LMFDB - Embedded newform 4032.2.v.e.3599.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("4032.1583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3599.3
Character	$\chi$	$=$	4032.3599
Dual form			4032.2.v.e.1583.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.51504 + 2.51504i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(-2.51504 + 2.51504i) q^{5} -1.00000 q^{7} +(0.984548 + 0.984548i) q^{11} +(-3.26086 + 3.26086i) q^{13} -5.50927i q^{17} +(1.21444 + 1.21444i) q^{19} -8.62021i q^{23} -7.65081i q^{25} +(-2.04663 - 2.04663i) q^{29} +0.164437i q^{31} +(2.51504 - 2.51504i) q^{35} +(-8.31105 - 8.31105i) q^{37} +9.56578 q^{41} +(-5.01867 + 5.01867i) q^{43} -3.37837 q^{47} +1.00000 q^{49} +(3.72717 - 3.72717i) q^{53} -4.95235 q^{55} +(10.0781 + 10.0781i) q^{59} +(-1.07042 + 1.07042i) q^{61} -16.4023i q^{65} +(3.12073 + 3.12073i) q^{67} +7.66257i q^{71} +8.40588i q^{73} +(-0.984548 - 0.984548i) q^{77} +13.8712i q^{79} +(7.31214 - 7.31214i) q^{83} +(13.8560 + 13.8560i) q^{85} +7.49158 q^{89} +(3.26086 - 3.26086i) q^{91} -6.10874 q^{95} -7.84085 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 40 q^{7}+O(q^{10}) $ 40 * q - 40 * q^7	$ 40 q - 40 q^{7} - 24 q^{13} + 32 q^{19} - 8 q^{37} - 32 q^{43} + 40 q^{49} - 48 q^{55} - 24 q^{61} + 64 q^{85} + 24 q^{91} + 64 q^{97}+O(q^{100}) $ 40 * q - 40 * q^7 - 24 * q^13 + 32 * q^19 - 8 * q^37 - 32 * q^43 + 40 * q^49 - 48 * q^55 - 24 * q^61 + 64 * q^85 + 24 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$5$	−2.51504	+	2.51504i	−1.12476	+	1.12476i	−0.133742	+	0.991016i	$0.542699\pi$
$5$	−2.51504	+	2.51504i	−1.12476	+	1.12476i	−0.991016	+	0.133742i	$0.957301\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.984548	+	0.984548i	0.296852	+	0.296852i	0.839780	−	0.542927i	$-0.182684\pi$
$11$	0.984548	+	0.984548i	0.296852	+	0.296852i	−0.542927	+	0.839780i	$0.682684\pi$
$12$		0			0
$13$	−3.26086	+	3.26086i	−0.904399	+	0.904399i	−0.995813	−	0.0914142i	$-0.970861\pi$
$13$	−3.26086	+	3.26086i	−0.904399	+	0.904399i	0.0914142	+	0.995813i	$0.470861\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.50927i		−	1.33619i	−0.744074	−	0.668097i	$-0.767109\pi$
$17$		−	5.50927i		−	1.33619i	0.744074	−	0.668097i	$-0.232891\pi$
$18$		0			0
$19$	1.21444	+	1.21444i	0.278613	+	0.278613i	0.832555	−	0.553942i	$-0.186877\pi$
$19$	1.21444	+	1.21444i	0.278613	+	0.278613i	−0.553942	+	0.832555i	$0.686877\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.62021i		−	1.79744i	−0.438526	−	0.898719i	$-0.644499\pi$
$23$		−	8.62021i		−	1.79744i	0.438526	−	0.898719i	$-0.355501\pi$
$24$		0			0
$25$		−	7.65081i		−	1.53016i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.04663	−	2.04663i	−0.380050	−	0.380050i	0.491070	−	0.871120i	$-0.336606\pi$
$29$	−2.04663	−	2.04663i	−0.380050	−	0.380050i	−0.871120	+	0.491070i	$0.836606\pi$
$30$		0			0
$31$			0.164437i			0.0295337i	0.999891	+	0.0147669i	$0.00470061\pi$
$31$			0.164437i			0.0295337i	−0.999891	+	0.0147669i	$0.995299\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.51504	−	2.51504i	0.425119	−	0.425119i
$36$		0			0
$37$	−8.31105	−	8.31105i	−1.36633	−	1.36633i	−0.865611	−	0.500717i	$-0.833070\pi$
$37$	−8.31105	−	8.31105i	−1.36633	−	1.36633i	−0.500717	−	0.865611i	$-0.666930\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	9.56578			1.49392			0.746962	−	0.664867i	$-0.231512\pi$
$41$	9.56578			1.49392			0.746962	+	0.664867i	$0.231512\pi$
$42$		0			0
$43$	−5.01867	+	5.01867i	−0.765340	+	0.765340i	−0.977282	−	0.211942i	$-0.932021\pi$
$43$	−5.01867	+	5.01867i	−0.765340	+	0.765340i	0.211942	+	0.977282i	$0.432021\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.37837			−0.492786			−0.246393	−	0.969170i	$-0.579245\pi$
$47$	−3.37837			−0.492786			−0.246393	+	0.969170i	$0.579245\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.72717	−	3.72717i	0.511966	−	0.511966i	−0.403162	−	0.915129i	$-0.632089\pi$
$53$	3.72717	−	3.72717i	0.511966	−	0.511966i	0.915129	+	0.403162i	$0.132089\pi$
$54$		0			0
$55$	−4.95235			−0.667774
$56$		0			0
$57$		0			0
$58$		0			0
$59$	10.0781	+	10.0781i	1.31206	+	1.31206i	0.919896	+	0.392162i	$0.128273\pi$
$59$	10.0781	+	10.0781i	1.31206	+	1.31206i	0.392162	+	0.919896i	$0.371727\pi$
$60$		0			0
$61$	−1.07042	+	1.07042i	−0.137054	+	0.137054i	−0.772305	−	0.635252i	$-0.780896\pi$
$61$	−1.07042	+	1.07042i	−0.137054	+	0.137054i	0.635252	+	0.772305i	$0.280896\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	16.4023i		−	2.03446i
$66$		0			0
$67$	3.12073	+	3.12073i	0.381258	+	0.381258i	0.871555	−	0.490297i	$-0.163112\pi$
$67$	3.12073	+	3.12073i	0.381258	+	0.381258i	−0.490297	+	0.871555i	$0.663112\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.66257i			0.909380i	0.890650	+	0.454690i	$0.150250\pi$
$71$			7.66257i			0.909380i	−0.890650	+	0.454690i	$0.849750\pi$
$72$		0			0
$73$			8.40588i			0.983834i	0.870642	+	0.491917i	$0.163704\pi$
$73$			8.40588i			0.983834i	−0.870642	+	0.491917i	$0.836296\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.984548	−	0.984548i	−0.112200	−	0.112200i
$78$		0			0
$79$			13.8712i			1.56063i	0.625387	+	0.780315i	$0.284941\pi$
$79$			13.8712i			1.56063i	−0.625387	+	0.780315i	$0.715059\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.31214	−	7.31214i	0.802611	−	0.802611i	−0.180892	−	0.983503i	$-0.557898\pi$
$83$	7.31214	−	7.31214i	0.802611	−	0.802611i	0.983503	+	0.180892i	$0.0578984\pi$
$84$		0			0
$85$	13.8560	+	13.8560i	1.50290	+	1.50290i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.49158			0.794106			0.397053	−	0.917796i	$-0.370033\pi$
$89$	7.49158			0.794106			0.397053	+	0.917796i	$0.370033\pi$
$90$		0			0
$91$	3.26086	−	3.26086i	0.341831	−	0.341831i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.10874			−0.626744
$96$		0			0
$97$	−7.84085			−0.796118			−0.398059	−	0.917360i	$-0.630316\pi$
$97$	−7.84085			−0.796118			−0.398059	+	0.917360i	$0.630316\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.748257	−	0.748257i	0.0744543	−	0.0744543i	−0.668899	−	0.743353i	$-0.733234\pi$
$101$	0.748257	−	0.748257i	0.0744543	−	0.0744543i	0.743353	+	0.668899i	$0.233234\pi$
$102$		0			0
$103$	11.8161			1.16428			0.582140	−	0.813089i	$-0.302216\pi$
$103$	11.8161			1.16428			0.582140	+	0.813089i	$0.302216\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.41189	+	4.41189i	0.426514	+	0.426514i	0.887439	−	0.460925i	$-0.152482\pi$
$107$	4.41189	+	4.41189i	0.426514	+	0.426514i	−0.460925	+	0.887439i	$0.652482\pi$
$108$		0			0
$109$	−0.128743	+	0.128743i	−0.0123314	+	0.0123314i	−0.713246	−	0.700914i	$-0.752776\pi$
$109$	−0.128743	+	0.128743i	−0.0123314	+	0.0123314i	0.700914	+	0.713246i	$0.252776\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	14.3042i		−	1.34563i	−0.739811	−	0.672815i	$-0.765085\pi$
$113$		−	14.3042i		−	1.34563i	0.739811	−	0.672815i	$-0.234915\pi$
$114$		0			0
$115$	21.6801	+	21.6801i	2.02168	+	2.02168i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			5.50927i			0.505034i
$120$		0			0
$121$		−	9.06133i		−	0.823757i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.66689	+	6.66689i	0.596304	+	0.596304i
$126$		0			0
$127$			1.31914i			0.117055i	0.998286	+	0.0585274i	$0.0186405\pi$
$127$			1.31914i			0.117055i	−0.998286	+	0.0585274i	$0.981359\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.84141	−	3.84141i	0.335626	−	0.335626i	−0.519093	−	0.854718i	$-0.673730\pi$
$131$	3.84141	−	3.84141i	0.335626	−	0.335626i	0.854718	+	0.519093i	$0.173730\pi$
$132$		0			0
$133$	−1.21444	−	1.21444i	−0.105306	−	0.105306i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.8262			−1.01038			−0.505191	−	0.863008i	$-0.668578\pi$
$137$	−11.8262			−1.01038			−0.505191	+	0.863008i	$0.668578\pi$
$138$		0			0
$139$	10.6322	−	10.6322i	0.901814	−	0.901814i	−0.0937788	−	0.995593i	$-0.529895\pi$
$139$	10.6322	−	10.6322i	0.901814	−	0.901814i	0.995593	+	0.0937788i	$0.0298947\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.42094			−0.536946
$144$		0			0
$145$	10.2947			0.854930
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.66703	−	9.66703i	0.791954	−	0.791954i	−0.189858	−	0.981812i	$-0.560803\pi$
$149$	9.66703	−	9.66703i	0.791954	−	0.791954i	0.981812	+	0.189858i	$0.0608027\pi$
$150$		0			0
$151$	5.05132			0.411070			0.205535	−	0.978650i	$-0.434107\pi$
$151$	5.05132			0.411070			0.205535	+	0.978650i	$0.434107\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.413565	−	0.413565i	−0.0332183	−	0.0332183i
$156$		0			0
$157$	11.1102	−	11.1102i	0.886692	−	0.886692i	−0.107512	−	0.994204i	$-0.534288\pi$
$157$	11.1102	−	11.1102i	0.886692	−	0.886692i	0.994204	+	0.107512i	$0.0342884\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			8.62021i			0.679367i
$162$		0			0
$163$	−0.936756	−	0.936756i	−0.0733724	−	0.0733724i	0.669468	−	0.742841i	$-0.266522\pi$
$163$	−0.936756	−	0.936756i	−0.0733724	−	0.0733724i	−0.742841	+	0.669468i	$0.766522\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	17.9147i		−	1.38628i	−0.720804	−	0.693139i	$-0.756227\pi$
$167$		−	17.9147i		−	1.38628i	0.720804	−	0.693139i	$-0.243773\pi$
$168$		0			0
$169$		−	8.26636i		−	0.635874i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.95689	−	1.95689i	−0.148780	−	0.148780i	0.628793	−	0.777573i	$-0.283549\pi$
$173$	−1.95689	−	1.95689i	−0.148780	−	0.148780i	−0.777573	+	0.628793i	$0.783549\pi$
$174$		0			0
$175$			7.65081i			0.578347i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16.4536	+	16.4536i	−1.22980	+	1.22980i	−0.265760	+	0.964039i	$0.585623\pi$
$179$	−16.4536	+	16.4536i	−1.22980	+	1.22980i	−0.964039	+	0.265760i	$0.914377\pi$
$180$		0			0
$181$	2.01639	+	2.01639i	0.149877	+	0.149877i	0.778063	−	0.628186i	$-0.216202\pi$
$181$	2.01639	+	2.01639i	0.149877	+	0.149877i	−0.628186	+	0.778063i	$0.716202\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	41.8052			3.07358
$186$		0			0
$187$	5.42414	−	5.42414i	0.396653	−	0.396653i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.72330			0.486481			0.243240	−	0.969966i	$-0.421790\pi$
$191$	6.72330			0.486481			0.243240	+	0.969966i	$0.421790\pi$
$192$		0			0
$193$	1.37543			0.0990058			0.0495029	−	0.998774i	$-0.484236\pi$
$193$	1.37543			0.0990058			0.0495029	+	0.998774i	$0.484236\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.2035	+	10.2035i	−0.726967	+	0.726967i	−0.970014	−	0.243048i	$-0.921853\pi$
$197$	−10.2035	+	10.2035i	−0.726967	+	0.726967i	0.243048	+	0.970014i	$0.421853\pi$
$198$		0			0
$199$	14.9391			1.05900			0.529501	−	0.848309i	$-0.322379\pi$
$199$	14.9391			1.05900			0.529501	+	0.848309i	$0.322379\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.04663	+	2.04663i	0.143646	+	0.143646i
$204$		0			0
$205$	−24.0583	+	24.0583i	−1.68030	+	1.68030i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			2.39136i			0.165414i
$210$		0			0
$211$	−4.93531	−	4.93531i	−0.339761	−	0.339761i	0.516516	−	0.856277i	$-0.327228\pi$
$211$	−4.93531	−	4.93531i	−0.339761	−	0.339761i	−0.856277	+	0.516516i	$0.827228\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	25.2443i		−	1.72164i
$216$		0			0
$217$		−	0.164437i		−	0.0111627i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	17.9649	+	17.9649i	1.20845	+	1.20845i
$222$		0			0
$223$		−	5.25747i		−	0.352066i	−0.984384	−	0.176033i	$-0.943673\pi$
$223$		−	5.25747i		−	0.352066i	0.984384	−	0.176033i	$-0.0563266\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.30931	−	6.30931i	0.418763	−	0.418763i	−0.466014	−	0.884777i	$-0.654310\pi$
$227$	6.30931	−	6.30931i	0.418763	−	0.418763i	0.884777	+	0.466014i	$0.154310\pi$
$228$		0			0
$229$	19.1785	+	19.1785i	1.26735	+	1.26735i	0.947452	+	0.319899i	$0.103649\pi$
$229$	19.1785	+	19.1785i	1.26735	+	1.26735i	0.319899	+	0.947452i	$0.396351\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.555029			−0.0363611			−0.0181806	−	0.999835i	$-0.505787\pi$
$233$	−0.555029			−0.0363611			−0.0181806	+	0.999835i	$0.505787\pi$
$234$		0			0
$235$	8.49672	−	8.49672i	0.554265	−	0.554265i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	23.0408			1.49039			0.745194	−	0.666848i	$-0.232357\pi$
$239$	23.0408			1.49039			0.745194	+	0.666848i	$0.232357\pi$
$240$		0			0
$241$	−7.47827			−0.481718			−0.240859	−	0.970560i	$-0.577429\pi$
$241$	−7.47827			−0.481718			−0.240859	+	0.970560i	$0.577429\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.51504	+	2.51504i	−0.160680	+	0.160680i
$246$		0			0
$247$	−7.92026			−0.503954
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.7049	−	18.7049i	−1.18064	−	1.18064i	−0.979578	−	0.201066i	$-0.935559\pi$
$251$	−18.7049	−	18.7049i	−1.18064	−	1.18064i	−0.201066	−	0.979578i	$-0.564441\pi$
$252$		0			0
$253$	8.48701	−	8.48701i	0.533574	−	0.533574i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			18.3076i			1.14200i	0.820951	+	0.570998i	$0.193444\pi$
$257$			18.3076i			1.14200i	−0.820951	+	0.570998i	$0.806556\pi$
$258$		0			0
$259$	8.31105	+	8.31105i	0.516423	+	0.516423i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	20.1306i		−	1.24130i	−0.784086	−	0.620652i	$-0.786868\pi$
$263$		−	20.1306i		−	1.24130i	0.784086	−	0.620652i	$-0.213132\pi$
$264$		0			0
$265$			18.7479i			1.15168i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.69779	−	8.69779i	−0.530313	−	0.530313i	0.390352	−	0.920666i	$-0.372353\pi$
$269$	−8.69779	−	8.69779i	−0.530313	−	0.530313i	−0.920666	+	0.390352i	$0.872353\pi$
$270$		0			0
$271$			11.6299i			0.706466i	0.935535	+	0.353233i	$0.114918\pi$
$271$			11.6299i			0.706466i	−0.935535	+	0.353233i	$0.885082\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.53259	−	7.53259i	0.454232	−	0.454232i
$276$		0			0
$277$	22.9876	+	22.9876i	1.38119	+	1.38119i	0.842511	+	0.538679i	$0.181077\pi$
$277$	22.9876	+	22.9876i	1.38119	+	1.38119i	0.538679	+	0.842511i	$0.318923\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	11.5187			0.687148			0.343574	−	0.939126i	$-0.388362\pi$
$281$	11.5187			0.687148			0.343574	+	0.939126i	$0.388362\pi$
$282$		0			0
$283$	15.6255	−	15.6255i	0.928839	−	0.928839i	−0.0687924	−	0.997631i	$-0.521915\pi$
$283$	15.6255	−	15.6255i	0.928839	−	0.928839i	0.997631	+	0.0687924i	$0.0219146\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.56578			−0.564650
$288$		0			0
$289$	−13.3521			−0.785417
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.77816	+	3.77816i	−0.220722	+	0.220722i	−0.808803	−	0.588080i	$-0.799884\pi$
$293$	−3.77816	+	3.77816i	−0.220722	+	0.220722i	0.588080	+	0.808803i	$0.299884\pi$
$294$		0			0
$295$	−50.6936			−2.95150
$296$		0			0
$297$		0			0
$298$		0			0
$299$	28.1093	+	28.1093i	1.62560	+	1.62560i
$300$		0			0
$301$	5.01867	−	5.01867i	0.289271	−	0.289271i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	5.38430i		−	0.308304i
$306$		0			0
$307$	−0.708546	−	0.708546i	−0.0404389	−	0.0404389i	0.686598	−	0.727037i	$-0.259103\pi$
$307$	−0.708546	−	0.708546i	−0.0404389	−	0.0404389i	−0.727037	+	0.686598i	$0.759103\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			21.6074i			1.22525i	0.790376	+	0.612623i	$0.209885\pi$
$311$			21.6074i			1.22525i	−0.790376	+	0.612623i	$0.790115\pi$
$312$		0			0
$313$		−	5.11637i		−	0.289194i	−0.989491	−	0.144597i	$-0.953811\pi$
$313$		−	5.11637i		−	0.289194i	0.989491	−	0.144597i	$-0.0461886\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.88622	−	7.88622i	−0.442934	−	0.442934i	0.450063	−	0.892997i	$-0.351402\pi$
$317$	−7.88622	−	7.88622i	−0.442934	−	0.442934i	−0.892997	+	0.450063i	$0.851402\pi$
$318$		0			0
$319$		−	4.03002i		−	0.225638i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.69070	−	6.69070i	0.372281	−	0.372281i
$324$		0			0
$325$	24.9482	+	24.9482i	1.38388	+	1.38388i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.37837			0.186256
$330$		0			0
$331$	−0.144453	+	0.144453i	−0.00793985	+	0.00793985i	−0.711066	−	0.703126i	$-0.751787\pi$
$331$	−0.144453	+	0.144453i	−0.00793985	+	0.00793985i	0.703126	+	0.711066i	$0.251787\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.6975			−0.857646
$336$		0			0
$337$	−18.3255			−0.998256			−0.499128	−	0.866528i	$-0.666346\pi$
$337$	−18.3255			−0.998256			−0.499128	+	0.866528i	$0.666346\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.161896	+	0.161896i	−0.00876716	+	0.00876716i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.70010	+	7.70010i	0.413363	+	0.413363i	0.882908	−	0.469545i	$-0.155582\pi$
$347$	7.70010	+	7.70010i	0.413363	+	0.413363i	−0.469545	+	0.882908i	$0.655582\pi$
$348$		0			0
$349$	9.23902	−	9.23902i	0.494553	−	0.494553i	−0.415184	−	0.909737i	$-0.636283\pi$
$349$	9.23902	−	9.23902i	0.494553	−	0.494553i	0.909737	+	0.415184i	$0.136283\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	13.9051i		−	0.740092i	−0.929013	−	0.370046i	$-0.879342\pi$
$353$		−	13.9051i		−	0.740092i	0.929013	−	0.370046i	$-0.120658\pi$
$354$		0			0
$355$	−19.2716	−	19.2716i	−1.02283	−	1.02283i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	20.3979i		−	1.07656i	−0.842766	−	0.538281i	$-0.819074\pi$
$359$		−	20.3979i		−	1.07656i	0.842766	−	0.538281i	$-0.180926\pi$
$360$		0			0
$361$		−	16.0503i		−	0.844750i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−21.1411	−	21.1411i	−1.10658	−	1.10658i
$366$		0			0
$367$			21.2755i			1.11057i	0.831660	+	0.555285i	$0.187391\pi$
$367$			21.2755i			1.11057i	−0.831660	+	0.555285i	$0.812609\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.72717	+	3.72717i	−0.193505	+	0.193505i
$372$		0			0
$373$	14.6527	+	14.6527i	0.758688	+	0.758688i	0.976084	−	0.217396i	$-0.0697562\pi$
$373$	14.6527	+	14.6527i	0.758688	+	0.758688i	−0.217396	+	0.976084i	$0.569756\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	13.3476			0.687434
$378$		0			0
$379$	8.85565	−	8.85565i	0.454884	−	0.454884i	−0.442088	−	0.896972i	$-0.645762\pi$
$379$	8.85565	−	8.85565i	0.454884	−	0.454884i	0.896972	+	0.442088i	$0.145762\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	33.3573			1.70448			0.852238	−	0.523154i	$-0.175245\pi$
$383$	33.3573			1.70448			0.852238	+	0.523154i	$0.175245\pi$
$384$		0			0
$385$	4.95235			0.252395
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.00738	−	2.00738i	0.101778	−	0.101778i	−0.654384	−	0.756162i	$-0.727072\pi$
$389$	2.00738	−	2.00738i	0.101778	−	0.101778i	0.756162	+	0.654384i	$0.227072\pi$
$390$		0			0
$391$	−47.4911			−2.40173
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−34.8865	−	34.8865i	−1.75533	−	1.75533i
$396$		0			0
$397$	−2.43985	+	2.43985i	−0.122453	+	0.122453i	−0.765677	−	0.643225i	$-0.777596\pi$
$397$	−2.43985	+	2.43985i	−0.122453	+	0.122453i	0.643225	+	0.765677i	$0.277596\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	11.6798i		−	0.583263i	−0.956531	−	0.291632i	$-0.905802\pi$
$401$		−	11.6798i		−	0.583263i	0.956531	−	0.291632i	$-0.0941981\pi$
$402$		0			0
$403$	−0.536205	−	0.536205i	−0.0267103	−	0.0267103i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	16.3653i		−	0.811195i
$408$		0			0
$409$		−	15.2951i		−	0.756292i	−0.925746	−	0.378146i	$-0.876562\pi$
$409$		−	15.2951i		−	0.756292i	0.925746	−	0.378146i	$-0.123438\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−10.0781	−	10.0781i	−0.495911	−	0.495911i
$414$		0			0
$415$			36.7806i			1.80549i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.1833	+	13.1833i	−0.644045	+	0.644045i	−0.951547	−	0.307502i	$-0.900507\pi$
$419$	−13.1833	+	13.1833i	−0.644045	+	0.644045i	0.307502	+	0.951547i	$0.400507\pi$
$420$		0			0
$421$	−22.2574	−	22.2574i	−1.08476	−	1.08476i	−0.996058	−	0.0887017i	$-0.971728\pi$
$421$	−22.2574	−	22.2574i	−1.08476	−	1.08476i	−0.0887017	−	0.996058i	$-0.528272\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−42.1504			−2.04459
$426$		0			0
$427$	1.07042	−	1.07042i	0.0518014	−	0.0518014i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	19.0326			0.916769			0.458385	−	0.888754i	$-0.348428\pi$
$431$	19.0326			0.916769			0.458385	+	0.888754i	$0.348428\pi$
$432$		0			0
$433$	−31.7120			−1.52398			−0.761992	−	0.647587i	$-0.775778\pi$
$433$	−31.7120			−1.52398			−0.761992	+	0.647587i	$0.775778\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10.4688	−	10.4688i	0.500789	−	0.500789i
$438$		0			0
$439$	−32.9521			−1.57272			−0.786359	−	0.617770i	$-0.788036\pi$
$439$	−32.9521			−1.57272			−0.786359	+	0.617770i	$0.788036\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.94776	+	4.94776i	0.235075	+	0.235075i	0.814807	−	0.579732i	$-0.196843\pi$
$443$	4.94776	+	4.94776i	0.235075	+	0.235075i	−0.579732	+	0.814807i	$0.696843\pi$
$444$		0			0
$445$	−18.8416	+	18.8416i	−0.893177	+	0.893177i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			31.5258i			1.48779i	0.668294	+	0.743897i	$0.267025\pi$
$449$			31.5258i			1.48779i	−0.668294	+	0.743897i	$0.732975\pi$
$450$		0			0
$451$	9.41796	+	9.41796i	0.443475	+	0.443475i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			16.4023i			0.768954i
$456$		0			0
$457$		−	35.2517i		−	1.64901i	−0.565857	−	0.824503i	$-0.691455\pi$
$457$		−	35.2517i		−	1.64901i	0.565857	−	0.824503i	$-0.308545\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.8769	−	15.8769i	−0.739462	−	0.739462i	0.233012	−	0.972474i	$-0.425142\pi$
$461$	−15.8769	−	15.8769i	−0.739462	−	0.739462i	−0.972474	+	0.233012i	$0.925142\pi$
$462$		0			0
$463$		−	15.9217i		−	0.739946i	−0.929043	−	0.369973i	$-0.879367\pi$
$463$		−	15.9217i		−	0.739946i	0.929043	−	0.369973i	$-0.120633\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.52420	−	6.52420i	0.301904	−	0.301904i	−0.539854	−	0.841758i	$-0.681521\pi$
$467$	6.52420	−	6.52420i	0.301904	−	0.301904i	0.841758	+	0.539854i	$0.181521\pi$
$468$		0			0
$469$	−3.12073	−	3.12073i	−0.144102	−	0.144102i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−9.88224			−0.454386
$474$		0			0
$475$	9.29148	−	9.29148i	0.426323	−	0.426323i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	31.5891			1.44334			0.721671	−	0.692237i	$-0.243375\pi$
$479$	31.5891			1.44334			0.721671	+	0.692237i	$0.243375\pi$
$480$		0			0
$481$	54.2023			2.47141
$482$		0			0
$483$		0			0
$484$		0			0
$485$	19.7200	−	19.7200i	0.895440	−	0.895440i
$486$		0			0
$487$	29.4810			1.33591			0.667955	−	0.744202i	$-0.267170\pi$
$487$	29.4810			1.33591			0.667955	+	0.744202i	$0.267170\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.06862	+	6.06862i	0.273873	+	0.273873i	0.830657	−	0.556784i	$-0.187965\pi$
$491$	6.06862	+	6.06862i	0.273873	+	0.273873i	−0.556784	+	0.830657i	$0.687965\pi$
$492$		0			0
$493$	−11.2755	+	11.2755i	−0.507821	+	0.507821i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	7.66257i		−	0.343713i
$498$		0			0
$499$	−15.8968	−	15.8968i	−0.711638	−	0.711638i	0.255240	−	0.966878i	$-0.417846\pi$
$499$	−15.8968	−	15.8968i	−0.711638	−	0.711638i	−0.966878	+	0.255240i	$0.917846\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			17.1978i			0.766813i	0.923580	+	0.383406i	$0.125249\pi$
$503$			17.1978i			0.766813i	−0.923580	+	0.383406i	$0.874751\pi$
$504$		0			0
$505$			3.76379i			0.167486i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.959431	+	0.959431i	0.0425260	+	0.0425260i	0.728050	−	0.685524i	$-0.240427\pi$
$509$	0.959431	+	0.959431i	0.0425260	+	0.0425260i	−0.685524	+	0.728050i	$0.740427\pi$
$510$		0			0
$511$		−	8.40588i		−	0.371854i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−29.7180	+	29.7180i	−1.30953	+	1.30953i
$516$		0			0
$517$	−3.32617	−	3.32617i	−0.146285	−	0.146285i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.04026			−0.396061			−0.198030	−	0.980196i	$-0.563454\pi$
$521$	−9.04026			−0.396061			−0.198030	+	0.980196i	$0.563454\pi$
$522$		0			0
$523$	5.32615	−	5.32615i	0.232896	−	0.232896i	−0.581004	−	0.813901i	$-0.697340\pi$
$523$	5.32615	−	5.32615i	0.232896	−	0.232896i	0.813901	+	0.581004i	$0.197340\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.905927			0.0394628
$528$		0			0
$529$	−51.3080			−2.23078
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−31.1926	+	31.1926i	−1.35110	+	1.35110i
$534$		0			0
$535$	−22.1921			−0.959449
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.984548	+	0.984548i	0.0424075	+	0.0424075i
$540$		0			0
$541$	12.7082	−	12.7082i	0.546367	−	0.546367i	−0.379021	−	0.925388i	$-0.623739\pi$
$541$	12.7082	−	12.7082i	0.546367	−	0.546367i	0.925388	+	0.379021i	$0.123739\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	0.647588i		−	0.0277396i
$546$		0			0
$547$	−19.7217	−	19.7217i	−0.843239	−	0.843239i	0.146040	−	0.989279i	$-0.453347\pi$
$547$	−19.7217	−	19.7217i	−0.843239	−	0.843239i	−0.989279	+	0.146040i	$0.953347\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	4.97105i		−	0.211774i
$552$		0			0
$553$		−	13.8712i		−	0.589863i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	32.2571	+	32.2571i	1.36678	+	1.36678i	0.864990	+	0.501789i	$0.167325\pi$
$557$	32.2571	+	32.2571i	1.36678	+	1.36678i	0.501789	+	0.864990i	$0.332675\pi$
$558$		0			0
$559$		−	32.7303i		−	1.38434i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	27.0814	−	27.0814i	1.14134	−	1.14134i	0.153138	−	0.988205i	$-0.451062\pi$
$563$	27.0814	−	27.0814i	1.14134	−	1.14134i	0.988205	−	0.153138i	$-0.0489380\pi$
$564$		0			0
$565$	35.9757	+	35.9757i	1.51351	+	1.51351i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	28.4823			1.19404			0.597020	−	0.802226i	$-0.296351\pi$
$569$	28.4823			1.19404			0.597020	+	0.802226i	$0.296351\pi$
$570$		0			0
$571$	−15.8844	+	15.8844i	−0.664743	+	0.664743i	−0.956494	−	0.291751i	$-0.905762\pi$
$571$	−15.8844	+	15.8844i	−0.664743	+	0.664743i	0.291751	+	0.956494i	$0.405762\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−65.9516			−2.75037
$576$		0			0
$577$	4.65713			0.193879			0.0969395	−	0.995290i	$-0.469095\pi$
$577$	4.65713			0.193879			0.0969395	+	0.995290i	$0.469095\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.31214	+	7.31214i	−0.303359	+	0.303359i
$582$		0			0
$583$	7.33916			0.303957
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.2034	+	13.2034i	0.544963	+	0.544963i	0.924980	−	0.380017i	$-0.124082\pi$
$587$	13.2034	+	13.2034i	0.544963	+	0.544963i	−0.380017	+	0.924980i	$0.624082\pi$
$588$		0			0
$589$	−0.199699	+	0.199699i	−0.00822847	+	0.00822847i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	8.00745i		−	0.328827i	−0.986392	−	0.164413i	$-0.947427\pi$
$593$		−	8.00745i		−	0.328827i	0.986392	−	0.164413i	$-0.0525731\pi$
$594$		0			0
$595$	−13.8560	−	13.8560i	−0.568041	−	0.568041i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	29.8782i		−	1.22079i	−0.792096	−	0.610396i	$-0.791010\pi$
$599$		−	29.8782i		−	1.22079i	0.792096	−	0.610396i	$-0.208990\pi$
$600$		0			0
$601$			28.4594i			1.16088i	0.814302	+	0.580441i	$0.197120\pi$
$601$			28.4594i			1.16088i	−0.814302	+	0.580441i	$0.802880\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	22.7896	+	22.7896i	0.926528	+	0.926528i
$606$		0			0
$607$			27.1606i			1.10242i	0.834368	+	0.551208i	$0.185833\pi$
$607$			27.1606i			1.10242i	−0.834368	+	0.551208i	$0.814167\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11.0164	−	11.0164i	0.445675	−	0.445675i
$612$		0			0
$613$	−25.1239	−	25.1239i	−1.01474	−	1.01474i	−0.999890	−	0.0148539i	$-0.995272\pi$
$613$	−25.1239	−	25.1239i	−1.01474	−	1.01474i	−0.0148539	−	0.999890i	$-0.504728\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−20.2712			−0.816089			−0.408044	−	0.912962i	$-0.633789\pi$
$617$	−20.2712			−0.816089			−0.408044	+	0.912962i	$0.633789\pi$
$618$		0			0
$619$	13.2344	−	13.2344i	0.531937	−	0.531937i	−0.389211	−	0.921148i	$-0.627252\pi$
$619$	13.2344	−	13.2344i	0.531937	−	0.531937i	0.921148	+	0.389211i	$0.127252\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.49158			−0.300144
$624$		0			0
$625$	4.71914			0.188766
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−45.7878	+	45.7878i	−1.82568	+	1.82568i
$630$		0			0
$631$	47.8911			1.90651			0.953257	−	0.302161i	$-0.0977080\pi$
$631$	47.8911			1.90651			0.953257	+	0.302161i	$0.0977080\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.31769	−	3.31769i	−0.131658	−	0.131658i
$636$		0			0
$637$	−3.26086	+	3.26086i	−0.129200	+	0.129200i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	23.5861i		−	0.931595i	−0.884891	−	0.465797i	$-0.845768\pi$
$641$		−	23.5861i		−	0.931595i	0.884891	−	0.465797i	$-0.154232\pi$
$642$		0			0
$643$	29.9073	+	29.9073i	1.17943	+	1.17943i	0.979890	+	0.199537i	$0.0639437\pi$
$643$	29.9073	+	29.9073i	1.17943	+	1.17943i	0.199537	+	0.979890i	$0.436056\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	35.8559i		−	1.40964i	−0.709385	−	0.704821i	$-0.751027\pi$
$647$		−	35.8559i		−	1.40964i	0.709385	−	0.704821i	$-0.248973\pi$
$648$		0			0
$649$			19.8448i			0.778975i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−26.7966	−	26.7966i	−1.04863	−	1.04863i	−0.998755	−	0.0498765i	$-0.984117\pi$
$653$	−26.7966	−	26.7966i	−1.04863	−	1.04863i	−0.0498765	−	0.998755i	$-0.515883\pi$
$654$		0			0
$655$			19.3226i			0.754995i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.9220	−	19.9220i	0.776050	−	0.776050i	−0.203107	−	0.979157i	$-0.565104\pi$
$659$	19.9220	−	19.9220i	0.776050	−	0.776050i	0.979157	+	0.203107i	$0.0651038\pi$
$660$		0			0
$661$	3.02969	+	3.02969i	0.117841	+	0.117841i	0.763568	−	0.645727i	$-0.223446\pi$
$661$	3.02969	+	3.02969i	0.117841	+	0.117841i	−0.645727	+	0.763568i	$0.723446\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.10874			0.236887
$666$		0			0
$667$	−17.6424	+	17.6424i	−0.683117	+	0.683117i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.10776			−0.0813693
$672$		0			0
$673$	41.8102			1.61166			0.805832	−	0.592144i	$-0.201718\pi$
$673$	41.8102			1.61166			0.805832	+	0.592144i	$0.201718\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−11.0622	+	11.0622i	−0.425154	+	0.425154i	−0.886974	−	0.461820i	$-0.847197\pi$
$677$	−11.0622	+	11.0622i	−0.425154	+	0.425154i	0.461820	+	0.886974i	$0.347197\pi$
$678$		0			0
$679$	7.84085			0.300904
$680$		0			0
$681$		0			0
$682$		0			0
$683$	25.6844	+	25.6844i	0.982786	+	0.982786i	0.999854	−	0.0170687i	$-0.00543339\pi$
$683$	25.6844	+	25.6844i	0.982786	+	0.982786i	−0.0170687	+	0.999854i	$0.505433\pi$
$684$		0			0
$685$	29.7434	−	29.7434i	1.13644	−	1.13644i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			24.3075i			0.926044i
$690$		0			0
$691$	−15.7276	−	15.7276i	−0.598306	−	0.598306i	0.341556	−	0.939862i	$-0.389046\pi$
$691$	−15.7276	−	15.7276i	−0.598306	−	0.598306i	−0.939862	+	0.341556i	$0.889046\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			53.4809i			2.02865i
$696$		0			0
$697$		−	52.7005i		−	1.99617i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4.35804	−	4.35804i	−0.164601	−	0.164601i	0.620000	−	0.784601i	$-0.287132\pi$
$701$	−4.35804	−	4.35804i	−0.164601	−	0.164601i	−0.784601	+	0.620000i	$0.787132\pi$
$702$		0			0
$703$		−	20.1866i		−	0.761352i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.748257	+	0.748257i	−0.0281411	+	0.0281411i
$708$		0			0
$709$	−2.68056	−	2.68056i	−0.100670	−	0.100670i	0.654978	−	0.755648i	$-0.272678\pi$
$709$	−2.68056	−	2.68056i	−0.100670	−	0.100670i	−0.755648	+	0.654978i	$0.772678\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.41748			0.0530850
$714$		0			0
$715$	16.1489	−	16.1489i	0.603934	−	0.603934i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−19.9117			−0.742581			−0.371290	−	0.928517i	$-0.621085\pi$
$719$	−19.9117			−0.742581			−0.371290	+	0.928517i	$0.621085\pi$
$720$		0			0
$721$	−11.8161			−0.440056
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.6584	+	15.6584i	−0.581539	+	0.581539i
$726$		0			0
$727$	−24.0642			−0.892491			−0.446246	−	0.894911i	$-0.647239\pi$
$727$	−24.0642			−0.892491			−0.446246	+	0.894911i	$0.647239\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	27.6492	+	27.6492i	1.02264	+	1.02264i
$732$		0			0
$733$	−3.18890	+	3.18890i	−0.117785	+	0.117785i	−0.763542	−	0.645758i	$-0.776542\pi$
$733$	−3.18890	+	3.18890i	−0.117785	+	0.117785i	0.645758	+	0.763542i	$0.276542\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			6.14502i			0.226355i
$738$		0			0
$739$	−20.8311	−	20.8311i	−0.766285	−	0.766285i	0.211165	−	0.977450i	$-0.432274\pi$
$739$	−20.8311	−	20.8311i	−0.766285	−	0.766285i	−0.977450	+	0.211165i	$0.932274\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			28.6291i			1.05030i	0.851010	+	0.525150i	$0.175991\pi$
$743$			28.6291i			1.05030i	−0.851010	+	0.525150i	$0.824009\pi$
$744$		0			0
$745$			48.6258i			1.78151i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.41189	−	4.41189i	−0.161207	−	0.161207i
$750$		0			0
$751$		−	28.1706i		−	1.02796i	−0.857802	−	0.513980i	$-0.828170\pi$
$751$		−	28.1706i		−	1.02796i	0.857802	−	0.513980i	$-0.171830\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−12.7042	+	12.7042i	−0.462355	+	0.462355i
$756$		0			0
$757$	−4.64056	−	4.64056i	−0.168664	−	0.168664i	0.617728	−	0.786392i	$-0.288053\pi$
$757$	−4.64056	−	4.64056i	−0.168664	−	0.168664i	−0.786392	+	0.617728i	$0.788053\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.29768			0.228291			0.114145	−	0.993464i	$-0.463587\pi$
$761$	6.29768			0.228291			0.114145	+	0.993464i	$0.463587\pi$
$762$		0			0
$763$	0.128743	−	0.128743i	0.00466082	−	0.00466082i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−65.7265			−2.37325
$768$		0			0
$769$	4.73382			0.170706			0.0853529	−	0.996351i	$-0.472798\pi$
$769$	4.73382			0.170706			0.0853529	+	0.996351i	$0.472798\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.44275	+	5.44275i	−0.195762	+	0.195762i	−0.798180	−	0.602418i	$-0.794204\pi$
$773$	−5.44275	+	5.44275i	−0.195762	+	0.195762i	0.602418	+	0.798180i	$0.294204\pi$
$774$		0			0
$775$	1.25808			0.0451914
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11.6171	+	11.6171i	0.416226	+	0.416226i
$780$		0			0
$781$	−7.54417	+	7.54417i	−0.269952	+	0.269952i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			55.8852i			1.99463i
$786$		0			0
$787$	27.1082	+	27.1082i	0.966302	+	0.966302i	0.999450	−	0.0331485i	$-0.0105534\pi$
$787$	27.1082	+	27.1082i	0.966302	+	0.966302i	−0.0331485	+	0.999450i	$0.510553\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			14.3042i			0.508600i
$792$		0			0
$793$		−	6.98099i		−	0.247902i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.7301	+	11.7301i	0.415501	+	0.415501i	0.883650	−	0.468149i	$-0.155079\pi$
$797$	11.7301	+	11.7301i	0.415501	+	0.415501i	−0.468149	+	0.883650i	$0.655079\pi$
$798$		0			0
$799$			18.6124i			0.658458i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−8.27599	+	8.27599i	−0.292053	+	0.292053i
$804$		0			0
$805$	−21.6801	−	21.6801i	−0.764124	−	0.764124i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.7126			−0.482111			−0.241055	−	0.970511i	$-0.577494\pi$
$809$	−13.7126			−0.482111			−0.241055	+	0.970511i	$0.577494\pi$
$810$		0			0
$811$	−22.2730	+	22.2730i	−0.782111	+	0.782111i	−0.980187	−	0.198076i	$-0.936531\pi$
$811$	−22.2730	+	22.2730i	−0.782111	+	0.782111i	0.198076	+	0.980187i	$0.436531\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.71195			0.165052
$816$		0			0
$817$	−12.1898			−0.426467
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.690329	+	0.690329i	−0.0240926	+	0.0240926i	−0.719050	−	0.694958i	$-0.755423\pi$
$821$	−0.690329	+	0.690329i	−0.0240926	+	0.0240926i	0.694958	+	0.719050i	$0.255423\pi$
$822$		0			0
$823$	1.43063			0.0498688			0.0249344	−	0.999689i	$-0.492062\pi$
$823$	1.43063			0.0498688			0.0249344	+	0.999689i	$0.492062\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.59770	+	9.59770i	0.333745	+	0.333745i	0.854007	−	0.520262i	$-0.174166\pi$
$827$	9.59770	+	9.59770i	0.333745	+	0.333745i	−0.520262	+	0.854007i	$0.674166\pi$
$828$		0			0
$829$	17.0260	−	17.0260i	0.591337	−	0.591337i	−0.346656	−	0.937992i	$-0.612683\pi$
$829$	17.0260	−	17.0260i	0.591337	−	0.591337i	0.937992	+	0.346656i	$0.112683\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	5.50927i		−	0.190885i
$834$		0			0
$835$	45.0560	+	45.0560i	1.55923	+	1.55923i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			29.0214i			1.00193i	0.865467	+	0.500965i	$0.167022\pi$
$839$			29.0214i			1.00193i	−0.865467	+	0.500965i	$0.832978\pi$
$840$		0			0
$841$		−	20.6226i		−	0.711123i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20.7902	+	20.7902i	0.715205	+	0.715205i
$846$		0			0
$847$			9.06133i			0.311351i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−71.6430	+	71.6430i	−2.45589	+	2.45589i
$852$		0			0
$853$	17.4096	+	17.4096i	0.596092	+	0.596092i	0.939270	−	0.343178i	$-0.111503\pi$
$853$	17.4096	+	17.4096i	0.596092	+	0.596092i	−0.343178	+	0.939270i	$0.611503\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	17.6637			0.603379			0.301690	−	0.953406i	$-0.402449\pi$
$857$	17.6637			0.603379			0.301690	+	0.953406i	$0.402449\pi$
$858$		0			0
$859$	−5.79823	+	5.79823i	−0.197833	+	0.197833i	−0.799070	−	0.601237i	$-0.794675\pi$
$859$	−5.79823	+	5.79823i	−0.197833	+	0.197833i	0.601237	+	0.799070i	$0.294675\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−39.0789			−1.33026			−0.665131	−	0.746727i	$-0.731624\pi$
$863$	−39.0789			−1.33026			−0.665131	+	0.746727i	$0.731624\pi$
$864$		0			0
$865$	9.84331			0.334683
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13.6568	+	13.6568i	−0.463277	+	0.463277i
$870$		0			0
$871$	−20.3525			−0.689618
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.66689	−	6.66689i	−0.225382	−	0.225382i
$876$		0			0
$877$	6.28950	−	6.28950i	0.212381	−	0.212381i	−0.592897	−	0.805278i	$-0.702016\pi$
$877$	6.28950	−	6.28950i	0.212381	−	0.212381i	0.805278	+	0.592897i	$0.202016\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	21.5545i		−	0.726190i	−0.931752	−	0.363095i	$-0.881720\pi$
$881$		−	21.5545i		−	0.726190i	0.931752	−	0.363095i	$-0.118280\pi$
$882$		0			0
$883$	−5.85864	−	5.85864i	−0.197159	−	0.197159i	0.601622	−	0.798781i	$-0.294521\pi$
$883$	−5.85864	−	5.85864i	−0.197159	−	0.197159i	−0.798781	+	0.601622i	$0.794521\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	28.0138i		−	0.940613i	−0.882503	−	0.470306i	$-0.844143\pi$
$887$		−	28.0138i		−	0.940613i	0.882503	−	0.470306i	$-0.155857\pi$
$888$		0			0
$889$		−	1.31914i		−	0.0442426i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.10284	−	4.10284i	−0.137296	−	0.137296i
$894$		0			0
$895$		−	82.7628i		−	2.76645i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	0.336542	−	0.336542i	0.0112243	−	0.0112243i
$900$		0			0
$901$	−20.5340	−	20.5340i	−0.684087	−	0.684087i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.1426			−0.337151
$906$		0			0
$907$	−3.74119	+	3.74119i	−0.124224	+	0.124224i	−0.766486	−	0.642262i	$-0.777996\pi$
$907$	−3.74119	+	3.74119i	−0.124224	+	0.124224i	0.642262	+	0.766486i	$0.277996\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−46.9910			−1.55688			−0.778441	−	0.627718i	$-0.783989\pi$
$911$	−46.9910			−1.55688			−0.778441	+	0.627718i	$0.783989\pi$
$912$		0			0
$913$	14.3983			0.476514
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.84141	+	3.84141i	−0.126855	+	0.126855i
$918$		0			0
$919$	28.7166			0.947272			0.473636	−	0.880721i	$-0.342941\pi$
$919$	28.7166			0.947272			0.473636	+	0.880721i	$0.342941\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−24.9866	−	24.9866i	−0.822442	−	0.822442i
$924$		0			0
$925$	−63.5863	+	63.5863i	−2.09070	+	2.09070i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	39.0905i		−	1.28252i	−0.767325	−	0.641259i	$-0.778413\pi$
$929$		−	39.0905i		−	1.28252i	0.767325	−	0.641259i	$-0.221587\pi$
$930$		0			0
$931$	1.21444	+	1.21444i	0.0398018	+	0.0398018i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			27.2838i			0.892277i
$936$		0			0
$937$		−	22.0325i		−	0.719770i	−0.932997	−	0.359885i	$-0.882816\pi$
$937$		−	22.0325i		−	0.719770i	0.932997	−	0.359885i	$-0.117184\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−28.7845	−	28.7845i	−0.938349	−	0.938349i	0.0598579	−	0.998207i	$-0.480935\pi$
$941$	−28.7845	−	28.7845i	−0.938349	−	0.938349i	−0.998207	+	0.0598579i	$0.980935\pi$
$942$		0			0
$943$		−	82.4590i		−	2.68523i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.81984	−	8.81984i	0.286606	−	0.286606i	−0.549130	−	0.835737i	$-0.685041\pi$
$947$	8.81984	−	8.81984i	0.286606	−	0.286606i	0.835737	+	0.549130i	$0.185041\pi$
$948$		0			0
$949$	−27.4104	−	27.4104i	−0.889778	−	0.889778i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.32312			0.0428599			0.0214300	−	0.999770i	$-0.493178\pi$
$953$	1.32312			0.0428599			0.0214300	+	0.999770i	$0.493178\pi$
$954$		0			0
$955$	−16.9093	+	16.9093i	−0.547173	+	0.547173i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	11.8262			0.381889
$960$		0			0
$961$	30.9730			0.999128
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.45926	+	3.45926i	−0.111358	+	0.111358i
$966$		0			0
$967$	−16.3028			−0.524263			−0.262132	−	0.965032i	$-0.584425\pi$
$967$	−16.3028			−0.524263			−0.262132	+	0.965032i	$0.584425\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−16.8788	−	16.8788i	−0.541666	−	0.541666i	0.382351	−	0.924017i	$-0.375114\pi$
$971$	−16.8788	−	16.8788i	−0.541666	−	0.541666i	−0.924017	+	0.382351i	$0.875114\pi$
$972$		0			0
$973$	−10.6322	+	10.6322i	−0.340854	+	0.340854i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.46807i		−	0.0469676i	−0.999724	−	0.0234838i	$-0.992524\pi$
$977$		−	1.46807i		−	0.0469676i	0.999724	−	0.0234838i	$-0.00747581\pi$
$978$		0			0
$979$	7.37582	+	7.37582i	0.235732	+	0.235732i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	34.4565i		−	1.09899i	−0.835496	−	0.549496i	$-0.814820\pi$
$983$		−	34.4565i		−	1.09899i	0.835496	−	0.549496i	$-0.185180\pi$
$984$		0			0
$985$		−	51.3241i		−	1.63532i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	43.2620	+	43.2620i	1.37565	+	1.37565i
$990$		0			0
$991$		−	14.4789i		−	0.459937i	−0.973198	−	0.229969i	$-0.926138\pi$
$991$		−	14.4789i		−	0.459937i	0.973198	−	0.229969i	$-0.0738624\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−37.5723	+	37.5723i	−1.19112	+	1.19112i
$996$		0			0
$997$	−32.5498	−	32.5498i	−1.03086	−	1.03086i	−0.999508	−	0.0313557i	$-0.990018\pi$
$997$	−32.5498	−	32.5498i	−1.03086	−	1.03086i	−0.0313557	−	0.999508i	$-0.509982\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.e.3599.3		40
3.2	odd	2	inner	4032.2.v.e.3599.18		40
4.3	odd	2		1008.2.v.e.827.2	yes	40
12.11	even	2		1008.2.v.e.827.19	yes	40
16.3	odd	4	inner	4032.2.v.e.1583.18		40
16.13	even	4		1008.2.v.e.323.19	yes	40
48.29	odd	4		1008.2.v.e.323.2	✓	40
48.35	even	4	inner	4032.2.v.e.1583.3		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.e.323.2	✓	40	48.29	odd	4
1008.2.v.e.323.19	yes	40	16.13	even	4
1008.2.v.e.827.2	yes	40	4.3	odd	2
1008.2.v.e.827.19	yes	40	12.11	even	2
4032.2.v.e.1583.3		40	48.35	even	4	inner
4032.2.v.e.1583.18		40	16.3	odd	4	inner
4032.2.v.e.3599.3		40	1.1	even	1	trivial
4032.2.v.e.3599.18		40	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.51504 + 2.51504i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(-2.51504 + 2.51504i) q^{5} -1.00000 q^{7} +(0.984548 + 0.984548i) q^{11} +(-3.26086 + 3.26086i) q^{13} -5.50927i q^{17} +(1.21444 + 1.21444i) q^{19} -8.62021i q^{23} -7.65081i q^{25} +(-2.04663 - 2.04663i) q^{29} +0.164437i q^{31} +(2.51504 - 2.51504i) q^{35} +(-8.31105 - 8.31105i) q^{37} +9.56578 q^{41} +(-5.01867 + 5.01867i) q^{43} -3.37837 q^{47} +1.00000 q^{49} +(3.72717 - 3.72717i) q^{53} -4.95235 q^{55} +(10.0781 + 10.0781i) q^{59} +(-1.07042 + 1.07042i) q^{61} -16.4023i q^{65} +(3.12073 + 3.12073i) q^{67} +7.66257i q^{71} +8.40588i q^{73} +(-0.984548 - 0.984548i) q^{77} +13.8712i q^{79} +(7.31214 - 7.31214i) q^{83} +(13.8560 + 13.8560i) q^{85} +7.49158 q^{89} +(3.26086 - 3.26086i) q^{91} -6.10874 q^{95} -7.84085 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{7}+O(q^{10}) \) 40 * q - 40 * q^7	\( 40 q - 40 q^{7} - 24 q^{13} + 32 q^{19} - 8 q^{37} - 32 q^{43} + 40 q^{49} - 48 q^{55} - 24 q^{61} + 64 q^{85} + 24 q^{91} + 64 q^{97}+O(q^{100}) \) 40 * q - 40 * q^7 - 24 * q^13 + 32 * q^19 - 8 * q^37 - 32 * q^43 + 40 * q^49 - 48 * q^55 - 24 * q^61 + 64 * q^85 + 24 * q^91 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more