LMFDB - Embedded newform 1080.2.s.a.593.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,2,Mod(377,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))

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

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

chi := DirichletCharacter("1080.377");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1080.s (of order $4$, degree $2$, 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:	$8.62384341830$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.2
Character	$\chi$	$=$	1080.593
Dual form			1080.2.s.a.377.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.03577 - 0.925011i) q^{5} +(-3.35220 + 3.35220i) q^{7} +O(q^{10})$	$q+(-2.03577 - 0.925011i) q^{5} +(-3.35220 + 3.35220i) q^{7} -1.49825i q^{11} +(2.20524 + 2.20524i) q^{13} +(0.0299519 + 0.0299519i) q^{17} -7.47881i q^{19} +(5.52391 - 5.52391i) q^{23} +(3.28871 + 3.76622i) q^{25} +9.41757 q^{29} -8.02681 q^{31} +(9.92513 - 3.72349i) q^{35} +(2.68969 - 2.68969i) q^{37} -10.3802i q^{41} +(4.44603 + 4.44603i) q^{43} +(-3.12030 - 3.12030i) q^{47} -15.4745i q^{49} +(0.0656692 - 0.0656692i) q^{53} +(-1.38590 + 3.05009i) q^{55} -4.96288 q^{59} -1.29978 q^{61} +(-2.44949 - 6.52923i) q^{65} +(3.91611 - 3.91611i) q^{67} -2.96933i q^{71} +(3.40354 + 3.40354i) q^{73} +(5.02243 + 5.02243i) q^{77} +11.8939i q^{79} +(0.625277 - 0.625277i) q^{83} +(-0.0332693 - 0.0886808i) q^{85} -1.51721 q^{89} -14.7848 q^{91} +(-6.91798 + 15.2251i) q^{95} +(6.30900 - 6.30900i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{7}+O(q^{10}) $ 24 * q - 4 * q^7	$ 24 q - 4 q^{7} - 8 q^{13} - 16 q^{25} - 16 q^{31} + 8 q^{37} - 28 q^{55} + 8 q^{67} - 20 q^{73} + 48 q^{85} - 64 q^{91} + 20 q^{97}+O(q^{100}) $ 24 * q - 4 * q^7 - 8 * q^13 - 16 * q^25 - 16 * q^31 + 8 * q^37 - 28 * q^55 + 8 * q^67 - 20 * q^73 + 48 * q^85 - 64 * q^91 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$	$-1$

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
$5$	−2.03577	−	0.925011i	−0.910424	−	0.413677i
$5$	−2.03577	−	0.925011i	−0.910424	−	0.413677i
$6$		0			0
$7$	−3.35220	+	3.35220i	−1.26701	+	1.26701i	−0.319390	+	0.947623i	$0.603478\pi$
$7$	−3.35220	+	3.35220i	−1.26701	+	1.26701i	−0.947623	+	0.319390i	$0.896522\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.49825i		−	0.451739i	−0.974158	−	0.225869i	$-0.927478\pi$
$11$		−	1.49825i		−	0.451739i	0.974158	−	0.225869i	$-0.0725223\pi$
$12$		0			0
$13$	2.20524	+	2.20524i	0.611624	+	0.611624i	0.943369	−	0.331745i	$-0.107637\pi$
$13$	2.20524	+	2.20524i	0.611624	+	0.611624i	−0.331745	+	0.943369i	$0.607637\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.0299519	+	0.0299519i	0.00726439	+	0.00726439i	0.710730	−	0.703465i	$-0.248365\pi$
$17$	0.0299519	+	0.0299519i	0.00726439	+	0.00726439i	−0.703465	+	0.710730i	$0.748365\pi$
$18$		0			0
$19$		−	7.47881i		−	1.71576i	−0.513852	−	0.857879i	$-0.671782\pi$
$19$		−	7.47881i		−	1.71576i	0.513852	−	0.857879i	$-0.328218\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.52391	−	5.52391i	1.15181	−	1.15181i	0.165626	−	0.986189i	$-0.447035\pi$
$23$	5.52391	−	5.52391i	1.15181	−	1.15181i	0.986189	−	0.165626i	$-0.0529645\pi$
$24$		0			0
$25$	3.28871	+	3.76622i	0.657742	+	0.753243i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	9.41757			1.74880			0.874399	−	0.485208i	$-0.161256\pi$
$29$	9.41757			1.74880			0.874399	+	0.485208i	$0.161256\pi$
$30$		0			0
$31$	−8.02681			−1.44166			−0.720829	−	0.693113i	$-0.756239\pi$
$31$	−8.02681			−1.44166			−0.720829	+	0.693113i	$0.756239\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9.92513	−	3.72349i	1.67765	−	0.629384i
$36$		0			0
$37$	2.68969	−	2.68969i	0.442183	−	0.442183i	−0.450562	−	0.892745i	$-0.648776\pi$
$37$	2.68969	−	2.68969i	0.442183	−	0.442183i	0.892745	+	0.450562i	$0.148776\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	10.3802i		−	1.62112i	−0.585655	−	0.810561i	$-0.699163\pi$
$41$		−	10.3802i		−	1.62112i	0.585655	−	0.810561i	$-0.300837\pi$
$42$		0			0
$43$	4.44603	+	4.44603i	0.678014	+	0.678014i	0.959551	−	0.281537i	$-0.0908441\pi$
$43$	4.44603	+	4.44603i	0.678014	+	0.678014i	−0.281537	+	0.959551i	$0.590844\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.12030	−	3.12030i	−0.455142	−	0.455142i	0.441915	−	0.897057i	$-0.354299\pi$
$47$	−3.12030	−	3.12030i	−0.455142	−	0.455142i	−0.897057	+	0.441915i	$0.854299\pi$
$48$		0			0
$49$		−	15.4745i		−	2.21065i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.0656692	−	0.0656692i	0.00902036	−	0.00902036i	−0.702582	−	0.711603i	$-0.747970\pi$
$53$	0.0656692	−	0.0656692i	0.00902036	−	0.00902036i	0.711603	+	0.702582i	$0.247970\pi$
$54$		0			0
$55$	−1.38590	+	3.05009i	−0.186874	+	0.411274i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.96288			−0.646111			−0.323056	−	0.946380i	$-0.604710\pi$
$59$	−4.96288			−0.646111			−0.323056	+	0.946380i	$0.604710\pi$
$60$		0			0
$61$	−1.29978			−0.166420			−0.0832100	−	0.996532i	$-0.526517\pi$
$61$	−1.29978			−0.166420			−0.0832100	+	0.996532i	$0.526517\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.44949	−	6.52923i	−0.303822	−	0.809852i
$66$		0			0
$67$	3.91611	−	3.91611i	0.478429	−	0.478429i	−0.426200	−	0.904629i	$-0.640148\pi$
$67$	3.91611	−	3.91611i	0.478429	−	0.478429i	0.904629	+	0.426200i	$0.140148\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.96933i		−	0.352395i	−0.984355	−	0.176197i	$-0.943620\pi$
$71$		−	2.96933i		−	0.352395i	0.984355	−	0.176197i	$-0.0563797\pi$
$72$		0			0
$73$	3.40354	+	3.40354i	0.398354	+	0.398354i	0.877652	−	0.479298i	$-0.159109\pi$
$73$	3.40354	+	3.40354i	0.398354	+	0.398354i	−0.479298	+	0.877652i	$0.659109\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.02243	+	5.02243i	0.572359	+	0.572359i
$78$		0			0
$79$			11.8939i			1.33816i	0.743189	+	0.669082i	$0.233313\pi$
$79$			11.8939i			1.33816i	−0.743189	+	0.669082i	$0.766687\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.625277	−	0.625277i	0.0686330	−	0.0686330i	−0.671957	−	0.740590i	$-0.734546\pi$
$83$	0.625277	−	0.625277i	0.0686330	−	0.0686330i	0.740590	+	0.671957i	$0.234546\pi$
$84$		0			0
$85$	−0.0332693	−	0.0886808i	−0.00360856	−	0.00961879i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.51721			−0.160824			−0.0804118	−	0.996762i	$-0.525624\pi$
$89$	−1.51721			−0.160824			−0.0804118	+	0.996762i	$0.525624\pi$
$90$		0			0
$91$	−14.7848			−1.54987
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.91798	+	15.2251i	−0.709770	+	1.56207i
$96$		0			0
$97$	6.30900	−	6.30900i	0.640582	−	0.640582i	−0.310117	−	0.950698i	$-0.600368\pi$
$97$	6.30900	−	6.30900i	0.640582	−	0.640582i	0.950698	+	0.310117i	$0.100368\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			3.69260i			0.367427i	0.982980	+	0.183714i	$0.0588119\pi$
$101$			3.69260i			0.367427i	−0.982980	+	0.183714i	$0.941188\pi$
$102$		0			0
$103$	−2.16694	−	2.16694i	−0.213515	−	0.213515i	0.592244	−	0.805759i	$-0.298242\pi$
$103$	−2.16694	−	2.16694i	−0.213515	−	0.213515i	−0.805759	+	0.592244i	$0.798242\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.89405	−	5.89405i	−0.569799	−	0.569799i	0.362273	−	0.932072i	$-0.382001\pi$
$107$	−5.89405	−	5.89405i	−0.569799	−	0.569799i	−0.932072	+	0.362273i	$0.882001\pi$
$108$		0			0
$109$			7.90748i			0.757399i	0.925520	+	0.378699i	$0.123629\pi$
$109$			7.90748i			0.757399i	−0.925520	+	0.378699i	$0.876371\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.5926	−	10.5926i	0.996465	−	0.996465i	−0.00352905	−	0.999994i	$-0.501123\pi$
$113$	10.5926	−	10.5926i	0.996465	−	0.996465i	0.999994	+	0.00352905i	$0.00112333\pi$
$114$		0			0
$115$	−16.3551	+	6.13573i	−1.52512	+	0.572160i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.200809			−0.0184082
$120$		0			0
$121$	8.75525			0.795932
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.21126	−	10.7092i	−0.287224	−	0.957863i
$126$		0			0
$127$	0.251916	−	0.251916i	0.0223539	−	0.0223539i	−0.695841	−	0.718195i	$-0.744968\pi$
$127$	0.251916	−	0.251916i	0.0223539	−	0.0223539i	0.718195	+	0.695841i	$0.244968\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	2.17734i		−	0.190235i	−0.995466	−	0.0951176i	$-0.969677\pi$
$131$		−	2.17734i		−	0.190235i	0.995466	−	0.0951176i	$-0.0303227\pi$
$132$		0			0
$133$	25.0705	+	25.0705i	2.17389	+	2.17389i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.07990	−	5.07990i	−0.434005	−	0.434005i	0.455984	−	0.889988i	$-0.349288\pi$
$137$	−5.07990	−	5.07990i	−0.434005	−	0.434005i	−0.889988	+	0.455984i	$0.849288\pi$
$138$		0			0
$139$		−	9.03544i		−	0.766376i	−0.923670	−	0.383188i	$-0.874826\pi$
$139$		−	9.03544i		−	0.766376i	0.923670	−	0.383188i	$-0.125174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.30400	−	3.30400i	0.276294	−	0.276294i
$144$		0			0
$145$	−19.1720	−	8.71135i	−1.59215	−	0.723438i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	12.3303			1.01014			0.505069	−	0.863079i	$-0.331467\pi$
$149$	12.3303			1.01014			0.505069	+	0.863079i	$0.331467\pi$
$150$		0			0
$151$	5.83688			0.474999			0.237499	−	0.971388i	$-0.423672\pi$
$151$	5.83688			0.474999			0.237499	+	0.971388i	$0.423672\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.3407	+	7.42488i	1.31252	+	0.596381i
$156$		0			0
$157$	7.70658	−	7.70658i	0.615052	−	0.615052i	−0.329206	−	0.944258i	$-0.606781\pi$
$157$	7.70658	−	7.70658i	0.615052	−	0.615052i	0.944258	+	0.329206i	$0.106781\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			37.0345i			2.91873i
$162$		0			0
$163$	2.59109	+	2.59109i	0.202950	+	0.202950i	0.801263	−	0.598313i	$-0.204162\pi$
$163$	2.59109	+	2.59109i	0.202950	+	0.202950i	−0.598313	+	0.801263i	$0.704162\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	14.5386	+	14.5386i	1.12503	+	1.12503i	0.990974	+	0.134057i	$0.0428006\pi$
$167$	14.5386	+	14.5386i	1.12503	+	1.12503i	0.134057	+	0.990974i	$0.457199\pi$
$168$		0			0
$169$		−	3.27382i		−	0.251832i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.3494	+	14.3494i	−1.09096	+	1.09096i	−0.0955372	+	0.995426i	$0.530457\pi$
$173$	−14.3494	+	14.3494i	−1.09096	+	1.09096i	−0.995426	+	0.0955372i	$0.969543\pi$
$174$		0			0
$175$	−23.6495	−	1.60070i	−1.78774	−	0.121001i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.97954			−0.745906			−0.372953	−	0.927850i	$-0.621655\pi$
$179$	−9.97954			−0.745906			−0.372953	+	0.927850i	$0.621655\pi$
$180$		0			0
$181$	15.8747			1.17996			0.589979	−	0.807419i	$-0.299136\pi$
$181$	15.8747			1.17996			0.589979	+	0.807419i	$0.299136\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.96359	+	2.98760i	−0.585495	+	0.219653i
$186$		0			0
$187$	0.0448753	−	0.0448753i	0.00328161	−	0.00328161i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	17.8329i		−	1.29034i	−0.764039	−	0.645171i	$-0.776786\pi$
$191$		−	17.8329i		−	1.29034i	0.764039	−	0.645171i	$-0.223214\pi$
$192$		0			0
$193$	−9.86357	−	9.86357i	−0.709996	−	0.709996i	0.256538	−	0.966534i	$-0.417418\pi$
$193$	−9.86357	−	9.86357i	−0.709996	−	0.709996i	−0.966534	+	0.256538i	$0.917418\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.89398	−	4.89398i	−0.348681	−	0.348681i	0.510937	−	0.859618i	$-0.329299\pi$
$197$	−4.89398	−	4.89398i	−0.348681	−	0.348681i	−0.859618	+	0.510937i	$0.829299\pi$
$198$		0			0
$199$		−	22.4640i		−	1.59243i	−0.605013	−	0.796215i	$-0.706832\pi$
$199$		−	22.4640i		−	1.59243i	0.605013	−	0.796215i	$-0.293168\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−31.5696	+	31.5696i	−2.21575	+	2.21575i
$204$		0			0
$205$	−9.60184	+	21.1318i	−0.670621	+	1.47591i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−11.2051			−0.775075
$210$		0			0
$211$	−2.71817			−0.187127			−0.0935634	−	0.995613i	$-0.529826\pi$
$211$	−2.71817			−0.187127			−0.0935634	+	0.995613i	$0.529826\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.93847	−	13.1637i	−0.336801	−	0.897759i
$216$		0			0
$217$	26.9075	−	26.9075i	1.82660	−	1.82660i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			0.132102i			0.00888615i
$222$		0			0
$223$	2.54453	+	2.54453i	0.170394	+	0.170394i	0.787153	−	0.616758i	$-0.211554\pi$
$223$	2.54453	+	2.54453i	0.170394	+	0.170394i	−0.616758	+	0.787153i	$0.711554\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.269165	−	0.269165i	−0.0178651	−	0.0178651i	0.698118	−	0.715983i	$-0.254021\pi$
$227$	−0.269165	−	0.269165i	−0.0178651	−	0.0178651i	−0.715983	+	0.698118i	$0.754021\pi$
$228$		0			0
$229$			6.59298i			0.435676i	0.975985	+	0.217838i	$0.0699005\pi$
$229$			6.59298i			0.435676i	−0.975985	+	0.217838i	$0.930099\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.10232	+	5.10232i	−0.334264	+	0.334264i	−0.854203	−	0.519939i	$-0.825955\pi$
$233$	−5.10232	+	5.10232i	−0.334264	+	0.334264i	0.519939	+	0.854203i	$0.325955\pi$
$234$		0			0
$235$	3.46589	+	9.23851i	0.226090	+	0.602654i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−22.2923			−1.44197			−0.720985	−	0.692951i	$-0.756310\pi$
$239$	−22.2923			−1.44197			−0.720985	+	0.692951i	$0.756310\pi$
$240$		0			0
$241$	−18.7735			−1.20931			−0.604653	−	0.796489i	$-0.706688\pi$
$241$	−18.7735			−1.20931			−0.604653	+	0.796489i	$0.706688\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−14.3141	+	31.5026i	−0.914495	+	2.01262i
$246$		0			0
$247$	16.4926	−	16.4926i	1.04940	−	1.04940i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			3.75368i			0.236930i	0.992958	+	0.118465i	$0.0377974\pi$
$251$			3.75368i			0.236930i	−0.992958	+	0.118465i	$0.962203\pi$
$252$		0			0
$253$	−8.27619	−	8.27619i	−0.520320	−	0.520320i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.70157	−	7.70157i	−0.480411	−	0.480411i	0.424852	−	0.905263i	$-0.360326\pi$
$257$	−7.70157	−	7.70157i	−0.480411	−	0.480411i	−0.905263	+	0.424852i	$0.860326\pi$
$258$		0			0
$259$			18.0328i			1.12050i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.46457	+	6.46457i	−0.398623	+	0.398623i	−0.877747	−	0.479124i	$-0.840954\pi$
$263$	−6.46457	+	6.46457i	−0.398623	+	0.398623i	0.479124	+	0.877747i	$0.340954\pi$
$264$		0			0
$265$	−0.194432	+	0.0729426i	−0.0119439	+	0.00448083i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.6135			−1.07391			−0.536957	−	0.843609i	$-0.680426\pi$
$269$	−17.6135			−1.07391			−0.536957	+	0.843609i	$0.680426\pi$
$270$		0			0
$271$	2.05362			0.124748			0.0623742	−	0.998053i	$-0.480133\pi$
$271$	2.05362			0.124748			0.0623742	+	0.998053i	$0.480133\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.64273	−	4.92730i	0.340269	−	0.297128i
$276$		0			0
$277$	−14.4871	+	14.4871i	−0.870443	+	0.870443i	−0.992521	−	0.122077i	$-0.961044\pi$
$277$	−14.4871	+	14.4871i	−0.870443	+	0.870443i	0.122077	+	0.992521i	$0.461044\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			16.8199i			1.00339i	0.865045	+	0.501695i	$0.167290\pi$
$281$			16.8199i			1.00339i	−0.865045	+	0.501695i	$0.832710\pi$
$282$		0			0
$283$	18.2981	+	18.2981i	1.08771	+	1.08771i	0.995764	+	0.0919450i	$0.0293084\pi$
$283$	18.2981	+	18.2981i	1.08771	+	1.08771i	0.0919450	+	0.995764i	$0.470692\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	34.7967	+	34.7967i	2.05398	+	2.05398i
$288$		0			0
$289$		−	16.9982i		−	0.999894i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.7989	−	14.7989i	0.864561	−	0.864561i	−0.127303	−	0.991864i	$-0.540632\pi$
$293$	14.7989	−	14.7989i	0.864561	−	0.864561i	0.991864	+	0.127303i	$0.0406321\pi$
$294$		0			0
$295$	10.1033	+	4.59071i	0.588235	+	0.267282i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	24.3631			1.40896
$300$		0			0
$301$	−29.8080			−1.71811
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.64606	+	1.20231i	0.151513	+	0.0688442i
$306$		0			0
$307$	−9.33544	+	9.33544i	−0.532802	+	0.532802i	−0.921405	−	0.388603i	$-0.872958\pi$
$307$	−9.33544	+	9.33544i	−0.532802	+	0.532802i	0.388603	+	0.921405i	$0.372958\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	3.49278i		−	0.198057i	−0.995085	−	0.0990286i	$-0.968426\pi$
$311$		−	3.49278i		−	0.198057i	0.995085	−	0.0990286i	$-0.0315735\pi$
$312$		0			0
$313$	9.44723	+	9.44723i	0.533989	+	0.533989i	0.921757	−	0.387768i	$-0.126754\pi$
$313$	9.44723	+	9.44723i	0.533989	+	0.533989i	−0.387768	+	0.921757i	$0.626754\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.3042	−	12.3042i	−0.691074	−	0.691074i	0.271394	−	0.962468i	$-0.412515\pi$
$317$	−12.3042	−	12.3042i	−0.691074	−	0.691074i	−0.962468	+	0.271394i	$0.912515\pi$
$318$		0			0
$319$		−	14.1099i		−	0.790000i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.224004	−	0.224004i	0.0124639	−	0.0124639i
$324$		0			0
$325$	−1.05302	+	15.5578i	−0.0584109	+	0.862992i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	20.9197			1.15334
$330$		0			0
$331$	9.61273			0.528363			0.264182	−	0.964473i	$-0.414898\pi$
$331$	9.61273			0.528363			0.264182	+	0.964473i	$0.414898\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.5947	+	4.34985i	−0.633488	+	0.237658i
$336$		0			0
$337$	−9.63500	+	9.63500i	−0.524852	+	0.524852i	−0.919033	−	0.394181i	$-0.871028\pi$
$337$	−9.63500	+	9.63500i	−0.524852	+	0.524852i	0.394181	+	0.919033i	$0.371028\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			12.0262i			0.651253i
$342$		0			0
$343$	28.4083	+	28.4083i	1.53391	+	1.53391i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.3880	−	17.3880i	−0.933437	−	0.933437i	0.0644816	−	0.997919i	$-0.479461\pi$
$347$	−17.3880	−	17.3880i	−0.933437	−	0.933437i	−0.997919	+	0.0644816i	$0.979461\pi$
$348$		0			0
$349$		−	24.7751i		−	1.32618i	−0.748539	−	0.663091i	$-0.769244\pi$
$349$		−	24.7751i		−	1.32618i	0.748539	−	0.663091i	$-0.230756\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	20.2988	−	20.2988i	1.08039	−	1.08039i	0.0839223	−	0.996472i	$-0.473255\pi$
$353$	20.2988	−	20.2988i	1.08039	−	1.08039i	0.996472	−	0.0839223i	$-0.0267448\pi$
$354$		0			0
$355$	−2.74666	+	6.04488i	−0.145778	+	0.320829i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−16.4950			−0.870572			−0.435286	−	0.900292i	$-0.643353\pi$
$359$	−16.4950			−0.870572			−0.435286	+	0.900292i	$0.643353\pi$
$360$		0			0
$361$	−36.9327			−1.94382
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.78051	−	10.0771i	−0.197881	−	0.527461i
$366$		0			0
$367$	17.2590	−	17.2590i	0.900912	−	0.900912i	−0.0946033	−	0.995515i	$-0.530158\pi$
$367$	17.2590	−	17.2590i	0.900912	−	0.900912i	0.995515	+	0.0946033i	$0.0301583\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			0.440273i			0.0228578i
$372$		0			0
$373$	25.5730	+	25.5730i	1.32412	+	1.32412i	0.910407	+	0.413715i	$0.135769\pi$
$373$	25.5730	+	25.5730i	1.32412	+	1.32412i	0.413715	+	0.910407i	$0.364231\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.7680	+	20.7680i	1.06961	+	1.06961i
$378$		0			0
$379$		−	17.8108i		−	0.914880i	−0.889240	−	0.457440i	$-0.848766\pi$
$379$		−	17.8108i		−	0.914880i	0.889240	−	0.457440i	$-0.151234\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−22.6572	+	22.6572i	−1.15773	+	1.15773i	−0.172767	+	0.984963i	$0.555271\pi$
$383$	−22.6572	+	22.6572i	−1.15773	+	1.15773i	−0.984963	+	0.172767i	$0.944729\pi$
$384$		0			0
$385$	−5.57871	−	14.8703i	−0.284317	−	0.757862i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	28.2404			1.43184			0.715922	−	0.698180i	$-0.246007\pi$
$389$	28.2404			1.43184			0.715922	+	0.698180i	$0.246007\pi$
$390$		0			0
$391$	0.330903			0.0167345
$392$		0			0
$393$		0			0
$394$		0			0
$395$	11.0020	−	24.2132i	0.553568	−	1.21830i
$396$		0			0
$397$	−22.8828	+	22.8828i	−1.14846	+	1.14846i	−0.161600	+	0.986856i	$0.551666\pi$
$397$	−22.8828	+	22.8828i	−1.14846	+	1.14846i	−0.986856	+	0.161600i	$0.948334\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			15.0060i			0.749362i	0.927154	+	0.374681i	$0.122248\pi$
$401$			15.0060i			0.749362i	−0.927154	+	0.374681i	$0.877752\pi$
$402$		0			0
$403$	−17.7011	−	17.7011i	−0.881752	−	0.881752i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.02983	−	4.02983i	−0.199751	−	0.199751i
$408$		0			0
$409$		−	19.8236i		−	0.980213i	−0.871663	−	0.490107i	$-0.836958\pi$
$409$		−	19.8236i		−	0.980213i	0.871663	−	0.490107i	$-0.163042\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	16.6366	−	16.6366i	0.818632	−	0.818632i
$414$		0			0
$415$	−1.85131	+	0.694531i	−0.0908771	+	0.0340932i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.8616			0.921450			0.460725	−	0.887543i	$-0.347590\pi$
$419$	18.8616			0.921450			0.460725	+	0.887543i	$0.347590\pi$
$420$		0			0
$421$	4.23591			0.206446			0.103223	−	0.994658i	$-0.467085\pi$
$421$	4.23591			0.206446			0.103223	+	0.994658i	$0.467085\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.0143022	+	0.211308i	−0.000693759	+	0.0102500i
$426$		0			0
$427$	4.35713	−	4.35713i	0.210856	−	0.210856i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	33.9505i		−	1.63534i	−0.575687	−	0.817670i	$-0.695265\pi$
$431$		−	33.9505i		−	1.63534i	0.575687	−	0.817670i	$-0.304735\pi$
$432$		0			0
$433$	7.97582	+	7.97582i	0.383294	+	0.383294i	0.872287	−	0.488994i	$-0.162636\pi$
$433$	7.97582	+	7.97582i	0.383294	+	0.383294i	−0.488994	+	0.872287i	$0.662636\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−41.3123	−	41.3123i	−1.97624	−	1.97624i
$438$		0			0
$439$		−	23.0060i		−	1.09802i	−0.835816	−	0.549009i	$-0.815005\pi$
$439$		−	23.0060i		−	1.09802i	0.835816	−	0.549009i	$-0.184995\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.9943	+	13.9943i	−0.664890	+	0.664890i	−0.956528	−	0.291639i	$-0.905799\pi$
$443$	−13.9943	+	13.9943i	−0.664890	+	0.664890i	0.291639	+	0.956528i	$0.405799\pi$
$444$		0			0
$445$	3.08868	+	1.40343i	0.146418	+	0.0665291i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−41.8993			−1.97735			−0.988675	−	0.150072i	$-0.952049\pi$
$449$	−41.8993			−1.97735			−0.988675	+	0.150072i	$0.952049\pi$
$450$		0			0
$451$	−15.5522			−0.732324
$452$		0			0
$453$		0			0
$454$		0			0
$455$	30.0985	+	13.6761i	1.41104	+	0.641147i
$456$		0			0
$457$	0.555163	−	0.555163i	0.0259694	−	0.0259694i	−0.694003	−	0.719972i	$-0.744154\pi$
$457$	0.555163	−	0.555163i	0.0259694	−	0.0259694i	0.719972	+	0.694003i	$0.244154\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	12.2854i		−	0.572186i	−0.958202	−	0.286093i	$-0.907643\pi$
$461$		−	12.2854i		−	0.572186i	0.958202	−	0.286093i	$-0.0923566\pi$
$462$		0			0
$463$	7.69806	+	7.69806i	0.357759	+	0.357759i	0.862986	−	0.505227i	$-0.168591\pi$
$463$	7.69806	+	7.69806i	0.357759	+	0.357759i	−0.505227	+	0.862986i	$0.668591\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.12215	+	2.12215i	0.0982013	+	0.0982013i	0.754501	−	0.656299i	$-0.227879\pi$
$467$	2.12215	+	2.12215i	0.0982013	+	0.0982013i	−0.656299	+	0.754501i	$0.727879\pi$
$468$		0			0
$469$			26.2552i			1.21235i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.66127	−	6.66127i	0.306285	−	0.306285i
$474$		0			0
$475$	28.1668	−	24.5957i	1.29238	−	1.12853i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−23.4849			−1.07305			−0.536527	−	0.843883i	$-0.680264\pi$
$479$	−23.4849			−1.07305			−0.536527	+	0.843883i	$0.680264\pi$
$480$		0			0
$481$	11.8628			0.540899
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−18.6796	+	7.00777i	−0.848195	+	0.318206i
$486$		0			0
$487$	7.41840	−	7.41840i	0.336160	−	0.336160i	−0.518760	−	0.854920i	$-0.673606\pi$
$487$	7.41840	−	7.41840i	0.336160	−	0.336160i	0.854920	+	0.518760i	$0.173606\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			3.08835i			0.139375i	0.997569	+	0.0696877i	$0.0222003\pi$
$491$			3.08835i			0.139375i	−0.997569	+	0.0696877i	$0.977800\pi$
$492$		0			0
$493$	0.282074	+	0.282074i	0.0127040	+	0.0127040i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.95381	+	9.95381i	0.446489	+	0.446489i
$498$		0			0
$499$		−	12.9305i		−	0.578847i	−0.957201	−	0.289424i	$-0.906536\pi$
$499$		−	12.9305i		−	0.578847i	0.957201	−	0.289424i	$-0.0934636\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	13.2063	−	13.2063i	0.588840	−	0.588840i	−0.348477	−	0.937317i	$-0.613301\pi$
$503$	13.2063	−	13.2063i	0.588840	−	0.588840i	0.937317	+	0.348477i	$0.113301\pi$
$504$		0			0
$505$	3.41570	−	7.51728i	0.151996	−	0.334515i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	27.1627			1.20397			0.601983	−	0.798509i	$-0.294378\pi$
$509$	27.1627			1.20397			0.601983	+	0.798509i	$0.294378\pi$
$510$		0			0
$511$	−22.8187			−1.00944
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.40694	+	6.41582i	0.106063	+	0.282715i
$516$		0			0
$517$	−4.67498	+	4.67498i	−0.205605	+	0.205605i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	6.44882i		−	0.282528i	−0.989972	−	0.141264i	$-0.954883\pi$
$521$		−	6.44882i		−	0.282528i	0.989972	−	0.141264i	$-0.0451166\pi$
$522$		0			0
$523$	9.24029	+	9.24029i	0.404050	+	0.404050i	0.879658	−	0.475608i	$-0.157772\pi$
$523$	9.24029	+	9.24029i	0.404050	+	0.404050i	−0.475608	+	0.879658i	$0.657772\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.240418	−	0.240418i	−0.0104728	−	0.0104728i
$528$		0			0
$529$		−	38.0272i		−	1.65335i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	22.8909	−	22.8909i	0.991517	−	0.991517i
$534$		0			0
$535$	6.54686	+	17.4510i	0.283046	+	0.754472i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−23.1847			−0.998635
$540$		0			0
$541$	24.2353			1.04196			0.520979	−	0.853570i	$-0.325567\pi$
$541$	24.2353			1.04196			0.520979	+	0.853570i	$0.325567\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.31450	−	16.0978i	0.313319	−	0.689554i
$546$		0			0
$547$	2.32827	−	2.32827i	0.0995497	−	0.0995497i	−0.655578	−	0.755128i	$-0.727575\pi$
$547$	2.32827	−	2.32827i	0.0995497	−	0.0995497i	0.755128	+	0.655578i	$0.227575\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	70.4322i		−	3.00051i
$552$		0			0
$553$	−39.8706	−	39.8706i	−1.69547	−	1.69547i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	23.6914	+	23.6914i	1.00384	+	1.00384i	0.999993	+	0.00384255i	$0.00122312\pi$
$557$	23.6914	+	23.6914i	1.00384	+	1.00384i	0.00384255	+	0.999993i	$0.498777\pi$
$558$		0			0
$559$			19.6092i			0.829379i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.1977	−	13.1977i	0.556215	−	0.556215i	−0.372013	−	0.928228i	$-0.621332\pi$
$563$	13.1977	−	13.1977i	0.556215	−	0.556215i	0.928228	+	0.372013i	$0.121332\pi$
$564$		0			0
$565$	−31.3623	+	11.7658i	−1.31942	+	0.494990i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	20.8504			0.874094			0.437047	−	0.899439i	$-0.356024\pi$
$569$	20.8504			0.874094			0.437047	+	0.899439i	$0.356024\pi$
$570$		0			0
$571$	37.5860			1.57293			0.786464	−	0.617637i	$-0.211910\pi$
$571$	37.5860			1.57293			0.786464	+	0.617637i	$0.211910\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	38.9708	+	2.63770i	1.62519	+	0.110000i
$576$		0			0
$577$	−15.5435	+	15.5435i	−0.647083	+	0.647083i	−0.952287	−	0.305204i	$-0.901275\pi$
$577$	−15.5435	+	15.5435i	−0.647083	+	0.647083i	0.305204	+	0.952287i	$0.401275\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			4.19211i			0.173918i
$582$		0			0
$583$	−0.0983887	−	0.0983887i	−0.00407485	−	0.00407485i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.98790	+	6.98790i	0.288421	+	0.288421i	0.836456	−	0.548034i	$-0.184624\pi$
$587$	6.98790	+	6.98790i	0.288421	+	0.288421i	−0.548034	+	0.836456i	$0.684624\pi$
$588$		0			0
$589$			60.0310i			2.47353i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	31.0512	−	31.0512i	1.27512	−	1.27512i	0.331752	−	0.943367i	$-0.392360\pi$
$593$	31.0512	−	31.0512i	1.27512	−	1.27512i	0.943367	−	0.331752i	$-0.107640\pi$
$594$		0			0
$595$	0.408801	+	0.185751i	0.0167592	+	0.00761504i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	46.7785			1.91132			0.955658	−	0.294478i	$-0.0951457\pi$
$599$	46.7785			1.91132			0.955658	+	0.294478i	$0.0951457\pi$
$600$		0			0
$601$	−22.9148			−0.934715			−0.467357	−	0.884069i	$-0.654794\pi$
$601$	−22.9148			−0.934715			−0.467357	+	0.884069i	$0.654794\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−17.8237	−	8.09870i	−0.724635	−	0.329259i
$606$		0			0
$607$	−21.6250	+	21.6250i	−0.877730	+	0.877730i	−0.993299	−	0.115569i	$-0.963131\pi$
$607$	−21.6250	+	21.6250i	−0.877730	+	0.877730i	0.115569	+	0.993299i	$0.463131\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	13.7620i		−	0.556752i
$612$		0			0
$613$	−17.6982	−	17.6982i	−0.714822	−	0.714822i	0.252718	−	0.967540i	$-0.418676\pi$
$613$	−17.6982	−	17.6982i	−0.714822	−	0.714822i	−0.967540	+	0.252718i	$0.918676\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.83787	+	4.83787i	0.194765	+	0.194765i	0.797752	−	0.602986i	$-0.206023\pi$
$617$	4.83787	+	4.83787i	0.194765	+	0.194765i	−0.602986	+	0.797752i	$0.706023\pi$
$618$		0			0
$619$			3.64191i			0.146381i	0.997318	+	0.0731904i	$0.0233181\pi$
$619$			3.64191i			0.146381i	−0.997318	+	0.0731904i	$0.976682\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.08598	−	5.08598i	0.203766	−	0.203766i
$624$		0			0
$625$	−3.36877	+	24.7720i	−0.134751	+	0.990879i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.161123			0.00642438
$630$		0			0
$631$	18.2896			0.728097			0.364048	−	0.931380i	$-0.381394\pi$
$631$	18.2896			0.728097			0.364048	+	0.931380i	$0.381394\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.745867	+	0.279817i	−0.0295988	+	0.0111042i
$636$		0			0
$637$	34.1251	−	34.1251i	1.35208	−	1.35208i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			10.2390i			0.404417i	0.979342	+	0.202208i	$0.0648119\pi$
$641$			10.2390i			0.404417i	−0.979342	+	0.202208i	$0.935188\pi$
$642$		0			0
$643$	10.2300	+	10.2300i	0.403434	+	0.403434i	0.879441	−	0.476008i	$-0.157917\pi$
$643$	10.2300	+	10.2300i	0.403434	+	0.403434i	−0.476008	+	0.879441i	$0.657917\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.21533	−	3.21533i	−0.126408	−	0.126408i	0.641073	−	0.767480i	$-0.278490\pi$
$647$	−3.21533	−	3.21533i	−0.126408	−	0.126408i	−0.767480	+	0.641073i	$0.778490\pi$
$648$		0			0
$649$			7.43562i			0.291874i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18.7537	+	18.7537i	−0.733890	+	0.733890i	−0.971388	−	0.237498i	$-0.923673\pi$
$653$	−18.7537	+	18.7537i	−0.733890	+	0.733890i	0.237498	+	0.971388i	$0.423673\pi$
$654$		0			0
$655$	−2.01406	+	4.43256i	−0.0786960	+	0.173195i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	43.1030			1.67905			0.839527	−	0.543318i	$-0.182832\pi$
$659$	43.1030			1.67905			0.839527	+	0.543318i	$0.182832\pi$
$660$		0			0
$661$	−24.6204			−0.957623			−0.478811	−	0.877918i	$-0.658932\pi$
$661$	−24.6204			−0.957623			−0.478811	+	0.877918i	$0.658932\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−27.8473	−	74.2282i	−1.07987	−	2.87845i
$666$		0			0
$667$	52.0218	−	52.0218i	2.01429	−	2.01429i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1.94740i			0.0751784i
$672$		0			0
$673$	−19.6536	−	19.6536i	−0.757590	−	0.757590i	0.218294	−	0.975883i	$-0.429951\pi$
$673$	−19.6536	−	19.6536i	−0.757590	−	0.757590i	−0.975883	+	0.218294i	$0.929951\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−27.5088	−	27.5088i	−1.05725	−	1.05725i	−0.998259	−	0.0589905i	$-0.981212\pi$
$677$	−27.5088	−	27.5088i	−1.05725	−	1.05725i	−0.0589905	−	0.998259i	$-0.518788\pi$
$678$		0			0
$679$			42.2981i			1.62325i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	22.0645	−	22.0645i	0.844275	−	0.844275i	−0.145137	−	0.989412i	$-0.546362\pi$
$683$	22.0645	−	22.0645i	0.844275	−	0.844275i	0.989412	+	0.145137i	$0.0463622\pi$
$684$		0			0
$685$	5.64253	+	15.0405i	0.215590	+	0.574666i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.289633			0.0110341
$690$		0			0
$691$	31.0777			1.18225			0.591125	−	0.806580i	$-0.298684\pi$
$691$	31.0777			1.18225			0.591125	+	0.806580i	$0.298684\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.35788	+	18.3941i	−0.317032	+	0.697727i
$696$		0			0
$697$	0.310907	−	0.310907i	0.0117765	−	0.0117765i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			30.0473i			1.13487i	0.823418	+	0.567435i	$0.192064\pi$
$701$			30.0473i			1.13487i	−0.823418	+	0.567435i	$0.807936\pi$
$702$		0			0
$703$	−20.1157	−	20.1157i	−0.758678	−	0.758678i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.3783	−	12.3783i	−0.465536	−	0.465536i
$708$		0			0
$709$			5.49520i			0.206377i	0.994662	+	0.103188i	$0.0329044\pi$
$709$			5.49520i			0.206377i	−0.994662	+	0.103188i	$0.967096\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−44.3394	+	44.3394i	−1.66052	+	1.66052i
$714$		0			0
$715$	−9.78242	+	3.66994i	−0.365842	+	0.137248i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−30.9265			−1.15336			−0.576682	−	0.816968i	$-0.695653\pi$
$719$	−30.9265			−1.15336			−0.576682	+	0.816968i	$0.695653\pi$
$720$		0			0
$721$	14.5280			0.541052
$722$		0			0
$723$		0			0
$724$		0			0
$725$	30.9716	+	35.4686i	1.15026	+	1.31727i
$726$		0			0
$727$	−22.6989	+	22.6989i	−0.841855	+	0.841855i	−0.989100	−	0.147245i	$-0.952959\pi$
$727$	−22.6989	+	22.6989i	−0.841855	+	0.841855i	0.147245	+	0.989100i	$0.452959\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			0.266334i			0.00985072i
$732$		0			0
$733$	19.3459	+	19.3459i	0.714556	+	0.714556i	0.967485	−	0.252929i	$-0.0813939\pi$
$733$	19.3459	+	19.3459i	0.714556	+	0.714556i	−0.252929	+	0.967485i	$0.581394\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.86730	−	5.86730i	−0.216125	−	0.216125i
$738$		0			0
$739$			29.0073i			1.06705i	0.845784	+	0.533526i	$0.179133\pi$
$739$			29.0073i			1.06705i	−0.845784	+	0.533526i	$0.820867\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.95681	+	1.95681i	−0.0717884	+	0.0717884i	−0.742089	−	0.670301i	$-0.766165\pi$
$743$	−1.95681	+	1.95681i	−0.0717884	+	0.0717884i	0.670301	+	0.742089i	$0.266165\pi$
$744$		0			0
$745$	−25.1016	−	11.4057i	−0.919653	−	0.417871i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	39.5161			1.44389
$750$		0			0
$751$	7.95955			0.290448			0.145224	−	0.989399i	$-0.453610\pi$
$751$	7.95955			0.290448			0.145224	+	0.989399i	$0.453610\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−11.8825	−	5.39918i	−0.432450	−	0.196496i
$756$		0			0
$757$	−14.3074	+	14.3074i	−0.520012	+	0.520012i	−0.917575	−	0.397563i	$-0.869856\pi$
$757$	−14.3074	+	14.3074i	−0.520012	+	0.520012i	0.397563	+	0.917575i	$0.369856\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1.59491i			0.0578154i	0.999582	+	0.0289077i	$0.00920289\pi$
$761$			1.59491i			0.0578154i	−0.999582	+	0.0289077i	$0.990797\pi$
$762$		0			0
$763$	−26.5075	−	26.5075i	−0.959634	−	0.959634i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10.9443	−	10.9443i	−0.395177	−	0.395177i
$768$		0			0
$769$			27.7745i			1.00158i	0.865570	+	0.500788i	$0.166956\pi$
$769$			27.7745i			1.00158i	−0.865570	+	0.500788i	$0.833044\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16.4399	+	16.4399i	−0.591303	+	0.591303i	−0.937983	−	0.346680i	$-0.887309\pi$
$773$	−16.4399	+	16.4399i	−0.591303	+	0.591303i	0.346680	+	0.937983i	$0.387309\pi$
$774$		0			0
$775$	−26.3978	−	30.2307i	−0.948239	−	1.08592i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−77.6319			−2.78145
$780$		0			0
$781$	−4.44880			−0.159191
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−22.8175	+	8.56014i	−0.814391	+	0.305525i
$786$		0			0
$787$	−9.98268	+	9.98268i	−0.355844	+	0.355844i	−0.862278	−	0.506434i	$-0.830963\pi$
$787$	−9.98268	+	9.98268i	−0.355844	+	0.355844i	0.506434	+	0.862278i	$0.330963\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			71.0169i			2.52507i
$792$		0			0
$793$	−2.86633	−	2.86633i	−0.101786	−	0.101786i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.84593	+	3.84593i	0.136230	+	0.136230i	0.771933	−	0.635704i	$-0.219290\pi$
$797$	3.84593	+	3.84593i	0.136230	+	0.136230i	−0.635704	+	0.771933i	$0.719290\pi$
$798$		0			0
$799$		−	0.186917i		−	0.00661266i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5.09935	−	5.09935i	0.179952	−	0.179952i
$804$		0			0
$805$	34.2573	−	75.3937i	1.20741	−	2.65728i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16.9419			0.595645			0.297823	−	0.954621i	$-0.403740\pi$
$809$	16.9419			0.595645			0.297823	+	0.954621i	$0.403740\pi$
$810$		0			0
$811$	38.0417			1.33582			0.667912	−	0.744240i	$-0.267188\pi$
$811$	38.0417			1.33582			0.667912	+	0.744240i	$0.267188\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.87807	−	7.67164i	−0.100814	−	0.268726i
$816$		0			0
$817$	33.2511	−	33.2511i	1.16331	−	1.16331i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	41.0129i		−	1.43136i	−0.698429	−	0.715679i	$-0.746117\pi$
$821$		−	41.0129i		−	1.43136i	0.698429	−	0.715679i	$-0.253883\pi$
$822$		0			0
$823$	4.77164	+	4.77164i	0.166329	+	0.166329i	0.785364	−	0.619035i	$-0.212476\pi$
$823$	4.77164	+	4.77164i	0.166329	+	0.166329i	−0.619035	+	0.785364i	$0.712476\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	28.2052	+	28.2052i	0.980790	+	0.980790i	0.999819	−	0.0190293i	$-0.00605757\pi$
$827$	28.2052	+	28.2052i	0.980790	+	0.980790i	−0.0190293	+	0.999819i	$0.506058\pi$
$828$		0			0
$829$		−	9.82288i		−	0.341163i	−0.985344	−	0.170581i	$-0.945435\pi$
$829$		−	9.82288i		−	0.341163i	0.985344	−	0.170581i	$-0.0545646\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.463491	−	0.463491i	0.0160590	−	0.0160590i
$834$		0			0
$835$	−16.1489	−	43.0456i	−0.558855	−	1.48965i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.9381			−0.791912			−0.395956	−	0.918269i	$-0.629587\pi$
$839$	−22.9381			−0.791912			−0.395956	+	0.918269i	$0.629587\pi$
$840$		0			0
$841$	59.6905			2.05829
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.02832	+	6.66474i	−0.104177	+	0.229274i
$846$		0			0
$847$	−29.3494	+	29.3494i	−1.00846	+	1.00846i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	29.7152i		−	1.01863i
$852$		0			0
$853$	−15.5429	−	15.5429i	−0.532177	−	0.532177i	0.389043	−	0.921220i	$-0.372806\pi$
$853$	−15.5429	−	15.5429i	−0.532177	−	0.532177i	−0.921220	+	0.389043i	$0.872806\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.164775	+	0.164775i	0.00562861	+	0.00562861i	0.709915	−	0.704287i	$-0.248733\pi$
$857$	0.164775	+	0.164775i	0.00562861	+	0.00562861i	−0.704287	+	0.709915i	$0.748733\pi$
$858$		0			0
$859$		−	7.73657i		−	0.263968i	−0.991252	−	0.131984i	$-0.957865\pi$
$859$		−	7.73657i		−	0.263968i	0.991252	−	0.131984i	$-0.0421348\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.50143	−	2.50143i	0.0851497	−	0.0851497i	−0.663249	−	0.748399i	$-0.730823\pi$
$863$	2.50143	−	2.50143i	0.0851497	−	0.0851497i	0.748399	+	0.663249i	$0.230823\pi$
$864$		0			0
$865$	42.4853	−	15.9387i	1.44455	−	0.541932i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	17.8200			0.604501
$870$		0			0
$871$	17.2719			0.585237
$872$		0			0
$873$		0			0
$874$		0			0
$875$	46.6643	+	25.1347i	1.57754	+	0.849709i
$876$		0			0
$877$	−28.7413	+	28.7413i	−0.970524	+	0.970524i	−0.999578	−	0.0290541i	$-0.990750\pi$
$877$	−28.7413	+	28.7413i	−0.970524	+	0.970524i	0.0290541	+	0.999578i	$0.490750\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			18.6638i			0.628799i	0.949291	+	0.314399i	$0.101803\pi$
$881$			18.6638i			0.628799i	−0.949291	+	0.314399i	$0.898197\pi$
$882$		0			0
$883$	−16.9868	−	16.9868i	−0.571653	−	0.571653i	0.360937	−	0.932590i	$-0.382457\pi$
$883$	−16.9868	−	16.9868i	−0.571653	−	0.571653i	−0.932590	+	0.360937i	$0.882457\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.50394	+	1.50394i	0.0504974	+	0.0504974i	0.731905	−	0.681407i	$-0.238632\pi$
$887$	1.50394	+	1.50394i	0.0504974	+	0.0504974i	−0.681407	+	0.731905i	$0.738632\pi$
$888$		0			0
$889$			1.68895i			0.0566454i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−23.3361	+	23.3361i	−0.780913	+	0.780913i
$894$		0			0
$895$	20.3160	+	9.23118i	0.679090	+	0.308564i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−75.5930			−2.52117
$900$		0			0
$901$	0.00393383			0.000131055
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−32.3172	−	14.6843i	−1.07426	−	0.488122i
$906$		0			0
$907$	−24.5926	+	24.5926i	−0.816584	+	0.816584i	−0.985611	−	0.169027i	$-0.945937\pi$
$907$	−24.5926	+	24.5926i	−0.816584	+	0.816584i	0.169027	+	0.985611i	$0.445937\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			2.16589i			0.0717592i	0.999356	+	0.0358796i	$0.0114233\pi$
$911$			2.16589i			0.0717592i	−0.999356	+	0.0358796i	$0.988577\pi$
$912$		0			0
$913$	−0.936820	−	0.936820i	−0.0310042	−	0.0310042i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.29889	+	7.29889i	0.241031	+	0.241031i
$918$		0			0
$919$			22.6962i			0.748678i	0.927292	+	0.374339i	$0.122130\pi$
$919$			22.6962i			0.748678i	−0.927292	+	0.374339i	$0.877870\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.54810	−	6.54810i	0.215533	−	0.215533i
$924$		0			0
$925$	18.9756	+	1.28435i	0.623913	+	0.0422290i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.32611			−0.109126			−0.0545631	−	0.998510i	$-0.517377\pi$
$929$	−3.32611			−0.109126			−0.0545631	+	0.998510i	$0.517377\pi$
$930$		0			0
$931$	−115.731			−3.79293
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.132866	+	0.0498456i	−0.00434518	+	0.00163013i
$936$		0			0
$937$	−14.3442	+	14.3442i	−0.468604	+	0.468604i	−0.901462	−	0.432858i	$-0.857505\pi$
$937$	−14.3442	+	14.3442i	−0.468604	+	0.468604i	0.432858	+	0.901462i	$0.357505\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	33.1192i		−	1.07966i	−0.841776	−	0.539828i	$-0.818489\pi$
$941$		−	33.1192i		−	1.07966i	0.841776	−	0.539828i	$-0.181511\pi$
$942$		0			0
$943$	−57.3395	−	57.3395i	−1.86723	−	1.86723i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	29.6786	+	29.6786i	0.964426	+	0.964426i	0.999389	−	0.0349629i	$-0.0111313\pi$
$947$	29.6786	+	29.6786i	0.964426	+	0.964426i	−0.0349629	+	0.999389i	$0.511131\pi$
$948$		0			0
$949$			15.0112i			0.487286i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	8.76619	−	8.76619i	0.283965	−	0.283965i	−0.550723	−	0.834688i	$-0.685648\pi$
$953$	8.76619	−	8.76619i	0.283965	−	0.283965i	0.834688	+	0.550723i	$0.185648\pi$
$954$		0			0
$955$	−16.4956	+	36.3036i	−0.533785	+	1.17476i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	34.0577			1.09978
$960$		0			0
$961$	33.4297			1.07838
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10.9560	+	29.2039i	0.352688	+	0.940106i
$966$		0			0
$967$	39.4708	−	39.4708i	1.26930	−	1.26930i	0.322843	−	0.946453i	$-0.395362\pi$
$967$	39.4708	−	39.4708i	1.26930	−	1.26930i	0.946453	−	0.322843i	$-0.104638\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			12.9056i			0.414161i	0.978324	+	0.207081i	$0.0663962\pi$
$971$			12.9056i			0.414161i	−0.978324	+	0.207081i	$0.933604\pi$
$972$		0			0
$973$	30.2886	+	30.2886i	0.971009	+	0.971009i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	21.9765	+	21.9765i	0.703091	+	0.703091i	0.965073	−	0.261982i	$-0.0843760\pi$
$977$	21.9765	+	21.9765i	0.703091	+	0.703091i	−0.261982	+	0.965073i	$0.584376\pi$
$978$		0			0
$979$			2.27315i			0.0726502i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−36.4802	+	36.4802i	−1.16354	+	1.16354i	−0.179841	+	0.983696i	$0.557558\pi$
$983$	−36.4802	+	36.4802i	−1.16354	+	1.16354i	−0.983696	+	0.179841i	$0.942442\pi$
$984$		0			0
$985$	5.43603	+	14.4900i	0.173206	+	0.461689i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	49.1190			1.56189
$990$		0			0
$991$	−24.8506			−0.789406			−0.394703	−	0.918809i	$-0.629152\pi$
$991$	−24.8506			−0.789406			−0.394703	+	0.918809i	$0.629152\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.7794	+	45.7315i	−0.658753	+	1.44979i
$996$		0			0
$997$	19.0522	−	19.0522i	0.603389	−	0.603389i	−0.337821	−	0.941210i	$-0.609690\pi$
$997$	19.0522	−	19.0522i	0.603389	−	0.603389i	0.941210	+	0.337821i	$0.109690\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.2.s.a.593.2	yes	24
3.2	odd	2	inner	1080.2.s.a.593.11	yes	24
4.3	odd	2		2160.2.w.h.593.2		24
5.2	odd	4	inner	1080.2.s.a.377.11	yes	24
12.11	even	2		2160.2.w.h.593.11		24
15.2	even	4	inner	1080.2.s.a.377.2	✓	24
20.7	even	4		2160.2.w.h.1457.11		24
60.47	odd	4		2160.2.w.h.1457.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.s.a.377.2	✓	24	15.2	even	4	inner
1080.2.s.a.377.11	yes	24	5.2	odd	4	inner
1080.2.s.a.593.2	yes	24	1.1	even	1	trivial
1080.2.s.a.593.11	yes	24	3.2	odd	2	inner
2160.2.w.h.593.2		24	4.3	odd	2
2160.2.w.h.593.11		24	12.11	even	2
2160.2.w.h.1457.2		24	60.47	odd	4
2160.2.w.h.1457.11		24	20.7	even	4

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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1080.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.03577 - 0.925011i) q^{5} +(-3.35220 + 3.35220i) q^{7} +O(q^{10})\)	\(q+(-2.03577 - 0.925011i) q^{5} +(-3.35220 + 3.35220i) q^{7} -1.49825i q^{11} +(2.20524 + 2.20524i) q^{13} +(0.0299519 + 0.0299519i) q^{17} -7.47881i q^{19} +(5.52391 - 5.52391i) q^{23} +(3.28871 + 3.76622i) q^{25} +9.41757 q^{29} -8.02681 q^{31} +(9.92513 - 3.72349i) q^{35} +(2.68969 - 2.68969i) q^{37} -10.3802i q^{41} +(4.44603 + 4.44603i) q^{43} +(-3.12030 - 3.12030i) q^{47} -15.4745i q^{49} +(0.0656692 - 0.0656692i) q^{53} +(-1.38590 + 3.05009i) q^{55} -4.96288 q^{59} -1.29978 q^{61} +(-2.44949 - 6.52923i) q^{65} +(3.91611 - 3.91611i) q^{67} -2.96933i q^{71} +(3.40354 + 3.40354i) q^{73} +(5.02243 + 5.02243i) q^{77} +11.8939i q^{79} +(0.625277 - 0.625277i) q^{83} +(-0.0332693 - 0.0886808i) q^{85} -1.51721 q^{89} -14.7848 q^{91} +(-6.91798 + 15.2251i) q^{95} +(6.30900 - 6.30900i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{7}+O(q^{10}) \) 24 * q - 4 * q^7	\( 24 q - 4 q^{7} - 8 q^{13} - 16 q^{25} - 16 q^{31} + 8 q^{37} - 28 q^{55} + 8 q^{67} - 20 q^{73} + 48 q^{85} - 64 q^{91} + 20 q^{97}+O(q^{100}) \) 24 * q - 4 * q^7 - 8 * q^13 - 16 * q^25 - 16 * q^31 + 8 * q^37 - 28 * q^55 + 8 * q^67 - 20 * q^73 + 48 * q^85 - 64 * q^91 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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