LMFDB - Embedded newform 384.3.h.a.65.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,3,Mod(65,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, 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("384.65");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	384.h (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:	yes
Analytic conductor:	$10.4632421514$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			65.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	384.65

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +9.79796 q^{5} +9.79796 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +9.79796 q^{5} +9.79796 q^{7} +9.00000 q^{9} -10.0000 q^{11} -29.3939 q^{15} -29.3939 q^{21} +71.0000 q^{25} -27.0000 q^{27} -29.3939 q^{29} +48.9898 q^{31} +30.0000 q^{33} +96.0000 q^{35} +88.1816 q^{45} +47.0000 q^{49} +48.9898 q^{53} -97.9796 q^{55} +10.0000 q^{59} +88.1816 q^{63} -50.0000 q^{73} -213.000 q^{75} -97.9796 q^{77} -146.969 q^{79} +81.0000 q^{81} +134.000 q^{83} +88.1816 q^{87} -146.969 q^{93} -190.000 q^{97} -90.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 18 * q^9	$ 2 q - 6 q^{3} + 18 q^{9} - 20 q^{11} + 142 q^{25} - 54 q^{27} + 60 q^{33} + 192 q^{35} + 94 q^{49} + 20 q^{59} - 100 q^{73} - 426 q^{75} + 162 q^{81} + 268 q^{83} - 380 q^{97} - 180 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 18 * q^9 - 20 * q^11 + 142 * q^25 - 54 * q^27 + 60 * q^33 + 192 * q^35 + 94 * q^49 + 20 * q^59 - 100 * q^73 - 426 * q^75 + 162 * q^81 + 268 * q^83 - 380 * q^97 - 180 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−3.00000	−1.00000
$3$	−3.00000	−1.00000
$4$	0	0
$5$	9.79796	1.95959	0.979796	−	0.200000i	$-0.0640942\pi$
$5$	9.79796	1.95959	0.979796	+	0.200000i	$0.0640942\pi$
$6$	0	0
$7$	9.79796	1.39971	0.699854	−	0.714286i	$-0.253248\pi$
$7$	9.79796	1.39971	0.699854	+	0.714286i	$0.253248\pi$
$8$	0	0
$9$	9.00000	1.00000
$10$	0	0
$11$	−10.0000	−0.909091	−0.454545	−	0.890724i	$-0.650198\pi$
$11$	−10.0000	−0.909091	−0.454545	+	0.890724i	$0.650198\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−29.3939	−1.95959
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	−29.3939	−1.39971
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	71.0000	2.84000
$26$	0	0
$27$	−27.0000	−1.00000
$28$	0	0
$29$	−29.3939	−1.01358	−0.506791	−	0.862069i	$-0.669168\pi$
$29$	−29.3939	−1.01358	−0.506791	+	0.862069i	$0.669168\pi$
$30$	0	0
$31$	48.9898	1.58032	0.790158	−	0.612903i	$-0.209998\pi$
$31$	48.9898	1.58032	0.790158	+	0.612903i	$0.209998\pi$
$32$	0	0
$33$	30.0000	0.909091
$34$	0	0
$35$	96.0000	2.74286
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	88.1816	1.95959
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	47.0000	0.959184
$50$	0	0
$51$	0	0
$52$	0	0
$53$	48.9898	0.924336	0.462168	−	0.886792i	$-0.347072\pi$
$53$	48.9898	0.924336	0.462168	+	0.886792i	$0.347072\pi$
$54$	0	0
$55$	−97.9796	−1.78145
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.0000	0.169492	0.0847458	−	0.996403i	$-0.472992\pi$
$59$	10.0000	0.169492	0.0847458	+	0.996403i	$0.472992\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	88.1816	1.39971
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−50.0000	−0.684932	−0.342466	−	0.939530i	$-0.611262\pi$
$73$	−50.0000	−0.684932	−0.342466	+	0.939530i	$0.611262\pi$
$74$	0	0
$75$	−213.000	−2.84000
$76$	0	0
$77$	−97.9796	−1.27246
$78$	0	0
$79$	−146.969	−1.86037	−0.930186	−	0.367089i	$-0.880355\pi$
$79$	−146.969	−1.86037	−0.930186	+	0.367089i	$0.880355\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	0	0
$83$	134.000	1.61446	0.807229	−	0.590238i	$-0.200966\pi$
$83$	134.000	1.61446	0.807229	+	0.590238i	$0.200966\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	88.1816	1.01358
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−146.969	−1.58032
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−190.000	−1.95876	−0.979381	−	0.202020i	$-0.935249\pi$
$97$	−190.000	−1.95876	−0.979381	+	0.202020i	$0.935249\pi$
$98$	0	0
$99$	−90.0000	−0.909091
$100$	0	0
$101$	−68.5857	−0.679066	−0.339533	−	0.940594i	$-0.610269\pi$
$101$	−68.5857	−0.679066	−0.339533	+	0.940594i	$0.610269\pi$
$102$	0	0
$103$	205.757	1.99764	0.998821	−	0.0485437i	$-0.0154580\pi$
$103$	205.757	1.99764	0.998821	+	0.0485437i	$0.0154580\pi$
$104$	0	0
$105$	−288.000	−2.74286
$106$	0	0
$107$	−86.0000	−0.803738	−0.401869	−	0.915697i	$-0.631639\pi$
$107$	−86.0000	−0.803738	−0.401869	+	0.915697i	$0.631639\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−21.0000	−0.173554
$122$	0	0
$123$	0	0
$124$	0	0
$125$	450.706	3.60565
$126$	0	0
$127$	−107.778	−0.848642	−0.424321	−	0.905512i	$-0.639487\pi$
$127$	−107.778	−0.848642	−0.424321	+	0.905512i	$0.639487\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	250.000	1.90840	0.954198	−	0.299174i	$-0.0967112\pi$
$131$	250.000	1.90840	0.954198	+	0.299174i	$0.0967112\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−264.545	−1.95959
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−288.000	−1.98621
$146$	0	0
$147$	−141.000	−0.959184
$148$	0	0
$149$	−68.5857	−0.460307	−0.230153	−	0.973154i	$-0.573923\pi$
$149$	−68.5857	−0.460307	−0.230153	+	0.973154i	$0.573923\pi$
$150$	0	0
$151$	48.9898	0.324436	0.162218	−	0.986755i	$-0.448135\pi$
$151$	48.9898	0.324436	0.162218	+	0.986755i	$0.448135\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	480.000	3.09677
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−146.969	−0.924336
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	293.939	1.78145
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−342.929	−1.98225	−0.991123	−	0.132948i	$-0.957556\pi$
$173$	−342.929	−1.98225	−0.991123	+	0.132948i	$0.957556\pi$
$174$	0	0
$175$	695.655	3.97517
$176$	0	0
$177$	−30.0000	−0.169492
$178$	0	0
$179$	−230.000	−1.28492	−0.642458	−	0.766321i	$-0.722085\pi$
$179$	−230.000	−1.28492	−0.642458	+	0.766321i	$0.722085\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−264.545	−1.39971
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	−290.000	−1.50259	−0.751295	−	0.659966i	$-0.770571\pi$
$193$	−290.000	−1.50259	−0.751295	+	0.659966i	$0.770571\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−342.929	−1.74075	−0.870377	−	0.492386i	$-0.836125\pi$
$197$	−342.929	−1.74075	−0.870377	+	0.492386i	$0.836125\pi$
$198$	0	0
$199$	−342.929	−1.72326	−0.861630	−	0.507538i	$-0.830556\pi$
$199$	−342.929	−1.72326	−0.861630	+	0.507538i	$0.830556\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−288.000	−1.41872
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	480.000	2.21198
$218$	0	0
$219$	150.000	0.684932
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−382.120	−1.71354	−0.856772	−	0.515695i	$-0.827534\pi$
$223$	−382.120	−1.71354	−0.856772	+	0.515695i	$0.827534\pi$
$224$	0	0
$225$	639.000	2.84000
$226$	0	0
$227$	−346.000	−1.52423	−0.762115	−	0.647442i	$-0.775839\pi$
$227$	−346.000	−1.52423	−0.762115	+	0.647442i	$0.775839\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	293.939	1.27246
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	440.908	1.86037
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	382.000	1.58506	0.792531	−	0.609831i	$-0.208763\pi$
$241$	382.000	1.58506	0.792531	+	0.609831i	$0.208763\pi$
$242$	0	0
$243$	−243.000	−1.00000
$244$	0	0
$245$	460.504	1.87961
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−402.000	−1.61446
$250$	0	0
$251$	470.000	1.87251	0.936255	−	0.351321i	$-0.114267\pi$
$251$	470.000	1.87251	0.936255	+	0.351321i	$0.114267\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−264.545	−1.01358
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	480.000	1.81132
$266$	0	0
$267$	0	0
$268$	0	0
$269$	323.333	1.20198	0.600990	−	0.799257i	$-0.294773\pi$
$269$	323.333	1.20198	0.600990	+	0.799257i	$0.294773\pi$
$270$	0	0
$271$	−538.888	−1.98852	−0.994258	−	0.107011i	$-0.965872\pi$
$271$	−538.888	−1.98852	−0.994258	+	0.107011i	$0.965872\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−710.000	−2.58182
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	440.908	1.58032
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	289.000	1.00000
$290$	0	0
$291$	570.000	1.95876
$292$	0	0
$293$	440.908	1.50481	0.752403	−	0.658703i	$-0.228895\pi$
$293$	440.908	1.50481	0.752403	+	0.658703i	$0.228895\pi$
$294$	0	0
$295$	97.9796	0.332134
$296$	0	0
$297$	270.000	0.909091
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	205.757	0.679066
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	−617.271	−1.99764
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−530.000	−1.69329	−0.846645	−	0.532158i	$-0.821381\pi$
$313$	−530.000	−1.69329	−0.846645	+	0.532158i	$0.821381\pi$
$314$	0	0
$315$	864.000	2.74286
$316$	0	0
$317$	−538.888	−1.69996	−0.849981	−	0.526814i	$-0.823386\pi$
$317$	−538.888	−1.69996	−0.849981	+	0.526814i	$0.823386\pi$
$318$	0	0
$319$	293.939	0.921438
$320$	0	0
$321$	258.000	0.803738
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	190.000	0.563798	0.281899	−	0.959444i	$-0.409036\pi$
$337$	190.000	0.563798	0.281899	+	0.959444i	$0.409036\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−489.898	−1.43665
$342$	0	0
$343$	−19.5959	−0.0571310
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−106.000	−0.305476	−0.152738	−	0.988267i	$-0.548809\pi$
$347$	−106.000	−0.305476	−0.152738	+	0.988267i	$0.548809\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	63.0000	0.173554
$364$	0	0
$365$	−489.898	−1.34219
$366$	0	0
$367$	−186.161	−0.507251	−0.253626	−	0.967302i	$-0.581623\pi$
$367$	−186.161	−0.507251	−0.253626	+	0.967302i	$0.581623\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	480.000	1.29380
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−1352.12	−3.60565
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	323.333	0.848642
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	−960.000	−2.49351
$386$	0	0
$387$	0	0
$388$	0	0
$389$	754.443	1.93944	0.969721	−	0.244216i	$-0.0785306\pi$
$389$	754.443	1.93944	0.969721	+	0.244216i	$0.0785306\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−750.000	−1.90840
$394$	0	0
$395$	−1440.00	−3.64557
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	793.635	1.95959
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−718.000	−1.75550	−0.877751	−	0.479118i	$-0.840957\pi$
$409$	−718.000	−1.75550	−0.877751	+	0.479118i	$0.840957\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	97.9796	0.237239
$414$	0	0
$415$	1312.93	3.16368
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−730.000	−1.74224	−0.871122	−	0.491067i	$-0.836607\pi$
$419$	−730.000	−1.74224	−0.871122	+	0.491067i	$0.836607\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−670.000	−1.54734	−0.773672	−	0.633586i	$-0.781582\pi$
$433$	−670.000	−1.54734	−0.773672	+	0.633586i	$0.781582\pi$
$434$	0	0
$435$	864.000	1.98621
$436$	0	0
$437$	0	0
$438$	0	0
$439$	832.827	1.89710	0.948550	−	0.316629i	$-0.102551\pi$
$439$	832.827	1.89710	0.948550	+	0.316629i	$0.102551\pi$
$440$	0	0
$441$	423.000	0.959184
$442$	0	0
$443$	86.0000	0.194131	0.0970655	−	0.995278i	$-0.469054\pi$
$443$	86.0000	0.194131	0.0970655	+	0.995278i	$0.469054\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	205.757	0.460307
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−146.969	−0.324436
$454$	0	0
$455$	0	0
$456$	0	0
$457$	530.000	1.15974	0.579869	−	0.814710i	$-0.303104\pi$
$457$	530.000	1.15974	0.579869	+	0.814710i	$0.303104\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	754.443	1.63654	0.818268	−	0.574837i	$-0.194935\pi$
$461$	754.443	1.63654	0.818268	+	0.574837i	$0.194935\pi$
$462$	0	0
$463$	−891.614	−1.92573	−0.962866	−	0.269978i	$-0.912983\pi$
$463$	−891.614	−1.92573	−0.962866	+	0.269978i	$0.912983\pi$
$464$	0	0
$465$	−1440.00	−3.09677
$466$	0	0
$467$	−634.000	−1.35760	−0.678801	−	0.734322i	$-0.737500\pi$
$467$	−634.000	−1.35760	−0.678801	+	0.734322i	$0.737500\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	440.908	0.924336
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1861.61	−3.83838
$486$	0	0
$487$	88.1816	0.181071	0.0905356	−	0.995893i	$-0.471142\pi$
$487$	88.1816	0.181071	0.0905356	+	0.995893i	$0.471142\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−470.000	−0.957230	−0.478615	−	0.878025i	$-0.658861\pi$
$491$	−470.000	−0.957230	−0.478615	+	0.878025i	$0.658861\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−881.816	−1.78145
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−672.000	−1.33069
$506$	0	0
$507$	−507.000	−1.00000
$508$	0	0
$509$	127.373	0.250243	0.125121	−	0.992141i	$-0.460068\pi$
$509$	127.373	0.250243	0.125121	+	0.992141i	$0.460068\pi$
$510$	0	0
$511$	−489.898	−0.958704
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2016.00	3.91456
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1028.79	1.98225
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	−2086.97	−3.97517
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	90.0000	0.169492
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−842.624	−1.57500
$536$	0	0
$537$	690.000	1.28492
$538$	0	0
$539$	−470.000	−0.871985
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−1440.00	−2.60398
$554$	0	0
$555$	0	0
$556$	0	0
$557$	636.867	1.14339	0.571694	−	0.820467i	$-0.306286\pi$
$557$	636.867	1.14339	0.571694	+	0.820467i	$0.306286\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	326.000	0.579041	0.289520	−	0.957172i	$-0.406504\pi$
$563$	326.000	0.579041	0.289520	+	0.957172i	$0.406504\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	793.635	1.39971
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−290.000	−0.502600	−0.251300	−	0.967909i	$-0.580858\pi$
$577$	−290.000	−0.502600	−0.251300	+	0.967909i	$0.580858\pi$
$578$	0	0
$579$	870.000	1.50259
$580$	0	0
$581$	1312.93	2.25977
$582$	0	0
$583$	−489.898	−0.840305
$584$	0	0
$585$	0	0
$586$	0	0
$587$	874.000	1.48893	0.744463	−	0.667663i	$-0.232705\pi$
$587$	874.000	1.48893	0.744463	+	0.667663i	$0.232705\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	1028.79	1.74075
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1028.79	1.72326
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	1198.00	1.99334	0.996672	−	0.0815138i	$-0.0259755\pi$
$601$	1198.00	1.99334	0.996672	+	0.0815138i	$0.0259755\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−205.757	−0.340094
$606$	0	0
$607$	−969.998	−1.59802	−0.799010	−	0.601318i	$-0.794643\pi$
$607$	−969.998	−1.59802	−0.799010	+	0.601318i	$0.794643\pi$
$608$	0	0
$609$	864.000	1.41872
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2641.00	4.22560
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	244.949	0.388192	0.194096	−	0.980983i	$-0.437823\pi$
$631$	244.949	0.388192	0.194096	+	0.980983i	$0.437823\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1056.00	−1.66299
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	−100.000	−0.154083
$650$	0	0
$651$	−1440.00	−2.21198
$652$	0	0
$653$	832.827	1.27539	0.637693	−	0.770291i	$-0.279889\pi$
$653$	832.827	1.27539	0.637693	+	0.770291i	$0.279889\pi$
$654$	0	0
$655$	2449.49	3.73968
$656$	0	0
$657$	−450.000	−0.684932
$658$	0	0
$659$	730.000	1.10774	0.553869	−	0.832603i	$-0.313151\pi$
$659$	730.000	1.10774	0.553869	+	0.832603i	$0.313151\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1146.36	1.71354
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−190.000	−0.282318	−0.141159	−	0.989987i	$-0.545083\pi$
$673$	−190.000	−0.282318	−0.141159	+	0.989987i	$0.545083\pi$
$674$	0	0
$675$	−1917.00	−2.84000
$676$	0	0
$677$	−146.969	−0.217089	−0.108545	−	0.994092i	$-0.534619\pi$
$677$	−146.969	−0.217089	−0.108545	+	0.994092i	$0.534619\pi$
$678$	0	0
$679$	−1861.61	−2.74170
$680$	0	0
$681$	1038.00	1.52423
$682$	0	0
$683$	1334.00	1.95315	0.976574	−	0.215182i	$-0.0690345\pi$
$683$	1334.00	1.95315	0.976574	+	0.215182i	$0.0690345\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	−881.816	−1.27246
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1401.11	−1.99873	−0.999364	−	0.0356633i	$-0.988646\pi$
$701$	−1401.11	−1.99873	−0.999364	+	0.0356633i	$0.988646\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−672.000	−0.950495
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−1322.72	−1.86037
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	2016.00	2.79612
$722$	0	0
$723$	−1146.00	−1.58506
$724$	0	0
$725$	−2086.97	−2.87857
$726$	0	0
$727$	−107.778	−0.148250	−0.0741249	−	0.997249i	$-0.523616\pi$
$727$	−107.778	−0.148250	−0.0741249	+	0.997249i	$0.523616\pi$
$728$	0	0
$729$	729.000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−1381.51	−1.87961
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−672.000	−0.902013
$746$	0	0
$747$	1206.00	1.61446
$748$	0	0
$749$	−842.624	−1.12500
$750$	0	0
$751$	−538.888	−0.717560	−0.358780	−	0.933422i	$-0.616807\pi$
$751$	−538.888	−0.717560	−0.358780	+	0.933422i	$0.616807\pi$
$752$	0	0
$753$	−1410.00	−1.87251
$754$	0	0
$755$	480.000	0.635762
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	862.000	1.12094	0.560468	−	0.828176i	$-0.310621\pi$
$769$	862.000	1.12094	0.560468	+	0.828176i	$0.310621\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1028.79	1.33090	0.665450	−	0.746442i	$-0.268240\pi$
$773$	1028.79	1.33090	0.665450	+	0.746442i	$0.268240\pi$
$774$	0	0
$775$	3478.28	4.48810
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	793.635	1.01358
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−1440.00	−1.81132
$796$	0	0
$797$	−930.806	−1.16789	−0.583944	−	0.811794i	$-0.698491\pi$
$797$	−930.806	−1.16789	−0.583944	+	0.811794i	$0.698491\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	500.000	0.622665
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−969.998	−1.20198
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	1616.66	1.98852
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1499.09	1.82593	0.912965	−	0.408039i	$-0.133787\pi$
$821$	1499.09	1.82593	0.912965	+	0.408039i	$0.133787\pi$
$822$	0	0
$823$	1577.47	1.91673	0.958367	−	0.285541i	$-0.0921732\pi$
$823$	1577.47	1.91673	0.958367	+	0.285541i	$0.0921732\pi$
$824$	0	0
$825$	2130.00	2.58182
$826$	0	0
$827$	1546.00	1.86941	0.934704	−	0.355428i	$-0.115665\pi$
$827$	1546.00	1.86941	0.934704	+	0.355428i	$0.115665\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1322.72	−1.58032
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	23.0000	0.0273484
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1655.86	1.95959
$846$	0	0
$847$	−205.757	−0.242925
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−3360.00	−3.88439
$866$	0	0
$867$	−867.000	−1.00000
$868$	0	0
$869$	1469.69	1.69125
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1710.00	−1.95876
$874$	0	0
$875$	4416.00	5.04686
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	−1322.72	−1.50481
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−293.939	−0.332134
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−1056.00	−1.18785
$890$	0	0
$891$	−810.000	−0.909091
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−2253.53	−2.51791
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1440.00	−1.60178
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−617.271	−0.679066
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−1340.00	−1.46769
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2449.49	2.67120
$918$	0	0
$919$	−1714.64	−1.86577	−0.932885	−	0.360174i	$-0.882717\pi$
$919$	−1714.64	−1.86577	−0.932885	+	0.360174i	$0.882717\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1851.81	1.99764
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1490.00	1.59018	0.795091	−	0.606491i	$-0.207423\pi$
$937$	1490.00	1.59018	0.795091	+	0.606491i	$0.207423\pi$
$938$	0	0
$939$	1590.00	1.69329
$940$	0	0
$941$	−1832.22	−1.94710	−0.973549	−	0.228480i	$-0.926624\pi$
$941$	−1832.22	−1.94710	−0.973549	+	0.228480i	$0.926624\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2592.00	−2.74286
$946$	0	0
$947$	1306.00	1.37909	0.689546	−	0.724242i	$-0.257810\pi$
$947$	1306.00	1.37909	0.689546	+	0.724242i	$0.257810\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	1616.66	1.69996
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−881.816	−0.921438
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1439.00	1.49740
$962$	0	0
$963$	−774.000	−0.803738
$964$	0	0
$965$	−2841.41	−2.94446
$966$	0	0
$967$	−303.737	−0.314102	−0.157051	−	0.987590i	$-0.550199\pi$
$967$	−303.737	−0.314102	−0.157051	+	0.987590i	$0.550199\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1930.00	−1.98764	−0.993821	−	0.110996i	$-0.964596\pi$
$971$	−1930.00	−1.98764	−0.993821	+	0.110996i	$0.964596\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−3360.00	−3.41117
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1420.70	1.43361	0.716803	−	0.697275i	$-0.245605\pi$
$991$	1420.70	1.43361	0.716803	+	0.697275i	$0.245605\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3360.00	−3.37688
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.h.a.65.2	yes	2
3.2	odd	2		384.3.h.d.65.1	yes	2
4.3	odd	2		384.3.h.d.65.2	yes	2
8.3	odd	2	inner	384.3.h.a.65.1	✓	2
8.5	even	2		384.3.h.d.65.1	yes	2
12.11	even	2	inner	384.3.h.a.65.1	✓	2
16.3	odd	4		768.3.e.j.257.3		4
16.5	even	4		768.3.e.j.257.4		4
16.11	odd	4		768.3.e.j.257.2		4
16.13	even	4		768.3.e.j.257.1		4
24.5	odd	2	CM	384.3.h.a.65.2	yes	2
24.11	even	2		384.3.h.d.65.2	yes	2
48.5	odd	4		768.3.e.j.257.1		4
48.11	even	4		768.3.e.j.257.3		4
48.29	odd	4		768.3.e.j.257.4		4
48.35	even	4		768.3.e.j.257.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.h.a.65.1	✓	2	8.3	odd	2	inner
384.3.h.a.65.1	✓	2	12.11	even	2	inner
384.3.h.a.65.2	yes	2	1.1	even	1	trivial
384.3.h.a.65.2	yes	2	24.5	odd	2	CM
384.3.h.d.65.1	yes	2	3.2	odd	2
384.3.h.d.65.1	yes	2	8.5	even	2
384.3.h.d.65.2	yes	2	4.3	odd	2
384.3.h.d.65.2	yes	2	24.11	even	2
768.3.e.j.257.1		4	16.13	even	4
768.3.e.j.257.1		4	48.5	odd	4
768.3.e.j.257.2		4	16.11	odd	4
768.3.e.j.257.2		4	48.35	even	4
768.3.e.j.257.3		4	16.3	odd	4
768.3.e.j.257.3		4	48.11	even	4
768.3.e.j.257.4		4	16.5	even	4
768.3.e.j.257.4		4	48.29	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +9.79796 q^{5} +9.79796 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +9.79796 q^{5} +9.79796 q^{7} +9.00000 q^{9} -10.0000 q^{11} -29.3939 q^{15} -29.3939 q^{21} +71.0000 q^{25} -27.0000 q^{27} -29.3939 q^{29} +48.9898 q^{31} +30.0000 q^{33} +96.0000 q^{35} +88.1816 q^{45} +47.0000 q^{49} +48.9898 q^{53} -97.9796 q^{55} +10.0000 q^{59} +88.1816 q^{63} -50.0000 q^{73} -213.000 q^{75} -97.9796 q^{77} -146.969 q^{79} +81.0000 q^{81} +134.000 q^{83} +88.1816 q^{87} -146.969 q^{93} -190.000 q^{97} -90.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 18 * q^9	\( 2 q - 6 q^{3} + 18 q^{9} - 20 q^{11} + 142 q^{25} - 54 q^{27} + 60 q^{33} + 192 q^{35} + 94 q^{49} + 20 q^{59} - 100 q^{73} - 426 q^{75} + 162 q^{81} + 268 q^{83} - 380 q^{97} - 180 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 18 * q^9 - 20 * q^11 + 142 * q^25 - 54 * q^27 + 60 * q^33 + 192 * q^35 + 94 * q^49 + 20 * q^59 - 100 * q^73 - 426 * q^75 + 162 * q^81 + 268 * q^83 - 380 * q^97 - 180 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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