LMFDB - Embedded newform 576.5.e.d.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,5,Mod(449,576)]

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([0, 0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("576.449");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	576.e (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:	$59.5410987363$
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 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	576.449
Dual form			576.5.e.d.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-29.6985i q^{5} -28.0000 q^{7} +O(q^{10})$	$q-29.6985i q^{5} -28.0000 q^{7} +16.9706i q^{11} +112.000 q^{13} +89.0955i q^{17} -560.000 q^{19} -797.616i q^{23} -257.000 q^{25} -988.535i q^{29} -364.000 q^{31} +831.558i q^{35} +826.000 q^{37} +1811.61i q^{41} -1736.00 q^{43} +1306.73i q^{47} -1617.00 q^{49} +1794.64i q^{53} +504.000 q^{55} +4514.17i q^{59} -2618.00 q^{61} -3326.23i q^{65} +3784.00 q^{67} +8604.08i q^{71} +6608.00 q^{73} -475.176i q^{77} -4276.00 q^{79} +118.794i q^{83} +2646.00 q^{85} -4365.68i q^{89} -3136.00 q^{91} +16631.2i q^{95} -5824.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 56 q^{7}+O(q^{10}) $ 2 * q - 56 * q^7	$ 2 q - 56 q^{7} + 224 q^{13} - 1120 q^{19} - 514 q^{25} - 728 q^{31} + 1652 q^{37} - 3472 q^{43} - 3234 q^{49} + 1008 q^{55} - 5236 q^{61} + 7568 q^{67} + 13216 q^{73} - 8552 q^{79} + 5292 q^{85} - 6272 q^{91} - 11648 q^{97}+O(q^{100}) $ 2 * q - 56 * q^7 + 224 * q^13 - 1120 * q^19 - 514 * q^25 - 728 * q^31 + 1652 * q^37 - 3472 * q^43 - 3234 * q^49 + 1008 * q^55 - 5236 * q^61 + 7568 * q^67 + 13216 * q^73 - 8552 * q^79 + 5292 * q^85 - 6272 * q^91 - 11648 * 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
$5$	− 29.6985i	− 1.18794i	−0.804487	−	0.593970i	$-0.797560\pi$
$5$	− 29.6985i	− 1.18794i	0.804487	−	0.593970i	$-0.202440\pi$
$6$	0	0
$7$	−28.0000	−0.571429	−0.285714	−	0.958315i	$-0.592231\pi$
$7$	−28.0000	−0.571429	−0.285714	+	0.958315i	$0.592231\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.9706i	0.140253i	0.997538	+	0.0701263i	$0.0223402\pi$
$11$	16.9706i	0.140253i	−0.997538	+	0.0701263i	$0.977660\pi$
$12$	0	0
$13$	112.000	0.662722	0.331361	−	0.943504i	$-0.392492\pi$
$13$	112.000	0.662722	0.331361	+	0.943504i	$0.392492\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	89.0955i	0.308289i	0.988048	+	0.154144i	$0.0492621\pi$
$17$	89.0955i	0.308289i	−0.988048	+	0.154144i	$0.950738\pi$
$18$	0	0
$19$	−560.000	−1.55125	−0.775623	−	0.631196i	$-0.782564\pi$
$19$	−560.000	−1.55125	−0.775623	+	0.631196i	$0.782564\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 797.616i	− 1.50778i	−0.657000	−	0.753891i	$-0.728175\pi$
$23$	− 797.616i	− 1.50778i	0.657000	−	0.753891i	$-0.271825\pi$
$24$	0	0
$25$	−257.000	−0.411200
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 988.535i	− 1.17543i	−0.809069	−	0.587714i	$-0.800028\pi$
$29$	− 988.535i	− 1.17543i	0.809069	−	0.587714i	$-0.199972\pi$
$30$	0	0
$31$	−364.000	−0.378772	−0.189386	−	0.981903i	$-0.560650\pi$
$31$	−364.000	−0.378772	−0.189386	+	0.981903i	$0.560650\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	831.558i	0.678823i
$36$	0	0
$37$	826.000	0.603360	0.301680	−	0.953409i	$-0.402453\pi$
$37$	826.000	0.603360	0.301680	+	0.953409i	$0.402453\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1811.61i	1.07770i	0.842403	+	0.538848i	$0.181140\pi$
$41$	1811.61i	1.07770i	−0.842403	+	0.538848i	$0.818860\pi$
$42$	0	0
$43$	−1736.00	−0.938886	−0.469443	−	0.882963i	$-0.655545\pi$
$43$	−1736.00	−0.938886	−0.469443	+	0.882963i	$0.655545\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1306.73i	0.591550i	0.955258	+	0.295775i	$0.0955778\pi$
$47$	1306.73i	0.591550i	−0.955258	+	0.295775i	$0.904422\pi$
$48$	0	0
$49$	−1617.00	−0.673469
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1794.64i	0.638888i	0.947605	+	0.319444i	$0.103496\pi$
$53$	1794.64i	0.638888i	−0.947605	+	0.319444i	$0.896504\pi$
$54$	0	0
$55$	504.000	0.166612
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4514.17i	1.29680i	0.761299	+	0.648401i	$0.224562\pi$
$59$	4514.17i	1.29680i	−0.761299	+	0.648401i	$0.775438\pi$
$60$	0	0
$61$	−2618.00	−0.703574	−0.351787	−	0.936080i	$-0.614426\pi$
$61$	−2618.00	−0.703574	−0.351787	+	0.936080i	$0.614426\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 3326.23i	− 0.787273i
$66$	0	0
$67$	3784.00	0.842949	0.421475	−	0.906840i	$-0.361513\pi$
$67$	3784.00	0.842949	0.421475	+	0.906840i	$0.361513\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8604.08i	1.70682i	0.521241	+	0.853410i	$0.325469\pi$
$71$	8604.08i	1.70682i	−0.521241	+	0.853410i	$0.674531\pi$
$72$	0	0
$73$	6608.00	1.24001	0.620004	−	0.784599i	$-0.287131\pi$
$73$	6608.00	1.24001	0.620004	+	0.784599i	$0.287131\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 475.176i	− 0.0801443i
$78$	0	0
$79$	−4276.00	−0.685147	−0.342573	−	0.939491i	$-0.611299\pi$
$79$	−4276.00	−0.685147	−0.342573	+	0.939491i	$0.611299\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	118.794i	0.0172440i	0.999963	+	0.00862200i	$0.00274450\pi$
$83$	118.794i	0.0172440i	−0.999963	+	0.00862200i	$0.997255\pi$
$84$	0	0
$85$	2646.00	0.366228
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 4365.68i	− 0.551152i	−0.961279	−	0.275576i	$-0.911131\pi$
$89$	− 4365.68i	− 0.551152i	0.961279	−	0.275576i	$-0.0888686\pi$
$90$	0	0
$91$	−3136.00	−0.378698
$92$	0	0
$93$	0	0
$94$	0	0
$95$	16631.2i	1.84279i
$96$	0	0
$97$	−5824.00	−0.618982	−0.309491	−	0.950902i	$-0.600159\pi$
$97$	−5824.00	−0.618982	−0.309491	+	0.950902i	$0.600159\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8048.29i	0.788971i	0.918902	+	0.394485i	$0.129077\pi$
$101$	8048.29i	0.788971i	−0.918902	+	0.394485i	$0.870923\pi$
$102$	0	0
$103$	−14980.0	−1.41201	−0.706004	−	0.708208i	$-0.749504\pi$
$103$	−14980.0	−1.41201	−0.706004	+	0.708208i	$0.749504\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9775.04i	0.853790i	0.904301	+	0.426895i	$0.140393\pi$
$107$	9775.04i	0.853790i	−0.904301	+	0.426895i	$0.859607\pi$
$108$	0	0
$109$	−10640.0	−0.895548	−0.447774	−	0.894147i	$-0.647783\pi$
$109$	−10640.0	−0.895548	−0.447774	+	0.894147i	$0.647783\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2872.27i	− 0.224941i	−0.993655	−	0.112470i	$-0.964124\pi$
$113$	− 2872.27i	− 0.224941i	0.993655	−	0.112470i	$-0.0358763\pi$
$114$	0	0
$115$	−23688.0	−1.79115
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 2494.67i	− 0.176165i
$120$	0	0
$121$	14353.0	0.980329
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 10929.0i	− 0.699459i
$126$	0	0
$127$	−19420.0	−1.20404	−0.602021	−	0.798480i	$-0.705638\pi$
$127$	−19420.0	−1.20404	−0.602021	+	0.798480i	$0.705638\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	20195.0i	1.17679i	0.808572	+	0.588397i	$0.200241\pi$
$131$	20195.0i	1.17679i	−0.808572	+	0.588397i	$0.799759\pi$
$132$	0	0
$133$	15680.0	0.886427
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17331.2i	0.923394i	0.887038	+	0.461697i	$0.152759\pi$
$137$	17331.2i	0.923394i	−0.887038	+	0.461697i	$0.847241\pi$
$138$	0	0
$139$	−18536.0	−0.959371	−0.479685	−	0.877441i	$-0.659249\pi$
$139$	−18536.0	−0.959371	−0.479685	+	0.877441i	$0.659249\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1900.70i	0.0929485i
$144$	0	0
$145$	−29358.0	−1.39634
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3152.28i	0.141988i	0.997477	+	0.0709941i	$0.0226172\pi$
$149$	3152.28i	0.141988i	−0.997477	+	0.0709941i	$0.977383\pi$
$150$	0	0
$151$	7028.00	0.308232	0.154116	−	0.988053i	$-0.450747\pi$
$151$	7028.00	0.308232	0.154116	+	0.988053i	$0.450747\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10810.2i	0.449958i
$156$	0	0
$157$	5530.00	0.224350	0.112175	−	0.993688i	$-0.464218\pi$
$157$	5530.00	0.224350	0.112175	+	0.993688i	$0.464218\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	22333.3i	0.861589i
$162$	0	0
$163$	−10856.0	−0.408596	−0.204298	−	0.978909i	$-0.565491\pi$
$163$	−10856.0	−0.408596	−0.204298	+	0.978909i	$0.565491\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	29698.5i	1.06488i	0.846467	+	0.532441i	$0.178725\pi$
$167$	29698.5i	1.06488i	−0.846467	+	0.532441i	$0.821275\pi$
$168$	0	0
$169$	−16017.0	−0.560800
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 27589.9i	− 0.921845i	−0.887440	−	0.460922i	$-0.847519\pi$
$173$	− 27589.9i	− 0.921845i	0.887440	−	0.460922i	$-0.152481\pi$
$174$	0	0
$175$	7196.00	0.234971
$176$	0	0
$177$	0	0
$178$	0	0
$179$	54781.0i	1.70971i	0.518863	+	0.854857i	$0.326356\pi$
$179$	54781.0i	1.70971i	−0.518863	+	0.854857i	$0.673644\pi$
$180$	0	0
$181$	−18704.0	−0.570923	−0.285461	−	0.958390i	$-0.592147\pi$
$181$	−18704.0	−0.570923	−0.285461	+	0.958390i	$0.592147\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 24530.9i	− 0.716755i
$186$	0	0
$187$	−1512.00	−0.0432383
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 8519.22i	− 0.233525i	−0.993160	−	0.116762i	$-0.962748\pi$
$191$	− 8519.22i	− 0.233525i	0.993160	−	0.116762i	$-0.0372516\pi$
$192$	0	0
$193$	48398.0	1.29931	0.649655	−	0.760229i	$-0.274913\pi$
$193$	48398.0	1.29931	0.649655	+	0.760229i	$0.274913\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10016.9i	0.258107i	0.991638	+	0.129054i	$0.0411939\pi$
$197$	10016.9i	0.258107i	−0.991638	+	0.129054i	$0.958806\pi$
$198$	0	0
$199$	−13300.0	−0.335850	−0.167925	−	0.985800i	$-0.553707\pi$
$199$	−13300.0	−0.335850	−0.167925	+	0.985800i	$0.553707\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	27679.0i	0.671673i
$204$	0	0
$205$	53802.0	1.28024
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 9503.52i	− 0.217566i
$210$	0	0
$211$	36904.0	0.828912	0.414456	−	0.910069i	$-0.363972\pi$
$211$	36904.0	0.828912	0.414456	+	0.910069i	$0.363972\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	51556.6i	1.11534i
$216$	0	0
$217$	10192.0	0.216441
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9978.69i	0.204310i
$222$	0	0
$223$	51380.0	1.03320	0.516600	−	0.856227i	$-0.327197\pi$
$223$	51380.0	1.03320	0.516600	+	0.856227i	$0.327197\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 90402.2i	− 1.75439i	−0.480130	−	0.877197i	$-0.659411\pi$
$227$	− 90402.2i	− 1.75439i	0.480130	−	0.877197i	$-0.340589\pi$
$228$	0	0
$229$	−78512.0	−1.49715	−0.748575	−	0.663051i	$-0.769261\pi$
$229$	−78512.0	−1.49715	−0.748575	+	0.663051i	$0.769261\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 52451.8i	− 0.966158i	−0.875577	−	0.483079i	$-0.839518\pi$
$233$	− 52451.8i	− 0.966158i	0.875577	−	0.483079i	$-0.160482\pi$
$234$	0	0
$235$	38808.0	0.702725
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22265.4i	0.389793i	0.980824	+	0.194897i	$0.0624371\pi$
$239$	22265.4i	0.389793i	−0.980824	+	0.194897i	$0.937563\pi$
$240$	0	0
$241$	−53200.0	−0.915962	−0.457981	−	0.888962i	$-0.651427\pi$
$241$	−53200.0	−0.915962	−0.457981	+	0.888962i	$0.651427\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	48022.4i	0.800041i
$246$	0	0
$247$	−62720.0	−1.02805
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 83037.0i	− 1.31803i	−0.752131	−	0.659013i	$-0.770974\pi$
$251$	− 83037.0i	− 1.31803i	0.752131	−	0.659013i	$-0.229026\pi$
$252$	0	0
$253$	13536.0	0.211470
$254$	0	0
$255$	0	0
$256$	0	0
$257$	84670.4i	1.28193i	0.767569	+	0.640966i	$0.221466\pi$
$257$	84670.4i	1.28193i	−0.767569	+	0.640966i	$0.778534\pi$
$258$	0	0
$259$	−23128.0	−0.344777
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 110343.i	− 1.59526i	−0.603146	−	0.797630i	$-0.706087\pi$
$263$	− 110343.i	− 1.59526i	0.603146	−	0.797630i	$-0.293913\pi$
$264$	0	0
$265$	53298.0	0.758960
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27530.5i	0.380460i	0.981740	+	0.190230i	$0.0609234\pi$
$269$	27530.5i	0.380460i	−0.981740	+	0.190230i	$0.939077\pi$
$270$	0	0
$271$	−73276.0	−0.997753	−0.498877	−	0.866673i	$-0.666254\pi$
$271$	−73276.0	−0.997753	−0.498877	+	0.866673i	$0.666254\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4361.43i	− 0.0576719i
$276$	0	0
$277$	−46736.0	−0.609105	−0.304552	−	0.952496i	$-0.598507\pi$
$277$	−46736.0	−0.609105	−0.304552	+	0.952496i	$0.598507\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 112774.i	− 1.42822i	−0.700034	−	0.714110i	$-0.746832\pi$
$281$	− 112774.i	− 1.42822i	0.700034	−	0.714110i	$-0.253168\pi$
$282$	0	0
$283$	24304.0	0.303462	0.151731	−	0.988422i	$-0.451515\pi$
$283$	24304.0	0.303462	0.151731	+	0.988422i	$0.451515\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 50725.0i	− 0.615826i
$288$	0	0
$289$	75583.0	0.904958
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 9473.82i	− 0.110354i	−0.998477	−	0.0551772i	$-0.982428\pi$
$293$	− 9473.82i	− 0.110354i	0.998477	−	0.0551772i	$-0.0175724\pi$
$294$	0	0
$295$	134064.	1.54052
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 89333.0i	− 0.999240i
$300$	0	0
$301$	48608.0	0.536506
$302$	0	0
$303$	0	0
$304$	0	0
$305$	77750.6i	0.835804i
$306$	0	0
$307$	−19208.0	−0.203801	−0.101900	−	0.994795i	$-0.532492\pi$
$307$	−19208.0	−0.203801	−0.101900	+	0.994795i	$0.532492\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 78641.6i	− 0.813077i	−0.913634	−	0.406538i	$-0.866736\pi$
$311$	− 78641.6i	− 0.813077i	0.913634	−	0.406538i	$-0.133264\pi$
$312$	0	0
$313$	−127918.	−1.30570	−0.652849	−	0.757488i	$-0.726427\pi$
$313$	−127918.	−1.30570	−0.652849	+	0.757488i	$0.726427\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 65875.5i	− 0.655549i	−0.944756	−	0.327775i	$-0.893701\pi$
$317$	− 65875.5i	− 0.655549i	0.944756	−	0.327775i	$-0.106299\pi$
$318$	0	0
$319$	16776.0	0.164857
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 49893.5i	− 0.478232i
$324$	0	0
$325$	−28784.0	−0.272511
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 36588.5i	− 0.338028i
$330$	0	0
$331$	−15848.0	−0.144650	−0.0723250	−	0.997381i	$-0.523042\pi$
$331$	−15848.0	−0.144650	−0.0723250	+	0.997381i	$0.523042\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 112379.i	− 1.00137i
$336$	0	0
$337$	−127120.	−1.11932	−0.559660	−	0.828722i	$-0.689068\pi$
$337$	−127120.	−1.11932	−0.559660	+	0.828722i	$0.689068\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 6177.28i	− 0.0531238i
$342$	0	0
$343$	112504.	0.956268
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21875.1i	0.181673i	0.995866	+	0.0908365i	$0.0289541\pi$
$347$	21875.1i	0.181673i	−0.995866	+	0.0908365i	$0.971046\pi$
$348$	0	0
$349$	200998.	1.65022	0.825108	−	0.564975i	$-0.191114\pi$
$349$	200998.	1.65022	0.825108	+	0.564975i	$0.191114\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	241775.i	1.94027i	0.242561	+	0.970136i	$0.422012\pi$
$353$	241775.i	1.94027i	−0.242561	+	0.970136i	$0.577988\pi$
$354$	0	0
$355$	255528.	2.02760
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 108187.i	− 0.839436i	−0.907655	−	0.419718i	$-0.862129\pi$
$359$	− 108187.i	− 0.839436i	0.907655	−	0.419718i	$-0.137871\pi$
$360$	0	0
$361$	183279.	1.40637
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 196248.i	− 1.47305i
$366$	0	0
$367$	221732.	1.64625	0.823126	−	0.567859i	$-0.192228\pi$
$367$	221732.	1.64625	0.823126	+	0.567859i	$0.192228\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 50249.8i	− 0.365079i
$372$	0	0
$373$	−87674.0	−0.630163	−0.315082	−	0.949065i	$-0.602032\pi$
$373$	−87674.0	−0.630163	−0.315082	+	0.949065i	$0.602032\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 110716.i	− 0.778982i
$378$	0	0
$379$	40768.0	0.283819	0.141909	−	0.989880i	$-0.454676\pi$
$379$	40768.0	0.283819	0.141909	+	0.989880i	$0.454676\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 75077.8i	− 0.511816i	−0.966701	−	0.255908i	$-0.917626\pi$
$383$	− 75077.8i	− 0.511816i	0.966701	−	0.255908i	$-0.0823744\pi$
$384$	0	0
$385$	−14112.0	−0.0952066
$386$	0	0
$387$	0	0
$388$	0	0
$389$	123236.i	0.814401i	0.913339	+	0.407201i	$0.133495\pi$
$389$	123236.i	0.814401i	−0.913339	+	0.407201i	$0.866505\pi$
$390$	0	0
$391$	71064.0	0.464832
$392$	0	0
$393$	0	0
$394$	0	0
$395$	126991.i	0.813913i
$396$	0	0
$397$	8134.00	0.0516087	0.0258044	−	0.999667i	$-0.491785\pi$
$397$	8134.00	0.0516087	0.0258044	+	0.999667i	$0.491785\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 133639.i	− 0.831083i	−0.909574	−	0.415541i	$-0.863592\pi$
$401$	− 133639.i	− 0.831083i	0.909574	−	0.415541i	$-0.136408\pi$
$402$	0	0
$403$	−40768.0	−0.251021
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14017.7i	0.0846228i
$408$	0	0
$409$	35168.0	0.210233	0.105117	−	0.994460i	$-0.466478\pi$
$409$	35168.0	0.210233	0.105117	+	0.994460i	$0.466478\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 126397.i	− 0.741030i
$414$	0	0
$415$	3528.00	0.0204848
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 157640.i	− 0.897919i	−0.893552	−	0.448959i	$-0.851795\pi$
$419$	− 157640.i	− 0.897919i	0.893552	−	0.448959i	$-0.148205\pi$
$420$	0	0
$421$	−18800.0	−0.106070	−0.0530351	−	0.998593i	$-0.516890\pi$
$421$	−18800.0	−0.106070	−0.0530351	+	0.998593i	$0.516890\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 22897.5i	− 0.126768i
$426$	0	0
$427$	73304.0	0.402042
$428$	0	0
$429$	0	0
$430$	0	0
$431$	72243.7i	0.388907i	0.980912	+	0.194453i	$0.0622933\pi$
$431$	72243.7i	0.388907i	−0.980912	+	0.194453i	$0.937707\pi$
$432$	0	0
$433$	−351106.	−1.87268	−0.936338	−	0.351101i	$-0.885807\pi$
$433$	−351106.	−1.87268	−0.936338	+	0.351101i	$0.885807\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	446665.i	2.33894i
$438$	0	0
$439$	−60508.0	−0.313967	−0.156983	−	0.987601i	$-0.550177\pi$
$439$	−60508.0	−0.313967	−0.156983	+	0.987601i	$0.550177\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 57105.9i	− 0.290987i	−0.989359	−	0.145494i	$-0.953523\pi$
$443$	− 57105.9i	− 0.290987i	0.989359	−	0.145494i	$-0.0464770\pi$
$444$	0	0
$445$	−129654.	−0.654736
$446$	0	0
$447$	0	0
$448$	0	0
$449$	210482.i	1.04405i	0.852930	+	0.522025i	$0.174823\pi$
$449$	210482.i	1.04405i	−0.852930	+	0.522025i	$0.825177\pi$
$450$	0	0
$451$	−30744.0	−0.151150
$452$	0	0
$453$	0	0
$454$	0	0
$455$	93134.4i	0.449871i
$456$	0	0
$457$	324800.	1.55519	0.777595	−	0.628765i	$-0.216439\pi$
$457$	324800.	1.55519	0.777595	+	0.628765i	$0.216439\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 165331.i	− 0.777954i	−0.921248	−	0.388977i	$-0.872829\pi$
$461$	− 165331.i	− 0.777954i	0.921248	−	0.388977i	$-0.127171\pi$
$462$	0	0
$463$	−117220.	−0.546814	−0.273407	−	0.961898i	$-0.588151\pi$
$463$	−117220.	−0.546814	−0.273407	+	0.961898i	$0.588151\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 170707.i	− 0.782740i	−0.920233	−	0.391370i	$-0.872001\pi$
$467$	− 170707.i	− 0.782740i	0.920233	−	0.391370i	$-0.127999\pi$
$468$	0	0
$469$	−105952.	−0.481685
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 29460.9i	− 0.131681i
$474$	0	0
$475$	143920.	0.637873
$476$	0	0
$477$	0	0
$478$	0	0
$479$	99668.1i	0.434395i	0.976128	+	0.217198i	$0.0696916\pi$
$479$	99668.1i	0.434395i	−0.976128	+	0.217198i	$0.930308\pi$
$480$	0	0
$481$	92512.0	0.399860
$482$	0	0
$483$	0	0
$484$	0	0
$485$	172964.i	0.735313i
$486$	0	0
$487$	−416500.	−1.75613	−0.878066	−	0.478540i	$-0.841166\pi$
$487$	−416500.	−1.75613	−0.878066	+	0.478540i	$0.841166\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 355024.i	− 1.47263i	−0.676637	−	0.736317i	$-0.736563\pi$
$491$	− 355024.i	− 1.47263i	0.676637	−	0.736317i	$-0.263437\pi$
$492$	0	0
$493$	88074.0	0.362371
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 240914.i	− 0.975325i
$498$	0	0
$499$	−135968.	−0.546054	−0.273027	−	0.962006i	$-0.588025\pi$
$499$	−135968.	−0.546054	−0.273027	+	0.962006i	$0.588025\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 170707.i	− 0.674707i	−0.941378	−	0.337353i	$-0.890468\pi$
$503$	− 170707.i	− 0.674707i	0.941378	−	0.337353i	$-0.109532\pi$
$504$	0	0
$505$	239022.	0.937249
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 193367.i	− 0.746357i	−0.927760	−	0.373178i	$-0.878268\pi$
$509$	− 193367.i	− 0.746357i	0.927760	−	0.373178i	$-0.121732\pi$
$510$	0	0
$511$	−185024.	−0.708576
$512$	0	0
$513$	0	0
$514$	0	0
$515$	444883.i	1.67738i
$516$	0	0
$517$	−22176.0	−0.0829664
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 391931.i	− 1.44389i	−0.691951	−	0.721945i	$-0.743248\pi$
$521$	− 391931.i	− 1.44389i	0.691951	−	0.721945i	$-0.256752\pi$
$522$	0	0
$523$	−410816.	−1.50191	−0.750955	−	0.660353i	$-0.770407\pi$
$523$	−410816.	−1.50191	−0.750955	+	0.660353i	$0.770407\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 32430.7i	− 0.116771i
$528$	0	0
$529$	−356351.	−1.27341
$530$	0	0
$531$	0	0
$532$	0	0
$533$	202900.i	0.714213i
$534$	0	0
$535$	290304.	1.01425
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 27441.4i	− 0.0944558i
$540$	0	0
$541$	57616.0	0.196856	0.0984280	−	0.995144i	$-0.468619\pi$
$541$	57616.0	0.196856	0.0984280	+	0.995144i	$0.468619\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	315992.i	1.06386i
$546$	0	0
$547$	−78920.0	−0.263762	−0.131881	−	0.991266i	$-0.542102\pi$
$547$	−78920.0	−0.263762	−0.131881	+	0.991266i	$0.542102\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	553580.i	1.82338i
$552$	0	0
$553$	119728.	0.391512
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 178025.i	− 0.573815i	−0.957958	−	0.286907i	$-0.907373\pi$
$557$	− 178025.i	− 0.573815i	0.957958	−	0.286907i	$-0.0926272\pi$
$558$	0	0
$559$	−194432.	−0.622220
$560$	0	0
$561$	0	0
$562$	0	0
$563$	431816.i	1.36233i	0.732131	+	0.681164i	$0.238526\pi$
$563$	431816.i	1.36233i	−0.732131	+	0.681164i	$0.761474\pi$
$564$	0	0
$565$	−85302.0	−0.267216
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 65756.7i	− 0.203103i	−0.994830	−	0.101551i	$-0.967619\pi$
$569$	− 65756.7i	− 0.203103i	0.994830	−	0.101551i	$-0.0323806\pi$
$570$	0	0
$571$	−111632.	−0.342386	−0.171193	−	0.985237i	$-0.554762\pi$
$571$	−111632.	−0.342386	−0.171193	+	0.985237i	$0.554762\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	204987.i	0.620000i
$576$	0	0
$577$	−164878.	−0.495235	−0.247617	−	0.968858i	$-0.579648\pi$
$577$	−164878.	−0.495235	−0.247617	+	0.968858i	$0.579648\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 3326.23i	− 0.00985372i
$582$	0	0
$583$	−30456.0	−0.0896057
$584$	0	0
$585$	0	0
$586$	0	0
$587$	568548.i	1.65003i	0.565114	+	0.825013i	$0.308832\pi$
$587$	568548.i	1.65003i	−0.565114	+	0.825013i	$0.691168\pi$
$588$	0	0
$589$	203840.	0.587569
$590$	0	0
$591$	0	0
$592$	0	0
$593$	580279.i	1.65016i	0.565013	+	0.825082i	$0.308871\pi$
$593$	580279.i	1.65016i	−0.565013	+	0.825082i	$0.691129\pi$
$594$	0	0
$595$	−74088.0	−0.209273
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5719.08i	0.0159394i	0.999968	+	0.00796971i	$0.00253686\pi$
$599$	5719.08i	0.0159394i	−0.999968	+	0.00796971i	$0.997463\pi$
$600$	0	0
$601$	233198.	0.645618	0.322809	−	0.946464i	$-0.395373\pi$
$601$	233198.	0.645618	0.322809	+	0.946464i	$0.395373\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 426262.i	− 1.16457i
$606$	0	0
$607$	154868.	0.420324	0.210162	−	0.977667i	$-0.432601\pi$
$607$	154868.	0.420324	0.210162	+	0.977667i	$0.432601\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	146354.i	0.392033i
$612$	0	0
$613$	333862.	0.888477	0.444238	−	0.895909i	$-0.353474\pi$
$613$	333862.	0.888477	0.444238	+	0.895909i	$0.353474\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19325.2i	0.0507638i	0.999678	+	0.0253819i	$0.00808018\pi$
$617$	19325.2i	0.0507638i	−0.999678	+	0.0253819i	$0.991920\pi$
$618$	0	0
$619$	473536.	1.23587	0.617933	−	0.786230i	$-0.287970\pi$
$619$	473536.	1.23587	0.617933	+	0.786230i	$0.287970\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	122239.i	0.314944i
$624$	0	0
$625$	−485201.	−1.24211
$626$	0	0
$627$	0	0
$628$	0	0
$629$	73592.8i	0.186009i
$630$	0	0
$631$	−384820.	−0.966493	−0.483247	−	0.875484i	$-0.660543\pi$
$631$	−384820.	−0.966493	−0.483247	+	0.875484i	$0.660543\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	576745.i	1.43033i
$636$	0	0
$637$	−181104.	−0.446323
$638$	0	0
$639$	0	0
$640$	0	0
$641$	145832.i	0.354926i	0.984127	+	0.177463i	$0.0567890\pi$
$641$	145832.i	0.354926i	−0.984127	+	0.177463i	$0.943211\pi$
$642$	0	0
$643$	−528584.	−1.27847	−0.639237	−	0.769010i	$-0.720750\pi$
$643$	−528584.	−1.27847	−0.639237	+	0.769010i	$0.720750\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	176409.i	0.421417i	0.977549	+	0.210709i	$0.0675771\pi$
$647$	176409.i	0.421417i	−0.977549	+	0.210709i	$0.932423\pi$
$648$	0	0
$649$	−76608.0	−0.181880
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 403250.i	− 0.945689i	−0.881146	−	0.472844i	$-0.843227\pi$
$653$	− 403250.i	− 0.945689i	0.881146	−	0.472844i	$-0.156773\pi$
$654$	0	0
$655$	599760.	1.39796
$656$	0	0
$657$	0	0
$658$	0	0
$659$	137020.i	0.315511i	0.987478	+	0.157755i	$0.0504257\pi$
$659$	137020.i	0.315511i	−0.987478	+	0.157755i	$0.949574\pi$
$660$	0	0
$661$	−286790.	−0.656389	−0.328194	−	0.944610i	$-0.606440\pi$
$661$	−286790.	−0.656389	−0.328194	+	0.944610i	$0.606440\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 465672.i	− 1.05302i
$666$	0	0
$667$	−788472.	−1.77229
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 44428.9i	− 0.0986781i
$672$	0	0
$673$	121058.	0.267278	0.133639	−	0.991030i	$-0.457334\pi$
$673$	121058.	0.267278	0.133639	+	0.991030i	$0.457334\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	377260.i	0.823120i	0.911383	+	0.411560i	$0.135016\pi$
$677$	377260.i	0.823120i	−0.911383	+	0.411560i	$0.864984\pi$
$678$	0	0
$679$	163072.	0.353704
$680$	0	0
$681$	0	0
$682$	0	0
$683$	335457.i	0.719110i	0.933124	+	0.359555i	$0.117072\pi$
$683$	335457.i	0.719110i	−0.933124	+	0.359555i	$0.882928\pi$
$684$	0	0
$685$	514710.	1.09694
$686$	0	0
$687$	0	0
$688$	0	0
$689$	200999.i	0.423405i
$690$	0	0
$691$	578872.	1.21235	0.606173	−	0.795333i	$-0.292704\pi$
$691$	578872.	1.21235	0.606173	+	0.795333i	$0.292704\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	550491.i	1.13967i
$696$	0	0
$697$	−161406.	−0.332242
$698$	0	0
$699$	0	0
$700$	0	0
$701$	654279.i	1.33146i	0.746194	+	0.665728i	$0.231879\pi$
$701$	654279.i	1.33146i	−0.746194	+	0.665728i	$0.768121\pi$
$702$	0	0
$703$	−462560.	−0.935960
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 225352.i	− 0.450840i
$708$	0	0
$709$	−779408.	−1.55050	−0.775251	−	0.631653i	$-0.782377\pi$
$709$	−779408.	−1.55050	−0.775251	+	0.631653i	$0.782377\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	290332.i	0.571106i
$714$	0	0
$715$	56448.0	0.110417
$716$	0	0
$717$	0	0
$718$	0	0
$719$	147898.i	0.286092i	0.989716	+	0.143046i	$0.0456897\pi$
$719$	147898.i	0.286092i	−0.989716	+	0.143046i	$0.954310\pi$
$720$	0	0
$721$	419440.	0.806862
$722$	0	0
$723$	0	0
$724$	0	0
$725$	254054.i	0.483336i
$726$	0	0
$727$	−713188.	−1.34938	−0.674691	−	0.738100i	$-0.735723\pi$
$727$	−713188.	−1.34938	−0.674691	+	0.738100i	$0.735723\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 154670.i	− 0.289448i
$732$	0	0
$733$	826672.	1.53860	0.769299	−	0.638889i	$-0.220606\pi$
$733$	826672.	1.53860	0.769299	+	0.638889i	$0.220606\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	64216.6i	0.118226i
$738$	0	0
$739$	−703280.	−1.28777	−0.643887	−	0.765121i	$-0.722679\pi$
$739$	−703280.	−1.28777	−0.643887	+	0.765121i	$0.722679\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 469643.i	− 0.850728i	−0.905022	−	0.425364i	$-0.860146\pi$
$743$	− 469643.i	− 0.850728i	0.905022	−	0.425364i	$-0.139854\pi$
$744$	0	0
$745$	93618.0	0.168673
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 273701.i	− 0.487880i
$750$	0	0
$751$	−963004.	−1.70745	−0.853725	−	0.520723i	$-0.825662\pi$
$751$	−963004.	−1.70745	−0.853725	+	0.520723i	$0.825662\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 208721.i	− 0.366161i
$756$	0	0
$757$	−899696.	−1.57002	−0.785008	−	0.619486i	$-0.787341\pi$
$757$	−899696.	−1.57002	−0.785008	+	0.619486i	$0.787341\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 886292.i	− 1.53041i	−0.643787	−	0.765204i	$-0.722638\pi$
$761$	− 886292.i	− 1.53041i	0.643787	−	0.765204i	$-0.277362\pi$
$762$	0	0
$763$	297920.	0.511741
$764$	0	0
$765$	0	0
$766$	0	0
$767$	505587.i	0.859419i
$768$	0	0
$769$	183218.	0.309824	0.154912	−	0.987928i	$-0.450491\pi$
$769$	183218.	0.309824	0.154912	+	0.987928i	$0.450491\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1.11521e6i	− 1.86637i	−0.359402	−	0.933183i	$-0.617019\pi$
$773$	− 1.11521e6i	− 1.86637i	0.359402	−	0.933183i	$-0.382981\pi$
$774$	0	0
$775$	93548.0	0.155751
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.01450e6i	− 1.67177i
$780$	0	0
$781$	−146016.	−0.239386
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 164233.i	− 0.266514i
$786$	0	0
$787$	276304.	0.446106	0.223053	−	0.974806i	$-0.428398\pi$
$787$	276304.	0.446106	0.223053	+	0.974806i	$0.428398\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	80423.5i	0.128538i
$792$	0	0
$793$	−293216.	−0.466274
$794$	0	0
$795$	0	0
$796$	0	0
$797$	249438.i	0.392686i	0.980535	+	0.196343i	$0.0629066\pi$
$797$	249438.i	0.392686i	−0.980535	+	0.196343i	$0.937093\pi$
$798$	0	0
$799$	−116424.	−0.182368
$800$	0	0
$801$	0	0
$802$	0	0
$803$	112141.i	0.173914i
$804$	0	0
$805$	663264.	1.02352
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 657868.i	− 1.00518i	−0.864526	−	0.502588i	$-0.832381\pi$
$809$	− 657868.i	− 1.00518i	0.864526	−	0.502588i	$-0.167619\pi$
$810$	0	0
$811$	−660800.	−1.00468	−0.502341	−	0.864670i	$-0.667528\pi$
$811$	−660800.	−1.00468	−0.502341	+	0.864670i	$0.667528\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	322407.i	0.485388i
$816$	0	0
$817$	972160.	1.45644
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.23522e6i	1.83256i	0.400533	+	0.916282i	$0.368825\pi$
$821$	1.23522e6i	1.83256i	−0.400533	+	0.916282i	$0.631175\pi$
$822$	0	0
$823$	1.07330e6	1.58461	0.792303	−	0.610127i	$-0.208882\pi$
$823$	1.07330e6	1.58461	0.792303	+	0.610127i	$0.208882\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 356687.i	− 0.521527i	−0.965403	−	0.260763i	$-0.916026\pi$
$827$	− 356687.i	− 0.521527i	0.965403	−	0.260763i	$-0.0839742\pi$
$828$	0	0
$829$	−91280.0	−0.132821	−0.0664105	−	0.997792i	$-0.521155\pi$
$829$	−91280.0	−0.132821	−0.0664105	+	0.997792i	$0.521155\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 144067.i	− 0.207623i
$834$	0	0
$835$	882000.	1.26501
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1.18473e6i	− 1.68305i	−0.540221	−	0.841523i	$-0.681659\pi$
$839$	− 1.18473e6i	− 1.68305i	0.540221	−	0.841523i	$-0.318341\pi$
$840$	0	0
$841$	−269921.	−0.381632
$842$	0	0
$843$	0	0
$844$	0	0
$845$	475681.i	0.666196i
$846$	0	0
$847$	−401884.	−0.560188
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 658831.i	− 0.909735i
$852$	0	0
$853$	−429254.	−0.589951	−0.294976	−	0.955505i	$-0.595312\pi$
$853$	−429254.	−0.589951	−0.294976	+	0.955505i	$0.595312\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	337286.i	0.459236i	0.973281	+	0.229618i	$0.0737477\pi$
$857$	337286.i	0.459236i	−0.973281	+	0.229618i	$0.926252\pi$
$858$	0	0
$859$	535864.	0.726220	0.363110	−	0.931746i	$-0.381715\pi$
$859$	535864.	0.726220	0.363110	+	0.931746i	$0.381715\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 894722.i	− 1.20134i	−0.799496	−	0.600671i	$-0.794900\pi$
$863$	− 894722.i	− 1.20134i	0.799496	−	0.600671i	$-0.205100\pi$
$864$	0	0
$865$	−819378.	−1.09510
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 72566.1i	− 0.0960936i
$870$	0	0
$871$	423808.	0.558641
$872$	0	0
$873$	0	0
$874$	0	0
$875$	306013.i	0.399691i
$876$	0	0
$877$	179878.	0.233872	0.116936	−	0.993139i	$-0.462693\pi$
$877$	179878.	0.233872	0.116936	+	0.993139i	$0.462693\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.27451e6i	1.64207i	0.570878	+	0.821035i	$0.306603\pi$
$881$	1.27451e6i	1.64207i	−0.570878	+	0.821035i	$0.693397\pi$
$882$	0	0
$883$	−383096.	−0.491345	−0.245672	−	0.969353i	$-0.579009\pi$
$883$	−383096.	−0.491345	−0.245672	+	0.969353i	$0.579009\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	450229.i	0.572251i	0.958192	+	0.286125i	$0.0923674\pi$
$887$	450229.i	0.572251i	−0.958192	+	0.286125i	$0.907633\pi$
$888$	0	0
$889$	543760.	0.688024
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 731771.i	− 0.917639i
$894$	0	0
$895$	1.62691e6	2.03104
$896$	0	0
$897$	0	0
$898$	0	0
$899$	359827.i	0.445219i
$900$	0	0
$901$	−159894.	−0.196962
$902$	0	0
$903$	0	0
$904$	0	0
$905$	555480.i	0.678222i
$906$	0	0
$907$	1.09900e6	1.33593	0.667964	−	0.744193i	$-0.267166\pi$
$907$	1.09900e6	1.33593	0.667964	+	0.744193i	$0.267166\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 284461.i	− 0.342756i	−0.985205	−	0.171378i	$-0.945178\pi$
$911$	− 284461.i	− 0.342756i	0.985205	−	0.171378i	$-0.0548220\pi$
$912$	0	0
$913$	−2016.00	−0.00241852
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 565459.i	− 0.672454i
$918$	0	0
$919$	−490588.	−0.580879	−0.290440	−	0.956893i	$-0.593802\pi$
$919$	−490588.	−0.580879	−0.290440	+	0.956893i	$0.593802\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	963656.i	1.13115i
$924$	0	0
$925$	−212282.	−0.248102
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 651852.i	− 0.755297i	−0.925949	−	0.377648i	$-0.876733\pi$
$929$	− 651852.i	− 0.755297i	0.925949	−	0.377648i	$-0.123267\pi$
$930$	0	0
$931$	905520.	1.04472
$932$	0	0
$933$	0	0
$934$	0	0
$935$	44904.1i	0.0513645i
$936$	0	0
$937$	−947842.	−1.07958	−0.539792	−	0.841798i	$-0.681497\pi$
$937$	−947842.	−1.07958	−0.539792	+	0.841798i	$0.681497\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 214750.i	− 0.242523i	−0.992621	−	0.121262i	$-0.961306\pi$
$941$	− 214750.i	− 0.242523i	0.992621	−	0.121262i	$-0.0386940\pi$
$942$	0	0
$943$	1.44497e6	1.62493
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.18220e6i	− 1.31823i	−0.752041	−	0.659117i	$-0.770930\pi$
$947$	− 1.18220e6i	− 1.31823i	0.752041	−	0.659117i	$-0.229070\pi$
$948$	0	0
$949$	740096.	0.821780
$950$	0	0
$951$	0	0
$952$	0	0
$953$	426729.i	0.469858i	0.972013	+	0.234929i	$0.0754857\pi$
$953$	426729.i	0.469858i	−0.972013	+	0.234929i	$0.924514\pi$
$954$	0	0
$955$	−253008.	−0.277413
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 485273.i	− 0.527654i
$960$	0	0
$961$	−791025.	−0.856532
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.43735e6i	− 1.54350i
$966$	0	0
$967$	392828.	0.420097	0.210048	−	0.977691i	$-0.432638\pi$
$967$	392828.	0.420097	0.210048	+	0.977691i	$0.432638\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 567716.i	− 0.602134i	−0.953603	−	0.301067i	$-0.902657\pi$
$971$	− 567716.i	− 0.602134i	0.953603	−	0.301067i	$-0.0973427\pi$
$972$	0	0
$973$	519008.	0.548212
$974$	0	0
$975$	0	0
$976$	0	0
$977$	122370.i	0.128200i	0.997943	+	0.0640999i	$0.0204176\pi$
$977$	122370.i	0.128200i	−0.997943	+	0.0640999i	$0.979582\pi$
$978$	0	0
$979$	74088.0	0.0773005
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.58495e6i	− 1.64024i	−0.572189	−	0.820121i	$-0.693906\pi$
$983$	− 1.58495e6i	− 1.64024i	0.572189	−	0.820121i	$-0.306094\pi$
$984$	0	0
$985$	297486.	0.306615
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.38466e6i	1.41563i
$990$	0	0
$991$	7364.00	0.00749836	0.00374918	−	0.999993i	$-0.498807\pi$
$991$	7364.00	0.00749836	0.00374918	+	0.999993i	$0.498807\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	394990.i	0.398970i
$996$	0	0
$997$	1.43479e6	1.44344	0.721719	−	0.692186i	$-0.243352\pi$
$997$	1.43479e6	1.44344	0.721719	+	0.692186i	$0.243352\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.5.e.d.449.1		2
3.2	odd	2	inner	576.5.e.d.449.2		2
4.3	odd	2		576.5.e.g.449.1		2
8.3	odd	2		144.5.e.c.17.2		2
8.5	even	2		9.5.b.a.8.1	✓	2
12.11	even	2		576.5.e.g.449.2		2
24.5	odd	2		9.5.b.a.8.2	yes	2
24.11	even	2		144.5.e.c.17.1		2
40.13	odd	4		225.5.d.a.224.2		4
40.29	even	2		225.5.c.a.26.2		2
40.37	odd	4		225.5.d.a.224.3		4
56.13	odd	2		441.5.b.a.197.1		2
72.5	odd	6		81.5.d.c.26.1		4
72.13	even	6		81.5.d.c.26.2		4
72.29	odd	6		81.5.d.c.53.2		4
72.61	even	6		81.5.d.c.53.1		4
120.29	odd	2		225.5.c.a.26.1		2
120.53	even	4		225.5.d.a.224.4		4
120.77	even	4		225.5.d.a.224.1		4
168.125	even	2		441.5.b.a.197.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.5.b.a.8.1	✓	2	8.5	even	2
9.5.b.a.8.2	yes	2	24.5	odd	2
81.5.d.c.26.1		4	72.5	odd	6
81.5.d.c.26.2		4	72.13	even	6
81.5.d.c.53.1		4	72.61	even	6
81.5.d.c.53.2		4	72.29	odd	6
144.5.e.c.17.1		2	24.11	even	2
144.5.e.c.17.2		2	8.3	odd	2
225.5.c.a.26.1		2	120.29	odd	2
225.5.c.a.26.2		2	40.29	even	2
225.5.d.a.224.1		4	120.77	even	4
225.5.d.a.224.2		4	40.13	odd	4
225.5.d.a.224.3		4	40.37	odd	4
225.5.d.a.224.4		4	120.53	even	4
441.5.b.a.197.1		2	56.13	odd	2
441.5.b.a.197.2		2	168.125	even	2
576.5.e.d.449.1		2	1.1	even	1	trivial
576.5.e.d.449.2		2	3.2	odd	2	inner
576.5.e.g.449.1		2	4.3	odd	2
576.5.e.g.449.2		2	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	576.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-29.6985i q^{5} -28.0000 q^{7} +O(q^{10})\)	\(q-29.6985i q^{5} -28.0000 q^{7} +16.9706i q^{11} +112.000 q^{13} +89.0955i q^{17} -560.000 q^{19} -797.616i q^{23} -257.000 q^{25} -988.535i q^{29} -364.000 q^{31} +831.558i q^{35} +826.000 q^{37} +1811.61i q^{41} -1736.00 q^{43} +1306.73i q^{47} -1617.00 q^{49} +1794.64i q^{53} +504.000 q^{55} +4514.17i q^{59} -2618.00 q^{61} -3326.23i q^{65} +3784.00 q^{67} +8604.08i q^{71} +6608.00 q^{73} -475.176i q^{77} -4276.00 q^{79} +118.794i q^{83} +2646.00 q^{85} -4365.68i q^{89} -3136.00 q^{91} +16631.2i q^{95} -5824.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 56 q^{7}+O(q^{10}) \) 2 * q - 56 * q^7	\( 2 q - 56 q^{7} + 224 q^{13} - 1120 q^{19} - 514 q^{25} - 728 q^{31} + 1652 q^{37} - 3472 q^{43} - 3234 q^{49} + 1008 q^{55} - 5236 q^{61} + 7568 q^{67} + 13216 q^{73} - 8552 q^{79} + 5292 q^{85} - 6272 q^{91} - 11648 q^{97}+O(q^{100}) \) 2 * q - 56 * q^7 + 224 * q^13 - 1120 * q^19 - 514 * q^25 - 728 * q^31 + 1652 * q^37 - 3472 * q^43 - 3234 * q^49 + 1008 * q^55 - 5236 * q^61 + 7568 * q^67 + 13216 * q^73 - 8552 * q^79 + 5292 * q^85 - 6272 * q^91 - 11648 * q^97

Introduction

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