LMFDB - Embedded newform 252.7.z.f.145.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,7,Mod(73,252)]

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, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("252.73");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	252.z (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$57.9736290722$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 6586 x^{10} - 4380 x^{9} + 29743412 x^{8} - 22316460 x^{7} + 72097469286 x^{6} + \cdots + 78\!\cdots\!61 $ x^12 + 6586x^10 - 4380x^9 + 29743412x^8 - 22316460x^7 + 72097469286x^6 - 74114131680x^5 + 127603094377722x^4 - 68863099720860x^3 + 120620660396154896x^2 + 69805069827473280x + 78212563531160784961
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{9}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.5
Root			$-23.8229 - 41.2625i$ of defining polynomial
Character	$\chi$	$=$	252.145
Dual form			252.7.z.f.73.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(136.035 + 78.5400i) q^{5} +(334.768 + 74.6948i) q^{7} +O(q^{10})$	$q+(136.035 + 78.5400i) q^{5} +(334.768 + 74.6948i) q^{7} +(1305.52 + 2261.23i) q^{11} +3315.82i q^{13} +(-445.584 + 257.258i) q^{17} +(-5768.65 - 3330.53i) q^{19} +(-5395.61 + 9345.47i) q^{23} +(4524.57 + 7836.78i) q^{25} +6779.56 q^{29} +(23619.9 - 13636.9i) q^{31} +(39673.7 + 36453.8i) q^{35} +(256.702 - 444.621i) q^{37} -69543.2i q^{41} -146154. q^{43} +(-150738. - 87028.8i) q^{47} +(106490. + 50010.9i) q^{49} +(-75787.0 - 131267. i) q^{53} +410143. i q^{55} +(202049. - 116653. i) q^{59} +(74423.8 + 42968.6i) q^{61} +(-260425. + 451069. i) q^{65} +(55530.5 + 96181.6i) q^{67} +501154. q^{71} +(-393637. + 227266. i) q^{73} +(268145. + 854505. i) q^{77} +(-136941. + 237189. i) q^{79} -186596. i q^{83} -80820.1 q^{85} +(-245350. - 141653. i) q^{89} +(-247675. + 1.11003e6i) q^{91} +(-523160. - 906140. i) q^{95} +892059. i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 714 q^{7}+O(q^{10}) $ 12 * q + 714 * q^7	$ 12 q + 714 q^{7} - 59466 q^{19} + 64086 q^{25} + 219834 q^{31} - 166242 q^{37} - 326916 q^{43} - 132006 q^{49} + 736008 q^{61} + 176238 q^{67} - 2931018 q^{73} + 449490 q^{79} + 2174592 q^{85} - 808542 q^{91}+O(q^{100}) $ 12 * q + 714 * q^7 - 59466 * q^19 + 64086 * q^25 + 219834 * q^31 - 166242 * q^37 - 326916 * q^43 - 132006 * q^49 + 736008 * q^61 + 176238 * q^67 - 2931018 * q^73 + 449490 * q^79 + 2174592 * q^85 - 808542 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$

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
$5$	136.035	+	78.5400i	1.08828	+	0.628320i	0.933119	−	0.359569i	$-0.117076\pi$
$5$	136.035	+	78.5400i	1.08828	+	0.628320i	0.155164	+	0.987889i	$0.450410\pi$
$6$		0			0
$7$	334.768	+	74.6948i	0.976000	+	0.217769i
$7$	334.768	+	74.6948i	0.976000	+	0.217769i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1305.52	+	2261.23i	0.980859	+	1.69890i	0.659061	+	0.752090i	$0.270954\pi$
$11$	1305.52	+	2261.23i	0.980859	+	1.69890i	0.321798	+	0.946808i	$0.395713\pi$
$12$		0			0
$13$			3315.82i			1.50925i	0.656157	+	0.754625i	$0.272181\pi$
$13$			3315.82i			1.50925i	−0.656157	+	0.754625i	$0.727819\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−445.584	+	257.258i	−0.0906948	+	0.0523627i	−0.544661	−	0.838656i	$-0.683342\pi$
$17$	−445.584	+	257.258i	−0.0906948	+	0.0523627i	0.453967	+	0.891019i	$0.350009\pi$
$18$		0			0
$19$	−5768.65	−	3330.53i	−0.841034	−	0.485571i	0.0165818	−	0.999863i	$-0.494722\pi$
$19$	−5768.65	−	3330.53i	−0.841034	−	0.485571i	−0.857615	+	0.514292i	$0.828055\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5395.61	+	9345.47i	−0.443462	+	0.768099i	−0.997944	−	0.0640968i	$-0.979583\pi$
$23$	−5395.61	+	9345.47i	−0.443462	+	0.768099i	0.554481	+	0.832196i	$0.312917\pi$
$24$		0			0
$25$	4524.57	+	7836.78i	0.289572	+	0.501554i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6779.56			0.277976			0.138988	−	0.990294i	$-0.455615\pi$
$29$	6779.56			0.277976			0.138988	+	0.990294i	$0.455615\pi$
$30$		0			0
$31$	23619.9	−	13636.9i	0.792853	−	0.457754i	−0.0481129	−	0.998842i	$-0.515321\pi$
$31$	23619.9	−	13636.9i	0.792853	−	0.457754i	0.840966	+	0.541088i	$0.181987\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	39673.7	+	36453.8i	0.925335	+	0.850235i
$36$		0			0
$37$	256.702	−	444.621i	0.00506785	−	0.00877778i	−0.863480	−	0.504382i	$-0.831720\pi$
$37$	256.702	−	444.621i	0.00506785	−	0.00877778i	0.868548	+	0.495605i	$0.165054\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	69543.2i		−	1.00903i	−0.863403	−	0.504514i	$-0.831672\pi$
$41$		−	69543.2i		−	1.00903i	0.863403	−	0.504514i	$-0.168328\pi$
$42$		0			0
$43$	−146154.			−1.83826			−0.919129	−	0.393956i	$-0.871106\pi$
$43$	−146154.			−1.83826			−0.919129	+	0.393956i	$0.871106\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−150738.	−	87028.8i	−1.45188	−	0.838242i	−0.453290	−	0.891363i	$-0.649750\pi$
$47$	−150738.	−	87028.8i	−1.45188	−	0.838242i	−0.998588	+	0.0531205i	$0.983083\pi$
$48$		0			0
$49$	106490.	+	50010.9i	0.905153	+	0.425085i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−75787.0	−	131267.i	−0.509057	−	0.881713i	−0.999945	−	0.0104904i	$-0.996661\pi$
$53$	−75787.0	−	131267.i	−0.509057	−	0.881713i	0.490887	−	0.871223i	$-0.336673\pi$
$54$		0			0
$55$			410143.i			2.46517i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	202049.	−	116653.i	0.983788	−	0.567990i	0.0803766	−	0.996765i	$-0.474388\pi$
$59$	202049.	−	116653.i	0.983788	−	0.567990i	0.903412	+	0.428774i	$0.141054\pi$
$60$		0			0
$61$	74423.8	+	42968.6i	0.327886	+	0.189305i	0.654902	−	0.755714i	$-0.272710\pi$
$61$	74423.8	+	42968.6i	0.327886	+	0.189305i	−0.327016	+	0.945019i	$0.606043\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−260425.	+	451069.i	−0.948292	+	1.64249i
$66$		0			0
$67$	55530.5	+	96181.6i	0.184632	+	0.319792i	0.943452	−	0.331508i	$-0.107557\pi$
$67$	55530.5	+	96181.6i	0.184632	+	0.319792i	−0.758820	+	0.651300i	$0.774224\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	501154.			1.40022			0.700110	−	0.714035i	$-0.253134\pi$
$71$	501154.			1.40022			0.700110	+	0.714035i	$0.253134\pi$
$72$		0			0
$73$	−393637.	+	227266.i	−1.01188	+	0.584207i	−0.911740	−	0.410767i	$-0.865261\pi$
$73$	−393637.	+	227266.i	−1.01188	+	0.584207i	−0.100135	+	0.994974i	$0.531928\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	268145.	+	854505.i	0.587351	+	1.87173i
$78$		0			0
$79$	−136941.	+	237189.i	−0.277749	+	0.481075i	−0.970825	−	0.239789i	$-0.922922\pi$
$79$	−136941.	+	237189.i	−0.277749	+	0.481075i	0.693076	+	0.720864i	$0.256255\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	186596.i		−	0.326338i	−0.986598	−	0.163169i	$-0.947828\pi$
$83$		−	186596.i		−	0.326338i	0.986598	−	0.163169i	$-0.0521716\pi$
$84$		0			0
$85$	−80820.1			−0.131602
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−245350.	−	141653.i	−0.348029	−	0.200935i	0.315788	−	0.948830i	$-0.397731\pi$
$89$	−245350.	−	141653.i	−0.348029	−	0.200935i	−0.663817	+	0.747895i	$0.731065\pi$
$90$		0			0
$91$	−247675.	+	1.11003e6i	−0.328668	+	1.47303i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−523160.	−	906140.i	−0.610188	−	1.05688i
$96$		0			0
$97$			892059.i			0.977413i	0.872448	+	0.488707i	$0.162531\pi$
$97$			892059.i			0.977413i	−0.872448	+	0.488707i	$0.837469\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.08150e6	+	624407.i	−1.04970	+	0.606043i	−0.922565	−	0.385843i	$-0.873911\pi$
$101$	−1.08150e6	+	624407.i	−1.04970	+	0.606043i	−0.127133	+	0.991886i	$0.540577\pi$
$102$		0			0
$103$	594482.	+	343225.i	0.544036	+	0.314099i	0.746713	−	0.665146i	$-0.231631\pi$
$103$	594482.	+	343225.i	0.544036	+	0.314099i	−0.202677	+	0.979246i	$0.564964\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	788845.	−	1.36632e6i	0.643932	−	1.11532i	−0.340615	−	0.940203i	$-0.610635\pi$
$107$	788845.	−	1.36632e6i	0.643932	−	1.11532i	0.984547	−	0.175120i	$-0.0560314\pi$
$108$		0			0
$109$	987760.	+	1.71085e6i	0.762732	+	1.32109i	0.941437	+	0.337188i	$0.109476\pi$
$109$	987760.	+	1.71085e6i	0.762732	+	1.32109i	−0.178706	+	0.983903i	$0.557191\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	646680.			0.448182			0.224091	−	0.974568i	$-0.428059\pi$
$113$	646680.			0.448182			0.224091	+	0.974568i	$0.428059\pi$
$114$		0			0
$115$	−1.46799e6	+	847542.i	−0.965225	+	0.557273i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−168383.	+	52838.9i	−0.0999211	+	0.0313555i
$120$		0			0
$121$	−2.52300e6	+	4.36997e6i	−1.42417	+	2.46673i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.03294e6i		−	0.528864i
$126$		0			0
$127$	1.81072e6			0.883977			0.441989	−	0.897021i	$-0.354273\pi$
$127$	1.81072e6			0.883977			0.441989	+	0.897021i	$0.354273\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.78012e6	+	1.60510e6i	1.23666	+	0.713984i	0.968409	−	0.249365i	$-0.0802220\pi$
$131$	2.78012e6	+	1.60510e6i	1.23666	+	0.713984i	0.268248	+	0.963350i	$0.413555\pi$
$132$		0			0
$133$	−1.68239e6	−	1.54584e6i	−0.715107	−	0.657069i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−997578.	−	1.72786e6i	−0.387958	−	0.671964i	0.604216	−	0.796820i	$-0.293486\pi$
$137$	−997578.	−	1.72786e6i	−0.387958	−	0.671964i	−0.992175	+	0.124857i	$0.960153\pi$
$138$		0			0
$139$		−	787049.i		−	0.293061i	−0.989206	−	0.146530i	$-0.953189\pi$
$139$		−	787049.i		−	0.293061i	0.989206	−	0.146530i	$-0.0468106\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.49784e6	+	4.32888e6i	−2.56406	+	1.48036i
$144$		0			0
$145$	922260.	+	532467.i	0.302517	+	0.174658i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.82668e6	−	4.89595e6i	0.854511	−	1.48006i	−0.0225871	−	0.999745i	$-0.507190\pi$
$149$	2.82668e6	−	4.89595e6i	0.854511	−	1.48006i	0.877098	−	0.480311i	$-0.159476\pi$
$150$		0			0
$151$	1.22152e6	+	2.11573e6i	0.354788	+	0.614510i	0.987082	−	0.160218i	$-0.0512198\pi$
$151$	1.22152e6	+	2.11573e6i	0.354788	+	0.614510i	−0.632294	+	0.774729i	$0.717886\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.28418e6			1.15046
$156$		0			0
$157$	1.46414e6	−	845323.i	0.378342	−	0.218436i	−0.298755	−	0.954330i	$-0.596571\pi$
$157$	1.46414e6	−	845323.i	0.378342	−	0.218436i	0.677097	+	0.735894i	$0.263238\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.50434e6	+	2.72554e6i	−0.600088	+	0.653093i
$162$		0			0
$163$	2.41898e6	−	4.18980e6i	0.558560	−	0.967454i	−0.439057	−	0.898459i	$-0.644687\pi$
$163$	2.41898e6	−	4.18980e6i	0.558560	−	0.967454i	0.997617	−	0.0689948i	$-0.0219792\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			6.91269e6i			1.48422i	0.670279	+	0.742109i	$0.266174\pi$
$167$			6.91269e6i			1.48422i	−0.670279	+	0.742109i	$0.733826\pi$
$168$		0			0
$169$	−6.16785e6			−1.27783
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.48120e6	+	4.89662e6i	1.63802	+	0.945711i	0.981514	+	0.191390i	$0.0612994\pi$
$173$	8.48120e6	+	4.89662e6i	1.63802	+	0.945711i	0.656505	+	0.754321i	$0.272034\pi$
$174$		0			0
$175$	929314.	+	2.96147e6i	0.173400	+	0.552577i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.28687e6	−	5.69302e6i	−0.573090	−	0.992621i	−0.996246	−	0.0865646i	$-0.972411\pi$
$179$	−3.28687e6	−	5.69302e6i	−0.573090	−	0.992621i	0.423156	−	0.906057i	$-0.360922\pi$
$180$		0			0
$181$			848673.i			0.143121i	0.997436	+	0.0715607i	$0.0227980\pi$
$181$			848673.i			0.143121i	−0.997436	+	0.0715607i	$0.977202\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	69841.1	−	40322.8i	0.0110305	−	0.00636847i
$186$		0			0
$187$	−1.16344e6	−	671712.i	−0.177918	−	0.102721i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	784687.	−	1.35912e6i	0.112615	−	0.195055i	−0.804209	−	0.594347i	$-0.797411\pi$
$191$	784687.	−	1.35912e6i	0.112615	−	0.195055i	0.916824	+	0.399292i	$0.130744\pi$
$192$		0			0
$193$	−3.76124e6	−	6.51466e6i	−0.523190	−	0.906191i	−0.999636	−	0.0269875i	$-0.991409\pi$
$193$	−3.76124e6	−	6.51466e6i	−0.523190	−	0.906191i	0.476446	−	0.879204i	$-0.341925\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.53944e6			−0.462952			−0.231476	−	0.972841i	$-0.574356\pi$
$197$	−3.53944e6			−0.462952			−0.231476	+	0.972841i	$0.574356\pi$
$198$		0			0
$199$	2.80694e6	−	1.62059e6i	0.356184	−	0.205643i	−0.311222	−	0.950337i	$-0.600738\pi$
$199$	2.80694e6	−	1.62059e6i	0.356184	−	0.205643i	0.667405	+	0.744695i	$0.267405\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.26958e6	+	506398.i	0.271305	+	0.0605346i
$204$		0			0
$205$	5.46193e6	−	9.46033e6i	0.633993	−	1.09811i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	1.73923e7i		−	1.90511i
$210$		0			0
$211$	6.69821e6			0.713036			0.356518	−	0.934288i	$-0.383964\pi$
$211$	6.69821e6			0.713036			0.356518	+	0.934288i	$0.383964\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.98822e7	−	1.14790e7i	−2.00054	−	1.15501i
$216$		0			0
$217$	8.92579e6	−	2.80093e6i	0.873510	−	0.274109i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−853021.	−	1.47748e6i	−0.0790283	−	0.136881i
$222$		0			0
$223$		−	1.16194e7i		−	1.04777i	−0.851788	−	0.523887i	$-0.824482\pi$
$223$		−	1.16194e7i		−	1.04777i	0.851788	−	0.523887i	$-0.175518\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.41747e6	−	1.97308e6i	0.292165	−	0.168681i	−0.346753	−	0.937956i	$-0.612716\pi$
$227$	3.41747e6	−	1.97308e6i	0.292165	−	0.168681i	0.638918	+	0.769275i	$0.279382\pi$
$228$		0			0
$229$	−7.09592e6	−	4.09683e6i	−0.590884	−	0.341147i	0.174563	−	0.984646i	$-0.444149\pi$
$229$	−7.09592e6	−	4.09683e6i	−0.590884	−	0.341147i	−0.765447	+	0.643499i	$0.777482\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−8.01788e6	+	1.38874e7i	−0.633857	+	1.09787i	0.352899	+	0.935662i	$0.385196\pi$
$233$	−8.01788e6	+	1.38874e7i	−0.633857	+	1.09787i	−0.986756	+	0.162212i	$0.948137\pi$
$234$		0			0
$235$	−1.36705e7	−	2.36780e7i	−1.05337	−	1.82449i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.92440e7			1.40962			0.704810	−	0.709396i	$-0.251032\pi$
$239$	1.92440e7			1.40962			0.704810	+	0.709396i	$0.251032\pi$
$240$		0			0
$241$	1.24933e7	−	7.21304e6i	0.892540	−	0.515308i	0.0177675	−	0.999842i	$-0.494344\pi$
$241$	1.24933e7	−	7.21304e6i	0.892540	−	0.515308i	0.874772	+	0.484534i	$0.161011\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.05586e7	+	1.51670e7i	0.717973	+	1.03134i
$246$		0			0
$247$	1.10434e7	−	1.91278e7i	0.732848	−	1.26933i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			1.10219e7i			0.697006i	0.937308	+	0.348503i	$0.113310\pi$
$251$			1.10219e7i			0.697006i	−0.937308	+	0.348503i	$0.886690\pi$
$252$		0			0
$253$	−2.81764e7			−1.73990
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.81591e6	+	1.04842e6i	0.106978	+	0.0617639i	0.552535	−	0.833490i	$-0.313661\pi$
$257$	1.81591e6	+	1.04842e6i	0.106978	+	0.0617639i	−0.445556	+	0.895254i	$0.646994\pi$
$258$		0			0
$259$	119147.	−	129671.i	0.00685776	−	0.00746349i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.04524e7	+	1.81040e7i	0.574576	+	0.995196i	0.996088	+	0.0883723i	$0.0281665\pi$
$263$	1.04524e7	+	1.81040e7i	0.574576	+	0.995196i	−0.421511	+	0.906823i	$0.638500\pi$
$264$		0			0
$265$		−	2.38092e7i		−	1.27940i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.24934e7	−	1.29866e7i	1.15557	−	0.667171i	0.205335	−	0.978692i	$-0.434172\pi$
$269$	2.24934e7	−	1.29866e7i	1.15557	−	0.667171i	0.950239	+	0.311520i	$0.100838\pi$
$270$		0			0
$271$	1.92603e7	+	1.11199e7i	0.967732	+	0.558720i	0.898544	−	0.438883i	$-0.144626\pi$
$271$	1.92603e7	+	1.11199e7i	0.967732	+	0.558720i	0.0691881	+	0.997604i	$0.477959\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.18139e7	+	2.04622e7i	−0.568059	+	0.983907i
$276$		0			0
$277$	1.81089e7	+	3.13655e7i	0.852024	+	1.47575i	0.879379	+	0.476123i	$0.157958\pi$
$277$	1.81089e7	+	3.13655e7i	0.852024	+	1.47575i	−0.0273547	+	0.999626i	$0.508708\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.23536e7			−1.90885			−0.954423	−	0.298457i	$-0.903528\pi$
$281$	−4.23536e7			−1.90885			−0.954423	+	0.298457i	$0.903528\pi$
$282$		0			0
$283$	1.16240e7	−	6.71110e6i	0.512855	−	0.296097i	−0.221151	−	0.975239i	$-0.570981\pi$
$283$	1.16240e7	−	6.71110e6i	0.512855	−	0.296097i	0.734006	+	0.679142i	$0.237648\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.19452e6	−	2.32809e7i	0.219735	−	0.984812i
$288$		0			0
$289$	−1.19364e7	+	2.06745e7i	−0.494516	+	0.856527i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	4.03857e6i		−	0.160555i	−0.996773	−	0.0802777i	$-0.974419\pi$
$293$		−	4.03857e6i		−	0.160555i	0.996773	−	0.0802777i	$-0.0255807\pi$
$294$		0			0
$295$	3.66478e7			1.42752
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.09879e7	−	1.78909e7i	−1.15925	−	0.669295i
$300$		0			0
$301$	−4.89278e7	−	1.09170e7i	−1.79414	−	0.400316i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.74951e6	+	1.16905e7i	0.237888	+	0.412034i
$306$		0			0
$307$		−	2.90517e7i		−	1.00405i	−0.864852	−	0.502027i	$-0.832588\pi$
$307$		−	2.90517e7i		−	1.00405i	0.864852	−	0.502027i	$-0.167412\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.72721e7	−	2.15191e7i	1.23909	−	0.715389i	0.270182	−	0.962809i	$-0.412916\pi$
$311$	3.72721e7	−	2.15191e7i	1.23909	−	0.715389i	0.968908	+	0.247420i	$0.0795829\pi$
$312$		0			0
$313$	−1.60034e7	−	9.23959e6i	−0.521891	−	0.301314i	0.215817	−	0.976434i	$-0.430759\pi$
$313$	−1.60034e7	−	9.23959e6i	−0.521891	−	0.301314i	−0.737708	+	0.675120i	$0.764092\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.37222e6	−	4.10881e6i	0.0744694	−	0.128985i	−0.826386	−	0.563104i	$-0.809607\pi$
$317$	2.37222e6	−	4.10881e6i	0.0744694	−	0.128985i	0.900855	+	0.434119i	$0.142940\pi$
$318$		0			0
$319$	8.85088e6	+	1.53302e7i	0.272656	+	0.472253i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.42722e6			0.101703
$324$		0			0
$325$	−2.59853e7	+	1.50026e7i	−0.756970	+	0.437037i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.39618e7	−	4.03939e7i	−1.23449	−	1.13430i
$330$		0			0
$331$	−3.02878e6	+	5.24600e6i	−0.0835187	+	0.144659i	−0.904759	−	0.425924i	$-0.859949\pi$
$331$	−3.02878e6	+	5.24600e6i	−0.0835187	+	0.144659i	0.821240	+	0.570582i	$0.193282\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			1.74455e7i			0.464032i
$336$		0			0
$337$	−3.38677e7			−0.884905			−0.442452	−	0.896792i	$-0.645891\pi$
$337$	−3.38677e7			−0.884905			−0.442452	+	0.896792i	$0.645891\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.16726e7	+	3.56067e7i	1.55535	+	0.897984i
$342$		0			0
$343$	3.19140e7	+	2.46963e7i	0.790859	+	0.611998i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.36100e6	+	9.28552e6i	0.128309	+	0.222238i	0.923022	−	0.384748i	$-0.125712\pi$
$347$	5.36100e6	+	9.28552e6i	0.128309	+	0.222238i	−0.794713	+	0.606986i	$0.792378\pi$
$348$		0			0
$349$			1.76316e7i			0.414779i	0.978258	+	0.207389i	$0.0664967\pi$
$349$			1.76316e7i			0.414779i	−0.978258	+	0.207389i	$0.933503\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.65852e7	+	1.53490e7i	−0.604387	+	0.348943i	−0.770766	−	0.637119i	$-0.780126\pi$
$353$	−2.65852e7	+	1.53490e7i	−0.604387	+	0.348943i	0.166378	+	0.986062i	$0.446793\pi$
$354$		0			0
$355$	6.81747e7	+	3.93607e7i	1.52384	+	0.879787i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.25102e7	−	5.63093e7i	0.702646	−	1.21702i	−0.264889	−	0.964279i	$-0.585335\pi$
$359$	3.25102e7	−	5.63093e7i	0.702646	−	1.21702i	0.967534	−	0.252739i	$-0.0813315\pi$
$360$		0			0
$361$	−1.33806e6	−	2.31759e6i	−0.0284417	−	0.0492624i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.13980e7			−1.46827
$366$		0			0
$367$	2.04636e7	−	1.18147e7i	0.413985	−	0.239014i	−0.278515	−	0.960432i	$-0.589842\pi$
$367$	2.04636e7	−	1.18147e7i	0.413985	−	0.239014i	0.692501	+	0.721417i	$0.256509\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.55661e7	−	4.96048e7i	−0.304830	−	0.971410i
$372$		0			0
$373$	−3.68038e7	+	6.37461e7i	−0.709197	+	1.22836i	0.255959	+	0.966688i	$0.417609\pi$
$373$	−3.68038e7	+	6.37461e7i	−0.709197	+	1.22836i	−0.965156	+	0.261677i	$0.915725\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2.24798e7i			0.419535i
$378$		0			0
$379$	−6.68460e7			−1.22788			−0.613942	−	0.789351i	$-0.710417\pi$
$379$	−6.68460e7			−1.22788			−0.613942	+	0.789351i	$0.710417\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.12881e7	+	6.51719e6i	0.200921	+	0.116002i	0.597085	−	0.802178i	$-0.296326\pi$
$383$	1.12881e7	+	6.51719e6i	0.200921	+	0.116002i	−0.396164	+	0.918180i	$0.629659\pi$
$384$		0			0
$385$	−3.06356e7	+	1.37303e8i	−0.536839	+	2.40601i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4.65342e7	+	8.05995e7i	0.790539	+	1.36925i	0.925634	+	0.378420i	$0.123533\pi$
$389$	4.65342e7	+	8.05995e7i	0.790539	+	1.36925i	−0.135095	+	0.990833i	$0.543134\pi$
$390$		0			0
$391$		−	5.55225e6i		−	0.0928835i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.72576e7	+	2.15107e7i	−0.604538	+	0.349030i
$396$		0			0
$397$	4.08901e7	+	2.36079e7i	0.653502	+	0.377300i	0.789797	−	0.613369i	$-0.210186\pi$
$397$	4.08901e7	+	2.36079e7i	0.653502	+	0.377300i	−0.136294	+	0.990668i	$0.543519\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.68261e7	+	2.91437e7i	−0.260946	+	0.451972i	−0.966494	−	0.256691i	$-0.917368\pi$
$401$	−1.68261e7	+	2.91437e7i	−0.260946	+	0.451972i	0.705547	+	0.708663i	$0.250701\pi$
$402$		0			0
$403$	4.52177e7	+	7.83193e7i	0.690865	+	1.19661i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.34052e6			0.0198834
$408$		0			0
$409$	−9.08078e7	+	5.24279e7i	−1.32725	+	0.766289i	−0.984874	−	0.173273i	$-0.944566\pi$
$409$	−9.08078e7	+	5.24279e7i	−1.32725	+	0.766289i	−0.342378	+	0.939562i	$0.611232\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.63531e7	−	2.39598e7i	1.08387	−	0.340120i
$414$		0			0
$415$	1.46553e7	−	2.53836e7i	0.205045	−	0.355148i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			7.27931e7i			0.989574i	0.869014	+	0.494787i	$0.164754\pi$
$419$			7.27931e7i			0.989574i	−0.869014	+	0.494787i	$0.835246\pi$
$420$		0			0
$421$	−8.33984e7			−1.11766			−0.558832	−	0.829281i	$-0.688750\pi$
$421$	−8.33984e7			−1.11766			−0.558832	+	0.829281i	$0.688750\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.03215e6	−	2.32796e6i	−0.0525254	−	0.0303256i
$426$		0			0
$427$	2.17052e7	+	1.99436e7i	0.278792	+	0.256165i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.49649e7	+	4.32405e7i	0.311816	+	0.540081i	0.978756	−	0.205031i	$-0.0657294\pi$
$431$	2.49649e7	+	4.32405e7i	0.311816	+	0.540081i	−0.666940	+	0.745112i	$0.732396\pi$
$432$		0			0
$433$			9.20837e7i			1.13428i	0.823623	+	0.567138i	$0.191950\pi$
$433$			9.20837e7i			1.13428i	−0.823623	+	0.567138i	$0.808050\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.22507e7	−	3.59405e7i	0.745934	−	0.430665i
$438$		0			0
$439$	2.97213e7	+	1.71596e7i	0.351297	+	0.202822i	0.665257	−	0.746615i	$-0.268322\pi$
$439$	2.97213e7	+	1.71596e7i	0.351297	+	0.202822i	−0.313959	+	0.949436i	$0.601656\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.55571e7	−	4.42661e7i	0.293968	−	0.509167i	−0.680776	−	0.732491i	$-0.738357\pi$
$443$	2.55571e7	−	4.42661e7i	0.293968	−	0.509167i	0.974744	+	0.223324i	$0.0716908\pi$
$444$		0			0
$445$	−2.22508e7	−	3.85395e7i	−0.252503	−	0.437347i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.46829e7			0.714580			0.357290	−	0.933994i	$-0.383701\pi$
$449$	6.46829e7			0.714580			0.357290	+	0.933994i	$0.383701\pi$
$450$		0			0
$451$	1.57253e8	−	9.07903e7i	1.71424	−	0.989714i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.20874e8	+	1.31551e8i	−1.28322	+	1.39656i
$456$		0			0
$457$	5.66510e7	−	9.81224e7i	0.593552	−	1.02806i	−0.400197	−	0.916429i	$-0.631058\pi$
$457$	5.66510e7	−	9.81224e7i	0.593552	−	1.02806i	0.993749	−	0.111633i	$-0.0356082\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	1.87367e8i		−	1.91245i	−0.292638	−	0.956223i	$-0.594533\pi$
$461$		−	1.87367e8i		−	1.91245i	0.292638	−	0.956223i	$-0.405467\pi$
$462$		0			0
$463$	−1.52237e8			−1.53383			−0.766914	−	0.641750i	$-0.778209\pi$
$463$	−1.52237e8			−1.53383			−0.766914	+	0.641750i	$0.778209\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.93113e7	−	5.15639e7i	−0.876912	−	0.506285i	−0.00727296	−	0.999974i	$-0.502315\pi$
$467$	−8.93113e7	−	5.15639e7i	−0.876912	−	0.506285i	−0.869639	+	0.493688i	$0.835648\pi$
$468$		0			0
$469$	1.14056e7	+	3.63464e7i	0.110560	+	0.352324i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.90808e8	−	3.30489e8i	−1.80307	−	3.12301i
$474$		0			0
$475$		−	6.02768e7i		−	0.562431i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.88344e7	+	2.81946e7i	−0.444344	+	0.256542i	−0.705439	−	0.708771i	$-0.749250\pi$
$479$	−4.88344e7	+	2.81946e7i	−0.444344	+	0.256542i	0.261094	+	0.965313i	$0.415917\pi$
$480$		0			0
$481$	1.47428e6	+	851178.i	0.0132479	+	0.00764865i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.00623e7	+	1.21351e8i	−0.614128	+	1.06370i
$486$		0			0
$487$	5.68754e7	+	9.85111e7i	0.492422	+	0.852900i	0.999962	−	0.00872796i	$-0.00277823\pi$
$487$	5.68754e7	+	9.85111e7i	0.492422	+	0.852900i	−0.507540	+	0.861628i	$0.669445\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.26412e7			0.529195			0.264598	−	0.964359i	$-0.414761\pi$
$491$	6.26412e7			0.529195			0.264598	+	0.964359i	$0.414761\pi$
$492$		0			0
$493$	−3.02086e6	+	1.74410e6i	−0.0252110	+	0.0145556i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.67770e8	+	3.74336e7i	1.36662	+	0.304925i
$498$		0			0
$499$	1.83524e6	−	3.17873e6i	0.0147704	−	0.0255830i	−0.858546	−	0.512737i	$-0.828632\pi$
$499$	1.83524e6	−	3.17873e6i	0.0147704	−	0.0255830i	0.873316	+	0.487154i	$0.161965\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8.33034e7i		−	0.654574i	−0.944925	−	0.327287i	$-0.893866\pi$
$503$		−	8.33034e7i		−	0.654574i	0.944925	−	0.327287i	$-0.106134\pi$
$504$		0			0
$505$	−1.96164e8			−1.52316
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.61652e6	−	2.66535e6i	−0.0350075	−	0.0202116i	0.482394	−	0.875954i	$-0.339767\pi$
$509$	−4.61652e6	−	2.66535e6i	−0.0350075	−	0.0202116i	−0.517402	+	0.855743i	$0.673101\pi$
$510$		0			0
$511$	−1.48753e8	+	4.66789e7i	−1.11481	+	0.349831i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.39137e7	+	9.33813e7i	0.394710	+	0.683657i
$516$		0			0
$517$		−	4.54473e8i		−	3.28879i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.72901e7	+	9.98245e6i	−0.122260	+	0.0705869i	−0.559883	−	0.828572i	$-0.689154\pi$
$521$	−1.72901e7	+	9.98245e6i	−0.122260	+	0.0705869i	0.437623	+	0.899159i	$0.355821\pi$
$522$		0			0
$523$	2.33047e8	+	1.34550e8i	1.62906	+	0.940541i	0.984372	+	0.176099i	$0.0563479\pi$
$523$	2.33047e8	+	1.34550e8i	1.62906	+	0.940541i	0.644693	+	0.764442i	$0.276985\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.01642e6	+	1.21528e7i	−0.0479384	+	0.0830318i
$528$		0			0
$529$	1.57928e7	+	2.73539e7i	0.106682	+	0.184779i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.30593e8			1.52287
$534$		0			0
$535$	2.14621e8	−	1.23912e8i	1.40156	−	0.809191i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.59394e7	+	3.06090e8i	0.165651	+	1.95471i
$540$		0			0
$541$	−7.53414e7	+	1.30495e8i	−0.475819	+	0.824143i	−0.999616	−	0.0277000i	$-0.991182\pi$
$541$	−7.53414e7	+	1.30495e8i	−0.475819	+	0.824143i	0.523797	+	0.851843i	$0.324515\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			3.10315e8i			1.91696i
$546$		0			0
$547$	1.63897e8			1.00140			0.500701	−	0.865620i	$-0.333076\pi$
$547$	1.63897e8			1.00140			0.500701	+	0.865620i	$0.333076\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.91089e7	−	2.25795e7i	−0.233787	−	0.134977i
$552$		0			0
$553$	−6.35602e7	+	6.91744e7i	−0.375846	+	0.409044i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.84031e7	−	1.01157e8i	−0.337964	−	0.585371i	0.646086	−	0.763265i	$-0.276405\pi$
$557$	−5.84031e7	−	1.01157e8i	−0.337964	−	0.585371i	−0.984050	+	0.177894i	$0.943072\pi$
$558$		0			0
$559$		−	4.84622e8i		−	2.77439i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.33717e8	−	7.72016e7i	0.749311	−	0.432615i	−0.0761341	−	0.997098i	$-0.524258\pi$
$563$	1.33717e8	−	7.72016e7i	0.749311	−	0.432615i	0.825445	+	0.564483i	$0.190924\pi$
$564$		0			0
$565$	8.79713e7	+	5.07903e7i	0.487748	+	0.281602i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.29301e7	+	1.26319e8i	−0.395886	+	0.685694i	−0.993214	−	0.116303i	$-0.962896\pi$
$569$	−7.29301e7	+	1.26319e8i	−0.395886	+	0.685694i	0.597328	+	0.801997i	$0.296229\pi$
$570$		0			0
$571$	−8.14722e7	−	1.41114e8i	−0.437624	−	0.757987i	0.559882	−	0.828572i	$-0.310846\pi$
$571$	−8.14722e7	−	1.41114e8i	−0.437624	−	0.757987i	−0.997506	+	0.0705858i	$0.977513\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−9.76511e7			−0.513658
$576$		0			0
$577$	−2.68759e7	+	1.55168e7i	−0.139906	+	0.0807746i	−0.568319	−	0.822808i	$-0.692406\pi$
$577$	−2.68759e7	+	1.55168e7i	−0.139906	+	0.0807746i	0.428413	+	0.903583i	$0.359073\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.39378e7	−	6.24664e7i	0.0710664	−	0.318506i
$582$		0			0
$583$	1.97883e8	−	3.42744e8i	0.998627	−	1.72967i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	3.59531e8i		−	1.77755i	−0.458344	−	0.888775i	$-0.651557\pi$
$587$		−	3.59531e8i		−	1.77755i	0.458344	−	0.888775i	$-0.348443\pi$
$588$		0			0
$589$	−1.81673e8			−0.889088
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.10349e7	+	5.25590e7i	0.436560	+	0.252048i	0.702137	−	0.712042i	$-0.252229\pi$
$593$	9.10349e7	+	5.25590e7i	0.436560	+	0.252048i	−0.265577	+	0.964090i	$0.585563\pi$
$594$		0			0
$595$	−2.70560e7	−	6.03684e6i	−0.128444	−	0.0286589i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.67005e7	−	1.15529e8i	−0.310348	−	0.537538i	0.668090	−	0.744081i	$-0.267112\pi$
$599$	−6.67005e7	−	1.15529e8i	−0.310348	−	0.537538i	−0.978438	+	0.206542i	$0.933779\pi$
$600$		0			0
$601$		−	2.19336e8i		−	1.01038i	−0.863007	−	0.505192i	$-0.831422\pi$
$601$		−	2.19336e8i		−	1.01038i	0.863007	−	0.505192i	$-0.168578\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.86435e8	+	3.96313e8i	−3.09980	+	1.78967i
$606$		0			0
$607$	−1.98185e8	−	1.14422e8i	−0.886143	−	0.511615i	−0.0134642	−	0.999909i	$-0.504286\pi$
$607$	−1.98185e8	−	1.14422e8i	−0.886143	−	0.511615i	−0.872679	+	0.488294i	$0.837619\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.88572e8	−	4.99821e8i	1.26512	−	2.19125i
$612$		0			0
$613$	−1.08106e8	−	1.87246e8i	−0.469321	−	0.812887i	0.530064	−	0.847957i	$-0.322168\pi$
$613$	−1.08106e8	−	1.87246e8i	−0.469321	−	0.812887i	−0.999385	+	0.0350704i	$0.988834\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.65273e8			0.703634			0.351817	−	0.936069i	$-0.385564\pi$
$617$	1.65273e8			0.703634			0.351817	+	0.936069i	$0.385564\pi$
$618$		0			0
$619$	2.14104e8	−	1.23613e8i	0.902718	−	0.521185i	0.0246372	−	0.999696i	$-0.492157\pi$
$619$	2.14104e8	−	1.23613e8i	0.902718	−	0.521185i	0.878081	+	0.478512i	$0.158824\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.15545e7	−	6.57471e7i	−0.295919	−	0.271902i
$624$		0			0
$625$	1.51823e8	−	2.62966e8i	0.621868	−	1.07711i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			264154.i			0.00106147i
$630$		0			0
$631$	−1.97554e8			−0.786317			−0.393158	−	0.919471i	$-0.628618\pi$
$631$	−1.97554e8			−0.786317			−0.393158	+	0.919471i	$0.628618\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.46322e8	+	1.42214e8i	0.962017	+	0.555421i
$636$		0			0
$637$	−1.65827e8	+	3.53103e8i	−0.641560	+	1.36610i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.13429e7	+	1.96465e7i	0.0430675	+	0.0745951i	0.886756	−	0.462239i	$-0.152954\pi$
$641$	1.13429e7	+	1.96465e7i	0.0430675	+	0.0745951i	−0.843688	+	0.536834i	$0.819620\pi$
$642$		0			0
$643$		−	3.36606e8i		−	1.26616i	−0.774085	−	0.633081i	$-0.781790\pi$
$643$		−	3.36606e8i		−	1.26616i	0.774085	−	0.633081i	$-0.218210\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.80996e7	+	2.19968e7i	−0.140672	+	0.0812171i	−0.568684	−	0.822556i	$-0.692547\pi$
$647$	−3.80996e7	+	2.19968e7i	−0.140672	+	0.0812171i	0.428012	+	0.903773i	$0.359214\pi$
$648$		0			0
$649$	5.27561e8	+	3.04587e8i	1.92992	+	1.11424i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.36808e7	−	9.29779e7i	0.192788	−	0.333918i	−0.753385	−	0.657579i	$-0.771580\pi$
$653$	5.36808e7	−	9.29779e7i	0.192788	−	0.333918i	0.946173	+	0.323661i	$0.104914\pi$
$654$		0			0
$655$	2.52129e8	+	4.36701e8i	0.897222	+	1.55403i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4.96679e7			−0.173548			−0.0867739	−	0.996228i	$-0.527656\pi$
$659$	−4.96679e7			−0.173548			−0.0867739	+	0.996228i	$0.527656\pi$
$660$		0			0
$661$	9.62080e7	−	5.55457e7i	0.333125	−	0.192330i	−0.324103	−	0.946022i	$-0.605062\pi$
$661$	9.62080e7	−	5.55457e7i	0.333125	−	0.192330i	0.657228	+	0.753692i	$0.271729\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.07453e8	−	3.42424e8i	−0.365389	−	1.16439i
$666$		0			0
$667$	−3.65799e7	+	6.33582e7i	−0.123272	+	0.213513i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			2.24386e8i			0.742726i
$672$		0			0
$673$	−4.29393e8			−1.40867			−0.704337	−	0.709866i	$-0.748755\pi$
$673$	−4.29393e8			−1.40867			−0.704337	+	0.709866i	$0.748755\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.11122e8	+	6.41564e7i	0.358125	+	0.206764i	0.668258	−	0.743930i	$-0.267040\pi$
$677$	1.11122e8	+	6.41564e7i	0.358125	+	0.206764i	−0.310133	+	0.950693i	$0.600374\pi$
$678$		0			0
$679$	−6.66322e7	+	2.98633e8i	−0.212850	+	0.953956i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.16965e7	+	1.58823e8i	0.287800	+	0.498484i	0.973284	−	0.229603i	$-0.0737428\pi$
$683$	9.16965e7	+	1.58823e8i	0.287800	+	0.498484i	−0.685484	+	0.728087i	$0.740410\pi$
$684$		0			0
$685$		−	3.13399e8i		−	0.975048i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.35257e8	−	2.51296e8i	1.33073	−	0.768295i
$690$		0			0
$691$	4.78059e8	+	2.76008e8i	1.44893	+	0.836540i	0.998418	−	0.0562298i	$-0.0179080\pi$
$691$	4.78059e8	+	2.76008e8i	1.44893	+	0.836540i	0.450512	+	0.892770i	$0.351241\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.18148e7	−	1.07066e8i	0.184136	−	0.318933i
$696$		0			0
$697$	1.78905e7	+	3.09873e7i	0.0528354	+	0.0915136i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.32955e8			−1.83746			−0.918732	−	0.394882i	$-0.870786\pi$
$701$	−6.32955e8			−1.83746			−0.918732	+	0.394882i	$0.870786\pi$
$702$		0			0
$703$	−2.96165e6	+	1.70991e6i	−0.00852447	+	0.00492161i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.08693e8	+	1.28249e8i	−1.15648	+	0.362906i
$708$		0			0
$709$	5.44154e6	−	9.42503e6i	0.0152680	−	0.0264450i	−0.858290	−	0.513164i	$-0.828473\pi$
$709$	5.44154e6	−	9.42503e6i	0.0152680	−	0.0264450i	0.873558	+	0.486719i	$0.161807\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			2.94318e8i			0.811987i
$714$		0			0
$715$	−1.35996e9			−3.72056
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.00684e8	−	5.81301e7i	−0.270879	−	0.156392i	0.358408	−	0.933565i	$-0.383320\pi$
$719$	−1.00684e8	−	5.81301e7i	−0.270879	−	0.156392i	−0.629287	+	0.777173i	$0.716653\pi$
$720$		0			0
$721$	1.73377e8	+	1.59305e8i	0.462578	+	0.425035i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.06746e7	+	5.31299e7i	0.0804942	+	0.139420i
$726$		0			0
$727$			5.51443e8i			1.43515i	0.696482	+	0.717575i	$0.254748\pi$
$727$			5.51443e8i			1.43515i	−0.696482	+	0.717575i	$0.745252\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.51240e7	−	3.75994e7i	0.166721	−	0.0962561i
$732$		0			0
$733$	−1.55500e8	−	8.97781e7i	−0.394838	−	0.227960i	0.289416	−	0.957203i	$-0.406539\pi$
$733$	−1.55500e8	−	8.97781e7i	−0.394838	−	0.227960i	−0.684254	+	0.729243i	$0.739872\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.44993e8	+	2.51135e8i	−0.362196	+	0.627342i
$738$		0			0
$739$	−8.19983e7	−	1.42025e8i	−0.203176	−	0.351911i	0.746374	−	0.665526i	$-0.231793\pi$
$739$	−8.19983e7	−	1.42025e8i	−0.203176	−	0.351911i	−0.949550	+	0.313616i	$0.898460\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.60076e8			0.390265			0.195133	−	0.980777i	$-0.437486\pi$
$743$	1.60076e8			0.390265			0.195133	+	0.980777i	$0.437486\pi$
$744$		0			0
$745$	7.69056e8	−	4.44015e8i	1.85990	−	1.07381i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.66137e8	−	3.98477e8i	0.871361	−	0.948327i
$750$		0			0
$751$	−2.23458e7	+	3.87040e7i	−0.0527564	+	0.0913768i	−0.891198	−	0.453615i	$-0.850134\pi$
$751$	−2.23458e7	+	3.87040e7i	−0.0527564	+	0.0913768i	0.838441	+	0.544992i	$0.183467\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			3.83752e8i			0.891681i
$756$		0			0
$757$	−2.30985e8			−0.532471			−0.266235	−	0.963908i	$-0.585780\pi$
$757$	−2.30985e8			−0.532471			−0.266235	+	0.963908i	$0.585780\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.03174e8	+	2.32773e8i	0.914826	+	0.528175i	0.881981	−	0.471286i	$-0.156210\pi$
$761$	4.03174e8	+	2.32773e8i	0.914826	+	0.528175i	0.0328451	+	0.999460i	$0.489543\pi$
$762$		0			0
$763$	2.02879e8	+	6.46519e8i	0.456734	+	1.45548i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.86801e8	+	6.69960e8i	0.857239	+	1.48478i
$768$		0			0
$769$		−	9.46426e7i		−	0.208117i	−0.994571	−	0.104059i	$-0.966817\pi$
$769$		−	9.46426e7i		−	0.208117i	0.994571	−	0.104059i	$-0.0331829\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8.89489e7	−	5.13547e7i	0.192576	−	0.111184i	−0.400612	−	0.916248i	$-0.631202\pi$
$773$	8.89489e7	−	5.13547e7i	0.192576	−	0.111184i	0.593188	+	0.805064i	$0.297869\pi$
$774$		0			0
$775$	2.13740e8	+	1.23403e8i	0.459177	+	0.265106i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.31616e8	+	4.01170e8i	−0.489955	+	0.848626i
$780$		0			0
$781$	6.54269e8	+	1.13323e9i	1.37342	+	2.37883i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.65567e8			0.548990
$786$		0			0
$787$	−7.07077e8	+	4.08231e8i	−1.45058	+	0.837495i	−0.998514	−	0.0544879i	$-0.982647\pi$
$787$	−7.07077e8	+	4.08231e8i	−1.45058	+	0.837495i	−0.452069	+	0.891983i	$0.649314\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.16488e8	+	4.83037e7i	0.437426	+	0.0976002i
$792$		0			0
$793$	−1.42476e8	+	2.46776e8i	−0.285708	+	0.494861i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	6.56026e8i		−	1.29582i	−0.761715	−	0.647912i	$-0.775642\pi$
$797$		−	6.56026e8i		−	1.29582i	0.761715	−	0.647912i	$-0.224358\pi$
$798$		0			0
$799$	8.95554e7			0.175570
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.02780e9	−	5.93403e8i	−1.98501	−	1.14605i
$804$		0			0
$805$	−5.54742e8	+	1.74079e8i	−1.06342	+	0.333702i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.57816e8	−	2.73345e8i	−0.298061	−	0.516257i	0.677631	−	0.735402i	$-0.263007\pi$
$809$	−1.57816e8	−	2.73345e8i	−0.298061	−	0.516257i	−0.975692	+	0.219145i	$0.929673\pi$
$810$		0			0
$811$		−	5.28556e8i		−	0.990898i	−0.868637	−	0.495449i	$-0.835004\pi$
$811$		−	5.28556e8i		−	0.990898i	0.868637	−	0.495449i	$-0.164996\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.58134e8	−	3.79974e8i	1.21574	−	0.701909i
$816$		0			0
$817$	8.43114e8	+	4.86772e8i	1.54604	+	0.892605i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2.53350e8	+	4.38816e8i	−0.457817	+	0.792963i	−0.998845	−	0.0480419i	$-0.984702\pi$
$821$	−2.53350e8	+	4.38816e8i	−0.457817	+	0.792963i	0.541028	+	0.841004i	$0.318035\pi$
$822$		0			0
$823$	−2.42861e8	−	4.20647e8i	−0.435670	−	0.754603i	0.561680	−	0.827355i	$-0.310155\pi$
$823$	−2.42861e8	−	4.20647e8i	−0.435670	−	0.754603i	−0.997350	+	0.0727515i	$0.976822\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5.86017e8			−1.03608			−0.518041	−	0.855356i	$-0.673338\pi$
$827$	−5.86017e8			−1.03608			−0.518041	+	0.855356i	$0.673338\pi$
$828$		0			0
$829$	2.32195e7	−	1.34058e7i	0.0407557	−	0.0235303i	−0.479484	−	0.877551i	$-0.659176\pi$
$829$	2.32195e7	−	1.34058e7i	0.0407557	−	0.0235303i	0.520239	+	0.854020i	$0.325843\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6.03161e7	+	5.11145e6i	−0.104351	+	0.00884320i
$834$		0			0
$835$	−5.42923e8	+	9.40370e8i	−0.932564	+	1.61525i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	8.56330e8i		−	1.44996i	−0.688771	−	0.724979i	$-0.741849\pi$
$839$		−	8.56330e8i		−	1.44996i	0.688771	−	0.724979i	$-0.258151\pi$
$840$		0			0
$841$	−5.48861e8			−0.922729
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.39046e8	−	4.84423e8i	−1.39064	−	0.802888i
$846$		0			0
$847$	−1.17103e9	+	1.27447e9i	−1.92717	+	2.09739i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.77013e6	+	4.79800e6i	0.00449481	+	0.00778523i
$852$		0			0
$853$		−	6.93083e8i		−	1.11670i	−0.829604	−	0.558352i	$-0.811434\pi$
$853$		−	6.93083e8i		−	1.11670i	0.829604	−	0.558352i	$-0.188566\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.03661e9	−	5.98489e8i	1.64693	−	0.950854i	0.668644	−	0.743583i	$-0.266875\pi$
$857$	1.03661e9	−	5.98489e8i	1.64693	−	0.950854i	0.978283	−	0.207271i	$-0.0664583\pi$
$858$		0			0
$859$	6.54077e8	+	3.77632e8i	1.03193	+	0.595784i	0.917536	−	0.397652i	$-0.130175\pi$
$859$	6.54077e8	+	3.77632e8i	1.03193	+	0.595784i	0.114392	+	0.993436i	$0.463508\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.37317e8	+	4.11045e8i	−0.369230	+	0.639525i	−0.989445	−	0.144907i	$-0.953712\pi$
$863$	−2.37317e8	+	4.11045e8i	−0.369230	+	0.639525i	0.620216	+	0.784431i	$0.287045\pi$
$864$		0			0
$865$	7.69162e8	+	1.33223e9i	1.18842	+	2.05840i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.15119e8			−1.08973
$870$		0			0
$871$	−3.18921e8	+	1.84129e8i	−0.482646	+	0.278656i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.71551e7	−	3.45794e8i	0.115170	−	0.516171i
$876$		0			0
$877$	4.10023e8	−	7.10181e8i	0.607869	−	1.05286i	−0.383723	−	0.923448i	$-0.625358\pi$
$877$	4.10023e8	−	7.10181e8i	0.607869	−	1.05286i	0.991591	−	0.129411i	$-0.0413086\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1.24754e8i			0.182443i	0.995831	+	0.0912216i	$0.0290772\pi$
$881$			1.24754e8i			0.182443i	−0.995831	+	0.0912216i	$0.970923\pi$
$882$		0			0
$883$	−8.94702e7			−0.129956			−0.0649780	−	0.997887i	$-0.520698\pi$
$883$	−8.94702e7			−0.129956			−0.0649780	+	0.997887i	$0.520698\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.75975e8	−	2.17069e8i	−0.538751	−	0.311048i	0.205821	−	0.978590i	$-0.434013\pi$
$887$	−3.75975e8	−	2.17069e8i	−0.538751	−	0.311048i	−0.744573	+	0.667541i	$0.767347\pi$
$888$		0			0
$889$	6.06172e8	+	1.35252e8i	0.862762	+	0.192503i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.79705e8	+	1.00408e9i	0.814052	+	1.40998i
$894$		0			0
$895$		−	1.03260e9i		−	1.44034i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.60133e8	−	9.24525e7i	0.220394	−	0.127245i
$900$		0			0
$901$	6.75388e7	+	3.89936e7i	0.0923377	+	0.0533112i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.66548e7	+	1.15450e8i	−0.0899261	+	0.155757i
$906$		0			0
$907$	−2.93196e8	−	5.07830e8i	−0.392949	−	0.680607i	0.599888	−	0.800084i	$-0.295212\pi$
$907$	−2.93196e8	−	5.07830e8i	−0.392949	−	0.680607i	−0.992837	+	0.119477i	$0.961878\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7.05258e8			0.932809			0.466405	−	0.884572i	$-0.345549\pi$
$911$	7.05258e8			0.932809			0.466405	+	0.884572i	$0.345549\pi$
$912$		0			0
$913$	4.21937e8	−	2.43605e8i	0.554416	−	0.320092i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.10802e8	+	7.44997e8i	1.05149	+	0.966155i
$918$		0			0
$919$	1.63656e8	−	2.83461e8i	0.210856	−	0.365213i	−0.741127	−	0.671365i	$-0.765708\pi$
$919$	1.63656e8	−	2.83461e8i	0.210856	−	0.365213i	0.951983	+	0.306152i	$0.0990416\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1.66174e9i			2.11328i
$924$		0			0
$925$	4.64586e6			0.00587004
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.04221e9	+	6.01721e8i	1.29990	+	0.750496i	0.980386	−	0.197087i	$-0.0631480\pi$
$929$	1.04221e9	+	6.01721e8i	1.29990	+	0.750496i	0.319511	+	0.947583i	$0.396481\pi$
$930$		0			0
$931$	−4.47743e8	−	6.43165e8i	−0.554855	−	0.797027i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.05513e8	−	1.82753e8i	−0.129083	−	0.223578i
$936$		0			0
$937$		−	3.77788e8i		−	0.459230i	−0.973282	−	0.229615i	$-0.926253\pi$
$937$		−	3.77788e8i		−	0.459230i	0.973282	−	0.229615i	$-0.0737466\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.66347e8	+	3.84716e8i	−0.799709	+	0.461712i	−0.843369	−	0.537334i	$-0.819431\pi$
$941$	−6.66347e8	+	3.84716e8i	−0.799709	+	0.461712i	0.0436607	+	0.999046i	$0.486098\pi$
$942$		0			0
$943$	6.49914e8	+	3.75228e8i	0.775034	+	0.447466i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.53429e8	+	2.65747e8i	−0.180658	+	0.312910i	−0.942105	−	0.335318i	$-0.891156\pi$
$947$	−1.53429e8	+	2.65747e8i	−0.180658	+	0.312910i	0.761447	+	0.648228i	$0.224490\pi$
$948$		0			0
$949$	−7.53574e8	−	1.30523e9i	−0.881713	−	1.52717i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.56965e7			0.0181352			0.00906761	−	0.999959i	$-0.497114\pi$
$953$	1.56965e7			0.0181352			0.00906761	+	0.999959i	$0.497114\pi$
$954$		0			0
$955$	2.13490e8	−	1.23259e8i	0.245114	−	0.141517i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.04895e8	−	6.52945e8i	−0.232315	−	0.740322i
$960$		0			0
$961$	−7.18191e7	+	1.24394e8i	−0.0809226	+	0.140162i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.18163e9i		−	1.31492i
$966$		0			0
$967$	−1.01121e9			−1.11831			−0.559157	−	0.829062i	$-0.688875\pi$
$967$	−1.01121e9			−1.11831			−0.559157	+	0.829062i	$0.688875\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.06493e9	+	6.14839e8i	1.16323	+	0.671590i	0.952075	−	0.305864i	$-0.0989452\pi$
$971$	1.06493e9	+	6.14839e8i	1.16323	+	0.671590i	0.211152	+	0.977453i	$0.432279\pi$
$972$		0			0
$973$	5.87885e7	−	2.63479e8i	0.0638195	−	0.286027i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.38788e8	−	7.60003e8i	−0.470512	−	0.814951i	0.528919	−	0.848672i	$-0.322598\pi$
$977$	−4.38788e8	−	7.60003e8i	−0.470512	−	0.814951i	−0.999431	+	0.0337210i	$0.989264\pi$
$978$		0			0
$979$		−	7.39724e8i		−	0.788354i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.20614e9	+	6.96367e8i	−1.26981	+	0.733125i	−0.974952	−	0.222417i	$-0.928605\pi$
$983$	−1.20614e9	+	6.96367e8i	−1.26981	+	0.733125i	−0.294857	+	0.955541i	$0.595272\pi$
$984$		0			0
$985$	−4.81489e8	−	2.77988e8i	−0.503823	−	0.290882i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.88592e8	−	1.36588e9i	0.815199	−	1.41197i
$990$		0			0
$991$	5.53175e8	+	9.58126e8i	0.568383	+	0.984469i	0.996726	+	0.0808523i	$0.0257642\pi$
$991$	5.53175e8	+	9.58126e8i	0.568383	+	0.984469i	−0.428343	+	0.903616i	$0.640902\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.09124e8			0.516838
$996$		0			0
$997$	8.44428e8	−	4.87530e8i	0.852073	−	0.491945i	−0.00927664	−	0.999957i	$-0.502953\pi$
$997$	8.44428e8	−	4.87530e8i	0.852073	−	0.491945i	0.861350	+	0.508012i	$0.169620\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.7.z.f.145.5	yes	12
3.2	odd	2	inner	252.7.z.f.145.2	yes	12
7.3	odd	6	inner	252.7.z.f.73.5	yes	12
21.17	even	6	inner	252.7.z.f.73.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.7.z.f.73.2	✓	12	21.17	even	6	inner
252.7.z.f.73.5	yes	12	7.3	odd	6	inner
252.7.z.f.145.2	yes	12	3.2	odd	2	inner
252.7.z.f.145.5	yes	12	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	252.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(136.035 + 78.5400i) q^{5} +(334.768 + 74.6948i) q^{7} +O(q^{10})\)	\(q+(136.035 + 78.5400i) q^{5} +(334.768 + 74.6948i) q^{7} +(1305.52 + 2261.23i) q^{11} +3315.82i q^{13} +(-445.584 + 257.258i) q^{17} +(-5768.65 - 3330.53i) q^{19} +(-5395.61 + 9345.47i) q^{23} +(4524.57 + 7836.78i) q^{25} +6779.56 q^{29} +(23619.9 - 13636.9i) q^{31} +(39673.7 + 36453.8i) q^{35} +(256.702 - 444.621i) q^{37} -69543.2i q^{41} -146154. q^{43} +(-150738. - 87028.8i) q^{47} +(106490. + 50010.9i) q^{49} +(-75787.0 - 131267. i) q^{53} +410143. i q^{55} +(202049. - 116653. i) q^{59} +(74423.8 + 42968.6i) q^{61} +(-260425. + 451069. i) q^{65} +(55530.5 + 96181.6i) q^{67} +501154. q^{71} +(-393637. + 227266. i) q^{73} +(268145. + 854505. i) q^{77} +(-136941. + 237189. i) q^{79} -186596. i q^{83} -80820.1 q^{85} +(-245350. - 141653. i) q^{89} +(-247675. + 1.11003e6i) q^{91} +(-523160. - 906140. i) q^{95} +892059. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 714 q^{7}+O(q^{10}) \) 12 * q + 714 * q^7	\( 12 q + 714 q^{7} - 59466 q^{19} + 64086 q^{25} + 219834 q^{31} - 166242 q^{37} - 326916 q^{43} - 132006 q^{49} + 736008 q^{61} + 176238 q^{67} - 2931018 q^{73} + 449490 q^{79} + 2174592 q^{85} - 808542 q^{91}+O(q^{100}) \) 12 * q + 714 * q^7 - 59466 * q^19 + 64086 * q^25 + 219834 * q^31 - 166242 * q^37 - 326916 * q^43 - 132006 * q^49 + 736008 * q^61 + 176238 * q^67 - 2931018 * q^73 + 449490 * q^79 + 2174592 * q^85 - 808542 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more