LMFDB - Embedded newform 2592.3.g.g.2431.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,3,Mod(2431,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2592.2431");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2592.g (of order $2$, degree $1$, 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:	$70.6268845222$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} + 2x^{8} + 8x^{7} + 33x^{6} - 184x^{5} + 165x^{4} + 200x^{3} + 250x^{2} - 2500x + 3125 $ x^10 - 4x^9 + 2x^8 + 8x^7 + 33x^6 - 184x^5 + 165x^4 + 200x^3 + 250x^2 - 2500*x + 3125
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.8
Root			$-1.50550 - 1.65332i$ of defining polynomial
Character	$\chi$	$=$	2592.2431
Dual form			2592.3.g.g.2431.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.93390 q^{5} +10.3494i q^{7} +O(q^{10})$	$q+1.93390 q^{5} +10.3494i q^{7} +21.0740i q^{11} +14.9419 q^{13} +5.89794 q^{17} +32.7678i q^{19} -4.96699i q^{23} -21.2600 q^{25} +18.6877 q^{29} +9.88195i q^{31} +20.0147i q^{35} +43.5347 q^{37} +15.6540 q^{41} +8.39046i q^{43} +49.9188i q^{47} -58.1102 q^{49} -75.8512 q^{53} +40.7551i q^{55} -67.9898i q^{59} -19.2884 q^{61} +28.8962 q^{65} -37.2966i q^{67} -95.4928i q^{71} +68.3482 q^{73} -218.104 q^{77} -84.5481i q^{79} -16.1810i q^{83} +11.4061 q^{85} +42.9444 q^{89} +154.640i q^{91} +63.3697i q^{95} -31.7977 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 14 q^{5}+O(q^{10}) $ 10 * q - 14 * q^5	$ 10 q - 14 q^{5} - 26 q^{13} + 36 q^{17} - 36 q^{25} + 134 q^{29} + 48 q^{37} + 26 q^{41} - 348 q^{49} - 96 q^{53} - 386 q^{61} + 106 q^{65} - 84 q^{73} - 58 q^{77} - 192 q^{85} - 120 q^{89} - 374 q^{97}+O(q^{100}) $ 10 * q - 14 * q^5 - 26 * q^13 + 36 * q^17 - 36 * q^25 + 134 * q^29 + 48 * q^37 + 26 * q^41 - 348 * q^49 - 96 * q^53 - 386 * q^61 + 106 * q^65 - 84 * q^73 - 58 * q^77 - 192 * q^85 - 120 * q^89 - 374 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	1.93390	0.386781	0.193390	−	0.981122i	$-0.438052\pi$
$5$	1.93390	0.386781	0.193390	+	0.981122i	$0.438052\pi$
$6$	0	0
$7$	10.3494i	1.47849i	0.673439	+	0.739243i	$0.264817\pi$
$7$	10.3494i	1.47849i	−0.673439	+	0.739243i	$0.735183\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	21.0740i	1.91582i	0.287068	+	0.957910i	$0.407319\pi$
$11$	21.0740i	1.91582i	−0.287068	+	0.957910i	$0.592681\pi$
$12$	0	0
$13$	14.9419	1.14938	0.574690	−	0.818372i	$-0.305123\pi$
$13$	14.9419	1.14938	0.574690	+	0.818372i	$0.305123\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.89794	0.346938	0.173469	−	0.984839i	$-0.444502\pi$
$17$	5.89794	0.346938	0.173469	+	0.984839i	$0.444502\pi$
$18$	0	0
$19$	32.7678i	1.72462i	0.506382	+	0.862309i	$0.330983\pi$
$19$	32.7678i	1.72462i	−0.506382	+	0.862309i	$0.669017\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.96699i	− 0.215956i	−0.994153	−	0.107978i	$-0.965562\pi$
$23$	− 4.96699i	− 0.215956i	0.994153	−	0.107978i	$-0.0344376\pi$
$24$	0	0
$25$	−21.2600	−0.850401
$26$	0	0
$27$	0	0
$28$	0	0
$29$	18.6877	0.644405	0.322202	−	0.946671i	$-0.395577\pi$
$29$	18.6877	0.644405	0.322202	+	0.946671i	$0.395577\pi$
$30$	0	0
$31$	9.88195i	0.318773i	0.987216	+	0.159386i	$0.0509515\pi$
$31$	9.88195i	0.318773i	−0.987216	+	0.159386i	$0.949048\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	20.0147i	0.571850i
$36$	0	0
$37$	43.5347	1.17661	0.588307	−	0.808638i	$-0.299795\pi$
$37$	43.5347	1.17661	0.588307	+	0.808638i	$0.299795\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	15.6540	0.381804	0.190902	−	0.981609i	$-0.438859\pi$
$41$	15.6540	0.381804	0.190902	+	0.981609i	$0.438859\pi$
$42$	0	0
$43$	8.39046i	0.195127i	0.995229	+	0.0975635i	$0.0311049\pi$
$43$	8.39046i	0.195127i	−0.995229	+	0.0975635i	$0.968895\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	49.9188i	1.06210i	0.847340	+	0.531051i	$0.178203\pi$
$47$	49.9188i	1.06210i	−0.847340	+	0.531051i	$0.821797\pi$
$48$	0	0
$49$	−58.1102	−1.18592
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−75.8512	−1.43115	−0.715577	−	0.698534i	$-0.753836\pi$
$53$	−75.8512	−1.43115	−0.715577	+	0.698534i	$0.753836\pi$
$54$	0	0
$55$	40.7551i	0.741002i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 67.9898i	− 1.15237i	−0.817320	−	0.576185i	$-0.804541\pi$
$59$	− 67.9898i	− 1.15237i	0.817320	−	0.576185i	$-0.195459\pi$
$60$	0	0
$61$	−19.2884	−0.316204	−0.158102	−	0.987423i	$-0.550537\pi$
$61$	−19.2884	−0.316204	−0.158102	+	0.987423i	$0.550537\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	28.8962	0.444558
$66$	0	0
$67$	− 37.2966i	− 0.556665i	−0.960485	−	0.278333i	$-0.910218\pi$
$67$	− 37.2966i	− 0.556665i	0.960485	−	0.278333i	$-0.0897818\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 95.4928i	− 1.34497i	−0.740111	−	0.672485i	$-0.765227\pi$
$71$	− 95.4928i	− 1.34497i	0.740111	−	0.672485i	$-0.234773\pi$
$72$	0	0
$73$	68.3482	0.936277	0.468138	−	0.883655i	$-0.344925\pi$
$73$	68.3482	0.936277	0.468138	+	0.883655i	$0.344925\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−218.104	−2.83251
$78$	0	0
$79$	− 84.5481i	− 1.07023i	−0.844779	−	0.535115i	$-0.820268\pi$
$79$	− 84.5481i	− 1.07023i	0.844779	−	0.535115i	$-0.179732\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 16.1810i	− 0.194952i	−0.995238	−	0.0974758i	$-0.968923\pi$
$83$	− 16.1810i	− 0.194952i	0.995238	−	0.0974758i	$-0.0310768\pi$
$84$	0	0
$85$	11.4061	0.134189
$86$	0	0
$87$	0	0
$88$	0	0
$89$	42.9444	0.482521	0.241260	−	0.970460i	$-0.422439\pi$
$89$	42.9444	0.482521	0.241260	+	0.970460i	$0.422439\pi$
$90$	0	0
$91$	154.640i	1.69934i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	63.3697i	0.667049i
$96$	0	0
$97$	−31.7977	−0.327811	−0.163905	−	0.986476i	$-0.552409\pi$
$97$	−31.7977	−0.327811	−0.163905	+	0.986476i	$0.552409\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	112.246	1.11134	0.555672	−	0.831402i	$-0.312461\pi$
$101$	112.246	1.11134	0.555672	+	0.831402i	$0.312461\pi$
$102$	0	0
$103$	55.7922i	0.541672i	0.962626	+	0.270836i	$0.0873001\pi$
$103$	55.7922i	0.541672i	−0.962626	+	0.270836i	$0.912700\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 89.4882i	− 0.836338i	−0.908369	−	0.418169i	$-0.862672\pi$
$107$	− 89.4882i	− 0.836338i	0.908369	−	0.418169i	$-0.137328\pi$
$108$	0	0
$109$	−158.842	−1.45727	−0.728634	−	0.684904i	$-0.759844\pi$
$109$	−158.842	−1.45727	−0.728634	+	0.684904i	$0.759844\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	50.8342	0.449860	0.224930	−	0.974375i	$-0.427785\pi$
$113$	50.8342	0.449860	0.224930	+	0.974375i	$0.427785\pi$
$114$	0	0
$115$	− 9.60568i	− 0.0835276i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	61.0402i	0.512943i
$120$	0	0
$121$	−323.114	−2.67037
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−89.4624	−0.715699
$126$	0	0
$127$	20.1417i	0.158596i	0.996851	+	0.0792979i	$0.0252678\pi$
$127$	20.1417i	0.158596i	−0.996851	+	0.0792979i	$0.974732\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 63.8071i	− 0.487077i	−0.969891	−	0.243539i	$-0.921692\pi$
$131$	− 63.8071i	− 0.487077i	0.969891	−	0.243539i	$-0.0783083\pi$
$132$	0	0
$133$	−339.127	−2.54982
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.9627	0.0800196	0.0400098	−	0.999199i	$-0.487261\pi$
$137$	10.9627	0.0800196	0.0400098	+	0.999199i	$0.487261\pi$
$138$	0	0
$139$	208.722i	1.50160i	0.660531	+	0.750799i	$0.270331\pi$
$139$	208.722i	1.50160i	−0.660531	+	0.750799i	$0.729669\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	314.887i	2.20200i
$144$	0	0
$145$	36.1403	0.249243
$146$	0	0
$147$	0	0
$148$	0	0
$149$	162.560	1.09101	0.545503	−	0.838109i	$-0.316339\pi$
$149$	162.560	1.09101	0.545503	+	0.838109i	$0.316339\pi$
$150$	0	0
$151$	111.271i	0.736895i	0.929649	+	0.368447i	$0.120111\pi$
$151$	111.271i	0.736895i	−0.929649	+	0.368447i	$0.879889\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	19.1107i	0.123295i
$156$	0	0
$157$	−94.1561	−0.599720	−0.299860	−	0.953983i	$-0.596940\pi$
$157$	−94.1561	−0.599720	−0.299860	+	0.953983i	$0.596940\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	51.4054	0.319288
$162$	0	0
$163$	− 151.014i	− 0.926468i	−0.886236	−	0.463234i	$-0.846689\pi$
$163$	− 151.014i	− 0.926468i	0.886236	−	0.463234i	$-0.153311\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 84.0612i	− 0.503360i	−0.967810	−	0.251680i	$-0.919017\pi$
$167$	− 84.0612i	− 0.503360i	0.967810	−	0.251680i	$-0.0809831\pi$
$168$	0	0
$169$	54.2612	0.321072
$170$	0	0
$171$	0	0
$172$	0	0
$173$	263.341	1.52220	0.761102	−	0.648632i	$-0.224658\pi$
$173$	263.341	1.52220	0.761102	+	0.648632i	$0.224658\pi$
$174$	0	0
$175$	− 220.028i	− 1.25731i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 215.534i	− 1.20410i	−0.798458	−	0.602050i	$-0.794351\pi$
$179$	− 215.534i	− 1.20410i	0.798458	−	0.602050i	$-0.205649\pi$
$180$	0	0
$181$	−332.798	−1.83866	−0.919332	−	0.393482i	$-0.871270\pi$
$181$	−332.798	−1.83866	−0.919332	+	0.393482i	$0.871270\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	84.1919	0.455092
$186$	0	0
$187$	124.293i	0.664671i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	132.056i	0.691392i	0.938347	+	0.345696i	$0.112357\pi$
$191$	132.056i	0.691392i	−0.938347	+	0.345696i	$0.887643\pi$
$192$	0	0
$193$	175.253	0.908046	0.454023	−	0.890990i	$-0.349988\pi$
$193$	175.253	0.908046	0.454023	+	0.890990i	$0.349988\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	308.985	1.56845	0.784227	−	0.620475i	$-0.213060\pi$
$197$	308.985	1.56845	0.784227	+	0.620475i	$0.213060\pi$
$198$	0	0
$199$	310.934i	1.56248i	0.624229	+	0.781241i	$0.285413\pi$
$199$	310.934i	1.56248i	−0.624229	+	0.781241i	$0.714587\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	193.407i	0.952743i
$204$	0	0
$205$	30.2733	0.147675
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−690.548	−3.30406
$210$	0	0
$211$	9.30228i	0.0440867i	0.999757	+	0.0220433i	$0.00701718\pi$
$211$	9.30228i	0.0440867i	−0.999757	+	0.0220433i	$0.992983\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.2263i	0.0754714i
$216$	0	0
$217$	−102.272	−0.471301
$218$	0	0
$219$	0	0
$220$	0	0
$221$	88.1267	0.398763
$222$	0	0
$223$	416.516i	1.86778i	0.357557	+	0.933891i	$0.383610\pi$
$223$	416.516i	1.86778i	−0.357557	+	0.933891i	$0.616390\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 356.641i	− 1.57111i	−0.618794	−	0.785553i	$-0.712378\pi$
$227$	− 356.641i	− 1.57111i	0.618794	−	0.785553i	$-0.287622\pi$
$228$	0	0
$229$	328.972	1.43656	0.718280	−	0.695754i	$-0.244930\pi$
$229$	328.972	1.43656	0.718280	+	0.695754i	$0.244930\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−65.3289	−0.280382	−0.140191	−	0.990124i	$-0.544772\pi$
$233$	−65.3289	−0.280382	−0.140191	+	0.990124i	$0.544772\pi$
$234$	0	0
$235$	96.5381i	0.410801i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	94.6208i	0.395903i	0.980212	+	0.197951i	$0.0634288\pi$
$239$	94.6208i	0.395903i	−0.980212	+	0.197951i	$0.936571\pi$
$240$	0	0
$241$	−84.5924	−0.351006	−0.175503	−	0.984479i	$-0.556155\pi$
$241$	−84.5924	−0.351006	−0.175503	+	0.984479i	$0.556155\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−112.379	−0.458692
$246$	0	0
$247$	489.613i	1.98224i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.43066i	0.00569986i	0.999996	+	0.00284993i	$0.000907162\pi$
$251$	1.43066i	0.00569986i	−0.999996	+	0.00284993i	$0.999093\pi$
$252$	0	0
$253$	104.674	0.413733
$254$	0	0
$255$	0	0
$256$	0	0
$257$	139.474	0.542702	0.271351	−	0.962480i	$-0.412530\pi$
$257$	139.474	0.542702	0.271351	+	0.962480i	$0.412530\pi$
$258$	0	0
$259$	450.558i	1.73961i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 270.638i	− 1.02904i	−0.857477	−	0.514522i	$-0.827970\pi$
$263$	− 270.638i	− 1.02904i	0.857477	−	0.514522i	$-0.172030\pi$
$264$	0	0
$265$	−146.689	−0.553543
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−176.888	−0.657577	−0.328788	−	0.944404i	$-0.606640\pi$
$269$	−176.888	−0.657577	−0.328788	+	0.944404i	$0.606640\pi$
$270$	0	0
$271$	− 205.187i	− 0.757147i	−0.925571	−	0.378574i	$-0.876415\pi$
$271$	− 205.187i	− 0.757147i	0.925571	−	0.378574i	$-0.123585\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 448.034i	− 1.62921i
$276$	0	0
$277$	241.990	0.873609	0.436805	−	0.899556i	$-0.356110\pi$
$277$	241.990	0.873609	0.436805	+	0.899556i	$0.356110\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−42.9124	−0.152713	−0.0763566	−	0.997081i	$-0.524329\pi$
$281$	−42.9124	−0.152713	−0.0763566	+	0.997081i	$0.524329\pi$
$282$	0	0
$283$	378.956i	1.33907i	0.742783	+	0.669533i	$0.233506\pi$
$283$	378.956i	1.33907i	−0.742783	+	0.669533i	$0.766494\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	162.009i	0.564492i
$288$	0	0
$289$	−254.214	−0.879634
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−112.533	−0.384070	−0.192035	−	0.981388i	$-0.561509\pi$
$293$	−112.533	−0.384070	−0.192035	+	0.981388i	$0.561509\pi$
$294$	0	0
$295$	− 131.486i	− 0.445714i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 74.2164i	− 0.248215i
$300$	0	0
$301$	−86.8363	−0.288493
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−37.3020	−0.122301
$306$	0	0
$307$	− 109.964i	− 0.358189i	−0.983832	−	0.179094i	$-0.942683\pi$
$307$	− 109.964i	− 0.358189i	0.983832	−	0.179094i	$-0.0573167\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	533.933i	1.71683i	0.512958	+	0.858413i	$0.328549\pi$
$311$	533.933i	1.71683i	−0.512958	+	0.858413i	$0.671451\pi$
$312$	0	0
$313$	157.746	0.503980	0.251990	−	0.967730i	$-0.418915\pi$
$313$	157.746	0.503980	0.251990	+	0.967730i	$0.418915\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−223.646	−0.705507	−0.352753	−	0.935716i	$-0.614755\pi$
$317$	−223.646	−0.705507	−0.352753	+	0.935716i	$0.614755\pi$
$318$	0	0
$319$	393.826i	1.23456i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	193.262i	0.598336i
$324$	0	0
$325$	−317.666	−0.977433
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−516.630	−1.57030
$330$	0	0
$331$	− 204.719i	− 0.618487i	−0.950983	−	0.309243i	$-0.899924\pi$
$331$	− 204.719i	− 0.618487i	0.950983	−	0.309243i	$-0.100076\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 72.1280i	− 0.215307i
$336$	0	0
$337$	−503.767	−1.49486	−0.747428	−	0.664343i	$-0.768712\pi$
$337$	−503.767	−1.49486	−0.747428	+	0.664343i	$0.768712\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−208.252	−0.610711
$342$	0	0
$343$	− 94.2847i	− 0.274883i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	295.678i	0.852097i	0.904700	+	0.426049i	$0.140095\pi$
$347$	295.678i	0.852097i	−0.904700	+	0.426049i	$0.859905\pi$
$348$	0	0
$349$	58.0930	0.166456	0.0832279	−	0.996531i	$-0.473477\pi$
$349$	58.0930	0.166456	0.0832279	+	0.996531i	$0.473477\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	500.315	1.41732	0.708662	−	0.705548i	$-0.249299\pi$
$353$	500.315	1.41732	0.708662	+	0.705548i	$0.249299\pi$
$354$	0	0
$355$	− 184.674i	− 0.520208i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 204.499i	− 0.569636i	−0.958582	−	0.284818i	$-0.908067\pi$
$359$	− 204.499i	− 0.569636i	0.958582	−	0.284818i	$-0.0919332\pi$
$360$	0	0
$361$	−712.726	−1.97431
$362$	0	0
$363$	0	0
$364$	0	0
$365$	132.179	0.362134
$366$	0	0
$367$	175.956i	0.479445i	0.970841	+	0.239723i	$0.0770565\pi$
$367$	175.956i	0.479445i	−0.970841	+	0.239723i	$0.922943\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 785.014i	− 2.11594i
$372$	0	0
$373$	−350.669	−0.940131	−0.470065	−	0.882632i	$-0.655770\pi$
$373$	−350.669	−0.940131	−0.470065	+	0.882632i	$0.655770\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	279.231	0.740665
$378$	0	0
$379$	79.8445i	0.210672i	0.994437	+	0.105336i	$0.0335917\pi$
$379$	79.8445i	0.210672i	−0.994437	+	0.105336i	$0.966408\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	296.521i	0.774206i	0.922036	+	0.387103i	$0.126524\pi$
$383$	296.521i	0.774206i	−0.922036	+	0.387103i	$0.873476\pi$
$384$	0	0
$385$	−421.791	−1.09556
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−126.050	−0.324036	−0.162018	−	0.986788i	$-0.551800\pi$
$389$	−126.050	−0.324036	−0.162018	+	0.986788i	$0.551800\pi$
$390$	0	0
$391$	− 29.2950i	− 0.0749233i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 163.508i	− 0.413944i
$396$	0	0
$397$	−1.66640	−0.00419748	−0.00209874	−	0.999998i	$-0.500668\pi$
$397$	−1.66640	−0.00419748	−0.00209874	+	0.999998i	$0.500668\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	558.042	1.39163	0.695813	−	0.718223i	$-0.255044\pi$
$401$	558.042	1.39163	0.695813	+	0.718223i	$0.255044\pi$
$402$	0	0
$403$	147.655i	0.366391i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	917.451i	2.25418i
$408$	0	0
$409$	199.468	0.487697	0.243849	−	0.969813i	$-0.421590\pi$
$409$	199.468	0.487697	0.243849	+	0.969813i	$0.421590\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	703.654	1.70376
$414$	0	0
$415$	− 31.2925i	− 0.0754035i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 346.158i	− 0.826153i	−0.910696	−	0.413076i	$-0.864454\pi$
$419$	− 346.158i	− 0.826153i	0.910696	−	0.413076i	$-0.135546\pi$
$420$	0	0
$421$	−321.186	−0.762913	−0.381456	−	0.924387i	$-0.624577\pi$
$421$	−321.186	−0.762913	−0.381456	+	0.924387i	$0.624577\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−125.390	−0.295036
$426$	0	0
$427$	− 199.624i	− 0.467503i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	734.565i	1.70433i	0.523276	+	0.852163i	$0.324710\pi$
$431$	734.565i	1.70433i	−0.523276	+	0.852163i	$0.675290\pi$
$432$	0	0
$433$	400.407	0.924729	0.462364	−	0.886690i	$-0.347001\pi$
$433$	400.407	0.924729	0.462364	+	0.886690i	$0.347001\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	162.757	0.372442
$438$	0	0
$439$	− 165.752i	− 0.377566i	−0.982019	−	0.188783i	$-0.939546\pi$
$439$	− 165.752i	− 0.377566i	0.982019	−	0.188783i	$-0.0604543\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	568.524i	1.28335i	0.766976	+	0.641675i	$0.221760\pi$
$443$	568.524i	1.28335i	−0.766976	+	0.641675i	$0.778240\pi$
$444$	0	0
$445$	83.0503	0.186630
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−183.191	−0.407999	−0.203999	−	0.978971i	$-0.565394\pi$
$449$	−183.191	−0.407999	−0.203999	+	0.978971i	$0.565394\pi$
$450$	0	0
$451$	329.892i	0.731468i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	299.059i	0.657272i
$456$	0	0
$457$	−389.680	−0.852692	−0.426346	−	0.904560i	$-0.640199\pi$
$457$	−389.680	−0.852692	−0.426346	+	0.904560i	$0.640199\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−468.673	−1.01664	−0.508322	−	0.861167i	$-0.669734\pi$
$461$	−468.673	−1.01664	−0.508322	+	0.861167i	$0.669734\pi$
$462$	0	0
$463$	− 440.274i	− 0.950917i	−0.879738	−	0.475458i	$-0.842282\pi$
$463$	− 440.274i	− 0.950917i	0.879738	−	0.475458i	$-0.157718\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 232.287i	− 0.497403i	−0.968580	−	0.248701i	$-0.919996\pi$
$467$	− 232.287i	− 0.497403i	0.968580	−	0.248701i	$-0.0800038\pi$
$468$	0	0
$469$	385.997	0.823022
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−176.821	−0.373828
$474$	0	0
$475$	− 696.643i	− 1.46662i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 63.2472i	− 0.132040i	−0.997818	−	0.0660200i	$-0.978970\pi$
$479$	− 63.2472i	− 0.132040i	0.997818	−	0.0660200i	$-0.0210301\pi$
$480$	0	0
$481$	650.492	1.35238
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−61.4936	−0.126791
$486$	0	0
$487$	− 65.1317i	− 0.133741i	−0.997762	−	0.0668704i	$-0.978699\pi$
$487$	− 65.1317i	− 0.133741i	0.997762	−	0.0668704i	$-0.0213014\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 684.941i	− 1.39499i	−0.716588	−	0.697496i	$-0.754297\pi$
$491$	− 684.941i	− 1.39499i	0.716588	−	0.697496i	$-0.245703\pi$
$492$	0	0
$493$	110.219	0.223568
$494$	0	0
$495$	0	0
$496$	0	0
$497$	988.294	1.98852
$498$	0	0
$499$	− 269.722i	− 0.540524i	−0.962787	−	0.270262i	$-0.912890\pi$
$499$	− 269.722i	− 0.540524i	0.962787	−	0.270262i	$-0.0871104\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	182.143i	0.362114i	0.983473	+	0.181057i	$0.0579518\pi$
$503$	182.143i	0.362114i	−0.983473	+	0.181057i	$0.942048\pi$
$504$	0	0
$505$	217.072	0.429846
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−243.743	−0.478866	−0.239433	−	0.970913i	$-0.576962\pi$
$509$	−243.743	−0.478866	−0.239433	+	0.970913i	$0.576962\pi$
$510$	0	0
$511$	707.363i	1.38427i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	107.897i	0.209508i
$516$	0	0
$517$	−1051.99	−2.03480
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−698.539	−1.34077	−0.670383	−	0.742016i	$-0.733870\pi$
$521$	−698.539	−1.34077	−0.670383	+	0.742016i	$0.733870\pi$
$522$	0	0
$523$	− 76.4187i	− 0.146116i	−0.997328	−	0.0730580i	$-0.976724\pi$
$523$	− 76.4187i	− 0.146116i	0.997328	−	0.0730580i	$-0.0232758\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	58.2832i	0.110594i
$528$	0	0
$529$	504.329	0.953363
$530$	0	0
$531$	0	0
$532$	0	0
$533$	233.901	0.438838
$534$	0	0
$535$	− 173.062i	− 0.323480i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1224.61i	− 2.27201i
$540$	0	0
$541$	319.631	0.590816	0.295408	−	0.955371i	$-0.404544\pi$
$541$	319.631	0.590816	0.295408	+	0.955371i	$0.404544\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−307.185	−0.563643
$546$	0	0
$547$	25.0779i	0.0458462i	0.999737	+	0.0229231i	$0.00729730\pi$
$547$	25.0779i	0.0458462i	−0.999737	+	0.0229231i	$0.992703\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	612.355i	1.11135i
$552$	0	0
$553$	875.023	1.58232
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−993.726	−1.78407	−0.892035	−	0.451967i	$-0.850722\pi$
$557$	−993.726	−1.78407	−0.892035	+	0.451967i	$0.850722\pi$
$558$	0	0
$559$	125.370i	0.224275i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 915.093i	− 1.62539i	−0.582692	−	0.812693i	$-0.698000\pi$
$563$	− 915.093i	− 1.62539i	0.582692	−	0.812693i	$-0.302000\pi$
$564$	0	0
$565$	98.3085	0.173997
$566$	0	0
$567$	0	0
$568$	0	0
$569$	173.487	0.304899	0.152449	−	0.988311i	$-0.451284\pi$
$569$	173.487	0.304899	0.152449	+	0.988311i	$0.451284\pi$
$570$	0	0
$571$	− 596.106i	− 1.04397i	−0.852955	−	0.521984i	$-0.825192\pi$
$571$	− 596.106i	− 1.04397i	0.852955	−	0.521984i	$-0.174808\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	105.598i	0.183649i
$576$	0	0
$577$	−121.875	−0.211222	−0.105611	−	0.994408i	$-0.533680\pi$
$577$	−121.875	−0.211222	−0.105611	+	0.994408i	$0.533680\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	167.463	0.288233
$582$	0	0
$583$	− 1598.49i	− 2.74183i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 324.844i	− 0.553397i	−0.960957	−	0.276698i	$-0.910760\pi$
$587$	− 324.844i	− 0.553397i	0.960957	−	0.276698i	$-0.0892403\pi$
$588$	0	0
$589$	−323.809	−0.549761
$590$	0	0
$591$	0	0
$592$	0	0
$593$	785.829	1.32518	0.662588	−	0.748984i	$-0.269458\pi$
$593$	785.829	1.32518	0.662588	+	0.748984i	$0.269458\pi$
$594$	0	0
$595$	118.046i	0.198396i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 720.109i	− 1.20219i	−0.799179	−	0.601093i	$-0.794732\pi$
$599$	− 720.109i	− 1.20219i	0.799179	−	0.601093i	$-0.205268\pi$
$600$	0	0
$601$	−566.907	−0.943273	−0.471637	−	0.881793i	$-0.656337\pi$
$601$	−566.907	−0.943273	−0.471637	+	0.881793i	$0.656337\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−624.872	−1.03285
$606$	0	0
$607$	− 79.1376i	− 0.130375i	−0.997873	−	0.0651875i	$-0.979235\pi$
$607$	− 79.1376i	− 0.130375i	0.997873	−	0.0651875i	$-0.0207645\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	745.883i	1.22076i
$612$	0	0
$613$	951.491	1.55219	0.776094	−	0.630617i	$-0.217198\pi$
$613$	951.491	1.55219	0.776094	+	0.630617i	$0.217198\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−304.870	−0.494117	−0.247059	−	0.969001i	$-0.579464\pi$
$617$	−304.870	−0.494117	−0.247059	+	0.969001i	$0.579464\pi$
$618$	0	0
$619$	51.2405i	0.0827794i	0.999143	+	0.0413897i	$0.0131785\pi$
$619$	51.2405i	0.0827794i	−0.999143	+	0.0413897i	$0.986821\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	444.449i	0.713401i
$624$	0	0
$625$	358.489	0.573582
$626$	0	0
$627$	0	0
$628$	0	0
$629$	256.765	0.408212
$630$	0	0
$631$	316.484i	0.501560i	0.968044	+	0.250780i	$0.0806871\pi$
$631$	316.484i	0.501560i	−0.968044	+	0.250780i	$0.919313\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	38.9521i	0.0613418i
$636$	0	0
$637$	−868.278	−1.36307
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1216.06	1.89714	0.948568	−	0.316572i	$-0.102532\pi$
$641$	1216.06	1.89714	0.948568	+	0.316572i	$0.102532\pi$
$642$	0	0
$643$	803.692i	1.24991i	0.780661	+	0.624955i	$0.214883\pi$
$643$	803.692i	1.24991i	−0.780661	+	0.624955i	$0.785117\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1091.43i	− 1.68690i	−0.537205	−	0.843452i	$-0.680520\pi$
$647$	− 1091.43i	− 1.68690i	0.537205	−	0.843452i	$-0.319480\pi$
$648$	0	0
$649$	1432.82	2.20773
$650$	0	0
$651$	0	0
$652$	0	0
$653$	51.6866	0.0791525	0.0395762	−	0.999217i	$-0.487399\pi$
$653$	51.6866	0.0791525	0.0395762	+	0.999217i	$0.487399\pi$
$654$	0	0
$655$	− 123.397i	− 0.188392i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 508.858i	− 0.772167i	−0.922464	−	0.386083i	$-0.873828\pi$
$659$	− 508.858i	− 0.772167i	0.922464	−	0.386083i	$-0.126172\pi$
$660$	0	0
$661$	327.890	0.496051	0.248026	−	0.968753i	$-0.420218\pi$
$661$	327.890	0.496051	0.248026	+	0.968753i	$0.420218\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−655.838	−0.986223
$666$	0	0
$667$	− 92.8218i	− 0.139163i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 406.485i	− 0.605789i
$672$	0	0
$673$	682.891	1.01470	0.507349	−	0.861741i	$-0.330626\pi$
$673$	682.891	1.01470	0.507349	+	0.861741i	$0.330626\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−356.947	−0.527249	−0.263624	−	0.964625i	$-0.584918\pi$
$677$	−356.947	−0.527249	−0.263624	+	0.964625i	$0.584918\pi$
$678$	0	0
$679$	− 329.087i	− 0.484664i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1099.06i	− 1.60917i	−0.593839	−	0.804584i	$-0.702388\pi$
$683$	− 1099.06i	− 1.60917i	0.593839	−	0.804584i	$-0.297612\pi$
$684$	0	0
$685$	21.2008	0.0309500
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1133.36	−1.64494
$690$	0	0
$691$	951.128i	1.37645i	0.725497	+	0.688226i	$0.241610\pi$
$691$	951.128i	1.37645i	−0.725497	+	0.688226i	$0.758390\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	403.648i	0.580789i
$696$	0	0
$697$	92.3263	0.132462
$698$	0	0
$699$	0	0
$700$	0	0
$701$	99.5247	0.141975	0.0709876	−	0.997477i	$-0.477385\pi$
$701$	99.5247	0.141975	0.0709876	+	0.997477i	$0.477385\pi$
$702$	0	0
$703$	1426.53i	2.02921i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1161.68i	1.64311i
$708$	0	0
$709$	1262.74	1.78101	0.890506	−	0.454972i	$-0.150351\pi$
$709$	1262.74	1.78101	0.890506	+	0.454972i	$0.150351\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	49.0835	0.0688409
$714$	0	0
$715$	608.960i	0.851693i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	875.064i	1.21706i	0.793532	+	0.608528i	$0.208240\pi$
$719$	875.064i	1.21706i	−0.793532	+	0.608528i	$0.791760\pi$
$720$	0	0
$721$	−577.416	−0.800854
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−397.302	−0.548002
$726$	0	0
$727$	− 149.447i	− 0.205567i	−0.994704	−	0.102784i	$-0.967225\pi$
$727$	− 149.447i	− 0.205567i	0.994704	−	0.102784i	$-0.0327749\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	49.4865i	0.0676970i
$732$	0	0
$733$	228.060	0.311133	0.155566	−	0.987825i	$-0.450280\pi$
$733$	228.060	0.311133	0.155566	+	0.987825i	$0.450280\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	785.989	1.06647
$738$	0	0
$739$	− 283.397i	− 0.383487i	−0.981445	−	0.191744i	$-0.938586\pi$
$739$	− 283.397i	− 0.383487i	0.981445	−	0.191744i	$-0.0614142\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	872.770i	1.17466i	0.809349	+	0.587328i	$0.199820\pi$
$743$	872.770i	1.17466i	−0.809349	+	0.587328i	$0.800180\pi$
$744$	0	0
$745$	314.375	0.421980
$746$	0	0
$747$	0	0
$748$	0	0
$749$	926.150	1.23651
$750$	0	0
$751$	− 807.936i	− 1.07581i	−0.843004	−	0.537907i	$-0.819215\pi$
$751$	− 807.936i	− 1.07581i	0.843004	−	0.537907i	$-0.180785\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	215.188i	0.285017i
$756$	0	0
$757$	447.804	0.591551	0.295775	−	0.955258i	$-0.404422\pi$
$757$	447.804	0.591551	0.295775	+	0.955258i	$0.404422\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1301.71	1.71052	0.855261	−	0.518198i	$-0.173397\pi$
$761$	1301.71	1.71052	0.855261	+	0.518198i	$0.173397\pi$
$762$	0	0
$763$	− 1643.92i	− 2.15455i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1015.90i	− 1.32451i
$768$	0	0
$769$	−117.540	−0.152848	−0.0764238	−	0.997075i	$-0.524350\pi$
$769$	−117.540	−0.152848	−0.0764238	+	0.997075i	$0.524350\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	748.233	0.967960	0.483980	−	0.875079i	$-0.339191\pi$
$773$	748.233	0.967960	0.483980	+	0.875079i	$0.339191\pi$
$774$	0	0
$775$	− 210.090i	− 0.271084i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	512.946i	0.658467i
$780$	0	0
$781$	2012.42	2.57672
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−182.089	−0.231960
$786$	0	0
$787$	− 637.848i	− 0.810481i	−0.914210	−	0.405240i	$-0.867188\pi$
$787$	− 637.848i	− 0.810481i	0.914210	−	0.405240i	$-0.132812\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	526.104i	0.665112i
$792$	0	0
$793$	−288.206	−0.363438
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−170.587	−0.214037	−0.107018	−	0.994257i	$-0.534130\pi$
$797$	−170.587	−0.214037	−0.107018	+	0.994257i	$0.534130\pi$
$798$	0	0
$799$	294.418i	0.368483i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1440.37i	1.79374i
$804$	0	0
$805$	99.4131	0.123494
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−969.433	−1.19831	−0.599155	−	0.800633i	$-0.704497\pi$
$809$	−969.433	−1.19831	−0.599155	+	0.800633i	$0.704497\pi$
$810$	0	0
$811$	− 230.645i	− 0.284396i	−0.989838	−	0.142198i	$-0.954583\pi$
$811$	− 230.645i	− 0.284396i	0.989838	−	0.142198i	$-0.0454170\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 292.047i	− 0.358340i
$816$	0	0
$817$	−274.937	−0.336520
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1255.18	1.52885	0.764424	−	0.644714i	$-0.223023\pi$
$821$	1255.18	1.52885	0.764424	+	0.644714i	$0.223023\pi$
$822$	0	0
$823$	398.088i	0.483704i	0.970313	+	0.241852i	$0.0777548\pi$
$823$	398.088i	0.483704i	−0.970313	+	0.241852i	$0.922245\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	406.686i	0.491760i	0.969300	+	0.245880i	$0.0790770\pi$
$827$	406.686i	0.491760i	−0.969300	+	0.245880i	$0.920923\pi$
$828$	0	0
$829$	1333.85	1.60899	0.804494	−	0.593961i	$-0.202437\pi$
$829$	1333.85	1.60899	0.804494	+	0.593961i	$0.202437\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−342.730	−0.411441
$834$	0	0
$835$	− 162.566i	− 0.194690i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 329.298i	− 0.392489i	−0.980555	−	0.196244i	$-0.937125\pi$
$839$	− 329.298i	− 0.392489i	0.980555	−	0.196244i	$-0.0628746\pi$
$840$	0	0
$841$	−491.769	−0.584743
$842$	0	0
$843$	0	0
$844$	0	0
$845$	104.936	0.124185
$846$	0	0
$847$	− 3344.04i	− 3.94810i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 216.236i	− 0.254097i
$852$	0	0
$853$	−491.822	−0.576579	−0.288290	−	0.957543i	$-0.593087\pi$
$853$	−491.822	−0.576579	−0.288290	+	0.957543i	$0.593087\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−586.968	−0.684910	−0.342455	−	0.939534i	$-0.611258\pi$
$857$	−586.968	−0.684910	−0.342455	+	0.939534i	$0.611258\pi$
$858$	0	0
$859$	255.780i	0.297765i	0.988855	+	0.148883i	$0.0475677\pi$
$859$	255.780i	0.297765i	−0.988855	+	0.148883i	$0.952432\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1123.57i	1.30193i	0.759107	+	0.650966i	$0.225636\pi$
$863$	1123.57i	1.30193i	−0.759107	+	0.650966i	$0.774364\pi$
$864$	0	0
$865$	509.277	0.588760
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1781.77	2.05037
$870$	0	0
$871$	− 557.283i	− 0.639820i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 925.883i	− 1.05815i
$876$	0	0
$877$	996.664	1.13645	0.568223	−	0.822874i	$-0.307631\pi$
$877$	996.664	1.13645	0.568223	+	0.822874i	$0.307631\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−804.052	−0.912659	−0.456329	−	0.889811i	$-0.650836\pi$
$881$	−804.052	−0.912659	−0.456329	+	0.889811i	$0.650836\pi$
$882$	0	0
$883$	− 852.030i	− 0.964926i	−0.875916	−	0.482463i	$-0.839742\pi$
$883$	− 852.030i	− 0.964926i	0.875916	−	0.482463i	$-0.160258\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	679.807i	0.766412i	0.923663	+	0.383206i	$0.125180\pi$
$887$	679.807i	0.766412i	−0.923663	+	0.383206i	$0.874820\pi$
$888$	0	0
$889$	−208.454	−0.234482
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1635.73	−1.83172
$894$	0	0
$895$	− 416.822i	− 0.465723i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	184.671i	0.205419i
$900$	0	0
$901$	−447.366	−0.496522
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−643.600	−0.711160
$906$	0	0
$907$	− 1193.04i	− 1.31537i	−0.753293	−	0.657685i	$-0.771536\pi$
$907$	− 1193.04i	− 1.31537i	0.753293	−	0.657685i	$-0.228464\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	538.731i	0.591362i	0.955287	+	0.295681i	$0.0955466\pi$
$911$	538.731i	0.591362i	−0.955287	+	0.295681i	$0.904453\pi$
$912$	0	0
$913$	340.998	0.373492
$914$	0	0
$915$	0	0
$916$	0	0
$917$	660.366	0.720137
$918$	0	0
$919$	− 1195.64i	− 1.30102i	−0.759498	−	0.650509i	$-0.774555\pi$
$919$	− 1195.64i	− 1.30102i	0.759498	−	0.650509i	$-0.225445\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1426.85i	− 1.54588i
$924$	0	0
$925$	−925.549	−1.00059
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1413.71	−1.52175	−0.760876	−	0.648897i	$-0.775230\pi$
$929$	−1413.71	−1.52175	−0.760876	+	0.648897i	$0.775230\pi$
$930$	0	0
$931$	− 1904.14i	− 2.04526i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	240.371i	0.257082i
$936$	0	0
$937$	−1275.33	−1.36108	−0.680541	−	0.732710i	$-0.738255\pi$
$937$	−1275.33	−1.36108	−0.680541	+	0.732710i	$0.738255\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	550.842	0.585379	0.292690	−	0.956207i	$-0.405450\pi$
$941$	550.842	0.585379	0.292690	+	0.956207i	$0.405450\pi$
$942$	0	0
$943$	− 77.7531i	− 0.0824530i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 804.423i	− 0.849444i	−0.905324	−	0.424722i	$-0.860372\pi$
$947$	− 804.423i	− 0.849444i	0.905324	−	0.424722i	$-0.139628\pi$
$948$	0	0
$949$	1021.25	1.07614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	721.612	0.757200	0.378600	−	0.925560i	$-0.376406\pi$
$953$	721.612	0.757200	0.378600	+	0.925560i	$0.376406\pi$
$954$	0	0
$955$	255.383i	0.267417i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	113.457i	0.118308i
$960$	0	0
$961$	863.347	0.898384
$962$	0	0
$963$	0	0
$964$	0	0
$965$	338.922	0.351215
$966$	0	0
$967$	− 1190.18i	− 1.23080i	−0.788215	−	0.615400i	$-0.788995\pi$
$967$	− 1190.18i	− 1.23080i	0.788215	−	0.615400i	$-0.211005\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1024.09i	− 1.05467i	−0.849657	−	0.527335i	$-0.823191\pi$
$971$	− 1024.09i	− 1.05467i	0.849657	−	0.527335i	$-0.176809\pi$
$972$	0	0
$973$	−2160.15	−2.22009
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1283.81	1.31404	0.657018	−	0.753875i	$-0.271818\pi$
$977$	1283.81	1.31404	0.657018	+	0.753875i	$0.271818\pi$
$978$	0	0
$979$	905.010i	0.924423i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 119.956i	− 0.122031i	−0.998137	−	0.0610154i	$-0.980566\pi$
$983$	− 119.956i	− 0.122031i	0.998137	−	0.0610154i	$-0.0194339\pi$
$984$	0	0
$985$	597.548	0.606647
$986$	0	0
$987$	0	0
$988$	0	0
$989$	41.6753	0.0421389
$990$	0	0
$991$	549.882i	0.554876i	0.960744	+	0.277438i	$0.0894853\pi$
$991$	549.882i	0.554876i	−0.960744	+	0.277438i	$0.910515\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	601.316i	0.604338i
$996$	0	0
$997$	40.5528	0.0406748	0.0203374	−	0.999793i	$-0.493526\pi$
$997$	40.5528	0.0406748	0.0203374	+	0.999793i	$0.493526\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.3.g.g.2431.8		10
3.2	odd	2		2592.3.g.h.2431.4		10
4.3	odd	2	inner	2592.3.g.g.2431.7		10
9.2	odd	6		864.3.o.b.415.8		20
9.4	even	3		288.3.o.b.223.7	yes	20
9.5	odd	6		864.3.o.b.127.7		20
9.7	even	3		288.3.o.b.31.4	✓	20
12.11	even	2		2592.3.g.h.2431.3		10
36.7	odd	6		288.3.o.b.31.7	yes	20
36.11	even	6		864.3.o.b.415.7		20
36.23	even	6		864.3.o.b.127.8		20
36.31	odd	6		288.3.o.b.223.4	yes	20
72.5	odd	6		1728.3.o.h.127.3		20
72.11	even	6		1728.3.o.h.1279.3		20
72.13	even	6		576.3.o.h.511.4		20
72.29	odd	6		1728.3.o.h.1279.4		20
72.43	odd	6		576.3.o.h.319.4		20
72.59	even	6		1728.3.o.h.127.4		20
72.61	even	6		576.3.o.h.319.7		20
72.67	odd	6		576.3.o.h.511.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.o.b.31.4	✓	20	9.7	even	3
288.3.o.b.31.7	yes	20	36.7	odd	6
288.3.o.b.223.4	yes	20	36.31	odd	6
288.3.o.b.223.7	yes	20	9.4	even	3
576.3.o.h.319.4		20	72.43	odd	6
576.3.o.h.319.7		20	72.61	even	6
576.3.o.h.511.4		20	72.13	even	6
576.3.o.h.511.7		20	72.67	odd	6
864.3.o.b.127.7		20	9.5	odd	6
864.3.o.b.127.8		20	36.23	even	6
864.3.o.b.415.7		20	36.11	even	6
864.3.o.b.415.8		20	9.2	odd	6
1728.3.o.h.127.3		20	72.5	odd	6
1728.3.o.h.127.4		20	72.59	even	6
1728.3.o.h.1279.3		20	72.11	even	6
1728.3.o.h.1279.4		20	72.29	odd	6
2592.3.g.g.2431.7		10	4.3	odd	2	inner
2592.3.g.g.2431.8		10	1.1	even	1	trivial
2592.3.g.h.2431.3		10	12.11	even	2
2592.3.g.h.2431.4		10	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2592.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.93390 q^{5} +10.3494i q^{7} +O(q^{10})\)	\(q+1.93390 q^{5} +10.3494i q^{7} +21.0740i q^{11} +14.9419 q^{13} +5.89794 q^{17} +32.7678i q^{19} -4.96699i q^{23} -21.2600 q^{25} +18.6877 q^{29} +9.88195i q^{31} +20.0147i q^{35} +43.5347 q^{37} +15.6540 q^{41} +8.39046i q^{43} +49.9188i q^{47} -58.1102 q^{49} -75.8512 q^{53} +40.7551i q^{55} -67.9898i q^{59} -19.2884 q^{61} +28.8962 q^{65} -37.2966i q^{67} -95.4928i q^{71} +68.3482 q^{73} -218.104 q^{77} -84.5481i q^{79} -16.1810i q^{83} +11.4061 q^{85} +42.9444 q^{89} +154.640i q^{91} +63.3697i q^{95} -31.7977 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 14 q^{5}+O(q^{10}) \) 10 * q - 14 * q^5	\( 10 q - 14 q^{5} - 26 q^{13} + 36 q^{17} - 36 q^{25} + 134 q^{29} + 48 q^{37} + 26 q^{41} - 348 q^{49} - 96 q^{53} - 386 q^{61} + 106 q^{65} - 84 q^{73} - 58 q^{77} - 192 q^{85} - 120 q^{89} - 374 q^{97}+O(q^{100}) \) 10 * q - 14 * q^5 - 26 * q^13 + 36 * q^17 - 36 * q^25 + 134 * q^29 + 48 * q^37 + 26 * q^41 - 348 * q^49 - 96 * q^53 - 386 * q^61 + 106 * q^65 - 84 * q^73 - 58 * q^77 - 192 * q^85 - 120 * q^89 - 374 * q^97

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