LMFDB - Embedded newform 576.7.e.b.449.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [576,7,Mod(449,576)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("576.449"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	576.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-968,0,0,0,0,0,-6736] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	576.449
Dual form			576.7.e.b.449.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+173.948i q^{5} -484.000 q^{7} +1340.67i q^{11} -3368.00 q^{13} -12.7279i q^{17} -5744.00 q^{19} +3377.14i q^{23} -14633.0 q^{25} +29354.8i q^{29} -39796.0 q^{31} -84191.0i q^{35} -52526.0 q^{37} +37042.5i q^{41} -3800.00 q^{43} -76791.8i q^{47} +116607. q^{49} +238738. i q^{53} -233208. q^{55} -249841. i q^{59} -13250.0 q^{61} -585858. i q^{65} -168968. q^{67} +531467. i q^{71} +236144. q^{73} -648886. i q^{77} -35116.0 q^{79} -10980.0i q^{83} +2214.00 q^{85} +129328. i q^{89} +1.63011e6 q^{91} -999159. i q^{95} -321424. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 968 q^{7} - 6736 q^{13} - 11488 q^{19} - 29266 q^{25} - 79592 q^{31} - 105052 q^{37} - 7600 q^{43} + 233214 q^{49} - 466416 q^{55} - 26500 q^{61} - 337936 q^{67} + 472288 q^{73} - 70232 q^{79} + 4428 q^{85}+ \cdots - 642848 q^{97}+O(q^{100}) $ 2 * q - 968 * q^7 - 6736 * q^13 - 11488 * q^19 - 29266 * q^25 - 79592 * q^31 - 105052 * q^37 - 7600 * q^43 + 233214 * q^49 - 466416 * q^55 - 26500 * q^61 - 337936 * q^67 + 472288 * q^73 - 70232 * q^79 + 4428 * q^85 + 3260224 * q^91 - 642848 * q^97

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
$5$	173.948i	1.39159i	0.718242	+	0.695793i	$0.244947\pi$
$5$	173.948i	1.39159i	−0.718242	+	0.695793i	$0.755053\pi$
$6$	0	0
$7$	−484.000	−1.41108	−0.705539	−	0.708671i	$-0.749295\pi$
$7$	−484.000	−1.41108	−0.705539	+	0.708671i	$0.749295\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1340.67i	1.00727i	0.863917	+	0.503634i	$0.168004\pi$
$11$	1340.67i	1.00727i	−0.863917	+	0.503634i	$0.831996\pi$
$12$	0	0
$13$	−3368.00	−1.53300	−0.766500	−	0.642245i	$-0.778003\pi$
$13$	−3368.00	−1.53300	−0.766500	+	0.642245i	$0.778003\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 12.7279i	− 0.00259066i	−0.999999	−	0.00129533i	$-0.999588\pi$
$17$	− 12.7279i	− 0.00259066i	0.999999	−	0.00129533i	$-0.000412317\pi$
$18$	0	0
$19$	−5744.00	−0.837440	−0.418720	−	0.908115i	$-0.637521\pi$
$19$	−5744.00	−0.837440	−0.418720	+	0.908115i	$0.637521\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3377.14i	0.277566i	0.990323	+	0.138783i	$0.0443190\pi$
$23$	3377.14i	0.277566i	−0.990323	+	0.138783i	$0.955681\pi$
$24$	0	0
$25$	−14633.0	−0.936512
$26$	0	0
$27$	0	0
$28$	0	0
$29$	29354.8i	1.20361i	0.798643	+	0.601805i	$0.205551\pi$
$29$	29354.8i	1.20361i	−0.798643	+	0.601805i	$0.794449\pi$
$30$	0	0
$31$	−39796.0	−1.33584	−0.667920	−	0.744233i	$-0.732815\pi$
$31$	−39796.0	−1.33584	−0.667920	+	0.744233i	$0.732815\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 84191.0i	− 1.96364i
$36$	0	0
$37$	−52526.0	−1.03698	−0.518489	−	0.855085i	$-0.673505\pi$
$37$	−52526.0	−1.03698	−0.518489	+	0.855085i	$0.673505\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	37042.5i	0.537463i	0.963215	+	0.268732i	$0.0866045\pi$
$41$	37042.5i	0.537463i	−0.963215	+	0.268732i	$0.913396\pi$
$42$	0	0
$43$	−3800.00	−0.0477945	−0.0238973	−	0.999714i	$-0.507607\pi$
$43$	−3800.00	−0.0477945	−0.0238973	+	0.999714i	$0.507607\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 76791.8i	− 0.739641i	−0.929103	−	0.369821i	$-0.879419\pi$
$47$	− 76791.8i	− 0.739641i	0.929103	−	0.369821i	$-0.120581\pi$
$48$	0	0
$49$	116607.	0.991143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	238738.i	1.60359i	0.597599	+	0.801795i	$0.296121\pi$
$53$	238738.i	1.60359i	−0.597599	+	0.801795i	$0.703879\pi$
$54$	0	0
$55$	−233208.	−1.40170
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 249841.i	− 1.21649i	−0.793751	−	0.608243i	$-0.791875\pi$
$59$	− 249841.i	− 1.21649i	0.793751	−	0.608243i	$-0.208125\pi$
$60$	0	0
$61$	−13250.0	−0.0583749	−0.0291875	−	0.999574i	$-0.509292\pi$
$61$	−13250.0	−0.0583749	−0.0291875	+	0.999574i	$0.509292\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 585858.i	− 2.13330i
$66$	0	0
$67$	−168968.	−0.561798	−0.280899	−	0.959737i	$-0.590633\pi$
$67$	−168968.	−0.561798	−0.280899	+	0.959737i	$0.590633\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	531467.i	1.48491i	0.669894	+	0.742457i	$0.266340\pi$
$71$	531467.i	1.48491i	−0.669894	+	0.742457i	$0.733660\pi$
$72$	0	0
$73$	236144.	0.607027	0.303514	−	0.952827i	$-0.401840\pi$
$73$	236144.	0.607027	0.303514	+	0.952827i	$0.401840\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 648886.i	− 1.42134i
$78$	0	0
$79$	−35116.0	−0.0712236	−0.0356118	−	0.999366i	$-0.511338\pi$
$79$	−35116.0	−0.0712236	−0.0356118	+	0.999366i	$0.511338\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 10980.0i	− 0.0192029i	−0.999954	−	0.00960144i	$-0.996944\pi$
$83$	− 10980.0i	− 0.0192029i	0.999954	−	0.00960144i	$-0.00305628\pi$
$84$	0	0
$85$	2214.00	0.00360513
$86$	0	0
$87$	0	0
$88$	0	0
$89$	129328.i	0.183453i	0.995784	+	0.0917263i	$0.0292385\pi$
$89$	129328.i	0.183453i	−0.995784	+	0.0917263i	$0.970762\pi$
$90$	0	0
$91$	1.63011e6	2.16318
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 999159.i	− 1.16537i
$96$	0	0
$97$	−321424.	−0.352179	−0.176089	−	0.984374i	$-0.556345\pi$
$97$	−321424.	−0.352179	−0.176089	+	0.984374i	$0.556345\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	668780.i	0.649111i	0.945867	+	0.324556i	$0.105215\pi$
$101$	668780.i	0.649111i	−0.945867	+	0.324556i	$0.894785\pi$
$102$	0	0
$103$	1.99341e6	1.82425	0.912127	−	0.409907i	$-0.134439\pi$
$103$	1.99341e6	1.82425	0.912127	+	0.409907i	$0.134439\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 260668.i	− 0.212783i	−0.994324	−	0.106391i	$-0.966070\pi$
$107$	− 260668.i	− 0.212783i	0.994324	−	0.106391i	$-0.0339296\pi$
$108$	0	0
$109$	−194456.	−0.150156	−0.0750779	−	0.997178i	$-0.523921\pi$
$109$	−194456.	−0.150156	−0.0750779	+	0.997178i	$0.523921\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	821897.i	0.569616i	0.958585	+	0.284808i	$0.0919298\pi$
$113$	821897.i	0.569616i	−0.958585	+	0.284808i	$0.908070\pi$
$114$	0	0
$115$	−587448.	−0.386257
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6160.31i	0.00365563i
$120$	0	0
$121$	−25847.0	−0.0145900
$122$	0	0
$123$	0	0
$124$	0	0
$125$	172557.i	0.0883490i
$126$	0	0
$127$	3.05721e6	1.49250	0.746250	−	0.665666i	$-0.231852\pi$
$127$	3.05721e6	1.49250	0.746250	+	0.665666i	$0.231852\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.07388e6i	1.36733i	0.729797	+	0.683664i	$0.239615\pi$
$131$	3.07388e6i	1.36733i	−0.729797	+	0.683664i	$0.760385\pi$
$132$	0	0
$133$	2.78010e6	1.18169
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.48412e6i	1.74388i	0.489617	+	0.871938i	$0.337137\pi$
$137$	4.48412e6i	1.74388i	−0.489617	+	0.871938i	$0.662863\pi$
$138$	0	0
$139$	1.09233e6	0.406732	0.203366	−	0.979103i	$-0.434812\pi$
$139$	1.09233e6	0.406732	0.203366	+	0.979103i	$0.434812\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 4.51539e6i	− 1.54414i
$144$	0	0
$145$	−5.10622e6	−1.67493
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2.22087e6i	− 0.671375i	−0.941973	−	0.335687i	$-0.891031\pi$
$149$	− 2.22087e6i	− 0.671375i	0.941973	−	0.335687i	$-0.108969\pi$
$150$	0	0
$151$	−4.07871e6	−1.18465	−0.592327	−	0.805697i	$-0.701791\pi$
$151$	−4.07871e6	−1.18465	−0.592327	+	0.805697i	$0.701791\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 6.92245e6i	− 1.85894i
$156$	0	0
$157$	−6.15568e6	−1.59066	−0.795329	−	0.606178i	$-0.792702\pi$
$157$	−6.15568e6	−1.59066	−0.795329	+	0.606178i	$0.792702\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.63454e6i	− 0.391667i
$162$	0	0
$163$	−800696.	−0.184886	−0.0924432	−	0.995718i	$-0.529468\pi$
$163$	−800696.	−0.184886	−0.0924432	+	0.995718i	$0.529468\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.80467e6i	− 1.03161i	−0.856707	−	0.515804i	$-0.827493\pi$
$167$	− 4.80467e6i	− 1.03161i	0.856707	−	0.515804i	$-0.172507\pi$
$168$	0	0
$169$	6.51661e6	1.35009
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.56992e6i	− 0.689478i	−0.938699	−	0.344739i	$-0.887967\pi$
$173$	− 3.56992e6i	− 0.689478i	0.938699	−	0.344739i	$-0.112033\pi$
$174$	0	0
$175$	7.08237e6	1.32149
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 7.43698e6i	− 1.29669i	−0.761345	−	0.648347i	$-0.775461\pi$
$179$	− 7.43698e6i	− 1.29669i	0.761345	−	0.648347i	$-0.224539\pi$
$180$	0	0
$181$	1.03812e7	1.75070	0.875350	−	0.483491i	$-0.160631\pi$
$181$	1.03812e7	1.75070	0.875350	+	0.483491i	$0.160631\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 9.13681e6i	− 1.44304i
$186$	0	0
$187$	17064.0	0.00260949
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.29941e7i	− 1.86485i	−0.361360	−	0.932426i	$-0.617687\pi$
$191$	− 1.29941e7i	− 1.86485i	0.361360	−	0.932426i	$-0.382313\pi$
$192$	0	0
$193$	−3.93195e6	−0.546936	−0.273468	−	0.961881i	$-0.588171\pi$
$193$	−3.93195e6	−0.546936	−0.273468	+	0.961881i	$0.588171\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.37967e6i	0.703651i	0.936066	+	0.351825i	$0.114439\pi$
$197$	5.37967e6i	0.703651i	−0.936066	+	0.351825i	$0.885561\pi$
$198$	0	0
$199$	−565900.	−0.0718093	−0.0359046	−	0.999355i	$-0.511431\pi$
$199$	−565900.	−0.0718093	−0.0359046	+	0.999355i	$0.511431\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.42077e7i	− 1.69839i
$204$	0	0
$205$	−6.44348e6	−0.747926
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 7.70083e6i	− 0.843527i
$210$	0	0
$211$	1.35165e7	1.43885	0.719427	−	0.694568i	$-0.244404\pi$
$211$	1.35165e7	1.43885	0.719427	+	0.694568i	$0.244404\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 661003.i	− 0.0665102i
$216$	0	0
$217$	1.92613e7	1.88497
$218$	0	0
$219$	0	0
$220$	0	0
$221$	42867.6i	0.00397148i
$222$	0	0
$223$	−5.35484e6	−0.482872	−0.241436	−	0.970417i	$-0.577618\pi$
$223$	−5.35484e6	−0.482872	−0.241436	+	0.970417i	$0.577618\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.36063e7i	− 1.16322i	−0.813466	−	0.581612i	$-0.802422\pi$
$227$	− 1.36063e7i	− 1.16322i	0.813466	−	0.581612i	$-0.197578\pi$
$228$	0	0
$229$	−4.34641e6	−0.361930	−0.180965	−	0.983490i	$-0.557922\pi$
$229$	−4.34641e6	−0.361930	−0.180965	+	0.983490i	$0.557922\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.02333e7i	1.59956i	0.600297	+	0.799778i	$0.295049\pi$
$233$	2.02333e7i	1.59956i	−0.600297	+	0.799778i	$0.704951\pi$
$234$	0	0
$235$	1.33578e7	1.02927
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.03947e7i	1.49391i	0.664877	+	0.746953i	$0.268484\pi$
$239$	2.03947e7i	1.49391i	−0.664877	+	0.746953i	$0.731516\pi$
$240$	0	0
$241$	−3.12093e6	−0.222963	−0.111481	−	0.993767i	$-0.535560\pi$
$241$	−3.12093e6	−0.222963	−0.111481	+	0.993767i	$0.535560\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.02836e7i	1.37926i
$246$	0	0
$247$	1.93458e7	1.28379
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.09519e6i	0.322210i	0.986937	+	0.161105i	$0.0515058\pi$
$251$	5.09519e6i	0.322210i	−0.986937	+	0.161105i	$0.948494\pi$
$252$	0	0
$253$	−4.52765e6	−0.279583
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.44374e7i	− 0.850529i	−0.905069	−	0.425264i	$-0.860181\pi$
$257$	− 1.44374e7i	− 0.850529i	0.905069	−	0.425264i	$-0.139819\pi$
$258$	0	0
$259$	2.54226e7	1.46326
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 3.12567e7i	− 1.71821i	−0.511801	−	0.859104i	$-0.671022\pi$
$263$	− 3.12567e7i	− 1.71821i	0.511801	−	0.859104i	$-0.328978\pi$
$264$	0	0
$265$	−4.15280e7	−2.23153
$266$	0	0
$267$	0	0
$268$	0	0
$269$	251338.i	0.0129122i	0.999979	+	0.00645612i	$0.00205506\pi$
$269$	251338.i	0.0129122i	−0.999979	+	0.00645612i	$0.997945\pi$
$270$	0	0
$271$	2.96399e7	1.48925	0.744627	−	0.667481i	$-0.232627\pi$
$271$	2.96399e7	1.48925	0.744627	+	0.667481i	$0.232627\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.96181e7i	− 0.943319i
$276$	0	0
$277$	−1.32213e7	−0.622062	−0.311031	−	0.950400i	$-0.600674\pi$
$277$	−1.32213e7	−0.622062	−0.311031	+	0.950400i	$0.600674\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.12360e6i	0.275987i	0.990433	+	0.137993i	$0.0440652\pi$
$281$	6.12360e6i	0.275987i	−0.990433	+	0.137993i	$0.955935\pi$
$282$	0	0
$283$	6.74325e6	0.297516	0.148758	−	0.988874i	$-0.452473\pi$
$283$	6.74325e6	0.297516	0.148758	+	0.988874i	$0.452473\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.79286e7i	− 0.758403i
$288$	0	0
$289$	2.41374e7	0.999993
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.00239e7i	− 0.398505i	−0.979948	−	0.199253i	$-0.936149\pi$
$293$	− 1.00239e7i	− 0.398505i	0.979948	−	0.199253i	$-0.0638514\pi$
$294$	0	0
$295$	4.34593e7	1.69284
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 1.13742e7i	− 0.425508i
$300$	0	0
$301$	1.83920e6	0.0674418
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.30481e6i	− 0.0812337i
$306$	0	0
$307$	5.23060e6	0.180774	0.0903871	−	0.995907i	$-0.471190\pi$
$307$	5.23060e6	0.180774	0.0903871	+	0.995907i	$0.471190\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 3.12221e7i	− 1.03796i	−0.854786	−	0.518981i	$-0.826312\pi$
$311$	− 3.12221e7i	− 1.03796i	0.854786	−	0.518981i	$-0.173688\pi$
$312$	0	0
$313$	2.24778e7	0.733029	0.366515	−	0.930412i	$-0.380551\pi$
$313$	2.24778e7	0.733029	0.366515	+	0.930412i	$0.380551\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.76211e7i	− 0.867088i	−0.901132	−	0.433544i	$-0.857263\pi$
$317$	− 2.76211e7i	− 0.867088i	0.901132	−	0.433544i	$-0.142737\pi$
$318$	0	0
$319$	−3.93553e7	−1.21236
$320$	0	0
$321$	0	0
$322$	0	0
$323$	73109.2i	0.00216952i
$324$	0	0
$325$	4.92839e7	1.43567
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.71672e7i	1.04369i
$330$	0	0
$331$	5.76138e6	0.158870	0.0794352	−	0.996840i	$-0.474688\pi$
$331$	5.76138e6	0.158870	0.0794352	+	0.996840i	$0.474688\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 2.93917e7i	− 0.781790i
$336$	0	0
$337$	−4.01052e7	−1.04788	−0.523939	−	0.851756i	$-0.675538\pi$
$337$	−4.01052e7	−1.04788	−0.523939	+	0.851756i	$0.675538\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 5.33535e7i	− 1.34555i
$342$	0	0
$343$	504328.	0.0124977
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.78127e7i	1.62302i	0.584341	+	0.811508i	$0.301353\pi$
$347$	6.78127e7i	1.62302i	−0.584341	+	0.811508i	$0.698647\pi$
$348$	0	0
$349$	4.20638e7	0.989538	0.494769	−	0.869024i	$-0.335253\pi$
$349$	4.20638e7	0.989538	0.494769	+	0.869024i	$0.335253\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.75976e7i	0.400063i	0.979789	+	0.200032i	$0.0641045\pi$
$353$	1.75976e7i	0.400063i	−0.979789	+	0.200032i	$0.935896\pi$
$354$	0	0
$355$	−9.24478e7	−2.06639
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.39920e7i	0.302410i	0.988502	+	0.151205i	$0.0483154\pi$
$359$	1.39920e7i	0.302410i	−0.988502	+	0.151205i	$0.951685\pi$
$360$	0	0
$361$	−1.40523e7	−0.298694
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.10768e7i	0.844731i
$366$	0	0
$367$	−2.65855e7	−0.537832	−0.268916	−	0.963164i	$-0.586665\pi$
$367$	−2.65855e7	−0.537832	−0.268916	+	0.963164i	$0.586665\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.15549e8i	− 2.26279i
$372$	0	0
$373$	−1.78829e7	−0.344598	−0.172299	−	0.985045i	$-0.555119\pi$
$373$	−1.78829e7	−0.344598	−0.172299	+	0.985045i	$0.555119\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 9.88671e7i	− 1.84513i
$378$	0	0
$379$	−7.20978e7	−1.32435	−0.662177	−	0.749347i	$-0.730367\pi$
$379$	−7.20978e7	−1.32435	−0.662177	+	0.749347i	$0.730367\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 8.68648e6i	− 0.154614i	−0.997007	−	0.0773068i	$-0.975368\pi$
$383$	− 8.68648e6i	− 0.154614i	0.997007	−	0.0773068i	$-0.0246321\pi$
$384$	0	0
$385$	1.12873e8	1.97791
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 4.94411e7i	− 0.839923i	−0.907542	−	0.419962i	$-0.862044\pi$
$389$	− 4.94411e7i	− 0.839923i	0.907542	−	0.419962i	$-0.137956\pi$
$390$	0	0
$391$	42984.0	0.000719079 0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 6.10837e6i	− 0.0991137i
$396$	0	0
$397$	−1.56911e7	−0.250774	−0.125387	−	0.992108i	$-0.540017\pi$
$397$	−1.56911e7	−0.250774	−0.125387	+	0.992108i	$0.540017\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.74514e7i	0.735895i	0.929847	+	0.367947i	$0.119939\pi$
$401$	4.74514e7i	0.735895i	−0.929847	+	0.367947i	$0.880061\pi$
$402$	0	0
$403$	1.34033e8	2.04784
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7.04203e7i	− 1.04451i
$408$	0	0
$409$	−1.15512e8	−1.68832	−0.844162	−	0.536088i	$-0.819901\pi$
$409$	−1.15512e8	−1.68832	−0.844162	+	0.536088i	$0.819901\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.20923e8i	1.71656i
$414$	0	0
$415$	1.90994e6	0.0267225
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.46693e8i	1.99420i	0.0761306	+	0.997098i	$0.475743\pi$
$419$	1.46693e8i	1.99420i	−0.0761306	+	0.997098i	$0.524257\pi$
$420$	0	0
$421$	−1.39239e8	−1.86601	−0.933005	−	0.359863i	$-0.882824\pi$
$421$	−1.39239e8	−1.86601	−0.933005	+	0.359863i	$0.882824\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	186248.i	0.00242619i
$426$	0	0
$427$	6.41300e6	0.0823716
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.00392e8i	1.25391i	0.779056	+	0.626954i	$0.215699\pi$
$431$	1.00392e8i	1.25391i	−0.779056	+	0.626954i	$0.784301\pi$
$432$	0	0
$433$	−4.00631e7	−0.493493	−0.246747	−	0.969080i	$-0.579362\pi$
$433$	−4.00631e7	−0.493493	−0.246747	+	0.969080i	$0.579362\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.93983e7i	− 0.232445i
$438$	0	0
$439$	−1.38592e8	−1.63811	−0.819057	−	0.573712i	$-0.805503\pi$
$439$	−1.38592e8	−1.63811	−0.819057	+	0.573712i	$0.805503\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 1.11443e8i	− 1.28186i	−0.767600	−	0.640929i	$-0.778549\pi$
$443$	− 1.11443e8i	− 1.28186i	0.767600	−	0.640929i	$-0.221451\pi$
$444$	0	0
$445$	−2.24965e7	−0.255290
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.11166e7i	0.675181i	0.941293	+	0.337591i	$0.109612\pi$
$449$	6.11166e7i	0.675181i	−0.941293	+	0.337591i	$0.890388\pi$
$450$	0	0
$451$	−4.96619e7	−0.541370
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.83555e8i	3.01026i
$456$	0	0
$457$	3.56665e7	0.373690	0.186845	−	0.982389i	$-0.440174\pi$
$457$	3.56665e7	0.373690	0.186845	+	0.982389i	$0.440174\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.51983e8i	1.55128i	0.631173	+	0.775642i	$0.282574\pi$
$461$	1.51983e8i	1.55128i	−0.631173	+	0.775642i	$0.717426\pi$
$462$	0	0
$463$	1.14978e8	1.15844	0.579218	−	0.815173i	$-0.303358\pi$
$463$	1.14978e8	1.15844	0.579218	+	0.815173i	$0.303358\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.81705e7i	0.865711i	0.901463	+	0.432855i	$0.142494\pi$
$467$	8.81705e7i	0.865711i	−0.901463	+	0.432855i	$0.857506\pi$
$468$	0	0
$469$	8.17805e7	0.792741
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 5.09456e6i	− 0.0481419i
$474$	0	0
$475$	8.40520e7	0.784272
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.94388e7i	0.813803i	0.913472	+	0.406902i	$0.133391\pi$
$479$	8.94388e7i	0.813803i	−0.913472	+	0.406902i	$0.866609\pi$
$480$	0	0
$481$	1.76908e8	1.58969
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 5.59111e7i	− 0.490087i
$486$	0	0
$487$	−7.51688e7	−0.650805	−0.325403	−	0.945576i	$-0.605500\pi$
$487$	−7.51688e7	−0.650805	−0.325403	+	0.945576i	$0.605500\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.50822e7i	0.380856i	0.981701	+	0.190428i	$0.0609876\pi$
$491$	4.50822e7i	0.380856i	−0.981701	+	0.190428i	$0.939012\pi$
$492$	0	0
$493$	373626.	0.00311815
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.57230e8i	− 2.09533i
$498$	0	0
$499$	−9.15458e7	−0.736778	−0.368389	−	0.929672i	$-0.620091\pi$
$499$	−9.15458e7	−0.736778	−0.368389	+	0.929672i	$0.620091\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.61043e8i	1.26543i	0.774386	+	0.632713i	$0.218059\pi$
$503$	1.61043e8i	1.26543i	−0.774386	+	0.632713i	$0.781941\pi$
$504$	0	0
$505$	−1.16333e8	−0.903295
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.39995e7i	0.181990i	0.995851	+	0.0909951i	$0.0290048\pi$
$509$	2.39995e7i	0.181990i	−0.995851	+	0.0909951i	$0.970995\pi$
$510$	0	0
$511$	−1.14294e8	−0.856564
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.46751e8i	2.53861i
$516$	0	0
$517$	1.02953e8	0.745018
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.00897e7i	0.637033i	0.947917	+	0.318517i	$0.103185\pi$
$521$	9.00897e7i	0.637033i	−0.947917	+	0.318517i	$0.896815\pi$
$522$	0	0
$523$	3.77691e7	0.264016	0.132008	−	0.991249i	$-0.457857\pi$
$523$	3.77691e7	0.264016	0.132008	+	0.991249i	$0.457857\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	506520.i	0.00346071i
$528$	0	0
$529$	1.36631e8	0.922957
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.24759e8i	− 0.823931i
$534$	0	0
$535$	4.53427e7	0.296105
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.56332e8i	0.998347i
$540$	0	0
$541$	−2.54800e7	−0.160919	−0.0804595	−	0.996758i	$-0.525639\pi$
$541$	−2.54800e7	−0.160919	−0.0804595	+	0.996758i	$0.525639\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.38253e7i	− 0.208955i
$546$	0	0
$547$	−2.05216e8	−1.25386	−0.626930	−	0.779076i	$-0.715689\pi$
$547$	−2.05216e8	−1.25386	−0.626930	+	0.779076i	$0.715689\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1.68614e8i	− 1.00795i
$552$	0	0
$553$	1.69961e7	0.100502
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.41143e8i	− 1.39543i	−0.716375	−	0.697715i	$-0.754200\pi$
$557$	− 2.41143e8i	− 1.39543i	0.716375	−	0.697715i	$-0.245800\pi$
$558$	0	0
$559$	1.27984e7	0.0732690
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 1.68877e8i	− 0.946337i	−0.880972	−	0.473168i	$-0.843110\pi$
$563$	− 1.68877e8i	− 0.946337i	0.880972	−	0.473168i	$-0.156890\pi$
$564$	0	0
$565$	−1.42968e8	−0.792670
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.43995e8i	1.32448i	0.749293	+	0.662238i	$0.230393\pi$
$569$	2.43995e8i	1.32448i	−0.749293	+	0.662238i	$0.769607\pi$
$570$	0	0
$571$	−2.41502e8	−1.29722	−0.648608	−	0.761123i	$-0.724648\pi$
$571$	−2.41502e8	−1.29722	−0.648608	+	0.761123i	$0.724648\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 4.94177e7i	− 0.259944i
$576$	0	0
$577$	−4.93979e7	−0.257147	−0.128573	−	0.991700i	$-0.541040\pi$
$577$	−4.93979e7	−0.257147	−0.128573	+	0.991700i	$0.541040\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.31430e6i	0.0270968i
$582$	0	0
$583$	−3.20069e8	−1.61525
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.72052e8i	− 0.850639i	−0.905043	−	0.425320i	$-0.860162\pi$
$587$	− 1.72052e8i	− 0.850639i	0.905043	−	0.425320i	$-0.139838\pi$
$588$	0	0
$589$	2.28588e8	1.11869
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 2.70643e8i	− 1.29788i	−0.760841	−	0.648938i	$-0.775213\pi$
$593$	− 2.70643e8i	− 1.29788i	0.760841	−	0.648938i	$-0.224787\pi$
$594$	0	0
$595$	−1.07158e6	−0.00508712
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.73299e8i	0.806337i	0.915126	+	0.403169i	$0.132091\pi$
$599$	1.73299e8i	0.806337i	−0.915126	+	0.403169i	$0.867909\pi$
$600$	0	0
$601$	−4.31090e8	−1.98584	−0.992921	−	0.118775i	$-0.962103\pi$
$601$	−4.31090e8	−1.98584	−0.992921	+	0.118775i	$0.962103\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.49604e6i	− 0.0203032i
$606$	0	0
$607$	1.66991e7	0.0746665	0.0373332	−	0.999303i	$-0.488114\pi$
$607$	1.66991e7	0.0746665	0.0373332	+	0.999303i	$0.488114\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.58635e8i	1.13387i
$612$	0	0
$613$	1.92321e8	0.834920	0.417460	−	0.908695i	$-0.362920\pi$
$613$	1.92321e8	0.834920	0.417460	+	0.908695i	$0.362920\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.87023e8i	0.796233i	0.917335	+	0.398117i	$0.130336\pi$
$617$	1.87023e8i	0.796233i	−0.917335	+	0.398117i	$0.869664\pi$
$618$	0	0
$619$	−2.54873e8	−1.07461	−0.537307	−	0.843387i	$-0.680558\pi$
$619$	−2.54873e8	−1.07461	−0.537307	+	0.843387i	$0.680558\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 6.25950e7i	− 0.258866i
$624$	0	0
$625$	−2.58657e8	−1.05946
$626$	0	0
$627$	0	0
$628$	0	0
$629$	668547.i	0.00268646i
$630$	0	0
$631$	9.23602e7	0.367618	0.183809	−	0.982962i	$-0.441157\pi$
$631$	9.23602e7	0.367618	0.183809	+	0.982962i	$0.441157\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.31797e8i	2.07694i
$636$	0	0
$637$	−3.92732e8	−1.51942
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.24666e8i	1.61240i	0.591643	+	0.806200i	$0.298480\pi$
$641$	4.24666e8i	1.61240i	−0.591643	+	0.806200i	$0.701520\pi$
$642$	0	0
$643$	−3.75946e8	−1.41414	−0.707071	−	0.707143i	$-0.749984\pi$
$643$	−3.75946e8	−1.41414	−0.707071	+	0.707143i	$0.749984\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.63747e7i	− 0.0973813i	−0.998814	−	0.0486906i	$-0.984495\pi$
$647$	− 2.63747e7i	− 0.0973813i	0.998814	−	0.0486906i	$-0.0155048\pi$
$648$	0	0
$649$	3.34955e8	1.22533
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.58756e8i	0.929291i	0.885497	+	0.464645i	$0.153818\pi$
$653$	2.58756e8i	0.929291i	−0.885497	+	0.464645i	$0.846182\pi$
$654$	0	0
$655$	−5.34696e8	−1.90275
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1.39345e8i	− 0.486895i	−0.969914	−	0.243447i	$-0.921722\pi$
$659$	− 1.39345e8i	− 0.486895i	0.969914	−	0.243447i	$-0.0782783\pi$
$660$	0	0
$661$	4.72545e8	1.63621	0.818104	−	0.575070i	$-0.195025\pi$
$661$	4.72545e8	1.63621	0.818104	+	0.575070i	$0.195025\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.83593e8i	1.64443i
$666$	0	0
$667$	−9.91354e7	−0.334081
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1.77639e7i	− 0.0587992i
$672$	0	0
$673$	5.48833e8	1.80051	0.900254	−	0.435364i	$-0.143380\pi$
$673$	5.48833e8	1.80051	0.900254	+	0.435364i	$0.143380\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.00760e8i	− 0.324731i	−0.986731	−	0.162365i	$-0.948088\pi$
$677$	− 1.00760e8i	− 0.324731i	0.986731	−	0.162365i	$-0.0519123\pi$
$678$	0	0
$679$	1.55569e8	0.496952
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 313056.i	0 0.000982562i	−1.00000		0.000491281i	$-0.999844\pi$
$683$	− 313056.i	0 0.000982562i	1.00000		0.000491281i	$-0.000156380\pi$
$684$	0	0
$685$	−7.80005e8	−2.42675
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 8.04068e8i	− 2.45830i
$690$	0	0
$691$	3.72812e8	1.12994	0.564971	−	0.825111i	$-0.308887\pi$
$691$	3.72812e8	1.12994	0.564971	+	0.825111i	$0.308887\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.90009e8i	0.566003i
$696$	0	0
$697$	471474.	0.00139239
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.21170e8i	1.80325i	0.432517	+	0.901626i	$0.357626\pi$
$701$	6.21170e8i	1.80325i	−0.432517	+	0.901626i	$0.642374\pi$
$702$	0	0
$703$	3.01709e8	0.868406
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.23690e8i	− 0.915947i
$708$	0	0
$709$	2.46510e8	0.691666	0.345833	−	0.938296i	$-0.387596\pi$
$709$	2.46510e8	0.691666	0.345833	+	0.938296i	$0.387596\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1.34397e8i	− 0.370783i
$714$	0	0
$715$	7.85445e8	2.14881
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.60389e7i	0.258381i	0.991620	+	0.129191i	$0.0412379\pi$
$719$	9.60389e7i	0.258381i	−0.991620	+	0.129191i	$0.958762\pi$
$720$	0	0
$721$	−9.64811e8	−2.57417
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 4.29549e8i	− 1.12719i
$726$	0	0
$727$	3.91371e8	1.01856	0.509278	−	0.860602i	$-0.329912\pi$
$727$	3.91371e8	1.01856	0.509278	+	0.860602i	$0.329912\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	48366.1i	0 0.000123819i
$732$	0	0
$733$	−3.49078e7	−0.0886361	−0.0443181	−	0.999017i	$-0.514111\pi$
$733$	−3.49078e7	−0.0886361	−0.0443181	+	0.999017i	$0.514111\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.26531e8i	− 0.565881i
$738$	0	0
$739$	3.02999e8	0.750773	0.375386	−	0.926868i	$-0.377510\pi$
$739$	3.02999e8	0.750773	0.375386	+	0.926868i	$0.377510\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 2.45628e8i	− 0.598842i	−0.954121	−	0.299421i	$-0.903207\pi$
$743$	− 2.45628e8i	− 0.598842i	0.954121	−	0.299421i	$-0.0967935\pi$
$744$	0	0
$745$	3.86317e8	0.934276
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.26163e8i	0.300253i
$750$	0	0
$751$	−8.23270e7	−0.194367	−0.0971835	−	0.995266i	$-0.530983\pi$
$751$	−8.23270e7	−0.194367	−0.0971835	+	0.995266i	$0.530983\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 7.09484e8i	− 1.64855i
$756$	0	0
$757$	6.03579e8	1.39138	0.695691	−	0.718341i	$-0.255098\pi$
$757$	6.03579e8	1.39138	0.695691	+	0.718341i	$0.255098\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.32982e8i	0.528651i	0.964434	+	0.264325i	$0.0851493\pi$
$761$	2.32982e8i	0.528651i	−0.964434	+	0.264325i	$0.914851\pi$
$762$	0	0
$763$	9.41167e7	0.211882
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.41463e8i	1.86487i
$768$	0	0
$769$	8.15796e8	1.79392	0.896958	−	0.442115i	$-0.145772\pi$
$769$	8.15796e8	1.79392	0.896958	+	0.442115i	$0.145772\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 3.66587e8i	− 0.793667i	−0.917891	−	0.396833i	$-0.870109\pi$
$773$	− 3.66587e8i	− 0.793667i	0.917891	−	0.396833i	$-0.129891\pi$
$774$	0	0
$775$	5.82335e8	1.25103
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.12772e8i	− 0.450093i
$780$	0	0
$781$	−7.12524e8	−1.49571
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.07077e9i	− 2.21354i
$786$	0	0
$787$	4.02462e8	0.825659	0.412830	−	0.910808i	$-0.364540\pi$
$787$	4.02462e8	0.825659	0.412830	+	0.910808i	$0.364540\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 3.97798e8i	− 0.803773i
$792$	0	0
$793$	4.46260e7	0.0894887
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 5.18940e8i	− 1.02504i	−0.858675	−	0.512521i	$-0.828712\pi$
$797$	− 5.18940e8i	− 1.02504i	0.858675	−	0.512521i	$-0.171288\pi$
$798$	0	0
$799$	−977400.	−0.00191616
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.16592e8i	0.611440i
$804$	0	0
$805$	2.84325e8	0.545038
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.04036e6i	0.00574221i	0.999996	+	0.00287110i	$0.000913902\pi$
$809$	3.04036e6i	0.00574221i	−0.999996	+	0.00287110i	$0.999086\pi$
$810$	0	0
$811$	−2.25521e8	−0.422790	−0.211395	−	0.977401i	$-0.567801\pi$
$811$	−2.25521e8	−0.422790	−0.211395	+	0.977401i	$0.567801\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1.39280e8i	− 0.257285i
$816$	0	0
$817$	2.18272e7	0.0400250
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.77035e8i	0.500617i	0.968166	+	0.250309i	$0.0805321\pi$
$821$	2.77035e8i	0.500617i	−0.968166	+	0.250309i	$0.919468\pi$
$822$	0	0
$823$	−7.07336e8	−1.26890	−0.634448	−	0.772965i	$-0.718773\pi$
$823$	−7.07336e8	−1.26890	−0.634448	+	0.772965i	$0.718773\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.66346e8i	− 0.470900i	−0.971886	−	0.235450i	$-0.924344\pi$
$827$	− 2.66346e8i	− 0.470900i	0.971886	−	0.235450i	$-0.0756564\pi$
$828$	0	0
$829$	−5.03826e8	−0.884336	−0.442168	−	0.896932i	$-0.645791\pi$
$829$	−5.03826e8	−0.884336	−0.442168	+	0.896932i	$0.645791\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1.48416e6i	− 0.00256772i
$834$	0	0
$835$	8.35764e8	1.43557
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 7.63364e8i	− 1.29255i	−0.763106	−	0.646273i	$-0.776327\pi$
$839$	− 7.63364e8i	− 1.29255i	0.763106	−	0.646273i	$-0.223673\pi$
$840$	0	0
$841$	−2.66883e8	−0.448676
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.13355e9i	1.87876i
$846$	0	0
$847$	1.25099e7	0.0205876
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.77388e8i	− 0.287829i
$852$	0	0
$853$	1.87985e7	0.0302884	0.0151442	−	0.999885i	$-0.495179\pi$
$853$	1.87985e7	0.0302884	0.0151442	+	0.999885i	$0.495179\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.86427e8i	1.09057i	0.838252	+	0.545283i	$0.183578\pi$
$857$	6.86427e8i	1.09057i	−0.838252	+	0.545283i	$0.816422\pi$
$858$	0	0
$859$	−5.51932e8	−0.870775	−0.435387	−	0.900243i	$-0.643389\pi$
$859$	−5.51932e8	−0.870775	−0.435387	+	0.900243i	$0.643389\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 3.65665e8i	− 0.568920i	−0.958688	−	0.284460i	$-0.908186\pi$
$863$	− 3.65665e8i	− 0.568920i	0.958688	−	0.284460i	$-0.0918142\pi$
$864$	0	0
$865$	6.20982e8	0.959468
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 4.70791e7i	− 0.0717413i
$870$	0	0
$871$	5.69084e8	0.861236
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 8.35174e7i	− 0.124667i
$876$	0	0
$877$	5.85387e8	0.867849	0.433925	−	0.900949i	$-0.357128\pi$
$877$	5.85387e8	0.867849	0.433925	+	0.900949i	$0.357128\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.29761e8i	0.628491i	0.949342	+	0.314246i	$0.101752\pi$
$881$	4.29761e8i	0.628491i	−0.949342	+	0.314246i	$0.898248\pi$
$882$	0	0
$883$	−2.20085e8	−0.319675	−0.159837	−	0.987143i	$-0.551097\pi$
$883$	−2.20085e8	−0.319675	−0.159837	+	0.987143i	$0.551097\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.17196e9i	− 1.67936i	−0.543084	−	0.839678i	$-0.682744\pi$
$887$	− 1.17196e9i	− 1.67936i	0.543084	−	0.839678i	$-0.317256\pi$
$888$	0	0
$889$	−1.47969e9	−2.10604
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.41092e8i	0.619405i
$894$	0	0
$895$	1.29365e9	1.80446
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.16820e9i	− 1.60783i
$900$	0	0
$901$	3.03863e6	0.00415436
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.80579e9i	2.43625i
$906$	0	0
$907$	−7.31614e8	−0.980529	−0.490264	−	0.871574i	$-0.663100\pi$
$907$	−7.31614e8	−0.980529	−0.490264	+	0.871574i	$0.663100\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 9.18595e8i	− 1.21498i	−0.794327	−	0.607490i	$-0.792177\pi$
$911$	− 9.18595e8i	− 1.21498i	0.794327	−	0.607490i	$-0.207823\pi$
$912$	0	0
$913$	1.47205e7	0.0193425
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.48776e9i	− 1.92941i
$918$	0	0
$919$	−2.15987e8	−0.278279	−0.139139	−	0.990273i	$-0.544434\pi$
$919$	−2.15987e8	−0.278279	−0.139139	+	0.990273i	$0.544434\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.78998e9i	− 2.27637i
$924$	0	0
$925$	7.68613e8	0.971141
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 3.10124e8i	− 0.386802i	−0.981120	−	0.193401i	$-0.938048\pi$
$929$	− 3.10124e8i	− 0.386802i	0.981120	−	0.193401i	$-0.0619518\pi$
$930$	0	0
$931$	−6.69791e8	−0.830023
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.96825e6i	0.00363133i
$936$	0	0
$937$	−7.42448e8	−0.902501	−0.451250	−	0.892397i	$-0.649022\pi$
$937$	−7.42448e8	−0.902501	−0.451250	+	0.892397i	$0.649022\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.81766e8i	0.218144i	0.994034	+	0.109072i	$0.0347879\pi$
$941$	1.81766e8i	0.218144i	−0.994034	+	0.109072i	$0.965212\pi$
$942$	0	0
$943$	−1.25098e8	−0.149181
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.59189e8i	1.01167i	0.862630	+	0.505835i	$0.168816\pi$
$947$	8.59189e8i	1.01167i	−0.862630	+	0.505835i	$0.831184\pi$
$948$	0	0
$949$	−7.95333e8	−0.930573
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 6.86819e8i	− 0.793530i	−0.917920	−	0.396765i	$-0.870133\pi$
$953$	− 6.86819e8i	− 0.793530i	0.917920	−	0.396765i	$-0.129867\pi$
$954$	0	0
$955$	2.26029e9	2.59510
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2.17031e9i	− 2.46075i
$960$	0	0
$961$	6.96218e8	0.784468
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 6.83957e8i	− 0.761109i
$966$	0	0
$967$	−1.09411e9	−1.20999	−0.604995	−	0.796230i	$-0.706825\pi$
$967$	−1.09411e9	−1.20999	−0.604995	+	0.796230i	$0.706825\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.43115e8i	0.484014i	0.970274	+	0.242007i	$0.0778058\pi$
$971$	4.43115e8i	0.484014i	−0.970274	+	0.242007i	$0.922194\pi$
$972$	0	0
$973$	−5.28687e8	−0.573931
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.19004e9i	− 1.27608i	−0.770001	−	0.638042i	$-0.779744\pi$
$977$	− 1.19004e9i	− 1.27608i	0.770001	−	0.638042i	$-0.220256\pi$
$978$	0	0
$979$	−1.73387e8	−0.184786
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.18187e9i	1.24425i	0.782918	+	0.622125i	$0.213730\pi$
$983$	1.18187e9i	1.24425i	−0.782918	+	0.622125i	$0.786270\pi$
$984$	0	0
$985$	−9.35785e8	−0.979191
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.28331e7i	− 0.0132661i
$990$	0	0
$991$	5.09602e8	0.523613	0.261806	−	0.965120i	$-0.415682\pi$
$991$	5.09602e8	0.523613	0.261806	+	0.965120i	$0.415682\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 9.84373e7i	− 0.0999288i
$996$	0	0
$997$	9.90780e8	0.999751	0.499875	−	0.866097i	$-0.333379\pi$
$997$	9.90780e8	0.999751	0.499875	+	0.866097i	$0.333379\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.7.e.b.449.2		2
3.2	odd	2	inner	576.7.e.b.449.1		2
4.3	odd	2		576.7.e.k.449.2		2
8.3	odd	2		144.7.e.d.17.1		2
8.5	even	2		18.7.b.a.17.1	✓	2
12.11	even	2		576.7.e.k.449.1		2
24.5	odd	2		18.7.b.a.17.2	yes	2
24.11	even	2		144.7.e.d.17.2		2
40.13	odd	4		450.7.b.a.449.2		4
40.29	even	2		450.7.d.a.251.2		2
40.37	odd	4		450.7.b.a.449.3		4
72.5	odd	6		162.7.d.d.107.1		4
72.13	even	6		162.7.d.d.107.2		4
72.29	odd	6		162.7.d.d.53.2		4
72.61	even	6		162.7.d.d.53.1		4
120.29	odd	2		450.7.d.a.251.1		2
120.53	even	4		450.7.b.a.449.4		4
120.77	even	4		450.7.b.a.449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.7.b.a.17.1	✓	2	8.5	even	2
18.7.b.a.17.2	yes	2	24.5	odd	2
144.7.e.d.17.1		2	8.3	odd	2
144.7.e.d.17.2		2	24.11	even	2
162.7.d.d.53.1		4	72.61	even	6
162.7.d.d.53.2		4	72.29	odd	6
162.7.d.d.107.1		4	72.5	odd	6
162.7.d.d.107.2		4	72.13	even	6
450.7.b.a.449.1		4	120.77	even	4
450.7.b.a.449.2		4	40.13	odd	4
450.7.b.a.449.3		4	40.37	odd	4
450.7.b.a.449.4		4	120.53	even	4
450.7.d.a.251.1		2	120.29	odd	2
450.7.d.a.251.2		2	40.29	even	2
576.7.e.b.449.1		2	3.2	odd	2	inner
576.7.e.b.449.2		2	1.1	even	1	trivial
576.7.e.k.449.1		2	12.11	even	2
576.7.e.k.449.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+173.948i q^{5} -484.000 q^{7} +1340.67i q^{11} -3368.00 q^{13} -12.7279i q^{17} -5744.00 q^{19} +3377.14i q^{23} -14633.0 q^{25} +29354.8i q^{29} -39796.0 q^{31} -84191.0i q^{35} -52526.0 q^{37} +37042.5i q^{41} -3800.00 q^{43} -76791.8i q^{47} +116607. q^{49} +238738. i q^{53} -233208. q^{55} -249841. i q^{59} -13250.0 q^{61} -585858. i q^{65} -168968. q^{67} +531467. i q^{71} +236144. q^{73} -648886. i q^{77} -35116.0 q^{79} -10980.0i q^{83} +2214.00 q^{85} +129328. i q^{89} +1.63011e6 q^{91} -999159. i q^{95} -321424. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 968 q^{7} - 6736 q^{13} - 11488 q^{19} - 29266 q^{25} - 79592 q^{31} - 105052 q^{37} - 7600 q^{43} + 233214 q^{49} - 466416 q^{55} - 26500 q^{61} - 337936 q^{67} + 472288 q^{73} - 70232 q^{79} + 4428 q^{85}+ \cdots - 642848 q^{97}+O(q^{100}) \) 2 * q - 968 * q^7 - 6736 * q^13 - 11488 * q^19 - 29266 * q^25 - 79592 * q^31 - 105052 * q^37 - 7600 * q^43 + 233214 * q^49 - 466416 * q^55 - 26500 * q^61 - 337936 * q^67 + 472288 * q^73 - 70232 * q^79 + 4428 * q^85 + 3260224 * q^91 - 642848 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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