LMFDB - Embedded newform 4320.2.k.d.2161.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(2161,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4320.2161");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.k (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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + 5x^{16} + 28x^{12} - 28x^{10} + 112x^{8} + 320x^{4} - 256x^{2} + 1024 $ x^20 - x^18 + 5x^16 + 28x^12 - 28x^10 + 112x^8 + 320x^4 - 256x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2161.6
Root			$0.662801 - 1.24928i$ of defining polynomial
Character	$\chi$	$=$	4320.2161
Dual form			4320.2.k.d.2161.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} +1.52861 q^{7} +O(q^{10})$	$q-1.00000i q^{5} +1.52861 q^{7} +5.34793i q^{11} +0.986544i q^{13} +5.84354 q^{17} +7.86987i q^{19} -4.41839 q^{23} -1.00000 q^{25} +9.95406i q^{29} -3.04016 q^{31} -1.52861i q^{35} -11.2485i q^{37} -8.75389 q^{41} -3.01287i q^{43} -9.74043 q^{47} -4.66335 q^{49} -5.83638i q^{53} +5.34793 q^{55} +5.85948i q^{59} -6.88333i q^{61} +0.986544 q^{65} +3.97298i q^{67} -10.2619 q^{71} -6.83638 q^{73} +8.17490i q^{77} +2.07752 q^{79} -4.30777i q^{83} -5.84354i q^{85} +3.01287 q^{89} +1.50804i q^{91} +7.86987 q^{95} -5.12708 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 20 q^{25} - 20 q^{31} + 36 q^{49} - 40 q^{73} + 4 q^{79} + 40 q^{97}+O(q^{100}) $ 20 * q - 20 * q^25 - 20 * q^31 + 36 * q^49 - 40 * q^73 + 4 * q^79 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0	0
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	1.52861	0.577761	0.288880	−	0.957365i	$-0.406717\pi$
$7$	1.52861	0.577761	0.288880	+	0.957365i	$0.406717\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.34793i	1.61246i	0.591601	+	0.806231i	$0.298496\pi$
$11$	5.34793i	1.61246i	−0.591601	+	0.806231i	$0.701504\pi$
$12$	0	0
$13$	0.986544i	0.273618i	0.990597	+	0.136809i	$0.0436847\pi$
$13$	0.986544i	0.273618i	−0.990597	+	0.136809i	$0.956315\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.84354	1.41727	0.708633	−	0.705577i	$-0.249312\pi$
$17$	5.84354	1.41727	0.708633	+	0.705577i	$0.249312\pi$
$18$	0	0
$19$	7.86987i	1.80547i	0.430195	+	0.902736i	$0.358445\pi$
$19$	7.86987i	1.80547i	−0.430195	+	0.902736i	$0.641555\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.41839	−0.921298	−0.460649	−	0.887582i	$-0.652383\pi$
$23$	−4.41839	−0.921298	−0.460649	+	0.887582i	$0.652383\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.95406i	1.84842i	0.381882	+	0.924211i	$0.375276\pi$
$29$	9.95406i	1.84842i	−0.381882	+	0.924211i	$0.624724\pi$
$30$	0	0
$31$	−3.04016	−0.546030	−0.273015	−	0.962010i	$-0.588021\pi$
$31$	−3.04016	−0.546030	−0.273015	+	0.962010i	$0.588021\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.52861i	− 0.258382i
$36$	0	0
$37$	− 11.2485i	− 1.84924i	−0.380894	−	0.924619i	$-0.624384\pi$
$37$	− 11.2485i	− 1.84924i	0.380894	−	0.924619i	$-0.375616\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.75389	−1.36713	−0.683564	−	0.729891i	$-0.739571\pi$
$41$	−8.75389	−1.36713	−0.683564	+	0.729891i	$0.739571\pi$
$42$	0	0
$43$	− 3.01287i	− 0.459459i	−0.973255	−	0.229729i	$-0.926216\pi$
$43$	− 3.01287i	− 0.459459i	0.973255	−	0.229729i	$-0.0737841\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.74043	−1.42079	−0.710394	−	0.703804i	$-0.751483\pi$
$47$	−9.74043	−1.42079	−0.710394	+	0.703804i	$0.751483\pi$
$48$	0	0
$49$	−4.66335	−0.666193
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 5.83638i	− 0.801688i	−0.916146	−	0.400844i	$-0.868717\pi$
$53$	− 5.83638i	− 0.801688i	0.916146	−	0.400844i	$-0.131283\pi$
$54$	0	0
$55$	5.34793	0.721115
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.85948i	0.762840i	0.924402	+	0.381420i	$0.124565\pi$
$59$	5.85948i	0.762840i	−0.924402	+	0.381420i	$0.875435\pi$
$60$	0	0
$61$	− 6.88333i	− 0.881320i	−0.897674	−	0.440660i	$-0.854745\pi$
$61$	− 6.88333i	− 0.881320i	0.897674	−	0.440660i	$-0.145255\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.986544	0.122366
$66$	0	0
$67$	3.97298i	0.485376i	0.970104	+	0.242688i	$0.0780292\pi$
$67$	3.97298i	0.485376i	−0.970104	+	0.242688i	$0.921971\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.2619	−1.21787	−0.608934	−	0.793221i	$-0.708402\pi$
$71$	−10.2619	−1.21787	−0.608934	+	0.793221i	$0.708402\pi$
$72$	0	0
$73$	−6.83638	−0.800137	−0.400069	−	0.916485i	$-0.631014\pi$
$73$	−6.83638	−0.800137	−0.400069	+	0.916485i	$0.631014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.17490i	0.931617i
$78$	0	0
$79$	2.07752	0.233739	0.116869	−	0.993147i	$-0.462714\pi$
$79$	2.07752	0.233739	0.116869	+	0.993147i	$0.462714\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.30777i	− 0.472839i	−0.971651	−	0.236419i	$-0.924026\pi$
$83$	− 4.30777i	− 0.472839i	0.971651	−	0.236419i	$-0.0759739\pi$
$84$	0	0
$85$	− 5.84354i	− 0.633821i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.01287	0.319364	0.159682	−	0.987169i	$-0.448953\pi$
$89$	3.01287	0.319364	0.159682	+	0.987169i	$0.448953\pi$
$90$	0	0
$91$	1.50804i	0.158086i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.86987	0.807432
$96$	0	0
$97$	−5.12708	−0.520576	−0.260288	−	0.965531i	$-0.583818\pi$
$97$	−5.12708	−0.520576	−0.260288	+	0.965531i	$0.583818\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.5399i	1.04876i	0.851485	+	0.524379i	$0.175703\pi$
$101$	10.5399i	1.04876i	−0.851485	+	0.524379i	$0.824297\pi$
$102$	0	0
$103$	3.84403	0.378764	0.189382	−	0.981904i	$-0.439352\pi$
$103$	3.84403	0.378764	0.189382	+	0.981904i	$0.439352\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.3592i	1.58150i	0.612136	+	0.790752i	$0.290310\pi$
$107$	16.3592i	1.58150i	−0.612136	+	0.790752i	$0.709690\pi$
$108$	0	0
$109$	6.90977i	0.661836i	0.943660	+	0.330918i	$0.107358\pi$
$109$	6.90977i	0.661836i	−0.943660	+	0.330918i	$0.892642\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.41100	0.791240	0.395620	−	0.918414i	$-0.370530\pi$
$113$	8.41100	0.791240	0.395620	+	0.918414i	$0.370530\pi$
$114$	0	0
$115$	4.41839i	0.412017i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.93250	0.818841
$120$	0	0
$121$	−17.6003	−1.60003
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−1.84403	−0.163631	−0.0818156	−	0.996647i	$-0.526072\pi$
$127$	−1.84403	−0.163631	−0.0818156	+	0.996647i	$0.526072\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.7810i	1.11668i	0.829611	+	0.558342i	$0.188562\pi$
$131$	12.7810i	1.11668i	−0.829611	+	0.558342i	$0.811438\pi$
$132$	0	0
$133$	12.0300i	1.04313i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.6540	−1.08110	−0.540552	−	0.841311i	$-0.681784\pi$
$137$	−12.6540	−1.08110	−0.540552	+	0.841311i	$0.681784\pi$
$138$	0	0
$139$	− 18.9626i	− 1.60838i	−0.594369	−	0.804192i	$-0.702598\pi$
$139$	− 18.9626i	− 1.60838i	0.594369	−	0.804192i	$-0.297402\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.27597	−0.441199
$144$	0	0
$145$	9.95406	0.826639
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.95406i	0.323929i	0.986797	+	0.161964i	$0.0517830\pi$
$149$	3.95406i	0.323929i	−0.986797	+	0.161964i	$0.948217\pi$
$150$	0	0
$151$	−2.76210	−0.224776	−0.112388	−	0.993664i	$-0.535850\pi$
$151$	−2.76210	−0.224776	−0.112388	+	0.993664i	$0.535850\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.04016i	0.244192i
$156$	0	0
$157$	− 7.85024i	− 0.626517i	−0.949668	−	0.313259i	$-0.898579\pi$
$157$	− 7.85024i	− 0.626517i	0.949668	−	0.313259i	$-0.101421\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.75400	−0.532290
$162$	0	0
$163$	14.5778i	1.14182i	0.821012	+	0.570911i	$0.193410\pi$
$163$	14.5778i	1.14182i	−0.821012	+	0.570911i	$0.806590\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.13307	−0.629356	−0.314678	−	0.949198i	$-0.601897\pi$
$167$	−8.13307	−0.629356	−0.314678	+	0.949198i	$0.601897\pi$
$168$	0	0
$169$	12.0267	0.925133
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 0.0170573i	− 0.00129684i	−1.00000		0.000648420i	$-0.999794\pi$
$173$	− 0.0170573i	− 0.00129684i	1.00000		0.000648420i	$-0.000206399\pi$
$174$	0	0
$175$	−1.52861	−0.115552
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 11.6386i	− 0.869913i	−0.900452	−	0.434956i	$-0.856764\pi$
$179$	− 11.6386i	− 0.869913i	0.900452	−	0.434956i	$-0.143236\pi$
$180$	0	0
$181$	− 4.41839i	− 0.328416i	−0.986426	−	0.164208i	$-0.947493\pi$
$181$	− 4.41839i	− 0.328416i	0.986426	−	0.164208i	$-0.0525069\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−11.2485	−0.827004
$186$	0	0
$187$	31.2508i	2.28529i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.1783	1.17062	0.585312	−	0.810808i	$-0.300972\pi$
$191$	16.1783	1.17062	0.585312	+	0.810808i	$0.300972\pi$
$192$	0	0
$193$	14.4052	1.03691	0.518453	−	0.855106i	$-0.326508\pi$
$193$	14.4052	1.03691	0.518453	+	0.855106i	$0.326508\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.4981i	0.890454i	0.895418	+	0.445227i	$0.146877\pi$
$197$	12.4981i	0.890454i	−0.895418	+	0.445227i	$0.853123\pi$
$198$	0	0
$199$	22.7242	1.61088	0.805438	−	0.592680i	$-0.201930\pi$
$199$	22.7242	1.61088	0.805438	+	0.592680i	$0.201930\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	15.2159i	1.06795i
$204$	0	0
$205$	8.75389i	0.611398i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−42.0875	−2.91125
$210$	0	0
$211$	− 8.29565i	− 0.571096i	−0.958364	−	0.285548i	$-0.907824\pi$
$211$	− 8.29565i	− 0.571096i	0.958364	−	0.285548i	$-0.0921756\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.01287	−0.205476
$216$	0	0
$217$	−4.64723	−0.315474
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.76491i	0.387790i
$222$	0	0
$223$	26.4139	1.76880	0.884402	−	0.466725i	$-0.154566\pi$
$223$	26.4139	1.76880	0.884402	+	0.466725i	$0.154566\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.75886i	− 0.315857i	−0.987451	−	0.157928i	$-0.949519\pi$
$227$	− 4.75886i	− 0.315857i	0.987451	−	0.157928i	$-0.0504815\pi$
$228$	0	0
$229$	14.3374i	0.947444i	0.880674	+	0.473722i	$0.157090\pi$
$229$	14.3374i	0.947444i	−0.880674	+	0.473722i	$0.842910\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.97287	0.391295	0.195648	−	0.980674i	$-0.437319\pi$
$233$	5.97287	0.391295	0.195648	+	0.980674i	$0.437319\pi$
$234$	0	0
$235$	9.74043i	0.635396i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.79379	−0.504138	−0.252069	−	0.967709i	$-0.581111\pi$
$239$	−7.79379	−0.504138	−0.252069	+	0.967709i	$0.581111\pi$
$240$	0	0
$241$	8.54891	0.550683	0.275342	−	0.961346i	$-0.411209\pi$
$241$	8.54891	0.550683	0.275342	+	0.961346i	$0.411209\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.66335i	0.297930i
$246$	0	0
$247$	−7.76397	−0.494010
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.9081i	1.25659i	0.777976	+	0.628294i	$0.216247\pi$
$251$	19.9081i	1.25659i	−0.777976	+	0.628294i	$0.783753\pi$
$252$	0	0
$253$	− 23.6292i	− 1.48556i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.4042	0.711377	0.355689	−	0.934605i	$-0.384246\pi$
$257$	11.4042	0.711377	0.355689	+	0.934605i	$0.384246\pi$
$258$	0	0
$259$	− 17.1945i	− 1.06842i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.46493	−0.151994	−0.0759972	−	0.997108i	$-0.524214\pi$
$263$	−2.46493	−0.151994	−0.0759972	+	0.997108i	$0.524214\pi$
$264$	0	0
$265$	−5.83638	−0.358526
$266$	0	0
$267$	0	0
$268$	0	0
$269$	12.1144i	0.738631i	0.929304	+	0.369315i	$0.120408\pi$
$269$	12.1144i	0.738631i	−0.929304	+	0.369315i	$0.879592\pi$
$270$	0	0
$271$	−6.01128	−0.365159	−0.182580	−	0.983191i	$-0.558445\pi$
$271$	−6.01128	−0.365159	−0.182580	+	0.983191i	$0.558445\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 5.34793i	− 0.322492i
$276$	0	0
$277$	− 13.1122i	− 0.787837i	−0.919145	−	0.393919i	$-0.871119\pi$
$277$	− 13.1122i	− 0.787837i	0.919145	−	0.393919i	$-0.128881\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.6730	0.994630	0.497315	−	0.867570i	$-0.334319\pi$
$281$	16.6730	0.994630	0.497315	+	0.867570i	$0.334319\pi$
$282$	0	0
$283$	− 9.79688i	− 0.582364i	−0.956668	−	0.291182i	$-0.905951\pi$
$283$	− 9.79688i	− 0.582364i	0.956668	−	0.291182i	$-0.0940486\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−13.3813	−0.789872
$288$	0	0
$289$	17.1470	1.00864
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.4981i	0.846989i	0.905899	+	0.423495i	$0.139197\pi$
$293$	14.4981i	0.846989i	−0.905899	+	0.423495i	$0.860803\pi$
$294$	0	0
$295$	5.85948	0.341152
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 4.35894i	− 0.252084i
$300$	0	0
$301$	− 4.60551i	− 0.265457i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.88333	−0.394138
$306$	0	0
$307$	− 11.7936i	− 0.673094i	−0.941667	−	0.336547i	$-0.890741\pi$
$307$	− 11.7936i	− 0.673094i	0.941667	−	0.336547i	$-0.109259\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.53163	−0.0868509	−0.0434255	−	0.999057i	$-0.513827\pi$
$311$	−1.53163	−0.0868509	−0.0434255	+	0.999057i	$0.513827\pi$
$312$	0	0
$313$	25.7415	1.45500	0.727498	−	0.686109i	$-0.240683\pi$
$313$	25.7415	1.45500	0.727498	+	0.686109i	$0.240683\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.59847i	0.370607i	0.982681	+	0.185304i	$0.0593268\pi$
$317$	6.59847i	0.370607i	−0.982681	+	0.185304i	$0.940673\pi$
$318$	0	0
$319$	−53.2336	−2.98051
$320$	0	0
$321$	0	0
$322$	0	0
$323$	45.9879i	2.55883i
$324$	0	0
$325$	− 0.986544i	− 0.0547236i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−14.8893	−0.820875
$330$	0	0
$331$	− 1.45480i	− 0.0799630i	−0.999200	−	0.0399815i	$-0.987270\pi$
$331$	− 1.45480i	− 0.0799630i	0.999200	−	0.0399815i	$-0.0127299\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.97298	0.217067
$336$	0	0
$337$	−27.9624	−1.52321	−0.761604	−	0.648043i	$-0.775588\pi$
$337$	−27.9624	−1.52321	−0.761604	+	0.648043i	$0.775588\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 16.2586i	− 0.880452i
$342$	0	0
$343$	−17.8287	−0.962660
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.79945i	− 0.203965i	−0.994786	−	0.101983i	$-0.967481\pi$
$347$	− 3.79945i	− 0.203965i	0.994786	−	0.101983i	$-0.0325186\pi$
$348$	0	0
$349$	− 0.0196324i	− 0.00105090i	−1.00000		0.000525448i	$-0.999833\pi$
$349$	− 0.0196324i	− 0.00105090i	1.00000		0.000525448i	$-0.000167255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.92987	−0.262390	−0.131195	−	0.991357i	$-0.541881\pi$
$353$	−4.92987	−0.262390	−0.131195	+	0.991357i	$0.541881\pi$
$354$	0	0
$355$	10.2619i	0.544647i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.3550	0.599292	0.299646	−	0.954050i	$-0.403131\pi$
$359$	11.3550	0.599292	0.299646	+	0.954050i	$0.403131\pi$
$360$	0	0
$361$	−42.9348	−2.25973
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.83638i	0.357832i
$366$	0	0
$367$	1.68458	0.0879344	0.0439672	−	0.999033i	$-0.486000\pi$
$367$	1.68458	0.0879344	0.0439672	+	0.999033i	$0.486000\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 8.92155i	− 0.463184i
$372$	0	0
$373$	20.9625i	1.08540i	0.839928	+	0.542698i	$0.182597\pi$
$373$	20.9625i	1.08540i	−0.839928	+	0.542698i	$0.817403\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.82012	−0.505762
$378$	0	0
$379$	5.84354i	0.300162i	0.988674	+	0.150081i	$0.0479535\pi$
$379$	5.84354i	0.300162i	−0.988674	+	0.150081i	$0.952046\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−29.3803	−1.50126	−0.750631	−	0.660722i	$-0.770250\pi$
$383$	−29.3803	−1.50126	−0.750631	+	0.660722i	$0.770250\pi$
$384$	0	0
$385$	8.17490	0.416632
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.2470i	1.22937i	0.788772	+	0.614687i	$0.210717\pi$
$389$	24.2470i	1.22937i	−0.788772	+	0.614687i	$0.789283\pi$
$390$	0	0
$391$	−25.8190	−1.30572
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 2.07752i	− 0.104531i
$396$	0	0
$397$	27.9219i	1.40136i	0.713476	+	0.700679i	$0.247120\pi$
$397$	27.9219i	1.40136i	−0.713476	+	0.700679i	$0.752880\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.01620	−0.0507464	−0.0253732	−	0.999678i	$-0.508077\pi$
$401$	−1.01620	−0.0507464	−0.0253732	+	0.999678i	$0.508077\pi$
$402$	0	0
$403$	− 2.99926i	− 0.149404i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	60.1560	2.98182
$408$	0	0
$409$	−29.4054	−1.45400	−0.727002	−	0.686635i	$-0.759087\pi$
$409$	−29.4054	−1.45400	−0.727002	+	0.686635i	$0.759087\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.95687i	0.440739i
$414$	0	0
$415$	−4.30777	−0.211460
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.07845i	0.248099i	0.992276	+	0.124049i	$0.0395881\pi$
$419$	5.07845i	0.248099i	−0.992276	+	0.124049i	$0.960412\pi$
$420$	0	0
$421$	16.8024i	0.818897i	0.912333	+	0.409449i	$0.134279\pi$
$421$	16.8024i	0.818897i	−0.912333	+	0.409449i	$0.865721\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.84354	−0.283453
$426$	0	0
$427$	− 10.5219i	− 0.509192i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.5116	−0.650829	−0.325415	−	0.945571i	$-0.605504\pi$
$431$	−13.5116	−0.650829	−0.325415	+	0.945571i	$0.605504\pi$
$432$	0	0
$433$	31.7335	1.52501	0.762507	−	0.646980i	$-0.223968\pi$
$433$	31.7335	1.52501	0.762507	+	0.646980i	$0.223968\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 34.7722i	− 1.66338i
$438$	0	0
$439$	21.7302	1.03713	0.518564	−	0.855039i	$-0.326467\pi$
$439$	21.7302	1.03713	0.518564	+	0.855039i	$0.326467\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.22744i	0.390897i	0.980714	+	0.195449i	$0.0626163\pi$
$443$	8.22744i	0.390897i	−0.980714	+	0.195449i	$0.937384\pi$
$444$	0	0
$445$	− 3.01287i	− 0.142824i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.83357	0.416882	0.208441	−	0.978035i	$-0.433161\pi$
$449$	8.83357	0.416882	0.208441	+	0.978035i	$0.433161\pi$
$450$	0	0
$451$	− 46.8152i	− 2.20444i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.50804	0.0706981
$456$	0	0
$457$	16.0924	0.752772	0.376386	−	0.926463i	$-0.377167\pi$
$457$	16.0924	0.752772	0.376386	+	0.926463i	$0.377167\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 22.1038i	− 1.02948i	−0.857347	−	0.514739i	$-0.827889\pi$
$461$	− 22.1038i	− 1.02948i	0.857347	−	0.514739i	$-0.172111\pi$
$462$	0	0
$463$	8.06502	0.374813	0.187407	−	0.982282i	$-0.439992\pi$
$463$	8.06502	0.374813	0.187407	+	0.982282i	$0.439992\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.53303i	− 0.209764i	−0.994485	−	0.104882i	$-0.966554\pi$
$467$	− 4.53303i	− 0.209764i	0.994485	−	0.104882i	$-0.0334464\pi$
$468$	0	0
$469$	6.07314i	0.280431i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.1126	0.740860
$474$	0	0
$475$	− 7.86987i	− 0.361094i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16.2348	0.741787	0.370893	−	0.928675i	$-0.379052\pi$
$479$	16.2348	0.741787	0.370893	+	0.928675i	$0.379052\pi$
$480$	0	0
$481$	11.0971	0.505985
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.12708i	0.232809i
$486$	0	0
$487$	−14.4136	−0.653143	−0.326572	−	0.945172i	$-0.605893\pi$
$487$	−14.4136	−0.653143	−0.326572	+	0.945172i	$0.605893\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 37.9033i	− 1.71055i	−0.518174	−	0.855275i	$-0.673388\pi$
$491$	− 37.9033i	− 1.71055i	0.518174	−	0.855275i	$-0.326612\pi$
$492$	0	0
$493$	58.1669i	2.61971i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.6865	−0.703636
$498$	0	0
$499$	− 29.8032i	− 1.33417i	−0.744980	−	0.667087i	$-0.767541\pi$
$499$	− 29.8032i	− 1.33417i	0.744980	−	0.667087i	$-0.232459\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−36.6293	−1.63322	−0.816611	−	0.577189i	$-0.804150\pi$
$503$	−36.6293	−1.63322	−0.816611	+	0.577189i	$0.804150\pi$
$504$	0	0
$505$	10.5399	0.469019
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.1891i	0.717571i	0.933420	+	0.358786i	$0.116809\pi$
$509$	16.1891i	0.717571i	−0.933420	+	0.358786i	$0.883191\pi$
$510$	0	0
$511$	−10.4502	−0.462288
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3.84403i	− 0.169388i
$516$	0	0
$517$	− 52.0911i	− 2.29097i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	40.1269	1.75799	0.878995	−	0.476830i	$-0.158214\pi$
$521$	40.1269	1.75799	0.878995	+	0.476830i	$0.158214\pi$
$522$	0	0
$523$	8.54884i	0.373815i	0.982378	+	0.186907i	$0.0598464\pi$
$523$	8.54884i	0.373815i	−0.982378	+	0.186907i	$0.940154\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.7653	−0.773869
$528$	0	0
$529$	−3.47782	−0.151210
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 8.63610i	− 0.374071i
$534$	0	0
$535$	16.3592	0.707270
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 24.9393i	− 1.07421i
$540$	0	0
$541$	18.2047i	0.782680i	0.920246	+	0.391340i	$0.127988\pi$
$541$	18.2047i	0.782680i	−0.920246	+	0.391340i	$0.872012\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.90977	0.295982
$546$	0	0
$547$	− 13.7431i	− 0.587611i	−0.955865	−	0.293805i	$-0.905078\pi$
$547$	− 13.7431i	− 0.587611i	0.955865	−	0.293805i	$-0.0949218\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−78.3371	−3.33727
$552$	0	0
$553$	3.17571	0.135045
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.6098i	− 0.619035i	−0.950894	−	0.309518i	$-0.899832\pi$
$557$	− 14.6098i	− 0.619035i	0.950894	−	0.309518i	$-0.100168\pi$
$558$	0	0
$559$	2.97233	0.125716
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.20995i	0.346008i	0.984921	+	0.173004i	$0.0553474\pi$
$563$	8.20995i	0.346008i	−0.984921	+	0.173004i	$0.944653\pi$
$564$	0	0
$565$	− 8.41100i	− 0.353853i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.72803	−0.198209	−0.0991046	−	0.995077i	$-0.531598\pi$
$569$	−4.72803	−0.198209	−0.0991046	+	0.995077i	$0.531598\pi$
$570$	0	0
$571$	− 14.2702i	− 0.597190i	−0.954380	−	0.298595i	$-0.903482\pi$
$571$	− 14.2702i	− 0.597190i	0.954380	−	0.298595i	$-0.0965180\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.41839	0.184260
$576$	0	0
$577$	39.2496	1.63398	0.816990	−	0.576651i	$-0.195641\pi$
$577$	39.2496	1.63398	0.816990	+	0.576651i	$0.195641\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 6.58490i	− 0.273188i
$582$	0	0
$583$	31.2125	1.29269
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.14493i	0.0472565i	0.999721	+	0.0236282i	$0.00752180\pi$
$587$	1.14493i	0.0472565i	−0.999721	+	0.0236282i	$0.992478\pi$
$588$	0	0
$589$	− 23.9257i	− 0.985841i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23.2964	0.956667	0.478334	−	0.878178i	$-0.341241\pi$
$593$	23.2964	0.956667	0.478334	+	0.878178i	$0.341241\pi$
$594$	0	0
$595$	− 8.93250i	− 0.366197i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	39.4568	1.61216	0.806080	−	0.591806i	$-0.201585\pi$
$599$	39.4568	1.61216	0.806080	+	0.591806i	$0.201585\pi$
$600$	0	0
$601$	0.728368	0.0297108	0.0148554	−	0.999890i	$-0.495271\pi$
$601$	0.728368	0.0297108	0.0148554	+	0.999890i	$0.495271\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	17.6003i	0.715556i
$606$	0	0
$607$	−18.0003	−0.730608	−0.365304	−	0.930888i	$-0.619035\pi$
$607$	−18.0003	−0.730608	−0.365304	+	0.930888i	$0.619035\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 9.60937i	− 0.388753i
$612$	0	0
$613$	43.1802i	1.74403i	0.489478	+	0.872015i	$0.337187\pi$
$613$	43.1802i	1.74403i	−0.489478	+	0.872015i	$0.662813\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	43.6681	1.75801	0.879005	−	0.476813i	$-0.158208\pi$
$617$	43.6681	1.75801	0.879005	+	0.476813i	$0.158208\pi$
$618$	0	0
$619$	28.2112i	1.13390i	0.823751	+	0.566951i	$0.191877\pi$
$619$	28.2112i	1.13390i	−0.823751	+	0.566951i	$0.808123\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.60551	0.184516
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 65.7309i	− 2.62086i
$630$	0	0
$631$	−42.6068	−1.69615	−0.848075	−	0.529877i	$-0.822238\pi$
$631$	−42.6068	−1.69615	−0.848075	+	0.529877i	$0.822238\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.84403i	0.0731781i
$636$	0	0
$637$	− 4.60060i	− 0.182282i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	29.1556	1.15158	0.575788	−	0.817599i	$-0.304695\pi$
$641$	29.1556	1.15158	0.575788	+	0.817599i	$0.304695\pi$
$642$	0	0
$643$	21.3586i	0.842301i	0.906991	+	0.421150i	$0.138374\pi$
$643$	21.3586i	0.842301i	−0.906991	+	0.421150i	$0.861626\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.33813	−0.327806	−0.163903	−	0.986476i	$-0.552408\pi$
$647$	−8.33813	−0.327806	−0.163903	+	0.986476i	$0.552408\pi$
$648$	0	0
$649$	−31.3361	−1.23005
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 38.3458i	− 1.50059i	−0.661105	−	0.750293i	$-0.729912\pi$
$653$	− 38.3458i	− 1.50059i	0.661105	−	0.750293i	$-0.270088\pi$
$654$	0	0
$655$	12.7810	0.499396
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.5080i	0.954698i	0.878714	+	0.477349i	$0.158402\pi$
$659$	24.5080i	0.954698i	−0.878714	+	0.477349i	$0.841598\pi$
$660$	0	0
$661$	− 27.2089i	− 1.05831i	−0.848527	−	0.529153i	$-0.822510\pi$
$661$	− 27.2089i	− 1.05831i	0.848527	−	0.529153i	$-0.177490\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.0300	0.466502
$666$	0	0
$667$	− 43.9809i	− 1.70295i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	36.8115	1.42109
$672$	0	0
$673$	38.4788	1.48325	0.741624	−	0.670816i	$-0.234056\pi$
$673$	38.4788	1.48325	0.741624	+	0.670816i	$0.234056\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.83595i	0.262727i	0.991334	+	0.131363i	$0.0419355\pi$
$677$	6.83595i	0.262727i	−0.991334	+	0.131363i	$0.958065\pi$
$678$	0	0
$679$	−7.83732	−0.300769
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 36.3056i	− 1.38920i	−0.719398	−	0.694598i	$-0.755582\pi$
$683$	− 36.3056i	− 1.38920i	0.719398	−	0.694598i	$-0.244418\pi$
$684$	0	0
$685$	12.6540i	0.483484i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.75784	0.219356
$690$	0	0
$691$	44.4920i	1.69256i	0.532741	+	0.846278i	$0.321162\pi$
$691$	44.4920i	1.69256i	−0.532741	+	0.846278i	$0.678838\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−18.9626	−0.719292
$696$	0	0
$697$	−51.1537	−1.93758
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.7347i	0.858676i	0.903144	+	0.429338i	$0.141253\pi$
$701$	22.7347i	0.858676i	−0.903144	+	0.429338i	$0.858747\pi$
$702$	0	0
$703$	88.5240	3.33875
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.1114i	0.605931i
$708$	0	0
$709$	47.2538i	1.77465i	0.461142	+	0.887326i	$0.347440\pi$
$709$	47.2538i	1.77465i	−0.461142	+	0.887326i	$0.652560\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	13.4326	0.503056
$714$	0	0
$715$	5.27597i	0.197310i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	48.8415	1.82148	0.910740	−	0.412979i	$-0.135512\pi$
$719$	48.8415	1.82148	0.910740	+	0.412979i	$0.135512\pi$
$720$	0	0
$721$	5.87603	0.218835
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 9.95406i	− 0.369684i
$726$	0	0
$727$	3.90811	0.144944	0.0724720	−	0.997370i	$-0.476911\pi$
$727$	3.90811	0.144944	0.0724720	+	0.997370i	$0.476911\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 17.6058i	− 0.651176i
$732$	0	0
$733$	− 0.332121i	− 0.0122672i	−0.999981	−	0.00613358i	$-0.998048\pi$
$733$	− 0.332121i	− 0.0122672i	0.999981	−	0.00613358i	$-0.00195239\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21.2472	−0.782651
$738$	0	0
$739$	14.6274i	0.538079i	0.963129	+	0.269039i	$0.0867062\pi$
$739$	14.6274i	0.538079i	−0.963129	+	0.269039i	$0.913294\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17.0424	0.625224	0.312612	−	0.949881i	$-0.398796\pi$
$743$	17.0424	0.625224	0.312612	+	0.949881i	$0.398796\pi$
$744$	0	0
$745$	3.95406	0.144865
$746$	0	0
$747$	0	0
$748$	0	0
$749$	25.0069i	0.913731i
$750$	0	0
$751$	−12.3129	−0.449303	−0.224652	−	0.974439i	$-0.572124\pi$
$751$	−12.3129	−0.449303	−0.224652	+	0.974439i	$0.572124\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.76210i	0.100523i
$756$	0	0
$757$	36.8680i	1.33999i	0.742366	+	0.669995i	$0.233704\pi$
$757$	36.8680i	1.33999i	−0.742366	+	0.669995i	$0.766296\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−16.6730	−0.604397	−0.302199	−	0.953245i	$-0.597721\pi$
$761$	−16.6730	−0.604397	−0.302199	+	0.953245i	$0.597721\pi$
$762$	0	0
$763$	10.5623i	0.382383i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.78064	−0.208727
$768$	0	0
$769$	24.0819	0.868417	0.434208	−	0.900812i	$-0.357028\pi$
$769$	24.0819	0.868417	0.434208	+	0.900812i	$0.357028\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 5.88458i	− 0.211654i	−0.994385	−	0.105827i	$-0.966251\pi$
$773$	− 5.88458i	− 0.211654i	0.994385	−	0.105827i	$-0.0337489\pi$
$774$	0	0
$775$	3.04016	0.109206
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 68.8920i	− 2.46831i
$780$	0	0
$781$	− 54.8801i	− 1.96376i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.85024	−0.280187
$786$	0	0
$787$	33.5218i	1.19492i	0.801897	+	0.597462i	$0.203824\pi$
$787$	33.5218i	1.19492i	−0.801897	+	0.597462i	$0.796176\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.8571	0.457147
$792$	0	0
$793$	6.79070	0.241145
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 8.39899i	− 0.297507i	−0.988874	−	0.148754i	$-0.952474\pi$
$797$	− 8.39899i	− 0.297507i	0.988874	−	0.148754i	$-0.0475262\pi$
$798$	0	0
$799$	−56.9186	−2.01364
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 36.5605i	− 1.29019i
$804$	0	0
$805$	6.75400i	0.238047i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−37.4680	−1.31731	−0.658653	−	0.752447i	$-0.728873\pi$
$809$	−37.4680	−1.31731	−0.658653	+	0.752447i	$0.728873\pi$
$810$	0	0
$811$	19.2209i	0.674936i	0.941337	+	0.337468i	$0.109570\pi$
$811$	19.2209i	0.674936i	−0.941337	+	0.337468i	$0.890430\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	14.5778	0.510638
$816$	0	0
$817$	23.7109	0.829540
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 26.3571i	− 0.919867i	−0.887953	−	0.459934i	$-0.847873\pi$
$821$	− 26.3571i	− 0.919867i	0.887953	−	0.459934i	$-0.152127\pi$
$822$	0	0
$823$	−20.2586	−0.706170	−0.353085	−	0.935591i	$-0.614867\pi$
$823$	−20.2586	−0.706170	−0.353085	+	0.935591i	$0.614867\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.3325i	0.359295i	0.983731	+	0.179648i	$0.0574958\pi$
$827$	10.3325i	0.359295i	−0.983731	+	0.179648i	$0.942504\pi$
$828$	0	0
$829$	− 53.2266i	− 1.84864i	−0.381621	−	0.924319i	$-0.624634\pi$
$829$	− 53.2266i	− 1.84864i	0.381621	−	0.924319i	$-0.375366\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−27.2505	−0.944173
$834$	0	0
$835$	8.13307i	0.281457i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.86638	0.202530	0.101265	−	0.994859i	$-0.467711\pi$
$839$	5.86638	0.202530	0.101265	+	0.994859i	$0.467711\pi$
$840$	0	0
$841$	−70.0832	−2.41666
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 12.0267i	− 0.413732i
$846$	0	0
$847$	−26.9041	−0.924435
$848$	0	0
$849$	0	0
$850$	0	0
$851$	49.7002i	1.70370i
$852$	0	0
$853$	− 48.3015i	− 1.65381i	−0.562341	−	0.826906i	$-0.690099\pi$
$853$	− 48.3015i	− 1.65381i	0.562341	−	0.826906i	$-0.309901\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.68171	0.0916054	0.0458027	−	0.998951i	$-0.485415\pi$
$857$	2.68171	0.0916054	0.0458027	+	0.998951i	$0.485415\pi$
$858$	0	0
$859$	− 3.19829i	− 0.109124i	−0.998510	−	0.0545622i	$-0.982624\pi$
$859$	− 3.19829i	− 0.109124i	0.998510	−	0.0545622i	$-0.0173763\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−10.4441	−0.355523	−0.177761	−	0.984074i	$-0.556885\pi$
$863$	−10.4441	−0.355523	−0.177761	+	0.984074i	$0.556885\pi$
$864$	0	0
$865$	−0.0170573	−0.000579964 0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.1104i	0.376895i
$870$	0	0
$871$	−3.91952	−0.132808
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.52861i	0.0516765i
$876$	0	0
$877$	− 2.26309i	− 0.0764192i	−0.999270	−	0.0382096i	$-0.987835\pi$
$877$	− 2.26309i	− 0.0764192i	0.999270	−	0.0382096i	$-0.0121655\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	15.6176	0.526170	0.263085	−	0.964773i	$-0.415260\pi$
$881$	15.6176	0.526170	0.263085	+	0.964773i	$0.415260\pi$
$882$	0	0
$883$	21.6433i	0.728356i	0.931329	+	0.364178i	$0.118650\pi$
$883$	21.6433i	0.728356i	−0.931329	+	0.364178i	$0.881350\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.2570	−1.08308	−0.541542	−	0.840674i	$-0.682159\pi$
$887$	−32.2570	−1.08308	−0.541542	+	0.840674i	$0.682159\pi$
$888$	0	0
$889$	−2.81880	−0.0945397
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 76.6559i	− 2.56519i
$894$	0	0
$895$	−11.6386	−0.389037
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 30.2620i	− 1.00929i
$900$	0	0
$901$	− 34.1051i	− 1.13621i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.41839	−0.146872
$906$	0	0
$907$	− 33.2207i	− 1.10308i	−0.834150	−	0.551538i	$-0.814041\pi$
$907$	− 33.2207i	− 1.10308i	0.834150	−	0.551538i	$-0.185959\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.56561	0.184397	0.0921985	−	0.995741i	$-0.470611\pi$
$911$	5.56561	0.184397	0.0921985	+	0.995741i	$0.470611\pi$
$912$	0	0
$913$	23.0376	0.762434
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.5372i	0.645176i
$918$	0	0
$919$	−2.80883	−0.0926549	−0.0463274	−	0.998926i	$-0.514752\pi$
$919$	−2.80883	−0.0926549	−0.0463274	+	0.998926i	$0.514752\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 10.1238i	− 0.333230i
$924$	0	0
$925$	11.2485i	0.369848i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.7121	0.679541	0.339771	−	0.940508i	$-0.389651\pi$
$929$	20.7121	0.679541	0.339771	+	0.940508i	$0.389651\pi$
$930$	0	0
$931$	− 36.6999i	− 1.20279i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	31.2508	1.02201
$936$	0	0
$937$	−41.6066	−1.35923	−0.679615	−	0.733569i	$-0.737853\pi$
$937$	−41.6066	−1.35923	−0.679615	+	0.733569i	$0.737853\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 16.6771i	− 0.543656i	−0.962346	−	0.271828i	$-0.912372\pi$
$941$	− 16.6771i	− 0.543656i	0.962346	−	0.271828i	$-0.0876282\pi$
$942$	0	0
$943$	38.6781	1.25953
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 38.5829i	− 1.25377i	−0.779110	−	0.626887i	$-0.784329\pi$
$947$	− 38.5829i	− 1.25377i	0.779110	−	0.626887i	$-0.215671\pi$
$948$	0	0
$949$	− 6.74439i	− 0.218932i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−31.6927	−1.02663	−0.513313	−	0.858202i	$-0.671582\pi$
$953$	−31.6927	−1.02663	−0.513313	+	0.858202i	$0.671582\pi$
$954$	0	0
$955$	− 16.1783i	− 0.523519i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19.3430	−0.624619
$960$	0	0
$961$	−21.7574	−0.701852
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 14.4052i	− 0.463718i
$966$	0	0
$967$	10.1657	0.326905	0.163453	−	0.986551i	$-0.447737\pi$
$967$	10.1657	0.326905	0.163453	+	0.986551i	$0.447737\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.4769i	0.625043i	0.949911	+	0.312522i	$0.101174\pi$
$971$	19.4769i	0.625043i	−0.949911	+	0.312522i	$0.898826\pi$
$972$	0	0
$973$	− 28.9864i	− 0.929261i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.99884	−0.255905	−0.127953	−	0.991780i	$-0.540841\pi$
$977$	−7.99884	−0.255905	−0.127953	+	0.991780i	$0.540841\pi$
$978$	0	0
$979$	16.1126i	0.514962i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26.1856	0.835190	0.417595	−	0.908633i	$-0.362873\pi$
$983$	26.1856	0.835190	0.417595	+	0.908633i	$0.362873\pi$
$984$	0	0
$985$	12.4981	0.398223
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.3121i	0.423299i
$990$	0	0
$991$	43.1761	1.37153	0.685767	−	0.727821i	$-0.259467\pi$
$991$	43.1761	1.37153	0.685767	+	0.727821i	$0.259467\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 22.7242i	− 0.720405i
$996$	0	0
$997$	14.4809i	0.458615i	0.973354	+	0.229308i	$0.0736462\pi$
$997$	14.4809i	0.458615i	−0.973354	+	0.229308i	$0.926354\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.k.d.2161.6		20
3.2	odd	2	inner	4320.2.k.d.2161.16		20
4.3	odd	2		1080.2.k.d.541.14	yes	20
8.3	odd	2		1080.2.k.d.541.13	yes	20
8.5	even	2	inner	4320.2.k.d.2161.15		20
12.11	even	2		1080.2.k.d.541.7	✓	20
24.5	odd	2	inner	4320.2.k.d.2161.5		20
24.11	even	2		1080.2.k.d.541.8	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.k.d.541.7	✓	20	12.11	even	2
1080.2.k.d.541.8	yes	20	24.11	even	2
1080.2.k.d.541.13	yes	20	8.3	odd	2
1080.2.k.d.541.14	yes	20	4.3	odd	2
4320.2.k.d.2161.5		20	24.5	odd	2	inner
4320.2.k.d.2161.6		20	1.1	even	1	trivial
4320.2.k.d.2161.15		20	8.5	even	2	inner
4320.2.k.d.2161.16		20	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} +1.52861 q^{7} +O(q^{10})\)	\(q-1.00000i q^{5} +1.52861 q^{7} +5.34793i q^{11} +0.986544i q^{13} +5.84354 q^{17} +7.86987i q^{19} -4.41839 q^{23} -1.00000 q^{25} +9.95406i q^{29} -3.04016 q^{31} -1.52861i q^{35} -11.2485i q^{37} -8.75389 q^{41} -3.01287i q^{43} -9.74043 q^{47} -4.66335 q^{49} -5.83638i q^{53} +5.34793 q^{55} +5.85948i q^{59} -6.88333i q^{61} +0.986544 q^{65} +3.97298i q^{67} -10.2619 q^{71} -6.83638 q^{73} +8.17490i q^{77} +2.07752 q^{79} -4.30777i q^{83} -5.84354i q^{85} +3.01287 q^{89} +1.50804i q^{91} +7.86987 q^{95} -5.12708 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 20 q^{25} - 20 q^{31} + 36 q^{49} - 40 q^{73} + 4 q^{79} + 40 q^{97}+O(q^{100}) \) 20 * q - 20 * q^25 - 20 * q^31 + 36 * q^49 - 40 * q^73 + 4 * q^79 + 40 * q^97

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