LMFDB - Embedded newform 4320.2.cc.b.3311.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(1871,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1, 0]))

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

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

chi := DirichletCharacter("4320.1871");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.cc (of order $6$, 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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			3311.23
Character	$\chi$	$=$	4320.3311
Dual form			4320.2.cc.b.1871.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(3.97204 - 2.29326i) q^{7} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{5} +(3.97204 - 2.29326i) q^{7} +(-4.08838 + 2.36043i) q^{11} +(1.87107 + 1.08026i) q^{13} -1.23675i q^{17} -4.35920 q^{19} +(0.117833 - 0.204092i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-2.84210 - 4.92266i) q^{29} +(-4.06804 - 2.34868i) q^{31} -4.58651i q^{35} -9.91926i q^{37} +(-6.34052 - 3.66070i) q^{41} +(2.62545 + 4.54741i) q^{43} +(1.05958 + 1.83524i) q^{47} +(7.01804 - 12.1556i) q^{49} +7.92028 q^{53} +4.72085i q^{55} +(-10.1979 - 5.88778i) q^{59} +(6.27586 - 3.62337i) q^{61} +(1.87107 - 1.08026i) q^{65} +(7.51856 - 13.0225i) q^{67} +0.851441 q^{71} +10.6769 q^{73} +(-10.8261 + 18.7514i) q^{77} +(-10.0179 + 5.78381i) q^{79} +(-2.35035 + 1.35698i) q^{83} +(-1.07106 - 0.618377i) q^{85} -12.9321i q^{89} +9.90928 q^{91} +(-2.17960 + 3.77518i) q^{95} +(-0.816409 - 1.41406i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 24 q^{5}+O(q^{10}) $ 48 * q + 24 * q^5	$ 48 q + 24 q^{5} - 24 q^{25} - 12 q^{41} + 12 q^{47} + 24 q^{49} - 36 q^{59} + 12 q^{61} - 60 q^{83}+O(q^{100}) $ 48 * q + 24 * q^5 - 24 * q^25 - 12 * q^41 + 12 * q^47 + 24 * q^49 - 36 * q^59 + 12 * q^61 - 60 * q^83

Display 100 coefficients 10 coefficients

Character values

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

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$	$-1$

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
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	3.97204	−	2.29326i	1.50129	−	0.866769i	0.501289	−	0.865280i	$-0.332859\pi$
$7$	3.97204	−	2.29326i	1.50129	−	0.866769i	0.999999	−	0.00148944i	$-0.000474104\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.08838	+	2.36043i	−1.23269	+	0.711695i	−0.967590	−	0.252525i	$-0.918739\pi$
$11$	−4.08838	+	2.36043i	−1.23269	+	0.711695i	−0.265102	+	0.964220i	$0.585406\pi$
$12$		0			0
$13$	1.87107	+	1.08026i	0.518942	+	0.299611i	0.736501	−	0.676436i	$-0.236476\pi$
$13$	1.87107	+	1.08026i	0.518942	+	0.299611i	−0.217560	+	0.976047i	$0.569810\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.23675i		−	0.299957i	−0.988689	−	0.149978i	$-0.952080\pi$
$17$		−	1.23675i		−	0.299957i	0.988689	−	0.149978i	$-0.0479204\pi$
$18$		0			0
$19$	−4.35920			−1.00007			−0.500034	−	0.866005i	$-0.666679\pi$
$19$	−4.35920			−1.00007			−0.500034	+	0.866005i	$0.666679\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.117833	−	0.204092i	0.0245698	−	0.0425562i	−0.853479	−	0.521127i	$-0.825512\pi$
$23$	0.117833	−	0.204092i	0.0245698	−	0.0425562i	0.878049	+	0.478571i	$0.158845\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.84210	−	4.92266i	−0.527764	−	0.914114i	−0.999476	−	0.0323617i	$-0.989697\pi$
$29$	−2.84210	−	4.92266i	−0.527764	−	0.914114i	0.471712	−	0.881753i	$-0.343636\pi$
$30$		0			0
$31$	−4.06804	−	2.34868i	−0.730641	−	0.421836i	0.0880155	−	0.996119i	$-0.471947\pi$
$31$	−4.06804	−	2.34868i	−0.730641	−	0.421836i	−0.818657	+	0.574283i	$0.805281\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	4.58651i		−	0.775262i
$36$		0			0
$37$		−	9.91926i		−	1.63072i	−0.578957	−	0.815358i	$-0.696540\pi$
$37$		−	9.91926i		−	1.63072i	0.578957	−	0.815358i	$-0.303460\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.34052	−	3.66070i	−0.990222	−	0.571705i	−0.0848815	−	0.996391i	$-0.527051\pi$
$41$	−6.34052	−	3.66070i	−0.990222	−	0.571705i	−0.905341	+	0.424686i	$0.860384\pi$
$42$		0			0
$43$	2.62545	+	4.54741i	0.400377	+	0.693473i	0.993771	−	0.111439i	$-0.0355458\pi$
$43$	2.62545	+	4.54741i	0.400377	+	0.693473i	−0.593394	+	0.804912i	$0.702212\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.05958	+	1.83524i	0.154555	+	0.267698i	0.932897	−	0.360143i	$-0.117272\pi$
$47$	1.05958	+	1.83524i	0.154555	+	0.267698i	−0.778342	+	0.627841i	$0.783939\pi$
$48$		0			0
$49$	7.01804	−	12.1556i	1.00258	−	1.73652i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.92028			1.08793			0.543967	−	0.839107i	$-0.316922\pi$
$53$	7.92028			1.08793			0.543967	+	0.839107i	$0.316922\pi$
$54$		0			0
$55$			4.72085i			0.636560i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.1979	−	5.88778i	−1.32766	−	0.766524i	−0.342722	−	0.939437i	$-0.611349\pi$
$59$	−10.1979	−	5.88778i	−1.32766	−	0.766524i	−0.984937	+	0.172913i	$0.944682\pi$
$60$		0			0
$61$	6.27586	−	3.62337i	0.803541	−	0.463925i	−0.0411667	−	0.999152i	$-0.513107\pi$
$61$	6.27586	−	3.62337i	0.803541	−	0.463925i	0.844708	+	0.535228i	$0.179774\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.87107	−	1.08026i	0.232078	−	0.133990i
$66$		0			0
$67$	7.51856	−	13.0225i	0.918538	−	1.59095i	0.116900	−	0.993144i	$-0.462704\pi$
$67$	7.51856	−	13.0225i	0.918538	−	1.59095i	0.801637	−	0.597811i	$-0.203962\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.851441			0.101047			0.0505237	−	0.998723i	$-0.483911\pi$
$71$	0.851441			0.101047			0.0505237	+	0.998723i	$0.483911\pi$
$72$		0			0
$73$	10.6769			1.24964			0.624819	−	0.780770i	$-0.285173\pi$
$73$	10.6769			1.24964			0.624819	+	0.780770i	$0.285173\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.8261	+	18.7514i	−1.23375	+	2.13692i
$78$		0			0
$79$	−10.0179	+	5.78381i	−1.12710	+	0.650730i	−0.943203	−	0.332217i	$-0.892204\pi$
$79$	−10.0179	+	5.78381i	−1.12710	+	0.650730i	−0.183894	+	0.982946i	$0.558870\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.35035	+	1.35698i	−0.257985	+	0.148948i	−0.623415	−	0.781891i	$-0.714255\pi$
$83$	−2.35035	+	1.35698i	−0.257985	+	0.148948i	0.365430	+	0.930839i	$0.380922\pi$
$84$		0			0
$85$	−1.07106	−	0.618377i	−0.116173	−	0.0670724i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	12.9321i		−	1.37080i	−0.728166	−	0.685401i	$-0.759627\pi$
$89$		−	12.9321i		−	1.37080i	0.728166	−	0.685401i	$-0.240373\pi$
$90$		0			0
$91$	9.90928			1.03877
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.17960	+	3.77518i	−0.223622	+	0.387325i
$96$		0			0
$97$	−0.816409	−	1.41406i	−0.0828938	−	0.143576i	0.821598	−	0.570067i	$-0.193083\pi$
$97$	−0.816409	−	1.41406i	−0.0828938	−	0.143576i	−0.904492	+	0.426491i	$0.859750\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.28871	−	2.23211i	−0.128231	−	0.222103i	0.794760	−	0.606924i	$-0.207597\pi$
$101$	−1.28871	−	2.23211i	−0.128231	−	0.222103i	−0.922991	+	0.384820i	$0.874263\pi$
$102$		0			0
$103$	−8.62567	−	4.98003i	−0.849913	−	0.490697i	0.0107087	−	0.999943i	$-0.496591\pi$
$103$	−8.62567	−	4.98003i	−0.849913	−	0.490697i	−0.860621	+	0.509245i	$0.829925\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			7.28403i			0.704173i	0.935967	+	0.352087i	$0.114528\pi$
$107$			7.28403i			0.704173i	−0.935967	+	0.352087i	$0.885472\pi$
$108$		0			0
$109$		−	2.56505i		−	0.245687i	−0.992426	−	0.122844i	$-0.960799\pi$
$109$		−	2.56505i		−	0.245687i	0.992426	−	0.122844i	$-0.0392014\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.311444	−	0.179812i	−0.0292981	−	0.0169153i	0.485279	−	0.874359i	$-0.338718\pi$
$113$	−0.311444	−	0.179812i	−0.0292981	−	0.0169153i	−0.514578	+	0.857444i	$0.672051\pi$
$114$		0			0
$115$	−0.117833	−	0.204092i	−0.0109880	−	0.0190317i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.83619	−	4.91243i	−0.259993	−	0.450322i
$120$		0			0
$121$	5.64322	−	9.77435i	0.513020	−	0.888577i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		−	9.02927i		−	0.801218i	−0.916249	−	0.400609i	$-0.868799\pi$
$127$		−	9.02927i		−	0.801218i	0.916249	−	0.400609i	$-0.131201\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.986098	−	0.569324i	−0.0861558	−	0.0497421i	0.456303	−	0.889824i	$-0.349173\pi$
$131$	−0.986098	−	0.569324i	−0.0861558	−	0.0497421i	−0.542459	+	0.840082i	$0.682507\pi$
$132$		0			0
$133$	−17.3149	+	9.99676i	−1.50139	+	0.866829i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.39260	−	1.95872i	0.289850	−	0.167345i	−0.348024	−	0.937485i	$-0.613147\pi$
$137$	3.39260	−	1.95872i	0.289850	−	0.167345i	0.637874	+	0.770141i	$0.279814\pi$
$138$		0			0
$139$	−5.70435	+	9.88022i	−0.483836	+	0.838029i	−0.999828	−	0.0185646i	$-0.994090\pi$
$139$	−5.70435	+	9.88022i	−0.483836	+	0.838029i	0.515991	+	0.856594i	$0.327424\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−10.1995			−0.852927
$144$		0			0
$145$	−5.68419			−0.472047
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.86802	+	17.0919i	−0.808420	+	1.40022i	0.105538	+	0.994415i	$0.466344\pi$
$149$	−9.86802	+	17.0919i	−0.808420	+	1.40022i	−0.913958	+	0.405809i	$0.866990\pi$
$150$		0			0
$151$	−5.32377	+	3.07368i	−0.433243	+	0.250133i	−0.700727	−	0.713429i	$-0.747141\pi$
$151$	−5.32377	+	3.07368i	−0.433243	+	0.250133i	0.267484	+	0.963562i	$0.413808\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.06804	+	2.34868i	−0.326753	+	0.188651i
$156$		0			0
$157$	0.339511	+	0.196017i	0.0270959	+	0.0156438i	0.513487	−	0.858098i	$-0.328354\pi$
$157$	0.339511	+	0.196017i	0.0270959	+	0.0156438i	−0.486391	+	0.873741i	$0.661687\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	1.08088i		−	0.0851855i
$162$		0			0
$163$	24.6306			1.92922			0.964609	−	0.263683i	$-0.0849374\pi$
$163$	24.6306			1.92922			0.964609	+	0.263683i	$0.0849374\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.50486	+	2.60649i	−0.116450	+	0.201697i	−0.918358	−	0.395750i	$-0.870485\pi$
$167$	−1.50486	+	2.60649i	−0.116450	+	0.201697i	0.801909	+	0.597447i	$0.203818\pi$
$168$		0			0
$169$	−4.16606	−	7.21583i	−0.320466	−	0.555064i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.98815	+	8.63974i	0.379242	+	0.656867i	0.990952	−	0.134215i	$-0.0428513\pi$
$173$	4.98815	+	8.63974i	0.379242	+	0.656867i	−0.611710	+	0.791082i	$0.709518\pi$
$174$		0			0
$175$	−3.97204	−	2.29326i	−0.300258	−	0.173354i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	11.1600i		−	0.834136i	−0.908875	−	0.417068i	$-0.863058\pi$
$179$		−	11.1600i		−	0.834136i	0.908875	−	0.417068i	$-0.136942\pi$
$180$		0			0
$181$			4.80407i			0.357083i	0.983932	+	0.178542i	$0.0571379\pi$
$181$			4.80407i			0.357083i	−0.983932	+	0.178542i	$0.942862\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.59033	−	4.95963i	−0.631574	−	0.364639i
$186$		0			0
$187$	2.91927	+	5.05632i	0.213478	+	0.369755i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.143662	+	0.248830i	0.0103950	+	0.0180047i	0.871176	−	0.490971i	$-0.163358\pi$
$191$	0.143662	+	0.248830i	0.0103950	+	0.0180047i	−0.860781	+	0.508975i	$0.830024\pi$
$192$		0			0
$193$	1.19503	−	2.06986i	0.0860204	−	0.148992i	−0.819805	−	0.572643i	$-0.805918\pi$
$193$	1.19503	−	2.06986i	0.0860204	−	0.148992i	0.905826	+	0.423651i	$0.139252\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.8177			0.770732			0.385366	−	0.922764i	$-0.374075\pi$
$197$	10.8177			0.770732			0.385366	+	0.922764i	$0.374075\pi$
$198$		0			0
$199$		−	5.10541i		−	0.361913i	−0.983491	−	0.180957i	$-0.942081\pi$
$199$		−	5.10541i		−	0.361913i	0.983491	−	0.180957i	$-0.0579193\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−22.5778	−	13.0353i	−1.58465	−	0.914899i
$204$		0			0
$205$	−6.34052	+	3.66070i	−0.442841	+	0.255674i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	17.8221	−	10.2896i	1.23278	−	0.711744i
$210$		0			0
$211$	−0.491584	+	0.851449i	−0.0338420	+	0.0586161i	−0.882450	−	0.470406i	$-0.844108\pi$
$211$	−0.491584	+	0.851449i	−0.0338420	+	0.0586161i	0.848608	+	0.529022i	$0.177441\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.25090			0.358108
$216$		0			0
$217$	−21.5445			−1.46254
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.33602	−	2.31405i	0.0898704	−	0.155660i
$222$		0			0
$223$	10.3531	−	5.97739i	0.693297	−	0.400275i	−0.111549	−	0.993759i	$-0.535581\pi$
$223$	10.3531	−	5.97739i	0.693297	−	0.400275i	0.804846	+	0.593484i	$0.202248\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17.0870	−	9.86519i	1.13410	−	0.654775i	0.189140	−	0.981950i	$-0.439430\pi$
$227$	17.0870	−	9.86519i	1.13410	−	0.654775i	0.944964	+	0.327175i	$0.106097\pi$
$228$		0			0
$229$	0.735791	+	0.424809i	0.0486225	+	0.0280722i	0.524114	−	0.851648i	$-0.324397\pi$
$229$	0.735791	+	0.424809i	0.0486225	+	0.0280722i	−0.475492	+	0.879720i	$0.657730\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			26.9773i			1.76734i	0.468111	+	0.883670i	$0.344935\pi$
$233$			26.9773i			1.76734i	−0.468111	+	0.883670i	$0.655065\pi$
$234$		0			0
$235$	2.11916			0.138239
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.98946	+	12.1061i	−0.452111	+	0.783079i	−0.998517	−	0.0544415i	$-0.982662\pi$
$239$	−6.98946	+	12.1061i	−0.452111	+	0.783079i	0.546406	+	0.837520i	$0.315995\pi$
$240$		0			0
$241$	7.14583	+	12.3769i	0.460303	+	0.797269i	0.998976	−	0.0452465i	$-0.0144073\pi$
$241$	7.14583	+	12.3769i	0.460303	+	0.797269i	−0.538673	+	0.842515i	$0.681074\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.01804	−	12.1556i	−0.448366	−	0.776593i
$246$		0			0
$247$	−8.15637	−	4.70908i	−0.518977	−	0.299632i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	9.70630i		−	0.612656i	−0.951926	−	0.306328i	$-0.900900\pi$
$251$		−	9.70630i		−	0.612656i	0.951926	−	0.306328i	$-0.0991004\pi$
$252$		0			0
$253$			1.11254i			0.0699449i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.4543	−	9.49991i	−1.02639	−	0.592588i	−0.110444	−	0.993882i	$-0.535227\pi$
$257$	−16.4543	−	9.49991i	−1.02639	−	0.592588i	−0.915949	+	0.401294i	$0.868561\pi$
$258$		0			0
$259$	−22.7474	−	39.3996i	−1.41345	−	2.44817i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.6788	−	21.9603i	−0.781806	−	1.35413i	−0.930889	−	0.365303i	$-0.880965\pi$
$263$	−12.6788	−	21.9603i	−0.781806	−	1.35413i	0.149083	−	0.988825i	$-0.452368\pi$
$264$		0			0
$265$	3.96014	−	6.85916i	0.243269	−	0.421355i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.6387			−0.648652			−0.324326	−	0.945945i	$-0.605138\pi$
$269$	−10.6387			−0.648652			−0.324326	+	0.945945i	$0.605138\pi$
$270$		0			0
$271$			22.9127i			1.39185i	0.718115	+	0.695924i	$0.245005\pi$
$271$			22.9127i			1.39185i	−0.718115	+	0.695924i	$0.754995\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.08838	+	2.36043i	0.246538	+	0.142339i
$276$		0			0
$277$	13.2767	−	7.66533i	0.797722	−	0.460565i	−0.0449517	−	0.998989i	$-0.514313\pi$
$277$	13.2767	−	7.66533i	0.797722	−	0.460565i	0.842674	+	0.538424i	$0.180980\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.30426	−	5.37182i	0.555046	−	0.320456i	−0.196109	−	0.980582i	$-0.562831\pi$
$281$	9.30426	−	5.37182i	0.555046	−	0.320456i	0.751155	+	0.660126i	$0.229497\pi$
$282$		0			0
$283$	−8.88802	+	15.3945i	−0.528338	+	0.915108i	0.471116	+	0.882071i	$0.343851\pi$
$283$	−8.88802	+	15.3945i	−0.528338	+	0.915108i	−0.999454	+	0.0330371i	$0.989482\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−33.5797			−1.98215
$288$		0			0
$289$	15.4704			0.910026
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.9870	−	24.2262i	0.817129	−	1.41531i	−0.0906593	−	0.995882i	$-0.528897\pi$
$293$	13.9870	−	24.2262i	0.817129	−	1.41531i	0.907789	−	0.419428i	$-0.137769\pi$
$294$		0			0
$295$	−10.1979	+	5.88778i	−0.593747	+	0.342800i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.440947	−	0.254581i	0.0255006	−	0.0147228i
$300$		0			0
$301$	20.8567	+	12.0416i	1.20216	+	0.694069i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	7.24674i		−	0.414947i
$306$		0			0
$307$	−15.3914			−0.878434			−0.439217	−	0.898381i	$-0.644744\pi$
$307$	−15.3914			−0.878434			−0.439217	+	0.898381i	$0.644744\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.87577	−	17.1053i	0.560004	−	0.969955i	−0.437492	−	0.899222i	$-0.644133\pi$
$311$	9.87577	−	17.1053i	0.560004	−	0.969955i	0.997495	−	0.0707323i	$-0.0225336\pi$
$312$		0			0
$313$	−8.85105	−	15.3305i	−0.500291	−	0.866529i	−1.00000		0.000335911i	$-0.999893\pi$
$313$	−8.85105	−	15.3305i	−0.500291	−	0.866529i	0.499709	−	0.866193i	$-0.333440\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.67910	−	4.64033i	−0.150473	−	0.260627i	0.780928	−	0.624621i	$-0.214746\pi$
$317$	−2.67910	−	4.64033i	−0.150473	−	0.260627i	−0.931401	+	0.363994i	$0.881413\pi$
$318$		0			0
$319$	23.2391	+	13.4171i	1.30114	+	0.751214i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			5.39126i			0.299978i
$324$		0			0
$325$		−	2.16053i		−	0.119844i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.41736	+	4.85977i	0.464064	+	0.267928i
$330$		0			0
$331$	−12.9222	−	22.3818i	−0.710266	−	1.23022i	−0.964757	−	0.263142i	$-0.915241\pi$
$331$	−12.9222	−	22.3818i	−0.710266	−	1.23022i	0.254491	−	0.967075i	$-0.418092\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.51856	−	13.0225i	−0.410783	−	0.711496i
$336$		0			0
$337$	8.23294	−	14.2599i	0.448477	−	0.776784i	−0.549810	−	0.835289i	$-0.685300\pi$
$337$	8.23294	−	14.2599i	0.448477	−	0.776784i	0.998287	+	0.0585051i	$0.0186334\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	22.1756			1.20087
$342$		0			0
$343$		−	32.2711i		−	1.74247i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	13.0391	+	7.52814i	0.699976	+	0.404132i	0.807339	−	0.590088i	$-0.200907\pi$
$347$	13.0391	+	7.52814i	0.699976	+	0.404132i	−0.107362	+	0.994220i	$0.534240\pi$
$348$		0			0
$349$	−7.00796	+	4.04605i	−0.375128	+	0.216580i	−0.675696	−	0.737180i	$-0.736157\pi$
$349$	−7.00796	+	4.04605i	−0.375128	+	0.216580i	0.300569	+	0.953760i	$0.402824\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−8.16936	+	4.71658i	−0.434811	+	0.251038i	−0.701394	−	0.712774i	$-0.747439\pi$
$353$	−8.16936	+	4.71658i	−0.434811	+	0.251038i	0.266583	+	0.963812i	$0.414105\pi$
$354$		0			0
$355$	0.425720	−	0.737369i	0.0225949	−	0.0391355i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	28.9175			1.52621			0.763103	−	0.646277i	$-0.223675\pi$
$359$	28.9175			1.52621			0.763103	+	0.646277i	$0.223675\pi$
$360$		0			0
$361$	0.00262185			0.000137992
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.33845	−	9.24648i	0.279428	−	0.483983i
$366$		0			0
$367$	27.0895	−	15.6401i	1.41406	−	0.816407i	0.418291	−	0.908313i	$-0.362629\pi$
$367$	27.0895	−	15.6401i	1.41406	−	0.816407i	0.995768	+	0.0919062i	$0.0292960\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	31.4596	−	18.1632i	1.63330	−	0.942988i
$372$		0			0
$373$	−17.8238	−	10.2906i	−0.922882	−	0.532826i	−0.0383285	−	0.999265i	$-0.512203\pi$
$373$	−17.8238	−	10.2906i	−0.922882	−	0.532826i	−0.884553	+	0.466439i	$0.845537\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	12.2808i		−	0.632496i
$378$		0			0
$379$	−1.15098			−0.0591219			−0.0295609	−	0.999563i	$-0.509411\pi$
$379$	−1.15098			−0.0591219			−0.0295609	+	0.999563i	$0.509411\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.0240645	−	0.0416809i	0.00122964	−	0.00212979i	−0.865410	−	0.501065i	$-0.832942\pi$
$383$	0.0240645	−	0.0416809i	0.00122964	−	0.00212979i	0.866640	+	0.498935i	$0.166275\pi$
$384$		0			0
$385$	10.8261	+	18.7514i	0.551750	+	0.955659i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.1379	−	17.5594i	−0.514012	−	0.890294i	−0.999868	−	0.0162556i	$-0.994825\pi$
$389$	−10.1379	−	17.5594i	−0.514012	−	0.890294i	0.485856	−	0.874039i	$-0.338508\pi$
$390$		0			0
$391$	−0.252412	−	0.145730i	−0.0127650	−	0.00736989i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			11.5676i			0.582030i
$396$		0			0
$397$			16.5809i			0.832169i	0.909326	+	0.416085i	$0.136598\pi$
$397$			16.5809i			0.832169i	−0.909326	+	0.416085i	$0.863402\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.1229	−	10.4632i	−0.905013	−	0.522510i	−0.0261899	−	0.999657i	$-0.508337\pi$
$401$	−18.1229	−	10.4632i	−0.905013	−	0.522510i	−0.878823	+	0.477147i	$0.841671\pi$
$402$		0			0
$403$	−5.07439	−	8.78910i	−0.252773	−	0.437816i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	23.4137	+	40.5537i	1.16057	+	2.01017i
$408$		0			0
$409$	−8.62868	+	14.9453i	−0.426661	+	0.738998i	−0.996574	−	0.0827069i	$-0.973643\pi$
$409$	−8.62868	+	14.9453i	−0.426661	+	0.738998i	0.569913	+	0.821705i	$0.306977\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−54.0088			−2.65760
$414$		0			0
$415$			2.71396i			0.133223i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.4910	−	7.21167i	−0.610224	−	0.352313i	0.162829	−	0.986654i	$-0.447938\pi$
$419$	−12.4910	−	7.21167i	−0.610224	−	0.352313i	−0.773053	+	0.634341i	$0.781271\pi$
$420$		0			0
$421$	−23.2632	+	13.4310i	−1.13378	+	0.654588i	−0.944883	−	0.327409i	$-0.893825\pi$
$421$	−23.2632	+	13.4310i	−1.13378	+	0.654588i	−0.188897	+	0.981997i	$0.560491\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.07106	+	0.618377i	−0.0519541	+	0.0299957i
$426$		0			0
$427$	16.6186	−	28.7843i	0.804231	−	1.39297i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.6429			0.608988			0.304494	−	0.952514i	$-0.401513\pi$
$431$	12.6429			0.608988			0.304494	+	0.952514i	$0.401513\pi$
$432$		0			0
$433$	−28.3673			−1.36324			−0.681622	−	0.731705i	$-0.738725\pi$
$433$	−28.3673			−1.36324			−0.681622	+	0.731705i	$0.738725\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.513656	+	0.889679i	−0.0245715	+	0.0425591i
$438$		0			0
$439$	−1.62013	+	0.935382i	−0.0773246	+	0.0446434i	−0.538164	−	0.842840i	$-0.680882\pi$
$439$	−1.62013	+	0.935382i	−0.0773246	+	0.0446434i	0.460839	+	0.887484i	$0.347548\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	32.9508	−	19.0242i	1.56554	−	0.903866i	0.568863	−	0.822433i	$-0.307384\pi$
$443$	32.9508	−	19.0242i	1.56554	−	0.903866i	0.996679	−	0.0814332i	$-0.0259497\pi$
$444$		0			0
$445$	−11.1995	−	6.46606i	−0.530909	−	0.306521i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			4.53199i			0.213878i	0.994266	+	0.106939i	$0.0341049\pi$
$449$			4.53199i			0.213878i	−0.994266	+	0.106939i	$0.965895\pi$
$450$		0			0
$451$	34.5632			1.62752
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.95464	−	8.58169i	0.232277	−	0.402316i
$456$		0			0
$457$	6.38333	+	11.0562i	0.298599	+	0.517189i	0.975816	−	0.218595i	$-0.0701472\pi$
$457$	6.38333	+	11.0562i	0.298599	+	0.517189i	−0.677216	+	0.735784i	$0.736814\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.8836	+	30.9753i	0.832923	+	1.44266i	0.895710	+	0.444638i	$0.146668\pi$
$461$	17.8836	+	30.9753i	0.832923	+	1.44266i	−0.0627876	+	0.998027i	$0.519999\pi$
$462$		0			0
$463$	−27.6653	−	15.9726i	−1.28572	−	0.742309i	−0.307829	−	0.951442i	$-0.599602\pi$
$463$	−27.6653	−	15.9726i	−1.28572	−	0.742309i	−0.977887	+	0.209133i	$0.932936\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			0.817062i			0.0378091i	0.999821	+	0.0189046i	$0.00601787\pi$
$467$			0.817062i			0.0378091i	−0.999821	+	0.0189046i	$0.993982\pi$
$468$		0			0
$469$		−	68.9679i		−	3.18464i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−21.4676	−	12.3944i	−0.987083	−	0.569893i
$474$		0			0
$475$	2.17960	+	3.77518i	0.100007	+	0.173217i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.3720	+	30.0891i	0.793746	+	1.37481i	0.923633	+	0.383279i	$0.125205\pi$
$479$	17.3720	+	30.0891i	0.793746	+	1.37481i	−0.129887	+	0.991529i	$0.541461\pi$
$480$		0			0
$481$	10.7154	−	18.5596i	0.488581	−	0.846246i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.63282			−0.0741425
$486$		0			0
$487$			39.8130i			1.80410i	0.431630	+	0.902051i	$0.357939\pi$
$487$			39.8130i			1.80410i	−0.431630	+	0.902051i	$0.642061\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−21.9098	−	12.6496i	−0.988775	−	0.570870i	−0.0838674	−	0.996477i	$-0.526727\pi$
$491$	−21.9098	−	12.6496i	−0.988775	−	0.570870i	−0.904908	+	0.425607i	$0.860061\pi$
$492$		0			0
$493$	−6.08812	+	3.51498i	−0.274195	+	0.158307i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.38195	−	1.95257i	0.151701	−	0.0875848i
$498$		0			0
$499$	−5.75232	+	9.96331i	−0.257509	+	0.446019i	−0.965574	−	0.260128i	$-0.916235\pi$
$499$	−5.75232	+	9.96331i	−0.257509	+	0.446019i	0.708065	+	0.706147i	$0.249568\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16.2502			−0.724561			−0.362281	−	0.932069i	$-0.618002\pi$
$503$	−16.2502			−0.724561			−0.362281	+	0.932069i	$0.618002\pi$
$504$		0			0
$505$	−2.57742			−0.114694
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.87628	−	4.98186i	0.127489	−	0.220817i	−0.795214	−	0.606328i	$-0.792642\pi$
$509$	2.87628	−	4.98186i	0.127489	−	0.220817i	0.922703	+	0.385512i	$0.125975\pi$
$510$		0			0
$511$	42.4091	−	24.4849i	1.87607	−	1.08315i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.62567	+	4.98003i	−0.380093	+	0.219447i
$516$		0			0
$517$	−8.66391	−	5.00211i	−0.381038	−	0.219993i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			30.4544i			1.33423i	0.744953	+	0.667117i	$0.232472\pi$
$521$			30.4544i			1.33423i	−0.744953	+	0.667117i	$0.767528\pi$
$522$		0			0
$523$	−18.7457			−0.819694			−0.409847	−	0.912154i	$-0.634418\pi$
$523$	−18.7457			−0.819694			−0.409847	+	0.912154i	$0.634418\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.90474	+	5.03116i	−0.126533	+	0.219161i
$528$		0			0
$529$	11.4722	+	19.8705i	0.498793	+	0.863934i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.90903	−	13.6988i	−0.342578	−	0.593363i
$534$		0			0
$535$	6.30815	+	3.64201i	0.272725	+	0.157458i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			66.2623i			2.85412i
$540$		0			0
$541$			28.8219i			1.23915i	0.784938	+	0.619575i	$0.212695\pi$
$541$			28.8219i			1.23915i	−0.784938	+	0.619575i	$0.787305\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.22140	−	1.28253i	−0.0951543	−	0.0549373i
$546$		0			0
$547$	8.43142	+	14.6036i	0.360502	+	0.624407i	0.988043	−	0.154176i	$-0.0492722\pi$
$547$	8.43142	+	14.6036i	0.360502	+	0.624407i	−0.627542	+	0.778583i	$0.715939\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.3893	+	21.4588i	0.527801	+	0.914177i
$552$		0			0
$553$	−26.5275	+	45.9470i	−1.12806	+	1.95387i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20.1743			0.854811			0.427406	−	0.904060i	$-0.359428\pi$
$557$	20.1743			0.854811			0.427406	+	0.904060i	$0.359428\pi$
$558$		0			0
$559$			11.3447i			0.479830i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.22595	+	3.01721i	0.220248	+	0.127160i	0.606065	−	0.795415i	$-0.292747\pi$
$563$	5.22595	+	3.01721i	0.220248	+	0.127160i	−0.385817	+	0.922575i	$0.626080\pi$
$564$		0			0
$565$	−0.311444	+	0.179812i	−0.0131025	+	0.00756475i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.8303	+	7.40759i	−0.537875	+	0.310543i	−0.744217	−	0.667937i	$-0.767177\pi$
$569$	−12.8303	+	7.40759i	−0.537875	+	0.310543i	0.206342	+	0.978480i	$0.433844\pi$
$570$		0			0
$571$	3.96582	−	6.86900i	0.165964	−	0.287459i	−0.771033	−	0.636795i	$-0.780260\pi$
$571$	3.96582	−	6.86900i	0.165964	−	0.287459i	0.936997	+	0.349337i	$0.113593\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.235665			−0.00982793
$576$		0			0
$577$	−9.19579			−0.382826			−0.191413	−	0.981510i	$-0.561307\pi$
$577$	−9.19579			−0.382826			−0.191413	+	0.981510i	$0.561307\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.22379	+	10.7799i	−0.258207	+	0.447227i
$582$		0			0
$583$	−32.3811	+	18.6952i	−1.34109	+	0.774278i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.45327	−	3.72580i	0.266355	−	0.153780i	−0.360875	−	0.932614i	$-0.617522\pi$
$587$	6.45327	−	3.72580i	0.266355	−	0.153780i	0.627230	+	0.778834i	$0.284189\pi$
$588$		0			0
$589$	17.7334	+	10.2384i	0.730692	+	0.421865i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			5.98713i			0.245862i	0.992415	+	0.122931i	$0.0392294\pi$
$593$			5.98713i			0.245862i	−0.992415	+	0.122931i	$0.960771\pi$
$594$		0			0
$595$	−5.67239			−0.232545
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.3740	+	31.8246i	−0.750740	+	1.30032i	0.196725	+	0.980459i	$0.436969\pi$
$599$	−18.3740	+	31.8246i	−0.750740	+	1.30032i	−0.947465	+	0.319861i	$0.896364\pi$
$600$		0			0
$601$	−5.67611	−	9.83130i	−0.231533	−	0.401027i	0.726726	−	0.686927i	$-0.241041\pi$
$601$	−5.67611	−	9.83130i	−0.231533	−	0.401027i	−0.958260	+	0.285900i	$0.907708\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.64322	−	9.77435i	−0.229430	−	0.397384i
$606$		0			0
$607$	5.06707	+	2.92548i	0.205666	+	0.118741i	0.599296	−	0.800528i	$-0.295447\pi$
$607$	5.06707	+	2.92548i	0.205666	+	0.118741i	−0.393630	+	0.919269i	$0.628781\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			4.57849i			0.185226i
$612$		0			0
$613$			18.4132i			0.743703i	0.928292	+	0.371851i	$0.121277\pi$
$613$			18.4132i			0.743703i	−0.928292	+	0.371851i	$0.878723\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.12123	−	4.11144i	−0.286690	−	0.165520i	0.349758	−	0.936840i	$-0.386264\pi$
$617$	−7.12123	−	4.11144i	−0.286690	−	0.165520i	−0.636448	+	0.771320i	$0.719597\pi$
$618$		0			0
$619$	22.2663	+	38.5663i	0.894957	+	1.55011i	0.833858	+	0.551979i	$0.186127\pi$
$619$	22.2663	+	38.5663i	0.894957	+	1.55011i	0.0610991	+	0.998132i	$0.480539\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−29.6567	−	51.3668i	−1.18817	−	2.05797i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.2677			−0.489145
$630$		0			0
$631$			2.02785i			0.0807273i	0.999185	+	0.0403637i	$0.0128516\pi$
$631$			2.02785i			0.0807273i	−0.999185	+	0.0403637i	$0.987148\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.81958	−	4.51464i	−0.310311	−	0.179158i
$636$		0			0
$637$	26.2625	−	15.1627i	1.04056	−	0.600767i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	22.2376	−	12.8389i	0.878332	−	0.507105i	0.00822364	−	0.999966i	$-0.497382\pi$
$641$	22.2376	−	12.8389i	0.878332	−	0.507105i	0.870108	+	0.492861i	$0.164049\pi$
$642$		0			0
$643$	−3.54072	+	6.13270i	−0.139632	+	0.241850i	−0.927357	−	0.374177i	$-0.877925\pi$
$643$	−3.54072	+	6.13270i	−0.139632	+	0.241850i	0.787725	+	0.616027i	$0.211259\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	48.6780			1.91373			0.956865	−	0.290533i	$-0.0938325\pi$
$647$	48.6780			1.91373			0.956865	+	0.290533i	$0.0938325\pi$
$648$		0			0
$649$	55.5907			2.18213
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.32043	+	2.28705i	−0.0516724	+	0.0894993i	−0.890705	−	0.454582i	$-0.849789\pi$
$653$	−1.32043	+	2.28705i	−0.0516724	+	0.0894993i	0.839032	+	0.544082i	$0.183122\pi$
$654$		0			0
$655$	−0.986098	+	0.569324i	−0.0385300	+	0.0222453i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.04662	−	1.75896i	0.118679	−	0.0685195i	−0.439485	−	0.898250i	$-0.644839\pi$
$659$	3.04662	−	1.75896i	0.118679	−	0.0685195i	0.558165	+	0.829730i	$0.311506\pi$
$660$		0			0
$661$	−3.44771	−	1.99053i	−0.134100	−	0.0774228i	0.431449	−	0.902137i	$-0.358002\pi$
$661$	−3.44771	−	1.99053i	−0.134100	−	0.0774228i	−0.565549	+	0.824715i	$0.691336\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			19.9935i			0.775315i
$666$		0			0
$667$	−1.33957			−0.0518683
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17.1054	+	29.6274i	−0.660346	+	1.14375i
$672$		0			0
$673$	8.00949	+	13.8728i	0.308743	+	0.534759i	0.978088	−	0.208193i	$-0.0667583\pi$
$673$	8.00949	+	13.8728i	0.308743	+	0.534759i	−0.669345	+	0.742952i	$0.733425\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.225619	−	0.390784i	−0.00867125	−	0.0150190i	0.861657	−	0.507491i	$-0.169427\pi$
$677$	−0.225619	−	0.390784i	−0.00867125	−	0.0150190i	−0.870328	+	0.492472i	$0.836094\pi$
$678$		0			0
$679$	−6.48561	−	3.74447i	−0.248895	−	0.143700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	23.3047i		−	0.891728i	−0.895101	−	0.445864i	$-0.852896\pi$
$683$		−	23.3047i		−	0.891728i	0.895101	−	0.445864i	$-0.147104\pi$
$684$		0			0
$685$		−	3.91744i		−	0.149678i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	14.8194	+	8.55598i	0.564574	+	0.325957i
$690$		0			0
$691$	4.50424	+	7.80158i	0.171349	+	0.296786i	0.938892	−	0.344212i	$-0.111854\pi$
$691$	4.50424	+	7.80158i	0.171349	+	0.296786i	−0.767542	+	0.640998i	$0.778521\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.70435	+	9.88022i	0.216378	+	0.374778i
$696$		0			0
$697$	−4.52738	+	7.84166i	−0.171487	+	0.297024i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10.3925			0.392520			0.196260	−	0.980552i	$-0.437120\pi$
$701$	10.3925			0.392520			0.196260	+	0.980552i	$0.437120\pi$
$702$		0			0
$703$			43.2400i			1.63083i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.2376	−	5.91068i	−0.385025	−	0.222294i
$708$		0			0
$709$	22.1577	−	12.7927i	0.832149	−	0.480442i	−0.0224386	−	0.999748i	$-0.507143\pi$
$709$	22.1577	−	12.7927i	0.832149	−	0.480442i	0.854588	+	0.519307i	$0.173810\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.958696	+	0.553503i	−0.0359035	+	0.0207289i
$714$		0			0
$715$	−5.09976	+	8.83305i	−0.190720	+	0.330337i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	36.4552			1.35955			0.679774	−	0.733422i	$-0.262078\pi$
$719$	36.4552			1.35955			0.679774	+	0.733422i	$0.262078\pi$
$720$		0			0
$721$	−45.6820			−1.70129
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.84210	+	4.92266i	−0.105553	+	0.182823i
$726$		0			0
$727$	−21.4834	+	12.4034i	−0.796774	+	0.460018i	−0.842342	−	0.538943i	$-0.818824\pi$
$727$	−21.4834	+	12.4034i	−0.796774	+	0.460018i	0.0455677	+	0.998961i	$0.485490\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.62403	−	3.24703i	0.208012	−	0.120096i
$732$		0			0
$733$	−27.3108	−	15.7679i	−1.00875	−	0.582400i	−0.0979219	−	0.995194i	$-0.531220\pi$
$733$	−27.3108	−	15.7679i	−1.00875	−	0.582400i	−0.910824	+	0.412794i	$0.864553\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			70.9880i			2.61488i
$738$		0			0
$739$	−20.1619			−0.741668			−0.370834	−	0.928699i	$-0.620928\pi$
$739$	−20.1619			−0.741668			−0.370834	+	0.928699i	$0.620928\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.498520	+	0.863461i	−0.0182889	+	0.0316773i	−0.875025	−	0.484078i	$-0.839155\pi$
$743$	−0.498520	+	0.863461i	−0.0182889	+	0.0316773i	0.856736	+	0.515755i	$0.172489\pi$
$744$		0			0
$745$	9.86802	+	17.0919i	0.361536	+	0.626199i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	16.7041	+	28.9324i	0.610356	+	1.05717i
$750$		0			0
$751$	37.7503	+	21.7951i	1.37753	+	0.795316i	0.991861	−	0.127323i	$-0.0406385\pi$
$751$	37.7503	+	21.7951i	1.37753	+	0.795316i	0.385666	+	0.922639i	$0.373972\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			6.14737i			0.223726i
$756$		0			0
$757$			5.54083i			0.201385i	0.994918	+	0.100692i	$0.0321058\pi$
$757$			5.54083i			0.201385i	−0.994918	+	0.100692i	$0.967894\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	23.1798	+	13.3829i	0.840267	+	0.485128i	0.857355	−	0.514726i	$-0.172106\pi$
$761$	23.1798	+	13.3829i	0.840267	+	0.485128i	−0.0170880	+	0.999854i	$0.505440\pi$
$762$		0			0
$763$	−5.88232	−	10.1885i	−0.212954	−	0.368847i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.7207	−	22.0329i	−0.459318	−	0.795562i
$768$		0			0
$769$	−1.93609	+	3.35341i	−0.0698174	+	0.120927i	−0.898821	−	0.438316i	$-0.855575\pi$
$769$	−1.93609	+	3.35341i	−0.0698174	+	0.120927i	0.829003	+	0.559244i	$0.188908\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	26.7361			0.961632			0.480816	−	0.876821i	$-0.340340\pi$
$773$	26.7361			0.961632			0.480816	+	0.876821i	$0.340340\pi$
$774$		0			0
$775$			4.69737i			0.168734i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	27.6396	+	15.9577i	0.990290	+	0.571744i
$780$		0			0
$781$	−3.48101	+	2.00976i	−0.124560	+	0.0719150i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.339511	−	0.196017i	0.0121177	−	0.00699614i
$786$		0			0
$787$	−4.82568	+	8.35833i	−0.172017	+	0.297942i	−0.939125	−	0.343576i	$-0.888362\pi$
$787$	−4.82568	+	8.35833i	−0.172017	+	0.297942i	0.767108	+	0.641518i	$0.221695\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.64942			−0.0586466
$792$		0			0
$793$	15.6568			0.555988
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.7189	+	22.0298i	−0.450528	+	0.780337i	−0.998419	−	0.0562130i	$-0.982097\pi$
$797$	−12.7189	+	22.0298i	−0.450528	+	0.780337i	0.547891	+	0.836550i	$0.315431\pi$
$798$		0			0
$799$	2.26974	−	1.31044i	0.0802978	−	0.0463600i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−43.6512	+	25.2021i	−1.54042	+	0.889361i
$804$		0			0
$805$	−0.936072	−	0.540441i	−0.0329922	−	0.0190481i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			10.7268i			0.377134i	0.982060	+	0.188567i	$0.0603843\pi$
$809$			10.7268i			0.377134i	−0.982060	+	0.188567i	$0.939616\pi$
$810$		0			0
$811$	−5.16201			−0.181263			−0.0906314	−	0.995885i	$-0.528889\pi$
$811$	−5.16201			−0.181263			−0.0906314	+	0.995885i	$0.528889\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.3153	−	21.3307i	0.431386	−	0.747183i
$816$		0			0
$817$	−11.4449	−	19.8231i	−0.400405	−	0.693521i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.71181	−	16.8213i	−0.338944	−	0.587069i	0.645290	−	0.763938i	$-0.276737\pi$
$821$	−9.71181	−	16.8213i	−0.338944	−	0.587069i	−0.984234	+	0.176869i	$0.943403\pi$
$822$		0			0
$823$	−29.5617	−	17.0675i	−1.03046	−	0.594935i	−0.113342	−	0.993556i	$-0.536155\pi$
$823$	−29.5617	−	17.0675i	−1.03046	−	0.594935i	−0.917116	+	0.398621i	$0.869489\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	21.1098i		−	0.734061i	−0.930209	−	0.367031i	$-0.880374\pi$
$827$		−	21.1098i		−	0.734061i	0.930209	−	0.367031i	$-0.119626\pi$
$828$		0			0
$829$		−	2.27907i		−	0.0791554i	−0.999216	−	0.0395777i	$-0.987399\pi$
$829$		−	2.27907i		−	0.0791554i	0.999216	−	0.0395777i	$-0.0126013\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−15.0335	−	8.67959i	−0.520880	−	0.300730i
$834$		0			0
$835$	1.50486	+	2.60649i	0.0520778	+	0.0902014i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−20.2338	−	35.0460i	−0.698548	−	1.20992i	−0.968970	−	0.247179i	$-0.920496\pi$
$839$	−20.2338	−	35.0460i	−0.698548	−	1.20992i	0.270421	−	0.962742i	$-0.412837\pi$
$840$		0			0
$841$	−1.65503	+	2.86660i	−0.0570700	+	0.0988481i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.33213			−0.286634
$846$		0			0
$847$		−	51.7654i		−	1.77868i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.02444	−	1.16881i	−0.0693971	−	0.0400664i
$852$		0			0
$853$	14.2703	−	8.23893i	0.488604	−	0.282096i	−0.235391	−	0.971901i	$-0.575637\pi$
$853$	14.2703	−	8.23893i	0.488604	−	0.282096i	0.723995	+	0.689805i	$0.242304\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	41.2192	−	23.7979i	1.40802	−	0.812921i	0.412824	−	0.910811i	$-0.364543\pi$
$857$	41.2192	−	23.7979i	1.40802	−	0.812921i	0.995197	+	0.0978896i	$0.0312092\pi$
$858$		0			0
$859$	15.2357	−	26.3890i	0.519836	−	0.900382i	−0.479898	−	0.877324i	$-0.659326\pi$
$859$	15.2357	−	26.3890i	0.519836	−	0.900382i	0.999734	−	0.0230580i	$-0.00734025\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	30.4421			1.03626			0.518131	−	0.855301i	$-0.326628\pi$
$863$	30.4421			1.03626			0.518131	+	0.855301i	$0.326628\pi$
$864$		0			0
$865$	9.97631			0.339205
$866$		0			0
$867$		0			0
$868$		0			0
$869$	27.3045	−	47.2928i	0.926242	−	1.60430i
$870$		0			0
$871$	28.1355	−	16.2440i	0.953335	−	0.550408i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.97204	+	2.29326i	−0.134279	+	0.0775262i
$876$		0			0
$877$	22.1778	+	12.8044i	0.748892	+	0.432373i	0.825293	−	0.564704i	$-0.191010\pi$
$877$	22.1778	+	12.8044i	0.748892	+	0.432373i	−0.0764015	+	0.997077i	$0.524343\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	43.3621i		−	1.46091i	−0.682963	−	0.730453i	$-0.739309\pi$
$881$		−	43.3621i		−	1.46091i	0.682963	−	0.730453i	$-0.260691\pi$
$882$		0			0
$883$	15.4645			0.520421			0.260210	−	0.965552i	$-0.416208\pi$
$883$	15.4645			0.520421			0.260210	+	0.965552i	$0.416208\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.66974	+	2.89207i	−0.0560642	+	0.0971061i	−0.892695	−	0.450661i	$-0.851188\pi$
$887$	−1.66974	+	2.89207i	−0.0560642	+	0.0971061i	0.836631	+	0.547767i	$0.184522\pi$
$888$		0			0
$889$	−20.7064	−	35.8646i	−0.694471	−	1.20286i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.61891	−	8.00019i	−0.154566	−	0.267716i
$894$		0			0
$895$	−9.66482	−	5.57999i	−0.323059	−	0.186518i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			26.7007i			0.890519i
$900$		0			0
$901$		−	9.79544i		−	0.326333i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.16044	+	2.40203i	0.138298	+	0.0798463i
$906$		0			0
$907$	−1.40422	−	2.43218i	−0.0466263	−	0.0807592i	0.841770	−	0.539836i	$-0.181514\pi$
$907$	−1.40422	−	2.43218i	−0.0466263	−	0.0807592i	−0.888397	+	0.459077i	$0.848180\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−15.6065	−	27.0312i	−0.517066	−	0.895585i	−0.999804	−	0.0198195i	$-0.993691\pi$
$911$	−15.6065	−	27.0312i	−0.517066	−	0.895585i	0.482738	−	0.875765i	$-0.339642\pi$
$912$		0			0
$913$	6.40609	−	11.0957i	0.212011	−	0.367213i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−5.22242			−0.172460
$918$		0			0
$919$		−	41.3010i		−	1.36240i	−0.732100	−	0.681198i	$-0.761459\pi$
$919$		−	41.3010i		−	1.36240i	0.732100	−	0.681198i	$-0.238541\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.59311	+	0.919780i	0.0524377	+	0.0302749i
$924$		0			0
$925$	−8.59033	+	4.95963i	−0.282448	+	0.163072i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.46762	+	1.42468i	−0.0809599	+	0.0467422i	−0.539933	−	0.841708i	$-0.681551\pi$
$929$	−2.46762	+	1.42468i	−0.0809599	+	0.0467422i	0.458974	+	0.888450i	$0.348217\pi$
$930$		0			0
$931$	−30.5931	+	52.9887i	−1.00265	+	1.73664i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.83853			0.190940
$936$		0			0
$937$	33.6983			1.10088			0.550438	−	0.834876i	$-0.314461\pi$
$937$	33.6983			1.10088			0.550438	+	0.834876i	$0.314461\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.1325	+	21.0141i	−0.395507	+	0.685039i	−0.993166	−	0.116712i	$-0.962765\pi$
$941$	−12.1325	+	21.0141i	−0.395507	+	0.685039i	0.597659	+	0.801751i	$0.296098\pi$
$942$		0			0
$943$	−1.49424	+	0.862700i	−0.0486592	+	0.0280934i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−31.5042	+	18.1890i	−1.02375	+	0.591062i	−0.915188	−	0.403028i	$-0.867958\pi$
$947$	−31.5042	+	18.1890i	−1.02375	+	0.591062i	−0.108561	+	0.994090i	$0.534624\pi$
$948$		0			0
$949$	19.9772	+	11.5339i	0.648489	+	0.374405i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	26.2586i		−	0.850598i	−0.905053	−	0.425299i	$-0.860169\pi$
$953$		−	26.2586i		−	0.850598i	0.905053	−	0.425299i	$-0.139831\pi$
$954$		0			0
$955$	0.287325			0.00929760
$956$		0			0
$957$		0			0
$958$		0			0
$959$	8.98369	−	15.5602i	0.290099	−	0.502466i
$960$		0			0
$961$	−4.46738	−	7.73773i	−0.144109	−	0.249604i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.19503	−	2.06986i	−0.0384695	−	0.0666311i
$966$		0			0
$967$	42.3417	+	24.4460i	1.36162	+	0.786130i	0.989839	−	0.142192i	$-0.0454152\pi$
$967$	42.3417	+	24.4460i	1.36162	+	0.786130i	0.371777	+	0.928322i	$0.378749\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			23.9838i			0.769678i	0.922984	+	0.384839i	$0.125743\pi$
$971$			23.9838i			0.769678i	−0.922984	+	0.384839i	$0.874257\pi$
$972$		0			0
$973$			52.3261i			1.67750i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	23.1131	+	13.3444i	0.739455	+	0.426924i	0.821871	−	0.569674i	$-0.192930\pi$
$977$	23.1131	+	13.3444i	0.739455	+	0.426924i	−0.0824164	+	0.996598i	$0.526264\pi$
$978$		0			0
$979$	30.5253	+	52.8714i	0.975593	+	1.68978i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−21.9816	−	38.0733i	−0.701104	−	1.21435i	−0.968079	−	0.250645i	$-0.919357\pi$
$983$	−21.9816	−	38.0733i	−0.701104	−	1.21435i	0.266975	−	0.963703i	$-0.413976\pi$
$984$		0			0
$985$	5.40887	−	9.36843i	0.172341	−	0.298503i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.23745			0.0393488
$990$		0			0
$991$		−	43.4855i		−	1.38136i	−0.723159	−	0.690682i	$-0.757311\pi$
$991$		−	43.4855i		−	1.38136i	0.723159	−	0.690682i	$-0.242689\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4.42142	−	2.55271i	−0.140168	−	0.0809262i
$996$		0			0
$997$	−5.50039	+	3.17565i	−0.174199	+	0.100574i	−0.584564	−	0.811347i	$-0.698735\pi$
$997$	−5.50039	+	3.17565i	−0.174199	+	0.100574i	0.410365	+	0.911921i	$0.365401\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.cc.b.3311.23		48
3.2	odd	2		1440.2.cc.a.1391.1		48
4.3	odd	2		1080.2.bm.b.611.12		48
8.3	odd	2		4320.2.cc.a.3311.2		48
8.5	even	2		1080.2.bm.a.611.20		48
9.2	odd	6		4320.2.cc.a.1871.2		48
9.7	even	3		1440.2.cc.b.911.1		48
12.11	even	2		360.2.bm.a.131.13	yes	48
24.5	odd	2		360.2.bm.b.131.5	yes	48
24.11	even	2		1440.2.cc.b.1391.1		48
36.7	odd	6		360.2.bm.b.11.5	yes	48
36.11	even	6		1080.2.bm.a.251.20		48
72.11	even	6	inner	4320.2.cc.b.1871.23		48
72.29	odd	6		1080.2.bm.b.251.12		48
72.43	odd	6		1440.2.cc.a.911.1		48
72.61	even	6		360.2.bm.a.11.13	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bm.a.11.13	✓	48	72.61	even	6
360.2.bm.a.131.13	yes	48	12.11	even	2
360.2.bm.b.11.5	yes	48	36.7	odd	6
360.2.bm.b.131.5	yes	48	24.5	odd	2
1080.2.bm.a.251.20		48	36.11	even	6
1080.2.bm.a.611.20		48	8.5	even	2
1080.2.bm.b.251.12		48	72.29	odd	6
1080.2.bm.b.611.12		48	4.3	odd	2
1440.2.cc.a.911.1		48	72.43	odd	6
1440.2.cc.a.1391.1		48	3.2	odd	2
1440.2.cc.b.911.1		48	9.7	even	3
1440.2.cc.b.1391.1		48	24.11	even	2
4320.2.cc.a.1871.2		48	9.2	odd	6
4320.2.cc.a.3311.2		48	8.3	odd	2
4320.2.cc.b.1871.23		48	72.11	even	6	inner
4320.2.cc.b.3311.23		48	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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.cc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(3.97204 - 2.29326i) q^{7} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{5} +(3.97204 - 2.29326i) q^{7} +(-4.08838 + 2.36043i) q^{11} +(1.87107 + 1.08026i) q^{13} -1.23675i q^{17} -4.35920 q^{19} +(0.117833 - 0.204092i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-2.84210 - 4.92266i) q^{29} +(-4.06804 - 2.34868i) q^{31} -4.58651i q^{35} -9.91926i q^{37} +(-6.34052 - 3.66070i) q^{41} +(2.62545 + 4.54741i) q^{43} +(1.05958 + 1.83524i) q^{47} +(7.01804 - 12.1556i) q^{49} +7.92028 q^{53} +4.72085i q^{55} +(-10.1979 - 5.88778i) q^{59} +(6.27586 - 3.62337i) q^{61} +(1.87107 - 1.08026i) q^{65} +(7.51856 - 13.0225i) q^{67} +0.851441 q^{71} +10.6769 q^{73} +(-10.8261 + 18.7514i) q^{77} +(-10.0179 + 5.78381i) q^{79} +(-2.35035 + 1.35698i) q^{83} +(-1.07106 - 0.618377i) q^{85} -12.9321i q^{89} +9.90928 q^{91} +(-2.17960 + 3.77518i) q^{95} +(-0.816409 - 1.41406i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 24 q^{5}+O(q^{10}) \) 48 * q + 24 * q^5	\( 48 q + 24 q^{5} - 24 q^{25} - 12 q^{41} + 12 q^{47} + 24 q^{49} - 36 q^{59} + 12 q^{61} - 60 q^{83}+O(q^{100}) \) 48 * q + 24 * q^5 - 24 * q^25 - 12 * q^41 + 12 * q^47 + 24 * q^49 - 36 * q^59 + 12 * q^61 - 60 * q^83

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