LMFDB - Embedded newform 252.8.k.e.37.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,8,Mod(37,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$78.7210264220$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 89566 x^{14} - 207320 x^{13} + 5161603375 x^{12} - 17143558340 x^{11} + 178819626045814 x^{10} + \cdots + 34\!\cdots\!16 $ x^16 + 89566x^14 - 207320x^13 + 5161603375x^12 - 17143558340x^11 + 178819626045814x^10 - 993183560713460x^9 + 4531523101103784409x^8 - 25038421018919762700x^7 + 77467785793463920028824x^6 - 352418565495722251364640x^5 + 967025470523891143640719680x^4 - 1651641278575894088053178880x^3 + 7200793191683961412674394484736x^2 + 21939035854717669821876654243840x + 34494438285389383231614325978300416
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{13}\cdot 7^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.8
Root			$95.9276 - 166.152i$ of defining polynomial
Character	$\chi$	$=$	252.37
Dual form			252.8.k.e.109.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(173.134 - 299.876i) q^{5} +(334.186 - 843.720i) q^{7} +O(q^{10})$	$q+(173.134 - 299.876i) q^{5} +(334.186 - 843.720i) q^{7} +(437.706 + 758.129i) q^{11} -507.478 q^{13} +(-9804.06 - 16981.1i) q^{17} +(17766.7 - 30772.7i) q^{19} +(48754.4 - 84445.0i) q^{23} +(-20888.0 - 36179.1i) q^{25} +56682.8 q^{29} +(26332.7 + 45609.6i) q^{31} +(-195153. - 246291. i) q^{35} +(-171941. + 297810. i) q^{37} +279731. q^{41} +73777.2 q^{43} +(-514934. + 891892. i) q^{47} +(-600182. - 563919. i) q^{49} +(186831. + 323600. i) q^{53} +303127. q^{55} +(520363. + 901296. i) q^{59} +(161336. - 279442. i) q^{61} +(-87861.6 + 152181. i) q^{65} +(-1.50274e6 - 2.60282e6i) q^{67} -2.97728e6 q^{71} +(-1.26966e6 - 2.19911e6i) q^{73} +(785924. - 115945. i) q^{77} +(936558. - 1.62217e6i) q^{79} +3.44787e6 q^{83} -6.78965e6 q^{85} +(5.03292e6 - 8.71726e6i) q^{89} +(-169592. + 428169. i) q^{91} +(-6.15201e6 - 1.06556e7i) q^{95} -1.52465e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 1680 q^{7}+O(q^{10}) $ 16 * q + 1680 * q^7	$ 16 q + 1680 q^{7} - 28280 q^{13} + 42224 q^{19} - 80460 q^{25} + 164752 q^{31} - 647980 q^{37} + 1341440 q^{43} + 230104 q^{49} - 323120 q^{55} - 4319336 q^{61} - 3905760 q^{67} + 6471780 q^{73} - 6093104 q^{79} + 456400 q^{85} + 15969856 q^{91} - 27141240 q^{97}+O(q^{100}) $ 16 * q + 1680 * q^7 - 28280 * q^13 + 42224 * q^19 - 80460 * q^25 + 164752 * q^31 - 647980 * q^37 + 1341440 * q^43 + 230104 * q^49 - 323120 * q^55 - 4319336 * q^61 - 3905760 * q^67 + 6471780 * q^73 - 6093104 * q^79 + 456400 * q^85 + 15969856 * q^91 - 27141240 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	173.134	−	299.876i	0.619422	−	1.07287i	−0.370170	−	0.928964i	$-0.620700\pi$
$5$	173.134	−	299.876i	0.619422	−	1.07287i	0.989591	−	0.143906i	$-0.0459662\pi$
$6$		0			0
$7$	334.186	−	843.720i	0.368252	−	0.929726i
$7$	334.186	−	843.720i	0.368252	−	0.929726i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	437.706	+	758.129i	0.0991535	+	0.171739i	0.911335	−	0.411667i	$-0.135053\pi$
$11$	437.706	+	758.129i	0.0991535	+	0.171739i	−0.812181	+	0.583406i	$0.801720\pi$
$12$		0			0
$13$	−507.478			−0.0640643			−0.0320321	−	0.999487i	$-0.510198\pi$
$13$	−507.478			−0.0640643			−0.0320321	+	0.999487i	$0.510198\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−9804.06	−	16981.1i	−0.483988	−	0.838292i	0.515843	−	0.856683i	$-0.327479\pi$
$17$	−9804.06	−	16981.1i	−0.483988	−	0.838292i	−0.999831	+	0.0183914i	$0.994146\pi$
$18$		0			0
$19$	17766.7	−	30772.7i	0.594248	−	1.02927i	−0.399404	−	0.916775i	$-0.630783\pi$
$19$	17766.7	−	30772.7i	0.594248	−	1.02927i	0.993652	−	0.112493i	$-0.0358837\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	48754.4	−	84445.0i	0.835538	−	1.44719i	−0.0580543	−	0.998313i	$-0.518490\pi$
$23$	48754.4	−	84445.0i	0.835538	−	1.44719i	0.893592	−	0.448880i	$-0.148177\pi$
$24$		0			0
$25$	−20888.0	−	36179.1i	−0.267367	−	0.463092i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	56682.8			0.431577			0.215789	−	0.976440i	$-0.430768\pi$
$29$	56682.8			0.431577			0.215789	+	0.976440i	$0.430768\pi$
$30$		0			0
$31$	26332.7	+	45609.6i	0.158756	+	0.274973i	0.934420	−	0.356172i	$-0.115918\pi$
$31$	26332.7	+	45609.6i	0.158756	+	0.274973i	−0.775664	+	0.631146i	$0.782585\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−195153.	−	246291.i	−0.769372	−	0.970979i
$36$		0			0
$37$	−171941.	+	297810.i	−0.558049	+	0.966569i	0.439610	+	0.898189i	$0.355117\pi$
$37$	−171941.	+	297810.i	−0.558049	+	0.966569i	−0.997659	+	0.0683807i	$0.978217\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	279731.			0.633865			0.316933	−	0.948448i	$-0.397347\pi$
$41$	279731.			0.633865			0.316933	+	0.948448i	$0.397347\pi$
$42$		0			0
$43$	73777.2			0.141509			0.0707543	−	0.997494i	$-0.477459\pi$
$43$	73777.2			0.141509			0.0707543	+	0.997494i	$0.477459\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−514934.	+	891892.i	−0.723451	+	1.25305i	0.236157	+	0.971715i	$0.424112\pi$
$47$	−514934.	+	891892.i	−0.723451	+	1.25305i	−0.959608	+	0.281339i	$0.909221\pi$
$48$		0			0
$49$	−600182.	−	563919.i	−0.728781	−	0.684747i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	186831.	+	323600.i	0.172378	+	0.298568i	0.939251	−	0.343231i	$-0.111522\pi$
$53$	186831.	+	323600.i	0.172378	+	0.298568i	−0.766873	+	0.641799i	$0.778188\pi$
$54$		0			0
$55$	303127.			0.245671
$56$		0			0
$57$		0			0
$58$		0			0
$59$	520363.	+	901296.i	0.329856	+	0.571328i	0.982483	−	0.186352i	$-0.0596664\pi$
$59$	520363.	+	901296.i	0.329856	+	0.571328i	−0.652627	+	0.757679i	$0.726333\pi$
$60$		0			0
$61$	161336.	−	279442.i	0.0910073	−	0.157629i	−0.816928	−	0.576740i	$-0.804325\pi$
$61$	161336.	−	279442.i	0.0910073	−	0.157629i	0.907935	+	0.419110i	$0.137658\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−87861.6	+	152181.i	−0.0396828	+	0.0687326i
$66$		0			0
$67$	−1.50274e6	−	2.60282e6i	−0.610410	−	1.05726i	−0.991171	−	0.132587i	$-0.957672\pi$
$67$	−1.50274e6	−	2.60282e6i	−0.610410	−	1.05726i	0.380762	−	0.924673i	$-0.375662\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.97728e6			−0.987224			−0.493612	−	0.869682i	$-0.664324\pi$
$71$	−2.97728e6			−0.987224			−0.493612	+	0.869682i	$0.664324\pi$
$72$		0			0
$73$	−1.26966e6	−	2.19911e6i	−0.381994	−	0.661633i	0.609353	−	0.792899i	$-0.291429\pi$
$73$	−1.26966e6	−	2.19911e6i	−0.381994	−	0.661633i	−0.991347	+	0.131266i	$0.958096\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	785924.	−	115945.i	0.196184	−	0.0289424i
$78$		0			0
$79$	936558.	−	1.62217e6i	0.213717	−	0.370169i	−0.739158	−	0.673532i	$-0.764776\pi$
$79$	936558.	−	1.62217e6i	0.213717	−	0.370169i	0.952875	+	0.303363i	$0.0981095\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.44787e6			0.661878			0.330939	−	0.943652i	$-0.392635\pi$
$83$	3.44787e6			0.661878			0.330939	+	0.943652i	$0.392635\pi$
$84$		0			0
$85$	−6.78965e6			−1.19917
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.03292e6	−	8.71726e6i	0.756754	−	1.31074i	−0.187744	−	0.982218i	$-0.560118\pi$
$89$	5.03292e6	−	8.71726e6i	0.756754	−	1.31074i	0.944498	−	0.328518i	$-0.106549\pi$
$90$		0			0
$91$	−169592.	+	428169.i	−0.0235918	+	0.0595622i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.15201e6	−	1.06556e7i	−0.736181	−	1.27510i
$96$		0			0
$97$	−1.52465e7			−1.69617			−0.848085	−	0.529860i	$-0.822244\pi$
$97$	−1.52465e7			−1.69617			−0.848085	+	0.529860i	$0.822244\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.13213e6	+	1.96091e6i	0.109338	+	0.189379i	0.915502	−	0.402313i	$-0.131794\pi$
$101$	1.13213e6	+	1.96091e6i	0.109338	+	0.189379i	−0.806164	+	0.591692i	$0.798460\pi$
$102$		0			0
$103$	281141.	−	486950.i	0.0253509	−	0.0439091i	−0.853072	−	0.521794i	$-0.825263\pi$
$103$	281141.	−	486950.i	0.0253509	−	0.0439091i	0.878423	+	0.477885i	$0.158596\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.95243e6	+	1.37740e7i	−0.627562	+	1.08697i	0.360477	+	0.932768i	$0.382614\pi$
$107$	−7.95243e6	+	1.37740e7i	−0.627562	+	1.08697i	−0.988039	+	0.154201i	$0.950720\pi$
$108$		0			0
$109$	−6.02223e6	−	1.04308e7i	−0.445415	−	0.771481i	0.552666	−	0.833403i	$-0.313610\pi$
$109$	−6.02223e6	−	1.04308e7i	−0.445415	−	0.771481i	−0.998081	+	0.0619216i	$0.980277\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.75416e7			1.14366			0.571828	−	0.820374i	$-0.306235\pi$
$113$	1.75416e7			1.14366			0.571828	+	0.820374i	$0.306235\pi$
$114$		0			0
$115$	−1.68820e7	−	2.92405e7i	−1.03510	−	1.79285i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.76037e7	+	2.59702e6i	−0.957611	+	0.141273i
$120$		0			0
$121$	9.36041e6	−	1.62127e7i	0.480337	−	0.831968i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.25865e7			0.576393
$126$		0			0
$127$	−1.98177e7			−0.858499			−0.429250	−	0.903186i	$-0.641222\pi$
$127$	−1.98177e7			−0.858499			−0.429250	+	0.903186i	$0.641222\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.15059e7	+	1.99289e7i	−0.447170	+	0.774521i	−0.998201	−	0.0599637i	$-0.980901\pi$
$131$	−1.15059e7	+	1.99289e7i	−0.447170	+	0.774521i	0.551030	+	0.834485i	$0.314235\pi$
$132$		0			0
$133$	−2.00262e7	−	2.52739e7i	−0.738104	−	0.931519i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.94076e6	+	6.82560e6i	0.130936	+	0.226787i	0.924038	−	0.382302i	$-0.124869\pi$
$137$	3.94076e6	+	6.82560e6i	0.130936	+	0.226787i	−0.793102	+	0.609089i	$0.791535\pi$
$138$		0			0
$139$	−3.33107e7			−1.05204			−0.526020	−	0.850472i	$-0.676316\pi$
$139$	−3.33107e7			−1.05204			−0.526020	+	0.850472i	$0.676316\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−222126.	−	384734.i	−0.00635220	−	0.0110023i
$144$		0			0
$145$	9.81371e6	−	1.69978e7i	0.267328	−	0.463026i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.85252e7	+	4.94071e7i	−0.706443	+	1.22360i	0.259725	+	0.965683i	$0.416368\pi$
$149$	−2.85252e7	+	4.94071e7i	−0.706443	+	1.22360i	−0.966168	+	0.257913i	$0.916965\pi$
$150$		0			0
$151$	−3.16251e7	−	5.47762e7i	−0.747502	−	1.29471i	−0.949017	−	0.315226i	$-0.897920\pi$
$151$	−3.16251e7	−	5.47762e7i	−0.747502	−	1.29471i	0.201515	−	0.979485i	$-0.435414\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.82363e7			0.393347
$156$		0			0
$157$	−5.92448e6	−	1.02615e7i	−0.122180	−	0.211623i	0.798447	−	0.602065i	$-0.205655\pi$
$157$	−5.92448e6	−	1.02615e7i	−0.122180	−	0.211623i	−0.920627	+	0.390443i	$0.872322\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5.49549e7	−	6.93554e7i	−1.03780	−	1.30975i
$162$		0			0
$163$	−2.57630e7	+	4.46228e7i	−0.465950	+	0.807049i	−0.999244	−	0.0388808i	$-0.987621\pi$
$163$	−2.57630e7	+	4.46228e7i	−0.465950	+	0.807049i	0.533294	+	0.845930i	$0.320954\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.95166e7			1.65344			0.826719	−	0.562615i	$-0.190205\pi$
$167$	9.95166e7			1.65344			0.826719	+	0.562615i	$0.190205\pi$
$168$		0			0
$169$	−6.24910e7			−0.995896
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.24458e7	−	9.08388e7i	0.770104	−	1.33386i	−0.167401	−	0.985889i	$-0.553538\pi$
$173$	5.24458e7	−	9.08388e7i	0.770104	−	1.33386i	0.937505	−	0.347971i	$-0.113129\pi$
$174$		0			0
$175$	−3.75055e7	+	5.53307e6i	−0.529007	+	0.0780428i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.92027e7	−	1.19863e8i	−0.901856	−	1.56206i	−0.825082	−	0.565012i	$-0.808871\pi$
$179$	−6.92027e7	−	1.19863e8i	−0.901856	−	1.56206i	−0.0767740	−	0.997049i	$-0.524462\pi$
$180$		0			0
$181$	3.73519e7			0.468206			0.234103	−	0.972212i	$-0.424785\pi$
$181$	3.73519e7			0.468206			0.234103	+	0.972212i	$0.424785\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.95374e7	+	1.03122e8i	0.691335	+	1.19743i
$186$		0			0
$187$	8.58259e6	−	1.48655e7i	0.0959782	−	0.166239i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.95067e7	+	6.84276e7i	−0.410255	+	0.710582i	−0.994917	−	0.100694i	$-0.967894\pi$
$191$	−3.95067e7	+	6.84276e7i	−0.410255	+	0.710582i	0.584663	+	0.811277i	$0.301227\pi$
$192$		0			0
$193$	−2.99903e7	−	5.19448e7i	−0.300283	−	0.520105i	0.675917	−	0.736978i	$-0.263748\pi$
$193$	−2.99903e7	−	5.19448e7i	−0.300283	−	0.520105i	−0.976200	+	0.216873i	$0.930414\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.03085e8			−0.960650			−0.480325	−	0.877091i	$-0.659481\pi$
$197$	−1.03085e8			−0.960650			−0.480325	+	0.877091i	$0.659481\pi$
$198$		0			0
$199$	−1.91440e7	−	3.31584e7i	−0.172205	−	0.298268i	0.766985	−	0.641665i	$-0.221756\pi$
$199$	−1.91440e7	−	3.31584e7i	−0.172205	−	0.298268i	−0.939191	+	0.343396i	$0.888423\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.89426e7	−	4.78244e7i	0.158929	−	0.401249i
$204$		0			0
$205$	4.84308e7	−	8.38847e7i	0.392630	−	0.680055i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.11063e7			0.235687
$210$		0			0
$211$	1.27808e8			0.936635			0.468318	−	0.883560i	$-0.344860\pi$
$211$	1.27808e8			0.936635			0.468318	+	0.883560i	$0.344860\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.27733e7	−	2.21240e7i	0.0876535	−	0.151820i
$216$		0			0
$217$	4.72817e7	−	6.97533e6i	0.314112	−	0.0463399i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.97535e6	+	8.61755e6i	0.0310063	+	0.0537045i
$222$		0			0
$223$	−6.36267e7			−0.384213			−0.192107	−	0.981374i	$-0.561532\pi$
$223$	−6.36267e7			−0.384213			−0.192107	+	0.981374i	$0.561532\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.87102e7	+	1.53651e8i	0.503365	+	0.871854i	0.999992	+	0.00388995i	$0.00123821\pi$
$227$	8.87102e7	+	1.53651e8i	0.503365	+	0.871854i	−0.496627	+	0.867964i	$0.665428\pi$
$228$		0			0
$229$	−4.96856e7	+	8.60580e7i	−0.273405	+	0.473551i	−0.969731	−	0.244174i	$-0.921483\pi$
$229$	−4.96856e7	+	8.60580e7i	−0.273405	+	0.473551i	0.696326	+	0.717725i	$0.254817\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.33649e8	+	2.31486e8i	−0.692180	+	1.19889i	0.278943	+	0.960308i	$0.410016\pi$
$233$	−1.33649e8	+	2.31486e8i	−0.692180	+	1.19889i	−0.971122	+	0.238582i	$0.923317\pi$
$234$		0			0
$235$	1.78305e8	+	3.08833e8i	0.896243	+	1.55234i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.52202e8			0.721152			0.360576	−	0.932730i	$-0.382580\pi$
$239$	1.52202e8			0.721152			0.360576	+	0.932730i	$0.382580\pi$
$240$		0			0
$241$	1.05229e8	+	1.82262e8i	0.484256	+	0.838757i	0.999836	−	0.0180846i	$-0.00575684\pi$
$241$	1.05229e8	+	1.82262e8i	0.484256	+	0.838757i	−0.515580	+	0.856841i	$0.672424\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.73018e8	+	8.23471e7i	−1.18607	+	0.357740i
$246$		0			0
$247$	−9.01619e6	+	1.56165e7i	−0.0380701	+	0.0659393i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.72017e8			0.686615			0.343307	−	0.939223i	$-0.388453\pi$
$251$	1.72017e8			0.686615			0.343307	+	0.939223i	$0.388453\pi$
$252$		0			0
$253$	8.53603e7			0.331386
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.04935e7	+	5.28163e7i	−0.112058	+	0.194089i	−0.916600	−	0.399806i	$-0.869077\pi$
$257$	−3.04935e7	+	5.28163e7i	−0.112058	+	0.194089i	0.804542	+	0.593896i	$0.202411\pi$
$258$		0			0
$259$	1.93808e8	+	2.44594e8i	0.693142	+	0.874774i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.37264e7	−	7.57363e7i	−0.148217	−	0.256720i	0.782351	−	0.622837i	$-0.214020\pi$
$263$	−4.37264e7	−	7.57363e7i	−0.148217	−	0.256720i	−0.930569	+	0.366118i	$0.880687\pi$
$264$		0			0
$265$	1.29387e8			0.427100
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.71329e8	−	4.69955e8i	−0.849890	−	1.47205i	−0.881306	−	0.472547i	$-0.843335\pi$
$269$	−2.71329e8	−	4.69955e8i	−0.849890	−	1.47205i	0.0314155	−	0.999506i	$-0.489999\pi$
$270$		0			0
$271$	−2.54907e7	+	4.41512e7i	−0.0778018	+	0.134757i	−0.902301	−	0.431106i	$-0.858123\pi$
$271$	−2.54907e7	+	4.41512e7i	−0.0778018	+	0.134757i	0.824499	+	0.565863i	$0.191457\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.82856e7	−	3.16716e7i	0.0530207	−	0.0918345i
$276$		0			0
$277$	1.11619e8	+	1.93330e8i	0.315543	+	0.546536i	0.979553	−	0.201187i	$-0.0644800\pi$
$277$	1.11619e8	+	1.93330e8i	0.315543	+	0.546536i	−0.664010	+	0.747724i	$0.731147\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.34774e8			1.16894			0.584469	−	0.811416i	$-0.301303\pi$
$281$	4.34774e8			1.16894			0.584469	+	0.811416i	$0.301303\pi$
$282$		0			0
$283$	3.24748e8	+	5.62480e8i	0.851714	+	1.47521i	0.879660	+	0.475602i	$0.157770\pi$
$283$	3.24748e8	+	5.62480e8i	0.851714	+	1.47521i	−0.0279467	+	0.999609i	$0.508897\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.34822e7	−	2.36014e8i	0.233422	−	0.589321i
$288$		0			0
$289$	1.29303e7	−	2.23959e7i	0.0315112	−	0.0545790i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.61039e7			0.153529			0.0767645	−	0.997049i	$-0.475541\pi$
$293$	6.61039e7			0.153529			0.0767645	+	0.997049i	$0.475541\pi$
$294$		0			0
$295$	3.60370e8			0.817280
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.47418e7	+	4.28540e7i	−0.0535281	+	0.0927134i
$300$		0			0
$301$	2.46553e7	−	6.22473e7i	0.0521108	−	0.131564i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.58653e7	−	9.67615e7i	−0.112744	−	0.195278i
$306$		0			0
$307$	5.80890e8			1.14580			0.572901	−	0.819625i	$-0.305818\pi$
$307$	5.80890e8			1.14580			0.572901	+	0.819625i	$0.305818\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.05179e7	−	3.55381e7i	−0.0386787	−	0.0669935i	0.846038	−	0.533123i	$-0.178982\pi$
$311$	−2.05179e7	−	3.55381e7i	−0.0386787	−	0.0669935i	−0.884717	+	0.466129i	$0.845648\pi$
$312$		0			0
$313$	1.58978e8	−	2.75359e8i	0.293044	−	0.507567i	−0.681484	−	0.731833i	$-0.738665\pi$
$313$	1.58978e8	−	2.75359e8i	0.293044	−	0.507567i	0.974528	+	0.224266i	$0.0719985\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.78346e8	−	6.55314e8i	0.667085	−	1.15543i	−0.311630	−	0.950203i	$-0.600875\pi$
$317$	3.78346e8	−	6.55314e8i	0.667085	−	1.15543i	0.978715	−	0.205222i	$-0.0657916\pi$
$318$		0			0
$319$	2.48104e7	+	4.29729e7i	0.0427924	+	0.0741186i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.96741e8			−1.15044
$324$		0			0
$325$	1.06002e7	+	1.83601e7i	0.0171286	+	0.0296677i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5.80423e8	+	7.32518e8i	0.898584	+	1.13405i
$330$		0			0
$331$	−2.07896e8	+	3.60086e8i	−0.315100	+	0.545769i	−0.979459	−	0.201645i	$-0.935371\pi$
$331$	−2.07896e8	+	3.60086e8i	−0.315100	+	0.545769i	0.664359	+	0.747414i	$0.268705\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.04070e9			−1.51240
$336$		0			0
$337$	7.71149e8			1.09757			0.548787	−	0.835962i	$-0.315090\pi$
$337$	7.71149e8			1.09757			0.548787	+	0.835962i	$0.315090\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.30520e7	+	3.99272e7i	−0.0314824	+	0.0545291i
$342$		0			0
$343$	−6.76362e8	+	3.17932e8i	−0.905002	+	0.425407i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.96630e8	+	8.60189e8i	0.638087	+	1.10520i	0.985852	+	0.167618i	$0.0536075\pi$
$347$	4.96630e8	+	8.60189e8i	0.638087	+	1.10520i	−0.347765	+	0.937582i	$0.613059\pi$
$348$		0			0
$349$	2.57730e8			0.324546			0.162273	−	0.986746i	$-0.448118\pi$
$349$	2.57730e8			0.324546			0.162273	+	0.986746i	$0.448118\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.84532e8	+	8.39235e8i	0.586288	+	1.01548i	0.994713	+	0.102689i	$0.0327448\pi$
$353$	4.84532e8	+	8.39235e8i	0.586288	+	1.01548i	−0.408425	+	0.912792i	$0.633922\pi$
$354$		0			0
$355$	−5.15468e8	+	8.92816e8i	−0.611508	+	1.05916i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.83984e7	+	3.18669e7i	−0.0209869	+	0.0363504i	−0.876328	−	0.481715i	$-0.840014\pi$
$359$	−1.83984e7	+	3.18669e7i	−0.0209869	+	0.0363504i	0.855341	+	0.518065i	$0.173348\pi$
$360$		0			0
$361$	−1.84372e8	−	3.19342e8i	−0.206262	−	0.357257i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.79282e8			−0.946462
$366$		0			0
$367$	5.21654e8	+	9.03531e8i	0.550872	+	0.954139i	0.998212	+	0.0597748i	$0.0190382\pi$
$367$	5.21654e8	+	9.03531e8i	0.550872	+	0.954139i	−0.447339	+	0.894364i	$0.647628\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.35464e8	−	4.94900e7i	0.341065	−	0.0503163i
$372$		0			0
$373$	3.17534e8	−	5.49986e8i	0.316818	−	0.548745i	−0.663004	−	0.748616i	$-0.730719\pi$
$373$	3.17534e8	−	5.49986e8i	0.316818	−	0.548745i	0.979822	+	0.199871i	$0.0640523\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.87653e7			−0.0276487
$378$		0			0
$379$	−1.23849e9			−1.16857			−0.584284	−	0.811550i	$-0.698624\pi$
$379$	−1.23849e9			−1.16857			−0.584284	+	0.811550i	$0.698624\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.63548e8	+	2.83273e8i	−0.148747	+	0.257638i	−0.930765	−	0.365619i	$-0.880857\pi$
$383$	−1.63548e8	+	2.83273e8i	−0.148747	+	0.257638i	0.782017	+	0.623257i	$0.214191\pi$
$384$		0			0
$385$	1.01301e8	−	2.55754e8i	0.0904690	−	0.228407i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.96563e7	+	1.03328e8i	0.0513846	+	0.0890008i	0.890574	−	0.454839i	$-0.150303\pi$
$389$	5.96563e7	+	1.03328e8i	0.0513846	+	0.0890008i	−0.839189	+	0.543840i	$0.816970\pi$
$390$		0			0
$391$	−1.91196e9			−1.61756
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.24299e8	−	5.61703e8i	−0.264762	−	0.458582i
$396$		0			0
$397$	1.15614e9	−	2.00250e9i	0.927353	−	1.60622i	0.139620	−	0.990205i	$-0.455412\pi$
$397$	1.15614e9	−	2.00250e9i	0.927353	−	1.60622i	0.787733	−	0.616017i	$-0.211255\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.43632e8	−	1.63442e9i	0.730798	−	1.26578i	−0.225745	−	0.974187i	$-0.572481\pi$
$401$	9.43632e8	−	1.63442e9i	0.730798	−	1.26578i	0.956543	−	0.291593i	$-0.0941852\pi$
$402$		0			0
$403$	−1.33633e7	−	2.31459e7i	−0.0101706	−	0.0176159i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.01038e8			−0.221330
$408$		0			0
$409$	−9.34476e8	−	1.61856e9i	−0.675362	−	1.16976i	−0.976363	−	0.216138i	$-0.930654\pi$
$409$	−9.34476e8	−	1.61856e9i	−0.675362	−	1.16976i	0.301001	−	0.953624i	$-0.402679\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.34339e8	−	1.37840e8i	0.652649	−	0.0962832i
$414$		0			0
$415$	5.96943e8	−	1.03394e9i	0.409982	−	0.710109i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.19342e9			−0.792584			−0.396292	−	0.918125i	$-0.629703\pi$
$419$	−1.19342e9			−0.792584			−0.396292	+	0.918125i	$0.629703\pi$
$420$		0			0
$421$	−4.94195e7			−0.0322783			−0.0161392	−	0.999870i	$-0.505137\pi$
$421$	−4.94195e7			−0.0322783			−0.0161392	+	0.999870i	$0.505137\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.09575e8	+	7.09404e8i	−0.258804	+	0.448262i
$426$		0			0
$427$	−1.81854e8	−	2.29508e8i	−0.113038	−	0.142659i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.26012e8	+	1.43069e9i	0.496954	+	0.860749i	0.999994	−	0.00351400i	$-0.00111854\pi$
$431$	8.26012e8	+	1.43069e9i	0.496954	+	0.860749i	−0.503040	+	0.864263i	$0.667785\pi$
$432$		0			0
$433$	−2.35353e9			−1.39320			−0.696598	−	0.717462i	$-0.745304\pi$
$433$	−2.35353e9			−1.39320			−0.696598	+	0.717462i	$0.745304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.73240e9	−	3.00061e9i	−0.993034	−	1.71999i
$438$		0			0
$439$	−6.32030e8	+	1.09471e9i	−0.356543	+	0.617551i	−0.987381	−	0.158364i	$-0.949378\pi$
$439$	−6.32030e8	+	1.09471e9i	−0.356543	+	0.617551i	0.630838	+	0.775915i	$0.282711\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.58299e9	−	2.74182e9i	0.865099	−	1.49840i	−0.00184942	−	0.999998i	$-0.500589\pi$
$443$	1.58299e9	−	2.74182e9i	0.865099	−	1.49840i	0.866949	−	0.498398i	$-0.166078\pi$
$444$		0			0
$445$	−1.74273e9	−	3.01850e9i	−0.937499	−	1.62380i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.54070e9			1.84598			0.922989	−	0.384827i	$-0.125739\pi$
$449$	3.54070e9			1.84598			0.922989	+	0.384827i	$0.125739\pi$
$450$		0			0
$451$	1.22440e8	+	2.12072e8i	0.0628500	+	0.108859i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	9.90357e7	+	1.24987e8i	0.0492892	+	0.0622051i
$456$		0			0
$457$	1.67457e9	−	2.90044e9i	0.820724	−	1.42154i	−0.0844204	−	0.996430i	$-0.526904\pi$
$457$	1.67457e9	−	2.90044e9i	0.820724	−	1.42154i	0.905144	−	0.425105i	$-0.139763\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.58651e9			−1.22959			−0.614795	−	0.788687i	$-0.710761\pi$
$461$	−2.58651e9			−1.22959			−0.614795	+	0.788687i	$0.710761\pi$
$462$		0			0
$463$	−1.17364e9			−0.549544			−0.274772	−	0.961509i	$-0.588602\pi$
$463$	−1.17364e9			−0.549544			−0.274772	+	0.961509i	$0.588602\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.66720e8	+	8.08383e8i	−0.212054	+	0.367289i	−0.952357	−	0.304984i	$-0.901349\pi$
$467$	−4.66720e8	+	8.08383e8i	−0.212054	+	0.367289i	0.740303	+	0.672273i	$0.234682\pi$
$468$		0			0
$469$	−2.69824e9	+	3.98063e8i	−1.20775	+	0.178175i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.22927e7	+	5.59327e7i	0.0140311	+	0.0243025i
$474$		0			0
$475$	−1.48444e9			−0.635529
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.06626e8	+	5.31091e8i	0.127478	+	0.220798i	0.922699	−	0.385522i	$-0.125979\pi$
$479$	3.06626e8	+	5.31091e8i	0.127478	+	0.220798i	−0.795221	+	0.606320i	$0.792645\pi$
$480$		0			0
$481$	8.72561e7	−	1.51132e8i	0.0357510	−	0.0619225i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.63968e9	+	4.57207e9i	−1.05064	+	1.81977i
$486$		0			0
$487$	1.79545e8	+	3.10981e8i	0.0704403	+	0.122006i	0.899094	−	0.437755i	$-0.144226\pi$
$487$	1.79545e8	+	3.10981e8i	0.0704403	+	0.122006i	−0.828654	+	0.559761i	$0.810893\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.24919e9			−0.857514			−0.428757	−	0.903420i	$-0.641048\pi$
$491$	−2.24919e9			−0.857514			−0.428757	+	0.903420i	$0.641048\pi$
$492$		0			0
$493$	−5.55722e8	−	9.62539e8i	−0.208878	−	0.361788i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.94967e8	+	2.51199e9i	−0.363547	+	0.917848i
$498$		0			0
$499$	−1.31886e9	+	2.28433e9i	−0.475168	+	0.823015i	−0.999595	−	0.0284401i	$-0.990946\pi$
$499$	−1.31886e9	+	2.28433e9i	−0.475168	+	0.823015i	0.524428	+	0.851455i	$0.324279\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4.36665e9			1.52989			0.764946	−	0.644094i	$-0.222766\pi$
$503$	4.36665e9			1.52989			0.764946	+	0.644094i	$0.222766\pi$
$504$		0			0
$505$	7.84040e8			0.270906
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.37515e8	−	1.45062e9i	0.281501	−	0.487575i	−0.690253	−	0.723568i	$-0.742501\pi$
$509$	8.37515e8	−	1.45062e9i	0.281501	−	0.487575i	0.971755	+	0.235993i	$0.0758343\pi$
$510$		0			0
$511$	−2.27974e9	+	3.36322e8i	−0.755808	+	0.111502i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.73498e7	−	1.68615e8i	−0.0314058	−	0.0543965i
$516$		0			0
$517$	−9.01559e8			−0.286931
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.53370e9	+	2.65645e9i	0.475125	+	0.822941i	0.999594	−	0.0284883i	$-0.00906933\pi$
$521$	1.53370e9	+	2.65645e9i	0.475125	+	0.822941i	−0.524469	+	0.851430i	$0.675736\pi$
$522$		0			0
$523$	3.71080e8	−	6.42729e8i	0.113426	−	0.196459i	−0.803724	−	0.595003i	$-0.797151\pi$
$523$	3.71080e8	−	6.42729e8i	0.113426	−	0.196459i	0.917149	+	0.398544i	$0.130484\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.16335e8	−	8.94318e8i	0.153672	−	0.266167i
$528$		0			0
$529$	−3.05156e9	−	5.28546e9i	−0.896246	−	1.55234i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.41957e8			−0.0406081
$534$		0			0
$535$	2.75367e9	+	4.76949e9i	0.777451	+	1.34658i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.64820e8	−	7.01846e8i	0.0453366	−	0.193055i
$540$		0			0
$541$	2.27287e9	−	3.93672e9i	0.617139	−	1.06892i	−0.372866	−	0.927885i	$-0.621625\pi$
$541$	2.27287e9	−	3.93672e9i	0.617139	−	1.06892i	0.990005	−	0.141031i	$-0.0450419\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.17060e9			−1.10360
$546$		0			0
$547$	1.93020e9			0.504252			0.252126	−	0.967694i	$-0.418870\pi$
$547$	1.93020e9			0.504252			0.252126	+	0.967694i	$0.418870\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.00706e9	−	1.74429e9i	0.256464	−	0.444209i
$552$		0			0
$553$	−1.05567e9	−	1.33230e9i	−0.265454	−	0.335014i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.03481e9	−	3.52440e9i	−0.498920	−	0.864155i	0.501079	−	0.865402i	$-0.332937\pi$
$557$	−2.03481e9	−	3.52440e9i	−0.498920	−	0.864155i	−0.999999	+	0.00124625i	$0.999603\pi$
$558$		0			0
$559$	−3.74403e7			−0.00906564
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.00511e9	+	3.47296e9i	0.473543	+	0.820201i	0.999541	−	0.0302848i	$-0.00964142\pi$
$563$	2.00511e9	+	3.47296e9i	0.473543	+	0.820201i	−0.525998	+	0.850486i	$0.676308\pi$
$564$		0			0
$565$	3.03704e9	−	5.26032e9i	0.708405	−	1.22699i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.19263e9	+	2.06570e9i	−0.271402	+	0.470083i	−0.969221	−	0.246192i	$-0.920821\pi$
$569$	−1.19263e9	+	2.06570e9i	−0.271402	+	0.470083i	0.697819	+	0.716274i	$0.254154\pi$
$570$		0			0
$571$	6.06291e8	+	1.05013e9i	0.136287	+	0.236056i	0.926088	−	0.377307i	$-0.123150\pi$
$571$	6.06291e8	+	1.05013e9i	0.136287	+	0.236056i	−0.789801	+	0.613363i	$0.789816\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.07353e9			−0.893579
$576$		0			0
$577$	3.05691e9	+	5.29472e9i	0.662471	+	1.14743i	0.979964	+	0.199173i	$0.0638256\pi$
$577$	3.05691e9	+	5.29472e9i	0.662471	+	1.14743i	−0.317493	+	0.948261i	$0.602841\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.15223e9	−	2.90904e9i	0.243738	−	0.615365i
$582$		0			0
$583$	−1.63554e8	+	2.83284e8i	−0.0341838	+	0.0592081i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8.73746e9			1.78300			0.891502	−	0.453018i	$-0.149653\pi$
$587$	8.73746e9			1.78300			0.891502	+	0.453018i	$0.149653\pi$
$588$		0			0
$589$	1.87138e9			0.377362
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.48503e9	−	2.57215e9i	0.292445	−	0.506529i	−0.681943	−	0.731406i	$-0.738865\pi$
$593$	1.48503e9	−	2.57215e9i	0.292445	−	0.506529i	0.974387	+	0.224877i	$0.0721980\pi$
$594$		0			0
$595$	−2.26901e9	+	5.72856e9i	−0.441597	+	1.11490i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.58591e9	−	4.47892e9i	−0.491608	−	0.851491i	0.508345	−	0.861154i	$-0.330257\pi$
$599$	−2.58591e9	−	4.47892e9i	−0.491608	−	0.851491i	−0.999953	+	0.00966288i	$0.996924\pi$
$600$		0			0
$601$	−2.35291e8			−0.0442125			−0.0221062	−	0.999756i	$-0.507037\pi$
$601$	−2.35291e8			−0.0442125			−0.0221062	+	0.999756i	$0.507037\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.24120e9	−	5.61393e9i	−0.595063	−	1.03068i
$606$		0			0
$607$	−3.12330e9	+	5.40971e9i	−0.566830	+	0.981778i	0.430047	+	0.902807i	$0.358497\pi$
$607$	−3.12330e9	+	5.40971e9i	−0.566830	+	0.981778i	−0.996877	+	0.0789718i	$0.974836\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.61318e8	−	4.52616e8i	0.0463474	−	0.0802760i
$612$		0			0
$613$	3.50079e9	+	6.06355e9i	0.613840	+	1.06320i	0.990587	+	0.136885i	$0.0437092\pi$
$613$	3.50079e9	+	6.06355e9i	0.613840	+	1.06320i	−0.376747	+	0.926316i	$0.622957\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.98807e9			1.02633			0.513167	−	0.858289i	$-0.328472\pi$
$617$	5.98807e9			1.02633			0.513167	+	0.858289i	$0.328472\pi$
$618$		0			0
$619$	5.02918e9	+	8.71079e9i	0.852275	+	1.47618i	0.879150	+	0.476544i	$0.158111\pi$
$619$	5.02918e9	+	8.71079e9i	0.852275	+	1.47618i	−0.0268758	+	0.999639i	$0.508556\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.67300e9	−	7.15956e9i	−0.939949	−	1.18626i
$624$		0			0
$625$	3.81102e9	−	6.60087e9i	0.624397	−	1.08149i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.74286e9			1.08036
$630$		0			0
$631$	9.39565e9			1.48876			0.744379	−	0.667757i	$-0.232746\pi$
$631$	9.39565e9			1.48876			0.744379	+	0.667757i	$0.232746\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.43111e9	+	5.94285e9i	−0.531773	+	0.921058i
$636$		0			0
$637$	3.04579e8	+	2.86177e8i	0.0466888	+	0.0438678i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.20822e9	+	3.82475e9i	0.331161	+	0.573588i	0.982740	−	0.184993i	$-0.0592263\pi$
$641$	2.20822e9	+	3.82475e9i	0.331161	+	0.573588i	−0.651579	+	0.758581i	$0.725893\pi$
$642$		0			0
$643$	−8.74949e9			−1.29791			−0.648954	−	0.760827i	$-0.724793\pi$
$643$	−8.74949e9			−1.29791			−0.648954	+	0.760827i	$0.724793\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.90059e9	+	8.48807e9i	0.711350	+	1.23209i	0.964351	+	0.264628i	$0.0852492\pi$
$647$	4.90059e9	+	8.48807e9i	0.711350	+	1.23209i	−0.253001	+	0.967466i	$0.581417\pi$
$648$		0			0
$649$	−4.55532e8	+	7.89005e8i	−0.0654128	+	0.113298i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.61621e9	+	7.99551e9i	−0.648768	+	1.12370i	0.334649	+	0.942343i	$0.391382\pi$
$653$	−4.61621e9	+	7.99551e9i	−0.648768	+	1.12370i	−0.983417	+	0.181357i	$0.941951\pi$
$654$		0			0
$655$	3.98413e9	+	6.90072e9i	0.553974	+	0.959511i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.45582e9			−0.334270			−0.167135	−	0.985934i	$-0.553452\pi$
$659$	−2.45582e9			−0.334270			−0.167135	+	0.985934i	$0.553452\pi$
$660$		0			0
$661$	4.69707e9	+	8.13557e9i	0.632590	+	1.09568i	0.987020	+	0.160595i	$0.0513414\pi$
$661$	4.69707e9	+	8.13557e9i	0.632590	+	1.09568i	−0.354431	+	0.935082i	$0.615325\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.10463e10	+	1.62962e9i	−1.45660	+	0.214887i
$666$		0			0
$667$	2.76354e9	−	4.78658e9i	0.360599	−	0.624576i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.82471e8			0.0360948
$672$		0			0
$673$	1.12505e10			1.42272			0.711362	−	0.702825i	$-0.248079\pi$
$673$	1.12505e10			1.42272			0.711362	+	0.702825i	$0.248079\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.57926e8	−	1.31277e9i	0.0938785	−	0.162602i	−0.815261	−	0.579093i	$-0.803407\pi$
$677$	7.57926e8	−	1.31277e9i	0.0938785	−	0.162602i	0.909140	+	0.416491i	$0.136740\pi$
$678$		0			0
$679$	−5.09517e9	+	1.28638e10i	−0.624618	+	1.57697i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.96753e9	−	3.40787e9i	−0.236293	−	0.409271i	0.723355	−	0.690476i	$-0.242599\pi$
$683$	−1.96753e9	−	3.40787e9i	−0.236293	−	0.409271i	−0.959648	+	0.281206i	$0.909266\pi$
$684$		0			0
$685$	2.72911e9			0.324418
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.48125e7	−	1.64220e8i	−0.0110433	−	0.0191275i
$690$		0			0
$691$	5.25151e8	−	9.09587e8i	0.0605495	−	0.104875i	−0.834162	−	0.551520i	$-0.814048\pi$
$691$	5.25151e8	−	9.09587e8i	0.0605495	−	0.104875i	0.894711	+	0.446645i	$0.147381\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.76720e9	+	9.98909e9i	−0.651656	+	1.12870i
$696$		0			0
$697$	−2.74250e9	−	4.75015e9i	−0.306783	−	0.531364i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.15758e10			1.26922			0.634609	−	0.772834i	$-0.281161\pi$
$701$	1.15758e10			1.26922			0.634609	+	0.772834i	$0.281161\pi$
$702$		0			0
$703$	6.10962e9	+	1.05822e10i	0.663240	+	1.14876i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.03280e9	−	2.99892e8i	0.216335	−	0.0319152i
$708$		0			0
$709$	8.50354e9	−	1.47286e10i	0.896061	−	1.55202i	0.0635758	−	0.997977i	$-0.479750\pi$
$709$	8.50354e9	−	1.47286e10i	0.896061	−	1.55202i	0.832486	−	0.554047i	$-0.186917\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.13534e9			0.530586
$714$		0			0
$715$	−1.53830e8			−0.0157388
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.12756e8	−	3.68504e8i	0.0213467	−	0.0369735i	−0.855155	−	0.518373i	$-0.826538\pi$
$719$	2.12756e8	−	3.68504e8i	0.0213467	−	0.0369735i	0.876501	+	0.481399i	$0.159871\pi$
$720$		0			0
$721$	−3.16896e8	−	3.99936e8i	−0.0314879	−	0.0397390i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.18399e9	−	2.05073e9i	−0.115389	−	0.199860i
$726$		0			0
$727$	3.28106e9			0.316697			0.158348	−	0.987383i	$-0.449383\pi$
$727$	3.28106e9			0.316697			0.158348	+	0.987383i	$0.449383\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.23316e8	−	1.25282e9i	−0.0684884	−	0.118625i
$732$		0			0
$733$	3.71639e9	−	6.43698e9i	0.348544	−	0.603696i	−0.637447	−	0.770494i	$-0.720010\pi$
$733$	3.71639e9	−	6.43698e9i	0.348544	−	0.603696i	0.985991	+	0.166798i	$0.0533429\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.31551e9	−	2.27854e9i	0.121049	−	0.209662i
$738$		0			0
$739$	−5.35283e9	−	9.27138e9i	−0.487897	−	0.845062i	0.512006	−	0.858982i	$-0.328902\pi$
$739$	−5.35283e9	−	9.27138e9i	−0.487897	−	0.845062i	−0.999903	+	0.0139196i	$0.995569\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.15663e9			−0.282333			−0.141167	−	0.989986i	$-0.545085\pi$
$743$	−3.15663e9			−0.282333			−0.141167	+	0.989986i	$0.545085\pi$
$744$		0			0
$745$	9.87735e9	+	1.71081e10i	0.875172	+	1.51584i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.96381e9	+	1.13127e10i	0.779482	+	0.983739i
$750$		0			0
$751$	−1.38864e9	+	2.40519e9i	−0.119632	+	0.207209i	−0.919622	−	0.392805i	$-0.871505\pi$
$751$	−1.38864e9	+	2.40519e9i	−0.119632	+	0.207209i	0.799990	+	0.600014i	$0.204838\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.19015e10			−1.85208
$756$		0			0
$757$	1.22142e9			0.102336			0.0511682	−	0.998690i	$-0.483706\pi$
$757$	1.22142e9			0.102336			0.0511682	+	0.998690i	$0.483706\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.08867e10	−	1.88563e10i	0.895466	−	1.55099i	0.0622399	−	0.998061i	$-0.480176\pi$
$761$	1.08867e10	−	1.88563e10i	0.895466	−	1.55099i	0.833226	−	0.552932i	$-0.186491\pi$
$762$		0			0
$763$	−1.08132e10	+	1.59524e9i	−0.881291	+	0.130014i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.64073e8	−	4.57388e8i	−0.0211320	−	0.0366017i
$768$		0			0
$769$	−1.87936e10			−1.49028			−0.745138	−	0.666910i	$-0.767617\pi$
$769$	−1.87936e10			−1.49028			−0.745138	+	0.666910i	$0.767617\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.59566e9	−	7.95992e9i	−0.357866	−	0.619841i	0.629738	−	0.776807i	$-0.283162\pi$
$773$	−4.59566e9	−	7.95992e9i	−0.357866	−	0.619841i	−0.987604	+	0.156966i	$0.949829\pi$
$774$		0			0
$775$	1.10008e9	−	1.90539e9i	0.0848920	−	0.147037i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.96988e9	−	8.60809e9i	0.376674	−	0.652418i
$780$		0			0
$781$	−1.30317e9	−	2.25716e9i	−0.0978868	−	0.169545i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.10291e9			−0.302725
$786$		0			0
$787$	8.75830e9	+	1.51698e10i	0.640484	+	1.10935i	0.985325	+	0.170690i	$0.0545995\pi$
$787$	8.75830e9	+	1.51698e10i	0.640484	+	1.10935i	−0.344841	+	0.938661i	$0.612067\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.86217e9	−	1.48002e10i	0.421154	−	1.06329i
$792$		0			0
$793$	−8.18744e7	+	1.41811e8i	−0.00583032	+	0.0100984i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.13809e10			0.796290			0.398145	−	0.917323i	$-0.369654\pi$
$797$	1.13809e10			0.796290			0.398145	+	0.917323i	$0.369654\pi$
$798$		0			0
$799$	2.01938e10			1.40057
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.11147e9	−	1.92513e9i	0.0757521	−	0.131207i
$804$		0			0
$805$	−3.03126e10	+	4.47192e9i	−2.04803	+	0.302140i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−8.27014e9	−	1.43243e10i	−0.549153	−	0.951160i	−0.998333	−	0.0577191i	$-0.981617\pi$
$809$	−8.27014e9	−	1.43243e10i	−0.549153	−	0.951160i	0.449180	−	0.893441i	$-0.351716\pi$
$810$		0			0
$811$	−1.76982e10			−1.16508			−0.582540	−	0.812802i	$-0.697941\pi$
$811$	−1.76982e10			−1.16508			−0.582540	+	0.812802i	$0.697941\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.92088e9	+	1.54514e10i	0.577239	+	0.999808i
$816$		0			0
$817$	1.31077e9	−	2.27033e9i	0.0840912	−	0.145650i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.88453e9	+	6.72821e9i	−0.244984	+	0.424325i	−0.962127	−	0.272601i	$-0.912116\pi$
$821$	−3.88453e9	+	6.72821e9i	−0.244984	+	0.424325i	0.717143	+	0.696926i	$0.245449\pi$
$822$		0			0
$823$	−6.92292e9	−	1.19908e10i	−0.432902	−	0.749809i	0.564220	−	0.825625i	$-0.309177\pi$
$823$	−6.92292e9	−	1.19908e10i	−0.432902	−	0.749809i	−0.997122	+	0.0758162i	$0.975844\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.07058e10			−0.658189			−0.329095	−	0.944297i	$-0.606743\pi$
$827$	−1.07058e10			−0.658189			−0.329095	+	0.944297i	$0.606743\pi$
$828$		0			0
$829$	9.39466e8	+	1.62720e9i	0.0572717	+	0.0991975i	0.893240	−	0.449581i	$-0.148427\pi$
$829$	9.39466e8	+	1.62720e9i	0.0572717	+	0.0991975i	−0.835968	+	0.548778i	$0.815093\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.69175e9	+	1.57205e10i	−0.221297	+	0.942340i
$834$		0			0
$835$	1.72297e10	−	2.98427e10i	1.02418	−	1.77392i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	9.56049e9			0.558873			0.279437	−	0.960164i	$-0.409852\pi$
$839$	9.56049e9			0.558873			0.279437	+	0.960164i	$0.409852\pi$
$840$		0			0
$841$	−1.40369e10			−0.813741
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.08193e10	+	1.87396e10i	−0.616879	+	1.06847i
$846$		0			0
$847$	−1.05509e10	−	1.33156e10i	−0.596617	−	0.752956i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.67657e10	+	2.90391e10i	0.932542	+	1.61521i
$852$		0			0
$853$	−8.45947e9			−0.466683			−0.233341	−	0.972395i	$-0.574966\pi$
$853$	−8.45947e9			−0.466683			−0.233341	+	0.972395i	$0.574966\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.17128e10	−	2.02872e10i	−0.635664	−	1.10100i	−0.986374	−	0.164519i	$-0.947393\pi$
$857$	−1.17128e10	−	2.02872e10i	−0.635664	−	1.10100i	0.350710	−	0.936484i	$-0.385940\pi$
$858$		0			0
$859$	4.32662e9	−	7.49393e9i	0.232902	−	0.403398i	−0.725759	−	0.687949i	$-0.758511\pi$
$859$	4.32662e9	−	7.49393e9i	0.232902	−	0.403398i	0.958661	+	0.284551i	$0.0918446\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.46757e10	−	2.54191e10i	0.777251	−	1.34624i	−0.156270	−	0.987714i	$-0.549947\pi$
$863$	1.46757e10	−	2.54191e10i	0.777251	−	1.34624i	0.933521	−	0.358523i	$-0.116720\pi$
$864$		0			0
$865$	−1.81603e10	−	3.14545e10i	−0.954039	−	1.65244i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.63975e9			0.0847633
$870$		0			0
$871$	7.62607e8	+	1.32087e9i	0.0391054	+	0.0677326i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.20622e9	−	1.06194e10i	0.212258	−	0.535887i
$876$		0			0
$877$	7.88074e9	−	1.36498e10i	0.394519	−	0.683328i	−0.598520	−	0.801108i	$-0.704244\pi$
$877$	7.88074e9	−	1.36498e10i	0.394519	−	0.683328i	0.993040	+	0.117780i	$0.0375778\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.80407e10			−1.87427			−0.937136	−	0.348966i	$-0.886533\pi$
$881$	−3.80407e10			−1.87427			−0.937136	+	0.348966i	$0.886533\pi$
$882$		0			0
$883$	5.42019e9			0.264943			0.132471	−	0.991187i	$-0.457709\pi$
$883$	5.42019e9			0.264943			0.132471	+	0.991187i	$0.457709\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.67590e10	+	2.90275e10i	−0.806337	+	1.39662i	0.109047	+	0.994037i	$0.465220\pi$
$887$	−1.67590e10	+	2.90275e10i	−0.806337	+	1.39662i	−0.915385	+	0.402580i	$0.868113\pi$
$888$		0			0
$889$	−6.62280e9	+	1.67206e10i	−0.316144	+	0.798169i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.82973e10	+	3.16919e10i	0.859819	+	1.48925i
$894$		0			0
$895$	−4.79253e10			−2.23452
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.49261e9	+	2.58528e9i	0.0685154	+	0.118672i
$900$		0			0
$901$	3.66340e9	−	6.34519e9i	0.166858	−	0.289007i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.46686e9	−	1.12009e10i	0.290017	−	0.502324i
$906$		0			0
$907$	−6.63551e9	−	1.14930e10i	−0.295290	−	0.511457i	0.679762	−	0.733433i	$-0.262083\pi$
$907$	−6.63551e9	−	1.14930e10i	−0.295290	−	0.511457i	−0.975052	+	0.221975i	$0.928750\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8.05061e9			−0.352788			−0.176394	−	0.984320i	$-0.556443\pi$
$911$	−8.05061e9			−0.352788			−0.176394	+	0.984320i	$0.556443\pi$
$912$		0			0
$913$	1.50916e9	+	2.61393e9i	0.0656276	+	0.113670i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.29693e10	+	1.63677e10i	0.555421	+	0.700965i
$918$		0			0
$919$	2.18095e10	−	3.77751e10i	0.926917	−	1.60547i	0.138467	−	0.990367i	$-0.455782\pi$
$919$	2.18095e10	−	3.77751e10i	0.926917	−	1.60547i	0.788449	−	0.615100i	$-0.210884\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.51091e9			0.0632458
$924$		0			0
$925$	1.43660e10			0.596815
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.54582e9	+	1.48018e10i	−0.349702	+	0.605702i	−0.986196	−	0.165579i	$-0.947051\pi$
$929$	−8.54582e9	+	1.48018e10i	−0.349702	+	0.605702i	0.636494	+	0.771282i	$0.280384\pi$
$930$		0			0
$931$	−2.80166e10	+	8.45031e9i	−1.13787	+	0.343201i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.97187e9	−	5.14743e9i	−0.118902	−	0.205944i
$936$		0			0
$937$	−4.55752e10			−1.80984			−0.904919	−	0.425583i	$-0.860069\pi$
$937$	−4.55752e10			−1.80984			−0.904919	+	0.425583i	$0.860069\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.19139e9	−	5.52765e9i	−0.124858	−	0.216260i	0.796819	−	0.604217i	$-0.206514\pi$
$941$	−3.19139e9	−	5.52765e9i	−0.124858	−	0.216260i	−0.921677	+	0.387957i	$0.873181\pi$
$942$		0			0
$943$	1.36381e10	−	2.36219e10i	0.529618	−	0.917326i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.37414e10	−	4.11214e10i	0.908410	−	1.57341i	0.0921376	−	0.995746i	$-0.470630\pi$
$947$	2.37414e10	−	4.11214e10i	0.908410	−	1.57341i	0.816273	−	0.577667i	$-0.196037\pi$
$948$		0			0
$949$	6.44324e8	+	1.11600e9i	0.0244722	+	0.0423870i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.56713e10			0.586517			0.293259	−	0.956033i	$-0.405260\pi$
$953$	1.56713e10			0.586517			0.293259	+	0.956033i	$0.405260\pi$
$954$		0			0
$955$	1.36799e10	+	2.36942e10i	0.508242	+	0.880300i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.07584e9	−	1.04388e9i	0.259067	−	0.0382194i
$960$		0			0
$961$	1.23695e10	−	2.14246e10i	0.449593	−	0.778718i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.07693e10			−0.744007
$966$		0			0
$967$	−5.24362e10			−1.86483			−0.932413	−	0.361395i	$-0.882300\pi$
$967$	−5.24362e10			−1.86483			−0.932413	+	0.361395i	$0.882300\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.58851e9	−	9.67959e9i	0.195898	−	0.339304i	−0.751297	−	0.659964i	$-0.770571\pi$
$971$	5.58851e9	−	9.67959e9i	0.195898	−	0.339304i	0.947194	+	0.320660i	$0.103905\pi$
$972$		0			0
$973$	−1.11320e10	+	2.81049e10i	−0.387416	+	0.978108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.68816e10	−	2.92399e10i	−0.579140	−	1.00310i	−0.995578	−	0.0939364i	$-0.970055\pi$
$977$	−1.68816e10	−	2.92399e10i	−0.579140	−	1.00310i	0.416438	−	0.909164i	$-0.363278\pi$
$978$		0			0
$979$	8.81175e9			0.300139
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.02803e9	−	1.04409e10i	−0.202413	−	0.350590i	0.746892	−	0.664945i	$-0.231545\pi$
$983$	−6.02803e9	−	1.04409e10i	−0.202413	−	0.350590i	−0.949305	+	0.314355i	$0.898212\pi$
$984$		0			0
$985$	−1.78475e10	+	3.09128e10i	−0.595047	+	1.03065i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.59696e9	−	6.23012e9i	0.118236	−	0.204790i
$990$		0			0
$991$	−1.67462e10	−	2.90053e10i	−0.546586	−	0.946715i	−0.998505	−	0.0546562i	$-0.982594\pi$
$991$	−1.67462e10	−	2.90053e10i	−0.546586	−	0.946715i	0.451919	−	0.892059i	$-0.350740\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.32579e10			−0.426671
$996$		0			0
$997$	2.76848e10	+	4.79515e10i	0.884725	+	1.53239i	0.846028	+	0.533138i	$0.178988\pi$
$997$	2.76848e10	+	4.79515e10i	0.884725	+	1.53239i	0.0386975	+	0.999251i	$0.487679\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.8.k.e.37.8	yes	16
3.2	odd	2	inner	252.8.k.e.37.1	✓	16
7.4	even	3	inner	252.8.k.e.109.8	yes	16
21.11	odd	6	inner	252.8.k.e.109.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.8.k.e.37.1	✓	16	3.2	odd	2	inner
252.8.k.e.37.8	yes	16	1.1	even	1	trivial
252.8.k.e.109.1	yes	16	21.11	odd	6	inner
252.8.k.e.109.8	yes	16	7.4	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(173.134 - 299.876i) q^{5} +(334.186 - 843.720i) q^{7} +O(q^{10})\)	\(q+(173.134 - 299.876i) q^{5} +(334.186 - 843.720i) q^{7} +(437.706 + 758.129i) q^{11} -507.478 q^{13} +(-9804.06 - 16981.1i) q^{17} +(17766.7 - 30772.7i) q^{19} +(48754.4 - 84445.0i) q^{23} +(-20888.0 - 36179.1i) q^{25} +56682.8 q^{29} +(26332.7 + 45609.6i) q^{31} +(-195153. - 246291. i) q^{35} +(-171941. + 297810. i) q^{37} +279731. q^{41} +73777.2 q^{43} +(-514934. + 891892. i) q^{47} +(-600182. - 563919. i) q^{49} +(186831. + 323600. i) q^{53} +303127. q^{55} +(520363. + 901296. i) q^{59} +(161336. - 279442. i) q^{61} +(-87861.6 + 152181. i) q^{65} +(-1.50274e6 - 2.60282e6i) q^{67} -2.97728e6 q^{71} +(-1.26966e6 - 2.19911e6i) q^{73} +(785924. - 115945. i) q^{77} +(936558. - 1.62217e6i) q^{79} +3.44787e6 q^{83} -6.78965e6 q^{85} +(5.03292e6 - 8.71726e6i) q^{89} +(-169592. + 428169. i) q^{91} +(-6.15201e6 - 1.06556e7i) q^{95} -1.52465e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 1680 q^{7}+O(q^{10}) \) 16 * q + 1680 * q^7	\( 16 q + 1680 q^{7} - 28280 q^{13} + 42224 q^{19} - 80460 q^{25} + 164752 q^{31} - 647980 q^{37} + 1341440 q^{43} + 230104 q^{49} - 323120 q^{55} - 4319336 q^{61} - 3905760 q^{67} + 6471780 q^{73} - 6093104 q^{79} + 456400 q^{85} + 15969856 q^{91} - 27141240 q^{97}+O(q^{100}) \) 16 * q + 1680 * q^7 - 28280 * q^13 + 42224 * q^19 - 80460 * q^25 + 164752 * q^31 - 647980 * q^37 + 1341440 * q^43 + 230104 * q^49 - 323120 * q^55 - 4319336 * q^61 - 3905760 * q^67 + 6471780 * q^73 - 6093104 * q^79 + 456400 * q^85 + 15969856 * q^91 - 27141240 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more