LMFDB - Embedded newform 2160.2.w.d.1457.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(593,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))

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

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

chi := DirichletCharacter("2160.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.w (of order $4$, degree $2$, 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:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.12745506816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.3
Root			$-1.54779 - 1.54779i$ of defining polynomial
Character	$\chi$	$=$	2160.1457
Dual form			2160.2.w.d.593.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 1.87083i) q^{5} +(-1.79129 - 1.79129i) q^{7} +O(q^{10})$	$q+(1.22474 - 1.87083i) q^{5} +(-1.79129 - 1.79129i) q^{7} -5.54506i q^{11} +(1.79129 - 1.79129i) q^{13} +(1.87083 - 1.87083i) q^{17} +3.00000i q^{19} +(4.32032 + 4.32032i) q^{23} +(-2.00000 - 4.58258i) q^{25} -4.38774 q^{29} +1.00000 q^{31} +(-5.54506 + 1.15732i) q^{35} +(-5.00000 - 5.00000i) q^{37} +5.54506i q^{41} +(-3.20871 + 3.20871i) q^{43} +(-0.646084 + 0.646084i) q^{47} -0.582576i q^{49} +(0.0674228 + 0.0674228i) q^{53} +(-10.3739 - 6.79129i) q^{55} -4.38774 q^{59} -1.00000 q^{61} +(-1.15732 - 5.54506i) q^{65} +(-8.58258 - 8.58258i) q^{67} -5.54506i q^{71} +(-10.3739 + 10.3739i) q^{73} +(-9.93280 + 9.93280i) q^{77} -10.5826i q^{79} +(8.70806 + 8.70806i) q^{83} +(-1.20871 - 5.79129i) q^{85} +16.6352 q^{89} -6.41742 q^{91} +(5.61249 + 3.67423i) q^{95} +(-3.58258 - 3.58258i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7}+O(q^{10}) $ 8 * q + 4 * q^7	$ 8 q + 4 q^{7} - 4 q^{13} - 16 q^{25} + 8 q^{31} - 40 q^{37} - 44 q^{43} - 28 q^{55} - 8 q^{61} - 32 q^{67} - 28 q^{73} - 28 q^{85} - 88 q^{91} + 8 q^{97}+O(q^{100}) $ 8 * q + 4 * q^7 - 4 * q^13 - 16 * q^25 + 8 * q^31 - 40 * q^37 - 44 * q^43 - 28 * q^55 - 8 * q^61 - 32 * q^67 - 28 * q^73 - 28 * q^85 - 88 * q^91 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.22474	−	1.87083i	0.547723	−	0.836660i
$5$	1.22474	−	1.87083i	0.547723	−	0.836660i
$6$		0			0
$7$	−1.79129	−	1.79129i	−0.677043	−	0.677043i	0.282287	−	0.959330i	$-0.408907\pi$
$7$	−1.79129	−	1.79129i	−0.677043	−	0.677043i	−0.959330	+	0.282287i	$0.908907\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.54506i		−	1.67190i	−0.548806	−	0.835950i	$-0.684917\pi$
$11$		−	5.54506i		−	1.67190i	0.548806	−	0.835950i	$-0.315083\pi$
$12$		0			0
$13$	1.79129	−	1.79129i	0.496814	−	0.496814i	−0.413631	−	0.910445i	$-0.635740\pi$
$13$	1.79129	−	1.79129i	0.496814	−	0.496814i	0.910445	+	0.413631i	$0.135740\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.87083	−	1.87083i	0.453743	−	0.453743i	−0.442852	−	0.896595i	$-0.646033\pi$
$17$	1.87083	−	1.87083i	0.453743	−	0.453743i	0.896595	+	0.442852i	$0.146033\pi$
$18$		0			0
$19$			3.00000i			0.688247i	0.938924	+	0.344124i	$0.111824\pi$
$19$			3.00000i			0.688247i	−0.938924	+	0.344124i	$0.888176\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.32032	+	4.32032i	0.900849	+	0.900849i	0.995510	−	0.0946609i	$-0.0301767\pi$
$23$	4.32032	+	4.32032i	0.900849	+	0.900849i	−0.0946609	+	0.995510i	$0.530177\pi$
$24$		0			0
$25$	−2.00000	−	4.58258i	−0.400000	−	0.916515i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.38774			−0.814783			−0.407392	−	0.913254i	$-0.633562\pi$
$29$	−4.38774			−0.814783			−0.407392	+	0.913254i	$0.633562\pi$
$30$		0			0
$31$	1.00000			0.179605			0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000			0.179605			0.0898027	+	0.995960i	$0.471376\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.54506	+	1.15732i	−0.937287	+	0.195623i
$36$		0			0
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.54506i			0.865993i	0.901396	+	0.432997i	$0.142544\pi$
$41$			5.54506i			0.865993i	−0.901396	+	0.432997i	$0.857456\pi$
$42$		0			0
$43$	−3.20871	+	3.20871i	−0.489324	+	0.489324i	−0.908093	−	0.418769i	$-0.862462\pi$
$43$	−3.20871	+	3.20871i	−0.489324	+	0.489324i	0.418769	+	0.908093i	$0.362462\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.646084	+	0.646084i	−0.0942410	+	0.0942410i	−0.752656	−	0.658415i	$-0.771227\pi$
$47$	−0.646084	+	0.646084i	−0.0942410	+	0.0942410i	0.658415	+	0.752656i	$0.271227\pi$
$48$		0			0
$49$		−	0.582576i		−	0.0832251i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.0674228	+	0.0674228i	0.00926123	+	0.00926123i	0.711722	−	0.702461i	$-0.247915\pi$
$53$	0.0674228	+	0.0674228i	0.00926123	+	0.00926123i	−0.702461	+	0.711722i	$0.747915\pi$
$54$		0			0
$55$	−10.3739	−	6.79129i	−1.39881	−	0.915737i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.38774			−0.571235			−0.285618	−	0.958344i	$-0.592199\pi$
$59$	−4.38774			−0.571235			−0.285618	+	0.958344i	$0.592199\pi$
$60$		0			0
$61$	−1.00000			−0.128037			−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000			−0.128037			−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.15732	−	5.54506i	−0.143548	−	0.687780i
$66$		0			0
$67$	−8.58258	−	8.58258i	−1.04853	−	1.04853i	−0.998761	−	0.0497677i	$-0.984152\pi$
$67$	−8.58258	−	8.58258i	−1.04853	−	1.04853i	−0.0497677	−	0.998761i	$-0.515848\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.54506i		−	0.658078i	−0.944316	−	0.329039i	$-0.893275\pi$
$71$		−	5.54506i		−	0.658078i	0.944316	−	0.329039i	$-0.106725\pi$
$72$		0			0
$73$	−10.3739	+	10.3739i	−1.21417	+	1.21417i	−0.244526	+	0.969643i	$0.578632\pi$
$73$	−10.3739	+	10.3739i	−1.21417	+	1.21417i	−0.969643	+	0.244526i	$0.921368\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.93280	+	9.93280i	−1.13195	+	1.13195i
$78$		0			0
$79$		−	10.5826i		−	1.19063i	−0.803491	−	0.595316i	$-0.797027\pi$
$79$		−	10.5826i		−	1.19063i	0.803491	−	0.595316i	$-0.202973\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.70806	+	8.70806i	0.955834	+	0.955834i	0.999065	−	0.0432314i	$-0.0137653\pi$
$83$	8.70806	+	8.70806i	0.955834	+	0.955834i	−0.0432314	+	0.999065i	$0.513765\pi$
$84$		0			0
$85$	−1.20871	−	5.79129i	−0.131103	−	0.628153i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	16.6352			1.76333			0.881663	−	0.471879i	$-0.156424\pi$
$89$	16.6352			1.76333			0.881663	+	0.471879i	$0.156424\pi$
$90$		0			0
$91$	−6.41742			−0.672729
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.61249	+	3.67423i	0.575829	+	0.376969i
$96$		0			0
$97$	−3.58258	−	3.58258i	−0.363755	−	0.363755i	0.501438	−	0.865194i	$-0.332805\pi$
$97$	−3.58258	−	3.58258i	−0.363755	−	0.363755i	−0.865194	+	0.501438i	$0.832805\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	11.0901i		−	1.10351i	−0.834007	−	0.551754i	$-0.813959\pi$
$101$		−	11.0901i		−	1.10351i	0.834007	−	0.551754i	$-0.186041\pi$
$102$		0			0
$103$	3.58258	−	3.58258i	0.353002	−	0.353002i	−0.508224	−	0.861225i	$-0.669698\pi$
$103$	3.58258	−	3.58258i	0.353002	−	0.353002i	0.861225	+	0.508224i	$0.169698\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.74166	−	3.74166i	0.361720	−	0.361720i	−0.502726	−	0.864446i	$-0.667670\pi$
$107$	3.74166	−	3.74166i	0.361720	−	0.361720i	0.864446	+	0.502726i	$0.167670\pi$
$108$		0			0
$109$		−	13.7477i		−	1.31679i	−0.752671	−	0.658397i	$-0.771235\pi$
$109$		−	13.7477i		−	1.31679i	0.752671	−	0.658397i	$-0.228765\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.64064	+	8.64064i	0.812843	+	0.812843i	0.985059	−	0.172216i	$-0.0550928\pi$
$113$	8.64064	+	8.64064i	0.812843	+	0.812843i	−0.172216	+	0.985059i	$0.555093\pi$
$114$		0			0
$115$	13.3739	−	2.79129i	1.24712	−	0.260289i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.70239			−0.614407
$120$		0			0
$121$	−19.7477			−1.79525
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.0227	−	1.87083i	−0.985901	−	0.167332i
$126$		0			0
$127$	5.00000	+	5.00000i	0.443678	+	0.443678i	0.893246	−	0.449568i	$-0.148422\pi$
$127$	5.00000	+	5.00000i	0.443678	+	0.443678i	−0.449568	+	0.893246i	$0.648422\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			7.61816i			0.665602i	0.942997	+	0.332801i	$0.107994\pi$
$131$			7.61816i			0.665602i	−0.942997	+	0.332801i	$0.892006\pi$
$132$		0			0
$133$	5.37386	−	5.37386i	0.465973	−	0.465973i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.3766	+	10.3766i	−0.886534	+	0.886534i	−0.994188	−	0.107654i	$-0.965666\pi$
$137$	−10.3766	+	10.3766i	−0.886534	+	0.886534i	0.107654	+	0.994188i	$0.465666\pi$
$138$		0			0
$139$		−	3.16515i		−	0.268465i	−0.990950	−	0.134232i	$-0.957143\pi$
$139$		−	3.16515i		−	0.268465i	0.990950	−	0.134232i	$-0.0428568\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−9.93280	−	9.93280i	−0.830623	−	0.830623i
$144$		0			0
$145$	−5.37386	+	8.20871i	−0.446275	+	0.681696i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.85971			0.643892			0.321946	−	0.946758i	$-0.395663\pi$
$149$	7.85971			0.643892			0.321946	+	0.946758i	$0.395663\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.22474	−	1.87083i	0.0983739	−	0.150269i
$156$		0			0
$157$	3.20871	+	3.20871i	0.256083	+	0.256083i	0.823459	−	0.567376i	$-0.192041\pi$
$157$	3.20871	+	3.20871i	0.256083	+	0.256083i	−0.567376	+	0.823459i	$0.692041\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	15.4779i		−	1.21983i
$162$		0			0
$163$	5.00000	−	5.00000i	0.391630	−	0.391630i	−0.483638	−	0.875268i	$-0.660685\pi$
$163$	5.00000	−	5.00000i	0.391630	−	0.391630i	0.875268	+	0.483638i	$0.160685\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.6463	+	10.6463i	−0.823836	+	0.823836i	−0.986656	−	0.162820i	$-0.947941\pi$
$167$	−10.6463	+	10.6463i	−0.823836	+	0.823836i	0.162820	+	0.986656i	$0.447941\pi$
$168$		0			0
$169$			6.58258i			0.506352i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.92713	+	7.92713i	0.602689	+	0.602689i	0.941025	−	0.338337i	$-0.109864\pi$
$173$	7.92713	+	7.92713i	0.602689	+	0.602689i	−0.338337	+	0.941025i	$0.609864\pi$
$174$		0			0
$175$	−4.62614	+	11.7913i	−0.349703	+	0.891338i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	9.74773			0.724543			0.362271	−	0.932073i	$-0.382001\pi$
$181$	9.74773			0.724543			0.362271	+	0.932073i	$0.382001\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−15.4779	+	3.23042i	−1.13796	+	0.237505i
$186$		0			0
$187$	−10.3739	−	10.3739i	−0.758612	−	0.758612i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	18.7083i		−	1.35368i	−0.736128	−	0.676842i	$-0.763348\pi$
$191$		−	18.7083i		−	1.35368i	0.736128	−	0.676842i	$-0.236652\pi$
$192$		0			0
$193$	1.79129	−	1.79129i	0.128940	−	0.128940i	−0.639692	−	0.768632i	$-0.720938\pi$
$193$	1.79129	−	1.79129i	0.128940	−	0.128940i	0.768632	+	0.639692i	$0.220938\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.87083	−	1.87083i	0.133291	−	0.133291i	−0.637314	−	0.770605i	$-0.719954\pi$
$197$	1.87083	−	1.87083i	0.133291	−	0.133291i	0.770605	+	0.637314i	$0.219954\pi$
$198$		0			0
$199$		−	16.7477i		−	1.18721i	−0.804755	−	0.593607i	$-0.797703\pi$
$199$		−	16.7477i		−	1.18721i	0.804755	−	0.593607i	$-0.202297\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.85971	+	7.85971i	0.551643	+	0.551643i
$204$		0			0
$205$	10.3739	+	6.79129i	0.724542	+	0.474324i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	16.6352			1.15068
$210$		0			0
$211$	−9.74773			−0.671061			−0.335531	−	0.942029i	$-0.608916\pi$
$211$	−9.74773			−0.671061			−0.335531	+	0.942029i	$0.608916\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.07310	+	9.93280i	0.141384	+	0.677412i
$216$		0			0
$217$	−1.79129	−	1.79129i	−0.121601	−	0.121601i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	6.70239i		−	0.450851i
$222$		0			0
$223$	−8.58258	+	8.58258i	−0.574732	+	0.574732i	−0.933447	−	0.358715i	$-0.883215\pi$
$223$	−8.58258	+	8.58258i	−0.574732	+	0.574732i	0.358715	+	0.933447i	$0.383215\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.90465	−	6.90465i	0.458278	−	0.458278i	−0.439812	−	0.898090i	$-0.644955\pi$
$227$	6.90465	−	6.90465i	0.458278	−	0.458278i	0.898090	+	0.439812i	$0.144955\pi$
$228$		0			0
$229$		−	5.83485i		−	0.385578i	−0.981240	−	0.192789i	$-0.938247\pi$
$229$		−	5.83485i		−	0.385578i	0.981240	−	0.192789i	$-0.0617532\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.7700	−	16.7700i	−1.09864	−	1.09864i	−0.994570	−	0.104072i	$-0.966813\pi$
$233$	−16.7700	−	16.7700i	−1.09864	−	1.09864i	−0.104072	−	0.994570i	$-0.533187\pi$
$234$		0			0
$235$	0.417424	+	2.00000i	0.0272298	+	0.130466i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.2474			0.792222			0.396111	−	0.918203i	$-0.370360\pi$
$239$	12.2474			0.792222			0.396111	+	0.918203i	$0.370360\pi$
$240$		0			0
$241$	−11.7477			−0.756738			−0.378369	−	0.925655i	$-0.623515\pi$
$241$	−11.7477			−0.756738			−0.378369	+	0.925655i	$0.623515\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.08990	−	0.713507i	−0.0696311	−	0.0455843i
$246$		0			0
$247$	5.37386	+	5.37386i	0.341931	+	0.341931i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	22.1803i		−	1.40001i	−0.714140	−	0.700003i	$-0.753182\pi$
$251$		−	22.1803i		−	1.40001i	0.714140	−	0.700003i	$-0.246818\pi$
$252$		0			0
$253$	23.9564	−	23.9564i	1.50613	−	1.50613i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.51691	+	2.51691i	−0.157001	+	0.157001i	−0.781236	−	0.624236i	$-0.785411\pi$
$257$	−2.51691	+	2.51691i	−0.157001	+	0.157001i	0.624236	+	0.781236i	$0.285411\pi$
$258$		0			0
$259$			17.9129i			1.11305i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.780929	−	0.780929i	−0.0481542	−	0.0481542i	0.682620	−	0.730774i	$-0.260841\pi$
$263$	−0.780929	−	0.780929i	−0.0481542	−	0.0481542i	−0.730774	+	0.682620i	$0.760841\pi$
$264$		0			0
$265$	0.208712	−	0.0435608i	0.0128211	−	0.00267592i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.47197			−0.211690			−0.105845	−	0.994383i	$-0.533755\pi$
$269$	−3.47197			−0.211690			−0.105845	+	0.994383i	$0.533755\pi$
$270$		0			0
$271$	1.00000			0.0607457			0.0303728	−	0.999539i	$-0.490331\pi$
$271$	1.00000			0.0607457			0.0303728	+	0.999539i	$0.490331\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−25.4107	+	11.0901i	−1.53232	+	0.668760i
$276$		0			0
$277$	−13.2087	−	13.2087i	−0.793635	−	0.793635i	0.188448	−	0.982083i	$-0.439654\pi$
$277$	−13.2087	−	13.2087i	−0.793635	−	0.793635i	−0.982083	+	0.188448i	$0.939654\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2.07310i			0.123671i	0.998086	+	0.0618353i	$0.0196954\pi$
$281$			2.07310i			0.123671i	−0.998086	+	0.0618353i	$0.980305\pi$
$282$		0			0
$283$	13.2087	−	13.2087i	0.785176	−	0.785176i	−0.195523	−	0.980699i	$-0.562640\pi$
$283$	13.2087	−	13.2087i	0.785176	−	0.785176i	0.980699	+	0.195523i	$0.0626403\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.93280	−	9.93280i	0.586315	−	0.586315i
$288$		0			0
$289$			10.0000i			0.588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.3149	+	12.3149i	0.719442	+	0.719442i	0.968491	−	0.249049i	$-0.0801179\pi$
$293$	12.3149	+	12.3149i	0.719442	+	0.719442i	−0.249049	+	0.968491i	$0.580118\pi$
$294$		0			0
$295$	−5.37386	+	8.20871i	−0.312878	+	0.477930i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	15.4779			0.895108
$300$		0			0
$301$	11.4955			0.662587
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.22474	+	1.87083i	−0.0701287	+	0.107123i
$306$		0			0
$307$	15.7477	+	15.7477i	0.898770	+	0.898770i	0.995327	−	0.0965572i	$-0.0307831\pi$
$307$	15.7477	+	15.7477i	0.898770	+	0.898770i	−0.0965572	+	0.995327i	$0.530783\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	9.01703i		−	0.511309i	−0.966768	−	0.255654i	$-0.917709\pi$
$311$		−	9.01703i		−	0.511309i	0.966768	−	0.255654i	$-0.0822909\pi$
$312$		0			0
$313$	−17.1652	+	17.1652i	−0.970232	+	0.970232i	−0.999570	−	0.0293378i	$-0.990660\pi$
$313$	−17.1652	+	17.1652i	−0.970232	+	0.970232i	0.0293378	+	0.999570i	$0.490660\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	22.8938	−	22.8938i	1.28584	−	1.28584i	0.348552	−	0.937289i	$-0.386673\pi$
$317$	22.8938	−	22.8938i	1.28584	−	1.28584i	0.937289	−	0.348552i	$-0.113327\pi$
$318$		0			0
$319$			24.3303i			1.36224i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.61249	+	5.61249i	0.312287	+	0.312287i
$324$		0			0
$325$	−11.7913	−	4.62614i	−0.654063	−	0.256612i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.31464			0.127610
$330$		0			0
$331$	−12.7477			−0.700678			−0.350339	−	0.936623i	$-0.613934\pi$
$331$	−12.7477			−0.700678			−0.350339	+	0.936623i	$0.613934\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−26.5680	+	5.54506i	−1.45156	+	0.302959i
$336$		0			0
$337$	18.2087	+	18.2087i	0.991892	+	0.991892i	0.999967	−	0.00807564i	$-0.00257058\pi$
$337$	18.2087	+	18.2087i	0.991892	+	0.991892i	−0.00807564	+	0.999967i	$0.502571\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	5.54506i		−	0.300282i
$342$		0			0
$343$	−13.5826	+	13.5826i	−0.733390	+	0.733390i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.50579	+	8.50579i	−0.456615	+	0.456615i	−0.897543	−	0.440928i	$-0.854649\pi$
$347$	−8.50579	+	8.50579i	−0.456615	+	0.456615i	0.440928	+	0.897543i	$0.354649\pi$
$348$		0			0
$349$		−	19.4174i		−	1.03939i	−0.854352	−	0.519695i	$-0.826045\pi$
$349$		−	19.4174i		−	1.03939i	0.854352	−	0.519695i	$-0.173955\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.3823	−	12.3823i	−0.659043	−	0.659043i	0.296111	−	0.955154i	$-0.404310\pi$
$353$	−12.3823	−	12.3823i	−0.659043	−	0.659043i	−0.955154	+	0.296111i	$0.904310\pi$
$354$		0			0
$355$	−10.3739	−	6.79129i	−0.550588	−	0.360444i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	29.7984			1.57270			0.786350	−	0.617781i	$-0.211968\pi$
$359$	29.7984			1.57270			0.786350	+	0.617781i	$0.211968\pi$
$360$		0			0
$361$	10.0000			0.526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.70239	+	32.1131i	0.350819	+	1.68087i
$366$		0			0
$367$	8.95644	+	8.95644i	0.467522	+	0.467522i	0.901111	−	0.433589i	$-0.142753\pi$
$367$	8.95644	+	8.95644i	0.467522	+	0.467522i	−0.433589	+	0.901111i	$0.642753\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	0.241547i		−	0.0125405i
$372$		0			0
$373$	−3.58258	+	3.58258i	−0.185499	+	0.185499i	−0.793747	−	0.608248i	$-0.791873\pi$
$373$	−3.58258	+	3.58258i	−0.185499	+	0.185499i	0.608248	+	0.793747i	$0.291873\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.85971	+	7.85971i	−0.404796	+	0.404796i
$378$		0			0
$379$		−	7.74773i		−	0.397974i	−0.980002	−	0.198987i	$-0.936235\pi$
$379$		−	7.74773i		−	0.397974i	0.980002	−	0.198987i	$-0.0637652\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	20.9555	+	20.9555i	1.07078	+	1.07078i	0.997297	+	0.0734797i	$0.0234104\pi$
$383$	20.9555	+	20.9555i	1.07078	+	1.07078i	0.0734797	+	0.997297i	$0.476590\pi$
$384$		0			0
$385$	6.41742	+	30.7477i	0.327062	+	1.56705i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.85971			−0.398503			−0.199251	−	0.979948i	$-0.563851\pi$
$389$	−7.85971			−0.398503			−0.199251	+	0.979948i	$0.563851\pi$
$390$		0			0
$391$	16.1652			0.817507
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−19.7982	−	12.9610i	−0.996155	−	0.652136i
$396$		0			0
$397$	26.1216	+	26.1216i	1.31101	+	1.31101i	0.920675	+	0.390330i	$0.127639\pi$
$397$	26.1216	+	26.1216i	1.31101	+	1.31101i	0.390330	+	0.920675i	$0.372361\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			18.7083i			0.934247i	0.884192	+	0.467124i	$0.154710\pi$
$401$			18.7083i			0.934247i	−0.884192	+	0.467124i	$0.845290\pi$
$402$		0			0
$403$	1.79129	−	1.79129i	0.0892304	−	0.0892304i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−27.7253	+	27.7253i	−1.37429	+	1.37429i
$408$		0			0
$409$			24.1652i			1.19489i	0.801910	+	0.597445i	$0.203817\pi$
$409$			24.1652i			1.19489i	−0.801910	+	0.597445i	$0.796183\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.85971	+	7.85971i	0.386751	+	0.386751i
$414$		0			0
$415$	26.9564	−	5.62614i	1.32324	−	0.276176i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.915775			−0.0447385			−0.0223693	−	0.999750i	$-0.507121\pi$
$419$	−0.915775			−0.0447385			−0.0223693	+	0.999750i	$0.507121\pi$
$420$		0			0
$421$	39.7477			1.93719			0.968593	−	0.248652i	$-0.0799876\pi$
$421$	39.7477			1.93719			0.968593	+	0.248652i	$0.0799876\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.3149	−	4.83156i	−0.597359	−	0.234365i
$426$		0			0
$427$	1.79129	+	1.79129i	0.0866865	+	0.0866865i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	9.01703i		−	0.434335i	−0.976134	−	0.217168i	$-0.930318\pi$
$431$		−	9.01703i		−	0.434335i	0.976134	−	0.217168i	$-0.0696818\pi$
$432$		0			0
$433$	4.62614	−	4.62614i	0.222318	−	0.222318i	−0.587156	−	0.809474i	$-0.699752\pi$
$433$	4.62614	−	4.62614i	0.222318	−	0.222318i	0.809474	+	0.587156i	$0.199752\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.9610	+	12.9610i	−0.620007	+	0.620007i
$438$		0			0
$439$		−	10.5826i		−	0.505079i	−0.967587	−	0.252539i	$-0.918734\pi$
$439$		−	10.5826i		−	0.505079i	0.967587	−	0.252539i	$-0.0812657\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.92713	−	7.92713i	−0.376629	−	0.376629i	0.493255	−	0.869885i	$-0.335807\pi$
$443$	−7.92713	−	7.92713i	−0.376629	−	0.376629i	−0.869885	+	0.493255i	$0.835807\pi$
$444$		0			0
$445$	20.3739	−	31.1216i	0.965814	−	1.47530i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13.1632			−0.621211			−0.310605	−	0.950539i	$-0.600532\pi$
$449$	−13.1632			−0.621211			−0.310605	+	0.950539i	$0.600532\pi$
$450$		0			0
$451$	30.7477			1.44785
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−7.85971	+	12.0059i	−0.368469	+	0.562845i
$456$		0			0
$457$	−3.58258	−	3.58258i	−0.167586	−	0.167586i	0.618332	−	0.785917i	$-0.287809\pi$
$457$	−3.58258	−	3.58258i	−0.167586	−	0.167586i	−0.785917	+	0.618332i	$0.787809\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	14.5621i		−	0.678224i	−0.940746	−	0.339112i	$-0.889873\pi$
$461$		−	14.5621i		−	0.678224i	0.940746	−	0.339112i	$-0.110127\pi$
$462$		0			0
$463$	7.83485	−	7.83485i	0.364116	−	0.364116i	−0.501210	−	0.865326i	$-0.667111\pi$
$463$	7.83485	−	7.83485i	0.364116	−	0.364116i	0.865326	+	0.501210i	$0.167111\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.34279	+	5.34279i	−0.247235	+	0.247235i	−0.819835	−	0.572600i	$-0.805935\pi$
$467$	−5.34279	+	5.34279i	−0.247235	+	0.247235i	0.572600	+	0.819835i	$0.305935\pi$
$468$		0			0
$469$			30.7477i			1.41980i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	17.7925	+	17.7925i	0.818101	+	0.818101i
$474$		0			0
$475$	13.7477	−	6.00000i	0.630789	−	0.275299i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.85971			0.359119			0.179560	−	0.983747i	$-0.442533\pi$
$479$	7.85971			0.359119			0.179560	+	0.983747i	$0.442533\pi$
$480$		0			0
$481$	−17.9129			−0.816757
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11.0901	+	2.31464i	−0.503577	+	0.105103i
$486$		0			0
$487$	−0.373864	−	0.373864i	−0.0169414	−	0.0169414i	0.698585	−	0.715527i	$-0.253813\pi$
$487$	−0.373864	−	0.373864i	−0.0169414	−	0.0169414i	−0.715527	+	0.698585i	$0.753813\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			14.5621i			0.657178i	0.944473	+	0.328589i	$0.106573\pi$
$491$			14.5621i			0.657178i	−0.944473	+	0.328589i	$0.893427\pi$
$492$		0			0
$493$	−8.20871	+	8.20871i	−0.369702	+	0.369702i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.93280	+	9.93280i	−0.445547	+	0.445547i
$498$		0			0
$499$			27.3303i			1.22347i	0.791062	+	0.611736i	$0.209529\pi$
$499$			27.3303i			1.22347i	−0.791062	+	0.611736i	$0.790471\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	16.5678	+	16.5678i	0.738720	+	0.738720i	0.972330	−	0.233610i	$-0.0750539\pi$
$503$	16.5678	+	16.5678i	0.738720	+	0.738720i	−0.233610	+	0.972330i	$0.575054\pi$
$504$		0			0
$505$	−20.7477	−	13.5826i	−0.923262	−	0.604417i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17.5510			0.777933			0.388966	−	0.921252i	$-0.372832\pi$
$509$	17.5510			0.777933			0.388966	+	0.921252i	$0.372832\pi$
$510$		0			0
$511$	37.1652			1.64409
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.31464	−	11.0901i	−0.101995	−	0.488689i
$516$		0			0
$517$	3.58258	+	3.58258i	0.157561	+	0.157561i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	27.7253i		−	1.21467i	−0.794447	−	0.607334i	$-0.792239\pi$
$521$		−	27.7253i		−	1.21467i	0.794447	−	0.607334i	$-0.207761\pi$
$522$		0			0
$523$	−15.3739	+	15.3739i	−0.672252	+	0.672252i	−0.958235	−	0.285983i	$-0.907680\pi$
$523$	−15.3739	+	15.3739i	−0.672252	+	0.672252i	0.285983	+	0.958235i	$0.407680\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.87083	−	1.87083i	0.0814946	−	0.0814946i
$528$		0			0
$529$			14.3303i			0.623057i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.93280	+	9.93280i	0.430238	+	0.430238i
$534$		0			0
$535$	−2.41742	−	11.5826i	−0.104514	−	0.500758i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.23042			−0.139144
$540$		0			0
$541$	−19.4955			−0.838175			−0.419088	−	0.907946i	$-0.637650\pi$
$541$	−19.4955			−0.838175			−0.419088	+	0.907946i	$0.637650\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−25.7196	−	16.8375i	−1.10171	−	0.721237i
$546$		0			0
$547$	23.9564	+	23.9564i	1.02430	+	1.02430i	0.999697	+	0.0246062i	$0.00783317\pi$
$547$	23.9564	+	23.9564i	1.02430	+	1.02430i	0.0246062	+	0.999697i	$0.492167\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	13.1632i		−	0.560772i
$552$		0			0
$553$	−18.9564	+	18.9564i	−0.806110	+	0.806110i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	26.0568	−	26.0568i	1.10406	−	1.10406i	0.110145	−	0.993916i	$-0.464869\pi$
$557$	26.0568	−	26.0568i	1.10406	−	1.10406i	0.993916	−	0.110145i	$-0.0351314\pi$
$558$		0			0
$559$			11.4955i			0.486206i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8.64064	−	8.64064i	−0.364159	−	0.364159i	0.501182	−	0.865342i	$-0.332899\pi$
$563$	−8.64064	−	8.64064i	−0.364159	−	0.364159i	−0.865342	+	0.501182i	$0.832899\pi$
$564$		0			0
$565$	26.7477	−	5.58258i	1.12529	−	0.234861i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	28.8826			1.21082			0.605412	−	0.795913i	$-0.293009\pi$
$569$	28.8826			1.21082			0.605412	+	0.795913i	$0.293009\pi$
$570$		0			0
$571$	22.4955			0.941405			0.470703	−	0.882292i	$-0.344000\pi$
$571$	22.4955			0.941405			0.470703	+	0.882292i	$0.344000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	11.1575	−	28.4388i	0.465302	−	1.18598i
$576$		0			0
$577$	−10.3739	−	10.3739i	−0.431870	−	0.431870i	0.457394	−	0.889264i	$-0.348783\pi$
$577$	−10.3739	−	10.3739i	−0.431870	−	0.431870i	−0.889264	+	0.457394i	$0.848783\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	31.1973i		−	1.29428i
$582$		0			0
$583$	0.373864	−	0.373864i	0.0154838	−	0.0154838i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.0118	−	27.0118i	1.11490	−	1.11490i	0.122418	−	0.992479i	$-0.460935\pi$
$587$	27.0118	−	27.0118i	1.11490	−	1.11490i	0.992479	−	0.122418i	$-0.0390649\pi$
$588$		0			0
$589$			3.00000i			0.123613i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	29.8658	+	29.8658i	1.22644	+	1.22644i	0.965300	+	0.261143i	$0.0840993\pi$
$593$	29.8658	+	29.8658i	1.22644	+	1.22644i	0.261143	+	0.965300i	$0.415901\pi$
$594$		0			0
$595$	−8.20871	+	12.5390i	−0.336524	+	0.514049i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	28.8826			1.18011			0.590056	−	0.807362i	$-0.299106\pi$
$599$	28.8826			1.18011			0.590056	+	0.807362i	$0.299106\pi$
$600$		0			0
$601$	39.7477			1.62134			0.810672	−	0.585501i	$-0.199102\pi$
$601$	39.7477			1.62134			0.810672	+	0.585501i	$0.199102\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−24.1859	+	36.9446i	−0.983298	+	1.50201i
$606$		0			0
$607$	−13.9564	−	13.9564i	−0.566474	−	0.566474i	0.364665	−	0.931139i	$-0.381184\pi$
$607$	−13.9564	−	13.9564i	−0.566474	−	0.566474i	−0.931139	+	0.364665i	$0.881184\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			2.31464i			0.0936405i
$612$		0			0
$613$	10.0000	−	10.0000i	0.403896	−	0.403896i	−0.475707	−	0.879604i	$-0.657808\pi$
$613$	10.0000	−	10.0000i	0.403896	−	0.403896i	0.879604	+	0.475707i	$0.157808\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14.1183	−	14.1183i	0.568380	−	0.568380i	−0.363294	−	0.931675i	$-0.618348\pi$
$617$	14.1183	−	14.1183i	0.568380	−	0.568380i	0.931675	+	0.363294i	$0.118348\pi$
$618$		0			0
$619$		−	19.5826i		−	0.787090i	−0.919305	−	0.393545i	$-0.871249\pi$
$619$		−	19.5826i		−	0.787090i	0.919305	−	0.393545i	$-0.128751\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−29.7984	−	29.7984i	−1.19385	−	1.19385i
$624$		0			0
$625$	−17.0000	+	18.3303i	−0.680000	+	0.733212i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−18.7083			−0.745948
$630$		0			0
$631$	−39.7477			−1.58233			−0.791166	−	0.611601i	$-0.790526\pi$
$631$	−39.7477			−1.58233			−0.791166	+	0.611601i	$0.790526\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15.4779	−	3.23042i	0.614220	−	0.128195i
$636$		0			0
$637$	−1.04356	−	1.04356i	−0.0413474	−	0.0413474i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	37.4166i		−	1.47787i	−0.673779	−	0.738933i	$-0.735330\pi$
$641$		−	37.4166i		−	1.47787i	0.673779	−	0.738933i	$-0.264670\pi$
$642$		0			0
$643$	−1.79129	+	1.79129i	−0.0706415	+	0.0706415i	−0.741545	−	0.670903i	$-0.765907\pi$
$643$	−1.79129	+	1.79129i	−0.0706415	+	0.0706415i	0.670903	+	0.741545i	$0.265907\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.5060	+	18.5060i	−0.727547	+	0.727547i	−0.970130	−	0.242584i	$-0.922005\pi$
$647$	−18.5060	+	18.5060i	−0.727547	+	0.727547i	0.242584	+	0.970130i	$0.422005\pi$
$648$		0			0
$649$			24.3303i			0.955048i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.4835	−	17.4835i	−0.684184	−	0.684184i	0.276756	−	0.960940i	$-0.410741\pi$
$653$	−17.4835	−	17.4835i	−0.684184	−	0.684184i	−0.960940	+	0.276756i	$0.910741\pi$
$654$		0			0
$655$	14.2523	+	9.33030i	0.556882	+	0.364565i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	37.6581			1.46695			0.733476	−	0.679715i	$-0.237897\pi$
$659$	37.6581			1.46695			0.733476	+	0.679715i	$0.237897\pi$
$660$		0			0
$661$	23.4955			0.913867			0.456934	−	0.889501i	$-0.348948\pi$
$661$	23.4955			0.913867			0.456934	+	0.889501i	$0.348948\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.47197	−	16.6352i	−0.134637	−	0.645085i
$666$		0			0
$667$	−18.9564	−	18.9564i	−0.733996	−	0.733996i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			5.54506i			0.214065i
$672$		0			0
$673$	−22.5390	+	22.5390i	−0.868815	+	0.868815i	−0.992341	−	0.123526i	$-0.960580\pi$
$673$	−22.5390	+	22.5390i	−0.868815	+	0.868815i	0.123526	+	0.992341i	$0.460580\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.5171	+	12.5171i	−0.481073	+	0.481073i	−0.905474	−	0.424401i	$-0.860485\pi$
$677$	−12.5171	+	12.5171i	−0.481073	+	0.481073i	0.424401	+	0.905474i	$0.360485\pi$
$678$		0			0
$679$			12.8348i			0.492556i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.0674228	−	0.0674228i	−0.00257986	−	0.00257986i	0.705816	−	0.708396i	$-0.250581\pi$
$683$	−0.0674228	−	0.0674228i	−0.00257986	−	0.00257986i	−0.708396	+	0.705816i	$0.750581\pi$
$684$		0			0
$685$	6.70417	+	32.1216i	0.256153	+	1.22730i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.241547			0.00920221
$690$		0			0
$691$	11.7477			0.446905			0.223452	−	0.974715i	$-0.428267\pi$
$691$	11.7477			0.446905			0.223452	+	0.974715i	$0.428267\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.92146	−	3.87650i	−0.224614	−	0.147044i
$696$		0			0
$697$	10.3739	+	10.3739i	0.392938	+	0.392938i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	47.8325i		−	1.80661i	−0.429001	−	0.903304i	$-0.641134\pi$
$701$		−	47.8325i		−	1.80661i	0.429001	−	0.903304i	$-0.358866\pi$
$702$		0			0
$703$	15.0000	−	15.0000i	0.565736	−	0.565736i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−19.8656	+	19.8656i	−0.747123	+	0.747123i
$708$		0			0
$709$		−	42.6606i		−	1.60215i	−0.598562	−	0.801076i	$-0.704261\pi$
$709$		−	42.6606i		−	1.60215i	0.598562	−	0.801076i	$-0.295739\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.32032	+	4.32032i	0.161797	+	0.161797i
$714$		0			0
$715$	−30.7477	+	6.41742i	−1.14990	+	0.239998i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−29.7984			−1.11129			−0.555647	−	0.831419i	$-0.687529\pi$
$719$	−29.7984			−1.11129			−0.555647	+	0.831419i	$0.687529\pi$
$720$		0			0
$721$	−12.8348			−0.477995
$722$		0			0
$723$		0			0
$724$		0			0
$725$	8.77548	+	20.1072i	0.325913	+	0.746761i
$726$		0			0
$727$	−37.1652	−	37.1652i	−1.37838	−	1.37838i	−0.847359	−	0.531020i	$-0.821809\pi$
$727$	−37.1652	−	37.1652i	−1.37838	−	1.37838i	−0.531020	−	0.847359i	$-0.678191\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			12.0059i			0.444054i
$732$		0			0
$733$	26.4174	−	26.4174i	0.975750	−	0.975750i	−0.0239630	−	0.999713i	$-0.507628\pi$
$733$	26.4174	−	26.4174i	0.975750	−	0.975750i	0.999713	+	0.0239630i	$0.00762839\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−47.5909	+	47.5909i	−1.75303	+	1.75303i
$738$		0			0
$739$			24.4955i			0.901080i	0.892756	+	0.450540i	$0.148769\pi$
$739$			24.4955i			0.901080i	−0.892756	+	0.450540i	$0.851231\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.1126	−	12.1126i	−0.444368	−	0.444368i	0.449109	−	0.893477i	$-0.351742\pi$
$743$	−12.1126	−	12.1126i	−0.444368	−	0.444368i	−0.893477	+	0.449109i	$0.851742\pi$
$744$		0			0
$745$	9.62614	−	14.7042i	0.352674	−	0.538719i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−13.4048			−0.489800
$750$		0			0
$751$	−9.74773			−0.355700			−0.177850	−	0.984058i	$-0.556914\pi$
$751$	−9.74773			−0.355700			−0.177850	+	0.984058i	$0.556914\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.44949	+	3.74166i	−0.0891461	+	0.136173i
$756$		0			0
$757$	−5.00000	−	5.00000i	−0.181728	−	0.181728i	0.610380	−	0.792108i	$-0.291017\pi$
$757$	−5.00000	−	5.00000i	−0.181728	−	0.181728i	−0.792108	+	0.610380i	$0.791017\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			9.01703i			0.326867i	0.986554	+	0.163434i	$0.0522569\pi$
$761$			9.01703i			0.326867i	−0.986554	+	0.163434i	$0.947743\pi$
$762$		0			0
$763$	−24.6261	+	24.6261i	−0.891526	+	0.891526i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7.85971	+	7.85971i	−0.283798	+	0.283798i
$768$		0			0
$769$		−	27.3303i		−	0.985556i	−0.870155	−	0.492778i	$-0.835981\pi$
$769$		−	27.3303i		−	0.985556i	0.870155	−	0.492778i	$-0.164019\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	36.8098	+	36.8098i	1.32396	+	1.32396i	0.910545	+	0.413411i	$0.135663\pi$
$773$	36.8098	+	36.8098i	1.32396	+	1.32396i	0.413411	+	0.910545i	$0.364337\pi$
$774$		0			0
$775$	−2.00000	−	4.58258i	−0.0718421	−	0.164611i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−16.6352			−0.596018
$780$		0			0
$781$	−30.7477			−1.10024
$782$		0			0
$783$		0			0
$784$		0			0
$785$	9.93280	−	2.07310i	0.354517	−	0.0739920i
$786$		0			0
$787$	37.5390	+	37.5390i	1.33812	+	1.33812i	0.897879	+	0.440242i	$0.145107\pi$
$787$	37.5390	+	37.5390i	1.33812	+	1.33812i	0.440242	+	0.897879i	$0.354893\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	30.9557i		−	1.10066i
$792$		0			0
$793$	−1.79129	+	1.79129i	−0.0636105	+	0.0636105i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.51691	+	2.51691i	−0.0891536	+	0.0891536i	−0.750277	−	0.661123i	$-0.770080\pi$
$797$	−2.51691	+	2.51691i	−0.0891536	+	0.0891536i	0.661123	+	0.750277i	$0.270080\pi$
$798$		0			0
$799$			2.41742i			0.0855223i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	57.5237	+	57.5237i	2.02997	+	2.02997i
$804$		0			0
$805$	−28.9564	−	18.9564i	−1.02058	−	0.668127i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	42.9616			1.51045			0.755225	−	0.655465i	$-0.227527\pi$
$809$	42.9616			1.51045			0.755225	+	0.655465i	$0.227527\pi$
$810$		0			0
$811$	−23.4955			−0.825037			−0.412518	−	0.910949i	$-0.635351\pi$
$811$	−23.4955			−0.825037			−0.412518	+	0.910949i	$0.635351\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.23042	−	15.4779i	−0.113157	−	0.542166i
$816$		0			0
$817$	−9.62614	−	9.62614i	−0.336776	−	0.336776i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	7.61816i		−	0.265876i	−0.991124	−	0.132938i	$-0.957559\pi$
$821$		−	7.61816i		−	0.265876i	0.991124	−	0.132938i	$-0.0424411\pi$
$822$		0			0
$823$	23.9564	−	23.9564i	0.835069	−	0.835069i	−0.153136	−	0.988205i	$-0.548937\pi$
$823$	23.9564	−	23.9564i	0.835069	−	0.835069i	0.988205	+	0.153136i	$0.0489373\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.6692	+	31.6692i	−1.10125	+	1.10125i	−0.106987	+	0.994260i	$0.534120\pi$
$827$	−31.6692	+	31.6692i	−1.10125	+	1.10125i	−0.994260	+	0.106987i	$0.965880\pi$
$828$		0			0
$829$		−	34.7477i		−	1.20684i	−0.797424	−	0.603419i	$-0.793805\pi$
$829$		−	34.7477i		−	1.20684i	0.797424	−	0.603419i	$-0.206195\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.08990	−	1.08990i	−0.0377628	−	0.0377628i
$834$		0			0
$835$	6.87841	+	32.9564i	0.238037	+	1.14050i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−35.1019			−1.21185			−0.605927	−	0.795521i	$-0.707197\pi$
$839$	−35.1019			−1.21185			−0.605927	+	0.795521i	$0.707197\pi$
$840$		0			0
$841$	−9.74773			−0.336129
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.3149	+	8.06198i	0.423644	+	0.277340i
$846$		0			0
$847$	35.3739	+	35.3739i	1.21546	+	1.21546i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	43.2032i		−	1.48099i
$852$		0			0
$853$	4.33030	−	4.33030i	0.148267	−	0.148267i	−0.629077	−	0.777343i	$-0.716567\pi$
$853$	4.33030	−	4.33030i	0.148267	−	0.148267i	0.777343	+	0.629077i	$0.216567\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.25857	−	6.25857i	0.213789	−	0.213789i	−0.592086	−	0.805875i	$-0.701695\pi$
$857$	6.25857	−	6.25857i	0.213789	−	0.213789i	0.805875	+	0.592086i	$0.201695\pi$
$858$		0			0
$859$		−	45.6606i		−	1.55792i	−0.627074	−	0.778960i	$-0.715748\pi$
$859$		−	45.6606i		−	1.55792i	0.627074	−	0.778960i	$-0.284252\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.2590	+	26.2590i	0.893868	+	0.893868i	0.994885	−	0.101017i	$-0.0322097\pi$
$863$	26.2590	+	26.2590i	0.893868	+	0.893868i	−0.101017	+	0.994885i	$0.532210\pi$
$864$		0			0
$865$	24.5390	−	5.12159i	0.834352	−	0.174139i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−58.6811			−1.99062
$870$		0			0
$871$	−30.7477			−1.04185
$872$		0			0
$873$		0			0
$874$		0			0
$875$	16.3936	+	23.0960i	0.554206	+	0.780788i
$876$		0			0
$877$	3.20871	+	3.20871i	0.108351	+	0.108351i	0.759204	−	0.650853i	$-0.225589\pi$
$877$	3.20871	+	3.20871i	0.108351	+	0.108351i	−0.650853	+	0.759204i	$0.725589\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			35.3435i			1.19075i	0.803447	+	0.595376i	$0.202997\pi$
$881$			35.3435i			1.19075i	−0.803447	+	0.595376i	$0.797003\pi$
$882$		0			0
$883$	25.3739	−	25.3739i	0.853898	−	0.853898i	−0.136712	−	0.990611i	$-0.543654\pi$
$883$	25.3739	−	25.3739i	0.853898	−	0.853898i	0.990611	+	0.136712i	$0.0436536\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	35.7873	−	35.7873i	1.20162	−	1.20162i	0.227946	−	0.973674i	$-0.426799\pi$
$887$	35.7873	−	35.7873i	1.20162	−	1.20162i	0.973674	−	0.227946i	$-0.0732009\pi$
$888$		0			0
$889$		−	17.9129i		−	0.600779i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.93825	−	1.93825i	−0.0648611	−	0.0648611i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.38774			−0.146339
$900$		0			0
$901$	0.252273			0.00840443
$902$		0			0
$903$		0			0
$904$		0			0
$905$	11.9385	−	18.2363i	0.396848	−	0.606196i
$906$		0			0
$907$	−17.9129	−	17.9129i	−0.594787	−	0.594787i	0.344133	−	0.938921i	$-0.388173\pi$
$907$	−17.9129	−	17.9129i	−0.594787	−	0.594787i	−0.938921	+	0.344133i	$0.888173\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			24.2534i			0.803549i	0.915739	+	0.401775i	$0.131606\pi$
$911$			24.2534i			0.803549i	−0.915739	+	0.401775i	$0.868394\pi$
$912$		0			0
$913$	48.2867	−	48.2867i	1.59806	−	1.59806i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	13.6463	−	13.6463i	0.450641	−	0.450641i
$918$		0			0
$919$			24.0000i			0.791687i	0.918318	+	0.395843i	$0.129548\pi$
$919$			24.0000i			0.791687i	−0.918318	+	0.395843i	$0.870452\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−9.93280	−	9.93280i	−0.326942	−	0.326942i
$924$		0			0
$925$	−12.9129	+	32.9129i	−0.424573	+	1.08217i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.915775			−0.0300456			−0.0150228	−	0.999887i	$-0.504782\pi$
$929$	−0.915775			−0.0300456			−0.0150228	+	0.999887i	$0.504782\pi$
$930$		0			0
$931$	1.74773			0.0572794
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−32.1131	+	6.70239i	−1.05021	+	0.219191i
$936$		0			0
$937$	−0.747727	−	0.747727i	−0.0244272	−	0.0244272i	0.694788	−	0.719215i	$-0.255498\pi$
$937$	−0.747727	−	0.747727i	−0.0244272	−	0.0244272i	−0.719215	+	0.694788i	$0.755498\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	14.5621i		−	0.474711i	−0.971423	−	0.237355i	$-0.923719\pi$
$941$		−	14.5621i		−	0.474711i	0.971423	−	0.237355i	$-0.0762806\pi$
$942$		0			0
$943$	−23.9564	+	23.9564i	−0.780129	+	0.780129i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.73054	+	9.73054i	−0.316200	+	0.316200i	−0.847306	−	0.531106i	$-0.821777\pi$
$947$	−9.73054	+	9.73054i	−0.316200	+	0.316200i	0.531106	+	0.847306i	$0.321777\pi$
$948$		0			0
$949$			37.1652i			1.20643i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−20.2420	−	20.2420i	−0.655703	−	0.655703i	0.298658	−	0.954360i	$-0.403461\pi$
$953$	−20.2420	−	20.2420i	−0.655703	−	0.655703i	−0.954360	+	0.298658i	$0.903461\pi$
$954$		0			0
$955$	−35.0000	−	22.9129i	−1.13257	−	0.741443i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	37.1750			1.20044
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.15732	−	5.54506i	−0.0372555	−	0.178502i
$966$		0			0
$967$	−19.3303	−	19.3303i	−0.621621	−	0.621621i	0.324325	−	0.945946i	$-0.394863\pi$
$967$	−19.3303	−	19.3303i	−0.621621	−	0.621621i	−0.945946	+	0.324325i	$0.894863\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	18.7083i		−	0.600377i	−0.953880	−	0.300189i	$-0.902950\pi$
$971$		−	18.7083i		−	0.600377i	0.953880	−	0.300189i	$-0.0970497\pi$
$972$		0			0
$973$	−5.66970	+	5.66970i	−0.181762	+	0.181762i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.56186	−	1.56186i	0.0499683	−	0.0499683i	−0.681681	−	0.731649i	$-0.738751\pi$
$977$	1.56186	−	1.56186i	0.0499683	−	0.0499683i	0.731649	+	0.681681i	$0.238751\pi$
$978$		0			0
$979$		−	92.2432i		−	2.94810i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	12.1800	+	12.1800i	0.388482	+	0.388482i	0.874146	−	0.485663i	$-0.161422\pi$
$983$	12.1800	+	12.1800i	0.388482	+	0.388482i	−0.485663	+	0.874146i	$0.661422\pi$
$984$		0			0
$985$	−1.20871	−	5.79129i	−0.0385128	−	0.184526i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−27.7253			−0.881614
$990$		0			0
$991$	−18.2523			−0.579803			−0.289901	−	0.957057i	$-0.593622\pi$
$991$	−18.2523			−0.579803			−0.289901	+	0.957057i	$0.593622\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−31.3321	−	20.5117i	−0.993295	−	0.650264i
$996$		0			0
$997$	16.7913	+	16.7913i	0.531785	+	0.531785i	0.921103	−	0.389318i	$-0.127289\pi$
$997$	16.7913	+	16.7913i	0.531785	+	0.531785i	−0.389318	+	0.921103i	$0.627289\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.w.d.1457.3		8
3.2	odd	2	inner	2160.2.w.d.1457.2		8
4.3	odd	2		135.2.f.a.107.2	yes	8
5.3	odd	4	inner	2160.2.w.d.593.1		8
12.11	even	2		135.2.f.a.107.3	yes	8
15.8	even	4	inner	2160.2.w.d.593.4		8
20.3	even	4		135.2.f.a.53.3	yes	8
20.7	even	4		675.2.f.i.593.2		8
20.19	odd	2		675.2.f.i.107.3		8
36.7	odd	6		405.2.m.c.377.3		16
36.11	even	6		405.2.m.c.377.2		16
36.23	even	6		405.2.m.c.107.3		16
36.31	odd	6		405.2.m.c.107.2		16
60.23	odd	4		135.2.f.a.53.2	✓	8
60.47	odd	4		675.2.f.i.593.3		8
60.59	even	2		675.2.f.i.107.2		8
180.23	odd	12		405.2.m.c.188.3		16
180.43	even	12		405.2.m.c.53.3		16
180.83	odd	12		405.2.m.c.53.2		16
180.103	even	12		405.2.m.c.188.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.f.a.53.2	✓	8	60.23	odd	4
135.2.f.a.53.3	yes	8	20.3	even	4
135.2.f.a.107.2	yes	8	4.3	odd	2
135.2.f.a.107.3	yes	8	12.11	even	2
405.2.m.c.53.2		16	180.83	odd	12
405.2.m.c.53.3		16	180.43	even	12
405.2.m.c.107.2		16	36.31	odd	6
405.2.m.c.107.3		16	36.23	even	6
405.2.m.c.188.2		16	180.103	even	12
405.2.m.c.188.3		16	180.23	odd	12
405.2.m.c.377.2		16	36.11	even	6
405.2.m.c.377.3		16	36.7	odd	6
675.2.f.i.107.2		8	60.59	even	2
675.2.f.i.107.3		8	20.19	odd	2
675.2.f.i.593.2		8	20.7	even	4
675.2.f.i.593.3		8	60.47	odd	4
2160.2.w.d.593.1		8	5.3	odd	4	inner
2160.2.w.d.593.4		8	15.8	even	4	inner
2160.2.w.d.1457.2		8	3.2	odd	2	inner
2160.2.w.d.1457.3		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.87083i) q^{5} +(-1.79129 - 1.79129i) q^{7} +O(q^{10})\)	\(q+(1.22474 - 1.87083i) q^{5} +(-1.79129 - 1.79129i) q^{7} -5.54506i q^{11} +(1.79129 - 1.79129i) q^{13} +(1.87083 - 1.87083i) q^{17} +3.00000i q^{19} +(4.32032 + 4.32032i) q^{23} +(-2.00000 - 4.58258i) q^{25} -4.38774 q^{29} +1.00000 q^{31} +(-5.54506 + 1.15732i) q^{35} +(-5.00000 - 5.00000i) q^{37} +5.54506i q^{41} +(-3.20871 + 3.20871i) q^{43} +(-0.646084 + 0.646084i) q^{47} -0.582576i q^{49} +(0.0674228 + 0.0674228i) q^{53} +(-10.3739 - 6.79129i) q^{55} -4.38774 q^{59} -1.00000 q^{61} +(-1.15732 - 5.54506i) q^{65} +(-8.58258 - 8.58258i) q^{67} -5.54506i q^{71} +(-10.3739 + 10.3739i) q^{73} +(-9.93280 + 9.93280i) q^{77} -10.5826i q^{79} +(8.70806 + 8.70806i) q^{83} +(-1.20871 - 5.79129i) q^{85} +16.6352 q^{89} -6.41742 q^{91} +(5.61249 + 3.67423i) q^{95} +(-3.58258 - 3.58258i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7}+O(q^{10}) \) 8 * q + 4 * q^7	\( 8 q + 4 q^{7} - 4 q^{13} - 16 q^{25} + 8 q^{31} - 40 q^{37} - 44 q^{43} - 28 q^{55} - 8 q^{61} - 32 q^{67} - 28 q^{73} - 28 q^{85} - 88 q^{91} + 8 q^{97}+O(q^{100}) \) 8 * q + 4 * q^7 - 4 * q^13 - 16 * q^25 + 8 * q^31 - 40 * q^37 - 44 * q^43 - 28 * q^55 - 8 * q^61 - 32 * q^67 - 28 * q^73 - 28 * q^85 - 88 * q^91 + 8 * q^97

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