LMFDB - Embedded newform 384.3.h.b.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:	no
Analytic conductor:	$10.4632421514$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			65.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	384.65
Dual form			384.3.h.b.65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 2.82843i) q^{3} +(-7.00000 - 5.65685i) q^{9} +O(q^{10})$	$q+(-1.00000 + 2.82843i) q^{3} +(-7.00000 - 5.65685i) q^{9} -14.0000 q^{11} -33.9411i q^{17} +16.9706i q^{19} -25.0000 q^{25} +(23.0000 - 14.1421i) q^{27} +(14.0000 - 39.5980i) q^{33} -67.8823i q^{41} -84.8528i q^{43} -49.0000 q^{49} +(96.0000 + 33.9411i) q^{51} +(-48.0000 - 16.9706i) q^{57} -82.0000 q^{59} +118.794i q^{67} +142.000 q^{73} +(25.0000 - 70.7107i) q^{75} +(17.0000 + 79.1960i) q^{81} -158.000 q^{83} +101.823i q^{89} -94.0000 q^{97} +(98.0000 + 79.1960i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 14 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 14 * q^9	$ 2 q - 2 q^{3} - 14 q^{9} - 28 q^{11} - 50 q^{25} + 46 q^{27} + 28 q^{33} - 98 q^{49} + 192 q^{51} - 96 q^{57} - 164 q^{59} + 284 q^{73} + 50 q^{75} + 34 q^{81} - 316 q^{83} - 188 q^{97} + 196 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 14 * q^9 - 28 * q^11 - 50 * q^25 + 46 * q^27 + 28 * q^33 - 98 * q^49 + 192 * q^51 - 96 * q^57 - 164 * q^59 + 284 * q^73 + 50 * q^75 + 34 * q^81 - 316 * q^83 - 188 * q^97 + 196 * 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
$3$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$3$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$4$		0			0
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$10$		0			0
$11$	−14.0000			−1.27273			−0.636364	−	0.771389i	$-0.719562\pi$
$11$	−14.0000			−1.27273			−0.636364	+	0.771389i	$0.719562\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	33.9411i		−	1.99654i	−0.0588235	−	0.998268i	$-0.518735\pi$
$17$		−	33.9411i		−	1.99654i	0.0588235	−	0.998268i	$-0.481265\pi$
$18$		0			0
$19$			16.9706i			0.893188i	0.894737	+	0.446594i	$0.147363\pi$
$19$			16.9706i			0.893188i	−0.894737	+	0.446594i	$0.852637\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−25.0000			−1.00000
$26$		0			0
$27$	23.0000	−	14.1421i	0.851852	−	0.523783i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$	14.0000	−	39.5980i	0.424242	−	1.19994i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	67.8823i		−	1.65566i	−0.560976	−	0.827832i	$-0.689574\pi$
$41$		−	67.8823i		−	1.65566i	0.560976	−	0.827832i	$-0.310426\pi$
$42$		0			0
$43$		−	84.8528i		−	1.97332i	−0.162791	−	0.986661i	$-0.552050\pi$
$43$		−	84.8528i		−	1.97332i	0.162791	−	0.986661i	$-0.447950\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−49.0000			−1.00000
$50$		0			0
$51$	96.0000	+	33.9411i	1.88235	+	0.665512i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−48.0000	−	16.9706i	−0.842105	−	0.297729i
$58$		0			0
$59$	−82.0000			−1.38983			−0.694915	−	0.719092i	$-0.744558\pi$
$59$	−82.0000			−1.38983			−0.694915	+	0.719092i	$0.744558\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$			118.794i			1.77304i	0.462687	+	0.886522i	$0.346886\pi$
$67$			118.794i			1.77304i	−0.462687	+	0.886522i	$0.653114\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	142.000			1.94521			0.972603	−	0.232473i	$-0.0746819\pi$
$73$	142.000			1.94521			0.972603	+	0.232473i	$0.0746819\pi$
$74$		0			0
$75$	25.0000	−	70.7107i	0.333333	−	0.942809i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	17.0000	+	79.1960i	0.209877	+	0.977728i
$82$		0			0
$83$	−158.000			−1.90361			−0.951807	−	0.306697i	$-0.900776\pi$
$83$	−158.000			−1.90361			−0.951807	+	0.306697i	$0.900776\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			101.823i			1.14408i	0.820225	+	0.572041i	$0.193848\pi$
$89$			101.823i			1.14408i	−0.820225	+	0.572041i	$0.806152\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−94.0000			−0.969072			−0.484536	−	0.874771i	$-0.661012\pi$
$97$	−94.0000			−0.969072			−0.484536	+	0.874771i	$0.661012\pi$
$98$		0			0
$99$	98.0000	+	79.1960i	0.989899	+	0.799959i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−178.000			−1.66355			−0.831776	−	0.555112i	$-0.812675\pi$
$107$	−178.000			−1.66355			−0.831776	+	0.555112i	$0.812675\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			203.647i			1.80218i	0.433628	+	0.901092i	$0.357233\pi$
$113$			203.647i			1.80218i	−0.433628	+	0.901092i	$0.642767\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	75.0000			0.619835
$122$		0			0
$123$	192.000	+	67.8823i	1.56098	+	0.551888i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$	240.000	+	84.8528i	1.86047	+	0.657774i
$130$		0			0
$131$	62.0000			0.473282			0.236641	−	0.971597i	$-0.423953\pi$
$131$	62.0000			0.473282			0.236641	+	0.971597i	$0.423953\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			135.765i			0.990982i	0.868613	+	0.495491i	$0.165012\pi$
$137$			135.765i			0.990982i	−0.868613	+	0.495491i	$0.834988\pi$
$138$		0			0
$139$		−	186.676i		−	1.34299i	−0.741007	−	0.671497i	$-0.765652\pi$
$139$		−	186.676i		−	1.34299i	0.741007	−	0.671497i	$-0.234348\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	49.0000	−	138.593i	0.333333	−	0.942809i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−192.000	+	237.588i	−1.25490	+	1.55286i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	50.9117i		−	0.312342i	−0.987730	−	0.156171i	$-0.950085\pi$
$163$		−	50.9117i		−	0.312342i	0.987730	−	0.156171i	$-0.0499150\pi$
$164$		0			0
$165$		0			0
$166$		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
$171$	96.0000	−	118.794i	0.561404	−	0.694701i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	82.0000	−	231.931i	0.463277	−	1.31034i
$178$		0			0
$179$	−34.0000			−0.189944			−0.0949721	−	0.995480i	$-0.530276\pi$
$179$	−34.0000			−0.189944			−0.0949721	+	0.995480i	$0.530276\pi$
$180$		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
$187$			475.176i			2.54105i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−98.0000			−0.507772			−0.253886	−	0.967234i	$-0.581709\pi$
$193$	−98.0000			−0.507772			−0.253886	+	0.967234i	$0.581709\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	−336.000	−	118.794i	−1.67164	−	0.591015i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	237.588i		−	1.13678i
$210$		0			0
$211$		−	356.382i		−	1.68901i	−0.535545	−	0.844507i	$-0.679894\pi$
$211$		−	356.382i		−	1.68901i	0.535545	−	0.844507i	$-0.320106\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−142.000	+	401.637i	−0.648402	+	1.83396i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	175.000	+	141.421i	0.777778	+	0.628539i
$226$		0			0
$227$	−446.000			−1.96476			−0.982379	−	0.186900i	$-0.940156\pi$
$227$	−446.000			−1.96476			−0.982379	+	0.186900i	$0.940156\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	169.706i		−	0.728350i	−0.931330	−	0.364175i	$-0.881351\pi$
$233$		−	169.706i		−	0.728350i	0.931330	−	0.364175i	$-0.118649\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−194.000			−0.804979			−0.402490	−	0.915425i	$-0.631855\pi$
$241$	−194.000			−0.804979			−0.402490	+	0.915425i	$0.631855\pi$
$242$		0			0
$243$	−241.000	−	31.1127i	−0.991770	−	0.128036i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	158.000	−	446.891i	0.634538	−	1.79474i
$250$		0			0
$251$	466.000			1.85657			0.928287	−	0.371865i	$-0.121282\pi$
$251$	466.000			1.85657			0.928287	+	0.371865i	$0.121282\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	339.411i		−	1.32067i	−0.750973	−	0.660333i	$-0.770415\pi$
$257$		−	339.411i		−	1.32067i	0.750973	−	0.660333i	$-0.229585\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−288.000	−	101.823i	−1.07865	−	0.381361i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	350.000			1.27273
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	509.117i		−	1.81180i	−0.423488	−	0.905902i	$-0.639194\pi$
$281$		−	509.117i		−	1.81180i	0.423488	−	0.905902i	$-0.360806\pi$
$282$		0			0
$283$			560.029i			1.97890i	0.144876	+	0.989450i	$0.453722\pi$
$283$			560.029i			1.97890i	−0.144876	+	0.989450i	$0.546278\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−863.000			−2.98616
$290$		0			0
$291$	94.0000	−	265.872i	0.323024	−	0.913650i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−322.000	+	197.990i	−1.08418	+	0.666633i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	288.500i		−	0.939738i	−0.882736	−	0.469869i	$-0.844301\pi$
$307$		−	288.500i		−	0.939738i	0.882736	−	0.469869i	$-0.155699\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	526.000			1.68051			0.840256	−	0.542191i	$-0.182405\pi$
$313$	526.000			1.68051			0.840256	+	0.542191i	$0.182405\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	178.000	−	503.460i	0.554517	−	1.56841i
$322$		0			0
$323$	576.000			1.78328
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	661.852i		−	1.99955i	−0.0211480	−	0.999776i	$-0.506732\pi$
$331$		−	661.852i		−	1.99955i	0.0211480	−	0.999776i	$-0.493268\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	478.000			1.41840			0.709199	−	0.705009i	$-0.249057\pi$
$337$	478.000			1.41840			0.709199	+	0.705009i	$0.249057\pi$
$338$		0			0
$339$	−576.000	−	203.647i	−1.69912	−	0.600728i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	658.000			1.89625			0.948127	−	0.317892i	$-0.102975\pi$
$347$	658.000			1.89625			0.948127	+	0.317892i	$0.102975\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			678.823i			1.92301i	0.274788	+	0.961505i	$0.411392\pi$
$353$			678.823i			1.92301i	−0.274788	+	0.961505i	$0.588608\pi$
$354$		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
$361$	73.0000			0.202216
$362$		0			0
$363$	−75.0000	+	212.132i	−0.206612	+	0.584386i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	−384.000	+	475.176i	−1.04065	+	1.28774i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			322.441i			0.850767i	0.905013	+	0.425383i	$0.139861\pi$
$379$			322.441i			0.850767i	−0.905013	+	0.425383i	$0.860139\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−480.000	+	593.970i	−1.24031	+	1.53481i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−62.0000	+	175.362i	−0.157761	+	0.446215i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	237.588i		−	0.592488i	−0.955112	−	0.296244i	$-0.904266\pi$
$401$		−	237.588i		−	0.592488i	0.955112	−	0.296244i	$-0.0957342\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−334.000			−0.816626			−0.408313	−	0.912842i	$-0.633883\pi$
$409$	−334.000			−0.816626			−0.408313	+	0.912842i	$0.633883\pi$
$410$		0			0
$411$	−384.000	−	135.765i	−0.934307	−	0.330327i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	528.000	+	186.676i	1.26619	+	0.447665i
$418$		0			0
$419$	514.000			1.22673			0.613365	−	0.789799i	$-0.289815\pi$
$419$	514.000			1.22673			0.613365	+	0.789799i	$0.289815\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			848.528i			1.99654i
$426$		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
$433$	578.000			1.33487			0.667436	−	0.744667i	$-0.267392\pi$
$433$	578.000			1.33487			0.667436	+	0.744667i	$0.267392\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	343.000	+	277.186i	0.777778	+	0.628539i
$442$		0			0
$443$	−878.000			−1.98194			−0.990971	−	0.134079i	$-0.957192\pi$
$443$	−878.000			−1.98194			−0.990971	+	0.134079i	$0.957192\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			237.588i			0.529149i	0.964365	+	0.264574i	$0.0852315\pi$
$449$			237.588i			0.529149i	−0.964365	+	0.264574i	$0.914769\pi$
$450$		0			0
$451$			950.352i			2.10721i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−238.000			−0.520788			−0.260394	−	0.965502i	$-0.583852\pi$
$457$	−238.000			−0.520788			−0.260394	+	0.965502i	$0.583852\pi$
$458$		0			0
$459$	−480.000	−	780.646i	−1.04575	−	1.70075i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	34.0000			0.0728051			0.0364026	−	0.999337i	$-0.488410\pi$
$467$	34.0000			0.0728051			0.0364026	+	0.999337i	$0.488410\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1187.94i			2.51150i
$474$		0			0
$475$		−	424.264i		−	0.893188i
$476$		0			0
$477$		0			0
$478$		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
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$	144.000	+	50.9117i	0.294479	+	0.104114i
$490$		0			0
$491$	782.000			1.59267			0.796334	−	0.604857i	$-0.206770\pi$
$491$	782.000			1.59267			0.796334	+	0.604857i	$0.206770\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			593.970i			1.19032i	0.803607	+	0.595160i	$0.202911\pi$
$499$			593.970i			1.19032i	−0.803607	+	0.595160i	$0.797089\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−169.000	+	478.004i	−0.333333	+	0.942809i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	240.000	+	390.323i	0.467836	+	0.760863i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	271.529i		−	0.521169i	−0.965451	−	0.260584i	$-0.916085\pi$
$521$		−	271.529i		−	0.521169i	0.965451	−	0.260584i	$-0.0839152\pi$
$522$		0			0
$523$		−	967.322i		−	1.84956i	−0.380497	−	0.924782i	$-0.624247\pi$
$523$		−	967.322i		−	1.84956i	0.380497	−	0.924782i	$-0.375753\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	529.000			1.00000
$530$		0			0
$531$	574.000	+	463.862i	1.08098	+	0.873563i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	34.0000	−	96.1665i	0.0633147	−	0.179081i
$538$		0			0
$539$	686.000			1.27273
$540$		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
$547$		−	390.323i		−	0.713570i	−0.934186	−	0.356785i	$-0.883873\pi$
$547$		−	390.323i		−	0.713570i	0.934186	−	0.356785i	$-0.116127\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1344.00	−	475.176i	−2.39572	−	0.847016i
$562$		0			0
$563$	226.000			0.401421			0.200710	−	0.979651i	$-0.435675\pi$
$563$	226.000			0.401421			0.200710	+	0.979651i	$0.435675\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	950.352i		−	1.67021i	−0.550088	−	0.835107i	$-0.685406\pi$
$569$		−	950.352i		−	1.67021i	0.550088	−	0.835107i	$-0.314594\pi$
$570$		0			0
$571$		−	933.381i		−	1.63464i	−0.576182	−	0.817321i	$-0.695458\pi$
$571$		−	933.381i		−	1.63464i	0.576182	−	0.817321i	$-0.304542\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−2.00000			−0.00346620			−0.00173310	−	0.999998i	$-0.500552\pi$
$577$	−2.00000			−0.00346620			−0.00173310	+	0.999998i	$0.500552\pi$
$578$		0			0
$579$	98.0000	−	277.186i	0.169257	−	0.478732i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1138.00			−1.93867			−0.969336	−	0.245741i	$-0.920969\pi$
$587$	−1138.00			−1.93867			−0.969336	+	0.245741i	$0.920969\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			814.587i			1.37367i	0.726813	+	0.686836i	$0.241001\pi$
$593$			814.587i			1.37367i	−0.726813	+	0.686836i	$0.758999\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−914.000			−1.52080			−0.760399	−	0.649456i	$-0.774997\pi$
$601$	−914.000			−1.52080			−0.760399	+	0.649456i	$0.774997\pi$
$602$		0			0
$603$	672.000	−	831.558i	1.11443	−	1.37903i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		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
$617$		−	1187.94i		−	1.92535i	−0.270665	−	0.962674i	$-0.587243\pi$
$617$		−	1187.94i		−	1.92535i	0.270665	−	0.962674i	$-0.412757\pi$
$618$		0			0
$619$			1103.09i			1.78205i	0.453958	+	0.891023i	$0.350012\pi$
$619$			1103.09i			1.78205i	−0.453958	+	0.891023i	$0.649988\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	625.000			1.00000
$626$		0			0
$627$	672.000	+	237.588i	1.07177	+	0.378928i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	1008.00	+	356.382i	1.59242	+	0.563004i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	1187.94i		−	1.85326i	−0.375975	−	0.926630i	$-0.622692\pi$
$641$		−	1187.94i		−	1.85326i	0.375975	−	0.926630i	$-0.377308\pi$
$642$		0			0
$643$			424.264i			0.659820i	0.944012	+	0.329910i	$0.107018\pi$
$643$			424.264i			0.659820i	−0.944012	+	0.329910i	$0.892982\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	1148.00			1.76888
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−994.000	−	803.273i	−1.51294	−	1.22264i
$658$		0			0
$659$	−994.000			−1.50835			−0.754173	−	0.656676i	$-0.771962\pi$
$659$	−994.000			−1.50835			−0.754173	+	0.656676i	$0.771962\pi$
$660$		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
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1246.00			−1.85141			−0.925706	−	0.378244i	$-0.876528\pi$
$673$	−1246.00			−1.85141			−0.925706	+	0.378244i	$0.876528\pi$
$674$		0			0
$675$	−575.000	+	353.553i	−0.851852	+	0.523783i
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	446.000	−	1261.48i	0.654919	−	1.85239i
$682$		0			0
$683$	−398.000			−0.582723			−0.291362	−	0.956613i	$-0.594108\pi$
$683$	−398.000			−0.582723			−0.291362	+	0.956613i	$0.594108\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			1170.97i			1.69460i	0.531114	+	0.847300i	$0.321773\pi$
$691$			1170.97i			1.69460i	−0.531114	+	0.847300i	$0.678227\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2304.00			−3.30560
$698$		0			0
$699$	480.000	+	169.706i	0.686695	+	0.242783i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		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
$721$		0			0
$722$		0			0
$723$	194.000	−	548.715i	0.268326	−	0.758942i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	329.000	−	650.538i	0.451303	−	0.892371i
$730$		0			0
$731$	−2880.00			−3.93981
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1663.12i		−	2.25660i
$738$		0			0
$739$		−	1442.50i		−	1.95196i	−0.217862	−	0.975980i	$-0.569908\pi$
$739$		−	1442.50i		−	1.95196i	0.217862	−	0.975980i	$-0.430092\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1106.00	+	893.783i	1.48059	+	1.19650i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$	−466.000	+	1318.05i	−0.618858	+	1.75039i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	610.940i		−	0.802812i	−0.915900	−	0.401406i	$-0.868522\pi$
$761$		−	610.940i		−	0.802812i	0.915900	−	0.401406i	$-0.131478\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1054.00			1.37061			0.685306	−	0.728256i	$-0.259669\pi$
$769$	1054.00			1.37061			0.685306	+	0.728256i	$0.259669\pi$
$770$		0			0
$771$	960.000	+	339.411i	1.24514	+	0.440222i
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1152.00			1.47882
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	1272.79i		−	1.61727i	−0.588310	−	0.808635i	$-0.700207\pi$
$787$		−	1272.79i		−	1.61727i	0.588310	−	0.808635i	$-0.299793\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	576.000	−	712.764i	0.719101	−	0.889842i
$802$		0			0
$803$	−1988.00			−2.47572
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			339.411i			0.419544i	0.977750	+	0.209772i	$0.0672722\pi$
$809$			339.411i			0.419544i	−0.977750	+	0.209772i	$0.932728\pi$
$810$		0			0
$811$			1612.20i			1.98792i	0.109741	+	0.993960i	$0.464998\pi$
$811$			1612.20i			1.98792i	−0.109741	+	0.993960i	$0.535002\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1440.00			1.76255
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$	−350.000	+	989.949i	−0.424242	+	1.19994i
$826$		0			0
$827$	1262.00			1.52600			0.762999	−	0.646400i	$-0.223726\pi$
$827$	1262.00			1.52600			0.762999	+	0.646400i	$0.223726\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1663.12i			1.99654i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−841.000			−1.00000
$842$		0			0
$843$	1440.00	+	509.117i	1.70819	+	0.603935i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1584.00	−	560.029i	−1.86572	−	0.659633i
$850$		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
$857$			1221.88i			1.42576i	0.701284	+	0.712882i	$0.252611\pi$
$857$			1221.88i			1.42576i	−0.701284	+	0.712882i	$0.747389\pi$
$858$		0			0
$859$		−	492.146i		−	0.572929i	−0.958091	−	0.286465i	$-0.907520\pi$
$859$		−	492.146i		−	0.572929i	0.958091	−	0.286465i	$-0.0924801\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	863.000	−	2440.93i	0.995386	−	2.81538i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	658.000	+	531.744i	0.753723	+	0.609100i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1018.23i		−	1.15577i	−0.816118	−	0.577885i	$-0.803878\pi$
$881$		−	1018.23i		−	1.15577i	0.816118	−	0.577885i	$-0.196122\pi$
$882$		0			0
$883$		−	118.794i		−	0.134534i	−0.997735	−	0.0672672i	$-0.978572\pi$
$883$		−	118.794i		−	0.134534i	0.997735	−	0.0672672i	$-0.0214280\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−238.000	−	1108.74i	−0.267116	−	1.24438i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$			593.970i			0.654873i	0.944873	+	0.327436i	$0.106185\pi$
$907$			593.970i			0.654873i	−0.944873	+	0.327436i	$0.893815\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	2212.00			2.42278
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	816.000	+	288.500i	0.885993	+	0.313246i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1527.35i			1.64408i	0.569429	+	0.822040i	$0.307164\pi$
$929$			1527.35i			1.64408i	−0.569429	+	0.822040i	$0.692836\pi$
$930$		0			0
$931$		−	831.558i		−	0.893188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−718.000			−0.766275			−0.383138	−	0.923691i	$-0.625157\pi$
$937$	−718.000			−0.766275			−0.383138	+	0.923691i	$0.625157\pi$
$938$		0			0
$939$	−526.000	+	1487.75i	−0.560170	+	1.58440i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−994.000			−1.04963			−0.524815	−	0.851216i	$-0.675866\pi$
$947$	−994.000			−1.04963			−0.524815	+	0.851216i	$0.675866\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1900.70i			1.99444i	0.0745016	+	0.997221i	$0.476263\pi$
$953$			1900.70i			1.99444i	−0.0745016	+	0.997221i	$0.523737\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−961.000			−1.00000
$962$		0			0
$963$	1246.00	+	1006.92i	1.29387	+	1.04561i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$	−576.000	+	1629.17i	−0.594427	+	1.68129i
$970$		0			0
$971$	−974.000			−1.00309			−0.501545	−	0.865132i	$-0.667235\pi$
$971$	−974.000			−1.00309			−0.501545	+	0.865132i	$0.667235\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			373.352i			0.382142i	0.981576	+	0.191071i	$0.0611960\pi$
$977$			373.352i			0.382142i	−0.981576	+	0.191071i	$0.938804\pi$
$978$		0			0
$979$		−	1425.53i		−	1.45611i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$	1872.00	+	661.852i	1.88520	+	0.666518i
$994$		0			0
$995$		0			0
$996$		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.b.65.2	yes	2
3.2	odd	2		384.3.h.c.65.2	yes	2
4.3	odd	2		384.3.h.c.65.1	yes	2
8.3	odd	2	CM	384.3.h.b.65.2	yes	2
8.5	even	2		384.3.h.c.65.1	yes	2
12.11	even	2	inner	384.3.h.b.65.1	✓	2
16.3	odd	4		768.3.e.k.257.4		4
16.5	even	4		768.3.e.k.257.4		4
16.11	odd	4		768.3.e.k.257.1		4
16.13	even	4		768.3.e.k.257.1		4
24.5	odd	2	inner	384.3.h.b.65.1	✓	2
24.11	even	2		384.3.h.c.65.2	yes	2
48.5	odd	4		768.3.e.k.257.3		4
48.11	even	4		768.3.e.k.257.2		4
48.29	odd	4		768.3.e.k.257.2		4
48.35	even	4		768.3.e.k.257.3		4

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

\(f(q)\)	\(=\)	\(q+(-1.00000 + 2.82843i) q^{3} +(-7.00000 - 5.65685i) q^{9} +O(q^{10})\)	\(q+(-1.00000 + 2.82843i) q^{3} +(-7.00000 - 5.65685i) q^{9} -14.0000 q^{11} -33.9411i q^{17} +16.9706i q^{19} -25.0000 q^{25} +(23.0000 - 14.1421i) q^{27} +(14.0000 - 39.5980i) q^{33} -67.8823i q^{41} -84.8528i q^{43} -49.0000 q^{49} +(96.0000 + 33.9411i) q^{51} +(-48.0000 - 16.9706i) q^{57} -82.0000 q^{59} +118.794i q^{67} +142.000 q^{73} +(25.0000 - 70.7107i) q^{75} +(17.0000 + 79.1960i) q^{81} -158.000 q^{83} +101.823i q^{89} -94.0000 q^{97} +(98.0000 + 79.1960i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 14 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 14 * q^9	\( 2 q - 2 q^{3} - 14 q^{9} - 28 q^{11} - 50 q^{25} + 46 q^{27} + 28 q^{33} - 98 q^{49} + 192 q^{51} - 96 q^{57} - 164 q^{59} + 284 q^{73} + 50 q^{75} + 34 q^{81} - 316 q^{83} - 188 q^{97} + 196 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 14 * q^9 - 28 * q^11 - 50 * q^25 + 46 * q^27 + 28 * q^33 - 98 * q^49 + 192 * q^51 - 96 * q^57 - 164 * q^59 + 284 * q^73 + 50 * q^75 + 34 * q^81 - 316 * q^83 - 188 * q^97 + 196 * q^99

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