LMFDB - Embedded newform 576.7.g.h.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,7,Mod(127,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("576.127");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	576.g (of order $2$, degree $1$, not 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:	$132.511152165$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-15}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 4 $ x^2 - x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$0.500000 - 1.93649i$ of defining polynomial
Character	$\chi$	$=$	576.127
Dual form			576.7.g.h.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.0000 q^{5} -309.839i q^{7} +O(q^{10})$	$q+10.0000 q^{5} -309.839i q^{7} +960.500i q^{11} -1466.00 q^{13} +4766.00 q^{17} +7529.08i q^{19} -10472.5i q^{23} -15525.0 q^{25} +25498.0 q^{29} -41890.2i q^{31} -3098.39i q^{35} -1994.00 q^{37} -29362.0 q^{41} -21533.8i q^{43} -7560.06i q^{47} +21649.0 q^{49} -192854. q^{53} +9605.00i q^{55} +78420.2i q^{59} +10918.0 q^{61} -14660.0 q^{65} -394146. i q^{67} +532241. i q^{71} +288626. q^{73} +297600. q^{77} +310706. i q^{79} +204153. i q^{83} +47660.0 q^{85} -310738. q^{89} +454223. i q^{91} +75290.8i q^{95} -1.45709e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{5}+O(q^{10}) $ 2 * q + 20 * q^5	$ 2 q + 20 q^{5} - 2932 q^{13} + 9532 q^{17} - 31050 q^{25} + 50996 q^{29} - 3988 q^{37} - 58724 q^{41} + 43298 q^{49} - 385708 q^{53} + 21836 q^{61} - 29320 q^{65} + 577252 q^{73} + 595200 q^{77} + 95320 q^{85} - 621476 q^{89} - 2914172 q^{97}+O(q^{100}) $ 2 * q + 20 * q^5 - 2932 * q^13 + 9532 * q^17 - 31050 * q^25 + 50996 * q^29 - 3988 * q^37 - 58724 * q^41 + 43298 * q^49 - 385708 * q^53 + 21836 * q^61 - 29320 * q^65 + 577252 * q^73 + 595200 * q^77 + 95320 * q^85 - 621476 * q^89 - 2914172 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	10.0000	0.0800000	0.0400000	−	0.999200i	$-0.487264\pi$
$5$	10.0000	0.0800000	0.0400000	+	0.999200i	$0.487264\pi$
$6$	0	0
$7$	− 309.839i	− 0.903320i	−0.892190	−	0.451660i	$-0.850832\pi$
$7$	− 309.839i	− 0.903320i	0.892190	−	0.451660i	$-0.149168\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	960.500i	0.721638i	0.932636	+	0.360819i	$0.117503\pi$
$11$	960.500i	0.721638i	−0.932636	+	0.360819i	$0.882497\pi$
$12$	0	0
$13$	−1466.00	−0.667274	−0.333637	−	0.942702i	$-0.608276\pi$
$13$	−1466.00	−0.667274	−0.333637	+	0.942702i	$0.608276\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4766.00	0.970079	0.485040	−	0.874492i	$-0.338805\pi$
$17$	4766.00	0.970079	0.485040	+	0.874492i	$0.338805\pi$
$18$	0	0
$19$	7529.08i	1.09769i	0.835923	+	0.548847i	$0.184933\pi$
$19$	7529.08i	1.09769i	−0.835923	+	0.548847i	$0.815067\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 10472.5i	− 0.860734i	−0.902654	−	0.430367i	$-0.858384\pi$
$23$	− 10472.5i	− 0.860734i	0.902654	−	0.430367i	$-0.141616\pi$
$24$	0	0
$25$	−15525.0	−0.993600
$26$	0	0
$27$	0	0
$28$	0	0
$29$	25498.0	1.04547	0.522736	−	0.852495i	$-0.324911\pi$
$29$	25498.0	1.04547	0.522736	+	0.852495i	$0.324911\pi$
$30$	0	0
$31$	− 41890.2i	− 1.40614i	−0.711123	−	0.703068i	$-0.751813\pi$
$31$	− 41890.2i	− 1.40614i	0.711123	−	0.703068i	$-0.248187\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3098.39i	− 0.0722656i
$36$	0	0
$37$	−1994.00	−0.0393659	−0.0196829	−	0.999806i	$-0.506266\pi$
$37$	−1994.00	−0.0393659	−0.0196829	+	0.999806i	$0.506266\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−29362.0	−0.426024	−0.213012	−	0.977050i	$-0.568327\pi$
$41$	−29362.0	−0.426024	−0.213012	+	0.977050i	$0.568327\pi$
$42$	0	0
$43$	− 21533.8i	− 0.270841i	−0.990788	−	0.135421i	$-0.956761\pi$
$43$	− 21533.8i	− 0.270841i	0.990788	−	0.135421i	$-0.0432386\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7560.06i	− 0.0728168i	−0.999337	−	0.0364084i	$-0.988408\pi$
$47$	− 7560.06i	− 0.0728168i	0.999337	−	0.0364084i	$-0.0115917\pi$
$48$	0	0
$49$	21649.0	0.184013
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−192854.	−1.29539	−0.647696	−	0.761899i	$-0.724267\pi$
$53$	−192854.	−1.29539	−0.647696	+	0.761899i	$0.724267\pi$
$54$	0	0
$55$	9605.00i	0.0577310i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	78420.2i	0.381831i	0.981606	+	0.190916i	$0.0611457\pi$
$59$	78420.2i	0.381831i	−0.981606	+	0.190916i	$0.938854\pi$
$60$	0	0
$61$	10918.0	0.0481009	0.0240505	−	0.999711i	$-0.492344\pi$
$61$	10918.0	0.0481009	0.0240505	+	0.999711i	$0.492344\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14660.0	−0.0533819
$66$	0	0
$67$	− 394146.i	− 1.31049i	−0.755418	−	0.655243i	$-0.772566\pi$
$67$	− 394146.i	− 1.31049i	0.755418	−	0.655243i	$-0.227434\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	532241.i	1.48708i	0.668694	+	0.743538i	$0.266854\pi$
$71$	532241.i	1.48708i	−0.668694	+	0.743538i	$0.733146\pi$
$72$	0	0
$73$	288626.	0.741937	0.370968	−	0.928646i	$-0.379026\pi$
$73$	288626.	0.741937	0.370968	+	0.928646i	$0.379026\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	297600.	0.651870
$78$	0	0
$79$	310706.i	0.630186i	0.949061	+	0.315093i	$0.102036\pi$
$79$	310706.i	0.630186i	−0.949061	+	0.315093i	$0.897964\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	204153.i	0.357043i	0.983936	+	0.178522i	$0.0571314\pi$
$83$	204153.i	0.357043i	−0.983936	+	0.178522i	$0.942869\pi$
$84$	0	0
$85$	47660.0	0.0776064
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−310738.	−0.440783	−0.220391	−	0.975412i	$-0.570733\pi$
$89$	−310738.	−0.440783	−0.220391	+	0.975412i	$0.570733\pi$
$90$	0	0
$91$	454223.i	0.602761i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	75290.8i	0.0878155i
$96$	0	0
$97$	−1.45709e6	−1.59650	−0.798252	−	0.602324i	$-0.794242\pi$
$97$	−1.45709e6	−1.59650	−0.798252	+	0.602324i	$0.794242\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−639158.	−0.620360	−0.310180	−	0.950678i	$-0.600389\pi$
$101$	−639158.	−0.620360	−0.310180	+	0.950678i	$0.600389\pi$
$102$	0	0
$103$	− 1.38913e6i	− 1.27125i	−0.771997	−	0.635626i	$-0.780742\pi$
$103$	− 1.38913e6i	− 1.27125i	0.771997	−	0.635626i	$-0.219258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.14935e6i	0.938209i	0.883143	+	0.469105i	$0.155423\pi$
$107$	1.14935e6i	0.938209i	−0.883143	+	0.469105i	$0.844577\pi$
$108$	0	0
$109$	−1.53574e6	−1.18587	−0.592936	−	0.805250i	$-0.702031\pi$
$109$	−1.53574e6	−1.18587	−0.592936	+	0.805250i	$0.702031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	601694.	0.417004	0.208502	−	0.978022i	$-0.433141\pi$
$113$	601694.	0.417004	0.208502	+	0.978022i	$0.433141\pi$
$114$	0	0
$115$	− 104725.i	− 0.0688587i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1.47669e6i	− 0.876292i
$120$	0	0
$121$	849001.	0.479239
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−311500.	−0.159488
$126$	0	0
$127$	− 1.67462e6i	− 0.817531i	−0.912640	−	0.408765i	$-0.865959\pi$
$127$	− 1.67462e6i	− 0.817531i	0.912640	−	0.408765i	$-0.134041\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.84454e6i	− 1.26531i	−0.774433	−	0.632656i	$-0.781965\pi$
$131$	− 2.84454e6i	− 1.26531i	0.774433	−	0.632656i	$-0.218035\pi$
$132$	0	0
$133$	2.33280e6	0.991568
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.81003e6	−1.48172	−0.740862	−	0.671658i	$-0.765583\pi$
$137$	−3.81003e6	−1.48172	−0.740862	+	0.671658i	$0.765583\pi$
$138$	0	0
$139$	138839.i	0.0516971i	0.999666	+	0.0258485i	$0.00822877\pi$
$139$	138839.i	0.0516971i	−0.999666	+	0.0258485i	$0.991771\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.40809e6i	− 0.481530i
$144$	0	0
$145$	254980.	0.0836377
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.27426e6	−0.989816	−0.494908	−	0.868945i	$-0.664798\pi$
$149$	−3.27426e6	−0.989816	−0.494908	+	0.868945i	$0.664798\pi$
$150$	0	0
$151$	5.59352e6i	1.62463i	0.583220	+	0.812314i	$0.301793\pi$
$151$	5.59352e6i	1.62463i	−0.583220	+	0.812314i	$0.698207\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 418902.i	− 0.112491i
$156$	0	0
$157$	−816794.	−0.211064	−0.105532	−	0.994416i	$-0.533655\pi$
$157$	−816794.	−0.211064	−0.105532	+	0.994416i	$0.533655\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.24480e6	−0.777518
$162$	0	0
$163$	− 1.84593e6i	− 0.426237i	−0.977026	−	0.213119i	$-0.931638\pi$
$163$	− 1.84593e6i	− 0.426237i	0.977026	−	0.213119i	$-0.0683621\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.96515e6i	1.71019i	0.518471	+	0.855095i	$0.326501\pi$
$167$	7.96515e6i	1.71019i	−0.518471	+	0.855095i	$0.673499\pi$
$168$	0	0
$169$	−2.67765e6	−0.554746
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.12653e6	−0.990115	−0.495057	−	0.868860i	$-0.664853\pi$
$173$	−5.12653e6	−0.990115	−0.495057	+	0.868860i	$0.664853\pi$
$174$	0	0
$175$	4.81025e6i	0.897538i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.33411e6i	0.406969i	0.979078	+	0.203485i	$0.0652267\pi$
$179$	2.33411e6i	0.406969i	−0.979078	+	0.203485i	$0.934773\pi$
$180$	0	0
$181$	−9.69156e6	−1.63440	−0.817199	−	0.576355i	$-0.804475\pi$
$181$	−9.69156e6	−1.63440	−0.817199	+	0.576355i	$0.804475\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−19940.0	−0.00314927
$186$	0	0
$187$	4.57774e6i	0.700046i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.14164e7i	− 1.63844i	−0.573479	−	0.819220i	$-0.694407\pi$
$191$	− 1.14164e7i	− 1.63844i	0.573479	−	0.819220i	$-0.305593\pi$
$192$	0	0
$193$	−2.43033e6	−0.338060	−0.169030	−	0.985611i	$-0.554064\pi$
$193$	−2.43033e6	−0.338060	−0.169030	+	0.985611i	$0.554064\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.23065e6	−0.291764	−0.145882	−	0.989302i	$-0.546602\pi$
$197$	−2.23065e6	−0.291764	−0.145882	+	0.989302i	$0.546602\pi$
$198$	0	0
$199$	− 4.89576e6i	− 0.621242i	−0.950534	−	0.310621i	$-0.899463\pi$
$199$	− 4.89576e6i	− 0.621242i	0.950534	−	0.310621i	$-0.100537\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 7.90027e6i	− 0.944395i
$204$	0	0
$205$	−293620.	−0.0340819
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.23168e6	−0.792137
$210$	0	0
$211$	3.90951e6i	0.416174i	0.978110	+	0.208087i	$0.0667238\pi$
$211$	3.90951e6i	0.416174i	−0.978110	+	0.208087i	$0.933276\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 215338.i	− 0.0216673i
$216$	0	0
$217$	−1.29792e7	−1.27019
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.98696e6	−0.647308
$222$	0	0
$223$	− 3.33114e6i	− 0.300385i	−0.988657	−	0.150192i	$-0.952011\pi$
$223$	− 3.33114e6i	− 0.300385i	0.988657	−	0.150192i	$-0.0479893\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.35033e7i	1.15442i	0.816597	+	0.577208i	$0.195858\pi$
$227$	1.35033e7i	1.15442i	−0.816597	+	0.577208i	$0.804142\pi$
$228$	0	0
$229$	−1.59598e6	−0.132899	−0.0664493	−	0.997790i	$-0.521167\pi$
$229$	−1.59598e6	−0.132899	−0.0664493	+	0.997790i	$0.521167\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.04383e6	−0.635909	−0.317954	−	0.948106i	$-0.602996\pi$
$233$	−8.04383e6	−0.635909	−0.317954	+	0.948106i	$0.602996\pi$
$234$	0	0
$235$	− 75600.6i	− 0.00582535i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.12532e7i	0.824296i	0.911117	+	0.412148i	$0.135221\pi$
$239$	1.12532e7i	0.824296i	−0.911117	+	0.412148i	$0.864779\pi$
$240$	0	0
$241$	5.05104e6	0.360853	0.180426	−	0.983589i	$-0.442252\pi$
$241$	5.05104e6	0.360853	0.180426	+	0.983589i	$0.442252\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	216490.	0.0147211
$246$	0	0
$247$	− 1.10376e7i	− 0.732462i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 4.71590e6i	− 0.298225i	−0.988820	−	0.149112i	$-0.952358\pi$
$251$	− 4.71590e6i	− 0.298225i	0.988820	−	0.149112i	$-0.0476416\pi$
$252$	0	0
$253$	1.00589e7	0.621138
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.34552e7	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$257$	−2.34552e7	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$258$	0	0
$259$	617818.i	0.0355600i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.16993e7i	− 1.19283i	−0.802676	−	0.596415i	$-0.796591\pi$
$263$	− 2.16993e7i	− 1.19283i	0.802676	−	0.596415i	$-0.203409\pi$
$264$	0	0
$265$	−1.92854e6	−0.103631
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.94278e7	−1.51182	−0.755911	−	0.654674i	$-0.772806\pi$
$269$	−2.94278e7	−1.51182	−0.755911	+	0.654674i	$0.772806\pi$
$270$	0	0
$271$	8.51474e6i	0.427822i	0.976853	+	0.213911i	$0.0686203\pi$
$271$	8.51474e6i	0.427822i	−0.976853	+	0.213911i	$0.931380\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.49118e7i	− 0.717019i
$276$	0	0
$277$	−2.76226e7	−1.29965	−0.649824	−	0.760085i	$-0.725157\pi$
$277$	−2.76226e7	−1.29965	−0.649824	+	0.760085i	$0.725157\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8.64008e6	−0.389403	−0.194701	−	0.980863i	$-0.562374\pi$
$281$	−8.64008e6	−0.389403	−0.194701	+	0.980863i	$0.562374\pi$
$282$	0	0
$283$	− 1.27350e7i	− 0.561873i	−0.959726	−	0.280937i	$-0.909355\pi$
$283$	− 1.27350e7i	− 0.561873i	0.959726	−	0.280937i	$-0.0906450\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.09748e6i	0.384836i
$288$	0	0
$289$	−1.42281e6	−0.0589460
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.45415e7	−1.77077	−0.885385	−	0.464859i	$-0.846105\pi$
$293$	−4.45415e7	−1.77077	−0.885385	+	0.464859i	$0.846105\pi$
$294$	0	0
$295$	784202.i	0.0305465i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.53528e7i	0.574345i
$300$	0	0
$301$	−6.67200e6	−0.244656
$302$	0	0
$303$	0	0
$304$	0	0
$305$	109180.	0.00384808
$306$	0	0
$307$	− 4.89051e7i	− 1.69020i	−0.534606	−	0.845102i	$-0.679540\pi$
$307$	− 4.89051e7i	− 1.69020i	0.534606	−	0.845102i	$-0.320460\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.00220e7i	1.66295i	0.555559	+	0.831477i	$0.312504\pi$
$311$	5.00220e7i	1.66295i	−0.555559	+	0.831477i	$0.687496\pi$
$312$	0	0
$313$	1.12719e6	0.0367589	0.0183795	−	0.999831i	$-0.494149\pi$
$313$	1.12719e6	0.0367589	0.0183795	+	0.999831i	$0.494149\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.44882e7	1.08266	0.541330	−	0.840810i	$-0.317921\pi$
$317$	3.44882e7	1.08266	0.541330	+	0.840810i	$0.317921\pi$
$318$	0	0
$319$	2.44908e7i	0.754452i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.58836e7i	1.06485i
$324$	0	0
$325$	2.27596e7	0.663003
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.34240e6	−0.0657769
$330$	0	0
$331$	4.02696e7i	1.11044i	0.831705	+	0.555218i	$0.187365\pi$
$331$	4.02696e7i	1.11044i	−0.831705	+	0.555218i	$0.812635\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 3.94146e6i	− 0.104839i
$336$	0	0
$337$	−3.42531e7	−0.894973	−0.447487	−	0.894291i	$-0.647681\pi$
$337$	−3.42531e7	−0.894973	−0.447487	+	0.894291i	$0.647681\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.02355e7	1.01472
$342$	0	0
$343$	− 4.31599e7i	− 1.06954i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5.45496e7i	− 1.30558i	−0.757539	−	0.652790i	$-0.773599\pi$
$347$	− 5.45496e7i	− 1.30558i	0.757539	−	0.652790i	$-0.226401\pi$
$348$	0	0
$349$	−4.70009e7	−1.10568	−0.552840	−	0.833287i	$-0.686456\pi$
$349$	−4.70009e7	−1.10568	−0.552840	+	0.833287i	$0.686456\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.27231e7	0.289248	0.144624	−	0.989487i	$-0.453803\pi$
$353$	1.27231e7	0.289248	0.144624	+	0.989487i	$0.453803\pi$
$354$	0	0
$355$	5.32241e6i	0.118966i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 2.02153e7i	− 0.436915i	−0.975846	−	0.218457i	$-0.929898\pi$
$359$	− 2.02153e7i	− 0.436915i	0.975846	−	0.218457i	$-0.0701025\pi$
$360$	0	0
$361$	−9.64116e6	−0.204931
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.88626e6	0.0593549
$366$	0	0
$367$	1.11057e7i	0.224672i	0.993670	+	0.112336i	$0.0358333\pi$
$367$	1.11057e7i	0.224672i	−0.993670	+	0.112336i	$0.964167\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.97536e7i	1.17015i
$372$	0	0
$373$	−687146.	−0.0132411	−0.00662053	−	0.999978i	$-0.502107\pi$
$373$	−687146.	−0.0132411	−0.00662053	+	0.999978i	$0.502107\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.73801e7	−0.697615
$378$	0	0
$379$	1.48499e7i	0.272775i	0.990656	+	0.136388i	$0.0435492\pi$
$379$	1.48499e7i	0.272775i	−0.990656	+	0.136388i	$0.956451\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 3.35885e7i	− 0.597853i	−0.954276	−	0.298926i	$-0.903372\pi$
$383$	− 3.35885e7i	− 0.597853i	0.954276	−	0.298926i	$-0.0966285\pi$
$384$	0	0
$385$	2.97600e6	0.0521496
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.01122e8	−1.71789	−0.858946	−	0.512066i	$-0.828880\pi$
$389$	−1.01122e8	−1.71789	−0.858946	+	0.512066i	$0.828880\pi$
$390$	0	0
$391$	− 4.99122e7i	− 0.834980i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.10706e6i	0.0504149i
$396$	0	0
$397$	3.48266e7	0.556595	0.278297	−	0.960495i	$-0.410230\pi$
$397$	3.48266e7	0.556595	0.278297	+	0.960495i	$0.410230\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.88398e7	1.06760	0.533798	−	0.845612i	$-0.320764\pi$
$401$	6.88398e7	1.06760	0.533798	+	0.845612i	$0.320764\pi$
$402$	0	0
$403$	6.14110e7i	0.938277i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.91524e6i	− 0.0284079i
$408$	0	0
$409$	4.59959e7	0.672278	0.336139	−	0.941812i	$-0.390879\pi$
$409$	4.59959e7	0.672278	0.336139	+	0.941812i	$0.390879\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.42976e7	0.344916
$414$	0	0
$415$	2.04153e6i	0.0285635i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2.71153e7i	− 0.368615i	−0.982869	−	0.184307i	$-0.940996\pi$
$419$	− 2.71153e7i	− 0.368615i	0.982869	−	0.184307i	$-0.0590042\pi$
$420$	0	0
$421$	9.42078e7	1.26253	0.631263	−	0.775569i	$-0.282537\pi$
$421$	9.42078e7	1.26253	0.631263	+	0.775569i	$0.282537\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.39922e7	−0.963871
$426$	0	0
$427$	− 3.38282e6i	− 0.0434505i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 5.19187e7i	− 0.648473i	−0.945976	−	0.324236i	$-0.894893\pi$
$431$	− 5.19187e7i	− 0.648473i	0.945976	−	0.324236i	$-0.105107\pi$
$432$	0	0
$433$	8.40210e7	1.03496	0.517481	−	0.855695i	$-0.326870\pi$
$433$	8.40210e7	1.03496	0.517481	+	0.855695i	$0.326870\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.88486e7	0.944822
$438$	0	0
$439$	1.48115e8i	1.75068i	0.483512	+	0.875338i	$0.339361\pi$
$439$	1.48115e8i	1.75068i	−0.483512	+	0.875338i	$0.660639\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 8.03735e7i	− 0.924489i	−0.886752	−	0.462245i	$-0.847044\pi$
$443$	− 8.03735e7i	− 0.924489i	0.886752	−	0.462245i	$-0.152956\pi$
$444$	0	0
$445$	−3.10738e6	−0.0352626
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.80925e7	0.973196	0.486598	−	0.873626i	$-0.338238\pi$
$449$	8.80925e7	0.973196	0.486598	+	0.873626i	$0.338238\pi$
$450$	0	0
$451$	− 2.82022e7i	− 0.307435i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.54223e6i	0.0482209i
$456$	0	0
$457$	−3.75423e7	−0.393344	−0.196672	−	0.980469i	$-0.563013\pi$
$457$	−3.75423e7	−0.393344	−0.196672	+	0.980469i	$0.563013\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.15260e8	1.17646	0.588228	−	0.808695i	$-0.299826\pi$
$461$	1.15260e8	1.17646	0.588228	+	0.808695i	$0.299826\pi$
$462$	0	0
$463$	1.03415e7i	0.104194i	0.998642	+	0.0520970i	$0.0165905\pi$
$463$	1.03415e7i	0.104194i	−0.998642	+	0.0520970i	$0.983410\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.64223e8i	− 1.61243i	−0.591620	−	0.806217i	$-0.701511\pi$
$467$	− 1.64223e8i	− 1.61243i	0.591620	−	0.806217i	$-0.298489\pi$
$468$	0	0
$469$	−1.22122e8	−1.18379
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.06832e7	0.195449
$474$	0	0
$475$	− 1.16889e8i	− 1.09067i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.76230e7i	0.706291i	0.935568	+	0.353146i	$0.114888\pi$
$479$	7.76230e7i	0.706291i	−0.935568	+	0.353146i	$0.885112\pi$
$480$	0	0
$481$	2.92320e6	0.0262678
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.45709e7	−0.127720
$486$	0	0
$487$	1.08071e7i	0.0935670i	0.998905	+	0.0467835i	$0.0148971\pi$
$487$	1.08071e7i	0.0935670i	−0.998905	+	0.0467835i	$0.985103\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.85067e8i	1.56345i	0.623624	+	0.781724i	$0.285660\pi$
$491$	1.85067e8i	1.56345i	−0.623624	+	0.781724i	$0.714340\pi$
$492$	0	0
$493$	1.21523e8	1.01419
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.64909e8	1.34331
$498$	0	0
$499$	− 6.83704e7i	− 0.550258i	−0.961407	−	0.275129i	$-0.911279\pi$
$499$	− 6.83704e7i	− 0.550258i	0.961407	−	0.275129i	$-0.0887206\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.31562e7i	0.103377i	0.998663	+	0.0516887i	$0.0164604\pi$
$503$	1.31562e7i	0.103377i	−0.998663	+	0.0516887i	$0.983540\pi$
$504$	0	0
$505$	−6.39158e6	−0.0496288
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.34186e8	1.01755	0.508775	−	0.860900i	$-0.330099\pi$
$509$	1.34186e8	1.01755	0.508775	+	0.860900i	$0.330099\pi$
$510$	0	0
$511$	− 8.94275e7i	− 0.670206i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1.38913e7i	− 0.101700i
$516$	0	0
$517$	7.26144e6	0.0525474
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.98565e8	1.40407	0.702036	−	0.712142i	$-0.252275\pi$
$521$	1.98565e8	1.40407	0.702036	+	0.712142i	$0.252275\pi$
$522$	0	0
$523$	2.15512e8i	1.50649i	0.657740	+	0.753245i	$0.271513\pi$
$523$	2.15512e8i	1.50649i	−0.657740	+	0.753245i	$0.728487\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.99649e8i	− 1.36406i
$528$	0	0
$529$	3.83616e7	0.259137
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.30447e7	0.284275
$534$	0	0
$535$	1.14935e7i	0.0750567i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.07939e7i	0.132791i
$540$	0	0
$541$	−1.44188e7	−0.0910623	−0.0455311	−	0.998963i	$-0.514498\pi$
$541$	−1.44188e7	−0.0910623	−0.0455311	+	0.998963i	$0.514498\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.53574e7	−0.0948697
$546$	0	0
$547$	4.24129e7i	0.259141i	0.991570	+	0.129571i	$0.0413599\pi$
$547$	4.24129e7i	0.259141i	−0.991570	+	0.129571i	$0.958640\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.91976e8i	1.14761i
$552$	0	0
$553$	9.62688e7	0.569259
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.90848e7	0.515511	0.257756	−	0.966210i	$-0.417017\pi$
$557$	8.90848e7	0.515511	0.257756	+	0.966210i	$0.417017\pi$
$558$	0	0
$559$	3.15685e7i	0.180725i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.41576e8i	1.35372i	0.736112	+	0.676860i	$0.236660\pi$
$563$	2.41576e8i	1.35372i	−0.736112	+	0.676860i	$0.763340\pi$
$564$	0	0
$565$	6.01694e6	0.0333603
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.56141e7	0.139041	0.0695203	−	0.997581i	$-0.477853\pi$
$569$	2.56141e7	0.139041	0.0695203	+	0.997581i	$0.477853\pi$
$570$	0	0
$571$	1.10781e8i	0.595057i	0.954713	+	0.297528i	$0.0961623\pi$
$571$	1.10781e8i	0.595057i	−0.954713	+	0.297528i	$0.903838\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.62586e8i	0.855225i
$576$	0	0
$577$	1.07272e8	0.558415	0.279208	−	0.960231i	$-0.409928\pi$
$577$	1.07272e8	0.558415	0.279208	+	0.960231i	$0.409928\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.32544e7	0.322524
$582$	0	0
$583$	− 1.85236e8i	− 0.934803i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.16397e7i	− 0.106988i	−0.998568	−	0.0534941i	$-0.982964\pi$
$587$	− 2.16397e7i	− 0.106988i	0.998568	−	0.0534941i	$-0.0170358\pi$
$588$	0	0
$589$	3.15395e8	1.54351
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.00341e8	−0.960738	−0.480369	−	0.877066i	$-0.659497\pi$
$593$	−2.00341e8	−0.960738	−0.480369	+	0.877066i	$0.659497\pi$
$594$	0	0
$595$	− 1.47669e7i	− 0.0701033i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1.37592e8i	− 0.640197i	−0.947384	−	0.320098i	$-0.896284\pi$
$599$	− 1.37592e8i	− 0.640197i	0.947384	−	0.320098i	$-0.103716\pi$
$600$	0	0
$601$	−1.90306e8	−0.876655	−0.438327	−	0.898815i	$-0.644429\pi$
$601$	−1.90306e8	−0.876655	−0.438327	+	0.898815i	$0.644429\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.49001e6	0.0383391
$606$	0	0
$607$	− 1.25461e8i	− 0.560974i	−0.959858	−	0.280487i	$-0.909504\pi$
$607$	− 1.25461e8i	− 0.560974i	0.959858	−	0.280487i	$-0.0904960\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.10831e7i	0.0485888i
$612$	0	0
$613$	9.91111e7	0.430270	0.215135	−	0.976584i	$-0.430981\pi$
$613$	9.91111e7	0.430270	0.215135	+	0.976584i	$0.430981\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.70827e7	−0.157876	−0.0789379	−	0.996880i	$-0.525153\pi$
$617$	−3.70827e7	−0.157876	−0.0789379	+	0.996880i	$0.525153\pi$
$618$	0	0
$619$	− 4.05274e8i	− 1.70874i	−0.519662	−	0.854372i	$-0.673942\pi$
$619$	− 4.05274e8i	− 1.70874i	0.519662	−	0.854372i	$-0.326058\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.62786e7i	0.398168i
$624$	0	0
$625$	2.39463e8	0.980841
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.50340e6	−0.0381880
$630$	0	0
$631$	2.52648e7i	0.100561i	0.998735	+	0.0502803i	$0.0160115\pi$
$631$	2.52648e7i	0.100561i	−0.998735	+	0.0502803i	$0.983989\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.67462e7i	− 0.0654025i
$636$	0	0
$637$	−3.17374e7	−0.122787
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.21293e8	1.59959	0.799797	−	0.600270i	$-0.204940\pi$
$641$	4.21293e8	1.59959	0.799797	+	0.600270i	$0.204940\pi$
$642$	0	0
$643$	8.17706e7i	0.307584i	0.988103	+	0.153792i	$0.0491486\pi$
$643$	8.17706e7i	0.307584i	−0.988103	+	0.153792i	$0.950851\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.84284e8i	− 0.680416i	−0.940350	−	0.340208i	$-0.889503\pi$
$647$	− 1.84284e8i	− 0.680416i	0.940350	−	0.340208i	$-0.110497\pi$
$648$	0	0
$649$	−7.53226e7	−0.275544
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.43842e7	0.231228	0.115614	−	0.993294i	$-0.463117\pi$
$653$	6.43842e7	0.231228	0.115614	+	0.993294i	$0.463117\pi$
$654$	0	0
$655$	− 2.84454e7i	− 0.101225i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 5.38099e8i	− 1.88021i	−0.340889	−	0.940103i	$-0.610728\pi$
$659$	− 5.38099e8i	− 1.88021i	0.340889	−	0.940103i	$-0.389272\pi$
$660$	0	0
$661$	−2.83897e8	−0.983008	−0.491504	−	0.870875i	$-0.663553\pi$
$661$	−2.83897e8	−0.983008	−0.491504	+	0.870875i	$0.663553\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.33280e7	0.0793255
$666$	0	0
$667$	− 2.67029e8i	− 0.899872i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.04867e7i	0.0347115i
$672$	0	0
$673$	2.77693e8	0.911002	0.455501	−	0.890235i	$-0.349460\pi$
$673$	2.77693e8	0.911002	0.455501	+	0.890235i	$0.349460\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.23026e7	−0.297473	−0.148737	−	0.988877i	$-0.547521\pi$
$677$	−9.23026e7	−0.297473	−0.148737	+	0.988877i	$0.547521\pi$
$678$	0	0
$679$	4.51462e8i	1.44215i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.70862e8i	0.850132i	0.905162	+	0.425066i	$0.139749\pi$
$683$	2.70862e8i	0.850132i	−0.905162	+	0.425066i	$0.860251\pi$
$684$	0	0
$685$	−3.81003e7	−0.118538
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.82724e8	0.864380
$690$	0	0
$691$	− 1.92568e7i	− 0.0583645i	−0.999574	−	0.0291823i	$-0.990710\pi$
$691$	− 1.92568e7i	− 0.0583645i	0.999574	−	0.0291823i	$-0.00929032\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.38839e6i	0.00413577i
$696$	0	0
$697$	−1.39939e8	−0.413277
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.52234e8	1.02253	0.511266	−	0.859423i	$-0.329177\pi$
$701$	3.52234e8	1.02253	0.511266	+	0.859423i	$0.329177\pi$
$702$	0	0
$703$	− 1.50130e7i	− 0.0432117i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.98036e8i	0.560384i
$708$	0	0
$709$	−4.62733e8	−1.29835	−0.649175	−	0.760639i	$-0.724886\pi$
$709$	−4.62733e8	−1.29835	−0.649175	+	0.760639i	$0.724886\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.38697e8	−1.21031
$714$	0	0
$715$	− 1.40809e7i	− 0.0385224i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.60385e8i	1.23861i	0.785150	+	0.619305i	$0.212586\pi$
$719$	4.60385e8i	1.23861i	−0.785150	+	0.619305i	$0.787414\pi$
$720$	0	0
$721$	−4.30406e8	−1.14835
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.95856e8	−1.03878
$726$	0	0
$727$	− 4.82173e8i	− 1.25487i	−0.778668	−	0.627437i	$-0.784104\pi$
$727$	− 4.82173e8i	− 1.25487i	0.778668	−	0.627437i	$-0.215896\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1.02630e8i	− 0.262738i
$732$	0	0
$733$	5.08270e8	1.29057	0.645287	−	0.763941i	$-0.276738\pi$
$733$	5.08270e8	1.29057	0.645287	+	0.763941i	$0.276738\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.78577e8	0.945696
$738$	0	0
$739$	1.27767e8i	0.316582i	0.987392	+	0.158291i	$0.0505984\pi$
$739$	1.27767e8i	0.316582i	−0.987392	+	0.158291i	$0.949402\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 2.83312e8i	− 0.690716i	−0.938471	−	0.345358i	$-0.887758\pi$
$743$	− 2.83312e8i	− 0.690716i	0.938471	−	0.345358i	$-0.112242\pi$
$744$	0	0
$745$	−3.27426e7	−0.0791853
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.56112e8	0.847503
$750$	0	0
$751$	2.15309e8i	0.508325i	0.967161	+	0.254163i	$0.0817999\pi$
$751$	2.15309e8i	0.508325i	−0.967161	+	0.254163i	$0.918200\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.59352e7i	0.129970i
$756$	0	0
$757$	3.03985e8	0.700753	0.350377	−	0.936609i	$-0.386054\pi$
$757$	3.03985e8	0.700753	0.350377	+	0.936609i	$0.386054\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8.63611e8	−1.95959	−0.979793	−	0.200014i	$-0.935901\pi$
$761$	−8.63611e8	−1.95959	−0.979793	+	0.200014i	$0.935901\pi$
$762$	0	0
$763$	4.75831e8i	1.07122i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.14964e8i	− 0.254786i
$768$	0	0
$769$	−1.97898e7	−0.0435174	−0.0217587	−	0.999763i	$-0.506927\pi$
$769$	−1.97898e7	−0.0435174	−0.0217587	+	0.999763i	$0.506927\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5.64048e8	−1.22117	−0.610587	−	0.791949i	$-0.709067\pi$
$773$	−5.64048e8	−1.22117	−0.610587	+	0.791949i	$0.709067\pi$
$774$	0	0
$775$	6.50345e8i	1.39714i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.21069e8i	− 0.467644i
$780$	0	0
$781$	−5.11217e8	−1.07313
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8.16794e6	−0.0168851
$786$	0	0
$787$	4.04198e8i	0.829220i	0.909999	+	0.414610i	$0.136082\pi$
$787$	4.04198e8i	0.829220i	−0.909999	+	0.414610i	$0.863918\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1.86428e8i	− 0.376688i
$792$	0	0
$793$	−1.60058e7	−0.0320965
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.15998e7	−0.0426654	−0.0213327	−	0.999772i	$-0.506791\pi$
$797$	−2.15998e7	−0.0426654	−0.0213327	+	0.999772i	$0.506791\pi$
$798$	0	0
$799$	− 3.60313e7i	− 0.0706381i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.77225e8i	0.535410i
$804$	0	0
$805$	−3.24480e7	−0.0622014
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.51310e8	−1.23011	−0.615053	−	0.788486i	$-0.710865\pi$
$809$	−6.51310e8	−1.23011	−0.615053	+	0.788486i	$0.710865\pi$
$810$	0	0
$811$	5.52891e8i	1.03652i	0.855223	+	0.518259i	$0.173420\pi$
$811$	5.52891e8i	1.03652i	−0.855223	+	0.518259i	$0.826580\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1.84593e7i	− 0.0340990i
$816$	0	0
$817$	1.62130e8	0.297301
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.94865e8	−1.61707	−0.808534	−	0.588450i	$-0.799738\pi$
$821$	−8.94865e8	−1.61707	−0.808534	+	0.588450i	$0.799738\pi$
$822$	0	0
$823$	− 8.44482e8i	− 1.51492i	−0.652879	−	0.757462i	$-0.726439\pi$
$823$	− 8.44482e8i	− 1.51492i	0.652879	−	0.757462i	$-0.273561\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.11099e7i	0.0550024i	0.999622	+	0.0275012i	$0.00875501\pi$
$827$	3.11099e7i	0.0550024i	−0.999622	+	0.0275012i	$0.991245\pi$
$828$	0	0
$829$	4.05444e7	0.0711652	0.0355826	−	0.999367i	$-0.488671\pi$
$829$	4.05444e7	0.0711652	0.0355826	+	0.999367i	$0.488671\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.03179e8	0.178508
$834$	0	0
$835$	7.96515e7i	0.136815i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 4.17820e8i	− 0.707463i	−0.935347	−	0.353731i	$-0.884913\pi$
$839$	− 4.17820e8i	− 0.707463i	0.935347	−	0.353731i	$-0.115087\pi$
$840$	0	0
$841$	5.53247e7	0.0930103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.67765e7	−0.0443797
$846$	0	0
$847$	− 2.63053e8i	− 0.432906i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.08823e7i	0.0338835i
$852$	0	0
$853$	4.28994e8	0.691201	0.345601	−	0.938382i	$-0.387675\pi$
$853$	4.28994e8	0.691201	0.345601	+	0.938382i	$0.387675\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.20598e8	−0.350476	−0.175238	−	0.984526i	$-0.556070\pi$
$857$	−2.20598e8	−0.350476	−0.175238	+	0.984526i	$0.556070\pi$
$858$	0	0
$859$	− 1.14768e9i	− 1.81068i	−0.424689	−	0.905339i	$-0.639617\pi$
$859$	− 1.14768e9i	− 1.81068i	0.424689	−	0.905339i	$-0.360383\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.44987e7i	0.100350i	0.998740	+	0.0501752i	$0.0159780\pi$
$863$	6.44987e7i	0.100350i	−0.998740	+	0.0501752i	$0.984022\pi$
$864$	0	0
$865$	−5.12653e7	−0.0792092
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.98433e8	−0.454766
$870$	0	0
$871$	5.77818e8i	0.874453i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.65147e7i	0.144069i
$876$	0	0
$877$	−5.15252e8	−0.763873	−0.381937	−	0.924189i	$-0.624743\pi$
$877$	−5.15252e8	−0.763873	−0.381937	+	0.924189i	$0.624743\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.18519e8	0.173324	0.0866622	−	0.996238i	$-0.472380\pi$
$881$	1.18519e8	0.173324	0.0866622	+	0.996238i	$0.472380\pi$
$882$	0	0
$883$	− 1.28669e8i	− 0.186893i	−0.995624	−	0.0934466i	$-0.970212\pi$
$883$	− 1.28669e8i	− 0.186893i	0.995624	−	0.0934466i	$-0.0297884\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.09948e8i	0.730727i	0.930865	+	0.365363i	$0.119055\pi$
$887$	5.09948e8i	0.730727i	−0.930865	+	0.365363i	$0.880945\pi$
$888$	0	0
$889$	−5.18861e8	−0.738492
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.69203e7	0.0799306
$894$	0	0
$895$	2.33411e7i	0.0325576i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.06812e9i	− 1.47007i
$900$	0	0
$901$	−9.19142e8	−1.25663
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.69156e7	−0.130752
$906$	0	0
$907$	− 5.43212e8i	− 0.728027i	−0.931394	−	0.364014i	$-0.881406\pi$
$907$	− 5.43212e8i	− 0.728027i	0.931394	−	0.364014i	$-0.118594\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 5.99585e8i	− 0.793041i	−0.918026	−	0.396520i	$-0.870218\pi$
$911$	− 5.99585e8i	− 0.793041i	0.918026	−	0.396520i	$-0.129782\pi$
$912$	0	0
$913$	−1.96089e8	−0.257656
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.81347e8	−1.14298
$918$	0	0
$919$	9.66486e8i	1.24523i	0.782529	+	0.622614i	$0.213929\pi$
$919$	9.66486e8i	1.24523i	−0.782529	+	0.622614i	$0.786071\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 7.80265e8i	− 0.992286i
$924$	0	0
$925$	3.09568e7	0.0391139
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7.22626e8	−0.901295	−0.450647	−	0.892702i	$-0.648807\pi$
$929$	−7.22626e8	−0.901295	−0.450647	+	0.892702i	$0.648807\pi$
$930$	0	0
$931$	1.62997e8i	0.201990i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.57774e7i	0.0560037i
$936$	0	0
$937$	−4.12016e8	−0.500836	−0.250418	−	0.968138i	$-0.580568\pi$
$937$	−4.12016e8	−0.500836	−0.250418	+	0.968138i	$0.580568\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7.87666e8	0.945308	0.472654	−	0.881248i	$-0.343296\pi$
$941$	7.87666e8	0.945308	0.472654	+	0.881248i	$0.343296\pi$
$942$	0	0
$943$	3.07495e8i	0.366693i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.29504e8i	0.623475i	0.950168	+	0.311738i	$0.100911\pi$
$947$	5.29504e8i	0.623475i	−0.950168	+	0.311738i	$0.899089\pi$
$948$	0	0
$949$	−4.23126e8	−0.495075
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.18323e9	−1.36706	−0.683532	−	0.729920i	$-0.739557\pi$
$953$	−1.18323e9	−1.36706	−0.683532	+	0.729920i	$0.739557\pi$
$954$	0	0
$955$	− 1.14164e8i	− 0.131075i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.18050e9i	1.33847i
$960$	0	0
$961$	−8.67284e8	−0.977218
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.43033e7	−0.0270448
$966$	0	0
$967$	− 4.78688e8i	− 0.529387i	−0.964333	−	0.264693i	$-0.914729\pi$
$967$	− 4.78688e8i	− 0.529387i	0.964333	−	0.264693i	$-0.0852708\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.02914e8i	0.767793i	0.923376	+	0.383897i	$0.125418\pi$
$971$	7.02914e8i	0.767793i	−0.923376	+	0.383897i	$0.874582\pi$
$972$	0	0
$973$	4.30176e7	0.0466990
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.21323e8	0.987934	0.493967	−	0.869481i	$-0.335546\pi$
$977$	9.21323e8	0.987934	0.493967	+	0.869481i	$0.335546\pi$
$978$	0	0
$979$	− 2.98464e8i	− 0.318085i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.44410e8i	0.257311i	0.991689	+	0.128655i	$0.0410661\pi$
$983$	2.44410e8i	0.257311i	−0.991689	+	0.128655i	$0.958934\pi$
$984$	0	0
$985$	−2.23065e7	−0.0233411
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.25514e8	−0.233122
$990$	0	0
$991$	− 1.78184e9i	− 1.83083i	−0.402516	−	0.915413i	$-0.631864\pi$
$991$	− 1.78184e9i	− 1.83083i	0.402516	−	0.915413i	$-0.368136\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 4.89576e7i	− 0.0496994i
$996$	0	0
$997$	−1.36790e9	−1.38029	−0.690143	−	0.723673i	$-0.742452\pi$
$997$	−1.36790e9	−1.38029	−0.690143	+	0.723673i	$0.742452\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.7.g.h.127.1		2
3.2	odd	2		64.7.c.c.63.1		2
4.3	odd	2	inner	576.7.g.h.127.2		2
8.3	odd	2		36.7.d.c.19.1		2
8.5	even	2		36.7.d.c.19.2		2
12.11	even	2		64.7.c.c.63.2		2
24.5	odd	2		4.7.b.a.3.1	✓	2
24.11	even	2		4.7.b.a.3.2	yes	2
48.5	odd	4		256.7.d.f.127.1		4
48.11	even	4		256.7.d.f.127.3		4
48.29	odd	4		256.7.d.f.127.4		4
48.35	even	4		256.7.d.f.127.2		4
120.29	odd	2		100.7.b.c.51.2		2
120.53	even	4		100.7.d.a.99.1		4
120.59	even	2		100.7.b.c.51.1		2
120.77	even	4		100.7.d.a.99.4		4
120.83	odd	4		100.7.d.a.99.3		4
120.107	odd	4		100.7.d.a.99.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.7.b.a.3.1	✓	2	24.5	odd	2
4.7.b.a.3.2	yes	2	24.11	even	2
36.7.d.c.19.1		2	8.3	odd	2
36.7.d.c.19.2		2	8.5	even	2
64.7.c.c.63.1		2	3.2	odd	2
64.7.c.c.63.2		2	12.11	even	2
100.7.b.c.51.1		2	120.59	even	2
100.7.b.c.51.2		2	120.29	odd	2
100.7.d.a.99.1		4	120.53	even	4
100.7.d.a.99.2		4	120.107	odd	4
100.7.d.a.99.3		4	120.83	odd	4
100.7.d.a.99.4		4	120.77	even	4
256.7.d.f.127.1		4	48.5	odd	4
256.7.d.f.127.2		4	48.35	even	4
256.7.d.f.127.3		4	48.11	even	4
256.7.d.f.127.4		4	48.29	odd	4
576.7.g.h.127.1		2	1.1	even	1	trivial
576.7.g.h.127.2		2	4.3	odd	2	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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.0000 q^{5} -309.839i q^{7} +O(q^{10})\)	\(q+10.0000 q^{5} -309.839i q^{7} +960.500i q^{11} -1466.00 q^{13} +4766.00 q^{17} +7529.08i q^{19} -10472.5i q^{23} -15525.0 q^{25} +25498.0 q^{29} -41890.2i q^{31} -3098.39i q^{35} -1994.00 q^{37} -29362.0 q^{41} -21533.8i q^{43} -7560.06i q^{47} +21649.0 q^{49} -192854. q^{53} +9605.00i q^{55} +78420.2i q^{59} +10918.0 q^{61} -14660.0 q^{65} -394146. i q^{67} +532241. i q^{71} +288626. q^{73} +297600. q^{77} +310706. i q^{79} +204153. i q^{83} +47660.0 q^{85} -310738. q^{89} +454223. i q^{91} +75290.8i q^{95} -1.45709e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{5}+O(q^{10}) \) 2 * q + 20 * q^5	\( 2 q + 20 q^{5} - 2932 q^{13} + 9532 q^{17} - 31050 q^{25} + 50996 q^{29} - 3988 q^{37} - 58724 q^{41} + 43298 q^{49} - 385708 q^{53} + 21836 q^{61} - 29320 q^{65} + 577252 q^{73} + 595200 q^{77} + 95320 q^{85} - 621476 q^{89} - 2914172 q^{97}+O(q^{100}) \) 2 * q + 20 * q^5 - 2932 * q^13 + 9532 * q^17 - 31050 * q^25 + 50996 * q^29 - 3988 * q^37 - 58724 * q^41 + 43298 * q^49 - 385708 * q^53 + 21836 * q^61 - 29320 * q^65 + 577252 * q^73 + 595200 * q^77 + 95320 * q^85 - 621476 * q^89 - 2914172 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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