0
$187$	4.28454e8	−	4.28454e8i	0.350379	−	0.350379i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.33136e8			0.325455			0.162727	−	0.986671i	$-0.447971\pi$
$191$	4.33136e8			0.325455			0.162727	+	0.986671i	$0.447971\pi$
$192$		0			0
$193$	−9.36884e8	−	9.36884e8i	−0.675237	−	0.675237i	0.283681	−	0.958919i	$-0.408444\pi$
$193$	−9.36884e8	−	9.36884e8i	−0.675237	−	0.675237i	−0.958919	+	0.283681i	$0.908444\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.12789e8	+	7.12789e8i	−0.473256	+	0.473256i	−0.902967	−	0.429711i	$-0.858616\pi$
$197$	−7.12789e8	+	7.12789e8i	−0.473256	+	0.473256i	0.429711	+	0.902967i	$0.358616\pi$
$198$		0			0
$199$			1.63094e9i			1.03998i	0.854173	+	0.519989i	$0.174064\pi$
$199$			1.63094e9i			1.03998i	−0.854173	+	0.519989i	$0.825936\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.13681e8	+	1.13681e8i	0.0669427	+	0.0669427i
$204$		0			0
$205$	2.48246e9	+	1.03034e9i	1.40562	+	0.583399i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	1.20640e9i		−	0.632274i
$210$		0			0
$211$	−2.87914e9			−1.45255			−0.726277	−	0.687402i	$-0.758751\pi$
$211$	−2.87914e9			−1.45255			−0.726277	+	0.687402i	$0.758751\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.21385e9	−	2.93641e9i	−0.568082	−	1.37424i
$216$		0			0
$217$	−4.75921e7	+	4.75921e7i	−0.0214633	+	0.0214633i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.25663e9			−0.526790
$222$		0			0
$223$	2.77849e9	+	2.77849e9i	1.12354	+	1.12354i	0.991205	+	0.132337i	$0.0422480\pi$
$223$	2.77849e9	+	2.77849e9i	1.12354	+	1.12354i	0.132337	+	0.991205i	$0.457752\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.66218e9	+	1.66218e9i	−0.626001	+	0.626001i	−0.947059	−	0.321059i	$-0.895961\pi$
$227$	−1.66218e9	+	1.66218e9i	−0.626001	+	0.626001i	0.321059	+	0.947059i	$0.395961\pi$
$228$		0			0
$229$			1.39534e9i			0.507384i	0.967285	+	0.253692i	$0.0816450\pi$
$229$			1.39534e9i			0.507384i	−0.967285	+	0.253692i	$0.918355\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.53597e9	+	1.53597e9i	0.521146	+	0.521146i	0.917918	−	0.396771i	$-0.129869\pi$
$233$	1.53597e9	+	1.53597e9i	0.521146	+	0.521146i	−0.396771	+	0.917918i	$0.629869\pi$
$234$		0			0
$235$	−5.05809e9	+	2.09091e9i	−1.65850	+	0.685589i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			5.45417e8i			0.167162i	0.996501	+	0.0835809i	$0.0266357\pi$
$239$			5.45417e8i			0.167162i	−0.996501	+	0.0835809i	$0.973364\pi$
$240$		0			0
$241$	−9.46365e8			−0.280537			−0.140269	−	0.990113i	$-0.544797\pi$
$241$	−9.46365e8			−0.280537			−0.140269	+	0.990113i	$0.544797\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.36372e9	+	3.28569e9i	−0.378496	+	0.911933i
$246$		0			0
$247$	−1.76914e9	+	1.76914e9i	−0.475308	+	0.475308i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.66712e9			1.17585			0.587927	−	0.808914i	$-0.299944\pi$
$251$	4.66712e9			1.17585			0.587927	+	0.808914i	$0.299944\pi$
$252$		0			0
$253$	−3.74502e9	−	3.74502e9i	−0.914053	−	0.914053i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.22158e8	+	4.22158e8i	−0.0967705	+	0.0967705i	−0.753835	−	0.657064i	$-0.771798\pi$
$257$	−4.22158e8	+	4.22158e8i	−0.0967705	+	0.0967705i	0.657064	+	0.753835i	$0.271798\pi$
$258$		0			0
$259$			8.76258e7i			0.0194730i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.92697e9	−	3.92697e9i	−0.820794	−	0.820794i	0.165428	−	0.986222i	$-0.447100\pi$
$263$	−3.92697e9	−	3.92697e9i	−0.820794	−	0.820794i	−0.986222	+	0.165428i	$0.947100\pi$
$264$		0			0
$265$	8.57420e7	+	2.07418e8i	0.0173864	+	0.0420593i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			1.88692e9i			0.360366i	0.983633	+	0.180183i	$0.0576691\pi$
$269$			1.88692e9i			0.360366i	−0.983633	+	0.180183i	$0.942331\pi$
$270$		0			0
$271$	3.82459e9			0.709100			0.354550	−	0.935037i	$-0.384634\pi$
$271$	3.82459e9			0.709100			0.354550	+	0.935037i	$0.384634\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.24795e9	+	5.26293e9i	0.917612	+	0.920231i
$276$		0			0
$277$	−6.08916e9	+	6.08916e9i	−1.03428	+	1.03428i	−0.0348898	+	0.999391i	$0.511108\pi$
$277$	−6.08916e9	+	6.08916e9i	−1.03428	+	1.03428i	−0.999391	+	0.0348898i	$0.988892\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.36217e9			−1.18081			−0.590406	−	0.807107i	$-0.701032\pi$
$281$	−7.36217e9			−1.18081			−0.590406	+	0.807107i	$0.701032\pi$
$282$		0			0
$283$	7.50823e9	+	7.50823e9i	1.17055	+	1.17055i	0.982078	+	0.188477i	$0.0603550\pi$
$283$	7.50823e9	+	7.50823e9i	1.17055	+	1.17055i	0.188477	+	0.982078i	$0.439645\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.20840e8	+	8.20840e8i	−0.120985	+	0.120985i
$288$		0			0
$289$			5.96159e9i			0.854616i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.53947e9	−	5.53947e9i	−0.751619	−	0.751619i	0.223163	−	0.974781i	$-0.428362\pi$
$293$	−5.53947e9	−	5.53947e9i	−0.751619	−	0.751619i	−0.974781	+	0.223163i	$0.928362\pi$
$294$		0			0
$295$	1.74335e9	−	4.20034e9i	0.230195	−	0.554622i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			1.09839e10i			1.37427i
$300$		0			0
$301$	1.37231e9			0.167181
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.87598e9	−	7.78623e8i	−0.216785	−	0.0899762i
$306$		0			0
$307$	−5.64460e8	+	5.64460e8i	−0.0635448	+	0.0635448i	−0.738165	−	0.674620i	$-0.764307\pi$
$307$	−5.64460e8	+	5.64460e8i	−0.0635448	+	0.0635448i	0.674620	+	0.738165i	$0.264307\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2.47892e9			0.264984			0.132492	−	0.991184i	$-0.457702\pi$
$311$	2.47892e9			0.264984			0.132492	+	0.991184i	$0.457702\pi$
$312$		0			0
$313$	−9.72227e9	−	9.72227e9i	−1.01296	−	1.01296i	−0.999915	−	0.0130407i	$-0.995849\pi$
$313$	−9.72227e9	−	9.72227e9i	−1.01296	−	1.01296i	−0.0130407	−	0.999915i	$-0.504151\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.03089e10	−	1.03089e10i	1.02088	−	1.02088i	0.0211068	−	0.999777i	$-0.493281\pi$
$317$	1.03089e10	−	1.03089e10i	1.02088	−	1.02088i	0.999777	−	0.0211068i	$-0.00671900\pi$
$318$		0			0
$319$		−	1.13321e10i		−	1.09432i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.42779e9	−	1.42779e9i	−0.131176	−	0.131176i
$324$		0			0
$325$	2.19694e7	−	1.54139e10i	0.00196917	−	1.38159i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	2.36386e9i		−	0.201762i
$330$		0			0
$331$	7.62056e9			0.634855			0.317428	−	0.948282i	$-0.397181\pi$
$331$	7.62056e9			0.634855			0.317428	+	0.948282i	$0.397181\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.19448e10	−	9.07153e9i	1.74242	−	0.720280i
$336$		0			0
$337$	−1.43147e9	+	1.43147e9i	−0.110985	+	0.110985i	−0.760418	−	0.649433i	$-0.775006\pi$
$337$	−1.43147e9	+	1.43147e9i	−0.110985	+	0.110985i	0.649433	+	0.760418i	$0.275006\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.74412e9			0.350864
$342$		0			0
$343$	−2.18678e9	−	2.18678e9i	−0.157989	−	0.157989i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.72187e9	+	7.72187e9i	−0.532604	+	0.532604i	−0.921346	−	0.388742i	$-0.872910\pi$
$347$	−7.72187e9	+	7.72187e9i	−0.532604	+	0.532604i	0.388742	+	0.921346i	$0.372910\pi$
$348$		0			0
$349$		−	3.20836e9i		−	0.216263i	−0.994137	−	0.108131i	$-0.965513\pi$
$349$		−	3.20836e9i		−	0.216263i	0.994137	−	0.108131i	$-0.0344867\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.06371e9	−	5.06371e9i	−0.326115	−	0.326115i	0.524992	−	0.851107i	$-0.324068\pi$
$353$	−5.06371e9	−	5.06371e9i	−0.326115	−	0.326115i	−0.851107	+	0.524992i	$0.824068\pi$
$354$		0			0
$355$	2.14137e10	+	8.88772e9i	1.34827	+	0.559599i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	6.18165e9i		−	0.372157i	−0.982535	−	0.186079i	$-0.940422\pi$
$359$		−	6.18165e9i		−	0.372157i	0.982535	−	0.186079i	$-0.0595779\pi$
$360$		0			0
$361$	1.29633e10			0.763287
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.34204e9	−	2.25992e10i	−0.526345	−	1.27327i
$366$		0			0
$367$	1.55537e10	−	1.55537e10i	0.857371	−	0.857371i	−0.133657	−	0.991028i	$-0.542672\pi$
$367$	1.55537e10	−	1.55537e10i	0.857371	−	0.857371i	0.991028	+	0.133657i	$0.0426720\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.69350e7			−0.00511664
$372$		0			0
$373$	−8.05344e9	−	8.05344e9i	−0.416051	−	0.416051i	0.467789	−	0.883840i	$-0.345051\pi$
$373$	−8.05344e9	−	8.05344e9i	−0.416051	−	0.416051i	−0.883840	+	0.467789i	$0.845051\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.66181e10	+	1.66181e10i	−0.822651	+	0.822651i
$378$		0			0
$379$			2.11955e10i			1.02727i	0.858007	+	0.513637i	$0.171702\pi$
$379$			2.11955e10i			1.02727i	−0.858007	+	0.513637i	$0.828298\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.49717e9	+	4.49717e9i	0.208999	+	0.208999i	0.803842	−	0.594843i	$-0.202786\pi$
$383$	4.49717e9	+	4.49717e9i	0.208999	+	0.208999i	−0.594843	+	0.803842i	$0.702786\pi$
$384$		0			0
$385$	−2.96652e9	+	1.22630e9i	−0.135022	+	0.0558152i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			1.17946e10i			0.515093i	0.966266	+	0.257547i	$0.0829141\pi$
$389$			1.17946e10i			0.515093i	−0.966266	+	0.257547i	$0.917086\pi$
$390$		0			0
$391$	−8.86459e9			−0.379273
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.87765e9	−	1.17520e10i	0.200365	−	0.482751i
$396$		0			0
$397$	2.96839e10	−	2.96839e10i	1.19498	−	1.19498i	0.219324	−	0.975652i	$-0.429615\pi$
$397$	2.96839e10	−	2.96839e10i	1.19498	−	1.19498i	0.975652	−	0.219324i	$-0.0703853\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.98937e10			1.15612			0.578059	−	0.815995i	$-0.303810\pi$
$401$	2.98937e10			1.15612			0.578059	+	0.815995i	$0.303810\pi$
$402$		0			0
$403$	−6.95710e9	−	6.95710e9i	−0.263760	−	0.263760i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.36740e9	−	4.36740e9i	0.159164	−	0.159164i
$408$		0			0
$409$		−	4.06334e10i		−	1.45208i	−0.687654	−	0.726039i	$-0.741359\pi$
$409$		−	4.06334e10i		−	1.45208i	0.687654	−	0.726039i	$-0.258641\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.38887e9	+	1.38887e9i	0.0477377	+	0.0477377i
$414$		0			0
$415$	7.54458e9	+	1.82510e10i	0.254356	+	0.615311i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	5.67792e9i		−	0.184218i	−0.995749	−	0.0921092i	$-0.970639\pi$
$419$		−	5.67792e9i		−	0.184218i	0.995749	−	0.0921092i	$-0.0293609\pi$
$420$		0			0
$421$	−3.08470e10			−0.981938			−0.490969	−	0.871177i	$-0.663357\pi$
$421$	−3.08470e10			−0.981938			−0.490969	+	0.871177i	$0.663357\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.24398e10	+	1.77305e7i	0.381293	+	0.000543456i
$426$		0			0
$427$	6.20304e8	−	6.20304e8i	0.0186592	−	0.0186592i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.27768e10			0.949856			0.474928	−	0.880025i	$-0.342474\pi$
$431$	3.27768e10			0.949856			0.474928	+	0.880025i	$0.342474\pi$
$432$		0			0
$433$	−7.60012e9	−	7.60012e9i	−0.216206	−	0.216206i	0.590691	−	0.806898i	$-0.298855\pi$
$433$	−7.60012e9	−	7.60012e9i	−0.216206	−	0.216206i	−0.806898	+	0.590691i	$0.798855\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.24800e10	+	1.24800e10i	−0.342207	+	0.342207i
$438$		0			0
$439$			1.35066e10i			0.363654i	0.983330	+	0.181827i	$0.0582012\pi$
$439$			1.35066e10i			0.363654i	−0.983330	+	0.181827i	$0.941799\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.65065e10	−	3.65065e10i	−0.947883	−	0.947883i	0.0508241	−	0.998708i	$-0.483815\pi$
$443$	−3.65065e10	−	3.65065e10i	−0.947883	−	0.947883i	−0.998708	+	0.0508241i	$0.983815\pi$
$444$		0			0
$445$	6.33618e9	−	1.52661e10i	0.161580	−	0.389303i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	2.93648e9i		−	0.0722507i	−0.999347	−	0.0361253i	$-0.988498\pi$
$449$		−	2.93648e9i		−	0.0722507i	0.999347	−	0.0361253i	$-0.0115016\pi$
$450$		0			0
$451$	8.18238e10			1.97776
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.14862e9	+	2.55198e9i	0.143460	+	0.0595431i
$456$		0			0
$457$	−4.06216e10	+	4.06216e10i	−0.931305	+	0.931305i	−0.997788	−	0.0664829i	$-0.978822\pi$
$457$	−4.06216e10	+	4.06216e10i	−0.931305	+	0.931305i	0.0664829	+	0.997788i	$0.478822\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.69744e10			−0.375830			−0.187915	−	0.982185i	$-0.560173\pi$
$461$	−1.69744e10			−0.375830			−0.187915	+	0.982185i	$0.560173\pi$
$462$		0			0
$463$	−1.02630e10	−	1.02630e10i	−0.223332	−	0.223332i	0.586568	−	0.809900i	$-0.300479\pi$
$463$	−1.02630e10	−	1.02630e10i	−0.223332	−	0.223332i	−0.809900	+	0.586568i	$0.800479\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.93031e10	+	1.93031e10i	−0.405845	+	0.405845i	−0.880287	−	0.474442i	$-0.842650\pi$
$467$	−1.93031e10	+	1.93031e10i	−0.405845	+	0.405845i	0.474442	+	0.880287i	$0.342650\pi$
$468$		0			0
$469$			1.02557e10i			0.211971i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.83979e10	−	6.83979e10i	−1.36646	−	1.36646i
$474$		0			0
$475$	1.75384e10	−	1.74884e10i	0.344520	−	0.343539i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			6.47760e10i			1.23047i	0.788343	+	0.615236i	$0.210939\pi$
$479$			6.47760e10i			1.23047i	−0.788343	+	0.615236i	$0.789061\pi$
$480$		0			0
$481$	−1.28093e10			−0.239301
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.99456e10	−	3.30478e10i	1.44487	−	0.597277i
$486$		0			0
$487$	−3.13580e10	+	3.13580e10i	−0.557483	+	0.557483i	−0.928590	−	0.371107i	$-0.878978\pi$
$487$	−3.13580e10	+	3.13580e10i	−0.557483	+	0.557483i	0.371107	+	0.928590i	$0.378978\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.72949e10			−0.469630			−0.234815	−	0.972040i	$-0.575448\pi$
$491$	−2.72949e10			−0.469630			−0.234815	+	0.972040i	$0.575448\pi$
$492$		0			0
$493$	−1.34117e10	−	1.34117e10i	−0.227037	−	0.227037i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.08056e9	+	7.08056e9i	−0.116049	+	0.116049i
$498$		0			0
$499$			8.55600e10i			1.37997i	0.723825	+	0.689984i	$0.242382\pi$
$499$			8.55600e10i			1.37997i	−0.723825	+	0.689984i	$0.757618\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4.20771e10	+	4.20771e10i	0.657315	+	0.657315i	0.954744	−	0.297429i	$-0.0961292\pi$
$503$	4.20771e10	+	4.20771e10i	0.657315	+	0.657315i	−0.297429	+	0.954744i	$0.596129\pi$
$504$		0			0
$505$	−6.05808e10	−	2.51440e10i	−0.931471	−	0.386606i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			1.19496e11i			1.78025i	0.455714	+	0.890126i	$0.349384\pi$
$509$			1.19496e11i			1.78025i	−0.455714	+	0.890126i	$0.650616\pi$
$510$		0			0
$511$	1.05616e10			0.154898
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.54502e10	+	8.57573e10i	0.503953	+	1.21911i
$516$		0			0
$517$	−1.17818e11	+	1.17818e11i	−1.64912	+	1.64912i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.31604e9			0.126439			0.0632194	−	0.998000i	$-0.479863\pi$
$521$	9.31604e9			0.126439			0.0632194	+	0.998000i	$0.479863\pi$
$522$		0			0
$523$	−4.01748e9	−	4.01748e9i	−0.0536967	−	0.0536967i	0.679749	−	0.733445i	$-0.262089\pi$
$523$	−4.01748e9	−	4.01748e9i	−0.0536967	−	0.0536967i	−0.733445	+	0.679749i	$0.762089\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.61476e9	−	5.61476e9i	0.0727928	−	0.0727928i
$528$		0			0
$529$		−	8.27726e8i		−	0.0105697i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.19992e11	−	1.19992e11i	−1.48677	−	1.48677i
$534$		0			0
$535$	9.74869e10	−	4.02990e10i	1.18996	−	0.491903i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1.08299e11i			1.28313i
$540$		0			0
$541$	−5.43998e10			−0.635050			−0.317525	−	0.948250i	$-0.602852\pi$
$541$	−5.43998e10			−0.635050			−0.317525	+	0.948250i	$0.602852\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.18278e10	−	1.00778e11i	0.474110	−	1.14230i
$546$		0			0
$547$	−6.69974e10	+	6.69974e10i	−0.748357	+	0.748357i	−0.974171	−	0.225814i	$-0.927496\pi$
$547$	−6.69974e10	+	6.69974e10i	−0.748357	+	0.748357i	0.225814	+	0.974171i	$0.427496\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.77633e10			−0.409697
$552$		0			0
$553$	3.88587e9	+	3.88587e9i	0.0415516	+	0.0415516i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.16300e11	+	1.16300e11i	−1.20825	+	1.20825i	−0.236660	+	0.971593i	$0.576053\pi$
$557$	−1.16300e11	+	1.16300e11i	−1.20825	+	1.20825i	−0.971593	+	0.236660i	$0.923947\pi$
$558$		0			0
$559$			2.00607e11i			2.05446i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.63107e10	−	4.63107e10i	−0.460944	−	0.460944i	0.438021	−	0.898965i	$-0.355680\pi$
$563$	−4.63107e10	−	4.63107e10i	−0.460944	−	0.460944i	−0.898965	+	0.438021i	$0.855680\pi$
$564$		0			0
$565$	−1.84345e9	−	4.45947e9i	−0.0180899	−	0.0437612i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			3.00951e10i			0.287109i	0.989642	+	0.143555i	$0.0458532\pi$
$569$			3.00951e10i			0.287109i	−0.989642	+	0.143555i	$0.954147\pi$
$570$		0			0
$571$	5.41650e10			0.509536			0.254768	−	0.967002i	$-0.418001\pi$
$571$	5.41650e10			0.509536			0.254768	+	0.967002i	$0.418001\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.54978e8	−	1.08734e11i	0.00141774	−	0.994700i
$576$		0			0
$577$	4.83913e10	−	4.83913e10i	0.436581	−	0.436581i	−0.454279	−	0.890859i	$-0.650103\pi$
$577$	4.83913e10	−	4.83913e10i	0.436581	−	0.436581i	0.890859	+	0.454279i	$0.150103\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8.52947e9			−0.0748544
$582$		0			0
$583$	4.83139e9	+	4.83139e9i	0.0418213	+	0.0418213i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.35625e10	−	5.35625e10i	0.451137	−	0.451137i	−0.444595	−	0.895732i	$-0.646652\pi$
$587$	5.35625e10	−	5.35625e10i	0.451137	−	0.451137i	0.895732	+	0.444595i	$0.146652\pi$
$588$		0			0
$589$		−	1.58095e10i		−	0.131358i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.01477e10	−	2.01477e10i	−0.162932	−	0.162932i	0.620932	−	0.783864i	$-0.286754\pi$
$593$	−2.01477e10	−	2.01477e10i	−0.162932	−	0.162932i	−0.783864	+	0.620932i	$0.786754\pi$
$594$		0			0
$595$	−2.05958e9	+	4.96227e9i	−0.0164328	+	0.0395925i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	7.82597e10i		−	0.607898i	−0.952688	−	0.303949i	$-0.901695\pi$
$599$		−	7.82597e10i		−	0.607898i	0.952688	−	0.303949i	$-0.0983053\pi$
$600$		0			0
$601$	−1.00940e11			−0.773686			−0.386843	−	0.922146i	$-0.626434\pi$
$601$	−1.00940e11			−0.773686			−0.386843	+	0.922146i	$0.626434\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8.52364e10	+	3.53773e10i	0.636215	+	0.264060i
$606$		0			0
$607$	6.85629e10	−	6.85629e10i	0.505050	−	0.505050i	−0.407953	−	0.913003i	$-0.633757\pi$
$607$	6.85629e10	−	6.85629e10i	0.505050	−	0.505050i	0.913003	+	0.407953i	$0.133757\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.45554e11			2.47942
$612$		0			0
$613$	−1.41547e11	−	1.41547e11i	−1.00244	−	1.00244i	−0.999997	−	0.00244188i	$-0.999223\pi$
$613$	−1.41547e11	−	1.41547e11i	−1.00244	−	1.00244i	−0.00244188	−	0.999997i	$-0.500777\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.48653e11	+	1.48653e11i	−1.02573	+	1.02573i	−0.0260699	+	0.999660i	$0.508299\pi$
$617$	−1.48653e11	+	1.48653e11i	−1.02573	+	1.02573i	−0.999660	+	0.0260699i	$0.991701\pi$
$618$		0			0
$619$		−	1.85869e10i		−	0.126603i	−0.997994	−	0.0633016i	$-0.979837\pi$
$619$		−	1.85869e10i		−	0.126603i	0.997994	−	0.0633016i	$-0.0201630\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.04783e9	+	5.04783e9i	0.0335083	+	0.0335083i
$624$		0			0
$625$	−4.34966e8	+	1.52587e11i	−0.00285059	+	0.999996i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.03378e10i		−	0.0660428i
$630$		0			0
$631$	−4.57215e10			−0.288405			−0.144203	−	0.989548i	$-0.546062\pi$
$631$	−4.57215e10			−0.288405			−0.144203	+	0.989548i	$0.546062\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.18800e9	−	2.14461e9i	0.0319084	−	0.0131903i
$636$		0			0
$637$	1.58817e11	−	1.58817e11i	0.964582	−	0.964582i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.01366e11			0.600426			0.300213	−	0.953872i	$-0.402942\pi$
$641$	1.01366e11			0.600426			0.300213	+	0.953872i	$0.402942\pi$
$642$		0			0
$643$	−7.00290e10	−	7.00290e10i	−0.409670	−	0.409670i	0.471953	−	0.881623i	$-0.343549\pi$
$643$	−7.00290e10	−	7.00290e10i	−0.409670	−	0.409670i	−0.881623	+	0.471953i	$0.843549\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2.02119e11	+	2.02119e11i	−1.15343	+	1.15343i	−0.167567	+	0.985861i	$0.553591\pi$
$647$	−2.02119e11	+	2.02119e11i	−1.15343	+	1.15343i	−0.985861	+	0.167567i	$0.946409\pi$
$648$		0			0
$649$		−	1.38447e11i		−	0.780376i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.44565e11	−	1.44565e11i	−0.795078	−	0.795078i	0.187237	−	0.982315i	$-0.440047\pi$
$653$	−1.44565e11	−	1.44565e11i	−0.795078	−	0.795078i	−0.982315	+	0.187237i	$0.940047\pi$
$654$		0			0
$655$	1.29660e11	+	5.38152e10i	0.704434	+	0.292374i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			6.86847e9i			0.0364182i	0.999834	+	0.0182091i	$0.00579645\pi$
$659$			6.86847e9i			0.0364182i	−0.999834	+	0.0182091i	$0.994204\pi$
$660$		0			0
$661$	3.42797e10			0.179569			0.0897844	−	0.995961i	$-0.471382\pi$
$661$	3.42797e10			0.179569			0.0897844	+	0.995961i	$0.471382\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.08654e9	+	9.88571e9i	0.0208963	+	0.0505500i
$666$		0			0
$667$	−1.17228e11	+	1.17228e11i	−0.592283	+	0.592283i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.18337e10			−0.305025
$672$		0			0
$673$	5.18977e10	+	5.18977e10i	0.252981	+	0.252981i	0.822192	−	0.569211i	$-0.192751\pi$
$673$	5.18977e10	+	5.18977e10i	0.252981	+	0.252981i	−0.569211	+	0.822192i	$0.692751\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.02064e9	+	5.02064e9i	−0.0239004	+	0.0239004i	−0.718956	−	0.695056i	$-0.755380\pi$
$677$	−5.02064e9	+	5.02064e9i	−0.0239004	+	0.0239004i	0.695056	+	0.718956i	$0.255380\pi$
$678$		0			0
$679$			3.73620e10i			0.175772i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.82161e10	−	3.82161e10i	−0.175616	−	0.175616i	0.613826	−	0.789441i	$-0.289630\pi$
$683$	−3.82161e10	−	3.82161e10i	−0.175616	−	0.175616i	−0.789441	+	0.613826i	$0.789630\pi$
$684$		0			0
$685$	2.76209e11	−	1.14179e11i	1.25452	−	0.518590i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1.41701e10i		−	0.0628778i
$690$		0			0
$691$	3.21018e11			1.40805			0.704023	−	0.710177i	$-0.251385\pi$
$691$	3.21018e11			1.40805			0.704023	+	0.710177i	$0.251385\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.31530e11	−	3.16903e11i	0.563749	−	1.35827i
$696$		0			0
$697$	9.68399e10	−	9.68399e10i	0.410321	−	0.410321i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2.92163e11			1.20991			0.604956	−	0.796259i	$-0.293191\pi$
$701$	2.92163e11			1.20991			0.604956	+	0.796259i	$0.293191\pi$
$702$		0			0
$703$	−1.45541e10	−	1.45541e10i	−0.0595886	−	0.0595886i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.00314e10	−	2.00314e10i	0.0801740	−	0.0801740i
$708$		0			0
$709$			1.84412e11i			0.729801i	0.931047	+	0.364900i	$0.118897\pi$
$709$			1.84412e11i			0.729801i	−0.931047	+	0.364900i	$0.881103\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.90773e10	−	4.90773e10i	−0.189899	−	0.189899i
$714$		0			0
$715$	−1.79262e11	−	4.33651e11i	−0.685905	−	1.65927i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			3.86759e11i			1.44719i	0.690226	+	0.723594i	$0.257511\pi$
$719$			3.86759e11i			1.44719i	−0.690226	+	0.723594i	$0.742489\pi$
$720$		0			0
$721$	−4.00780e10			−0.148308
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.64743e11	−	1.64274e11i	0.596286	−	0.594589i
$726$		0			0
$727$	1.08489e11	−	1.08489e11i	0.388373	−	0.388373i	−0.485734	−	0.874107i	$-0.661448\pi$
$727$	1.08489e11	−	1.08489e11i	0.388373	−	0.388373i	0.874107	+	0.485734i	$0.161448\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.61900e11			−0.566994
$732$		0			0
$733$	−1.81338e11	−	1.81338e11i	−0.628165	−	0.628165i	0.319441	−	0.947606i	$-0.396505\pi$
$733$	−1.81338e11	−	1.81338e11i	−0.628165	−	0.628165i	−0.947606	+	0.319441i	$0.896505\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.11162e11	−	5.11162e11i	1.73256	−	1.73256i
$738$		0			0
$739$		−	5.46351e11i		−	1.83187i	−0.401330	−	0.915933i	$-0.631452\pi$
$739$		−	5.46351e11i		−	1.83187i	0.401330	−	0.915933i	$-0.368548\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.08189e11	−	3.08189e11i	−1.01126	−	1.01126i	−0.999936	−	0.0113233i	$-0.996396\pi$
$743$	−3.08189e11	−	3.08189e11i	−1.01126	−	1.01126i	−0.0113233	−	0.999936i	$-0.503604\pi$
$744$		0			0
$745$	1.35724e11	−	3.27007e11i	0.440586	−	1.06153i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			4.55598e10i			0.144762i
$750$		0			0
$751$	1.35575e9			0.00426205			0.00213103	−	0.999998i	$-0.499322\pi$
$751$	1.35575e9			0.00426205			0.00213103	+	0.999998i	$0.499322\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.44716e11	+	2.26084e11i	1.67642	+	0.695795i
$756$		0			0
$757$	−2.95475e11	+	2.95475e11i	−0.899781	+	0.899781i	−0.995416	−	0.0956356i	$-0.969512\pi$
$757$	−2.95475e11	+	2.95475e11i	−0.899781	+	0.899781i	0.0956356	+	0.995416i	$0.469512\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.50018e11			−1.04364			−0.521822	−	0.853055i	$-0.674747\pi$
$761$	−3.50018e11			−1.04364			−0.521822	+	0.853055i	$0.674747\pi$
$762$		0			0
$763$	3.33229e10	+	3.33229e10i	0.0983205	+	0.0983205i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.03028e11	+	2.03028e11i	−0.586642	+	0.586642i
$768$		0			0
$769$			3.37337e11i			0.964624i	0.875999	+	0.482312i	$0.160203\pi$
$769$			3.37337e11i			0.964624i	−0.875999	+	0.482312i	$0.839797\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.80219e10	+	5.80219e10i	0.162508	+	0.162508i	0.783677	−	0.621169i	$-0.213342\pi$
$773$	5.80219e10	+	5.80219e10i	0.162508	+	0.162508i	−0.621169	+	0.783677i	$0.713342\pi$
$774$		0			0
$775$	6.87728e10	+	6.89691e10i	0.190638	+	0.191182i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	2.72672e11i		−	0.740442i
$780$		0			0
$781$	7.05812e11			1.89708
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.12601e11	+	4.65470e10i	−0.296528	+	0.122578i
$786$		0			0
$787$	1.28984e10	−	1.28984e10i	0.0336231	−	0.0336231i	−0.690095	−	0.723718i	$-0.742431\pi$
$787$	1.28984e10	−	1.28984e10i	0.0336231	−	0.0336231i	0.723718	+	0.690095i	$0.242431\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.08410e9			0.00532368
$792$		0			0
$793$	9.06772e10	+	9.06772e10i	0.229301	+	0.229301i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.41350e10	+	1.41350e10i	−0.0350318	+	0.0350318i	−0.724406	−	0.689374i	$-0.757886\pi$
$797$	−1.41350e10	+	1.41350e10i	−0.0350318	+	0.0350318i	0.689374	+	0.724406i	$0.257886\pi$
$798$		0			0
$799$			2.78881e11i			0.684276i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5.26404e11	−	5.26404e11i	−1.26607	−	1.26607i
$804$		0			0
$805$	4.33740e10	+	1.80023e10i	0.103287	+	0.0428692i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			6.90397e11i			1.61178i	0.592067	+	0.805889i	$0.298312\pi$
$809$			6.90397e11i			1.61178i	−0.592067	+	0.805889i	$0.701688\pi$
$810$		0			0
$811$	−8.75326e10			−0.202342			−0.101171	−	0.994869i	$-0.532259\pi$
$811$	−8.75326e10			−0.202342			−0.101171	+	0.994869i	$0.532259\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.31179e10	−	1.52688e11i	−0.143061	−	0.346078i
$816$		0			0
$817$	−2.27931e11	+	2.27931e11i	−0.511583	+	0.511583i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.30471e11			0.507275			0.253638	−	0.967299i	$-0.418373\pi$
$821$	2.30471e11			0.507275			0.253638	+	0.967299i	$0.418373\pi$
$822$		0			0
$823$	−8.64879e10	−	8.64879e10i	−0.188519	−	0.188519i	0.606536	−	0.795056i	$-0.292558\pi$
$823$	−8.64879e10	−	8.64879e10i	−0.188519	−	0.188519i	−0.795056	+	0.606536i	$0.792558\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.31886e11	−	2.31886e11i	0.495739	−	0.495739i	−0.414370	−	0.910109i	$-0.635998\pi$
$827$	2.31886e11	−	2.31886e11i	0.495739	−	0.495739i	0.910109	+	0.414370i	$0.135998\pi$
$828$		0			0
$829$		−	1.24583e11i		−	0.263779i	−0.991264	−	0.131890i	$-0.957896\pi$
$829$		−	1.24583e11i		−	0.263779i	0.991264	−	0.131890i	$-0.0421044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.28174e11	+	1.28174e11i	0.266207	+	0.266207i
$834$		0			0
$835$	−4.39438e11	+	1.81654e11i	−0.903965	+	0.373680i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1.32482e11i			0.267367i	0.991024	+	0.133684i	$0.0426806\pi$
$839$			1.32482e11i			0.267367i	−0.991024	+	0.133684i	$0.957319\pi$
$840$		0			0
$841$	1.45525e11			0.290906
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.77613e11	+	4.27933e11i	−0.348376	+	0.839361i
$846$		0			0
$847$	−2.81840e10	+	2.81840e10i	−0.0547606	+	0.0547606i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−9.03602e10			−0.172290
$852$		0			0
$853$	−1.92442e11	−	1.92442e11i	−0.363499	−	0.363499i	0.501600	−	0.865099i	$-0.332745\pi$
$853$	−1.92442e11	−	1.92442e11i	−0.363499	−	0.363499i	−0.865099	+	0.501600i	$0.832745\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5.33628e11	+	5.33628e11i	−0.989272	+	0.989272i	−0.999943	−	0.0106713i	$-0.996603\pi$
$857$	−5.33628e11	+	5.33628e11i	−0.989272	+	0.989272i	0.0106713	+	0.999943i	$0.496603\pi$
$858$		0			0
$859$		−	3.48989e11i		−	0.640972i	−0.947253	−	0.320486i	$-0.896154\pi$
$859$		−	3.48989e11i		−	0.640972i	0.947253	−	0.320486i	$-0.103846\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	7.36194e11	+	7.36194e11i	1.32724	+	1.32724i	0.907769	+	0.419470i	$0.137784\pi$
$863$	7.36194e11	+	7.36194e11i	1.32724	+	1.32724i	0.419470	+	0.907769i	$0.362216\pi$
$864$		0			0
$865$	1.61947e11	+	3.91764e11i	0.289273	+	0.699777i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	3.87355e11i		−	0.679251i
$870$		0			0
$871$	−1.49920e12			−2.60488
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.08315e10	−	2.53497e10i	−0.103776	−	0.0432455i
$876$		0			0
$877$	−5.69351e10	+	5.69351e10i	−0.0962458	+	0.0962458i	−0.753590	−	0.657344i	$-0.771680\pi$
$877$	−5.69351e10	+	5.69351e10i	−0.0962458	+	0.0962458i	0.657344	+	0.753590i	$0.271680\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	4.10972e11			0.682196			0.341098	−	0.940028i	$-0.389201\pi$
$881$	4.10972e11			0.682196			0.341098	+	0.940028i	$0.389201\pi$
$882$		0			0
$883$	3.49612e11	+	3.49612e11i	0.575100	+	0.575100i	0.933549	−	0.358449i	$-0.116694\pi$
$883$	3.49612e11	+	3.49612e11i	0.575100	+	0.575100i	−0.358449	+	0.933549i	$0.616694\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.64244e11	+	5.64244e11i	−0.911533	+	0.911533i	−0.996393	−	0.0848602i	$-0.972956\pi$
$887$	−5.64244e11	+	5.64244e11i	−0.911533	+	0.911533i	0.0848602	+	0.996393i	$0.472956\pi$
$888$		0			0
$889$			2.42457e9i			0.00388175i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.92622e11	+	3.92622e11i	0.617403	+	0.617403i
$894$		0			0
$895$	−3.33210e11	+	8.02822e11i	−0.519310	+	1.25120i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1.48503e11i		−	0.227351i
$900$		0			0
$901$	1.14361e10			0.0173531
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.41147e11	−	3.49117e11i	−1.25394	−	0.520447i
$906$		0			0
$907$	−9.44245e8	+	9.44245e8i	−0.00139526	+	0.00139526i	−0.707804	−	0.706409i	$-0.750314\pi$
$907$	−9.44245e8	+	9.44245e8i	−0.00139526	+	0.00139526i	0.706409	+	0.707804i	$0.250314\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.02795e11			0.584804			0.292402	−	0.956295i	$-0.405545\pi$
$911$	4.02795e11			0.584804			0.292402	+	0.956295i	$0.405545\pi$
$912$		0			0
$913$	4.25122e11	+	4.25122e11i	0.611829	+	0.611829i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.28728e10	+	4.28728e10i	−0.0606324	+	0.0606324i
$918$		0			0
$919$			5.71953e11i			0.801860i	0.916109	+	0.400930i	$0.131313\pi$
$919$			5.71953e11i			0.801860i	−0.916109	+	0.400930i	$0.868687\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.03505e12	−	1.03505e12i	−1.42611	−	1.42611i
$924$		0			0
$925$	1.26804e11	+	1.80734e8i	0.173207	+	0.000246872i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	9.36530e11i		−	1.25736i	−0.777665	−	0.628679i	$-0.783596\pi$
$929$		−	9.36530e11i		−	1.25736i	0.777665	−	0.628679i	$-0.216404\pi$
$930$		0			0
$931$	3.60899e11			0.480382
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.49980e11	−	1.44674e11i	0.457927	−	0.189297i
$936$		0			0
$937$	2.93070e11	−	2.93070e11i	0.380200	−	0.380200i	−0.490974	−	0.871174i	$-0.663359\pi$
$937$	2.93070e11	−	2.93070e11i	0.380200	−	0.380200i	0.871174	+	0.490974i	$0.163359\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.88370e11			−1.13301			−0.566507	−	0.824057i	$-0.691706\pi$
$941$	−8.88370e11			−1.13301			−0.566507	+	0.824057i	$0.691706\pi$
$942$		0			0
$943$	−8.46455e11	−	8.46455e11i	−1.07043	−	1.07043i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.47904e10	−	5.47904e10i	0.0681247	−	0.0681247i	−0.672224	−	0.740348i	$-0.734661\pi$
$947$	5.47904e10	−	5.47904e10i	0.0681247	−	0.0681247i	0.740348	+	0.672224i	$0.234661\pi$
$948$		0			0
$949$			1.54391e12i			1.90352i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.59818e11	+	9.59818e11i	1.16364	+	1.16364i	0.983673	+	0.179963i	$0.0575979\pi$
$953$	9.59818e11	+	9.59818e11i	1.16364	+	1.16364i	0.179963	+	0.983673i	$0.442402\pi$
$954$		0			0
$955$	2.50029e11	+	1.03774e11i	0.300592	+	0.124760i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1.29084e11i			0.152616i
$960$		0			0
$961$	−7.90721e11			−0.927106
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.16353e11	−	7.65287e11i	−0.364807	−	0.882500i
$966$		0			0
$967$	5.73546e11	−	5.73546e11i	0.655938	−	0.655938i	−0.298479	−	0.954416i	$-0.596479\pi$
$967$	5.73546e11	−	5.73546e11i	0.655938	−	0.655938i	0.954416	+	0.298479i	$0.0964792\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.59341e11			−0.741708			−0.370854	−	0.928691i	$-0.620935\pi$
$971$	−6.59341e11			−0.741708			−0.370854	+	0.928691i	$0.620935\pi$
$972$		0			0
$973$	1.04786e11	+	1.04786e11i	0.116910	+	0.116910i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.80295e11	−	4.80295e11i	0.527144	−	0.527144i	−0.392576	−	0.919720i	$-0.628416\pi$
$977$	4.80295e11	−	4.80295e11i	0.527144	−	0.527144i	0.919720	+	0.392576i	$0.128416\pi$
$978$		0			0
$979$		−	5.03183e11i		−	0.547766i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.05351e12	+	1.05351e12i	1.12830	+	1.12830i	0.990453	+	0.137852i	$0.0440198\pi$
$983$	1.05351e12	+	1.05351e12i	1.12830	+	1.12830i	0.137852	+	0.990453i	$0.455980\pi$
$984$		0			0
$985$	−5.82236e11	+	2.40684e11i	−0.618521	+	0.255684i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1.41513e12i			1.47915i
$990$		0			0
$991$	−6.86106e11			−0.711372			−0.355686	−	0.934606i	$-0.615753\pi$
$991$	−6.86106e11			−0.711372			−0.355686	+	0.934606i	$0.615753\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.90754e11	+	9.41464e11i	−0.398667	+	0.960531i
$996$		0			0
$997$	−9.70841e11	+	9.70841e11i	−0.982579	+	0.982579i	−0.999851	−	0.0172719i	$-0.994502\pi$
$997$	−9.70841e11	+	9.70841e11i	−0.982579	+	0.982579i	0.0172719	+	0.999851i	$0.494502\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.9.l.a.73.4		8
3.2	odd	2		20.9.f.a.13.3	✓	8
5.2	odd	4	inner	180.9.l.a.37.4		8
12.11	even	2		80.9.p.d.33.2		8
15.2	even	4		20.9.f.a.17.3	yes	8
15.8	even	4		100.9.f.b.57.2		8
15.14	odd	2		100.9.f.b.93.2		8
60.47	odd	4		80.9.p.d.17.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.9.f.a.13.3	✓	8	3.2	odd	2
20.9.f.a.17.3	yes	8	15.2	even	4
80.9.p.d.17.2		8	60.47	odd	4
80.9.p.d.33.2		8	12.11	even	2
100.9.f.b.57.2		8	15.8	even	4
100.9.f.b.93.2		8	15.14	odd	2
180.9.l.a.37.4		8	5.2	odd	4	inner
180.9.l.a.73.4		8	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	180.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(577.254 + 239.589i) q^{5} +(-190.873 + 190.873i) q^{7} +19026.8 q^{11} +(-27902.1 - 27902.1i) q^{13} +(22518.5 - 22518.5i) q^{17} -63405.3i q^{19} +(-196829. - 196829. i) q^{23} +(275820. + 276607. i) q^{25} -595585. i q^{29} +249340. q^{31} +(-155913. + 64451.1i) q^{35} +(229540. - 229540. i) q^{37} +4.30046e6 q^{41} +(-3.59483e6 - 3.59483e6i) q^{43} +(-6.19225e6 + 6.19225e6i) q^{47} +5.69194e6i q^{49} +(253926. + 253926. i) q^{53} +(1.09833e7 + 4.55859e6i) q^{55} -7.27642e6i q^{59} -3.24983e6 q^{61} +(-9.42158e6 - 2.27916e7i) q^{65} +(2.68654e7 - 2.68654e7i) q^{67} +3.70958e7 q^{71} +(-2.76665e7 - 2.76665e7i) q^{73} +(-3.63169e6 + 3.63169e6i) q^{77} -2.03584e7i q^{79} +(2.23434e7 + 2.23434e7i) q^{83} +(1.83941e7 - 7.60373e6i) q^{85} -2.64461e7i q^{89} +1.06515e7 q^{91} +(1.51912e7 - 3.66010e7i) q^{95} +(9.78714e7 - 9.78714e7i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 894 q^{5} - 2030 q^{7} + 420 q^{11} + 33180 q^{13} - 43620 q^{17} + 663270 q^{23} + 163396 q^{25} - 3178492 q^{31} - 2571618 q^{35} + 5344080 q^{37} + 10185252 q^{41} - 10342710 q^{43} - 19232250 q^{47}+ \cdots - 179570760 q^{97}+O(q^{100}) \) 8 * q - 894 * q^5 - 2030 * q^7 + 420 * q^11 + 33180 * q^13 - 43620 * q^17 + 663270 * q^23 + 163396 * q^25 - 3178492 * q^31 - 2571618 * q^35 + 5344080 * q^37 + 10185252 * q^41 - 10342710 * q^43 - 19232250 * q^47 + 24320640 * q^53 + 21483180 * q^55 - 82515684 * q^61 - 72045768 * q^65 + 100675930 * q^67 + 99290076 * q^71 - 93528520 * q^73 - 134199660 * q^77 + 10450350 * q^83 + 51676156 * q^85 - 130681068 * q^91 - 84367944 * q^95 - 179570760 * q^97

Introduction

L-functions

Modular forms

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