LMFDB - Embedded newform 2320.2.j.b.289.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(289,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2320.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2320 = 2^{4} \cdot 5 \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2320.j (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:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 3x^{2} + 1$ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{5}$
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	2320.289
Dual form			2320.2.j.b.289.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 2.00000i) q^{5} +2.00000i q^{7} -3.00000 q^{9} -4.47214i q^{11} +4.00000i q^{13} -4.47214 q^{17} -4.47214i q^{19} -6.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +(3.00000 - 4.47214i) q^{29} +4.47214i q^{31} +(-4.00000 - 2.00000i) q^{35} +4.47214 q^{37} +(3.00000 - 6.00000i) q^{45} +8.94427 q^{47} +3.00000 q^{49} -4.00000i q^{53} +(8.94427 + 4.47214i) q^{55} -4.00000 q^{59} +8.94427i q^{61} -6.00000i q^{63} +(-8.00000 - 4.00000i) q^{65} -2.00000i q^{67} +13.4164 q^{73} +8.94427 q^{77} -13.4164i q^{79} +9.00000 q^{81} +6.00000i q^{83} +(4.47214 - 8.94427i) q^{85} -17.8885i q^{89} -8.00000 q^{91} +(8.94427 + 4.47214i) q^{95} -4.47214 q^{97} +13.4164i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{5} - 12 q^{9} - 12 q^{25} + 12 q^{29} - 16 q^{35} + 12 q^{45} + 12 q^{49} - 16 q^{59} - 32 q^{65} + 36 q^{81} - 32 q^{91}+O(q^{100})$ 4 * q - 4 * q^5 - 12 * q^9 - 12 * q^25 + 12 * q^29 - 16 * q^35 + 12 * q^45 + 12 * q^49 - 16 * q^59 - 32 * q^65 + 36 * q^81 - 32 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$ .

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$		0			0
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$		−	4.47214i		−	1.34840i	−0.738549	−	0.674200i	$-0.764489\pi$
$11$		−	4.47214i		−	1.34840i	0.738549	−	0.674200i	$-0.235511\pi$
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.47214			−1.08465			−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214			−1.08465			−0.542326	+	0.840168i	$0.682456\pi$
$18$		0			0
$19$		−	4.47214i		−	1.02598i	−0.858395	−	0.512989i	$-0.828538\pi$
$19$		−	4.47214i		−	1.02598i	0.858395	−	0.512989i	$-0.171462\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.00000	−	4.47214i	0.557086	−	0.830455i
$29$	3.00000	−	4.47214i	0.557086	−	0.830455i
$30$		0			0
$31$			4.47214i			0.803219i	0.915811	+	0.401610i	$0.131549\pi$
$31$			4.47214i			0.803219i	−0.915811	+	0.401610i	$0.868451\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000	−	2.00000i	−0.676123	−	0.338062i
$36$		0			0
$37$	4.47214			0.735215			0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214			0.735215			0.367607	+	0.929981i	$0.380177\pi$
$38$		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			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	3.00000	−	6.00000i	0.447214	−	0.894427i
$46$		0			0
$47$	8.94427			1.30466			0.652328	−	0.757937i	$-0.273792\pi$
$47$	8.94427			1.30466			0.652328	+	0.757937i	$0.273792\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4.00000i		−	0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$		−	4.00000i		−	0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$		0			0
$55$	8.94427	+	4.47214i	1.20605	+	0.603023i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$			8.94427i			1.14520i	0.819836	+	0.572598i	$0.194065\pi$
$61$			8.94427i			1.14520i	−0.819836	+	0.572598i	$0.805935\pi$
$62$		0			0
$63$		−	6.00000i		−	0.755929i
$64$		0			0
$65$	−8.00000	−	4.00000i	−0.992278	−	0.496139i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	13.4164			1.57027			0.785136	−	0.619324i	$-0.212593\pi$
$73$	13.4164			1.57027			0.785136	+	0.619324i	$0.212593\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.94427			1.01929
$78$		0			0
$79$		−	13.4164i		−	1.50946i	−0.656033	−	0.754732i	$-0.727767\pi$
$79$		−	13.4164i		−	1.50946i	0.656033	−	0.754732i	$-0.272233\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$		0			0
$85$	4.47214	−	8.94427i	0.485071	−	0.970143i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	17.8885i		−	1.89618i	−0.317999	−	0.948091i	$-0.603011\pi$
$89$		−	17.8885i		−	1.89618i	0.317999	−	0.948091i	$-0.396989\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.94427	+	4.47214i	0.917663	+	0.458831i
$96$		0			0
$97$	−4.47214			−0.454077			−0.227038	−	0.973886i	$-0.572904\pi$
$97$	−4.47214			−0.454077			−0.227038	+	0.973886i	$0.572904\pi$
$98$		0			0
$99$			13.4164i			1.34840i
$100$		0			0
$101$			8.94427i			0.889988i	0.895533	+	0.444994i	$0.146794\pi$
$101$			8.94427i			0.889988i	−0.895533	+	0.444994i	$0.853206\pi$
$102$		0			0
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	2.00000i		−	0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$		−	2.00000i		−	0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.47214			−0.420703			−0.210352	−	0.977626i	$-0.567461\pi$
$113$	−4.47214			−0.420703			−0.210352	+	0.977626i	$0.567461\pi$
$114$		0			0
$115$	12.0000	+	6.00000i	1.11901	+	0.559503i
$116$		0			0
$117$		−	12.0000i		−	1.10940i
$118$		0			0
$119$		−	8.94427i		−	0.819920i
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$	−8.94427			−0.793676			−0.396838	−	0.917889i	$-0.629892\pi$
$127$	−8.94427			−0.793676			−0.396838	+	0.917889i	$0.629892\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	4.47214i		−	0.390732i	−0.980730	−	0.195366i	$-0.937410\pi$
$131$		−	4.47214i		−	0.390732i	0.980730	−	0.195366i	$-0.0625895\pi$
$132$		0			0
$133$	8.94427			0.775567
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.4164			1.14624			0.573121	−	0.819471i	$-0.305733\pi$
$137$	13.4164			1.14624			0.573121	+	0.819471i	$0.305733\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	17.8885			1.49592
$144$		0			0
$145$	5.94427	+	10.4721i	0.493645	+	0.869664i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	13.4164			1.08465
$154$		0			0
$155$	−8.94427	−	4.47214i	−0.718421	−	0.359211i
$156$		0			0
$157$	−13.4164			−1.07075			−0.535373	−	0.844616i	$-0.679829\pi$
$157$	−13.4164			−1.07075			−0.535373	+	0.844616i	$0.679829\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.0000			0.945732
$162$		0			0
$163$	−17.8885			−1.40114			−0.700569	−	0.713584i	$-0.747071\pi$
$163$	−17.8885			−1.40114			−0.700569	+	0.713584i	$0.747071\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	22.0000i		−	1.70241i	−0.524832	−	0.851206i	$-0.675872\pi$
$167$		−	22.0000i		−	1.70241i	0.524832	−	0.851206i	$-0.324128\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$			13.4164i			1.02598i
$172$		0			0
$173$			4.00000i			0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$			4.00000i			0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$	8.00000	−	6.00000i	0.604743	−	0.453557i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.47214	+	8.94427i	−0.328798	+	0.657596i
$186$		0			0
$187$			20.0000i			1.46254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	13.4164i		−	0.970777i	−0.874299	−	0.485389i	$-0.838678\pi$
$191$		−	13.4164i		−	0.970777i	0.874299	−	0.485389i	$-0.161322\pi$
$192$		0			0
$193$	−4.47214			−0.321911			−0.160956	−	0.986962i	$-0.551458\pi$
$193$	−4.47214			−0.321911			−0.160956	+	0.986962i	$0.551458\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.94427	+	6.00000i	0.627765	+	0.421117i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			18.0000i			1.25109i
$208$		0			0
$209$	−20.0000			−1.38343
$210$		0			0
$211$		−	22.3607i		−	1.53937i	−0.638422	−	0.769686i	$-0.720413\pi$
$211$		−	22.3607i		−	1.53937i	0.638422	−	0.769686i	$-0.279587\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−8.94427			−0.607177
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	17.8885i		−	1.20331i
$222$		0			0
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$		0			0
$225$	9.00000	+	12.0000i	0.600000	+	0.800000i
$226$		0			0
$227$		−	18.0000i		−	1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$		−	18.0000i		−	1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$		0			0
$229$		−	26.8328i		−	1.77316i	−0.462573	−	0.886581i	$-0.653074\pi$
$229$		−	26.8328i		−	1.77316i	0.462573	−	0.886581i	$-0.346926\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	24.0000i		−	1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$		−	24.0000i		−	1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$		0			0
$235$	−8.94427	+	17.8885i	−0.583460	+	1.16692i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.00000	+	6.00000i	−0.191663	+	0.383326i
$246$		0			0
$247$	17.8885			1.13822
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	4.47214i		−	0.282279i	−0.989990	−	0.141139i	$-0.954923\pi$
$251$		−	4.47214i		−	0.282279i	0.989990	−	0.141139i	$-0.0450766\pi$
$252$		0			0
$253$	−26.8328			−1.68696
$254$		0			0
$255$		0			0
$256$		0			0
$257$			8.00000i			0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$			8.00000i			0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$		0			0
$259$			8.94427i			0.555770i
$260$		0			0
$261$	−9.00000	+	13.4164i	−0.557086	+	0.830455i
$262$		0			0
$263$	26.8328			1.65458			0.827291	−	0.561773i	$-0.189881\pi$
$263$	26.8328			1.65458			0.827291	+	0.561773i	$0.189881\pi$
$264$		0			0
$265$	8.00000	+	4.00000i	0.491436	+	0.245718i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			8.94427i			0.545342i	0.962107	+	0.272671i	$0.0879070\pi$
$269$			8.94427i			0.545342i	−0.962107	+	0.272671i	$0.912093\pi$
$270$		0			0
$271$		−	13.4164i		−	0.814989i	−0.913208	−	0.407494i	$-0.866403\pi$
$271$		−	13.4164i		−	0.814989i	0.913208	−	0.407494i	$-0.133597\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−17.8885	+	13.4164i	−1.07872	+	0.809040i
$276$		0			0
$277$			28.0000i			1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$			28.0000i			1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$		0			0
$279$		−	13.4164i		−	0.803219i
$280$		0			0
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$			6.00000i			0.356663i	0.983970	+	0.178331i	$0.0570699\pi$
$283$			6.00000i			0.356663i	−0.983970	+	0.178331i	$0.942930\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	3.00000			0.176471
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.47214			0.261265			0.130632	−	0.991431i	$-0.458299\pi$
$293$	4.47214			0.261265			0.130632	+	0.991431i	$0.458299\pi$
$294$		0			0
$295$	4.00000	−	8.00000i	0.232889	−	0.465778i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	24.0000			1.38796
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−17.8885	−	8.94427i	−1.02430	−	0.512148i
$306$		0			0
$307$	−17.8885			−1.02095			−0.510477	−	0.859892i	$-0.670531\pi$
$307$	−17.8885			−1.02095			−0.510477	+	0.859892i	$0.670531\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			22.3607i			1.26796i	0.773350	+	0.633979i	$0.218579\pi$
$311$			22.3607i			1.26796i	−0.773350	+	0.633979i	$0.781421\pi$
$312$		0			0
$313$		−	16.0000i		−	0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$		−	16.0000i		−	0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$		0			0
$315$	12.0000	+	6.00000i	0.676123	+	0.338062i
$316$		0			0
$317$	22.3607			1.25590			0.627950	−	0.778253i	$-0.283894\pi$
$317$	22.3607			1.25590			0.627950	+	0.778253i	$0.283894\pi$
$318$		0			0
$319$	−20.0000	−	13.4164i	−1.11979	−	0.751175i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			20.0000i			1.11283i
$324$		0			0
$325$	16.0000	−	12.0000i	0.887520	−	0.665640i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			17.8885i			0.986227i
$330$		0			0
$331$			13.4164i			0.737432i	0.929542	+	0.368716i	$0.120203\pi$
$331$			13.4164i			0.737432i	−0.929542	+	0.368716i	$0.879797\pi$
$332$		0			0
$333$	−13.4164			−0.735215
$334$		0			0
$335$	4.00000	+	2.00000i	0.218543	+	0.109272i
$336$		0			0
$337$	−4.47214			−0.243613			−0.121806	−	0.992554i	$-0.538869\pi$
$337$	−4.47214			−0.243613			−0.121806	+	0.992554i	$0.538869\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	20.0000			1.08306
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	2.00000i		−	0.107366i	−0.998558	−	0.0536828i	$-0.982904\pi$
$347$		−	2.00000i		−	0.107366i	0.998558	−	0.0536828i	$-0.0170960\pi$
$348$		0			0
$349$	6.00000			0.321173			0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000			0.321173			0.160586	+	0.987022i	$0.448662\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			24.0000i			1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$			24.0000i			1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.4164i		−	0.708091i	−0.935228	−	0.354045i	$-0.884806\pi$
$359$		−	13.4164i		−	0.708091i	0.935228	−	0.354045i	$-0.115194\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−13.4164	+	26.8328i	−0.702247	+	1.40449i
$366$		0			0
$367$	8.94427			0.466887			0.233444	−	0.972370i	$-0.425001\pi$
$367$	8.94427			0.466887			0.233444	+	0.972370i	$0.425001\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.00000			0.415339
$372$		0			0
$373$			4.00000i			0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$			4.00000i			0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	17.8885	+	12.0000i	0.921307	+	0.618031i
$378$		0			0
$379$		−	4.47214i		−	0.229718i	−0.993382	−	0.114859i	$-0.963358\pi$
$379$		−	4.47214i		−	0.229718i	0.993382	−	0.114859i	$-0.0366417\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	6.00000i		−	0.306586i	−0.988181	−	0.153293i	$-0.951012\pi$
$383$		−	6.00000i		−	0.306586i	0.988181	−	0.153293i	$-0.0489878\pi$
$384$		0			0
$385$	−8.94427	+	17.8885i	−0.455842	+	0.911685i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	8.94427i		−	0.453493i	−0.973954	−	0.226746i	$-0.927191\pi$
$389$		−	8.94427i		−	0.453493i	0.973954	−	0.226746i	$-0.0728088\pi$
$390$		0			0
$391$			26.8328i			1.35699i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	26.8328	+	13.4164i	1.35011	+	0.675053i
$396$		0			0
$397$			12.0000i			0.602263i	0.953583	+	0.301131i	$0.0973643\pi$
$397$			12.0000i			0.602263i	−0.953583	+	0.301131i	$0.902636\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−30.0000			−1.49813			−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000			−1.49813			−0.749064	+	0.662497i	$0.769497\pi$
$402$		0			0
$403$	−17.8885			−0.891092
$404$		0			0
$405$	−9.00000	+	18.0000i	−0.447214	+	0.894427i
$406$		0			0
$407$		−	20.0000i		−	0.991363i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$	−12.0000	−	6.00000i	−0.589057	−	0.294528i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.00000			0.195413			0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000			0.195413			0.0977064	+	0.995215i	$0.468849\pi$
$420$		0			0
$421$		−	8.94427i		−	0.435917i	−0.975958	−	0.217959i	$-0.930060\pi$
$421$		−	8.94427i		−	0.435917i	0.975958	−	0.217959i	$-0.0699398\pi$
$422$		0			0
$423$	−26.8328			−1.30466
$424$		0			0
$425$	13.4164	+	17.8885i	0.650791	+	0.867722i
$426$		0			0
$427$	−17.8885			−0.865687
$428$		0			0
$429$		0			0
$430$		0			0
$431$	32.0000			1.54139			0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000			1.54139			0.770693	+	0.637207i	$0.219910\pi$
$432$		0			0
$433$	−40.2492			−1.93425			−0.967127	−	0.254293i	$-0.918157\pi$
$433$	−40.2492			−1.93425			−0.967127	+	0.254293i	$0.918157\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−26.8328			−1.28359
$438$		0			0
$439$	16.0000			0.763638			0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$		0			0
$441$	−9.00000			−0.428571
$442$		0			0
$443$	−17.8885			−0.849910			−0.424955	−	0.905214i	$-0.639710\pi$
$443$	−17.8885			−0.849910			−0.424955	+	0.905214i	$0.639710\pi$
$444$		0			0
$445$	35.7771	+	17.8885i	1.69600	+	0.847998i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	17.8885i		−	0.844213i	−0.906546	−	0.422106i	$-0.861291\pi$
$449$		−	17.8885i		−	0.844213i	0.906546	−	0.422106i	$-0.138709\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.00000	−	16.0000i	0.375046	−	0.750092i
$456$		0			0
$457$		−	32.0000i		−	1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$		−	32.0000i		−	1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			26.8328i			1.24973i	0.780733	+	0.624864i	$0.214846\pi$
$461$			26.8328i			1.24973i	−0.780733	+	0.624864i	$0.785154\pi$
$462$		0			0
$463$			34.0000i			1.58011i	0.613033	+	0.790057i	$0.289949\pi$
$463$			34.0000i			1.58011i	−0.613033	+	0.790057i	$0.710051\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	17.8885			0.827783			0.413892	−	0.910326i	$-0.364169\pi$
$467$	17.8885			0.827783			0.413892	+	0.910326i	$0.364169\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−17.8885	+	13.4164i	−0.820783	+	0.615587i
$476$		0			0
$477$			12.0000i			0.549442i
$478$		0			0
$479$		−	13.4164i		−	0.613011i	−0.951869	−	0.306506i	$-0.900840\pi$
$479$		−	13.4164i		−	0.613011i	0.951869	−	0.306506i	$-0.0991598\pi$
$480$		0			0
$481$			17.8885i			0.815647i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.47214	−	8.94427i	0.203069	−	0.406138i
$486$		0			0
$487$			2.00000i			0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$			2.00000i			0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			13.4164i			0.605474i	0.953074	+	0.302737i	$0.0979004\pi$
$491$			13.4164i			0.605474i	−0.953074	+	0.302737i	$0.902100\pi$
$492$		0			0
$493$	−13.4164	+	20.0000i	−0.604245	+	0.900755i
$494$		0			0
$495$	−26.8328	−	13.4164i	−1.20605	−	0.603023i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−8.94427			−0.398805			−0.199403	−	0.979918i	$-0.563900\pi$
$503$	−8.94427			−0.398805			−0.199403	+	0.979918i	$0.563900\pi$
$504$		0			0
$505$	−17.8885	−	8.94427i	−0.796030	−	0.398015i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.0000			−0.443242			−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000			−0.443242			−0.221621	+	0.975133i	$0.571135\pi$
$510$		0			0
$511$			26.8328i			1.18701i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.0000	+	6.00000i	0.528783	+	0.264392i
$516$		0			0
$517$		−	40.0000i		−	1.75920i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$		−	26.0000i		−	1.13690i	−0.822718	−	0.568450i	$-0.807543\pi$
$523$		−	26.0000i		−	1.13690i	0.822718	−	0.568450i	$-0.192457\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	20.0000i		−	0.871214i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	12.0000			0.520756
$532$		0			0
$533$		0			0
$534$		0			0
$535$	4.00000	+	2.00000i	0.172935	+	0.0864675i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	13.4164i		−	0.577886i
$540$		0			0
$541$		−	8.94427i		−	0.384544i	−0.981342	−	0.192272i	$-0.938414\pi$
$541$		−	8.94427i		−	0.384544i	0.981342	−	0.192272i	$-0.0615856\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.0000	−	20.0000i	0.428353	−	0.856706i
$546$		0			0
$547$			38.0000i			1.62476i	0.583127	+	0.812381i	$0.301829\pi$
$547$			38.0000i			1.62476i	−0.583127	+	0.812381i	$0.698171\pi$
$548$		0			0
$549$		−	26.8328i		−	1.14520i
$550$		0			0
$551$	−20.0000	−	13.4164i	−0.852029	−	0.571558i
$552$		0			0
$553$	26.8328			1.14105
$554$		0			0
$555$		0			0
$556$		0			0
$557$			12.0000i			0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$			12.0000i			0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17.8885			−0.753912			−0.376956	−	0.926231i	$-0.623029\pi$
$563$	−17.8885			−0.753912			−0.376956	+	0.926231i	$0.623029\pi$
$564$		0			0
$565$	4.47214	−	8.94427i	0.188144	−	0.376288i
$566$		0			0
$567$			18.0000i			0.755929i
$568$		0			0
$569$		−	35.7771i		−	1.49985i	−0.661521	−	0.749927i	$-0.730089\pi$
$569$		−	35.7771i		−	1.49985i	0.661521	−	0.749927i	$-0.269911\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.0000	+	18.0000i	−1.00087	+	0.750652i
$576$		0			0
$577$	−4.47214			−0.186177			−0.0930887	−	0.995658i	$-0.529674\pi$
$577$	−4.47214			−0.186177			−0.0930887	+	0.995658i	$0.529674\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.0000			−0.497844
$582$		0			0
$583$	−17.8885			−0.740868
$584$		0			0
$585$	24.0000	+	12.0000i	0.992278	+	0.496139i
$586$		0			0
$587$			38.0000i			1.56843i	0.620491	+	0.784214i	$0.286934\pi$
$587$			38.0000i			1.56843i	−0.620491	+	0.784214i	$0.713066\pi$
$588$		0			0
$589$	20.0000			0.824086
$590$		0			0
$591$		0			0
$592$		0			0
$593$			24.0000i			0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$			24.0000i			0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$		0			0
$595$	17.8885	+	8.94427i	0.733359	+	0.366679i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	31.3050i		−	1.27909i	−0.768755	−	0.639543i	$-0.779124\pi$
$599$		−	31.3050i		−	1.27909i	0.768755	−	0.639543i	$-0.220876\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$			6.00000i			0.244339i
$604$		0			0
$605$	9.00000	−	18.0000i	0.365902	−	0.731804i
$606$		0			0
$607$	−26.8328			−1.08911			−0.544555	−	0.838725i	$-0.683302\pi$
$607$	−26.8328			−1.08911			−0.544555	+	0.838725i	$0.683302\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			35.7771i			1.44739i
$612$		0			0
$613$			4.00000i			0.161558i	0.996732	+	0.0807792i	$0.0257409\pi$
$613$			4.00000i			0.161558i	−0.996732	+	0.0807792i	$0.974259\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13.4164			0.540124			0.270062	−	0.962843i	$-0.412956\pi$
$617$	13.4164			0.540124			0.270062	+	0.962843i	$0.412956\pi$
$618$		0			0
$619$		−	4.47214i		−	0.179750i	−0.995953	−	0.0898752i	$-0.971353\pi$
$619$		−	4.47214i		−	0.179750i	0.995953	−	0.0898752i	$-0.0286468\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	35.7771			1.43338
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−20.0000			−0.797452
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.94427	−	17.8885i	0.354943	−	0.709885i
$636$		0			0
$637$			12.0000i			0.475457i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	17.8885i		−	0.706555i	−0.935519	−	0.353278i	$-0.885067\pi$
$641$		−	17.8885i		−	0.706555i	0.935519	−	0.353278i	$-0.114933\pi$
$642$		0			0
$643$			14.0000i			0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$			14.0000i			0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			2.00000i			0.0786281i	0.999227	+	0.0393141i	$0.0125173\pi$
$647$			2.00000i			0.0786281i	−0.999227	+	0.0393141i	$0.987483\pi$
$648$		0			0
$649$			17.8885i			0.702187i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22.3607			0.875041			0.437521	−	0.899208i	$-0.355857\pi$
$653$	22.3607			0.875041			0.437521	+	0.899208i	$0.355857\pi$
$654$		0			0
$655$	8.94427	+	4.47214i	0.349482	+	0.174741i
$656$		0			0
$657$	−40.2492			−1.57027
$658$		0			0
$659$			31.3050i			1.21947i	0.792606	+	0.609734i	$0.208724\pi$
$659$			31.3050i			1.21947i	−0.792606	+	0.609734i	$0.791276\pi$
$660$		0			0
$661$	−10.0000			−0.388955			−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000			−0.388955			−0.194477	+	0.980907i	$0.562301\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.94427	+	17.8885i	−0.346844	+	0.693688i
$666$		0			0
$667$	−26.8328	−	18.0000i	−1.03897	−	0.696963i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	40.0000			1.54418
$672$		0			0
$673$			16.0000i			0.616755i	0.951264	+	0.308377i	$0.0997859\pi$
$673$			16.0000i			0.616755i	−0.951264	+	0.308377i	$0.900214\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.47214			0.171878			0.0859391	−	0.996300i	$-0.472611\pi$
$677$	4.47214			0.171878			0.0859391	+	0.996300i	$0.472611\pi$
$678$		0			0
$679$		−	8.94427i		−	0.343250i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			6.00000i			0.229584i	0.993390	+	0.114792i	$0.0366201\pi$
$683$			6.00000i			0.229584i	−0.993390	+	0.114792i	$0.963380\pi$
$684$		0			0
$685$	−13.4164	+	26.8328i	−0.512615	+	1.02523i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.0000			0.609551
$690$		0			0
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$		0			0
$693$	−26.8328			−1.01929
$694$		0			0
$695$	−20.0000	+	40.0000i	−0.758643	+	1.51729i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$		−	20.0000i		−	0.754314i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−17.8885			−0.672768
$708$		0			0
$709$	30.0000			1.12667			0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000			1.12667			0.563337	+	0.826227i	$0.309517\pi$
$710$		0			0
$711$			40.2492i			1.50946i
$712$		0			0
$713$	26.8328			1.00490
$714$		0			0
$715$	−17.8885	+	35.7771i	−0.668994	+	1.33799i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−26.8885	+	1.41641i	−0.998615	+	0.0526041i
$726$		0			0
$727$	26.8328			0.995174			0.497587	−	0.867414i	$-0.334220\pi$
$727$	26.8328			0.995174			0.497587	+	0.867414i	$0.334220\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−13.4164			−0.495546			−0.247773	−	0.968818i	$-0.579699\pi$
$733$	−13.4164			−0.495546			−0.247773	+	0.968818i	$0.579699\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.94427			−0.329466
$738$		0			0
$739$		−	40.2492i		−	1.48059i	−0.672281	−	0.740296i	$-0.734685\pi$
$739$		−	40.2492i		−	1.48059i	0.672281	−	0.740296i	$-0.265315\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.94427			−0.328134			−0.164067	−	0.986449i	$-0.552461\pi$
$743$	−8.94427			−0.328134			−0.164067	+	0.986449i	$0.552461\pi$
$744$		0			0
$745$	10.0000	−	20.0000i	0.366372	−	0.732743i
$746$		0			0
$747$		−	18.0000i		−	0.658586i
$748$		0			0
$749$	4.00000			0.146157
$750$		0			0
$751$			40.2492i			1.46872i	0.678763	+	0.734358i	$0.262516\pi$
$751$			40.2492i			1.46872i	−0.678763	+	0.734358i	$0.737484\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−31.3050			−1.13780			−0.568899	−	0.822407i	$-0.692630\pi$
$757$	−31.3050			−1.13780			−0.568899	+	0.822407i	$0.692630\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$		0			0
$763$		−	20.0000i		−	0.724049i
$764$		0			0
$765$	−13.4164	+	26.8328i	−0.485071	+	0.970143i
$766$		0			0
$767$		−	16.0000i		−	0.577727i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	40.2492			1.44766			0.723832	−	0.689976i	$-0.242379\pi$
$773$	40.2492			1.44766			0.723832	+	0.689976i	$0.242379\pi$
$774$		0			0
$775$	17.8885	−	13.4164i	0.642575	−	0.481932i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	13.4164	−	26.8328i	0.478852	−	0.957704i
$786$		0			0
$787$			22.0000i			0.784215i	0.919919	+	0.392108i	$0.128254\pi$
$787$			22.0000i			0.784215i	−0.919919	+	0.392108i	$0.871746\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	8.94427i		−	0.318022i
$792$		0			0
$793$	−35.7771			−1.27048
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.4164			−0.475234			−0.237617	−	0.971359i	$-0.576366\pi$
$797$	−13.4164			−0.475234			−0.237617	+	0.971359i	$0.576366\pi$
$798$		0			0
$799$	−40.0000			−1.41510
$800$		0			0
$801$			53.6656i			1.89618i
$802$		0			0
$803$		−	60.0000i		−	2.11735i
$804$		0			0
$805$	−12.0000	+	24.0000i	−0.422944	+	0.845889i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	17.8885i		−	0.628928i	−0.949269	−	0.314464i	$-0.898175\pi$
$809$		−	17.8885i		−	0.628928i	0.949269	−	0.314464i	$-0.101825\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	17.8885	−	35.7771i	0.626608	−	1.25322i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	24.0000			0.838628
$820$		0			0
$821$	22.0000			0.767805			0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000			0.767805			0.383903	+	0.923374i	$0.374580\pi$
$822$		0			0
$823$	26.8328			0.935333			0.467667	−	0.883905i	$-0.345095\pi$
$823$	26.8328			0.935333			0.467667	+	0.883905i	$0.345095\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−53.6656			−1.86614			−0.933068	−	0.359699i	$-0.882879\pi$
$827$	−53.6656			−1.86614			−0.933068	+	0.359699i	$0.882879\pi$
$828$		0			0
$829$			8.94427i			0.310647i	0.987864	+	0.155324i	$0.0496421\pi$
$829$			8.94427i			0.310647i	−0.987864	+	0.155324i	$0.950358\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−13.4164			−0.464851
$834$		0			0
$835$	44.0000	+	22.0000i	1.52268	+	0.761341i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	31.3050i		−	1.08077i	−0.841419	−	0.540383i	$-0.818279\pi$
$839$		−	31.3050i		−	1.08077i	0.841419	−	0.540383i	$-0.181721\pi$
$840$		0			0
$841$	−11.0000	−	26.8328i	−0.379310	−	0.925270i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.00000	−	6.00000i	0.103203	−	0.206406i
$846$		0			0
$847$		−	18.0000i		−	0.618487i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	26.8328i		−	0.919817i
$852$		0			0
$853$	40.2492			1.37811			0.689054	−	0.724710i	$-0.258026\pi$
$853$	40.2492			1.37811			0.689054	+	0.724710i	$0.258026\pi$
$854$		0			0
$855$	−26.8328	−	13.4164i	−0.917663	−	0.458831i
$856$		0			0
$857$			48.0000i			1.63965i	0.572615	+	0.819824i	$0.305929\pi$
$857$			48.0000i			1.63965i	−0.572615	+	0.819824i	$0.694071\pi$
$858$		0			0
$859$		−	4.47214i		−	0.152587i	−0.997085	−	0.0762937i	$-0.975691\pi$
$859$		−	4.47214i		−	0.152587i	0.997085	−	0.0762937i	$-0.0243086\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	54.0000i		−	1.83818i	−0.394046	−	0.919091i	$-0.628925\pi$
$863$		−	54.0000i		−	1.83818i	0.394046	−	0.919091i	$-0.371075\pi$
$864$		0			0
$865$	−8.00000	−	4.00000i	−0.272008	−	0.136004i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−60.0000			−2.03536
$870$		0			0
$871$	8.00000			0.271070
$872$		0			0
$873$	13.4164			0.454077
$874$		0			0
$875$	4.00000	+	22.0000i	0.135225	+	0.743736i
$876$		0			0
$877$		−	12.0000i		−	0.405211i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$		−	12.0000i		−	0.405211i	0.979260	−	0.202606i	$-0.0649409\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	17.8885i		−	0.602680i	−0.953517	−	0.301340i	$-0.902566\pi$
$881$		−	17.8885i		−	0.602680i	0.953517	−	0.301340i	$-0.0974340\pi$
$882$		0			0
$883$			54.0000i			1.81724i	0.417619	+	0.908622i	$0.362865\pi$
$883$			54.0000i			1.81724i	−0.417619	+	0.908622i	$0.637135\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	44.7214			1.50160			0.750798	−	0.660532i	$-0.229669\pi$
$887$	44.7214			1.50160			0.750798	+	0.660532i	$0.229669\pi$
$888$		0			0
$889$		−	17.8885i		−	0.599963i
$890$		0			0
$891$		−	40.2492i		−	1.34840i
$892$		0			0
$893$		−	40.0000i		−	1.33855i
$894$		0			0
$895$	20.0000	−	40.0000i	0.668526	−	1.33705i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	20.0000	+	13.4164i	0.667037	+	0.447462i
$900$		0			0
$901$			17.8885i			0.595954i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10.0000	−	20.0000i	0.332411	−	0.664822i
$906$		0			0
$907$	−17.8885			−0.593979			−0.296990	−	0.954881i	$-0.595983\pi$
$907$	−17.8885			−0.593979			−0.296990	+	0.954881i	$0.595983\pi$
$908$		0			0
$909$		−	26.8328i		−	0.889988i
$910$		0			0
$911$			4.47214i			0.148168i	0.997252	+	0.0740842i	$0.0236034\pi$
$911$			4.47214i			0.148168i	−0.997252	+	0.0740842i	$0.976397\pi$
$912$		0			0
$913$	26.8328			0.888037
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.94427			0.295366
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−13.4164	−	17.8885i	−0.441129	−	0.588172i
$926$		0			0
$927$			18.0000i			0.591198i
$928$		0			0
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$		−	13.4164i		−	0.439705i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−40.0000	−	20.0000i	−1.30814	−	0.654070i
$936$		0			0
$937$			32.0000i			1.04539i	0.852518	+	0.522697i	$0.175074\pi$
$937$			32.0000i			1.04539i	−0.852518	+	0.522697i	$0.824926\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−18.0000			−0.586783			−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000			−0.586783			−0.293392	+	0.955992i	$0.594784\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.8885			0.581300			0.290650	−	0.956830i	$-0.406129\pi$
$947$	17.8885			0.581300			0.290650	+	0.956830i	$0.406129\pi$
$948$		0			0
$949$			53.6656i			1.74206i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	24.0000i		−	0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$		−	24.0000i		−	0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$		0			0
$955$	26.8328	+	13.4164i	0.868290	+	0.434145i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			26.8328i			0.866477i
$960$		0			0
$961$	11.0000			0.354839
$962$		0			0
$963$			6.00000i			0.193347i
$964$		0			0
$965$	4.47214	−	8.94427i	0.143963	−	0.287926i
$966$		0			0
$967$	44.7214			1.43814			0.719071	−	0.694937i	$-0.244568\pi$
$967$	44.7214			1.43814			0.719071	+	0.694937i	$0.244568\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	22.3607i		−	0.717588i	−0.933417	−	0.358794i	$-0.883188\pi$
$971$		−	22.3607i		−	0.717588i	0.933417	−	0.358794i	$-0.116812\pi$
$972$		0			0
$973$			40.0000i			1.28234i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	8.00000i		−	0.255943i	−0.991778	−	0.127971i	$-0.959153\pi$
$977$		−	8.00000i		−	0.255943i	0.991778	−	0.127971i	$-0.0408466\pi$
$978$		0			0
$979$	−80.0000			−2.55681
$980$		0			0
$981$	30.0000			0.957826
$982$		0			0
$983$	26.8328			0.855834			0.427917	−	0.903818i	$-0.359248\pi$
$983$	26.8328			0.855834			0.427917	+	0.903818i	$0.359248\pi$
$984$		0			0
$985$	24.0000	+	12.0000i	0.764704	+	0.382352i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	40.0000			1.27064			0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000			1.27064			0.635321	+	0.772248i	$0.280868\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.0000	−	32.0000i	0.507234	−	1.01447i
$996$		0			0
$997$	40.2492			1.27471			0.637353	−	0.770572i	$-0.280029\pi$
$997$	40.2492			1.27471			0.637353	+	0.770572i	$0.280029\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.j.b.289.3		4
4.3	odd	2		145.2.d.b.144.2	yes	4
5.4	even	2	inner	2320.2.j.b.289.1		4
12.11	even	2		1305.2.f.h.289.3		4
20.3	even	4		725.2.c.a.376.2		2
20.7	even	4		725.2.c.b.376.1		2
20.19	odd	2		145.2.d.b.144.3	yes	4
29.28	even	2	inner	2320.2.j.b.289.4		4
60.59	even	2		1305.2.f.h.289.2		4
116.115	odd	2		145.2.d.b.144.4	yes	4
145.144	even	2	inner	2320.2.j.b.289.2		4
348.347	even	2		1305.2.f.h.289.1		4
580.347	even	4		725.2.c.b.376.2		2
580.463	even	4		725.2.c.a.376.1		2
580.579	odd	2		145.2.d.b.144.1	✓	4
1740.1739	even	2		1305.2.f.h.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.d.b.144.1	✓	4	580.579	odd	2
145.2.d.b.144.2	yes	4	4.3	odd	2
145.2.d.b.144.3	yes	4	20.19	odd	2
145.2.d.b.144.4	yes	4	116.115	odd	2
725.2.c.a.376.1		2	580.463	even	4
725.2.c.a.376.2		2	20.3	even	4
725.2.c.b.376.1		2	20.7	even	4
725.2.c.b.376.2		2	580.347	even	4
1305.2.f.h.289.1		4	348.347	even	2
1305.2.f.h.289.2		4	60.59	even	2
1305.2.f.h.289.3		4	12.11	even	2
1305.2.f.h.289.4		4	1740.1739	even	2
2320.2.j.b.289.1		4	5.4	even	2	inner
2320.2.j.b.289.2		4	145.144	even	2	inner
2320.2.j.b.289.3		4	1.1	even	1	trivial
2320.2.j.b.289.4		4	29.28	even	2	inner

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2320.2.j.b.289.3

Level	$2320$
Weight	$2$
Character	2320.289
Analytic conductor	$18.525$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$