LMFDB - Embedded newform 252.9.bk.a.53.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(53,252)]

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, 3, 4]))

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

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

chi := DirichletCharacter("252.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.bk (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:	$102.659409735$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$22$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.6
Character	$\chi$	$=$	252.53
Dual form			252.9.bk.a.233.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-517.869 + 298.992i) q^{5} +(2338.25 - 545.349i) q^{7} +O(q^{10})$	$q+(-517.869 + 298.992i) q^{5} +(2338.25 - 545.349i) q^{7} +(-1572.34 - 907.792i) q^{11} -35456.9 q^{13} +(-15328.2 - 8849.72i) q^{17} +(61792.0 + 107027. i) q^{19} +(351614. - 203005. i) q^{23} +(-16520.0 + 28613.4i) q^{25} -1.28998e6i q^{29} +(-574013. + 994219. i) q^{31} +(-1.04785e6 + 981537. i) q^{35} +(-639192. - 1.10711e6i) q^{37} +1.15174e6i q^{41} -4.49924e6 q^{43} +(-588772. + 339928. i) q^{47} +(5.16999e6 - 2.55032e6i) q^{49} +(-8.08110e6 - 4.66562e6i) q^{53} +1.08569e6 q^{55} +(1.14373e7 + 6.60332e6i) q^{59} +(7.72820e6 + 1.33856e7i) q^{61} +(1.83620e7 - 1.06013e7i) q^{65} +(8.52255e6 - 1.47615e7i) q^{67} +2.31454e7i q^{71} +(1.04308e7 - 1.80667e7i) q^{73} +(-4.17159e6 - 1.26517e6i) q^{77} +(2.77913e7 + 4.81360e7i) q^{79} -4.09134e7i q^{83} +1.05840e7 q^{85} +(8.29143e7 - 4.78706e7i) q^{89} +(-8.29069e7 + 1.93364e7i) q^{91} +(-6.40004e7 - 3.69507e7i) q^{95} +5.71573e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 1230 q^{7}+O(q^{10}) $ 44 * q + 1230 * q^7	$ 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97}+O(q^{100}) $ 44 * q + 1230 * q^7 - 101380 * q^13 + 62770 * q^19 + 2247666 * q^25 + 1389254 * q^31 - 2136026 * q^37 + 11510140 * q^43 - 3824398 * q^49 - 42646528 * q^55 + 27346232 * q^61 + 14239194 * q^67 - 64344138 * q^73 + 7061786 * q^79 - 54198208 * q^85 - 45697066 * q^91 + 476543496 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	−517.869	+	298.992i	−0.828591	+	0.478387i	−0.853370	−	0.521306i	$-0.825445\pi$
$5$	−517.869	+	298.992i	−0.828591	+	0.478387i	0.0247789	+	0.999693i	$0.492112\pi$
$6$		0			0
$7$	2338.25	−	545.349i	0.973863	−	0.227134i
$7$	2338.25	−	545.349i	0.973863	−	0.227134i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1572.34	−	907.792i	−0.107393	−	0.0620034i	0.445341	−	0.895361i	$-0.353082\pi$
$11$	−1572.34	−	907.792i	−0.107393	−	0.0620034i	−0.552735	+	0.833357i	$0.686416\pi$
$12$		0			0
$13$	−35456.9			−1.24144			−0.620722	−	0.784031i	$-0.713161\pi$
$13$	−35456.9			−1.24144			−0.620722	+	0.784031i	$0.713161\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15328.2	−	8849.72i	−0.183525	−	0.105958i	0.405423	−	0.914129i	$-0.367124\pi$
$17$	−15328.2	−	8849.72i	−0.183525	−	0.105958i	−0.588948	+	0.808171i	$0.700458\pi$
$18$		0			0
$19$	61792.0	+	107027.i	0.474153	+	0.821256i	0.999562	−	0.0295932i	$-0.00942118\pi$
$19$	61792.0	+	107027.i	0.474153	+	0.821256i	−0.525409	+	0.850850i	$0.676088\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	351614.	−	203005.i	1.25648	−	0.725428i	0.284091	−	0.958797i	$-0.408308\pi$
$23$	351614.	−	203005.i	1.25648	−	0.725428i	0.972388	+	0.233369i	$0.0749750\pi$
$24$		0			0
$25$	−16520.0	+	28613.4i	−0.0422911	+	0.0732504i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	1.28998e6i		−	1.82386i	−0.410348	−	0.911929i	$-0.634593\pi$
$29$		−	1.28998e6i		−	1.82386i	0.410348	−	0.911929i	$-0.365407\pi$
$30$		0			0
$31$	−574013.	+	994219.i	−0.621548	+	1.07655i	0.367649	+	0.929965i	$0.380163\pi$
$31$	−574013.	+	994219.i	−0.621548	+	1.07655i	−0.989198	+	0.146589i	$0.953171\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.04785e6	+	981537.i	−0.698277	+	0.654085i
$36$		0			0
$37$	−639192.	−	1.10711e6i	−0.341055	−	0.590725i	0.643574	−	0.765384i	$-0.277451\pi$
$37$	−639192.	−	1.10711e6i	−0.341055	−	0.590725i	−0.984629	+	0.174659i	$0.944118\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.15174e6i			0.407584i	0.979014	+	0.203792i	$0.0653267\pi$
$41$			1.15174e6i			0.407584i	−0.979014	+	0.203792i	$0.934673\pi$
$42$		0			0
$43$	−4.49924e6			−1.31603			−0.658014	−	0.753006i	$-0.728603\pi$
$43$	−4.49924e6			−1.31603			−0.658014	+	0.753006i	$0.728603\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−588772.	+	339928.i	−0.120658	+	0.0696619i	−0.559114	−	0.829091i	$-0.688859\pi$
$47$	−588772.	+	339928.i	−0.120658	+	0.0696619i	0.438456	+	0.898752i	$0.355525\pi$
$48$		0			0
$49$	5.16999e6	−	2.55032e6i	0.896820	−	0.442395i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.08110e6	−	4.66562e6i	−1.02416	−	0.591298i	−0.108852	−	0.994058i	$-0.534717\pi$
$53$	−8.08110e6	−	4.66562e6i	−1.02416	−	0.591298i	−0.915305	+	0.402760i	$0.868051\pi$
$54$		0			0
$55$	1.08569e6			0.118647
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.14373e7	+	6.60332e6i	0.943876	+	0.544947i	0.891173	−	0.453663i	$-0.149883\pi$
$59$	1.14373e7	+	6.60332e6i	0.943876	+	0.544947i	0.0527028	+	0.998610i	$0.483216\pi$
$60$		0			0
$61$	7.72820e6	+	1.33856e7i	0.558161	+	0.966763i	0.997650	+	0.0685150i	$0.0218261\pi$
$61$	7.72820e6	+	1.33856e7i	0.558161	+	0.966763i	−0.439489	+	0.898248i	$0.644841\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.83620e7	−	1.06013e7i	1.02865	−	0.593891i
$66$		0			0
$67$	8.52255e6	−	1.47615e7i	0.422932	−	0.732540i	−0.573293	−	0.819351i	$-0.694334\pi$
$67$	8.52255e6	−	1.47615e7i	0.422932	−	0.732540i	0.996225	+	0.0868109i	$0.0276676\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.31454e7i			0.910817i	0.890283	+	0.455409i	$0.150507\pi$
$71$			2.31454e7i			0.910817i	−0.890283	+	0.455409i	$0.849493\pi$
$72$		0			0
$73$	1.04308e7	−	1.80667e7i	0.367306	−	0.636192i	−0.621838	−	0.783146i	$-0.713614\pi$
$73$	1.04308e7	−	1.80667e7i	0.367306	−	0.636192i	0.989143	+	0.146954i	$0.0469470\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.17159e6	−	1.26517e6i	−0.118669	−	0.0359902i
$78$		0			0
$79$	2.77913e7	+	4.81360e7i	0.713512	+	1.23584i	0.963531	+	0.267598i	$0.0862299\pi$
$79$	2.77913e7	+	4.81360e7i	0.713512	+	1.23584i	−0.250019	+	0.968241i	$0.580437\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	4.09134e7i		−	0.862090i	−0.902330	−	0.431045i	$-0.858145\pi$
$83$		−	4.09134e7i		−	0.862090i	0.902330	−	0.431045i	$-0.141855\pi$
$84$		0			0
$85$	1.05840e7			0.202756
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.29143e7	−	4.78706e7i	1.32151	−	0.762972i	0.337538	−	0.941312i	$-0.390406\pi$
$89$	8.29143e7	−	4.78706e7i	1.32151	−	0.762972i	0.983969	+	0.178340i	$0.0570726\pi$
$90$		0			0
$91$	−8.29069e7	+	1.93364e7i	−1.20900	+	0.281975i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.40004e7	−	3.69507e7i	−0.785757	−	0.453657i
$96$		0			0
$97$	5.71573e7			0.645632			0.322816	−	0.946462i	$-0.395371\pi$
$97$	5.71573e7			0.645632			0.322816	+	0.946462i	$0.395371\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.11673e7	+	6.44744e6i	0.107316	+	0.0619586i	0.552697	−	0.833382i	$-0.313599\pi$
$101$	1.11673e7	+	6.44744e6i	0.107316	+	0.0619586i	−0.445382	+	0.895341i	$0.646932\pi$
$102$		0			0
$103$	2.44198e7	+	4.22964e7i	0.216967	+	0.375798i	0.953879	−	0.300190i	$-0.0970502\pi$
$103$	2.44198e7	+	4.22964e7i	0.216967	+	0.375798i	−0.736912	+	0.675989i	$0.763717\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.00639e7	−	3.46779e7i	0.458225	−	0.264556i	−0.253073	−	0.967447i	$-0.581441\pi$
$107$	6.00639e7	−	3.46779e7i	0.458225	−	0.264556i	0.711298	+	0.702891i	$0.248108\pi$
$108$		0			0
$109$	1.45824e7	−	2.52574e7i	0.103305	−	0.178930i	−0.809739	−	0.586790i	$-0.800391\pi$
$109$	1.45824e7	−	2.52574e7i	0.103305	−	0.178930i	0.913045	+	0.407860i	$0.133725\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.09275e8i		−	0.670205i	−0.942182	−	0.335103i	$-0.891229\pi$
$113$		−	1.09275e8i		−	0.670205i	0.942182	−	0.335103i	$-0.108771\pi$
$114$		0			0
$115$	−1.21394e8	+	2.10260e8i	−0.694072	+	1.20217i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.06672e7	−	1.23336e7i	−0.202795	−	0.0615039i
$120$		0			0
$121$	−1.05531e8	−	1.82786e8i	−0.492311	−	0.852708i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	2.53345e8i		−	1.03770i
$126$		0			0
$127$	2.52517e8			0.970681			0.485340	−	0.874325i	$-0.338696\pi$
$127$	2.52517e8			0.970681			0.485340	+	0.874325i	$0.338696\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.70826e8	−	2.14096e8i	1.25917	−	0.726983i	0.286258	−	0.958153i	$-0.407588\pi$
$131$	3.70826e8	−	2.14096e8i	1.25917	−	0.726983i	0.972914	+	0.231169i	$0.0742551\pi$
$132$		0			0
$133$	2.02852e8	+	2.16557e8i	0.648295	+	0.692095i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.74792e8	+	1.58651e8i	0.780050	+	0.450362i	0.836448	−	0.548046i	$-0.184628\pi$
$137$	2.74792e8	+	1.58651e8i	0.780050	+	0.450362i	−0.0563980	+	0.998408i	$0.517962\pi$
$138$		0			0
$139$	6.61026e8			1.77076			0.885380	−	0.464868i	$-0.153898\pi$
$139$	6.61026e8			1.77076			0.885380	+	0.464868i	$0.153898\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.57504e7	+	3.21875e7i	0.133323	+	0.0769738i
$144$		0			0
$145$	3.85694e8	+	6.68041e8i	0.872511	+	1.51123i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.25970e8	−	4.76874e8i	1.67579	−	0.967517i	0.711491	−	0.702695i	$-0.248020\pi$
$149$	8.25970e8	−	4.76874e8i	1.67579	−	0.967517i	0.964297	−	0.264822i	$-0.0853132\pi$
$150$		0			0
$151$	−3.02229e8	+	5.23477e8i	−0.581338	+	1.00691i	0.413983	+	0.910285i	$0.364137\pi$
$151$	−3.02229e8	+	5.23477e8i	−0.581338	+	1.00691i	−0.995321	+	0.0966229i	$0.969196\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	6.86501e8i		−	1.18936i
$156$		0			0
$157$	5.56925e8	−	9.64623e8i	0.916639	−	1.58767i	0.112155	−	0.993691i	$-0.464225\pi$
$157$	5.56925e8	−	9.64623e8i	0.916639	−	1.58767i	0.804484	−	0.593975i	$-0.202442\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.11452e8	−	6.66427e8i	1.05887	−	0.991858i
$162$		0			0
$163$	2.77579e8	+	4.80780e8i	0.393220	+	0.681077i	0.992872	−	0.119184i	$-0.0380277\pi$
$163$	2.77579e8	+	4.80780e8i	0.393220	+	0.681077i	−0.599652	+	0.800261i	$0.704694\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1.08481e8i			0.139473i	0.997565	+	0.0697364i	$0.0222158\pi$
$167$			1.08481e8i			0.139473i	−0.997565	+	0.0697364i	$0.977784\pi$
$168$		0			0
$169$	4.41461e8			0.541184
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.77710e8	+	4.49011e8i	−0.868227	+	0.501271i	−0.866759	−	0.498728i	$-0.833801\pi$
$173$	−7.77710e8	+	4.49011e8i	−0.868227	+	0.501271i	−0.00146854	+	0.999999i	$0.500467\pi$
$174$		0			0
$175$	−2.30234e7	+	7.59144e7i	−0.0245481	+	0.0809416i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.60713e8	−	2.65993e8i	−0.448764	−	0.259094i	0.258544	−	0.965999i	$-0.416757\pi$
$179$	−4.60713e8	−	2.65993e8i	−0.448764	−	0.259094i	−0.707308	+	0.706905i	$0.750091\pi$
$180$		0			0
$181$	−9.05192e7			−0.0843386			−0.0421693	−	0.999110i	$-0.513427\pi$
$181$	−9.05192e7			−0.0843386			−0.0421693	+	0.999110i	$0.513427\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.62036e8	+	3.82227e8i	0.565191	+	0.326313i
$186$		0			0
$187$	1.60674e7	+	2.78296e7i	0.0131395	+	0.0227583i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.57013e9	−	9.06517e8i	1.17979	−	0.681150i	0.223821	−	0.974630i	$-0.428147\pi$
$191$	1.57013e9	−	9.06517e8i	1.17979	−	0.681150i	0.955965	+	0.293481i	$0.0948136\pi$
$192$		0			0
$193$	−7.96214e8	+	1.37908e9i	−0.573853	+	0.993942i	0.422312	+	0.906450i	$0.361219\pi$
$193$	−7.96214e8	+	1.37908e9i	−0.573853	+	0.993942i	−0.996165	+	0.0874919i	$0.972115\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.10470e9i		−	0.733466i	−0.930326	−	0.366733i	$-0.880476\pi$
$197$		−	1.10470e9i		−	0.733466i	0.930326	−	0.366733i	$-0.119524\pi$
$198$		0			0
$199$	−4.02682e8	+	6.97466e8i	−0.256774	+	0.444745i	−0.965376	−	0.260863i	$-0.915993\pi$
$199$	−4.02682e8	+	6.97466e8i	−0.256774	+	0.444745i	0.708602	+	0.705608i	$0.249326\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.03490e8	−	3.01629e9i	−0.414261	−	1.77619i
$204$		0			0
$205$	−3.44360e8	−	5.96449e8i	−0.194983	−	0.337721i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	2.24377e8i		−	0.117596i
$210$		0			0
$211$	1.98359e9			1.00074			0.500371	−	0.865811i	$-0.333197\pi$
$211$	1.98359e9			1.00074			0.500371	+	0.865811i	$0.333197\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.33002e9	−	1.34524e9i	1.09045	−	0.629571i
$216$		0			0
$217$	−7.99987e8	+	2.63777e9i	−0.360781	+	1.18959i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.43489e8	+	3.13784e8i	0.227836	+	0.131541i
$222$		0			0
$223$	3.73592e9			1.51070			0.755351	−	0.655321i	$-0.227467\pi$
$223$	3.73592e9			1.51070			0.755351	+	0.655321i	$0.227467\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.02521e8	+	3.47865e8i	0.226918	+	0.131011i	0.609149	−	0.793056i	$-0.291511\pi$
$227$	6.02521e8	+	3.47865e8i	0.226918	+	0.131011i	−0.382232	+	0.924067i	$0.624844\pi$
$228$		0			0
$229$	2.15791e8	+	3.73761e8i	0.0784678	+	0.135910i	0.902589	−	0.430503i	$-0.141664\pi$
$229$	2.15791e8	+	3.73761e8i	0.0784678	+	0.135910i	−0.824121	+	0.566413i	$0.808331\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.32117e9	−	7.62775e8i	0.448264	−	0.258805i	−0.258833	−	0.965922i	$-0.583338\pi$
$233$	1.32117e9	−	7.62775e8i	0.448264	−	0.258805i	0.707097	+	0.707117i	$0.250005\pi$
$234$		0			0
$235$	2.03271e8	−	3.52076e8i	0.0666507	−	0.115442i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2.62327e8i		−	0.0803990i	−0.999192	−	0.0401995i	$-0.987201\pi$
$239$		−	2.62327e8i		−	0.0803990i	0.999192	−	0.0401995i	$-0.0127994\pi$
$240$		0			0
$241$	−2.27912e8	+	3.94755e8i	−0.0675615	+	0.117020i	−0.897827	−	0.440348i	$-0.854855\pi$
$241$	−2.27912e8	+	3.94755e8i	−0.0675615	+	0.117020i	0.830266	+	0.557368i	$0.188189\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.91485e9	+	2.86652e9i	−0.531461	+	0.795592i
$246$		0			0
$247$	−2.19095e9	−	3.79484e9i	−0.588634	−	1.01954i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			5.71873e9i			1.44080i	0.693556	+	0.720402i	$0.256043\pi$
$251$			5.71873e9i			1.44080i	−0.693556	+	0.720402i	$0.743957\pi$
$252$		0			0
$253$	−7.37144e8			−0.179916
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.14707e8	−	2.39431e8i	0.0950623	−	0.0548843i	−0.451715	−	0.892162i	$-0.649188\pi$
$257$	4.14707e8	−	2.39431e8i	0.0950623	−	0.0548843i	0.546778	+	0.837278i	$0.315854\pi$
$258$		0			0
$259$	−2.09835e9	−	2.24012e9i	−0.466315	−	0.497820i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.27146e9	−	1.88878e9i	−0.683784	−	0.394783i	0.117495	−	0.993073i	$-0.462514\pi$
$263$	−3.27146e9	−	1.88878e9i	−0.683784	−	0.394783i	−0.801279	+	0.598290i	$0.795847\pi$
$264$		0			0
$265$	5.57994e9			1.13148
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.07652e9	−	1.19888e9i	−0.396577	−	0.228964i	0.288429	−	0.957501i	$-0.406867\pi$
$269$	−2.07652e9	−	1.19888e9i	−0.396577	−	0.228964i	−0.685006	+	0.728537i	$0.740200\pi$
$270$		0			0
$271$	1.35974e9	+	2.35513e9i	0.252103	+	0.436655i	0.964105	−	0.265523i	$-0.0855445\pi$
$271$	1.35974e9	+	2.35513e9i	0.252103	+	0.436655i	−0.712002	+	0.702178i	$0.752211\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.19501e7	−	2.99934e7i	0.00908355	−	0.00524439i
$276$		0			0
$277$	−3.64025e8	+	6.30509e8i	−0.0618318	+	0.107096i	−0.895284	−	0.445495i	$-0.853028\pi$
$277$	−3.64025e8	+	6.30509e8i	−0.0618318	+	0.107096i	0.833452	+	0.552591i	$0.186361\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1.18211e10i			1.89598i	0.318299	+	0.947990i	$0.396888\pi$
$281$			1.18211e10i			1.89598i	−0.318299	+	0.947990i	$0.603112\pi$
$282$		0			0
$283$	3.44075e9	−	5.95956e9i	0.536423	−	0.929113i	−0.462670	−	0.886531i	$-0.653108\pi$
$283$	3.44075e9	−	5.95956e9i	0.536423	−	0.929113i	0.999093	−	0.0425819i	$-0.0135583\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.28099e8	+	2.69304e9i	0.0925764	+	0.396932i
$288$		0			0
$289$	−3.33124e9	−	5.76988e9i	−0.477546	−	0.827134i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	8.35666e9i		−	1.13387i	−0.823764	−	0.566933i	$-0.808130\pi$
$293$		−	8.35666e9i		−	1.13387i	0.823764	−	0.566933i	$-0.191870\pi$
$294$		0			0
$295$	−7.89736e9			−1.04278
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.24672e10	+	7.19791e9i	−1.55985	+	0.900579i
$300$		0			0
$301$	−1.05203e10	+	2.45366e9i	−1.28163	+	0.298915i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.00440e9	−	4.62134e9i	−0.924974	−	0.534034i
$306$		0			0
$307$	−1.15726e10			−1.30280			−0.651400	−	0.758735i	$-0.725818\pi$
$307$	−1.15726e10			−1.30280			−0.651400	+	0.758735i	$0.725818\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.04592e10	+	6.03861e9i	1.11804	+	0.645499i	0.940899	−	0.338687i	$-0.109983\pi$
$311$	1.04592e10	+	6.03861e9i	1.11804	+	0.645499i	0.177138	+	0.984186i	$0.443316\pi$
$312$		0			0
$313$	7.14939e8	+	1.23831e9i	0.0744889	+	0.129019i	0.900864	−	0.434102i	$-0.142934\pi$
$313$	7.14939e8	+	1.23831e9i	0.0744889	+	0.129019i	−0.826375	+	0.563120i	$0.809601\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.17505e10	+	6.78415e9i	−1.16364	+	0.671828i	−0.952174	−	0.305557i	$-0.901157\pi$
$317$	−1.17505e10	+	6.78415e9i	−1.16364	+	0.671828i	−0.211467	+	0.977385i	$0.567824\pi$
$318$		0			0
$319$	−1.17103e9	+	2.02829e9i	−0.113085	+	0.195870i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.18737e9i		−	0.200961i
$324$		0			0
$325$	5.85747e8	−	1.01454e9i	0.0525021	−	0.0909362i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.19132e9	+	1.11592e9i	−0.101682	+	0.0952467i
$330$		0			0
$331$	−2.46041e9	−	4.26156e9i	−0.204972	−	0.355023i	0.745151	−	0.666895i	$-0.232377\pi$
$331$	−2.46041e9	−	4.26156e9i	−0.204972	−	0.355023i	−0.950124	+	0.311872i	$0.899044\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			1.01927e10i			0.809301i
$336$		0			0
$337$	−6.63479e9			−0.514408			−0.257204	−	0.966357i	$-0.582801\pi$
$337$	−6.63479e9			−0.514408			−0.257204	+	0.966357i	$0.582801\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.80509e9	−	1.04217e9i	0.133500	−	0.0770763i
$342$		0			0
$343$	1.06979e10	−	8.78273e9i	0.772897	−	0.634531i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.51787e9	−	2.03104e9i	−0.242639	−	0.140088i	0.373750	−	0.927530i	$-0.378072\pi$
$347$	−3.51787e9	−	2.03104e9i	−0.242639	−	0.140088i	−0.616389	+	0.787442i	$0.711405\pi$
$348$		0			0
$349$	−4.59026e9			−0.309411			−0.154705	−	0.987961i	$-0.549443\pi$
$349$	−4.59026e9			−0.309411			−0.154705	+	0.987961i	$0.549443\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.26530e9	−	3.03992e9i	−0.339097	−	0.195778i	0.320775	−	0.947155i	$-0.396057\pi$
$353$	−5.26530e9	−	3.03992e9i	−0.339097	−	0.195778i	−0.659873	+	0.751377i	$0.729390\pi$
$354$		0			0
$355$	−6.92029e9	−	1.19863e10i	−0.435723	−	0.754695i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.76858e9	−	2.17579e9i	0.226882	−	0.130990i	−0.382251	−	0.924059i	$-0.624851\pi$
$359$	3.76858e9	−	2.17579e9i	0.226882	−	0.130990i	0.609133	+	0.793068i	$0.291518\pi$
$360$		0			0
$361$	8.55270e8	−	1.48137e9i	0.0503587	−	0.0872238i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1.24749e10i			0.702857i
$366$		0			0
$367$	−9.24771e9	+	1.60175e10i	−0.509765	+	0.882939i	0.490171	+	0.871626i	$0.336934\pi$
$367$	−9.24771e9	+	1.60175e10i	−0.509765	+	0.882939i	−0.999936	+	0.0113124i	$0.996399\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.14400e10	−	6.50236e9i	−1.13169	−	0.343222i
$372$		0			0
$373$	−3.96707e9	−	6.87116e9i	−0.204943	−	0.354973i	0.745171	−	0.666873i	$-0.232368\pi$
$373$	−3.96707e9	−	6.87116e9i	−0.204943	−	0.354973i	−0.950115	+	0.311901i	$0.899034\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			4.57387e10i			2.26422i
$378$		0			0
$379$	1.63069e10			0.790342			0.395171	−	0.918608i	$-0.370685\pi$
$379$	1.63069e10			0.790342			0.395171	+	0.918608i	$0.370685\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.96738e10	+	1.13587e10i	−0.914309	+	0.527876i	−0.881815	−	0.471595i	$-0.843678\pi$
$383$	−1.96738e10	+	1.13587e10i	−0.914309	+	0.527876i	−0.0324938	+	0.999472i	$0.510345\pi$
$384$		0			0
$385$	2.53861e9	−	5.92081e8i	0.115546	−	0.0269487i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.88603e10	+	2.24360e10i	1.69710	+	0.979821i	0.948489	+	0.316810i	$0.102612\pi$
$389$	3.88603e10	+	2.24360e10i	1.69710	+	0.979821i	0.748610	+	0.663010i	$0.230721\pi$
$390$		0			0
$391$	−7.18614e9			−0.307460
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.87846e10	−	1.66188e10i	−1.18242	−	0.682670i
$396$		0			0
$397$	−2.24361e10	−	3.88605e10i	−0.903204	−	1.56439i	−0.823311	−	0.567591i	$-0.807875\pi$
$397$	−2.24361e10	−	3.88605e10i	−0.903204	−	1.56439i	−0.0798931	−	0.996803i	$-0.525458\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.60799e10	+	2.08307e10i	−1.39536	+	0.805614i	−0.993903	−	0.110261i	$-0.964831\pi$
$401$	−3.60799e10	+	2.08307e10i	−1.39536	+	0.805614i	−0.401462	+	0.915876i	$0.631498\pi$
$402$		0			0
$403$	2.03527e10	−	3.52519e10i	0.771618	−	1.33648i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.32102e9i			0.0845864i
$408$		0			0
$409$	−1.72902e8	+	2.99476e8i	−0.00617885	+	0.0107021i	−0.869098	−	0.494639i	$-0.835300\pi$
$409$	−1.72902e8	+	2.99476e8i	−0.00617885	+	0.0107021i	0.862919	+	0.505341i	$0.168633\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.03443e10	+	9.20287e9i	1.04298	+	0.316317i
$414$		0			0
$415$	1.22328e10	+	2.11878e10i	0.412413	+	0.714321i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	5.40269e10i		−	1.75289i	−0.481503	−	0.876444i	$-0.659909\pi$
$419$		−	5.40269e10i		−	1.75289i	0.481503	−	0.876444i	$-0.340091\pi$
$420$		0			0
$421$	−1.75550e10			−0.558820			−0.279410	−	0.960172i	$-0.590139\pi$
$421$	−1.75550e10			−0.558820			−0.279410	+	0.960172i	$0.590139\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.06442e8	−	2.92394e8i	0.0155229	−	0.00896217i
$426$		0			0
$427$	2.53703e10	+	2.70844e10i	0.763157	+	0.814718i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.28336e10	−	7.40949e9i	−0.371912	−	0.214723i	0.302382	−	0.953187i	$-0.402218\pi$
$431$	−1.28336e10	−	7.40949e9i	−0.371912	−	0.214723i	−0.674293	+	0.738464i	$0.735552\pi$
$432$		0			0
$433$	−1.02897e10			−0.292720			−0.146360	−	0.989231i	$-0.546756\pi$
$433$	−1.02897e10			−0.292720			−0.146360	+	0.989231i	$0.546756\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.34539e10	+	2.50881e10i	1.19153	+	0.687927i
$438$		0			0
$439$	1.12804e10	+	1.95382e10i	0.303715	+	0.526049i	0.976974	−	0.213357i	$-0.0684397\pi$
$439$	1.12804e10	+	1.95382e10i	0.303715	+	0.526049i	−0.673260	+	0.739406i	$0.735106\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.71575e9	−	5.60939e9i	0.252268	−	0.145647i	−0.368535	−	0.929614i	$-0.620140\pi$
$443$	9.71575e9	−	5.60939e9i	0.252268	−	0.145647i	0.620802	+	0.783967i	$0.286807\pi$
$444$		0			0
$445$	−2.86259e10	+	4.95814e10i	−0.729993	+	1.26438i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	4.44507e10i		−	1.09369i	−0.837235	−	0.546844i	$-0.815829\pi$
$449$		−	4.44507e10i		−	1.09369i	0.837235	−	0.546844i	$-0.184171\pi$
$450$		0			0
$451$	1.04554e9	−	1.81092e9i	0.0252716	−	0.0437718i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.71535e10	−	3.48022e10i	0.866871	−	0.812011i
$456$		0			0
$457$	1.66755e10	+	2.88828e10i	0.382308	+	0.662178i	0.991392	−	0.130929i	$-0.0417958\pi$
$457$	1.66755e10	+	2.88828e10i	0.382308	+	0.662178i	−0.609083	+	0.793106i	$0.708463\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	6.58638e10i		−	1.45829i	−0.684361	−	0.729144i	$-0.739919\pi$
$461$		−	6.58638e10i		−	1.45829i	0.684361	−	0.729144i	$-0.260081\pi$
$462$		0			0
$463$	−5.17003e9			−0.112504			−0.0562521	−	0.998417i	$-0.517915\pi$
$463$	−5.17003e9			−0.112504			−0.0562521	+	0.998417i	$0.517915\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.75868e10	−	2.17008e10i	0.790256	−	0.456255i	−0.0497964	−	0.998759i	$-0.515857\pi$
$467$	3.75868e10	−	2.17008e10i	0.790256	−	0.456255i	0.840053	+	0.542505i	$0.182524\pi$
$468$		0			0
$469$	1.18777e10	−	3.91638e10i	0.245493	−	0.809456i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.07434e9	+	4.08437e9i	0.141332	+	0.0815982i
$474$		0			0
$475$	−4.08321e9			−0.0802098
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.31609e10	−	1.33719e10i	−0.439959	−	0.254011i	0.263621	−	0.964626i	$-0.415083\pi$
$479$	−2.31609e10	−	1.33719e10i	−0.439959	−	0.254011i	−0.703580	+	0.710616i	$0.748416\pi$
$480$		0			0
$481$	2.26638e10	+	3.92548e10i	0.423401	+	0.733352i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.96000e10	+	1.70896e10i	−0.534965	+	0.308862i
$486$		0			0
$487$	1.02541e10	−	1.77606e10i	0.182298	−	0.315749i	−0.760365	−	0.649496i	$-0.774980\pi$
$487$	1.02541e10	−	1.77606e10i	0.182298	−	0.315749i	0.942663	+	0.333747i	$0.108313\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			3.23018e9i			0.0555777i	0.999614	+	0.0277888i	$0.00884660\pi$
$491$			3.23018e9i			0.0555777i	−0.999614	+	0.0277888i	$0.991153\pi$
$492$		0			0
$493$	−1.14160e10	+	1.97730e10i	−0.193252	+	0.334723i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.26223e10	+	5.41196e10i	0.206878	+	0.887012i
$498$		0			0
$499$	2.67911e10	+	4.64035e10i	0.432104	+	0.748426i	0.997054	−	0.0766989i	$-0.0244380\pi$
$499$	2.67911e10	+	4.64035e10i	0.432104	+	0.748426i	−0.564950	+	0.825125i	$0.691105\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	7.69019e10i		−	1.20134i	−0.799498	−	0.600669i	$-0.794901\pi$
$503$		−	7.69019e10i		−	1.20134i	0.799498	−	0.600669i	$-0.205099\pi$
$504$		0			0
$505$	−7.71094e9			−0.118561
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.01771e10	−	2.31963e10i	0.598560	−	0.345579i	−0.169915	−	0.985459i	$-0.554349\pi$
$509$	4.01771e10	−	2.31963e10i	0.598560	−	0.345579i	0.768475	+	0.639880i	$0.221016\pi$
$510$		0			0
$511$	1.45372e10	−	4.79329e10i	0.213205	−	0.702992i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.52926e10	−	1.46027e10i	−0.359554	−	0.207589i
$516$		0			0
$517$	1.23434e9			0.0172771
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.23241e10	+	3.59828e10i	0.845872	+	0.488365i	0.859256	−	0.511546i	$-0.170927\pi$
$521$	6.23241e10	+	3.59828e10i	0.845872	+	0.488365i	−0.0133836	+	0.999910i	$0.504260\pi$
$522$		0			0
$523$	2.22903e10	+	3.86079e10i	0.297926	+	0.516023i	0.975661	−	0.219283i	$-0.0703717\pi$
$523$	2.22903e10	+	3.86079e10i	0.297926	+	0.516023i	−0.677735	+	0.735306i	$0.737038\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.75971e10	−	1.01597e10i	0.228139	−	0.131716i
$528$		0			0
$529$	4.32663e10	−	7.49394e10i	0.552493	−	0.956946i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	4.08370e10i		−	0.505993i
$534$		0			0
$535$	−2.07369e10	+	3.59173e10i	−0.253121	+	0.438418i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.04442e10	−	6.83298e8i	−0.123742	−	0.00809571i
$540$		0			0
$541$	−5.21947e9	−	9.04039e9i	−0.0609309	−	0.105535i	0.833951	−	0.551839i	$-0.186074\pi$
$541$	−5.21947e9	−	9.04039e9i	−0.0609309	−	0.105535i	−0.894882	+	0.446303i	$0.852740\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1.74401e10i			0.197680i
$546$		0			0
$547$	9.16157e10			1.02334			0.511671	−	0.859182i	$-0.329027\pi$
$547$	9.16157e10			1.02334			0.511671	+	0.859182i	$0.329027\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.38063e11	−	7.97105e10i	1.49785	−	0.864787i
$552$		0			0
$553$	9.12340e10	+	9.73979e10i	0.975565	+	1.04148i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.47004e10	+	2.58078e10i	0.464398	+	0.268120i	0.713892	−	0.700256i	$-0.246931\pi$
$557$	4.47004e10	+	2.58078e10i	0.464398	+	0.268120i	−0.249494	+	0.968376i	$0.580264\pi$
$558$		0			0
$559$	1.59529e11			1.63377
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.38401e9	+	3.68581e9i	0.0635419	+	0.0366859i	0.531434	−	0.847099i	$-0.321653\pi$
$563$	6.38401e9	+	3.68581e9i	0.0635419	+	0.0366859i	−0.467892	+	0.883785i	$0.654987\pi$
$564$		0			0
$565$	3.26724e10	+	5.65903e10i	0.320618	+	0.555326i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.48763e10	−	4.90033e10i	0.809724	−	0.467495i	−0.0371359	−	0.999310i	$-0.511823\pi$
$569$	8.48763e10	−	4.90033e10i	0.809724	−	0.467495i	0.846860	+	0.531816i	$0.178490\pi$
$570$		0			0
$571$	6.08919e10	−	1.05468e11i	0.572816	−	0.992147i	−0.423459	−	0.905915i	$-0.639184\pi$
$571$	6.08919e10	−	1.05468e11i	0.572816	−	0.992147i	0.996275	−	0.0862317i	$-0.0274825\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1.34145e10i			0.122717i
$576$		0			0
$577$	−9.81452e10	+	1.69993e11i	−0.885454	+	1.53365i	−0.0402615	+	0.999189i	$0.512819\pi$
$577$	−9.81452e10	+	1.69993e11i	−0.885454	+	1.53365i	−0.845192	+	0.534462i	$0.820514\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.23121e10	−	9.56655e10i	−0.195810	−	0.839558i
$582$		0			0
$583$	8.47083e9	+	1.46719e10i	0.0733250	+	0.127003i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.26503e11i		−	1.06549i	−0.846277	−	0.532743i	$-0.821161\pi$
$587$		−	1.26503e11i		−	1.06549i	0.846277	−	0.532743i	$-0.178839\pi$
$588$		0			0
$589$	−1.41878e11			−1.17883
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.65960e10	+	2.11287e10i	−0.295947	+	0.170865i	−0.640621	−	0.767857i	$-0.721323\pi$
$593$	−3.65960e10	+	2.11287e10i	−0.295947	+	0.170865i	0.344674	+	0.938723i	$0.387990\pi$
$594$		0			0
$595$	2.47480e10	−	5.77197e9i	0.197457	−	0.0460528i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.48720e11	−	8.58634e10i	−1.15521	−	0.666962i	−0.205060	−	0.978749i	$-0.565739\pi$
$599$	−1.48720e11	−	8.58634e10i	−1.15521	−	0.666962i	−0.950152	+	0.311788i	$0.899072\pi$
$600$		0			0
$601$	−3.98855e10			−0.305715			−0.152857	−	0.988248i	$-0.548848\pi$
$601$	−3.98855e10			−0.305715			−0.152857	+	0.988248i	$0.548848\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.09303e11	+	6.31060e10i	0.815849	+	0.471031i
$606$		0			0
$607$	−1.14485e11	−	1.98293e11i	−0.843320	−	1.46067i	−0.887072	−	0.461631i	$-0.847264\pi$
$607$	−1.14485e11	−	1.98293e11i	−0.843320	−	1.46067i	0.0437514	−	0.999042i	$-0.486069\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.08760e10	−	1.20528e10i	0.149790	−	0.0864814i
$612$		0			0
$613$	−3.39859e10	+	5.88654e10i	−0.240690	+	0.416887i	−0.960911	−	0.276858i	$-0.910707\pi$
$613$	−3.39859e10	+	5.88654e10i	−0.240690	+	0.416887i	0.720221	+	0.693744i	$0.244040\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2.24449e11i			1.54874i	0.632735	+	0.774368i	$0.281932\pi$
$617$			2.24449e11i			1.54874i	−0.632735	+	0.774368i	$0.718068\pi$
$618$		0			0
$619$	−1.15809e10	+	2.00586e10i	−0.0788821	+	0.136628i	−0.902768	−	0.430128i	$-0.858468\pi$
$619$	−1.15809e10	+	2.00586e10i	−0.0788821	+	0.136628i	0.823886	+	0.566756i	$0.191802\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.67768e11	−	1.57150e11i	1.11367	−	1.04319i
$624$		0			0
$625$	6.92950e10	+	1.20022e11i	0.454132	+	0.786579i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			2.26267e10i			0.144550i
$630$		0			0
$631$	1.69026e11			1.06619			0.533097	−	0.846054i	$-0.321028\pi$
$631$	1.69026e11			1.06619			0.533097	+	0.846054i	$0.321028\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.30771e11	+	7.55007e10i	−0.804298	+	0.464361i
$636$		0			0
$637$	−1.83312e11	+	9.04265e10i	−1.11335	+	0.549209i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.06968e9	+	1.19493e9i	0.0122595	+	0.00707800i	0.506117	−	0.862465i	$-0.331080\pi$
$641$	2.06968e9	+	1.19493e9i	0.0122595	+	0.00707800i	−0.493858	+	0.869543i	$0.664414\pi$
$642$		0			0
$643$	1.39269e10			0.0814723			0.0407361	−	0.999170i	$-0.487030\pi$
$643$	1.39269e10			0.0814723			0.0407361	+	0.999170i	$0.487030\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.77460e10	−	2.75662e10i	−0.272471	−	0.157311i	0.357539	−	0.933898i	$-0.383616\pi$
$647$	−4.77460e10	−	2.75662e10i	−0.272471	−	0.157311i	−0.630010	+	0.776587i	$0.716949\pi$
$648$		0			0
$649$	−1.19889e10	−	2.07654e10i	−0.0675772	−	0.117047i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.01074e11	+	1.16090e11i	−1.10587	+	0.638472i	−0.937755	−	0.347296i	$-0.887100\pi$
$653$	−2.01074e11	+	1.16090e11i	−1.10587	+	0.638472i	−0.168110	+	0.985768i	$0.553766\pi$
$654$		0			0
$655$	−1.28026e11	+	2.21748e11i	−0.695559	+	1.20474i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	2.27443e11i		−	1.20595i	−0.797759	−	0.602977i	$-0.793981\pi$
$659$		−	2.27443e11i		−	1.20595i	0.797759	−	0.602977i	$-0.206019\pi$
$660$		0			0
$661$	1.52803e11	−	2.64662e11i	0.800434	−	1.38639i	−0.118897	−	0.992907i	$-0.537936\pi$
$661$	1.52803e11	−	2.64662e11i	0.800434	−	1.38639i	0.919331	−	0.393486i	$-0.128731\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.69800e11	−	5.14972e10i	−0.868261	−	0.263328i
$666$		0			0
$667$	−2.61872e11	−	4.53575e11i	−1.32308	−	2.29164i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	2.80624e10i		−	0.138432i
$672$		0			0
$673$	−2.36129e11			−1.15104			−0.575519	−	0.817788i	$-0.695200\pi$
$673$	−2.36129e11			−1.15104			−0.575519	+	0.817788i	$0.695200\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.65227e11	+	9.53939e10i	−0.786551	+	0.454115i	−0.838747	−	0.544522i	$-0.816711\pi$
$677$	−1.65227e11	+	9.53939e10i	−0.786551	+	0.454115i	0.0521961	+	0.998637i	$0.483378\pi$
$678$		0			0
$679$	1.33648e11	−	3.11707e10i	0.628757	−	0.146645i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.09400e11	+	6.31623e10i	0.502731	+	0.290252i	0.729841	−	0.683617i	$-0.239594\pi$
$683$	1.09400e11	+	6.31623e10i	0.502731	+	0.290252i	−0.227109	+	0.973869i	$0.572928\pi$
$684$		0			0
$685$	−1.89742e11			−0.861790
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.86531e11	+	1.65428e11i	1.27143	+	0.734063i
$690$		0			0
$691$	1.77438e11	+	3.07332e11i	0.778278	+	1.34802i	0.932934	+	0.360048i	$0.117240\pi$
$691$	1.77438e11	+	3.07332e11i	0.778278	+	1.34802i	−0.154656	+	0.987968i	$0.549427\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.42325e11	+	1.97642e11i	−1.46724	+	0.847109i
$696$		0			0
$697$	1.01925e10	−	1.76540e10i	0.0431869	−	0.0748018i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			2.68942e11i			1.11375i	0.830597	+	0.556874i	$0.187999\pi$
$701$			2.68942e11i			1.11375i	−0.830597	+	0.556874i	$0.812001\pi$
$702$		0			0
$703$	7.89940e10	−	1.36822e11i	0.323424	−	0.560188i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.96280e10	+	8.98563e9i	0.118584	+	0.0359642i
$708$		0			0
$709$	−1.85843e11	−	3.21890e11i	−0.735466	−	1.27386i	−0.954519	−	0.298151i	$-0.903630\pi$
$709$	−1.85843e11	−	3.21890e11i	−0.735466	−	1.27386i	0.219053	−	0.975713i	$-0.429703\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			4.66109e11i			1.80356i
$714$		0			0
$715$	−3.84952e10			−0.147293
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.11727e11	+	2.37710e11i	−1.54061	+	0.889473i	−0.541812	+	0.840499i	$0.682262\pi$
$719$	−4.11727e11	+	2.37710e11i	−1.54061	+	0.889473i	−0.998800	+	0.0489736i	$0.984405\pi$
$720$		0			0
$721$	8.01659e10	+	8.55821e10i	0.296653	+	0.316695i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.69107e10	+	2.13104e10i	0.133598	+	0.0771330i
$726$		0			0
$727$	5.14164e11			1.84062			0.920309	−	0.391192i	$-0.127937\pi$
$727$	5.14164e11			1.84062			0.920309	+	0.391192i	$0.127937\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.89651e10	+	3.98170e10i	0.241524	+	0.139444i
$732$		0			0
$733$	−2.15362e11	−	3.73017e11i	−0.746024	−	1.29215i	−0.949715	−	0.313116i	$-0.898627\pi$
$733$	−2.15362e11	−	3.73017e11i	−0.746024	−	1.29215i	0.203691	−	0.979035i	$-0.434706\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.68007e10	+	1.54734e10i	−0.0908400	+	0.0524465i
$738$		0			0
$739$	−1.90588e10	+	3.30108e10i	−0.0639025	+	0.110682i	−0.896207	−	0.443637i	$-0.853688\pi$
$739$	−1.90588e10	+	3.30108e10i	−0.0639025	+	0.110682i	0.832304	+	0.554319i	$0.187021\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	5.19643e10i		−	0.170510i	−0.996359	−	0.0852550i	$-0.972830\pi$
$743$		−	5.19643e10i		−	0.170510i	0.996359	−	0.0852550i	$-0.0271705\pi$
$744$		0			0
$745$	−2.85163e11	+	4.93917e11i	−0.925696	+	1.60335i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.21533e11	−	1.13841e11i	0.386159	−	0.361720i
$750$		0			0
$751$	−2.02878e11	−	3.51395e11i	−0.637787	−	1.10468i	−0.985917	−	0.167233i	$-0.946517\pi$
$751$	−2.02878e11	−	3.51395e11i	−0.637787	−	1.10468i	0.348131	−	0.937446i	$-0.386817\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	3.61457e11i		−	1.11242i
$756$		0			0
$757$	−6.05947e10			−0.184523			−0.0922616	−	0.995735i	$-0.529410\pi$
$757$	−6.05947e10			−0.184523			−0.0922616	+	0.995735i	$0.529410\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.30256e9	−	3.63879e9i	0.0187922	−	0.0108497i	−0.490574	−	0.871399i	$-0.663213\pi$
$761$	6.30256e9	−	3.63879e9i	0.0187922	−	0.0108497i	0.509367	+	0.860550i	$0.329880\pi$
$762$		0			0
$763$	2.03231e10	−	6.70106e10i	0.0599641	−	0.197718i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.05531e11	−	2.34133e11i	−1.17177	−	0.676521i
$768$		0			0
$769$	3.64380e11			1.04196			0.520978	−	0.853570i	$-0.325568\pi$
$769$	3.64380e11			1.04196			0.520978	+	0.853570i	$0.325568\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.13050e11	+	2.38475e11i	1.15687	+	0.667920i	0.950552	−	0.310565i	$-0.100518\pi$
$773$	4.13050e11	+	2.38475e11i	1.15687	+	0.667920i	0.206319	+	0.978485i	$0.433852\pi$
$774$		0			0
$775$	−1.89653e10	−	3.28489e10i	−0.0525719	−	0.0910573i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.23267e11	+	7.11681e10i	−0.334731	+	0.193257i
$780$		0			0
$781$	2.10112e10	−	3.63925e10i	0.0564738	−	0.0978155i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			6.66065e11i			1.75403i
$786$		0			0
$787$	−1.68255e11	+	2.91425e11i	−0.438599	+	0.759676i	−0.997582	−	0.0695034i	$-0.977859\pi$
$787$	−1.68255e11	+	2.91425e11i	−0.438599	+	0.759676i	0.558983	+	0.829179i	$0.311192\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5.95932e10	−	2.55512e11i	−0.152227	−	0.652688i
$792$		0			0
$793$	−2.74018e11	−	4.74613e11i	−0.692926	−	1.20018i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			3.92820e11i			0.973554i	0.873526	+	0.486777i	$0.161828\pi$
$797$			3.92820e11i			0.973554i	−0.873526	+	0.486777i	$0.838172\pi$
$798$		0			0
$799$	1.20331e10			0.0295250
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.28017e10	+	1.89381e10i	−0.0788922	+	0.0455484i
$804$		0			0
$805$	−1.69183e11	+	5.57841e11i	−0.402878	+	1.32839i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.36083e11	−	1.94038e11i	−0.784609	−	0.452994i	0.0534525	−	0.998570i	$-0.482977\pi$
$809$	−3.36083e11	−	1.94038e11i	−0.784609	−	0.452994i	−0.838061	+	0.545576i	$0.816311\pi$
$810$		0			0
$811$	−2.05867e11			−0.475886			−0.237943	−	0.971279i	$-0.576473\pi$
$811$	−2.05867e11			−0.475886			−0.237943	+	0.971279i	$0.576473\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.87499e11	−	1.65988e11i	−0.651637	−	0.376223i
$816$		0			0
$817$	−2.78017e11	−	4.81539e11i	−0.623998	−	1.08080i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.27601e11	+	1.89140e11i	−0.721061	+	0.416305i	−0.815143	−	0.579260i	$-0.803342\pi$
$821$	−3.27601e11	+	1.89140e11i	−0.721061	+	0.416305i	0.0940820	+	0.995564i	$0.470008\pi$
$822$		0			0
$823$	8.94336e10	−	1.54904e11i	0.194940	−	0.337646i	−0.751941	−	0.659231i	$-0.770882\pi$
$823$	8.94336e10	−	1.54904e11i	0.194940	−	0.337646i	0.946881	+	0.321584i	$0.104215\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			6.35168e11i			1.35790i	0.734186	+	0.678948i	$0.237564\pi$
$827$			6.35168e11i			1.35790i	−0.734186	+	0.678948i	$0.762436\pi$
$828$		0			0
$829$	7.61479e10	−	1.31892e11i	0.161228	−	0.279255i	−0.774082	−	0.633086i	$-0.781788\pi$
$829$	7.61479e10	−	1.31892e11i	0.161228	−	0.279255i	0.935309	+	0.353831i	$0.115121\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.01816e11	−	6.66121e9i	−0.211464	−	0.0138348i
$834$		0			0
$835$	−3.24351e10	−	5.61792e10i	−0.0667220	−	0.115566i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	4.77973e11i		−	0.964618i	−0.876001	−	0.482309i	$-0.839798\pi$
$839$		−	4.77973e11i		−	0.964618i	0.876001	−	0.482309i	$-0.160202\pi$
$840$		0			0
$841$	−1.16380e12			−2.32646
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.28619e11	+	1.31993e11i	−0.448420	+	0.258896i
$846$		0			0
$847$	−3.46440e11	−	3.69846e11i	−0.673123	−	0.718600i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.49498e11	−	2.59518e11i	−0.857057	−	0.494822i
$852$		0			0
$853$	−9.21672e10			−0.174093			−0.0870463	−	0.996204i	$-0.527743\pi$
$853$	−9.21672e10			−0.174093			−0.0870463	+	0.996204i	$0.527743\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.00433e11	+	3.46660e11i	1.11312	+	0.642659i	0.939635	−	0.342178i	$-0.111165\pi$
$857$	6.00433e11	+	3.46660e11i	1.11312	+	0.642659i	0.173483	+	0.984837i	$0.444498\pi$
$858$		0			0
$859$	2.47173e11	+	4.28116e11i	0.453971	+	0.786300i	0.998628	−	0.0523580i	$-0.0166737\pi$
$859$	2.47173e11	+	4.28116e11i	0.453971	+	0.786300i	−0.544658	+	0.838659i	$0.683340\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.38014e11	+	7.96827e10i	−0.248818	+	0.143655i	−0.619223	−	0.785215i	$-0.712552\pi$
$863$	−1.38014e11	+	7.96827e10i	−0.248818	+	0.143655i	0.370405	+	0.928870i	$0.379219\pi$
$864$		0			0
$865$	2.68502e11	−	4.65058e11i	0.479604	−	0.830698i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1.00915e11i		−	0.176961i
$870$		0			0
$871$	−3.02183e11	+	5.23397e11i	−0.525047	+	0.909407i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.38161e11	−	5.92383e11i	−0.235697	−	1.01058i
$876$		0			0
$877$	5.49820e11	+	9.52316e11i	0.929441	+	1.60984i	0.784258	+	0.620435i	$0.213044\pi$
$877$	5.49820e11	+	9.52316e11i	0.929441	+	1.60984i	0.145183	+	0.989405i	$0.453623\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	6.13784e11i		−	1.01885i	−0.860514	−	0.509427i	$-0.829857\pi$
$881$		−	6.13784e11i		−	1.01885i	0.860514	−	0.509427i	$-0.170143\pi$
$882$		0			0
$883$	9.78359e11			1.60937			0.804684	−	0.593704i	$-0.202335\pi$
$883$	9.78359e11			1.60937			0.804684	+	0.593704i	$0.202335\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.31068e11	−	3.64347e11i	1.01949	−	0.588601i	0.105532	−	0.994416i	$-0.466346\pi$
$887$	6.31068e11	−	3.64347e11i	1.01949	−	0.588601i	0.913955	+	0.405815i	$0.133012\pi$
$888$		0			0
$889$	5.90448e11	−	1.37710e11i	0.945311	−	0.220475i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.27629e10	−	4.20097e10i	−0.114421	−	0.0660607i
$894$		0			0
$895$	3.18119e11			0.495789
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.28252e12	+	7.40465e11i	1.96348	+	1.13362i
$900$		0			0
$901$	8.25790e10	+	1.43031e11i	0.125306	+	0.217036i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.68771e10	−	2.70645e10i	0.0698822	−	0.0403465i
$906$		0			0
$907$	1.43764e11	−	2.49006e11i	0.212432	−	0.367943i	−0.740043	−	0.672560i	$-0.765195\pi$
$907$	1.43764e11	−	2.49006e11i	0.212432	−	0.367943i	0.952475	+	0.304616i	$0.0985282\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			7.16133e11i			1.03973i	0.854249	+	0.519864i	$0.174017\pi$
$911$			7.16133e11i			1.03973i	−0.854249	+	0.519864i	$0.825983\pi$
$912$		0			0
$913$	−3.71408e10	+	6.43298e10i	−0.0534526	+	0.0925826i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.50325e11	−	7.02840e11i	1.06114	−	0.993983i
$918$		0			0
$919$	−3.55721e11	−	6.16127e11i	−0.498710	−	0.863791i	0.501289	−	0.865280i	$-0.332859\pi$
$919$	−3.55721e11	−	6.16127e11i	−0.498710	−	0.863791i	−0.999999	+	0.00148915i	$0.999526\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	8.20664e11i		−	1.13073i
$924$		0			0
$925$	4.22377e10			0.0576944
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.17850e11	−	1.25776e11i	0.292479	−	0.168863i	−0.346580	−	0.938020i	$-0.612657\pi$
$929$	2.17850e11	−	1.25776e11i	0.292479	−	0.168863i	0.639059	+	0.769157i	$0.279324\pi$
$930$		0			0
$931$	5.92417e11	+	3.95739e11i	0.788550	+	0.526756i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.66417e10	−	9.60807e9i	−0.0217746	−	0.0125716i
$936$		0			0
$937$	1.36704e12			1.77347			0.886735	−	0.462278i	$-0.152968\pi$
$937$	1.36704e12			1.77347			0.886735	+	0.462278i	$0.152968\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8.16618e11	+	4.71475e11i	1.04150	+	0.601312i	0.920259	−	0.391311i	$-0.127978\pi$
$941$	8.16618e11	+	4.71475e11i	1.04150	+	0.601312i	0.121244	+	0.992623i	$0.461312\pi$
$942$		0			0
$943$	2.33808e11	+	4.04967e11i	0.295673	+	0.512121i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.71459e11	+	3.87667e11i	−0.834872	+	0.482013i	−0.855518	−	0.517773i	$-0.826761\pi$
$947$	−6.71459e11	+	3.87667e11i	−0.834872	+	0.482013i	0.0206461	+	0.999787i	$0.493428\pi$
$948$		0			0
$949$	−3.69845e11	+	6.40590e11i	−0.455990	+	0.789797i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1.55037e12i		−	1.87959i	−0.341732	−	0.939797i	$-0.611014\pi$
$953$		−	1.55037e12i		−	1.87959i	0.341732	−	0.939797i	$-0.388986\pi$
$954$		0			0
$955$	−5.42083e11	+	9.38915e11i	−0.651707	+	1.12879i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.29053e11	+	2.21108e11i	0.861955	+	0.261415i
$960$		0			0
$961$	−2.32536e11	−	4.02764e11i	−0.272644	−	0.472234i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	9.52247e11i		−	1.09810i
$966$		0			0
$967$	−5.12610e11			−0.586248			−0.293124	−	0.956074i	$-0.594695\pi$
$967$	−5.12610e11			−0.586248			−0.293124	+	0.956074i	$0.594695\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.00319e10	+	1.15654e10i	−0.0225344	+	0.0130102i	−0.511225	−	0.859447i	$-0.670808\pi$
$971$	−2.00319e10	+	1.15654e10i	−0.0225344	+	0.0130102i	0.488691	+	0.872457i	$0.337475\pi$
$972$		0			0
$973$	1.54564e12	−	3.60490e11i	1.72448	−	0.402200i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.45252e11	+	1.99331e11i	0.378929	+	0.218775i	0.677352	−	0.735659i	$-0.263127\pi$
$977$	3.45252e11	+	1.99331e11i	0.378929	+	0.218775i	−0.298423	+	0.954434i	$0.596461\pi$
$978$		0			0
$979$	−1.73826e11			−0.189228
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.29743e10	+	5.36787e10i	0.0995747	+	0.0574895i	0.548960	−	0.835848i	$-0.315024\pi$
$983$	9.29743e10	+	5.36787e10i	0.0995747	+	0.0574895i	−0.449386	+	0.893338i	$0.648357\pi$
$984$		0			0
$985$	3.30297e11	+	5.72091e11i	0.350881	+	0.607743i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.58200e12	+	9.13366e11i	−1.65356	+	0.954684i
$990$		0			0
$991$	8.75960e11	−	1.51721e12i	0.908217	−	1.57308i	0.0916773	−	0.995789i	$-0.470777\pi$
$991$	8.75960e11	−	1.51721e12i	0.908217	−	1.57308i	0.816540	−	0.577289i	$-0.195889\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	4.81595e11i		−	0.491349i
$996$		0			0
$997$	−4.08634e11	+	7.07775e11i	−0.413575	+	0.716332i	−0.995278	−	0.0970689i	$-0.969053\pi$
$997$	−4.08634e11	+	7.07775e11i	−0.413575	+	0.716332i	0.581703	+	0.813401i	$0.302387\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bk.a.53.6	✓	44
3.2	odd	2	inner	252.9.bk.a.53.17	yes	44
7.2	even	3	inner	252.9.bk.a.233.17	yes	44
21.2	odd	6	inner	252.9.bk.a.233.6	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bk.a.53.6	✓	44	1.1	even	1	trivial
252.9.bk.a.53.17	yes	44	3.2	odd	2	inner
252.9.bk.a.233.6	yes	44	21.2	odd	6	inner
252.9.bk.a.233.17	yes	44	7.2	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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-517.869 + 298.992i) q^{5} +(2338.25 - 545.349i) q^{7} +O(q^{10})\)	\(q+(-517.869 + 298.992i) q^{5} +(2338.25 - 545.349i) q^{7} +(-1572.34 - 907.792i) q^{11} -35456.9 q^{13} +(-15328.2 - 8849.72i) q^{17} +(61792.0 + 107027. i) q^{19} +(351614. - 203005. i) q^{23} +(-16520.0 + 28613.4i) q^{25} -1.28998e6i q^{29} +(-574013. + 994219. i) q^{31} +(-1.04785e6 + 981537. i) q^{35} +(-639192. - 1.10711e6i) q^{37} +1.15174e6i q^{41} -4.49924e6 q^{43} +(-588772. + 339928. i) q^{47} +(5.16999e6 - 2.55032e6i) q^{49} +(-8.08110e6 - 4.66562e6i) q^{53} +1.08569e6 q^{55} +(1.14373e7 + 6.60332e6i) q^{59} +(7.72820e6 + 1.33856e7i) q^{61} +(1.83620e7 - 1.06013e7i) q^{65} +(8.52255e6 - 1.47615e7i) q^{67} +2.31454e7i q^{71} +(1.04308e7 - 1.80667e7i) q^{73} +(-4.17159e6 - 1.26517e6i) q^{77} +(2.77913e7 + 4.81360e7i) q^{79} -4.09134e7i q^{83} +1.05840e7 q^{85} +(8.29143e7 - 4.78706e7i) q^{89} +(-8.29069e7 + 1.93364e7i) q^{91} +(-6.40004e7 - 3.69507e7i) q^{95} +5.71573e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 1230 q^{7}+O(q^{10}) \) 44 * q + 1230 * q^7	\( 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97}+O(q^{100}) \) 44 * q + 1230 * q^7 - 101380 * q^13 + 62770 * q^19 + 2247666 * q^25 + 1389254 * q^31 - 2136026 * q^37 + 11510140 * q^43 - 3824398 * q^49 - 42646528 * q^55 + 27346232 * q^61 + 14239194 * q^67 - 64344138 * q^73 + 7061786 * q^79 - 54198208 * q^85 - 45697066 * q^91 + 476543496 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more