LMFDB - Embedded newform 4032.2.v.e.3599.6

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			3599.6
Character	$\chi$	$=$	4032.3599
Dual form			4032.2.v.e.1583.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.17902 + 1.17902i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(-1.17902 + 1.17902i) q^{5} -1.00000 q^{7} +(-4.54895 - 4.54895i) q^{11} +(2.56073 - 2.56073i) q^{13} -2.05280i q^{17} +(3.64422 + 3.64422i) q^{19} +2.27140i q^{23} +2.21982i q^{25} +(-0.544898 - 0.544898i) q^{29} -10.1006i q^{31} +(1.17902 - 1.17902i) q^{35} +(4.71698 + 4.71698i) q^{37} +0.487549 q^{41} +(-7.56607 + 7.56607i) q^{43} +0.768184 q^{47} +1.00000 q^{49} +(0.269015 - 0.269015i) q^{53} +10.7266 q^{55} +(-0.0979540 - 0.0979540i) q^{59} +(-7.41725 + 7.41725i) q^{61} +6.03831i q^{65} +(-6.83972 - 6.83972i) q^{67} -8.66316i q^{71} +13.6358i q^{73} +(4.54895 + 4.54895i) q^{77} +9.29142i q^{79} +(-9.76640 + 9.76640i) q^{83} +(2.42029 + 2.42029i) q^{85} +7.27355 q^{89} +(-2.56073 + 2.56073i) q^{91} -8.59324 q^{95} +10.4508 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.17902	+	1.17902i	−0.527275	+	0.527275i	−0.919759	−	0.392484i	$-0.871616\pi$
$5$	−1.17902	+	1.17902i	−0.527275	+	0.527275i	0.392484	+	0.919759i	$0.371616\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.54895	−	4.54895i	−1.37156	−	1.37156i	−0.858133	−	0.513428i	$-0.828375\pi$
$11$	−4.54895	−	4.54895i	−1.37156	−	1.37156i	−0.513428	−	0.858133i	$-0.671625\pi$
$12$		0			0
$13$	2.56073	−	2.56073i	0.710218	−	0.710218i	−0.256363	−	0.966581i	$-0.582524\pi$
$13$	2.56073	−	2.56073i	0.710218	−	0.710218i	0.966581	+	0.256363i	$0.0825241\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.05280i		−	0.497876i	−0.968519	−	0.248938i	$-0.919918\pi$
$17$		−	2.05280i		−	0.497876i	0.968519	−	0.248938i	$-0.0800815\pi$
$18$		0			0
$19$	3.64422	+	3.64422i	0.836042	+	0.836042i	0.988335	−	0.152293i	$-0.0486657\pi$
$19$	3.64422	+	3.64422i	0.836042	+	0.836042i	−0.152293	+	0.988335i	$0.548666\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.27140i			0.473620i	0.971556	+	0.236810i	$0.0761019\pi$
$23$			2.27140i			0.473620i	−0.971556	+	0.236810i	$0.923898\pi$
$24$		0			0
$25$			2.21982i			0.443963i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.544898	−	0.544898i	−0.101185	−	0.101185i	0.654702	−	0.755887i	$-0.272794\pi$
$29$	−0.544898	−	0.544898i	−0.101185	−	0.101185i	−0.755887	+	0.654702i	$0.772794\pi$
$30$		0			0
$31$		−	10.1006i		−	1.81412i	−0.421004	−	0.907059i	$-0.638322\pi$
$31$		−	10.1006i		−	1.81412i	0.421004	−	0.907059i	$-0.361678\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.17902	−	1.17902i	0.199291	−	0.199291i
$36$		0			0
$37$	4.71698	+	4.71698i	0.775467	+	0.775467i	0.979056	−	0.203590i	$-0.0652608\pi$
$37$	4.71698	+	4.71698i	0.775467	+	0.775467i	−0.203590	+	0.979056i	$0.565261\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.487549			0.0761424			0.0380712	−	0.999275i	$-0.487879\pi$
$41$	0.487549			0.0761424			0.0380712	+	0.999275i	$0.487879\pi$
$42$		0			0
$43$	−7.56607	+	7.56607i	−1.15381	+	1.15381i	−0.168033	+	0.985781i	$0.553742\pi$
$43$	−7.56607	+	7.56607i	−1.15381	+	1.15381i	−0.985781	+	0.168033i	$0.946258\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.768184			0.112051			0.0560256	−	0.998429i	$-0.482157\pi$
$47$	0.768184			0.112051			0.0560256	+	0.998429i	$0.482157\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.269015	−	0.269015i	0.0369520	−	0.0369520i	−0.688389	−	0.725341i	$-0.741682\pi$
$53$	0.269015	−	0.269015i	0.0369520	−	0.0369520i	0.725341	+	0.688389i	$0.241682\pi$
$54$		0			0
$55$	10.7266			1.44638
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.0979540	−	0.0979540i	−0.0127525	−	0.0127525i	0.700702	−	0.713454i	$-0.252870\pi$
$59$	−0.0979540	−	0.0979540i	−0.0127525	−	0.0127525i	−0.713454	+	0.700702i	$0.752870\pi$
$60$		0			0
$61$	−7.41725	+	7.41725i	−0.949682	+	0.949682i	−0.998793	−	0.0491117i	$-0.984361\pi$
$61$	−7.41725	+	7.41725i	−0.949682	+	0.949682i	0.0491117	+	0.998793i	$0.484361\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			6.03831i			0.748960i
$66$		0			0
$67$	−6.83972	−	6.83972i	−0.835604	−	0.835604i	0.152672	−	0.988277i	$-0.451212\pi$
$67$	−6.83972	−	6.83972i	−0.835604	−	0.835604i	−0.988277	+	0.152672i	$0.951212\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.66316i		−	1.02813i	−0.857752	−	0.514064i	$-0.828139\pi$
$71$		−	8.66316i		−	1.02813i	0.857752	−	0.514064i	$-0.171861\pi$
$72$		0			0
$73$			13.6358i			1.59594i	0.602694	+	0.797972i	$0.294094\pi$
$73$			13.6358i			1.59594i	−0.602694	+	0.797972i	$0.705906\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.54895	+	4.54895i	0.518401	+	0.518401i
$78$		0			0
$79$			9.29142i			1.04537i	0.852527	+	0.522683i	$0.175069\pi$
$79$			9.29142i			1.04537i	−0.852527	+	0.522683i	$0.824931\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.76640	+	9.76640i	−1.07200	+	1.07200i	−0.0748037	+	0.997198i	$0.523833\pi$
$83$	−9.76640	+	9.76640i	−1.07200	+	1.07200i	−0.997198	+	0.0748037i	$0.976167\pi$
$84$		0			0
$85$	2.42029	+	2.42029i	0.262517	+	0.262517i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.27355			0.770995			0.385497	−	0.922709i	$-0.374030\pi$
$89$	7.27355			0.770995			0.385497	+	0.922709i	$0.374030\pi$
$90$		0			0
$91$	−2.56073	+	2.56073i	−0.268437	+	0.268437i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.59324			−0.881648
$96$		0			0
$97$	10.4508			1.06112			0.530561	−	0.847647i	$-0.321981\pi$
$97$	10.4508			1.06112			0.530561	+	0.847647i	$0.321981\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.75000	+	5.75000i	−0.572147	+	0.572147i	−0.932728	−	0.360581i	$-0.882578\pi$
$101$	−5.75000	+	5.75000i	−0.572147	+	0.572147i	0.360581	+	0.932728i	$0.382578\pi$
$102$		0			0
$103$	−7.11905			−0.701460			−0.350730	−	0.936477i	$-0.614067\pi$
$103$	−7.11905			−0.701460			−0.350730	+	0.936477i	$0.614067\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.13593	−	6.13593i	−0.593182	−	0.593182i	0.345307	−	0.938490i	$-0.387775\pi$
$107$	−6.13593	−	6.13593i	−0.593182	−	0.593182i	−0.938490	+	0.345307i	$0.887775\pi$
$108$		0			0
$109$	1.54344	−	1.54344i	0.147835	−	0.147835i	−0.629315	−	0.777150i	$-0.716665\pi$
$109$	1.54344	−	1.54344i	0.147835	−	0.147835i	0.777150	+	0.629315i	$0.216665\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			14.0023i			1.31723i	0.752481	+	0.658614i	$0.228857\pi$
$113$			14.0023i			1.31723i	−0.752481	+	0.658614i	$0.771143\pi$
$114$		0			0
$115$	−2.67803	−	2.67803i	−0.249728	−	0.249728i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.05280i			0.188179i
$120$		0			0
$121$			30.3859i			2.76236i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.51232	−	8.51232i	−0.761365	−	0.761365i
$126$		0			0
$127$		−	5.32939i		−	0.472906i	−0.971643	−	0.236453i	$-0.924015\pi$
$127$		−	5.32939i		−	0.472906i	0.971643	−	0.236453i	$-0.0759850\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−13.4364	+	13.4364i	−1.17395	+	1.17395i	−0.192685	+	0.981261i	$0.561720\pi$
$131$	−13.4364	+	13.4364i	−1.17395	+	1.17395i	−0.981261	+	0.192685i	$0.938280\pi$
$132$		0			0
$133$	−3.64422	−	3.64422i	−0.315994	−	0.315994i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.1704			−0.868914			−0.434457	−	0.900693i	$-0.643060\pi$
$137$	−10.1704			−0.868914			−0.434457	+	0.900693i	$0.643060\pi$
$138$		0			0
$139$	6.90089	−	6.90089i	0.585326	−	0.585326i	−0.351036	−	0.936362i	$-0.614171\pi$
$139$	6.90089	−	6.90089i	0.585326	−	0.585326i	0.936362	+	0.351036i	$0.114171\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−23.2973			−1.94821
$144$		0			0
$145$	1.28489			0.106705
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.134093	−	0.134093i	0.0109853	−	0.0109853i	−0.701593	−	0.712578i	$-0.747527\pi$
$149$	0.134093	−	0.134093i	0.0109853	−	0.0109853i	0.712578	+	0.701593i	$0.247527\pi$
$150$		0			0
$151$	13.7321			1.11750			0.558750	−	0.829336i	$-0.311281\pi$
$151$	13.7321			1.11750			0.558750	+	0.829336i	$0.311281\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.9088	+	11.9088i	0.956538	+	0.956538i
$156$		0			0
$157$	−2.49258	+	2.49258i	−0.198929	+	0.198929i	−0.799541	−	0.600612i	$-0.794924\pi$
$157$	−2.49258	+	2.49258i	−0.198929	+	0.198929i	0.600612	+	0.799541i	$0.294924\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.27140i		−	0.179012i
$162$		0			0
$163$	−6.43203	−	6.43203i	−0.503796	−	0.503796i	0.408820	−	0.912615i	$-0.365941\pi$
$163$	−6.43203	−	6.43203i	−0.503796	−	0.503796i	−0.912615	+	0.408820i	$0.865941\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.38595i		−	0.339395i	−0.985496	−	0.169697i	$-0.945721\pi$
$167$		−	4.38595i		−	0.339395i	0.985496	−	0.169697i	$-0.0542790\pi$
$168$		0			0
$169$		−	0.114651i		−	0.00881930i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−18.2846	−	18.2846i	−1.39015	−	1.39015i	−0.824960	−	0.565191i	$-0.808803\pi$
$173$	−18.2846	−	18.2846i	−1.39015	−	1.39015i	−0.565191	−	0.824960i	$-0.691197\pi$
$174$		0			0
$175$		−	2.21982i		−	0.167802i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.18124	−	3.18124i	0.237777	−	0.237777i	−0.578152	−	0.815929i	$-0.696226\pi$
$179$	3.18124	−	3.18124i	0.237777	−	0.237777i	0.815929	+	0.578152i	$0.196226\pi$
$180$		0			0
$181$	−18.4064	−	18.4064i	−1.36813	−	1.36813i	−0.863098	−	0.505036i	$-0.831479\pi$
$181$	−18.4064	−	18.4064i	−1.36813	−	1.36813i	−0.505036	−	0.863098i	$-0.668521\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−11.1228			−0.817768
$186$		0			0
$187$	−9.33807	+	9.33807i	−0.682867	+	0.682867i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−22.1964			−1.60607			−0.803037	−	0.595929i	$-0.796784\pi$
$191$	−22.1964			−1.60607			−0.803037	+	0.595929i	$0.796784\pi$
$192$		0			0
$193$	−6.93243			−0.499007			−0.249503	−	0.968374i	$-0.580267\pi$
$193$	−6.93243			−0.499007			−0.249503	+	0.968374i	$0.580267\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.07895	+	6.07895i	−0.433107	+	0.433107i	−0.889684	−	0.456577i	$-0.849075\pi$
$197$	−6.07895	+	6.07895i	−0.433107	+	0.433107i	0.456577	+	0.889684i	$0.349075\pi$
$198$		0			0
$199$	10.9223			0.774265			0.387132	−	0.922024i	$-0.373466\pi$
$199$	10.9223			0.774265			0.387132	+	0.922024i	$0.373466\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.544898	+	0.544898i	0.0382444	+	0.0382444i
$204$		0			0
$205$	−0.574831	+	0.574831i	−0.0401480	+	0.0401480i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	33.1548i		−	2.29337i
$210$		0			0
$211$	9.67612	+	9.67612i	0.666131	+	0.666131i	0.956818	−	0.290687i	$-0.0938838\pi$
$211$	9.67612	+	9.67612i	0.666131	+	0.666131i	−0.290687	+	0.956818i	$0.593884\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	17.8411i		−	1.21675i
$216$		0			0
$217$			10.1006i			0.685672i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.25665	−	5.25665i	−0.353601	−	0.353601i
$222$		0			0
$223$			4.17535i			0.279602i	0.990180	+	0.139801i	$0.0446463\pi$
$223$			4.17535i			0.279602i	−0.990180	+	0.139801i	$0.955354\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−21.2251	+	21.2251i	−1.40876	+	1.40876i	−0.642348	+	0.766413i	$0.722039\pi$
$227$	−21.2251	+	21.2251i	−1.40876	+	1.40876i	−0.766413	+	0.642348i	$0.777961\pi$
$228$		0			0
$229$	−6.46169	−	6.46169i	−0.427000	−	0.427000i	0.460605	−	0.887605i	$-0.347633\pi$
$229$	−6.46169	−	6.46169i	−0.427000	−	0.427000i	−0.887605	+	0.460605i	$0.847633\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.6456			0.697413			0.348707	−	0.937232i	$-0.386621\pi$
$233$	10.6456			0.697413			0.348707	+	0.937232i	$0.386621\pi$
$234$		0			0
$235$	−0.905705	+	0.905705i	−0.0590817	+	0.0590817i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	10.2774			0.664792			0.332396	−	0.943140i	$-0.392143\pi$
$239$	10.2774			0.664792			0.332396	+	0.943140i	$0.392143\pi$
$240$		0			0
$241$	−0.195675			−0.0126045			−0.00630227	−	0.999980i	$-0.502006\pi$
$241$	−0.195675			−0.0126045			−0.00630227	+	0.999980i	$0.502006\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.17902	+	1.17902i	−0.0753249	+	0.0753249i
$246$		0			0
$247$	18.6637			1.18754
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.95377	+	8.95377i	0.565157	+	0.565157i	0.930768	−	0.365611i	$-0.119140\pi$
$251$	8.95377	+	8.95377i	0.565157	+	0.565157i	−0.365611	+	0.930768i	$0.619140\pi$
$252$		0			0
$253$	10.3325	−	10.3325i	0.649599	−	0.649599i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			21.3928i			1.33445i	0.744858	+	0.667223i	$0.232517\pi$
$257$			21.3928i			1.33445i	−0.744858	+	0.667223i	$0.767483\pi$
$258$		0			0
$259$	−4.71698	−	4.71698i	−0.293099	−	0.293099i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			31.7659i			1.95877i	0.202006	+	0.979384i	$0.435254\pi$
$263$			31.7659i			1.95877i	−0.202006	+	0.979384i	$0.564746\pi$
$264$		0			0
$265$			0.634348i			0.0389677i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	11.1414	+	11.1414i	0.679301	+	0.679301i	0.959842	−	0.280541i	$-0.0905140\pi$
$269$	11.1414	+	11.1414i	0.679301	+	0.679301i	−0.280541	+	0.959842i	$0.590514\pi$
$270$		0			0
$271$			19.1289i			1.16200i	0.813904	+	0.580999i	$0.197338\pi$
$271$			19.1289i			1.16200i	−0.813904	+	0.580999i	$0.802662\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.0978	−	10.0978i	0.608922	−	0.608922i
$276$		0			0
$277$	−1.64734	−	1.64734i	−0.0989791	−	0.0989791i	0.655883	−	0.754862i	$-0.272296\pi$
$277$	−1.64734	−	1.64734i	−0.0989791	−	0.0989791i	−0.754862	+	0.655883i	$0.772296\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.73918			0.282716			0.141358	−	0.989959i	$-0.454853\pi$
$281$	4.73918			0.282716			0.141358	+	0.989959i	$0.454853\pi$
$282$		0			0
$283$	−9.70607	+	9.70607i	−0.576966	+	0.576966i	−0.934066	−	0.357100i	$-0.883766\pi$
$283$	−9.70607	+	9.70607i	−0.576966	+	0.576966i	0.357100	+	0.934066i	$0.383766\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.487549			−0.0287791
$288$		0			0
$289$	12.7860			0.752119
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.58419	+	4.58419i	−0.267811	+	0.267811i	−0.828218	−	0.560406i	$-0.810645\pi$
$293$	−4.58419	+	4.58419i	−0.267811	+	0.267811i	0.560406	+	0.828218i	$0.310645\pi$
$294$		0			0
$295$	0.230980			0.0134482
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.81644	+	5.81644i	0.336373	+	0.336373i
$300$		0			0
$301$	7.56607	−	7.56607i	0.436101	−	0.436101i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	17.4902i		−	1.00149i
$306$		0			0
$307$	12.3272	+	12.3272i	0.703552	+	0.703552i	0.965171	−	0.261619i	$-0.0842565\pi$
$307$	12.3272	+	12.3272i	0.703552	+	0.703552i	−0.261619	+	0.965171i	$0.584256\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	1.24831i		−	0.0707852i	−0.999373	−	0.0353926i	$-0.988732\pi$
$311$		−	1.24831i		−	0.0707852i	0.999373	−	0.0353926i	$-0.0112682\pi$
$312$		0			0
$313$		−	23.1106i		−	1.30629i	−0.757234	−	0.653144i	$-0.773450\pi$
$313$		−	23.1106i		−	1.30629i	0.757234	−	0.653144i	$-0.226550\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.92113	+	6.92113i	0.388729	+	0.388729i	0.874234	−	0.485505i	$-0.161364\pi$
$317$	6.92113	+	6.92113i	0.388729	+	0.388729i	−0.485505	+	0.874234i	$0.661364\pi$
$318$		0			0
$319$			4.95743i			0.277563i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.48085	−	7.48085i	0.416245	−	0.416245i
$324$		0			0
$325$	5.68434	+	5.68434i	0.315311	+	0.315311i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.768184			−0.0423513
$330$		0			0
$331$	−14.3780	+	14.3780i	−0.790288	+	0.790288i	−0.981541	−	0.191253i	$-0.938745\pi$
$331$	−14.3780	+	14.3780i	−0.790288	+	0.790288i	0.191253	+	0.981541i	$0.438745\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	16.1283			0.881186
$336$		0			0
$337$	−5.99526			−0.326583			−0.163291	−	0.986578i	$-0.552211\pi$
$337$	−5.99526			−0.326583			−0.163291	+	0.986578i	$0.552211\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−45.9471	+	45.9471i	−2.48817	+	2.48817i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.6602	+	11.6602i	0.625954	+	0.625954i	0.947047	−	0.321094i	$-0.104050\pi$
$347$	11.6602	+	11.6602i	0.625954	+	0.625954i	−0.321094	+	0.947047i	$0.604050\pi$
$348$		0			0
$349$	−18.7098	+	18.7098i	−1.00151	+	1.00151i	−0.00151266	+	0.999999i	$0.500481\pi$
$349$	−18.7098	+	18.7098i	−1.00151	+	1.00151i	−0.999999	+	0.00151266i	$0.999519\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	4.65833i		−	0.247938i	−0.992286	−	0.123969i	$-0.960438\pi$
$353$		−	4.65833i		−	0.247938i	0.992286	−	0.123969i	$-0.0395624\pi$
$354$		0			0
$355$	10.2141	+	10.2141i	0.542106	+	0.542106i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.3794i		−	0.706140i	−0.935597	−	0.353070i	$-0.885138\pi$
$359$		−	13.3794i		−	0.706140i	0.935597	−	0.353070i	$-0.114862\pi$
$360$		0			0
$361$			7.56074i			0.397934i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.0769	−	16.0769i	−0.841501	−	0.841501i
$366$		0			0
$367$			20.7376i			1.08249i	0.840864	+	0.541247i	$0.182047\pi$
$367$			20.7376i			1.08249i	−0.840864	+	0.541247i	$0.817953\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.269015	+	0.269015i	−0.0139665	+	0.0139665i
$372$		0			0
$373$	6.84888	+	6.84888i	0.354622	+	0.354622i	0.861826	−	0.507204i	$-0.169321\pi$
$373$	6.84888	+	6.84888i	0.354622	+	0.354622i	−0.507204	+	0.861826i	$0.669321\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.79067			−0.143727
$378$		0			0
$379$	−17.1046	+	17.1046i	−0.878602	+	0.878602i	−0.993390	−	0.114788i	$-0.963381\pi$
$379$	−17.1046	+	17.1046i	−0.878602	+	0.878602i	0.114788	+	0.993390i	$0.463381\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.7106			0.751678			0.375839	−	0.926685i	$-0.377355\pi$
$383$	14.7106			0.751678			0.375839	+	0.926685i	$0.377355\pi$
$384$		0			0
$385$	−10.7266			−0.546680
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4.86662	−	4.86662i	0.246747	−	0.246747i	−0.572887	−	0.819634i	$-0.694177\pi$
$389$	4.86662	−	4.86662i	0.246747	−	0.246747i	0.819634	+	0.572887i	$0.194177\pi$
$390$		0			0
$391$	4.66272			0.235804
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10.9548	−	10.9548i	−0.551195	−	0.551195i
$396$		0			0
$397$	25.4535	−	25.4535i	1.27747	−	1.27747i	0.335396	−	0.942077i	$-0.391130\pi$
$397$	25.4535	−	25.4535i	1.27747	−	1.27747i	0.942077	−	0.335396i	$-0.108870\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	21.0632i		−	1.05184i	−0.850533	−	0.525922i	$-0.823720\pi$
$401$		−	21.0632i		−	1.05184i	0.850533	−	0.525922i	$-0.176280\pi$
$402$		0			0
$403$	−25.8648	−	25.8648i	−1.28842	−	1.28842i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	42.9146i		−	2.12720i
$408$		0			0
$409$			36.0796i			1.78402i	0.452016	+	0.892010i	$0.350705\pi$
$409$			36.0796i			1.78402i	−0.452016	+	0.892010i	$0.649295\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.0979540	+	0.0979540i	0.00482000	+	0.00482000i
$414$		0			0
$415$		−	23.0296i		−	1.13048i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.95680	+	4.95680i	−0.242155	+	0.242155i	−0.817741	−	0.575586i	$-0.804774\pi$
$419$	−4.95680	+	4.95680i	−0.242155	+	0.242155i	0.575586	+	0.817741i	$0.304774\pi$
$420$		0			0
$421$	−7.80250	−	7.80250i	−0.380271	−	0.380271i	0.490929	−	0.871200i	$-0.336657\pi$
$421$	−7.80250	−	7.80250i	−0.380271	−	0.380271i	−0.871200	+	0.490929i	$0.836657\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.55683			0.221039
$426$		0			0
$427$	7.41725	−	7.41725i	0.358946	−	0.358946i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.41852			0.212832			0.106416	−	0.994322i	$-0.466062\pi$
$431$	4.41852			0.212832			0.106416	+	0.994322i	$0.466062\pi$
$432$		0			0
$433$	−3.06712			−0.147396			−0.0736981	−	0.997281i	$-0.523480\pi$
$433$	−3.06712			−0.147396			−0.0736981	+	0.997281i	$0.523480\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.27750	+	8.27750i	−0.395966	+	0.395966i
$438$		0			0
$439$	4.24215			0.202467			0.101233	−	0.994863i	$-0.467721\pi$
$439$	4.24215			0.202467			0.101233	+	0.994863i	$0.467721\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.34161	−	2.34161i	−0.111253	−	0.111253i	0.649289	−	0.760542i	$-0.275067\pi$
$443$	−2.34161	−	2.34161i	−0.111253	−	0.111253i	−0.760542	+	0.649289i	$0.775067\pi$
$444$		0			0
$445$	−8.57567	+	8.57567i	−0.406526	+	0.406526i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	8.86849i		−	0.418530i	−0.977859	−	0.209265i	$-0.932893\pi$
$449$		−	8.86849i		−	0.418530i	0.977859	−	0.209265i	$-0.0671071\pi$
$450$		0			0
$451$	−2.21784	−	2.21784i	−0.104434	−	0.104434i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	6.03831i		−	0.283080i
$456$		0			0
$457$		−	7.14847i		−	0.334392i	−0.985924	−	0.167196i	$-0.946529\pi$
$457$		−	7.14847i		−	0.334392i	0.985924	−	0.167196i	$-0.0534712\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.8346	+	17.8346i	0.830638	+	0.830638i	0.987604	−	0.156966i	$-0.0501714\pi$
$461$	17.8346	+	17.8346i	0.830638	+	0.830638i	−0.156966	+	0.987604i	$0.550171\pi$
$462$		0			0
$463$		−	16.1618i		−	0.751102i	−0.926802	−	0.375551i	$-0.877453\pi$
$463$		−	16.1618i		−	0.751102i	0.926802	−	0.375551i	$-0.122547\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	13.5356	−	13.5356i	0.626352	−	0.626352i	−0.320797	−	0.947148i	$-0.603951\pi$
$467$	13.5356	−	13.5356i	0.626352	−	0.626352i	0.947148	+	0.320797i	$0.103951\pi$
$468$		0			0
$469$	6.83972	+	6.83972i	0.315829	+	0.315829i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	68.8354			3.16505
$474$		0			0
$475$	−8.08951	+	8.08951i	−0.371172	+	0.371172i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.85872			−0.450456			−0.225228	−	0.974306i	$-0.572313\pi$
$479$	−9.85872			−0.450456			−0.225228	+	0.974306i	$0.572313\pi$
$480$		0			0
$481$	24.1578			1.10150
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.3218	+	12.3218i	−0.559503	+	0.559503i
$486$		0			0
$487$	29.4055			1.33249			0.666245	−	0.745733i	$-0.267900\pi$
$487$	29.4055			1.33249			0.666245	+	0.745733i	$0.267900\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.21529	+	5.21529i	0.235363	+	0.235363i	0.814927	−	0.579564i	$-0.196777\pi$
$491$	5.21529	+	5.21529i	0.235363	+	0.235363i	−0.579564	+	0.814927i	$0.696777\pi$
$492$		0			0
$493$	−1.11856	+	1.11856i	−0.0503776	+	0.0503776i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			8.66316i			0.388596i
$498$		0			0
$499$	11.8610	+	11.8610i	0.530973	+	0.530973i	0.920862	−	0.389889i	$-0.127487\pi$
$499$	11.8610	+	11.8610i	0.530973	+	0.530973i	−0.389889	+	0.920862i	$0.627487\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9.18051i		−	0.409339i	−0.978831	−	0.204669i	$-0.934388\pi$
$503$		−	9.18051i		−	0.409339i	0.978831	−	0.204669i	$-0.0656119\pi$
$504$		0			0
$505$		−	13.5588i		−	0.603357i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15.7454	−	15.7454i	−0.697902	−	0.697902i	0.266056	−	0.963958i	$-0.414279\pi$
$509$	−15.7454	−	15.7454i	−0.697902	−	0.697902i	−0.963958	+	0.266056i	$0.914279\pi$
$510$		0			0
$511$		−	13.6358i		−	0.603210i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.39351	−	8.39351i	0.369862	−	0.369862i
$516$		0			0
$517$	−3.49443	−	3.49443i	−0.153685	−	0.153685i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−35.2555			−1.54457			−0.772286	−	0.635275i	$-0.780887\pi$
$521$	−35.2555			−1.54457			−0.772286	+	0.635275i	$0.780887\pi$
$522$		0			0
$523$	−13.3455	+	13.3455i	−0.583559	+	0.583559i	−0.935880	−	0.352320i	$-0.885393\pi$
$523$	−13.3455	+	13.3455i	−0.583559	+	0.583559i	0.352320	+	0.935880i	$0.385393\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−20.7344			−0.903206
$528$		0			0
$529$	17.8407			0.775684
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.24848	−	1.24848i	0.0540777	−	0.0540777i
$534$		0			0
$535$	14.4688			0.625540
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.54895	−	4.54895i	−0.195937	−	0.195937i
$540$		0			0
$541$	22.4225	−	22.4225i	0.964020	−	0.964020i	−0.0353552	−	0.999375i	$-0.511256\pi$
$541$	22.4225	−	22.4225i	0.964020	−	0.964020i	0.999375	+	0.0353552i	$0.0112563\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			3.63950i			0.155899i
$546$		0			0
$547$	6.02624	+	6.02624i	0.257664	+	0.257664i	0.824103	−	0.566440i	$-0.191680\pi$
$547$	6.02624	+	6.02624i	0.257664	+	0.257664i	−0.566440	+	0.824103i	$0.691680\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	3.97146i		−	0.169190i
$552$		0			0
$553$		−	9.29142i		−	0.395111i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	19.9169	+	19.9169i	0.843905	+	0.843905i	0.989364	−	0.145459i	$-0.0464658\pi$
$557$	19.9169	+	19.9169i	0.843905	+	0.843905i	−0.145459	+	0.989364i	$0.546466\pi$
$558$		0			0
$559$			38.7493i			1.63892i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.0269	+	18.0269i	−0.759743	+	0.759743i	−0.976275	−	0.216533i	$-0.930525\pi$
$563$	−18.0269	+	18.0269i	−0.759743	+	0.759743i	0.216533	+	0.976275i	$0.430525\pi$
$564$		0			0
$565$	−16.5090	−	16.5090i	−0.694541	−	0.694541i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−28.6582			−1.20142			−0.600708	−	0.799469i	$-0.705115\pi$
$569$	−28.6582			−1.20142			−0.600708	+	0.799469i	$0.705115\pi$
$570$		0			0
$571$	4.74392	−	4.74392i	0.198527	−	0.198527i	−0.600841	−	0.799368i	$-0.705168\pi$
$571$	4.74392	−	4.74392i	0.198527	−	0.198527i	0.799368	+	0.600841i	$0.205168\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.04209			−0.210270
$576$		0			0
$577$	−27.7361			−1.15467			−0.577335	−	0.816507i	$-0.695907\pi$
$577$	−27.7361			−1.15467			−0.577335	+	0.816507i	$0.695907\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	9.76640	−	9.76640i	0.405179	−	0.405179i
$582$		0			0
$583$	−2.44747			−0.101364
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.8086	+	19.8086i	0.817587	+	0.817587i	0.985758	−	0.168171i	$-0.0537860\pi$
$587$	19.8086	+	19.8086i	0.817587	+	0.817587i	−0.168171	+	0.985758i	$0.553786\pi$
$588$		0			0
$589$	36.8088	−	36.8088i	1.51668	−	1.51668i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			18.8296i			0.773239i	0.922239	+	0.386620i	$0.126357\pi$
$593$			18.8296i			0.773239i	−0.922239	+	0.386620i	$0.873643\pi$
$594$		0			0
$595$	−2.42029	−	2.42029i	−0.0992222	−	0.0992222i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	24.0141i		−	0.981188i	−0.871388	−	0.490594i	$-0.836780\pi$
$599$		−	24.0141i		−	0.981188i	0.871388	−	0.490594i	$-0.163220\pi$
$600$		0			0
$601$			14.3602i			0.585766i	0.956148	+	0.292883i	$0.0946147\pi$
$601$			14.3602i			0.585766i	−0.956148	+	0.292883i	$0.905385\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−35.8257	−	35.8257i	−1.45652	−	1.45652i
$606$		0			0
$607$		−	6.44642i		−	0.261652i	−0.991405	−	0.130826i	$-0.958237\pi$
$607$		−	6.44642i		−	0.261652i	0.991405	−	0.130826i	$-0.0417629\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.96711	−	1.96711i	0.0795807	−	0.0795807i
$612$		0			0
$613$	−31.4106	−	31.4106i	−1.26866	−	1.26866i	−0.946779	−	0.321884i	$-0.895684\pi$
$613$	−31.4106	−	31.4106i	−1.26866	−	1.26866i	−0.321884	−	0.946779i	$-0.604316\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	30.2797			1.21902			0.609508	−	0.792780i	$-0.291367\pi$
$617$	30.2797			1.21902			0.609508	+	0.792780i	$0.291367\pi$
$618$		0			0
$619$	−6.97556	+	6.97556i	−0.280372	+	0.280372i	−0.833257	−	0.552886i	$-0.813527\pi$
$619$	−6.97556	+	6.97556i	−0.280372	+	0.280372i	0.552886	+	0.833257i	$0.313527\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.27355			−0.291409
$624$		0			0
$625$	8.97334			0.358934
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9.68300	−	9.68300i	0.386086	−	0.386086i
$630$		0			0
$631$	−14.0268			−0.558398			−0.279199	−	0.960233i	$-0.590069\pi$
$631$	−14.0268			−0.558398			−0.279199	+	0.960233i	$0.590069\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.28346	+	6.28346i	0.249352	+	0.249352i
$636$		0			0
$637$	2.56073	−	2.56073i	0.101460	−	0.101460i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			20.9884i			0.828992i	0.910051	+	0.414496i	$0.136042\pi$
$641$			20.9884i			0.828992i	−0.910051	+	0.414496i	$0.863958\pi$
$642$		0			0
$643$	−32.1818	−	32.1818i	−1.26913	−	1.26913i	−0.946539	−	0.322589i	$-0.895447\pi$
$643$	−32.1818	−	32.1818i	−1.26913	−	1.26913i	−0.322589	−	0.946539i	$-0.604553\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			0.647703i			0.0254638i	0.999919	+	0.0127319i	$0.00405280\pi$
$647$			0.647703i			0.0254638i	−0.999919	+	0.0127319i	$0.995947\pi$
$648$		0			0
$649$			0.891176i			0.0349817i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−25.7373	−	25.7373i	−1.00718	−	1.00718i	−0.999974	−	0.00720262i	$-0.997707\pi$
$653$	−25.7373	−	25.7373i	−1.00718	−	1.00718i	−0.00720262	−	0.999974i	$-0.502293\pi$
$654$		0			0
$655$		−	31.6837i		−	1.23798i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	17.0862	−	17.0862i	0.665583	−	0.665583i	−0.291108	−	0.956690i	$-0.594024\pi$
$659$	17.0862	−	17.0862i	0.665583	−	0.665583i	0.956690	+	0.291108i	$0.0940238\pi$
$660$		0			0
$661$	−8.34196	−	8.34196i	−0.324465	−	0.324465i	0.526012	−	0.850477i	$-0.323687\pi$
$661$	−8.34196	−	8.34196i	−0.324465	−	0.324465i	−0.850477	+	0.526012i	$0.823687\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.59324			0.333232
$666$		0			0
$667$	1.23768	−	1.23768i	0.0479233	−	0.0479233i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	67.4814			2.60509
$672$		0			0
$673$	−1.69237			−0.0652361			−0.0326180	−	0.999468i	$-0.510384\pi$
$673$	−1.69237			−0.0652361			−0.0326180	+	0.999468i	$0.510384\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	36.3935	−	36.3935i	1.39872	−	1.39872i	0.594968	−	0.803749i	$-0.297165\pi$
$677$	36.3935	−	36.3935i	1.39872	−	1.39872i	0.803749	−	0.594968i	$-0.202835\pi$
$678$		0			0
$679$	−10.4508			−0.401066
$680$		0			0
$681$		0			0
$682$		0			0
$683$	13.6522	+	13.6522i	0.522388	+	0.522388i	0.918292	−	0.395904i	$-0.129569\pi$
$683$	13.6522	+	13.6522i	0.522388	+	0.522388i	−0.395904	+	0.918292i	$0.629569\pi$
$684$		0			0
$685$	11.9911	−	11.9911i	0.458156	−	0.458156i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1.37775i		−	0.0524880i
$690$		0			0
$691$	2.36989	+	2.36989i	0.0901550	+	0.0901550i	0.750746	−	0.660591i	$-0.229694\pi$
$691$	2.36989	+	2.36989i	0.0901550	+	0.0901550i	−0.660591	+	0.750746i	$0.729694\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			16.2726i			0.617254i
$696$		0			0
$697$		−	1.00084i		−	0.0379095i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27.6655	+	27.6655i	1.04491	+	1.04491i	0.998943	+	0.0459668i	$0.0146369\pi$
$701$	27.6655	+	27.6655i	1.04491	+	1.04491i	0.0459668	+	0.998943i	$0.485363\pi$
$702$		0			0
$703$			34.3795i			1.29665i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.75000	−	5.75000i	0.216251	−	0.216251i
$708$		0			0
$709$	4.38704	+	4.38704i	0.164759	+	0.164759i	0.784671	−	0.619912i	$-0.212832\pi$
$709$	4.38704	+	4.38704i	0.164759	+	0.164759i	−0.619912	+	0.784671i	$0.712832\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	22.9425			0.859202
$714$		0			0
$715$	27.4680	−	27.4680i	1.02724	−	1.02724i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−18.2313			−0.679915			−0.339957	−	0.940441i	$-0.610413\pi$
$719$	−18.2313			−0.679915			−0.339957	+	0.940441i	$0.610413\pi$
$720$		0			0
$721$	7.11905			0.265127
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.20957	−	1.20957i	0.0449224	−	0.0449224i
$726$		0			0
$727$	−17.1985			−0.637857			−0.318928	−	0.947779i	$-0.603323\pi$
$727$	−17.1985			−0.637857			−0.318928	+	0.947779i	$0.603323\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	15.5316	+	15.5316i	0.574457	+	0.574457i
$732$		0			0
$733$	1.76630	−	1.76630i	0.0652396	−	0.0652396i	−0.673734	−	0.738974i	$-0.735311\pi$
$733$	1.76630	−	1.76630i	0.0652396	−	0.0652396i	0.738974	+	0.673734i	$0.235311\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			62.2271i			2.29216i
$738$		0			0
$739$	−20.8217	−	20.8217i	−0.765939	−	0.765939i	0.211450	−	0.977389i	$-0.432182\pi$
$739$	−20.8217	−	20.8217i	−0.765939	−	0.765939i	−0.977389	+	0.211450i	$0.932182\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			39.1460i			1.43613i	0.695977	+	0.718064i	$0.254971\pi$
$743$			39.1460i			1.43613i	−0.695977	+	0.718064i	$0.745029\pi$
$744$		0			0
$745$			0.316196i			0.0115845i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.13593	+	6.13593i	0.224202	+	0.224202i
$750$		0			0
$751$		−	44.9147i		−	1.63896i	−0.573106	−	0.819481i	$-0.694262\pi$
$751$		−	44.9147i		−	1.63896i	0.573106	−	0.819481i	$-0.305738\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.1904	+	16.1904i	−0.589229	+	0.589229i
$756$		0			0
$757$	−35.8730	−	35.8730i	−1.30383	−	1.30383i	−0.925791	−	0.378036i	$-0.876600\pi$
$757$	−35.8730	−	35.8730i	−1.30383	−	1.30383i	−0.378036	−	0.925791i	$-0.623400\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−47.4451			−1.71988			−0.859941	−	0.510393i	$-0.829500\pi$
$761$	−47.4451			−1.71988			−0.859941	+	0.510393i	$0.829500\pi$
$762$		0			0
$763$	−1.54344	+	1.54344i	−0.0558764	+	0.0558764i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.501667			−0.0181141
$768$		0			0
$769$	4.59306			0.165630			0.0828150	−	0.996565i	$-0.473609\pi$
$769$	4.59306			0.165630			0.0828150	+	0.996565i	$0.473609\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.66000	+	1.66000i	−0.0597062	+	0.0597062i	−0.736329	−	0.676623i	$-0.763443\pi$
$773$	−1.66000	+	1.66000i	−0.0597062	+	0.0597062i	0.676623	+	0.736329i	$0.263443\pi$
$774$		0			0
$775$	22.4214			0.805401
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.77674	+	1.77674i	0.0636583	+	0.0636583i
$780$		0			0
$781$	−39.4083	+	39.4083i	−1.41014	+	1.41014i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	5.87761i		−	0.209781i
$786$		0			0
$787$	16.7673	+	16.7673i	0.597688	+	0.597688i	0.939697	−	0.342009i	$-0.111107\pi$
$787$	16.7673	+	16.7673i	0.597688	+	0.597688i	−0.342009	+	0.939697i	$0.611107\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	14.0023i		−	0.497865i
$792$		0			0
$793$			37.9871i			1.34896i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	24.6258	+	24.6258i	0.872290	+	0.872290i	0.992722	−	0.120431i	$-0.0384277\pi$
$797$	24.6258	+	24.6258i	0.872290	+	0.872290i	−0.120431	+	0.992722i	$0.538428\pi$
$798$		0			0
$799$		−	1.57692i		−	0.0557876i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	62.0284	−	62.0284i	2.18894	−	2.18894i
$804$		0			0
$805$	2.67803	+	2.67803i	0.0943882	+	0.0943882i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	41.1357			1.44626			0.723128	−	0.690714i	$-0.242704\pi$
$809$	41.1357			1.44626			0.723128	+	0.690714i	$0.242704\pi$
$810$		0			0
$811$	3.55287	−	3.55287i	0.124758	−	0.124758i	−0.641971	−	0.766729i	$-0.721883\pi$
$811$	3.55287	−	3.55287i	0.124758	−	0.124758i	0.766729	+	0.641971i	$0.221883\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	15.1670			0.531277
$816$		0			0
$817$	−55.1449			−1.92928
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.376144	+	0.376144i	−0.0131275	+	0.0131275i	−0.713640	−	0.700513i	$-0.752955\pi$
$821$	−0.376144	+	0.376144i	−0.0131275	+	0.0131275i	0.700513	+	0.713640i	$0.252955\pi$
$822$		0			0
$823$	−6.94476			−0.242079			−0.121039	−	0.992648i	$-0.538623\pi$
$823$	−6.94476			−0.242079			−0.121039	+	0.992648i	$0.538623\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−7.45823	−	7.45823i	−0.259348	−	0.259348i	0.565441	−	0.824789i	$-0.308706\pi$
$827$	−7.45823	−	7.45823i	−0.259348	−	0.259348i	−0.824789	+	0.565441i	$0.808706\pi$
$828$		0			0
$829$	−15.4119	+	15.4119i	−0.535279	+	0.535279i	−0.922139	−	0.386860i	$-0.873560\pi$
$829$	−15.4119	+	15.4119i	−0.535279	+	0.535279i	0.386860	+	0.922139i	$0.373560\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	2.05280i		−	0.0711251i
$834$		0			0
$835$	5.17113	+	5.17113i	0.178954	+	0.178954i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	17.2950i		−	0.597091i	−0.954395	−	0.298545i	$-0.903499\pi$
$839$		−	17.2950i		−	0.597091i	0.954395	−	0.298545i	$-0.0965014\pi$
$840$		0			0
$841$		−	28.4062i		−	0.979523i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.135176	+	0.135176i	0.00465019	+	0.00465019i
$846$		0			0
$847$		−	30.3859i		−	1.04407i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−10.7142	+	10.7142i	−0.367277	+	0.367277i
$852$		0			0
$853$	−17.7845	−	17.7845i	−0.608930	−	0.608930i	0.333736	−	0.942666i	$-0.391690\pi$
$853$	−17.7845	−	17.7845i	−0.608930	−	0.608930i	−0.942666	+	0.333736i	$0.891690\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.25624			0.145390			0.0726952	−	0.997354i	$-0.476840\pi$
$857$	4.25624			0.145390			0.0726952	+	0.997354i	$0.476840\pi$
$858$		0			0
$859$	−27.8297	+	27.8297i	−0.949539	+	0.949539i	−0.998787	−	0.0492479i	$-0.984318\pi$
$859$	−27.8297	+	27.8297i	−0.949539	+	0.949539i	0.0492479	+	0.998787i	$0.484318\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−8.41542			−0.286464			−0.143232	−	0.989689i	$-0.545750\pi$
$863$	−8.41542			−0.286464			−0.143232	+	0.989689i	$0.545750\pi$
$864$		0			0
$865$	43.1158			1.46598
$866$		0			0
$867$		0			0
$868$		0			0
$869$	42.2662	−	42.2662i	1.43378	−	1.43378i
$870$		0			0
$871$	−35.0293			−1.18692
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.51232	+	8.51232i	0.287769	+	0.287769i
$876$		0			0
$877$	−14.9657	+	14.9657i	−0.505356	+	0.505356i	−0.913097	−	0.407742i	$-0.866316\pi$
$877$	−14.9657	+	14.9657i	−0.505356	+	0.505356i	0.407742	+	0.913097i	$0.366316\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	16.9109i		−	0.569743i	−0.958566	−	0.284871i	$-0.908049\pi$
$881$		−	16.9109i		−	0.569743i	0.958566	−	0.284871i	$-0.0919509\pi$
$882$		0			0
$883$	−21.0835	−	21.0835i	−0.709518	−	0.709518i	0.256916	−	0.966434i	$-0.417294\pi$
$883$	−21.0835	−	21.0835i	−0.709518	−	0.709518i	−0.966434	+	0.256916i	$0.917294\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			41.4937i			1.39322i	0.717450	+	0.696610i	$0.245309\pi$
$887$			41.4937i			1.39322i	−0.717450	+	0.696610i	$0.754691\pi$
$888$		0			0
$889$			5.32939i			0.178742i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.79943	+	2.79943i	0.0936795	+	0.0936795i
$894$		0			0
$895$			7.50150i			0.250748i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.50379	+	5.50379i	−0.183562	+	0.183562i
$900$		0			0
$901$	−0.552232	−	0.552232i	−0.0183975	−	0.0183975i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	43.4030			1.44276
$906$		0			0
$907$	3.24476	−	3.24476i	0.107740	−	0.107740i	−0.651182	−	0.758922i	$-0.725726\pi$
$907$	3.24476	−	3.24476i	0.107740	−	0.107740i	0.758922	+	0.651182i	$0.225726\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	47.2047			1.56396			0.781980	−	0.623303i	$-0.214210\pi$
$911$	47.2047			1.56396			0.781980	+	0.623303i	$0.214210\pi$
$912$		0			0
$913$	88.8538			2.94063
$914$		0			0
$915$		0			0
$916$		0			0
$917$	13.4364	−	13.4364i	0.443710	−	0.443710i
$918$		0			0
$919$	−21.1617			−0.698061			−0.349030	−	0.937111i	$-0.613489\pi$
$919$	−21.1617			−0.698061			−0.349030	+	0.937111i	$0.613489\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−22.1840	−	22.1840i	−0.730195	−	0.730195i
$924$		0			0
$925$	−10.4708	+	10.4708i	−0.344279	+	0.344279i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	54.4109i		−	1.78516i	−0.450886	−	0.892581i	$-0.648892\pi$
$929$		−	54.4109i		−	1.78516i	0.450886	−	0.892581i	$-0.351108\pi$
$930$		0			0
$931$	3.64422	+	3.64422i	0.119435	+	0.119435i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	22.0196i		−	0.720117i
$936$		0			0
$937$		−	40.2919i		−	1.31628i	−0.752895	−	0.658140i	$-0.771343\pi$
$937$		−	40.2919i		−	1.31628i	0.752895	−	0.658140i	$-0.228657\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21.2053	+	21.2053i	0.691272	+	0.691272i	0.962512	−	0.271239i	$-0.0874335\pi$
$941$	21.2053	+	21.2053i	0.691272	+	0.691272i	−0.271239	+	0.962512i	$0.587434\pi$
$942$		0			0
$943$			1.10742i			0.0360626i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−37.6027	+	37.6027i	−1.22192	+	1.22192i	−0.254977	+	0.966947i	$0.582068\pi$
$947$	−37.6027	+	37.6027i	−1.22192	+	1.22192i	−0.966947	+	0.254977i	$0.917932\pi$
$948$		0			0
$949$	34.9175	+	34.9175i	1.13347	+	1.13347i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−19.9135			−0.645062			−0.322531	−	0.946559i	$-0.604534\pi$
$953$	−19.9135			−0.645062			−0.322531	+	0.946559i	$0.604534\pi$
$954$		0			0
$955$	26.1700	−	26.1700i	0.846842	−	0.846842i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	10.1704			0.328418
$960$		0			0
$961$	−71.0217			−2.29102
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.17348	−	8.17348i	0.263114	−	0.263114i
$966$		0			0
$967$	24.5899			0.790759			0.395380	−	0.918518i	$-0.370613\pi$
$967$	24.5899			0.790759			0.395380	+	0.918518i	$0.370613\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.9255	+	19.9255i	0.639438	+	0.639438i	0.950417	−	0.310979i	$-0.100657\pi$
$971$	19.9255	+	19.9255i	0.639438	+	0.639438i	−0.310979	+	0.950417i	$0.600657\pi$
$972$		0			0
$973$	−6.90089	+	6.90089i	−0.221232	+	0.221232i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	50.0455i		−	1.60110i	−0.599267	−	0.800549i	$-0.704541\pi$
$977$		−	50.0455i		−	1.60110i	0.599267	−	0.800549i	$-0.295459\pi$
$978$		0			0
$979$	−33.0870	−	33.0870i	−1.05747	−	1.05747i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			17.3760i			0.554207i	0.960840	+	0.277104i	$0.0893746\pi$
$983$			17.3760i			0.554207i	−0.960840	+	0.277104i	$0.910625\pi$
$984$		0			0
$985$		−	14.3344i		−	0.456733i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−17.1856	−	17.1856i	−0.546470	−	0.546470i
$990$		0			0
$991$		−	44.1839i		−	1.40355i	−0.712399	−	0.701774i	$-0.752391\pi$
$991$		−	44.1839i		−	1.40355i	0.712399	−	0.701774i	$-0.247609\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−12.8777	+	12.8777i	−0.408250	+	0.408250i
$996$		0			0
$997$	1.76515	+	1.76515i	0.0559030	+	0.0559030i	0.734506	−	0.678603i	$-0.237414\pi$
$997$	1.76515	+	1.76515i	0.0559030	+	0.0559030i	−0.678603	+	0.734506i	$0.737414\pi$
$998$		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.3599.6		40
3.2	odd	2	inner	4032.2.v.e.3599.15		40
4.3	odd	2		1008.2.v.e.827.1	yes	40
12.11	even	2		1008.2.v.e.827.20	yes	40
16.3	odd	4	inner	4032.2.v.e.1583.15		40
16.13	even	4		1008.2.v.e.323.20	yes	40
48.29	odd	4		1008.2.v.e.323.1	✓	40
48.35	even	4	inner	4032.2.v.e.1583.6		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.e.323.1	✓	40	48.29	odd	4
1008.2.v.e.323.20	yes	40	16.13	even	4
1008.2.v.e.827.1	yes	40	4.3	odd	2
1008.2.v.e.827.20	yes	40	12.11	even	2
4032.2.v.e.1583.6		40	48.35	even	4	inner
4032.2.v.e.1583.15		40	16.3	odd	4	inner
4032.2.v.e.3599.6		40	1.1	even	1	trivial
4032.2.v.e.3599.15		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)

\(f(q)\)	\(=\)	\(q+(-1.17902 + 1.17902i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(-1.17902 + 1.17902i) q^{5} -1.00000 q^{7} +(-4.54895 - 4.54895i) q^{11} +(2.56073 - 2.56073i) q^{13} -2.05280i q^{17} +(3.64422 + 3.64422i) q^{19} +2.27140i q^{23} +2.21982i q^{25} +(-0.544898 - 0.544898i) q^{29} -10.1006i q^{31} +(1.17902 - 1.17902i) q^{35} +(4.71698 + 4.71698i) q^{37} +0.487549 q^{41} +(-7.56607 + 7.56607i) q^{43} +0.768184 q^{47} +1.00000 q^{49} +(0.269015 - 0.269015i) q^{53} +10.7266 q^{55} +(-0.0979540 - 0.0979540i) q^{59} +(-7.41725 + 7.41725i) q^{61} +6.03831i q^{65} +(-6.83972 - 6.83972i) q^{67} -8.66316i q^{71} +13.6358i q^{73} +(4.54895 + 4.54895i) q^{77} +9.29142i q^{79} +(-9.76640 + 9.76640i) q^{83} +(2.42029 + 2.42029i) q^{85} +7.27355 q^{89} +(-2.56073 + 2.56073i) q^{91} -8.59324 q^{95} +10.4508 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{7}+O(q^{10}) \) 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

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