LMFDB - Embedded newform 144.3.e.a.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,3,Mod(17,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("144.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	144.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:	$3.92371580679$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	144.17
Dual form			144.3.e.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.07107i q^{5} -12.0000 q^{7} +O(q^{10})$	$q+7.07107i q^{5} -12.0000 q^{7} +5.65685i q^{11} -8.00000 q^{13} +9.89949i q^{17} +16.0000 q^{19} +39.5980i q^{23} -25.0000 q^{25} -29.6985i q^{29} +4.00000 q^{31} -84.8528i q^{35} +30.0000 q^{37} -21.2132i q^{41} +8.00000 q^{43} +16.9706i q^{47} +95.0000 q^{49} +49.4975i q^{53} -40.0000 q^{55} -79.1960i q^{59} -14.0000 q^{61} -56.5685i q^{65} +88.0000 q^{67} +28.2843i q^{71} -80.0000 q^{73} -67.8823i q^{77} -100.000 q^{79} +130.108i q^{83} -70.0000 q^{85} +148.492i q^{89} +96.0000 q^{91} +113.137i q^{95} -112.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 24 q^{7}+O(q^{10}) $ 2 * q - 24 * q^7	$ 2 q - 24 q^{7} - 16 q^{13} + 32 q^{19} - 50 q^{25} + 8 q^{31} + 60 q^{37} + 16 q^{43} + 190 q^{49} - 80 q^{55} - 28 q^{61} + 176 q^{67} - 160 q^{73} - 200 q^{79} - 140 q^{85} + 192 q^{91} - 224 q^{97}+O(q^{100}) $ 2 * q - 24 * q^7 - 16 * q^13 + 32 * q^19 - 50 * q^25 + 8 * q^31 + 60 * q^37 + 16 * q^43 + 190 * q^49 - 80 * q^55 - 28 * q^61 + 176 * q^67 - 160 * q^73 - 200 * q^79 - 140 * q^85 + 192 * q^91 - 224 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	7.07107i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$5$	7.07107i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$6$	0	0
$7$	−12.0000	−1.71429	−0.857143	−	0.515079i	$-0.827763\pi$
$7$	−12.0000	−1.71429	−0.857143	+	0.515079i	$0.827763\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.65685i	0.514259i	0.966377	+	0.257130i	$0.0827768\pi$
$11$	5.65685i	0.514259i	−0.966377	+	0.257130i	$0.917223\pi$
$12$	0	0
$13$	−8.00000	−0.615385	−0.307692	−	0.951486i	$-0.599557\pi$
$13$	−8.00000	−0.615385	−0.307692	+	0.951486i	$0.599557\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	9.89949i	0.582323i	0.956674	+	0.291162i	$0.0940417\pi$
$17$	9.89949i	0.582323i	−0.956674	+	0.291162i	$0.905958\pi$
$18$	0	0
$19$	16.0000	0.842105	0.421053	−	0.907036i	$-0.361661\pi$
$19$	16.0000	0.842105	0.421053	+	0.907036i	$0.361661\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	39.5980i	1.72165i	0.508900	+	0.860826i	$0.330052\pi$
$23$	39.5980i	1.72165i	−0.508900	+	0.860826i	$0.669948\pi$
$24$	0	0
$25$	−25.0000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 29.6985i	− 1.02409i	−0.858960	−	0.512043i	$-0.828889\pi$
$29$	− 29.6985i	− 1.02409i	0.858960	−	0.512043i	$-0.171111\pi$
$30$	0	0
$31$	4.00000	0.129032	0.0645161	−	0.997917i	$-0.479450\pi$
$31$	4.00000	0.129032	0.0645161	+	0.997917i	$0.479450\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 84.8528i	− 2.42437i
$36$	0	0
$37$	30.0000	0.810811	0.405405	−	0.914137i	$-0.367130\pi$
$37$	30.0000	0.810811	0.405405	+	0.914137i	$0.367130\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 21.2132i	− 0.517395i	−0.965958	−	0.258698i	$-0.916707\pi$
$41$	− 21.2132i	− 0.517395i	0.965958	−	0.258698i	$-0.0832933\pi$
$42$	0	0
$43$	8.00000	0.186047	0.0930233	−	0.995664i	$-0.470347\pi$
$43$	8.00000	0.186047	0.0930233	+	0.995664i	$0.470347\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	16.9706i	0.361076i	0.983568	+	0.180538i	$0.0577838\pi$
$47$	16.9706i	0.361076i	−0.983568	+	0.180538i	$0.942216\pi$
$48$	0	0
$49$	95.0000	1.93878
$50$	0	0
$51$	0	0
$52$	0	0
$53$	49.4975i	0.933915i	0.884280	+	0.466957i	$0.154650\pi$
$53$	49.4975i	0.933915i	−0.884280	+	0.466957i	$0.845350\pi$
$54$	0	0
$55$	−40.0000	−0.727273
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 79.1960i	− 1.34230i	−0.741320	−	0.671152i	$-0.765800\pi$
$59$	− 79.1960i	− 1.34230i	0.741320	−	0.671152i	$-0.234200\pi$
$60$	0	0
$61$	−14.0000	−0.229508	−0.114754	−	0.993394i	$-0.536608\pi$
$61$	−14.0000	−0.229508	−0.114754	+	0.993394i	$0.536608\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 56.5685i	− 0.870285i
$66$	0	0
$67$	88.0000	1.31343	0.656716	−	0.754138i	$-0.271945\pi$
$67$	88.0000	1.31343	0.656716	+	0.754138i	$0.271945\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	28.2843i	0.398370i	0.979962	+	0.199185i	$0.0638295\pi$
$71$	28.2843i	0.398370i	−0.979962	+	0.199185i	$0.936171\pi$
$72$	0	0
$73$	−80.0000	−1.09589	−0.547945	−	0.836514i	$-0.684590\pi$
$73$	−80.0000	−1.09589	−0.547945	+	0.836514i	$0.684590\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 67.8823i	− 0.881588i
$78$	0	0
$79$	−100.000	−1.26582	−0.632911	−	0.774224i	$-0.718140\pi$
$79$	−100.000	−1.26582	−0.632911	+	0.774224i	$0.718140\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	130.108i	1.56756i	0.621037	+	0.783781i	$0.286712\pi$
$83$	130.108i	1.56756i	−0.621037	+	0.783781i	$0.713288\pi$
$84$	0	0
$85$	−70.0000	−0.823529
$86$	0	0
$87$	0	0
$88$	0	0
$89$	148.492i	1.66845i	0.551421	+	0.834227i	$0.314086\pi$
$89$	148.492i	1.66845i	−0.551421	+	0.834227i	$0.685914\pi$
$90$	0	0
$91$	96.0000	1.05495
$92$	0	0
$93$	0	0
$94$	0	0
$95$	113.137i	1.19092i
$96$	0	0
$97$	−112.000	−1.15464	−0.577320	−	0.816518i	$-0.695901\pi$
$97$	−112.000	−1.15464	−0.577320	+	0.816518i	$0.695901\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	63.6396i	0.630095i	0.949076	+	0.315048i	$0.102020\pi$
$101$	63.6396i	0.630095i	−0.949076	+	0.315048i	$0.897980\pi$
$102$	0	0
$103$	44.0000	0.427184	0.213592	−	0.976923i	$-0.431484\pi$
$103$	44.0000	0.427184	0.213592	+	0.976923i	$0.431484\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	136.000	1.24771	0.623853	−	0.781542i	$-0.285566\pi$
$109$	136.000	1.24771	0.623853	+	0.781542i	$0.285566\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 21.2132i	− 0.187727i	−0.995585	−	0.0938637i	$-0.970078\pi$
$113$	− 21.2132i	− 0.187727i	0.995585	−	0.0938637i	$-0.0299218\pi$
$114$	0	0
$115$	−280.000	−2.43478
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 118.794i	− 0.998268i
$120$	0	0
$121$	89.0000	0.735537
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.0000	0.157480	0.0787402	−	0.996895i	$-0.474910\pi$
$127$	20.0000	0.157480	0.0787402	+	0.996895i	$0.474910\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 56.5685i	− 0.431821i	−0.976413	−	0.215910i	$-0.930728\pi$
$131$	− 56.5685i	− 0.431821i	0.976413	−	0.215910i	$-0.0692719\pi$
$132$	0	0
$133$	−192.000	−1.44361
$134$	0	0
$135$	0	0
$136$	0	0
$137$	89.0955i	0.650332i	0.945657	+	0.325166i	$0.105420\pi$
$137$	89.0955i	0.650332i	−0.945657	+	0.325166i	$0.894580\pi$
$138$	0	0
$139$	−120.000	−0.863309	−0.431655	−	0.902039i	$-0.642070\pi$
$139$	−120.000	−0.863309	−0.431655	+	0.902039i	$0.642070\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 45.2548i	− 0.316467i
$144$	0	0
$145$	210.000	1.44828
$146$	0	0
$147$	0	0
$148$	0	0
$149$	128.693i	0.863714i	0.901942	+	0.431857i	$0.142142\pi$
$149$	128.693i	0.863714i	−0.901942	+	0.431857i	$0.857858\pi$
$150$	0	0
$151$	196.000	1.29801	0.649007	−	0.760783i	$-0.275185\pi$
$151$	196.000	1.29801	0.649007	+	0.760783i	$0.275185\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	28.2843i	0.182479i
$156$	0	0
$157$	238.000	1.51592	0.757962	−	0.652299i	$-0.226195\pi$
$157$	238.000	1.51592	0.757962	+	0.652299i	$0.226195\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 475.176i	− 2.95140i
$162$	0	0
$163$	168.000	1.03067	0.515337	−	0.856987i	$-0.327667\pi$
$163$	168.000	1.03067	0.515337	+	0.856987i	$0.327667\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 169.706i	− 1.01620i	−0.861298	−	0.508101i	$-0.830348\pi$
$167$	− 169.706i	− 1.01620i	0.861298	−	0.508101i	$-0.169652\pi$
$168$	0	0
$169$	−105.000	−0.621302
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 207.889i	− 1.20167i	−0.799372	−	0.600836i	$-0.794834\pi$
$173$	− 207.889i	− 1.20167i	0.799372	−	0.600836i	$-0.205166\pi$
$174$	0	0
$175$	300.000	1.71429
$176$	0	0
$177$	0	0
$178$	0	0
$179$	203.647i	1.13769i	0.822444	+	0.568846i	$0.192610\pi$
$179$	203.647i	1.13769i	−0.822444	+	0.568846i	$0.807390\pi$
$180$	0	0
$181$	−216.000	−1.19337	−0.596685	−	0.802475i	$-0.703516\pi$
$181$	−216.000	−1.19337	−0.596685	+	0.802475i	$0.703516\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	212.132i	1.14666i
$186$	0	0
$187$	−56.0000	−0.299465
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 373.352i	− 1.95472i	−0.211573	−	0.977362i	$-0.567859\pi$
$191$	− 373.352i	− 1.95472i	0.211573	−	0.977362i	$-0.432141\pi$
$192$	0	0
$193$	−178.000	−0.922280	−0.461140	−	0.887327i	$-0.652559\pi$
$193$	−178.000	−0.922280	−0.461140	+	0.887327i	$0.652559\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 207.889i	− 1.05528i	−0.849469	−	0.527638i	$-0.823078\pi$
$197$	− 207.889i	− 1.05528i	0.849469	−	0.527638i	$-0.176922\pi$
$198$	0	0
$199$	−196.000	−0.984925	−0.492462	−	0.870334i	$-0.663903\pi$
$199$	−196.000	−0.984925	−0.492462	+	0.870334i	$0.663903\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	356.382i	1.75558i
$204$	0	0
$205$	150.000	0.731707
$206$	0	0
$207$	0	0
$208$	0	0
$209$	90.5097i	0.433061i
$210$	0	0
$211$	56.0000	0.265403	0.132701	−	0.991156i	$-0.457635\pi$
$211$	56.0000	0.265403	0.132701	+	0.991156i	$0.457635\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	56.5685i	0.263109i
$216$	0	0
$217$	−48.0000	−0.221198
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 79.1960i	− 0.358353i
$222$	0	0
$223$	−28.0000	−0.125561	−0.0627803	−	0.998027i	$-0.519997\pi$
$223$	−28.0000	−0.125561	−0.0627803	+	0.998027i	$0.519997\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 84.8528i	− 0.373801i	−0.982379	−	0.186900i	$-0.940156\pi$
$227$	− 84.8528i	− 0.373801i	0.982379	−	0.186900i	$-0.0598442\pi$
$228$	0	0
$229$	184.000	0.803493	0.401747	−	0.915751i	$-0.368403\pi$
$229$	184.000	0.803493	0.401747	+	0.915751i	$0.368403\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	304.056i	1.30496i	0.757806	+	0.652481i	$0.226272\pi$
$233$	304.056i	1.30496i	−0.757806	+	0.652481i	$0.773728\pi$
$234$	0	0
$235$	−120.000	−0.510638
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 226.274i	− 0.946754i	−0.880860	−	0.473377i	$-0.843035\pi$
$239$	− 226.274i	− 0.946754i	0.880860	−	0.473377i	$-0.156965\pi$
$240$	0	0
$241$	384.000	1.59336	0.796680	−	0.604401i	$-0.206587\pi$
$241$	384.000	1.59336	0.796680	+	0.604401i	$0.206587\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	671.751i	2.74184i
$246$	0	0
$247$	−128.000	−0.518219
$248$	0	0
$249$	0	0
$250$	0	0
$251$	197.990i	0.788804i	0.918938	+	0.394402i	$0.129048\pi$
$251$	197.990i	0.788804i	−0.918938	+	0.394402i	$0.870952\pi$
$252$	0	0
$253$	−224.000	−0.885375
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 91.9239i	− 0.357680i	−0.983878	−	0.178840i	$-0.942765\pi$
$257$	− 91.9239i	− 0.357680i	0.983878	−	0.178840i	$-0.0572345\pi$
$258$	0	0
$259$	−360.000	−1.38996
$260$	0	0
$261$	0	0
$262$	0	0
$263$	395.980i	1.50563i	0.658234	+	0.752813i	$0.271304\pi$
$263$	395.980i	1.50563i	−0.658234	+	0.752813i	$0.728696\pi$
$264$	0	0
$265$	−350.000	−1.32075
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 83.4386i	− 0.310181i	−0.987900	−	0.155090i	$-0.950433\pi$
$269$	− 83.4386i	− 0.310181i	0.987900	−	0.155090i	$-0.0495669\pi$
$270$	0	0
$271$	−140.000	−0.516605	−0.258303	−	0.966064i	$-0.583163\pi$
$271$	−140.000	−0.516605	−0.258303	+	0.966064i	$0.583163\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 141.421i	− 0.514259i
$276$	0	0
$277$	−248.000	−0.895307	−0.447653	−	0.894207i	$-0.647740\pi$
$277$	−248.000	−0.895307	−0.447653	+	0.894207i	$0.647740\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 168.291i	− 0.598902i	−0.954112	−	0.299451i	$-0.903197\pi$
$281$	− 168.291i	− 0.598902i	0.954112	−	0.299451i	$-0.0968035\pi$
$282$	0	0
$283$	−368.000	−1.30035	−0.650177	−	0.759783i	$-0.725305\pi$
$283$	−368.000	−1.30035	−0.650177	+	0.759783i	$0.725305\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	254.558i	0.886963i
$288$	0	0
$289$	191.000	0.660900
$290$	0	0
$291$	0	0
$292$	0	0
$293$	207.889i	0.709520i	0.934957	+	0.354760i	$0.115437\pi$
$293$	207.889i	0.709520i	−0.934957	+	0.354760i	$0.884563\pi$
$294$	0	0
$295$	560.000	1.89831
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 316.784i	− 1.05948i
$300$	0	0
$301$	−96.0000	−0.318937
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 98.9949i	− 0.324574i
$306$	0	0
$307$	552.000	1.79805	0.899023	−	0.437902i	$-0.144278\pi$
$307$	552.000	1.79805	0.899023	+	0.437902i	$0.144278\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 33.9411i	− 0.109135i	−0.998510	−	0.0545677i	$-0.982622\pi$
$311$	− 33.9411i	− 0.109135i	0.998510	−	0.0545677i	$-0.0173781\pi$
$312$	0	0
$313$	−222.000	−0.709265	−0.354633	−	0.935006i	$-0.615394\pi$
$313$	−222.000	−0.709265	−0.354633	+	0.935006i	$0.615394\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 445.477i	− 1.40529i	−0.711540	−	0.702646i	$-0.752002\pi$
$317$	− 445.477i	− 1.40529i	0.711540	−	0.702646i	$-0.247998\pi$
$318$	0	0
$319$	168.000	0.526646
$320$	0	0
$321$	0	0
$322$	0	0
$323$	158.392i	0.490377i
$324$	0	0
$325$	200.000	0.615385
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 203.647i	− 0.618987i
$330$	0	0
$331$	520.000	1.57100	0.785498	−	0.618864i	$-0.212407\pi$
$331$	520.000	1.57100	0.785498	+	0.618864i	$0.212407\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	622.254i	1.85747i
$336$	0	0
$337$	48.0000	0.142433	0.0712166	−	0.997461i	$-0.477312\pi$
$337$	48.0000	0.142433	0.0712166	+	0.997461i	$0.477312\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	22.6274i	0.0663561i
$342$	0	0
$343$	−552.000	−1.60933
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 356.382i	− 1.02704i	−0.858079	−	0.513518i	$-0.828342\pi$
$347$	− 356.382i	− 1.02704i	0.858079	−	0.513518i	$-0.171658\pi$
$348$	0	0
$349$	210.000	0.601719	0.300860	−	0.953668i	$-0.402726\pi$
$349$	210.000	0.601719	0.300860	+	0.953668i	$0.402726\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	272.943i	0.773210i	0.922245	+	0.386605i	$0.126352\pi$
$353$	272.943i	0.773210i	−0.922245	+	0.386605i	$0.873648\pi$
$354$	0	0
$355$	−200.000	−0.563380
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 593.970i	− 1.65451i	−0.561825	−	0.827256i	$-0.689901\pi$
$359$	− 593.970i	− 1.65451i	0.561825	−	0.827256i	$-0.310099\pi$
$360$	0	0
$361$	−105.000	−0.290859
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 565.685i	− 1.54982i
$366$	0	0
$367$	−172.000	−0.468665	−0.234332	−	0.972157i	$-0.575290\pi$
$367$	−172.000	−0.468665	−0.234332	+	0.972157i	$0.575290\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 593.970i	− 1.60100i
$372$	0	0
$373$	2.00000	0.00536193	0.00268097	−	0.999996i	$-0.499147\pi$
$373$	2.00000	0.00536193	0.00268097	+	0.999996i	$0.499147\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	237.588i	0.630207i
$378$	0	0
$379$	736.000	1.94195	0.970976	−	0.239176i	$-0.0768773\pi$
$379$	736.000	1.94195	0.970976	+	0.239176i	$0.0768773\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	480.000	1.24675
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 459.619i	− 1.18154i	−0.806840	−	0.590770i	$-0.798824\pi$
$389$	− 459.619i	− 1.18154i	0.806840	−	0.590770i	$-0.201176\pi$
$390$	0	0
$391$	−392.000	−1.00256
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 707.107i	− 1.79014i
$396$	0	0
$397$	338.000	0.851385	0.425693	−	0.904868i	$-0.360030\pi$
$397$	338.000	0.851385	0.425693	+	0.904868i	$0.360030\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 148.492i	− 0.370305i	−0.982710	−	0.185153i	$-0.940722\pi$
$401$	− 148.492i	− 0.370305i	0.982710	−	0.185153i	$-0.0592780\pi$
$402$	0	0
$403$	−32.0000	−0.0794045
$404$	0	0
$405$	0	0
$406$	0	0
$407$	169.706i	0.416967i
$408$	0	0
$409$	144.000	0.352078	0.176039	−	0.984383i	$-0.443671\pi$
$409$	144.000	0.352078	0.176039	+	0.984383i	$0.443671\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	950.352i	2.30109i
$414$	0	0
$415$	−920.000	−2.21687
$416$	0	0
$417$	0	0
$418$	0	0
$419$	763.675i	1.82261i	0.411727	+	0.911307i	$0.364926\pi$
$419$	763.675i	1.82261i	−0.411727	+	0.911307i	$0.635074\pi$
$420$	0	0
$421$	680.000	1.61520	0.807601	−	0.589729i	$-0.200766\pi$
$421$	680.000	1.61520	0.807601	+	0.589729i	$0.200766\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 247.487i	− 0.582323i
$426$	0	0
$427$	168.000	0.393443
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 5.65685i	− 0.0131250i	−0.999978	−	0.00656248i	$-0.997911\pi$
$431$	− 5.65685i	− 0.0131250i	0.999978	−	0.00656248i	$-0.00208892\pi$
$432$	0	0
$433$	−482.000	−1.11316	−0.556582	−	0.830793i	$-0.687887\pi$
$433$	−482.000	−1.11316	−0.556582	+	0.830793i	$0.687887\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	633.568i	1.44981i
$438$	0	0
$439$	756.000	1.72210	0.861048	−	0.508524i	$-0.169809\pi$
$439$	756.000	1.72210	0.861048	+	0.508524i	$0.169809\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	197.990i	0.446930i	0.974712	+	0.223465i	$0.0717368\pi$
$443$	197.990i	0.446930i	−0.974712	+	0.223465i	$0.928263\pi$
$444$	0	0
$445$	−1050.00	−2.35955
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 77.7817i	− 0.173233i	−0.996242	−	0.0866166i	$-0.972394\pi$
$449$	− 77.7817i	− 0.173233i	0.996242	−	0.0866166i	$-0.0276055\pi$
$450$	0	0
$451$	120.000	0.266075
$452$	0	0
$453$	0	0
$454$	0	0
$455$	678.823i	1.49192i
$456$	0	0
$457$	464.000	1.01532	0.507659	−	0.861558i	$-0.330511\pi$
$457$	464.000	1.01532	0.507659	+	0.861558i	$0.330511\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	49.4975i	0.107370i	0.998558	+	0.0536849i	$0.0170967\pi$
$461$	49.4975i	0.107370i	−0.998558	+	0.0536849i	$0.982903\pi$
$462$	0	0
$463$	−20.0000	−0.0431965	−0.0215983	−	0.999767i	$-0.506875\pi$
$463$	−20.0000	−0.0431965	−0.0215983	+	0.999767i	$0.506875\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 695.793i	− 1.48992i	−0.667109	−	0.744960i	$-0.732468\pi$
$467$	− 695.793i	− 1.48992i	0.667109	−	0.744960i	$-0.267532\pi$
$468$	0	0
$469$	−1056.00	−2.25160
$470$	0	0
$471$	0	0
$472$	0	0
$473$	45.2548i	0.0956762i
$474$	0	0
$475$	−400.000	−0.842105
$476$	0	0
$477$	0	0
$478$	0	0
$479$	673.166i	1.40536i	0.711508	+	0.702678i	$0.248013\pi$
$479$	673.166i	1.40536i	−0.711508	+	0.702678i	$0.751987\pi$
$480$	0	0
$481$	−240.000	−0.498960
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 791.960i	− 1.63291i
$486$	0	0
$487$	−388.000	−0.796715	−0.398357	−	0.917230i	$-0.630420\pi$
$487$	−388.000	−0.796715	−0.398357	+	0.917230i	$0.630420\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	294.000	0.596349
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 339.411i	− 0.682920i
$498$	0	0
$499$	−96.0000	−0.192385	−0.0961924	−	0.995363i	$-0.530666\pi$
$499$	−96.0000	−0.192385	−0.0961924	+	0.995363i	$0.530666\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	424.264i	0.843467i	0.906720	+	0.421734i	$0.138578\pi$
$503$	424.264i	0.843467i	−0.906720	+	0.421734i	$0.861422\pi$
$504$	0	0
$505$	−450.000	−0.891089
$506$	0	0
$507$	0	0
$508$	0	0
$509$	309.713i	0.608473i	0.952597	+	0.304237i	$0.0984013\pi$
$509$	309.713i	0.608473i	−0.952597	+	0.304237i	$0.901599\pi$
$510$	0	0
$511$	960.000	1.87867
$512$	0	0
$513$	0	0
$514$	0	0
$515$	311.127i	0.604130i
$516$	0	0
$517$	−96.0000	−0.185687
$518$	0	0
$519$	0	0
$520$	0	0
$521$	722.663i	1.38707i	0.720423	+	0.693535i	$0.243948\pi$
$521$	722.663i	1.38707i	−0.720423	+	0.693535i	$0.756052\pi$
$522$	0	0
$523$	−352.000	−0.673040	−0.336520	−	0.941676i	$-0.609250\pi$
$523$	−352.000	−0.673040	−0.336520	+	0.941676i	$0.609250\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	39.5980i	0.0751385i
$528$	0	0
$529$	−1039.00	−1.96408
$530$	0	0
$531$	0	0
$532$	0	0
$533$	169.706i	0.318397i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	537.401i	0.997034i
$540$	0	0
$541$	936.000	1.73013	0.865065	−	0.501660i	$-0.167277\pi$
$541$	936.000	1.73013	0.865065	+	0.501660i	$0.167277\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	961.665i	1.76452i
$546$	0	0
$547$	−56.0000	−0.102377	−0.0511883	−	0.998689i	$-0.516301\pi$
$547$	−56.0000	−0.102377	−0.0511883	+	0.998689i	$0.516301\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 475.176i	− 0.862388i
$552$	0	0
$553$	1200.00	2.16998
$554$	0	0
$555$	0	0
$556$	0	0
$557$	321.026i	0.576349i	0.957578	+	0.288175i	$0.0930483\pi$
$557$	321.026i	0.576349i	−0.957578	+	0.288175i	$0.906952\pi$
$558$	0	0
$559$	−64.0000	−0.114490
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 469.519i	− 0.833959i	−0.908916	−	0.416979i	$-0.863089\pi$
$563$	− 469.519i	− 0.833959i	0.908916	−	0.416979i	$-0.136911\pi$
$564$	0	0
$565$	150.000	0.265487
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 408.708i	− 0.718291i	−0.933282	−	0.359146i	$-0.883068\pi$
$569$	− 408.708i	− 0.718291i	0.933282	−	0.359146i	$-0.116932\pi$
$570$	0	0
$571$	−816.000	−1.42907	−0.714536	−	0.699599i	$-0.753362\pi$
$571$	−816.000	−1.42907	−0.714536	+	0.699599i	$0.753362\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 989.949i	− 1.72165i
$576$	0	0
$577$	242.000	0.419411	0.209705	−	0.977765i	$-0.432750\pi$
$577$	242.000	0.419411	0.209705	+	0.977765i	$0.432750\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1561.29i	− 2.68725i
$582$	0	0
$583$	−280.000	−0.480274
$584$	0	0
$585$	0	0
$586$	0	0
$587$	395.980i	0.674582i	0.941400	+	0.337291i	$0.109511\pi$
$587$	395.980i	0.674582i	−0.941400	+	0.337291i	$0.890489\pi$
$588$	0	0
$589$	64.0000	0.108659
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 205.061i	− 0.345803i	−0.984939	−	0.172901i	$-0.944686\pi$
$593$	− 205.061i	− 0.345803i	0.984939	−	0.172901i	$-0.0553142\pi$
$594$	0	0
$595$	840.000	1.41176
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 593.970i	− 0.991602i	−0.868436	−	0.495801i	$-0.834875\pi$
$599$	− 593.970i	− 0.991602i	0.868436	−	0.495801i	$-0.165125\pi$
$600$	0	0
$601$	286.000	0.475874	0.237937	−	0.971281i	$-0.423529\pi$
$601$	286.000	0.475874	0.237937	+	0.971281i	$0.423529\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	629.325i	1.04021i
$606$	0	0
$607$	100.000	0.164745	0.0823723	−	0.996602i	$-0.473750\pi$
$607$	100.000	0.164745	0.0823723	+	0.996602i	$0.473750\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 135.765i	− 0.222200i
$612$	0	0
$613$	770.000	1.25612	0.628059	−	0.778166i	$-0.283850\pi$
$613$	770.000	1.25612	0.628059	+	0.778166i	$0.283850\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 869.741i	− 1.40963i	−0.709391	−	0.704815i	$-0.751030\pi$
$617$	− 869.741i	− 1.40963i	0.709391	−	0.704815i	$-0.248970\pi$
$618$	0	0
$619$	−256.000	−0.413570	−0.206785	−	0.978386i	$-0.566300\pi$
$619$	−256.000	−0.413570	−0.206785	+	0.978386i	$0.566300\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1781.91i	− 2.86021i
$624$	0	0
$625$	−625.000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	296.985i	0.472154i
$630$	0	0
$631$	−196.000	−0.310618	−0.155309	−	0.987866i	$-0.549637\pi$
$631$	−196.000	−0.310618	−0.155309	+	0.987866i	$0.549637\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	141.421i	0.222711i
$636$	0	0
$637$	−760.000	−1.19309
$638$	0	0
$639$	0	0
$640$	0	0
$641$	643.467i	1.00385i	0.864911	+	0.501924i	$0.167374\pi$
$641$	643.467i	1.00385i	−0.864911	+	0.501924i	$0.832626\pi$
$642$	0	0
$643$	−120.000	−0.186625	−0.0933126	−	0.995637i	$-0.529746\pi$
$643$	−120.000	−0.186625	−0.0933126	+	0.995637i	$0.529746\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	707.107i	1.09290i	0.837491	+	0.546450i	$0.184021\pi$
$647$	707.107i	1.09290i	−0.837491	+	0.546450i	$0.815979\pi$
$648$	0	0
$649$	448.000	0.690293
$650$	0	0
$651$	0	0
$652$	0	0
$653$	131.522i	0.201412i	0.994916	+	0.100706i	$0.0321101\pi$
$653$	131.522i	0.201412i	−0.994916	+	0.100706i	$0.967890\pi$
$654$	0	0
$655$	400.000	0.610687
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1187.94i	1.80264i	0.433154	+	0.901320i	$0.357400\pi$
$659$	1187.94i	1.80264i	−0.433154	+	0.901320i	$0.642600\pi$
$660$	0	0
$661$	−130.000	−0.196672	−0.0983359	−	0.995153i	$-0.531352\pi$
$661$	−130.000	−0.196672	−0.0983359	+	0.995153i	$0.531352\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1357.65i	− 2.04157i
$666$	0	0
$667$	1176.00	1.76312
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 79.1960i	− 0.118027i
$672$	0	0
$673$	610.000	0.906389	0.453195	−	0.891412i	$-0.350284\pi$
$673$	610.000	0.906389	0.453195	+	0.891412i	$0.350284\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 603.869i	− 0.891978i	−0.895038	−	0.445989i	$-0.852852\pi$
$677$	− 603.869i	− 0.891978i	0.895038	−	0.445989i	$-0.147148\pi$
$678$	0	0
$679$	1344.00	1.97938
$680$	0	0
$681$	0	0
$682$	0	0
$683$	989.949i	1.44941i	0.689057	+	0.724707i	$0.258025\pi$
$683$	989.949i	1.44941i	−0.689057	+	0.724707i	$0.741975\pi$
$684$	0	0
$685$	−630.000	−0.919708
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 395.980i	− 0.574717i
$690$	0	0
$691$	−632.000	−0.914616	−0.457308	−	0.889308i	$-0.651186\pi$
$691$	−632.000	−0.914616	−0.457308	+	0.889308i	$0.651186\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 848.528i	− 1.22090i
$696$	0	0
$697$	210.000	0.301291
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 29.6985i	− 0.0423659i	−0.999776	−	0.0211829i	$-0.993257\pi$
$701$	− 29.6985i	− 0.0423659i	0.999776	−	0.0211829i	$-0.00674324\pi$
$702$	0	0
$703$	480.000	0.682788
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 763.675i	− 1.08016i
$708$	0	0
$709$	56.0000	0.0789845	0.0394922	−	0.999220i	$-0.487426\pi$
$709$	56.0000	0.0789845	0.0394922	+	0.999220i	$0.487426\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	158.392i	0.222149i
$714$	0	0
$715$	320.000	0.447552
$716$	0	0
$717$	0	0
$718$	0	0
$719$	616.597i	0.857576i	0.903405	+	0.428788i	$0.141059\pi$
$719$	616.597i	0.857576i	−0.903405	+	0.428788i	$0.858941\pi$
$720$	0	0
$721$	−528.000	−0.732316
$722$	0	0
$723$	0	0
$724$	0	0
$725$	742.462i	1.02409i
$726$	0	0
$727$	−372.000	−0.511692	−0.255846	−	0.966718i	$-0.582354\pi$
$727$	−372.000	−0.511692	−0.255846	+	0.966718i	$0.582354\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	79.1960i	0.108339i
$732$	0	0
$733$	−520.000	−0.709413	−0.354707	−	0.934978i	$-0.615419\pi$
$733$	−520.000	−0.709413	−0.354707	+	0.934978i	$0.615419\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	497.803i	0.675445i
$738$	0	0
$739$	−1040.00	−1.40731	−0.703654	−	0.710543i	$-0.748449\pi$
$739$	−1040.00	−1.40731	−0.703654	+	0.710543i	$0.748449\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 622.254i	− 0.837489i	−0.908104	−	0.418744i	$-0.862470\pi$
$743$	− 622.254i	− 0.837489i	0.908104	−	0.418744i	$-0.137530\pi$
$744$	0	0
$745$	−910.000	−1.22148
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	180.000	0.239680	0.119840	−	0.992793i	$-0.461762\pi$
$751$	180.000	0.239680	0.119840	+	0.992793i	$0.461762\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1385.93i	1.83567i
$756$	0	0
$757$	152.000	0.200793	0.100396	−	0.994948i	$-0.467989\pi$
$757$	152.000	0.200793	0.100396	+	0.994948i	$0.467989\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	544.472i	0.715469i	0.933823	+	0.357735i	$0.116451\pi$
$761$	544.472i	0.715469i	−0.933823	+	0.357735i	$0.883549\pi$
$762$	0	0
$763$	−1632.00	−2.13893
$764$	0	0
$765$	0	0
$766$	0	0
$767$	633.568i	0.826033i
$768$	0	0
$769$	−526.000	−0.684005	−0.342003	−	0.939699i	$-0.611105\pi$
$769$	−526.000	−0.684005	−0.342003	+	0.939699i	$0.611105\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1421.28i	− 1.83866i	−0.393487	−	0.919330i	$-0.628731\pi$
$773$	− 1421.28i	− 1.83866i	0.393487	−	0.919330i	$-0.371269\pi$
$774$	0	0
$775$	−100.000	−0.129032
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 339.411i	− 0.435701i
$780$	0	0
$781$	−160.000	−0.204866
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1682.91i	2.14384i
$786$	0	0
$787$	−1072.00	−1.36213	−0.681067	−	0.732221i	$-0.738484\pi$
$787$	−1072.00	−1.36213	−0.681067	+	0.732221i	$0.738484\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	254.558i	0.321819i
$792$	0	0
$793$	112.000	0.141236
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1534.42i	1.92525i	0.270843	+	0.962623i	$0.412697\pi$
$797$	1534.42i	1.92525i	−0.270843	+	0.962623i	$0.587303\pi$
$798$	0	0
$799$	−168.000	−0.210263
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 452.548i	− 0.563572i
$804$	0	0
$805$	3360.00	4.17391
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 134.350i	− 0.166070i	−0.996547	−	0.0830348i	$-0.973539\pi$
$809$	− 134.350i	− 0.166070i	0.996547	−	0.0830348i	$-0.0264613\pi$
$810$	0	0
$811$	−544.000	−0.670777	−0.335388	−	0.942080i	$-0.608868\pi$
$811$	−544.000	−0.670777	−0.335388	+	0.942080i	$0.608868\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1187.94i	1.45759i
$816$	0	0
$817$	128.000	0.156671
$818$	0	0
$819$	0	0
$820$	0	0
$821$	878.227i	1.06970i	0.844946	+	0.534852i	$0.179633\pi$
$821$	878.227i	1.06970i	−0.844946	+	0.534852i	$0.820367\pi$
$822$	0	0
$823$	868.000	1.05468	0.527339	−	0.849655i	$-0.323190\pi$
$823$	868.000	1.05468	0.527339	+	0.849655i	$0.323190\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1414.21i	− 1.71005i	−0.518585	−	0.855026i	$-0.673541\pi$
$827$	− 1414.21i	− 1.71005i	0.518585	−	0.855026i	$-0.326459\pi$
$828$	0	0
$829$	−1208.00	−1.45718	−0.728589	−	0.684952i	$-0.759823\pi$
$829$	−1208.00	−1.45718	−0.728589	+	0.684952i	$0.759823\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	940.452i	1.12899i
$834$	0	0
$835$	1200.00	1.43713
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1069.15i	− 1.27431i	−0.770736	−	0.637155i	$-0.780111\pi$
$839$	− 1069.15i	− 1.27431i	0.770736	−	0.637155i	$-0.219889\pi$
$840$	0	0
$841$	−41.0000	−0.0487515
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 742.462i	− 0.878653i
$846$	0	0
$847$	−1068.00	−1.26092
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1187.94i	1.39593i
$852$	0	0
$853$	−1282.00	−1.50293	−0.751465	−	0.659772i	$-0.770653\pi$
$853$	−1282.00	−1.50293	−0.751465	+	0.659772i	$0.770653\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1039.45i	− 1.21289i	−0.795125	−	0.606445i	$-0.792595\pi$
$857$	− 1039.45i	− 1.21289i	0.795125	−	0.606445i	$-0.207405\pi$
$858$	0	0
$859$	1032.00	1.20140	0.600698	−	0.799476i	$-0.294889\pi$
$859$	1032.00	1.20140	0.600698	+	0.799476i	$0.294889\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	56.5685i	0.0655487i	0.999463	+	0.0327744i	$0.0104343\pi$
$863$	56.5685i	0.0655487i	−0.999463	+	0.0327744i	$0.989566\pi$
$864$	0	0
$865$	1470.00	1.69942
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 565.685i	− 0.650961i
$870$	0	0
$871$	−704.000	−0.808266
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−462.000	−0.526796	−0.263398	−	0.964687i	$-0.584843\pi$
$877$	−462.000	−0.526796	−0.263398	+	0.964687i	$0.584843\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	247.487i	0.280916i	0.990087	+	0.140458i	$0.0448576\pi$
$881$	247.487i	0.280916i	−0.990087	+	0.140458i	$0.955142\pi$
$882$	0	0
$883$	−8.00000	−0.00906002	−0.00453001	−	0.999990i	$-0.501442\pi$
$883$	−8.00000	−0.00906002	−0.00453001	+	0.999990i	$0.501442\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	893.783i	1.00765i	0.863807	+	0.503824i	$0.168074\pi$
$887$	893.783i	1.00765i	−0.863807	+	0.503824i	$0.831926\pi$
$888$	0	0
$889$	−240.000	−0.269966
$890$	0	0
$891$	0	0
$892$	0	0
$893$	271.529i	0.304064i
$894$	0	0
$895$	−1440.00	−1.60894
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 118.794i	− 0.132140i
$900$	0	0
$901$	−490.000	−0.543840
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1527.35i	− 1.68768i
$906$	0	0
$907$	200.000	0.220507	0.110254	−	0.993903i	$-0.464834\pi$
$907$	200.000	0.220507	0.110254	+	0.993903i	$0.464834\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 79.1960i	− 0.0869330i	−0.999055	−	0.0434665i	$-0.986160\pi$
$911$	− 79.1960i	− 0.0869330i	0.999055	−	0.0434665i	$-0.0138402\pi$
$912$	0	0
$913$	−736.000	−0.806134
$914$	0	0
$915$	0	0
$916$	0	0
$917$	678.823i	0.740264i
$918$	0	0
$919$	884.000	0.961915	0.480958	−	0.876744i	$-0.340289\pi$
$919$	884.000	0.961915	0.480958	+	0.876744i	$0.340289\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 226.274i	− 0.245151i
$924$	0	0
$925$	−750.000	−0.810811
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1076.22i	− 1.15847i	−0.815161	−	0.579234i	$-0.803352\pi$
$929$	− 1076.22i	− 1.15847i	0.815161	−	0.579234i	$-0.196648\pi$
$930$	0	0
$931$	1520.00	1.63265
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 395.980i	− 0.423508i
$936$	0	0
$937$	78.0000	0.0832444	0.0416222	−	0.999133i	$-0.486747\pi$
$937$	78.0000	0.0832444	0.0416222	+	0.999133i	$0.486747\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 267.286i	− 0.284045i	−0.989863	−	0.142023i	$-0.954639\pi$
$941$	− 267.286i	− 0.284045i	0.989863	−	0.142023i	$-0.0453605\pi$
$942$	0	0
$943$	840.000	0.890774
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 576.999i	− 0.609292i	−0.952466	−	0.304646i	$-0.901462\pi$
$947$	− 576.999i	− 0.609292i	0.952466	−	0.304646i	$-0.0985381\pi$
$948$	0	0
$949$	640.000	0.674394
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1336.43i	− 1.40234i	−0.712993	−	0.701171i	$-0.752661\pi$
$953$	− 1336.43i	− 1.40234i	0.712993	−	0.701171i	$-0.247339\pi$
$954$	0	0
$955$	2640.00	2.76440
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1069.15i	− 1.11485i
$960$	0	0
$961$	−945.000	−0.983351
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1258.65i	− 1.30430i
$966$	0	0
$967$	−20.0000	−0.0206825	−0.0103413	−	0.999947i	$-0.503292\pi$
$967$	−20.0000	−0.0206825	−0.0103413	+	0.999947i	$0.503292\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 118.794i	− 0.122342i	−0.998127	−	0.0611709i	$-0.980517\pi$
$971$	− 118.794i	− 0.122342i	0.998127	−	0.0611709i	$-0.0194835\pi$
$972$	0	0
$973$	1440.00	1.47996
$974$	0	0
$975$	0	0
$976$	0	0
$977$	420.021i	0.429909i	0.976624	+	0.214955i	$0.0689604\pi$
$977$	420.021i	0.429909i	−0.976624	+	0.214955i	$0.931040\pi$
$978$	0	0
$979$	−840.000	−0.858018
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1187.94i	1.20848i	0.796801	+	0.604242i	$0.206524\pi$
$983$	1187.94i	1.20848i	−0.796801	+	0.604242i	$0.793476\pi$
$984$	0	0
$985$	1470.00	1.49239
$986$	0	0
$987$	0	0
$988$	0	0
$989$	316.784i	0.320307i
$990$	0	0
$991$	116.000	0.117053	0.0585267	−	0.998286i	$-0.481360\pi$
$991$	116.000	0.117053	0.0585267	+	0.998286i	$0.481360\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1385.93i	− 1.39289i
$996$	0	0
$997$	130.000	0.130391	0.0651956	−	0.997873i	$-0.479233\pi$
$997$	130.000	0.130391	0.0651956	+	0.997873i	$0.479233\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.3.e.a.17.2		2
3.2	odd	2	inner	144.3.e.a.17.1		2
4.3	odd	2		72.3.e.a.17.2	yes	2
5.2	odd	4		3600.3.c.c.449.2		4
5.3	odd	4		3600.3.c.c.449.4		4
5.4	even	2		3600.3.l.l.1601.2		2
8.3	odd	2		576.3.e.h.449.1		2
8.5	even	2		576.3.e.a.449.1		2
9.2	odd	6		1296.3.q.k.1025.2		4
9.4	even	3		1296.3.q.k.593.2		4
9.5	odd	6		1296.3.q.k.593.1		4
9.7	even	3		1296.3.q.k.1025.1		4
12.11	even	2		72.3.e.a.17.1	✓	2
15.2	even	4		3600.3.c.c.449.1		4
15.8	even	4		3600.3.c.c.449.3		4
15.14	odd	2		3600.3.l.l.1601.1		2
16.3	odd	4		2304.3.h.a.2177.3		4
16.5	even	4		2304.3.h.h.2177.2		4
16.11	odd	4		2304.3.h.a.2177.2		4
16.13	even	4		2304.3.h.h.2177.3		4
20.3	even	4		1800.3.c.a.449.1		4
20.7	even	4		1800.3.c.a.449.3		4
20.19	odd	2		1800.3.l.a.1601.1		2
24.5	odd	2		576.3.e.a.449.2		2
24.11	even	2		576.3.e.h.449.2		2
28.27	even	2		3528.3.d.a.1961.1		2
36.7	odd	6		648.3.m.a.377.1		4
36.11	even	6		648.3.m.a.377.2		4
36.23	even	6		648.3.m.a.593.1		4
36.31	odd	6		648.3.m.a.593.2		4
48.5	odd	4		2304.3.h.h.2177.4		4
48.11	even	4		2304.3.h.a.2177.4		4
48.29	odd	4		2304.3.h.h.2177.1		4
48.35	even	4		2304.3.h.a.2177.1		4
60.23	odd	4		1800.3.c.a.449.2		4
60.47	odd	4		1800.3.c.a.449.4		4
60.59	even	2		1800.3.l.a.1601.2		2
84.83	odd	2		3528.3.d.a.1961.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.e.a.17.1	✓	2	12.11	even	2
72.3.e.a.17.2	yes	2	4.3	odd	2
144.3.e.a.17.1		2	3.2	odd	2	inner
144.3.e.a.17.2		2	1.1	even	1	trivial
576.3.e.a.449.1		2	8.5	even	2
576.3.e.a.449.2		2	24.5	odd	2
576.3.e.h.449.1		2	8.3	odd	2
576.3.e.h.449.2		2	24.11	even	2
648.3.m.a.377.1		4	36.7	odd	6
648.3.m.a.377.2		4	36.11	even	6
648.3.m.a.593.1		4	36.23	even	6
648.3.m.a.593.2		4	36.31	odd	6
1296.3.q.k.593.1		4	9.5	odd	6
1296.3.q.k.593.2		4	9.4	even	3
1296.3.q.k.1025.1		4	9.7	even	3
1296.3.q.k.1025.2		4	9.2	odd	6
1800.3.c.a.449.1		4	20.3	even	4
1800.3.c.a.449.2		4	60.23	odd	4
1800.3.c.a.449.3		4	20.7	even	4
1800.3.c.a.449.4		4	60.47	odd	4
1800.3.l.a.1601.1		2	20.19	odd	2
1800.3.l.a.1601.2		2	60.59	even	2
2304.3.h.a.2177.1		4	48.35	even	4
2304.3.h.a.2177.2		4	16.11	odd	4
2304.3.h.a.2177.3		4	16.3	odd	4
2304.3.h.a.2177.4		4	48.11	even	4
2304.3.h.h.2177.1		4	48.29	odd	4
2304.3.h.h.2177.2		4	16.5	even	4
2304.3.h.h.2177.3		4	16.13	even	4
2304.3.h.h.2177.4		4	48.5	odd	4
3528.3.d.a.1961.1		2	28.27	even	2
3528.3.d.a.1961.2		2	84.83	odd	2
3600.3.c.c.449.1		4	15.2	even	4
3600.3.c.c.449.2		4	5.2	odd	4
3600.3.c.c.449.3		4	15.8	even	4
3600.3.c.c.449.4		4	5.3	odd	4
3600.3.l.l.1601.1		2	15.14	odd	2
3600.3.l.l.1601.2		2	5.4	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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	144.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.07107i q^{5} -12.0000 q^{7} +O(q^{10})\)	\(q+7.07107i q^{5} -12.0000 q^{7} +5.65685i q^{11} -8.00000 q^{13} +9.89949i q^{17} +16.0000 q^{19} +39.5980i q^{23} -25.0000 q^{25} -29.6985i q^{29} +4.00000 q^{31} -84.8528i q^{35} +30.0000 q^{37} -21.2132i q^{41} +8.00000 q^{43} +16.9706i q^{47} +95.0000 q^{49} +49.4975i q^{53} -40.0000 q^{55} -79.1960i q^{59} -14.0000 q^{61} -56.5685i q^{65} +88.0000 q^{67} +28.2843i q^{71} -80.0000 q^{73} -67.8823i q^{77} -100.000 q^{79} +130.108i q^{83} -70.0000 q^{85} +148.492i q^{89} +96.0000 q^{91} +113.137i q^{95} -112.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 24 q^{7}+O(q^{10}) \) 2 * q - 24 * q^7	\( 2 q - 24 q^{7} - 16 q^{13} + 32 q^{19} - 50 q^{25} + 8 q^{31} + 60 q^{37} + 16 q^{43} + 190 q^{49} - 80 q^{55} - 28 q^{61} + 176 q^{67} - 160 q^{73} - 200 q^{79} - 140 q^{85} + 192 q^{91} - 224 q^{97}+O(q^{100}) \) 2 * q - 24 * q^7 - 16 * q^13 + 32 * q^19 - 50 * q^25 + 8 * q^31 + 60 * q^37 + 16 * q^43 + 190 * q^49 - 80 * q^55 - 28 * q^61 + 176 * q^67 - 160 * q^73 - 200 * q^79 - 140 * q^85 + 192 * q^91 - 224 * q^97

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