LMFDB - Embedded newform 2304.3.h.a.2177.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(2177,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2304.2177");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.h (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:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2177.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.2177
Dual form			2304.3.h.a.2177.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.07107 q^{5} -12.0000 q^{7} +O(q^{10})$	$q+7.07107 q^{5} -12.0000 q^{7} -5.65685 q^{11} -8.00000i q^{13} +9.89949i q^{17} -16.0000i q^{19} +39.5980i q^{23} +25.0000 q^{25} +29.6985 q^{29} -4.00000 q^{31} -84.8528 q^{35} -30.0000i q^{37} +21.2132i q^{41} +8.00000i q^{43} -16.9706i q^{47} +95.0000 q^{49} +49.4975 q^{53} -40.0000 q^{55} +79.1960 q^{59} -14.0000i q^{61} -56.5685i q^{65} -88.0000i q^{67} +28.2843i q^{71} +80.0000 q^{73} +67.8823 q^{77} +100.000 q^{79} +130.108 q^{83} +70.0000i q^{85} -148.492i q^{89} +96.0000i q^{91} -113.137i q^{95} -112.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 48 q^{7}+O(q^{10}) $ 4 * q - 48 * q^7	$ 4 q - 48 q^{7} + 100 q^{25} - 16 q^{31} + 380 q^{49} - 160 q^{55} + 320 q^{73} + 400 q^{79} - 448 q^{97}+O(q^{100}) $ 4 * q - 48 * q^7 + 100 * q^25 - 16 * q^31 + 380 * q^49 - 160 * q^55 + 320 * q^73 + 400 * q^79 - 448 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	7.07107	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$5$	7.07107	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$6$	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
$11$	−5.65685	−0.514259	−0.257130	−	0.966377i	$-0.582777\pi$
$11$	−5.65685	−0.514259	−0.257130	+	0.966377i	$0.582777\pi$
$12$	0	0
$13$	− 8.00000i	− 0.615385i	−0.951486	−	0.307692i	$-0.900443\pi$
$13$	− 8.00000i	− 0.615385i	0.951486	−	0.307692i	$-0.0995567\pi$
$14$	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
$19$	− 16.0000i	− 0.842105i	−0.907036	−	0.421053i	$-0.861661\pi$
$19$	− 16.0000i	− 0.842105i	0.907036	−	0.421053i	$-0.138339\pi$
$20$	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
$29$	29.6985	1.02409	0.512043	−	0.858960i	$-0.328889\pi$
$29$	29.6985	1.02409	0.512043	+	0.858960i	$0.328889\pi$
$30$	0	0
$31$	−4.00000	−0.129032	−0.0645161	−	0.997917i	$-0.520550\pi$
$31$	−4.00000	−0.129032	−0.0645161	+	0.997917i	$0.520550\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−84.8528	−2.42437
$36$	0	0
$37$	− 30.0000i	− 0.810811i	−0.914137	−	0.405405i	$-0.867130\pi$
$37$	− 30.0000i	− 0.810811i	0.914137	−	0.405405i	$-0.132870\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	21.2132i	0.517395i	0.965958	+	0.258698i	$0.0832933\pi$
$41$	21.2132i	0.517395i	−0.965958	+	0.258698i	$0.916707\pi$
$42$	0	0
$43$	8.00000i	0.186047i	0.995664	+	0.0930233i	$0.0296531\pi$
$43$	8.00000i	0.186047i	−0.995664	+	0.0930233i	$0.970347\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 16.9706i	− 0.361076i	−0.983568	−	0.180538i	$-0.942216\pi$
$47$	− 16.9706i	− 0.361076i	0.983568	−	0.180538i	$-0.0577838\pi$
$48$	0	0
$49$	95.0000	1.93878
$50$	0	0
$51$	0	0
$52$	0	0
$53$	49.4975	0.933915	0.466957	−	0.884280i	$-0.345350\pi$
$53$	49.4975	0.933915	0.466957	+	0.884280i	$0.345350\pi$
$54$	0	0
$55$	−40.0000	−0.727273
$56$	0	0
$57$	0	0
$58$	0	0
$59$	79.1960	1.34230	0.671152	−	0.741320i	$-0.265800\pi$
$59$	79.1960	1.34230	0.671152	+	0.741320i	$0.265800\pi$
$60$	0	0
$61$	− 14.0000i	− 0.229508i	−0.993394	−	0.114754i	$-0.963392\pi$
$61$	− 14.0000i	− 0.229508i	0.993394	−	0.114754i	$-0.0366080\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 56.5685i	− 0.870285i
$66$	0	0
$67$	− 88.0000i	− 1.31343i	−0.754138	−	0.656716i	$-0.771945\pi$
$67$	− 88.0000i	− 1.31343i	0.754138	−	0.656716i	$-0.228055\pi$
$68$	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	$-0.315410\pi$
$73$	80.0000	1.09589	0.547945	+	0.836514i	$0.315410\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	67.8823	0.881588
$78$	0	0
$79$	100.000	1.26582	0.632911	−	0.774224i	$-0.281860\pi$
$79$	100.000	1.26582	0.632911	+	0.774224i	$0.281860\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	130.108	1.56756	0.783781	−	0.621037i	$-0.213288\pi$
$83$	130.108	1.56756	0.783781	+	0.621037i	$0.213288\pi$
$84$	0	0
$85$	70.0000i	0.823529i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 148.492i	− 1.66845i	−0.551421	−	0.834227i	$-0.685914\pi$
$89$	− 148.492i	− 1.66845i	0.551421	−	0.834227i	$-0.314086\pi$
$90$	0	0
$91$	96.0000i	1.05495i
$92$	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
$101$	63.6396	0.630095	0.315048	−	0.949076i	$-0.397980\pi$
$101$	63.6396	0.630095	0.315048	+	0.949076i	$0.397980\pi$
$102$	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		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	136.000i	1.24771i	0.781542	+	0.623853i	$0.214434\pi$
$109$	136.000i	1.24771i	−0.781542	+	0.623853i	$0.785566\pi$
$110$	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
$115$	280.000i	2.43478i
$116$	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	$-0.525090\pi$
$127$	−20.0000	−0.157480	−0.0787402	+	0.996895i	$0.525090\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−56.5685	−0.431821	−0.215910	−	0.976413i	$-0.569272\pi$
$131$	−56.5685	−0.431821	−0.215910	+	0.976413i	$0.569272\pi$
$132$	0	0
$133$	192.000i	1.44361i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 89.0955i	− 0.650332i	−0.945657	−	0.325166i	$-0.894580\pi$
$137$	− 89.0955i	− 0.650332i	0.945657	−	0.325166i	$-0.105420\pi$
$138$	0	0
$139$	− 120.000i	− 0.863309i	−0.902039	−	0.431655i	$-0.857930\pi$
$139$	− 120.000i	− 0.863309i	0.902039	−	0.431655i	$-0.142070\pi$
$140$	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
$149$	128.693	0.863714	0.431857	−	0.901942i	$-0.357858\pi$
$149$	128.693	0.863714	0.431857	+	0.901942i	$0.357858\pi$
$150$	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
$155$	−28.2843	−0.182479
$156$	0	0
$157$	238.000i	1.51592i	0.652299	+	0.757962i	$0.273805\pi$
$157$	238.000i	1.51592i	−0.652299	+	0.757962i	$0.726195\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 475.176i	− 2.95140i
$162$	0	0
$163$	− 168.000i	− 1.03067i	−0.856987	−	0.515337i	$-0.827667\pi$
$163$	− 168.000i	− 1.03067i	0.856987	−	0.515337i	$-0.172333\pi$
$164$	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
$173$	207.889	1.20167	0.600836	−	0.799372i	$-0.294834\pi$
$173$	207.889	1.20167	0.600836	+	0.799372i	$0.294834\pi$
$174$	0	0
$175$	−300.000	−1.71429
$176$	0	0
$177$	0	0
$178$	0	0
$179$	203.647	1.13769	0.568846	−	0.822444i	$-0.307390\pi$
$179$	203.647	1.13769	0.568846	+	0.822444i	$0.307390\pi$
$180$	0	0
$181$	216.000i	1.19337i	0.802475	+	0.596685i	$0.203516\pi$
$181$	216.000i	1.19337i	−0.802475	+	0.596685i	$0.796484\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 212.132i	− 1.14666i
$186$	0	0
$187$	− 56.0000i	− 0.299465i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	373.352i	1.95472i	0.211573	+	0.977362i	$0.432141\pi$
$191$	373.352i	1.95472i	−0.211573	+	0.977362i	$0.567859\pi$
$192$	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
$197$	−207.889	−1.05528	−0.527638	−	0.849469i	$-0.676922\pi$
$197$	−207.889	−1.05528	−0.527638	+	0.849469i	$0.676922\pi$
$198$	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
$203$	−356.382	−1.75558
$204$	0	0
$205$	150.000i	0.731707i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	90.5097i	0.433061i
$210$	0	0
$211$	− 56.0000i	− 0.265403i	−0.991156	−	0.132701i	$-0.957635\pi$
$211$	− 56.0000i	− 0.265403i	0.991156	−	0.132701i	$-0.0423651\pi$
$212$	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
$221$	79.1960	0.358353
$222$	0	0
$223$	28.0000	0.125561	0.0627803	−	0.998027i	$-0.480003\pi$
$223$	28.0000	0.125561	0.0627803	+	0.998027i	$0.480003\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−84.8528	−0.373801	−0.186900	−	0.982379i	$-0.559844\pi$
$227$	−84.8528	−0.373801	−0.186900	+	0.982379i	$0.559844\pi$
$228$	0	0
$229$	− 184.000i	− 0.803493i	−0.915751	−	0.401747i	$-0.868403\pi$
$229$	− 184.000i	− 0.803493i	0.915751	−	0.401747i	$-0.131597\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 304.056i	− 1.30496i	−0.757806	−	0.652481i	$-0.773728\pi$
$233$	− 304.056i	− 1.30496i	0.757806	−	0.652481i	$-0.226272\pi$
$234$	0	0
$235$	− 120.000i	− 0.510638i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	226.274i	0.946754i	0.880860	+	0.473377i	$0.156965\pi$
$239$	226.274i	0.946754i	−0.880860	+	0.473377i	$0.843035\pi$
$240$	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
$245$	671.751	2.74184
$246$	0	0
$247$	−128.000	−0.518219
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−197.990	−0.788804	−0.394402	−	0.918938i	$-0.629048\pi$
$251$	−197.990	−0.788804	−0.394402	+	0.918938i	$0.629048\pi$
$252$	0	0
$253$	− 224.000i	− 0.885375i
$254$	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
$259$	360.000i	1.38996i
$260$	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
$269$	83.4386	0.310181	0.155090	−	0.987900i	$-0.450433\pi$
$269$	83.4386	0.310181	0.155090	+	0.987900i	$0.450433\pi$
$270$	0	0
$271$	140.000	0.516605	0.258303	−	0.966064i	$-0.416837\pi$
$271$	140.000	0.516605	0.258303	+	0.966064i	$0.416837\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−141.421	−0.514259
$276$	0	0
$277$	248.000i	0.895307i	0.894207	+	0.447653i	$0.147740\pi$
$277$	248.000i	0.895307i	−0.894207	+	0.447653i	$0.852260\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	168.291i	0.598902i	0.954112	+	0.299451i	$0.0968035\pi$
$281$	168.291i	0.598902i	−0.954112	+	0.299451i	$0.903197\pi$
$282$	0	0
$283$	− 368.000i	− 1.30035i	−0.759783	−	0.650177i	$-0.774695\pi$
$283$	− 368.000i	− 1.30035i	0.759783	−	0.650177i	$-0.225305\pi$
$284$	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
$293$	207.889	0.709520	0.354760	−	0.934957i	$-0.384563\pi$
$293$	207.889	0.709520	0.354760	+	0.934957i	$0.384563\pi$
$294$	0	0
$295$	560.000	1.89831
$296$	0	0
$297$	0	0
$298$	0	0
$299$	316.784	1.05948
$300$	0	0
$301$	− 96.0000i	− 0.318937i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 98.9949i	− 0.324574i
$306$	0	0
$307$	− 552.000i	− 1.79805i	−0.437902	−	0.899023i	$-0.644278\pi$
$307$	− 552.000i	− 1.79805i	0.437902	−	0.899023i	$-0.355722\pi$
$308$	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	$-0.384606\pi$
$313$	222.000	0.709265	0.354633	+	0.935006i	$0.384606\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	445.477	1.40529	0.702646	−	0.711540i	$-0.252002\pi$
$317$	445.477	1.40529	0.702646	+	0.711540i	$0.252002\pi$
$318$	0	0
$319$	−168.000	−0.526646
$320$	0	0
$321$	0	0
$322$	0	0
$323$	158.392	0.490377
$324$	0	0
$325$	− 200.000i	− 0.615385i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	203.647i	0.618987i
$330$	0	0
$331$	520.000i	1.57100i	0.618864	+	0.785498i	$0.287593\pi$
$331$	520.000i	1.57100i	−0.618864	+	0.785498i	$0.712407\pi$
$332$	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
$341$	22.6274	0.0663561
$342$	0	0
$343$	−552.000	−1.60933
$344$	0	0
$345$	0	0
$346$	0	0
$347$	356.382	1.02704	0.513518	−	0.858079i	$-0.328342\pi$
$347$	356.382	1.02704	0.513518	+	0.858079i	$0.328342\pi$
$348$	0	0
$349$	210.000i	0.601719i	0.953668	+	0.300860i	$0.0972736\pi$
$349$	210.000i	0.601719i	−0.953668	+	0.300860i	$0.902726\pi$
$350$	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
$355$	200.000i	0.563380i
$356$	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
$365$	565.685	1.54982
$366$	0	0
$367$	172.000	0.468665	0.234332	−	0.972157i	$-0.424710\pi$
$367$	172.000	0.468665	0.234332	+	0.972157i	$0.424710\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−593.970	−1.60100
$372$	0	0
$373$	− 2.00000i	− 0.00536193i	−0.999996	−	0.00268097i	$-0.999147\pi$
$373$	− 2.00000i	− 0.00536193i	0.999996	−	0.00268097i	$-0.000853379\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 237.588i	− 0.630207i
$378$	0	0
$379$	736.000i	1.94195i	0.239176	+	0.970976i	$0.423123\pi$
$379$	736.000i	1.94195i	−0.239176	+	0.970976i	$0.576877\pi$
$380$	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
$389$	−459.619	−1.18154	−0.590770	−	0.806840i	$-0.701176\pi$
$389$	−459.619	−1.18154	−0.590770	+	0.806840i	$0.701176\pi$
$390$	0	0
$391$	−392.000	−1.00256
$392$	0	0
$393$	0	0
$394$	0	0
$395$	707.107	1.79014
$396$	0	0
$397$	338.000i	0.851385i	0.904868	+	0.425693i	$0.139970\pi$
$397$	338.000i	0.851385i	−0.904868	+	0.425693i	$0.860030\pi$
$398$	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
$403$	32.0000i	0.0794045i
$404$	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	$-0.556329\pi$
$409$	−144.000	−0.352078	−0.176039	+	0.984383i	$0.556329\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−950.352	−2.30109
$414$	0	0
$415$	920.000	2.21687
$416$	0	0
$417$	0	0
$418$	0	0
$419$	763.675	1.82261	0.911307	−	0.411727i	$-0.135074\pi$
$419$	763.675	1.82261	0.911307	+	0.411727i	$0.135074\pi$
$420$	0	0
$421$	− 680.000i	− 1.61520i	−0.589729	−	0.807601i	$-0.700766\pi$
$421$	− 680.000i	− 1.61520i	0.589729	−	0.807601i	$-0.299234\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	247.487i	0.582323i
$426$	0	0
$427$	168.000i	0.393443i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.65685i	0.0131250i	0.999978	+	0.00656248i	$0.00208892\pi$
$431$	5.65685i	0.0131250i	−0.999978	+	0.00656248i	$0.997911\pi$
$432$	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
$437$	633.568	1.44981
$438$	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
$443$	−197.990	−0.446930	−0.223465	−	0.974712i	$-0.571737\pi$
$443$	−197.990	−0.446930	−0.223465	+	0.974712i	$0.571737\pi$
$444$	0	0
$445$	− 1050.00i	− 2.35955i
$446$	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
$451$	− 120.000i	− 0.266075i
$452$	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	$-0.669489\pi$
$457$	−464.000	−1.01532	−0.507659	+	0.861558i	$0.669489\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−49.4975	−0.107370	−0.0536849	−	0.998558i	$-0.517097\pi$
$461$	−49.4975	−0.107370	−0.0536849	+	0.998558i	$0.517097\pi$
$462$	0	0
$463$	20.0000	0.0431965	0.0215983	−	0.999767i	$-0.493125\pi$
$463$	20.0000	0.0431965	0.0215983	+	0.999767i	$0.493125\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−695.793	−1.48992	−0.744960	−	0.667109i	$-0.767532\pi$
$467$	−695.793	−1.48992	−0.744960	+	0.667109i	$0.767532\pi$
$468$	0	0
$469$	1056.00i	2.25160i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 45.2548i	− 0.0956762i
$474$	0	0
$475$	− 400.000i	− 0.842105i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 673.166i	− 1.40536i	−0.711508	−	0.702678i	$-0.751987\pi$
$479$	− 673.166i	− 1.40536i	0.711508	−	0.702678i	$-0.248013\pi$
$480$	0	0
$481$	−240.000	−0.498960
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−791.960	−1.63291
$486$	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		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	294.000i	0.596349i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 339.411i	− 0.682920i
$498$	0	0
$499$	96.0000i	0.192385i	0.995363	+	0.0961924i	$0.0306664\pi$
$499$	96.0000i	0.192385i	−0.995363	+	0.0961924i	$0.969334\pi$
$500$	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
$509$	−309.713	−0.608473	−0.304237	−	0.952597i	$-0.598401\pi$
$509$	−309.713	−0.608473	−0.304237	+	0.952597i	$0.598401\pi$
$510$	0	0
$511$	−960.000	−1.87867
$512$	0	0
$513$	0	0
$514$	0	0
$515$	311.127	0.604130
$516$	0	0
$517$	96.0000i	0.185687i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 722.663i	− 1.38707i	−0.720423	−	0.693535i	$-0.756052\pi$
$521$	− 722.663i	− 1.38707i	0.720423	−	0.693535i	$-0.243948\pi$
$522$	0	0
$523$	− 352.000i	− 0.673040i	−0.941676	−	0.336520i	$-0.890750\pi$
$523$	− 352.000i	− 0.673040i	0.941676	−	0.336520i	$-0.109250\pi$
$524$	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
$533$	169.706	0.318397
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−537.401	−0.997034
$540$	0	0
$541$	936.000i	1.73013i	0.501660	+	0.865065i	$0.332723\pi$
$541$	936.000i	1.73013i	−0.501660	+	0.865065i	$0.667277\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	961.665i	1.76452i
$546$	0	0
$547$	56.0000i	0.102377i	0.998689	+	0.0511883i	$0.0163009\pi$
$547$	56.0000i	0.102377i	−0.998689	+	0.0511883i	$0.983699\pi$
$548$	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
$557$	−321.026	−0.576349	−0.288175	−	0.957578i	$-0.593048\pi$
$557$	−321.026	−0.576349	−0.288175	+	0.957578i	$0.593048\pi$
$558$	0	0
$559$	64.0000	0.114490
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−469.519	−0.833959	−0.416979	−	0.908916i	$-0.636911\pi$
$563$	−469.519	−0.833959	−0.416979	+	0.908916i	$0.636911\pi$
$564$	0	0
$565$	− 150.000i	− 0.265487i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	408.708i	0.718291i	0.933282	+	0.359146i	$0.116932\pi$
$569$	408.708i	0.718291i	−0.933282	+	0.359146i	$0.883068\pi$
$570$	0	0
$571$	− 816.000i	− 1.42907i	−0.699599	−	0.714536i	$-0.746638\pi$
$571$	− 816.000i	− 1.42907i	0.699599	−	0.714536i	$-0.253362\pi$
$572$	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
$581$	−1561.29	−2.68725
$582$	0	0
$583$	−280.000	−0.480274
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−395.980	−0.674582	−0.337291	−	0.941400i	$-0.609511\pi$
$587$	−395.980	−0.674582	−0.337291	+	0.941400i	$0.609511\pi$
$588$	0	0
$589$	64.0000i	0.108659i
$590$	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
$595$	− 840.000i	− 1.41176i
$596$	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	$-0.576471\pi$
$601$	−286.000	−0.475874	−0.237937	+	0.971281i	$0.576471\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−629.325	−1.04021
$606$	0	0
$607$	−100.000	−0.164745	−0.0823723	−	0.996602i	$-0.526250\pi$
$607$	−100.000	−0.164745	−0.0823723	+	0.996602i	$0.526250\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−135.765	−0.222200
$612$	0	0
$613$	− 770.000i	− 1.25612i	−0.778166	−	0.628059i	$-0.783850\pi$
$613$	− 770.000i	− 1.25612i	0.778166	−	0.628059i	$-0.216150\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	869.741i	1.40963i	0.709391	+	0.704815i	$0.248970\pi$
$617$	869.741i	1.40963i	−0.709391	+	0.704815i	$0.751030\pi$
$618$	0	0
$619$	− 256.000i	− 0.413570i	−0.978386	−	0.206785i	$-0.933700\pi$
$619$	− 256.000i	− 0.413570i	0.978386	−	0.206785i	$-0.0663001\pi$
$620$	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
$629$	296.985	0.472154
$630$	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
$635$	−141.421	−0.222711
$636$	0	0
$637$	− 760.000i	− 1.19309i
$638$	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
$643$	120.000i	0.186625i	0.995637	+	0.0933126i	$0.0297456\pi$
$643$	120.000i	0.186625i	−0.995637	+	0.0933126i	$0.970254\pi$
$644$	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
$653$	−131.522	−0.201412	−0.100706	−	0.994916i	$-0.532110\pi$
$653$	−131.522	−0.201412	−0.100706	+	0.994916i	$0.532110\pi$
$654$	0	0
$655$	−400.000	−0.610687
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1187.94	1.80264	0.901320	−	0.433154i	$-0.142600\pi$
$659$	1187.94	1.80264	0.901320	+	0.433154i	$0.142600\pi$
$660$	0	0
$661$	130.000i	0.196672i	0.995153	+	0.0983359i	$0.0313519\pi$
$661$	130.000i	0.196672i	−0.995153	+	0.0983359i	$0.968648\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1357.65i	2.04157i
$666$	0	0
$667$	1176.00i	1.76312i
$668$	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
$677$	−603.869	−0.891978	−0.445989	−	0.895038i	$-0.647148\pi$
$677$	−603.869	−0.891978	−0.445989	+	0.895038i	$0.647148\pi$
$678$	0	0
$679$	1344.00	1.97938
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−989.949	−1.44941	−0.724707	−	0.689057i	$-0.758025\pi$
$683$	−989.949	−1.44941	−0.724707	+	0.689057i	$0.758025\pi$
$684$	0	0
$685$	− 630.000i	− 0.919708i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 395.980i	− 0.574717i
$690$	0	0
$691$	632.000i	0.914616i	0.889308	+	0.457308i	$0.151186\pi$
$691$	632.000i	0.914616i	−0.889308	+	0.457308i	$0.848814\pi$
$692$	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
$701$	29.6985	0.0423659	0.0211829	−	0.999776i	$-0.493257\pi$
$701$	29.6985	0.0423659	0.0211829	+	0.999776i	$0.493257\pi$
$702$	0	0
$703$	−480.000	−0.682788
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−763.675	−1.08016
$708$	0	0
$709$	− 56.0000i	− 0.0789845i	−0.999220	−	0.0394922i	$-0.987426\pi$
$709$	− 56.0000i	− 0.0789845i	0.999220	−	0.0394922i	$-0.0125740\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 158.392i	− 0.222149i
$714$	0	0
$715$	320.000i	0.447552i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 616.597i	− 0.857576i	−0.903405	−	0.428788i	$-0.858941\pi$
$719$	− 616.597i	− 0.857576i	0.903405	−	0.428788i	$-0.141059\pi$
$720$	0	0
$721$	−528.000	−0.732316
$722$	0	0
$723$	0	0
$724$	0	0
$725$	742.462	1.02409
$726$	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
$731$	−79.1960	−0.108339
$732$	0	0
$733$	− 520.000i	− 0.709413i	−0.934978	−	0.354707i	$-0.884581\pi$
$733$	− 520.000i	− 0.709413i	0.934978	−	0.354707i	$-0.115419\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	497.803i	0.675445i
$738$	0	0
$739$	1040.00i	1.40731i	0.710543	+	0.703654i	$0.248449\pi$
$739$	1040.00i	1.40731i	−0.710543	+	0.703654i	$0.751551\pi$
$740$	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	$-0.538238\pi$
$751$	−180.000	−0.239680	−0.119840	+	0.992793i	$0.538238\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1385.93	1.83567
$756$	0	0
$757$	− 152.000i	− 0.200793i	−0.994948	−	0.100396i	$-0.967989\pi$
$757$	− 152.000i	− 0.200793i	0.994948	−	0.100396i	$-0.0320111\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 544.472i	− 0.715469i	−0.933823	−	0.357735i	$-0.883549\pi$
$761$	− 544.472i	− 0.715469i	0.933823	−	0.357735i	$-0.116451\pi$
$762$	0	0
$763$	− 1632.00i	− 2.13893i
$764$	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
$773$	−1421.28	−1.83866	−0.919330	−	0.393487i	$-0.871269\pi$
$773$	−1421.28	−1.83866	−0.919330	+	0.393487i	$0.871269\pi$
$774$	0	0
$775$	−100.000	−0.129032
$776$	0	0
$777$	0	0
$778$	0	0
$779$	339.411	0.435701
$780$	0	0
$781$	− 160.000i	− 0.204866i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1682.91i	2.14384i
$786$	0	0
$787$	1072.00i	1.36213i	0.732221	+	0.681067i	$0.238484\pi$
$787$	1072.00i	1.36213i	−0.732221	+	0.681067i	$0.761516\pi$
$788$	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
$797$	−1534.42	−1.92525	−0.962623	−	0.270843i	$-0.912697\pi$
$797$	−1534.42	−1.92525	−0.962623	+	0.270843i	$0.912697\pi$
$798$	0	0
$799$	168.000	0.210263
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−452.548	−0.563572
$804$	0	0
$805$	− 3360.00i	− 4.17391i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	134.350i	0.166070i	0.996547	+	0.0830348i	$0.0264613\pi$
$809$	134.350i	0.166070i	−0.996547	+	0.0830348i	$0.973539\pi$
$810$	0	0
$811$	− 544.000i	− 0.670777i	−0.942080	−	0.335388i	$-0.891132\pi$
$811$	− 544.000i	− 0.670777i	0.942080	−	0.335388i	$-0.108868\pi$
$812$	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
$821$	878.227	1.06970	0.534852	−	0.844946i	$-0.320367\pi$
$821$	878.227	1.06970	0.534852	+	0.844946i	$0.320367\pi$
$822$	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
$827$	1414.21	1.71005	0.855026	−	0.518585i	$-0.173541\pi$
$827$	1414.21	1.71005	0.855026	+	0.518585i	$0.173541\pi$
$828$	0	0
$829$	− 1208.00i	− 1.45718i	−0.684952	−	0.728589i	$-0.740177\pi$
$829$	− 1208.00i	− 1.45718i	0.684952	−	0.728589i	$-0.259823\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	940.452i	1.12899i
$834$	0	0
$835$	− 1200.00i	− 1.43713i
$836$	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
$845$	742.462	0.878653
$846$	0	0
$847$	1068.00	1.26092
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1187.94	1.39593
$852$	0	0
$853$	1282.00i	1.50293i	0.659772	+	0.751465i	$0.270653\pi$
$853$	1282.00i	1.50293i	−0.659772	+	0.751465i	$0.729347\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1039.45i	1.21289i	0.795125	+	0.606445i	$0.207405\pi$
$857$	1039.45i	1.21289i	−0.795125	+	0.606445i	$0.792595\pi$
$858$	0	0
$859$	1032.00i	1.20140i	0.799476	+	0.600698i	$0.205111\pi$
$859$	1032.00i	1.20140i	−0.799476	+	0.600698i	$0.794889\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 56.5685i	− 0.0655487i	−0.999463	−	0.0327744i	$-0.989566\pi$
$863$	− 56.5685i	− 0.0655487i	0.999463	−	0.0327744i	$-0.0104343\pi$
$864$	0	0
$865$	1470.00	1.69942
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−565.685	−0.650961
$870$	0	0
$871$	−704.000	−0.808266
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 462.000i	− 0.526796i	−0.964687	−	0.263398i	$-0.915157\pi$
$877$	− 462.000i	− 0.526796i	0.964687	−	0.263398i	$-0.0848432\pi$
$878$	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
$883$	8.00000i	0.00906002i	0.999990	+	0.00453001i	$0.00144195\pi$
$883$	8.00000i	0.00906002i	−0.999990	+	0.00453001i	$0.998558\pi$
$884$	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
$893$	−271.529	−0.304064
$894$	0	0
$895$	1440.00	1.60894
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−118.794	−0.132140
$900$	0	0
$901$	490.000i	0.543840i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1527.35i	1.68768i
$906$	0	0
$907$	200.000i	0.220507i	0.993903	+	0.110254i	$0.0351663\pi$
$907$	200.000i	0.220507i	−0.993903	+	0.110254i	$0.964834\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	79.1960i	0.0869330i	0.999055	+	0.0434665i	$0.0138402\pi$
$911$	79.1960i	0.0869330i	−0.999055	+	0.0434665i	$0.986160\pi$
$912$	0	0
$913$	−736.000	−0.806134
$914$	0	0
$915$	0	0
$916$	0	0
$917$	678.823	0.740264
$918$	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
$923$	226.274	0.245151
$924$	0	0
$925$	− 750.000i	− 0.810811i
$926$	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
$931$	− 1520.00i	− 1.63265i
$932$	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	$-0.513253\pi$
$937$	−78.0000	−0.0832444	−0.0416222	+	0.999133i	$0.513253\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	267.286	0.284045	0.142023	−	0.989863i	$-0.454639\pi$
$941$	267.286	0.284045	0.142023	+	0.989863i	$0.454639\pi$
$942$	0	0
$943$	−840.000	−0.890774
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−576.999	−0.609292	−0.304646	−	0.952466i	$-0.598538\pi$
$947$	−576.999	−0.609292	−0.304646	+	0.952466i	$0.598538\pi$
$948$	0	0
$949$	− 640.000i	− 0.674394i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1336.43i	1.40234i	0.712993	+	0.701171i	$0.247339\pi$
$953$	1336.43i	1.40234i	−0.712993	+	0.701171i	$0.752661\pi$
$954$	0	0
$955$	2640.00i	2.76440i
$956$	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
$965$	−1258.65	−1.30430
$966$	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
$971$	118.794	0.122342	0.0611709	−	0.998127i	$-0.480517\pi$
$971$	118.794	0.122342	0.0611709	+	0.998127i	$0.480517\pi$
$972$	0	0
$973$	1440.00i	1.47996i
$974$	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
$979$	840.000i	0.858018i
$980$	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
$989$	−316.784	−0.320307
$990$	0	0
$991$	−116.000	−0.117053	−0.0585267	−	0.998286i	$-0.518640\pi$
$991$	−116.000	−0.117053	−0.0585267	+	0.998286i	$0.518640\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1385.93	−1.39289
$996$	0	0
$997$	− 130.000i	− 0.130391i	−0.997873	−	0.0651956i	$-0.979233\pi$
$997$	− 130.000i	− 0.130391i	0.997873	−	0.0651956i	$-0.0207671\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.h.a.2177.3		4
3.2	odd	2	inner	2304.3.h.a.2177.1		4
4.3	odd	2		2304.3.h.h.2177.3		4
8.3	odd	2		2304.3.h.h.2177.2		4
8.5	even	2	inner	2304.3.h.a.2177.2		4
12.11	even	2		2304.3.h.h.2177.1		4
16.3	odd	4		576.3.e.a.449.1		2
16.5	even	4		72.3.e.a.17.2	yes	2
16.11	odd	4		144.3.e.a.17.2		2
16.13	even	4		576.3.e.h.449.1		2
24.5	odd	2	inner	2304.3.h.a.2177.4		4
24.11	even	2		2304.3.h.h.2177.4		4
48.5	odd	4		72.3.e.a.17.1	✓	2
48.11	even	4		144.3.e.a.17.1		2
48.29	odd	4		576.3.e.h.449.2		2
48.35	even	4		576.3.e.a.449.2		2
80.27	even	4		3600.3.c.c.449.2		4
80.37	odd	4		1800.3.c.a.449.3		4
80.43	even	4		3600.3.c.c.449.4		4
80.53	odd	4		1800.3.c.a.449.1		4
80.59	odd	4		3600.3.l.l.1601.2		2
80.69	even	4		1800.3.l.a.1601.1		2
112.69	odd	4		3528.3.d.a.1961.1		2
144.5	odd	12		648.3.m.a.593.1		4
144.11	even	12		1296.3.q.k.1025.2		4
144.43	odd	12		1296.3.q.k.1025.1		4
144.59	even	12		1296.3.q.k.593.1		4
144.85	even	12		648.3.m.a.593.2		4
144.101	odd	12		648.3.m.a.377.2		4
144.133	even	12		648.3.m.a.377.1		4
144.139	odd	12		1296.3.q.k.593.2		4
240.53	even	4		1800.3.c.a.449.2		4
240.59	even	4		3600.3.l.l.1601.1		2
240.107	odd	4		3600.3.c.c.449.1		4
240.149	odd	4		1800.3.l.a.1601.2		2
240.197	even	4		1800.3.c.a.449.4		4
240.203	odd	4		3600.3.c.c.449.3		4
336.293	even	4		3528.3.d.a.1961.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.e.a.17.1	✓	2	48.5	odd	4
72.3.e.a.17.2	yes	2	16.5	even	4
144.3.e.a.17.1		2	48.11	even	4
144.3.e.a.17.2		2	16.11	odd	4
576.3.e.a.449.1		2	16.3	odd	4
576.3.e.a.449.2		2	48.35	even	4
576.3.e.h.449.1		2	16.13	even	4
576.3.e.h.449.2		2	48.29	odd	4
648.3.m.a.377.1		4	144.133	even	12
648.3.m.a.377.2		4	144.101	odd	12
648.3.m.a.593.1		4	144.5	odd	12
648.3.m.a.593.2		4	144.85	even	12
1296.3.q.k.593.1		4	144.59	even	12
1296.3.q.k.593.2		4	144.139	odd	12
1296.3.q.k.1025.1		4	144.43	odd	12
1296.3.q.k.1025.2		4	144.11	even	12
1800.3.c.a.449.1		4	80.53	odd	4
1800.3.c.a.449.2		4	240.53	even	4
1800.3.c.a.449.3		4	80.37	odd	4
1800.3.c.a.449.4		4	240.197	even	4
1800.3.l.a.1601.1		2	80.69	even	4
1800.3.l.a.1601.2		2	240.149	odd	4
2304.3.h.a.2177.1		4	3.2	odd	2	inner
2304.3.h.a.2177.2		4	8.5	even	2	inner
2304.3.h.a.2177.3		4	1.1	even	1	trivial
2304.3.h.a.2177.4		4	24.5	odd	2	inner
2304.3.h.h.2177.1		4	12.11	even	2
2304.3.h.h.2177.2		4	8.3	odd	2
2304.3.h.h.2177.3		4	4.3	odd	2
2304.3.h.h.2177.4		4	24.11	even	2
3528.3.d.a.1961.1		2	112.69	odd	4
3528.3.d.a.1961.2		2	336.293	even	4
3600.3.c.c.449.1		4	240.107	odd	4
3600.3.c.c.449.2		4	80.27	even	4
3600.3.c.c.449.3		4	240.203	odd	4
3600.3.c.c.449.4		4	80.43	even	4
3600.3.l.l.1601.1		2	240.59	even	4
3600.3.l.l.1601.2		2	80.59	odd	4

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.07107 q^{5} -12.0000 q^{7} +O(q^{10})\)	\(q+7.07107 q^{5} -12.0000 q^{7} -5.65685 q^{11} -8.00000i q^{13} +9.89949i q^{17} -16.0000i q^{19} +39.5980i q^{23} +25.0000 q^{25} +29.6985 q^{29} -4.00000 q^{31} -84.8528 q^{35} -30.0000i q^{37} +21.2132i q^{41} +8.00000i q^{43} -16.9706i q^{47} +95.0000 q^{49} +49.4975 q^{53} -40.0000 q^{55} +79.1960 q^{59} -14.0000i q^{61} -56.5685i q^{65} -88.0000i q^{67} +28.2843i q^{71} +80.0000 q^{73} +67.8823 q^{77} +100.000 q^{79} +130.108 q^{83} +70.0000i q^{85} -148.492i q^{89} +96.0000i q^{91} -113.137i q^{95} -112.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 48 q^{7}+O(q^{10}) \) 4 * q - 48 * q^7	\( 4 q - 48 q^{7} + 100 q^{25} - 16 q^{31} + 380 q^{49} - 160 q^{55} + 320 q^{73} + 400 q^{79} - 448 q^{97}+O(q^{100}) \) 4 * q - 48 * q^7 + 100 * q^25 - 16 * q^31 + 380 * q^49 - 160 * q^55 + 320 * q^73 + 400 * q^79 - 448 * q^97

Introduction

L-functions

Modular forms

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