LMFDB - Embedded newform 2880.2.bl.c.431.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(431,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, 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("2880.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bl (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:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			431.9
Character	$\chi$	$=$	2880.431
Dual form			2880.2.bl.c.1871.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +3.46834 q^{7} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{5} +3.46834 q^{7} +(2.99622 - 2.99622i) q^{11} +(4.77434 + 4.77434i) q^{13} -2.39050i q^{17} +(3.53066 - 3.53066i) q^{19} -0.278836i q^{23} +1.00000i q^{25} +(3.01517 - 3.01517i) q^{29} -0.996340i q^{31} +(2.45249 + 2.45249i) q^{35} +(-7.22519 + 7.22519i) q^{37} -2.39969 q^{41} +(-6.32983 - 6.32983i) q^{43} +4.37750 q^{47} +5.02938 q^{49} +(-7.49769 - 7.49769i) q^{53} +4.23729 q^{55} +(7.26952 - 7.26952i) q^{59} +(-0.978845 - 0.978845i) q^{61} +6.75194i q^{65} +(-4.87547 + 4.87547i) q^{67} +13.1542i q^{71} +14.2580i q^{73} +(10.3919 - 10.3919i) q^{77} -2.98407i q^{79} +(-11.2372 - 11.2372i) q^{83} +(1.69034 - 1.69034i) q^{85} -5.54970 q^{89} +(16.5590 + 16.5590i) q^{91} +4.99310 q^{95} +13.0596 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 8 q^{19} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 16 q^{67} + 16 q^{85} + 16 q^{91}+O(q^{100}) $ 32 * q + 8 * q^19 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 16 * q^67 + 16 * q^85 + 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$	3.46834			1.31091			0.655455	−	0.755234i	$-0.272477\pi$
$7$	3.46834			1.31091			0.655455	+	0.755234i	$0.272477\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.99622	−	2.99622i	0.903393	−	0.903393i	−0.0923348	−	0.995728i	$-0.529433\pi$
$11$	2.99622	−	2.99622i	0.903393	−	0.903393i	0.995728	+	0.0923348i	$0.0294330\pi$
$12$		0			0
$13$	4.77434	+	4.77434i	1.32416	+	1.32416i	0.910371	+	0.413793i	$0.135796\pi$
$13$	4.77434	+	4.77434i	1.32416	+	1.32416i	0.413793	+	0.910371i	$0.364204\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.39050i		−	0.579782i	−0.957060	−	0.289891i	$-0.906381\pi$
$17$		−	2.39050i		−	0.579782i	0.957060	−	0.289891i	$-0.0936190\pi$
$18$		0			0
$19$	3.53066	−	3.53066i	0.809988	−	0.809988i	−0.174644	−	0.984632i	$-0.555877\pi$
$19$	3.53066	−	3.53066i	0.809988	−	0.809988i	0.984632	+	0.174644i	$0.0558773\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.278836i		−	0.0581414i	−0.999577	−	0.0290707i	$-0.990745\pi$
$23$		−	0.278836i		−	0.0581414i	0.999577	−	0.0290707i	$-0.00925480\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.01517	−	3.01517i	0.559903	−	0.559903i	−0.369377	−	0.929280i	$-0.620429\pi$
$29$	3.01517	−	3.01517i	0.559903	−	0.559903i	0.929280	+	0.369377i	$0.120429\pi$
$30$		0			0
$31$		−	0.996340i		−	0.178948i	−0.995989	−	0.0894740i	$-0.971481\pi$
$31$		−	0.996340i		−	0.178948i	0.995989	−	0.0894740i	$-0.0285186\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.45249	+	2.45249i	0.414546	+	0.414546i
$36$		0			0
$37$	−7.22519	+	7.22519i	−1.18781	+	1.18781i	−0.210142	+	0.977671i	$0.567393\pi$
$37$	−7.22519	+	7.22519i	−1.18781	+	1.18781i	−0.977671	+	0.210142i	$0.932607\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.39969			−0.374769			−0.187384	−	0.982287i	$-0.560001\pi$
$41$	−2.39969			−0.374769			−0.187384	+	0.982287i	$0.560001\pi$
$42$		0			0
$43$	−6.32983	−	6.32983i	−0.965290	−	0.965290i	0.0341275	−	0.999417i	$-0.489135\pi$
$43$	−6.32983	−	6.32983i	−0.965290	−	0.965290i	−0.999417	+	0.0341275i	$0.989135\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.37750			0.638524			0.319262	−	0.947666i	$-0.396565\pi$
$47$	4.37750			0.638524			0.319262	+	0.947666i	$0.396565\pi$
$48$		0			0
$49$	5.02938			0.718483
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.49769	−	7.49769i	−1.02989	−	1.02989i	−0.999539	−	0.0303470i	$-0.990339\pi$
$53$	−7.49769	−	7.49769i	−1.02989	−	1.02989i	−0.0303470	−	0.999539i	$-0.509661\pi$
$54$		0			0
$55$	4.23729			0.571356
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.26952	−	7.26952i	0.946411	−	0.946411i	−0.0522241	−	0.998635i	$-0.516631\pi$
$59$	7.26952	−	7.26952i	0.946411	−	0.946411i	0.998635	+	0.0522241i	$0.0166310\pi$
$60$		0			0
$61$	−0.978845	−	0.978845i	−0.125328	−	0.125328i	0.641661	−	0.766989i	$-0.278246\pi$
$61$	−0.978845	−	0.978845i	−0.125328	−	0.125328i	−0.766989	+	0.641661i	$0.778246\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			6.75194i			0.837475i
$66$		0			0
$67$	−4.87547	+	4.87547i	−0.595634	+	0.595634i	−0.939148	−	0.343514i	$-0.888383\pi$
$67$	−4.87547	+	4.87547i	−0.595634	+	0.595634i	0.343514	+	0.939148i	$0.388383\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			13.1542i			1.56111i	0.625086	+	0.780556i	$0.285064\pi$
$71$			13.1542i			1.56111i	−0.625086	+	0.780556i	$0.714936\pi$
$72$		0			0
$73$			14.2580i			1.66877i	0.551183	+	0.834385i	$0.314177\pi$
$73$			14.2580i			1.66877i	−0.551183	+	0.834385i	$0.685823\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.3919	−	10.3919i	1.18427	−	1.18427i
$78$		0			0
$79$		−	2.98407i		−	0.335734i	−0.985810	−	0.167867i	$-0.946312\pi$
$79$		−	2.98407i		−	0.335734i	0.985810	−	0.167867i	$-0.0536879\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−11.2372	−	11.2372i	−1.23344	−	1.23344i	−0.962632	−	0.270813i	$-0.912708\pi$
$83$	−11.2372	−	11.2372i	−1.23344	−	1.23344i	−0.270813	−	0.962632i	$-0.587292\pi$
$84$		0			0
$85$	1.69034	−	1.69034i	0.183343	−	0.183343i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.54970			−0.588267			−0.294133	−	0.955764i	$-0.595031\pi$
$89$	−5.54970			−0.588267			−0.294133	+	0.955764i	$0.595031\pi$
$90$		0			0
$91$	16.5590	+	16.5590i	1.73586	+	1.73586i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.99310			0.512282
$96$		0			0
$97$	13.0596			1.32600			0.663002	−	0.748617i	$-0.269282\pi$
$97$	13.0596			1.32600			0.663002	+	0.748617i	$0.269282\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.983266	+	0.983266i	0.0978386	+	0.0978386i	0.754332	−	0.656493i	$-0.227961\pi$
$101$	0.983266	+	0.983266i	0.0978386	+	0.0978386i	−0.656493	+	0.754332i	$0.727961\pi$
$102$		0			0
$103$	3.49106			0.343984			0.171992	−	0.985098i	$-0.444980\pi$
$103$	3.49106			0.343984			0.171992	+	0.985098i	$0.444980\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.27786	+	5.27786i	−0.510230	+	0.510230i	−0.914597	−	0.404367i	$-0.867492\pi$
$107$	−5.27786	+	5.27786i	−0.510230	+	0.510230i	0.404367	+	0.914597i	$0.367492\pi$
$108$		0			0
$109$	−12.0086	−	12.0086i	−1.15021	−	1.15021i	−0.986510	−	0.163701i	$-0.947657\pi$
$109$	−12.0086	−	12.0086i	−1.15021	−	1.15021i	−0.163701	−	0.986510i	$-0.552343\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.37948i			0.129771i	0.997893	+	0.0648855i	$0.0206682\pi$
$113$			1.37948i			0.129771i	−0.997893	+	0.0648855i	$0.979332\pi$
$114$		0			0
$115$	0.197167	−	0.197167i	0.0183859	−	0.0183859i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	8.29107i		−	0.760041i
$120$		0			0
$121$		−	6.95463i		−	0.632239i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$		−	5.21404i		−	0.462671i	−0.972874	−	0.231336i	$-0.925690\pi$
$127$		−	5.21404i		−	0.462671i	0.972874	−	0.231336i	$-0.0743095\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.38061	−	2.38061i	−0.207995	−	0.207995i	0.595420	−	0.803415i	$-0.296986\pi$
$131$	−2.38061	−	2.38061i	−0.207995	−	0.207995i	−0.803415	+	0.595420i	$0.796986\pi$
$132$		0			0
$133$	12.2455	−	12.2455i	1.06182	−	1.06182i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.0060			−0.854867			−0.427433	−	0.904047i	$-0.640582\pi$
$137$	−10.0060			−0.854867			−0.427433	+	0.904047i	$0.640582\pi$
$138$		0			0
$139$	11.3877	+	11.3877i	0.965894	+	0.965894i	0.999437	−	0.0335432i	$-0.0106791\pi$
$139$	11.3877	+	11.3877i	0.965894	+	0.965894i	−0.0335432	+	0.999437i	$0.510679\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	28.6099			2.39248
$144$		0			0
$145$	4.26409			0.354114
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.32263	−	5.32263i	−0.436047	−	0.436047i	0.454632	−	0.890679i	$-0.349771\pi$
$149$	−5.32263	−	5.32263i	−0.436047	−	0.436047i	−0.890679	+	0.454632i	$0.849771\pi$
$150$		0			0
$151$	17.0551			1.38793			0.693963	−	0.720011i	$-0.255863\pi$
$151$	17.0551			1.38793			0.693963	+	0.720011i	$0.255863\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.704519	−	0.704519i	0.0565883	−	0.0565883i
$156$		0			0
$157$	15.2614	+	15.2614i	1.21799	+	1.21799i	0.968335	+	0.249655i	$0.0803173\pi$
$157$	15.2614	+	15.2614i	1.21799	+	1.21799i	0.249655	+	0.968335i	$0.419683\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.967100i		−	0.0762181i
$162$		0			0
$163$	−2.79575	+	2.79575i	−0.218980	+	0.218980i	−0.808069	−	0.589088i	$-0.799487\pi$
$163$	−2.79575	+	2.79575i	−0.218980	+	0.218980i	0.589088	+	0.808069i	$0.299487\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			7.44962i			0.576469i	0.957560	+	0.288234i	$0.0930683\pi$
$167$			7.44962i			0.576469i	−0.957560	+	0.288234i	$0.906932\pi$
$168$		0			0
$169$			32.5887i			2.50682i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.51910	−	1.51910i	0.115495	−	0.115495i	−0.646997	−	0.762492i	$-0.723975\pi$
$173$	1.51910	−	1.51910i	0.115495	−	0.115495i	0.762492	+	0.646997i	$0.223975\pi$
$174$		0			0
$175$			3.46834i			0.262182i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.00538	−	8.00538i	−0.598350	−	0.598350i	0.341523	−	0.939873i	$-0.389057\pi$
$179$	−8.00538	−	8.00538i	−0.598350	−	0.598350i	−0.939873	+	0.341523i	$0.889057\pi$
$180$		0			0
$181$	5.30352	−	5.30352i	0.394208	−	0.394208i	−0.481976	−	0.876184i	$-0.660081\pi$
$181$	5.30352	−	5.30352i	0.394208	−	0.394208i	0.876184	+	0.481976i	$0.160081\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.2180			−0.751239
$186$		0			0
$187$	−7.16246	−	7.16246i	−0.523771	−	0.523771i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	14.8433			1.07402			0.537011	−	0.843575i	$-0.319553\pi$
$191$	14.8433			1.07402			0.537011	+	0.843575i	$0.319553\pi$
$192$		0			0
$193$	−6.80232			−0.489642			−0.244821	−	0.969568i	$-0.578729\pi$
$193$	−6.80232			−0.489642			−0.244821	+	0.969568i	$0.578729\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.0580	+	18.0580i	1.28658	+	1.28658i	0.936849	+	0.349733i	$0.113728\pi$
$197$	18.0580	+	18.0580i	1.28658	+	1.28658i	0.349733	+	0.936849i	$0.386272\pi$
$198$		0			0
$199$	−13.9233			−0.986999			−0.493499	−	0.869746i	$-0.664282\pi$
$199$	−13.9233			−0.986999			−0.493499	+	0.869746i	$0.664282\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.4576	−	10.4576i	0.733982	−	0.733982i
$204$		0			0
$205$	−1.69684	−	1.69684i	−0.118512	−	0.118512i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	21.1572i		−	1.46348i
$210$		0			0
$211$	−5.22810	+	5.22810i	−0.359917	+	0.359917i	−0.863782	−	0.503865i	$-0.831911\pi$
$211$	−5.22810	+	5.22810i	−0.359917	+	0.359917i	0.503865	+	0.863782i	$0.331911\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	8.95173i		−	0.610503i
$216$		0			0
$217$		−	3.45565i		−	0.234585i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	11.4131	−	11.4131i	0.767726	−	0.767726i
$222$		0			0
$223$			7.54327i			0.505135i	0.967579	+	0.252567i	$0.0812750\pi$
$223$			7.54327i			0.505135i	−0.967579	+	0.252567i	$0.918725\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.99787	−	3.99787i	−0.265348	−	0.265348i	0.561875	−	0.827222i	$-0.310080\pi$
$227$	−3.99787	−	3.99787i	−0.265348	−	0.265348i	−0.827222	+	0.561875i	$0.810080\pi$
$228$		0			0
$229$	−6.02600	+	6.02600i	−0.398209	+	0.398209i	−0.877601	−	0.479392i	$-0.840857\pi$
$229$	−6.02600	+	6.02600i	−0.398209	+	0.398209i	0.479392	+	0.877601i	$0.340857\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.82845			0.185298			0.0926490	−	0.995699i	$-0.470467\pi$
$233$	2.82845			0.185298			0.0926490	+	0.995699i	$0.470467\pi$
$234$		0			0
$235$	3.09536	+	3.09536i	0.201919	+	0.201919i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.4810			−1.38949			−0.694746	−	0.719256i	$-0.744483\pi$
$239$	−21.4810			−1.38949			−0.694746	+	0.719256i	$0.744483\pi$
$240$		0			0
$241$	4.45343			0.286871			0.143435	−	0.989660i	$-0.454185\pi$
$241$	4.45343			0.286871			0.143435	+	0.989660i	$0.454185\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.55631	+	3.55631i	0.227204	+	0.227204i
$246$		0			0
$247$	33.7131			2.14511
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.19432	+	3.19432i	−0.201623	+	0.201623i	−0.800695	−	0.599072i	$-0.795536\pi$
$251$	−3.19432	+	3.19432i	−0.201623	+	0.201623i	0.599072	+	0.800695i	$0.295536\pi$
$252$		0			0
$253$	−0.835454	−	0.835454i	−0.0525246	−	0.0525246i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			15.4024i			0.960773i	0.877057	+	0.480386i	$0.159504\pi$
$257$			15.4024i			0.960773i	−0.877057	+	0.480386i	$0.840496\pi$
$258$		0			0
$259$	−25.0594	+	25.0594i	−1.55712	+	1.55712i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	2.18463i		−	0.134710i	−0.997729	−	0.0673551i	$-0.978544\pi$
$263$		−	2.18463i		−	0.134710i	0.997729	−	0.0673551i	$-0.0214560\pi$
$264$		0			0
$265$		−	10.6033i		−	0.651357i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	19.8892	−	19.8892i	1.21267	−	1.21267i	0.242521	−	0.970146i	$-0.422026\pi$
$269$	19.8892	−	19.8892i	1.21267	−	1.21267i	0.970146	−	0.242521i	$-0.0779745\pi$
$270$		0			0
$271$			20.1953i			1.22678i	0.789782	+	0.613388i	$0.210194\pi$
$271$			20.1953i			1.22678i	−0.789782	+	0.613388i	$0.789806\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.99622	+	2.99622i	0.180679	+	0.180679i
$276$		0			0
$277$	16.8275	−	16.8275i	1.01106	−	1.01106i	0.0111260	−	0.999938i	$-0.496458\pi$
$277$	16.8275	−	16.8275i	1.01106	−	1.01106i	0.999938	−	0.0111260i	$-0.00354158\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.46129			−0.325793			−0.162897	−	0.986643i	$-0.552084\pi$
$281$	−5.46129			−0.325793			−0.162897	+	0.986643i	$0.552084\pi$
$282$		0			0
$283$	−0.298980	−	0.298980i	−0.0177725	−	0.0177725i	0.698165	−	0.715937i	$-0.254000\pi$
$283$	−0.298980	−	0.298980i	−0.0177725	−	0.0177725i	−0.715937	+	0.698165i	$0.754000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.32294			−0.491288
$288$		0			0
$289$	11.2855			0.663853
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.11089	+	5.11089i	0.298582	+	0.298582i	0.840458	−	0.541877i	$-0.182286\pi$
$293$	5.11089	+	5.11089i	0.298582	+	0.298582i	−0.541877	+	0.840458i	$0.682286\pi$
$294$		0			0
$295$	10.2807			0.598563
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.33126	−	1.33126i	0.0769888	−	0.0769888i
$300$		0			0
$301$	−21.9540	−	21.9540i	−1.26541	−	1.26541i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	1.38430i		−	0.0792645i
$306$		0			0
$307$	9.69036	−	9.69036i	0.553058	−	0.553058i	−0.374264	−	0.927322i	$-0.622105\pi$
$307$	9.69036	−	9.69036i	0.553058	−	0.553058i	0.927322	+	0.374264i	$0.122105\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	16.8682i		−	0.956510i	−0.878221	−	0.478255i	$-0.841269\pi$
$311$		−	16.8682i		−	0.956510i	0.878221	−	0.478255i	$-0.158731\pi$
$312$		0			0
$313$		−	2.81354i		−	0.159031i	−0.996834	−	0.0795153i	$-0.974663\pi$
$313$		−	2.81354i		−	0.159031i	0.996834	−	0.0795153i	$-0.0253373\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.85773	−	8.85773i	0.497500	−	0.497500i	−0.413159	−	0.910659i	$-0.635575\pi$
$317$	8.85773	−	8.85773i	0.497500	−	0.497500i	0.910659	+	0.413159i	$0.135575\pi$
$318$		0			0
$319$		−	18.0682i		−	1.01163i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.44004	−	8.44004i	−0.469616	−	0.469616i
$324$		0			0
$325$	−4.77434	+	4.77434i	−0.264833	+	0.264833i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	15.1827			0.837047
$330$		0			0
$331$	−12.4995	−	12.4995i	−0.687035	−	0.687035i	0.274540	−	0.961576i	$-0.411474\pi$
$331$	−12.4995	−	12.4995i	−0.687035	−	0.687035i	−0.961576	+	0.274540i	$0.911474\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.89496			−0.376712
$336$		0			0
$337$	14.6032			0.795485			0.397742	−	0.917497i	$-0.369794\pi$
$337$	14.6032			0.795485			0.397742	+	0.917497i	$0.369794\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.98525	−	2.98525i	−0.161660	−	0.161660i
$342$		0			0
$343$	−6.83477			−0.369043
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−24.0390	+	24.0390i	−1.29048	+	1.29048i	−0.355994	+	0.934488i	$0.615858\pi$
$347$	−24.0390	+	24.0390i	−1.29048	+	1.29048i	−0.934488	+	0.355994i	$0.884142\pi$
$348$		0			0
$349$	−7.53550	−	7.53550i	−0.403366	−	0.403366i	0.476051	−	0.879417i	$-0.342068\pi$
$349$	−7.53550	−	7.53550i	−0.403366	−	0.403366i	−0.879417	+	0.476051i	$0.842068\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			33.8522i			1.80177i	0.434059	+	0.900884i	$0.357081\pi$
$353$			33.8522i			1.80177i	−0.434059	+	0.900884i	$0.642919\pi$
$354$		0			0
$355$	−9.30140	+	9.30140i	−0.493667	+	0.493667i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			27.0426i			1.42726i	0.700525	+	0.713628i	$0.252949\pi$
$359$			27.0426i			1.42726i	−0.700525	+	0.713628i	$0.747051\pi$
$360$		0			0
$361$		−	5.93107i		−	0.312162i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.0819	+	10.0819i	−0.527711	+	0.527711i
$366$		0			0
$367$		−	26.8371i		−	1.40089i	−0.713708	−	0.700444i	$-0.752985\pi$
$367$		−	26.8371i		−	1.40089i	0.713708	−	0.700444i	$-0.247015\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−26.0045	−	26.0045i	−1.35009	−	1.35009i
$372$		0			0
$373$	−12.2243	+	12.2243i	−0.632952	+	0.632952i	−0.948807	−	0.315855i	$-0.897709\pi$
$373$	−12.2243	+	12.2243i	−0.632952	+	0.632952i	0.315855	+	0.948807i	$0.397709\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	28.7909			1.48281
$378$		0			0
$379$	−14.2988	−	14.2988i	−0.734479	−	0.734479i	0.237024	−	0.971504i	$-0.423828\pi$
$379$	−14.2988	−	14.2988i	−0.734479	−	0.734479i	−0.971504	+	0.237024i	$0.923828\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−18.0479			−0.922206			−0.461103	−	0.887347i	$-0.652546\pi$
$383$	−18.0479			−0.922206			−0.461103	+	0.887347i	$0.652546\pi$
$384$		0			0
$385$	14.6964			0.748996
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.78439	−	4.78439i	−0.242578	−	0.242578i	0.575338	−	0.817916i	$-0.304871\pi$
$389$	−4.78439	−	4.78439i	−0.242578	−	0.242578i	−0.817916	+	0.575338i	$0.804871\pi$
$390$		0			0
$391$	−0.666559			−0.0337093
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.11005	−	2.11005i	0.106168	−	0.106168i
$396$		0			0
$397$	−5.31563	−	5.31563i	−0.266784	−	0.266784i	0.561019	−	0.827803i	$-0.310409\pi$
$397$	−5.31563	−	5.31563i	−0.266784	−	0.266784i	−0.827803	+	0.561019i	$0.810409\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			26.4395i			1.32033i	0.751122	+	0.660163i	$0.229513\pi$
$401$			26.4395i			1.32033i	−0.751122	+	0.660163i	$0.770487\pi$
$402$		0			0
$403$	4.75687	−	4.75687i	0.236957	−	0.236957i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			43.2964i			2.14612i
$408$		0			0
$409$			0.127003i			0.00627990i	0.999995	+	0.00313995i	$0.000999479\pi$
$409$			0.127003i			0.00627990i	−0.999995	+	0.00313995i	$0.999001\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	25.2132	−	25.2132i	1.24066	−	1.24066i
$414$		0			0
$415$		−	15.8918i		−	0.780099i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−24.5009	−	24.5009i	−1.19695	−	1.19695i	−0.975076	−	0.221871i	$-0.928784\pi$
$419$	−24.5009	−	24.5009i	−1.19695	−	1.19695i	−0.221871	−	0.975076i	$-0.571216\pi$
$420$		0			0
$421$	−4.32945	+	4.32945i	−0.211005	+	0.211005i	−0.804694	−	0.593690i	$-0.797671\pi$
$421$	−4.32945	+	4.32945i	−0.211005	+	0.211005i	0.593690	+	0.804694i	$0.297671\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.39050			0.115956
$426$		0			0
$427$	−3.39497	−	3.39497i	−0.164294	−	0.164294i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	19.3446			0.931798			0.465899	−	0.884838i	$-0.345731\pi$
$431$	19.3446			0.931798			0.465899	+	0.884838i	$0.345731\pi$
$432$		0			0
$433$	−3.03637			−0.145919			−0.0729594	−	0.997335i	$-0.523244\pi$
$433$	−3.03637			−0.145919			−0.0729594	+	0.997335i	$0.523244\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.984476	−	0.984476i	−0.0470939	−	0.0470939i
$438$		0			0
$439$	−20.3087			−0.969280			−0.484640	−	0.874714i	$-0.661049\pi$
$439$	−20.3087			−0.969280			−0.484640	+	0.874714i	$0.661049\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.4093	+	10.4093i	−0.494561	+	0.494561i	−0.909740	−	0.415179i	$-0.863719\pi$
$443$	−10.4093	+	10.4093i	−0.494561	+	0.494561i	0.415179	+	0.909740i	$0.363719\pi$
$444$		0			0
$445$	−3.92423	−	3.92423i	−0.186026	−	0.186026i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1.30849i			0.0617514i	0.999523	+	0.0308757i	$0.00982961\pi$
$449$			1.30849i			0.0617514i	−0.999523	+	0.0308757i	$0.990170\pi$
$450$		0			0
$451$	−7.18999	+	7.18999i	−0.338563	+	0.338563i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			23.4180i			1.09785i
$456$		0			0
$457$			18.5516i			0.867809i	0.900959	+	0.433904i	$0.142864\pi$
$457$			18.5516i			0.867809i	−0.900959	+	0.433904i	$0.857136\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.77732	+	3.77732i	−0.175927	+	0.175927i	−0.789578	−	0.613651i	$-0.789700\pi$
$461$	−3.77732	+	3.77732i	−0.175927	+	0.175927i	0.613651	+	0.789578i	$0.289700\pi$
$462$		0			0
$463$		−	7.43640i		−	0.345599i	−0.984957	−	0.172799i	$-0.944719\pi$
$463$		−	7.43640i		−	0.345599i	0.984957	−	0.172799i	$-0.0552812\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	26.8849	+	26.8849i	1.24408	+	1.24408i	0.958292	+	0.285790i	$0.0922561\pi$
$467$	26.8849	+	26.8849i	1.24408	+	1.24408i	0.285790	+	0.958292i	$0.407744\pi$
$468$		0			0
$469$	−16.9098	+	16.9098i	−0.780822	+	0.780822i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−37.9311			−1.74407
$474$		0			0
$475$	3.53066	+	3.53066i	0.161998	+	0.161998i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.3250			0.517451			0.258725	−	0.965951i	$-0.416698\pi$
$479$	11.3250			0.517451			0.258725	+	0.965951i	$0.416698\pi$
$480$		0			0
$481$	−68.9910			−3.14572
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9.23456	+	9.23456i	0.419320	+	0.419320i
$486$		0			0
$487$	−3.51600			−0.159325			−0.0796625	−	0.996822i	$-0.525384\pi$
$487$	−3.51600			−0.159325			−0.0796625	+	0.996822i	$0.525384\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	15.2692	−	15.2692i	0.689089	−	0.689089i	−0.272941	−	0.962031i	$-0.587997\pi$
$491$	15.2692	−	15.2692i	0.689089	−	0.689089i	0.962031	+	0.272941i	$0.0879965\pi$
$492$		0			0
$493$	−7.20777	−	7.20777i	−0.324622	−	0.324622i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			45.6231i			2.04648i
$498$		0			0
$499$	−11.4734	+	11.4734i	−0.513620	+	0.513620i	−0.915634	−	0.402014i	$-0.868310\pi$
$499$	−11.4734	+	11.4734i	−0.513620	+	0.513620i	0.402014	+	0.915634i	$0.368310\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			18.7681i			0.836828i	0.908256	+	0.418414i	$0.137414\pi$
$503$			18.7681i			0.836828i	−0.908256	+	0.418414i	$0.862586\pi$
$504$		0			0
$505$			1.39055i			0.0618785i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−26.2642	+	26.2642i	−1.16414	+	1.16414i	−0.180579	+	0.983560i	$0.557797\pi$
$509$	−26.2642	+	26.2642i	−1.16414	+	1.16414i	−0.983560	+	0.180579i	$0.942203\pi$
$510$		0			0
$511$			49.4515i			2.18761i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.46855	+	2.46855i	0.108777	+	0.108777i
$516$		0			0
$517$	13.1159	−	13.1159i	0.576838	−	0.576838i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−22.3421			−0.978827			−0.489413	−	0.872052i	$-0.662789\pi$
$521$	−22.3421			−0.978827			−0.489413	+	0.872052i	$0.662789\pi$
$522$		0			0
$523$	7.09669	+	7.09669i	0.310317	+	0.310317i	0.845032	−	0.534715i	$-0.179581\pi$
$523$	7.09669	+	7.09669i	0.310317	+	0.310317i	−0.534715	+	0.845032i	$0.679581\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.38175			−0.103751
$528$		0			0
$529$	22.9223			0.996620
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−11.4569	−	11.4569i	−0.496255	−	0.496255i
$534$		0			0
$535$	−7.46402			−0.322698
$536$		0			0
$537$		0			0
$538$		0			0
$539$	15.0691	−	15.0691i	0.649073	−	0.649073i
$540$		0			0
$541$	−10.9131	−	10.9131i	−0.469192	−	0.469192i	0.432461	−	0.901653i	$-0.357645\pi$
$541$	−10.9131	−	10.9131i	−0.469192	−	0.469192i	−0.901653	+	0.432461i	$0.857645\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	16.9827i		−	0.727458i
$546$		0			0
$547$	13.8438	−	13.8438i	0.591920	−	0.591920i	−0.346230	−	0.938150i	$-0.612538\pi$
$547$	13.8438	−	13.8438i	0.591920	−	0.591920i	0.938150	+	0.346230i	$0.112538\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	21.2911i		−	0.907030i
$552$		0			0
$553$		−	10.3498i		−	0.440117i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.0369	+	25.0369i	−1.06085	+	1.06085i	−0.0628224	+	0.998025i	$0.520010\pi$
$557$	−25.0369	+	25.0369i	−1.06085	+	1.06085i	−0.998025	+	0.0628224i	$0.979990\pi$
$558$		0			0
$559$		−	60.4415i		−	2.55640i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−24.8835	−	24.8835i	−1.04871	−	1.04871i	−0.998751	−	0.0499629i	$-0.984090\pi$
$563$	−24.8835	−	24.8835i	−1.04871	−	1.04871i	−0.0499629	−	0.998751i	$-0.515910\pi$
$564$		0			0
$565$	−0.975443	+	0.975443i	−0.0410372	+	0.0410372i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.85220			−0.0776483			−0.0388242	−	0.999246i	$-0.512361\pi$
$569$	−1.85220			−0.0776483			−0.0388242	+	0.999246i	$0.512361\pi$
$570$		0			0
$571$	−18.8986	−	18.8986i	−0.790884	−	0.790884i	0.190754	−	0.981638i	$-0.438907\pi$
$571$	−18.8986	−	18.8986i	−0.790884	−	0.790884i	−0.981638	+	0.190754i	$0.938907\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.278836			0.0116283
$576$		0			0
$577$	17.3812			0.723587			0.361793	−	0.932258i	$-0.382165\pi$
$577$	17.3812			0.723587			0.361793	+	0.932258i	$0.382165\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−38.9745	−	38.9745i	−1.61693	−	1.61693i
$582$		0			0
$583$	−44.9294			−1.86078
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.846887	+	0.846887i	−0.0349548	+	0.0349548i	−0.724368	−	0.689413i	$-0.757868\pi$
$587$	−0.846887	+	0.846887i	−0.0349548	+	0.0349548i	0.689413	+	0.724368i	$0.257868\pi$
$588$		0			0
$589$	−3.51774	−	3.51774i	−0.144946	−	0.144946i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	32.0715i		−	1.31702i	−0.752573	−	0.658509i	$-0.771188\pi$
$593$		−	32.0715i		−	1.31702i	0.752573	−	0.658509i	$-0.228812\pi$
$594$		0			0
$595$	5.86267	−	5.86267i	0.240346	−	0.240346i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	37.1178i		−	1.51659i	−0.651911	−	0.758296i	$-0.726032\pi$
$599$		−	37.1178i		−	1.51659i	0.651911	−	0.758296i	$-0.273968\pi$
$600$		0			0
$601$			4.56408i			0.186173i	0.995658	+	0.0930863i	$0.0296733\pi$
$601$			4.56408i			0.186173i	−0.995658	+	0.0930863i	$0.970327\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.91766	−	4.91766i	0.199931	−	0.199931i
$606$		0			0
$607$			29.3122i			1.18975i	0.803820	+	0.594873i	$0.202798\pi$
$607$			29.3122i			1.18975i	−0.803820	+	0.594873i	$0.797202\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	20.8997	+	20.8997i	0.845510	+	0.845510i
$612$		0			0
$613$	−15.0647	+	15.0647i	−0.608458	+	0.608458i	−0.942543	−	0.334085i	$-0.891573\pi$
$613$	−15.0647	+	15.0647i	−0.608458	+	0.608458i	0.334085	+	0.942543i	$0.391573\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−20.3033			−0.817382			−0.408691	−	0.912673i	$-0.634015\pi$
$617$	−20.3033			−0.817382			−0.408691	+	0.912673i	$0.634015\pi$
$618$		0			0
$619$	4.50618	+	4.50618i	0.181119	+	0.181119i	0.791843	−	0.610725i	$-0.209122\pi$
$619$	4.50618	+	4.50618i	0.181119	+	0.181119i	−0.610725	+	0.791843i	$0.709122\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−19.2482			−0.771164
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	17.2718	+	17.2718i	0.688673	+	0.688673i
$630$		0			0
$631$	−6.12680			−0.243904			−0.121952	−	0.992536i	$-0.538915\pi$
$631$	−6.12680			−0.243904			−0.121952	+	0.992536i	$0.538915\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.68688	−	3.68688i	0.146309	−	0.146309i
$636$		0			0
$637$	24.0120	+	24.0120i	0.951390	+	0.951390i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	3.51936i		−	0.139006i	−0.997582	−	0.0695031i	$-0.977859\pi$
$641$		−	3.51936i		−	0.139006i	0.997582	−	0.0695031i	$-0.0221414\pi$
$642$		0			0
$643$	15.6415	−	15.6415i	0.616840	−	0.616840i	−0.327880	−	0.944719i	$-0.606334\pi$
$643$	15.6415	−	15.6415i	0.616840	−	0.616840i	0.944719	+	0.327880i	$0.106334\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			24.5413i			0.964817i	0.875946	+	0.482408i	$0.160238\pi$
$647$			24.5413i			0.964817i	−0.875946	+	0.482408i	$0.839762\pi$
$648$		0			0
$649$		−	43.5621i		−	1.70996i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.47579	−	9.47579i	0.370816	−	0.370816i	−0.496958	−	0.867775i	$-0.665550\pi$
$653$	9.47579	−	9.47579i	0.370816	−	0.370816i	0.867775	+	0.496958i	$0.165550\pi$
$654$		0			0
$655$		−	3.36670i		−	0.131548i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	34.0933	+	34.0933i	1.32809	+	1.32809i	0.907037	+	0.421050i	$0.138338\pi$
$659$	34.0933	+	34.0933i	1.32809	+	1.32809i	0.421050	+	0.907037i	$0.361662\pi$
$660$		0			0
$661$	29.2420	−	29.2420i	1.13738	−	1.13738i	0.148464	−	0.988918i	$-0.452567\pi$
$661$	29.2420	−	29.2420i	1.13738	−	1.13738i	0.988918	−	0.148464i	$-0.0474328\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	17.3178			0.671555
$666$		0			0
$667$	−0.840739	−	0.840739i	−0.0325536	−	0.0325536i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.86566			−0.226441
$672$		0			0
$673$	−36.3379			−1.40072			−0.700362	−	0.713788i	$-0.746978\pi$
$673$	−36.3379			−1.40072			−0.700362	+	0.713788i	$0.746978\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	24.7889	+	24.7889i	0.952714	+	0.952714i	0.998931	−	0.0462176i	$-0.0147168\pi$
$677$	24.7889	+	24.7889i	0.952714	+	0.952714i	−0.0462176	+	0.998931i	$0.514717\pi$
$678$		0			0
$679$	45.2953			1.73827
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.48314	+	1.48314i	−0.0567508	+	0.0567508i	−0.734913	−	0.678162i	$-0.762777\pi$
$683$	−1.48314	+	1.48314i	−0.0567508	+	0.0567508i	0.678162	+	0.734913i	$0.262777\pi$
$684$		0			0
$685$	−7.07528	−	7.07528i	−0.270333	−	0.270333i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	71.5930i		−	2.72748i
$690$		0			0
$691$	12.4070	−	12.4070i	0.471986	−	0.471986i	−0.430571	−	0.902557i	$-0.641688\pi$
$691$	12.4070	−	12.4070i	0.471986	−	0.471986i	0.902557	+	0.430571i	$0.141688\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			16.1047i			0.610885i
$696$		0			0
$697$			5.73646i			0.217284i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−8.56355	+	8.56355i	−0.323441	+	0.323441i	−0.850086	−	0.526645i	$-0.823450\pi$
$701$	−8.56355	+	8.56355i	−0.323441	+	0.323441i	0.526645	+	0.850086i	$0.323450\pi$
$702$		0			0
$703$			51.0193i			1.92423i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.41030	+	3.41030i	0.128258	+	0.128258i
$708$		0			0
$709$	−10.9420	+	10.9420i	−0.410937	+	0.410937i	−0.882065	−	0.471128i	$-0.843847\pi$
$709$	−10.9420	+	10.9420i	−0.410937	+	0.410937i	0.471128	+	0.882065i	$0.343847\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.277816			−0.0104043
$714$		0			0
$715$	20.2303	+	20.2303i	0.756569	+	0.756569i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−28.0718			−1.04690			−0.523451	−	0.852056i	$-0.675356\pi$
$719$	−28.0718			−1.04690			−0.523451	+	0.852056i	$0.675356\pi$
$720$		0			0
$721$	12.1082			0.450932
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.01517	+	3.01517i	0.111981	+	0.111981i
$726$		0			0
$727$	−27.0563			−1.00346			−0.501732	−	0.865023i	$-0.667304\pi$
$727$	−27.0563			−1.00346			−0.501732	+	0.865023i	$0.667304\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−15.1315	+	15.1315i	−0.559658	+	0.559658i
$732$		0			0
$733$	−3.02612	−	3.02612i	−0.111772	−	0.111772i	0.649009	−	0.760781i	$-0.275184\pi$
$733$	−3.02612	−	3.02612i	−0.111772	−	0.111772i	−0.760781	+	0.649009i	$0.775184\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			29.2159i			1.07618i
$738$		0			0
$739$	−32.2327	+	32.2327i	−1.18570	+	1.18570i	−0.207454	+	0.978245i	$0.566518\pi$
$739$	−32.2327	+	32.2327i	−1.18570	+	1.18570i	−0.978245	+	0.207454i	$0.933482\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	32.9344i		−	1.20825i	−0.796891	−	0.604123i	$-0.793524\pi$
$743$		−	32.9344i		−	1.20825i	0.796891	−	0.604123i	$-0.206476\pi$
$744$		0			0
$745$		−	7.52734i		−	0.275780i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−18.3054	+	18.3054i	−0.668866	+	0.668866i
$750$		0			0
$751$		−	50.2835i		−	1.83487i	−0.397883	−	0.917436i	$-0.630255\pi$
$751$		−	50.2835i		−	1.83487i	0.397883	−	0.917436i	$-0.369745\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0598	+	12.0598i	0.438900	+	0.438900i
$756$		0			0
$757$	−7.29842	+	7.29842i	−0.265266	+	0.265266i	−0.827189	−	0.561924i	$-0.810062\pi$
$757$	−7.29842	+	7.29842i	−0.265266	+	0.265266i	0.561924	+	0.827189i	$0.310062\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−49.3132			−1.78760			−0.893800	−	0.448465i	$-0.851971\pi$
$761$	−49.3132			−1.78760			−0.893800	+	0.448465i	$0.851971\pi$
$762$		0			0
$763$	−41.6498	−	41.6498i	−1.50782	−	1.50782i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	69.4144			2.50641
$768$		0			0
$769$	−26.4849			−0.955069			−0.477534	−	0.878613i	$-0.658469\pi$
$769$	−26.4849			−0.955069			−0.477534	+	0.878613i	$0.658469\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.53754	−	9.53754i	−0.343042	−	0.343042i	0.514468	−	0.857510i	$-0.327989\pi$
$773$	−9.53754	−	9.53754i	−0.343042	−	0.343042i	−0.857510	+	0.514468i	$0.827989\pi$
$774$		0			0
$775$	0.996340			0.0357896
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.47248	+	8.47248i	−0.303558	+	0.303558i
$780$		0			0
$781$	39.4127	+	39.4127i	1.41030	+	1.41030i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			21.5828i			0.770325i
$786$		0			0
$787$	16.0619	−	16.0619i	0.572544	−	0.572544i	−0.360295	−	0.932839i	$-0.617324\pi$
$787$	16.0619	−	16.0619i	0.572544	−	0.572544i	0.932839	+	0.360295i	$0.117324\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			4.78452i			0.170118i
$792$		0			0
$793$		−	9.34668i		−	0.331910i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−23.8740	+	23.8740i	−0.845662	+	0.845662i	−0.989588	−	0.143926i	$-0.954027\pi$
$797$	−23.8740	+	23.8740i	−0.845662	+	0.845662i	0.143926	+	0.989588i	$0.454027\pi$
$798$		0			0
$799$		−	10.4644i		−	0.370205i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	42.7200	+	42.7200i	1.50755	+	1.50755i
$804$		0			0
$805$	0.683843	−	0.683843i	0.0241023	−	0.0241023i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−32.5913			−1.14585			−0.572925	−	0.819608i	$-0.694191\pi$
$809$	−32.5913			−1.14585			−0.572925	+	0.819608i	$0.694191\pi$
$810$		0			0
$811$	12.1454	+	12.1454i	0.426481	+	0.426481i	0.887428	−	0.460947i	$-0.152490\pi$
$811$	12.1454	+	12.1454i	0.426481	+	0.426481i	−0.460947	+	0.887428i	$0.652490\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.95379			−0.138495
$816$		0			0
$817$	−44.6969			−1.56375
$818$		0			0
$819$		0			0
$820$		0			0
$821$	35.0856	+	35.0856i	1.22450	+	1.22450i	0.966016	+	0.258481i	$0.0832219\pi$
$821$	35.0856	+	35.0856i	1.22450	+	1.22450i	0.258481	+	0.966016i	$0.416778\pi$
$822$		0			0
$823$	29.4327			1.02596			0.512979	−	0.858401i	$-0.328542\pi$
$823$	29.4327			1.02596			0.512979	+	0.858401i	$0.328542\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9.47432	+	9.47432i	−0.329455	+	0.329455i	−0.852379	−	0.522924i	$-0.824841\pi$
$827$	−9.47432	+	9.47432i	−0.329455	+	0.329455i	0.522924	+	0.852379i	$0.324841\pi$
$828$		0			0
$829$	−36.2984	−	36.2984i	−1.26070	−	1.26070i	−0.950756	−	0.309940i	$-0.899691\pi$
$829$	−36.2984	−	36.2984i	−1.26070	−	1.26070i	−0.309940	−	0.950756i	$-0.600309\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	12.0228i		−	0.416564i
$834$		0			0
$835$	−5.26768	+	5.26768i	−0.182295	+	0.182295i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	9.14591i		−	0.315752i	−0.987459	−	0.157876i	$-0.949535\pi$
$839$		−	9.14591i		−	0.315752i	0.987459	−	0.157876i	$-0.0504646\pi$
$840$		0			0
$841$			10.8175i			0.373017i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−23.0437	+	23.0437i	−0.792726	+	0.792726i
$846$		0			0
$847$		−	24.1210i		−	0.828808i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.01465	+	2.01465i	0.0690611	+	0.0690611i
$852$		0			0
$853$	27.5798	−	27.5798i	0.944315	−	0.944315i	−0.0542139	−	0.998529i	$-0.517265\pi$
$853$	27.5798	−	27.5798i	0.944315	−	0.944315i	0.998529	+	0.0542139i	$0.0172653\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.6566			0.500661			0.250330	−	0.968160i	$-0.419461\pi$
$857$	14.6566			0.500661			0.250330	+	0.968160i	$0.419461\pi$
$858$		0			0
$859$	−23.3717	−	23.3717i	−0.797433	−	0.797433i	0.185257	−	0.982690i	$-0.440688\pi$
$859$	−23.3717	−	23.3717i	−0.797433	−	0.797433i	−0.982690	+	0.185257i	$0.940688\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	34.5520			1.17616			0.588082	−	0.808801i	$-0.299883\pi$
$863$	34.5520			1.17616			0.588082	+	0.808801i	$0.299883\pi$
$864$		0			0
$865$	2.14834			0.0730457
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8.94091	−	8.94091i	−0.303300	−	0.303300i
$870$		0			0
$871$	−46.5543			−1.57743
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.45249	+	2.45249i	−0.0829092	+	0.0829092i
$876$		0			0
$877$	−27.4491	−	27.4491i	−0.926889	−	0.926889i	0.0706146	−	0.997504i	$-0.477504\pi$
$877$	−27.4491	−	27.4491i	−0.926889	−	0.926889i	−0.997504	+	0.0706146i	$0.977504\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			45.7633i			1.54180i	0.636954	+	0.770902i	$0.280194\pi$
$881$			45.7633i			1.54180i	−0.636954	+	0.770902i	$0.719806\pi$
$882$		0			0
$883$	5.31320	−	5.31320i	0.178803	−	0.178803i	−0.612031	−	0.790834i	$-0.709647\pi$
$883$	5.31320	−	5.31320i	0.178803	−	0.178803i	0.790834	+	0.612031i	$0.209647\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1.11175i			0.0373289i	0.999826	+	0.0186645i	$0.00594142\pi$
$887$			1.11175i			0.0373289i	−0.999826	+	0.0186645i	$0.994059\pi$
$888$		0			0
$889$		−	18.0841i		−	0.606520i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	15.4554	−	15.4554i	0.517197	−	0.517197i
$894$		0			0
$895$		−	11.3213i		−	0.378430i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.00414	−	3.00414i	−0.100194	−	0.100194i
$900$		0			0
$901$	−17.9232	+	17.9232i	−0.597109	+	0.597109i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.50032			0.249319
$906$		0			0
$907$	16.1623	+	16.1623i	0.536662	+	0.536662i	0.922547	−	0.385885i	$-0.126104\pi$
$907$	16.1623	+	16.1623i	0.536662	+	0.536662i	−0.385885	+	0.922547i	$0.626104\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	13.4331			0.445058			0.222529	−	0.974926i	$-0.428569\pi$
$911$	13.4331			0.445058			0.222529	+	0.974926i	$0.428569\pi$
$912$		0			0
$913$	−67.3383			−2.22857
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.25678	−	8.25678i	−0.272663	−	0.272663i
$918$		0			0
$919$	−25.6681			−0.846711			−0.423356	−	0.905964i	$-0.639148\pi$
$919$	−25.6681			−0.846711			−0.423356	+	0.905964i	$0.639148\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−62.8024	+	62.8024i	−2.06717	+	2.06717i
$924$		0			0
$925$	−7.22519	−	7.22519i	−0.237563	−	0.237563i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	56.1941i		−	1.84367i	−0.387583	−	0.921835i	$-0.626690\pi$
$929$		−	56.1941i		−	1.84367i	0.387583	−	0.921835i	$-0.373310\pi$
$930$		0			0
$931$	17.7570	−	17.7570i	0.581963	−	0.581963i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	10.1292i		−	0.331262i
$936$		0			0
$937$		−	3.04310i		−	0.0994137i	−0.998764	−	0.0497069i	$-0.984171\pi$
$937$		−	3.04310i		−	0.0994137i	0.998764	−	0.0497069i	$-0.0158287\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.0250	+	10.0250i	−0.326807	+	0.326807i	−0.851371	−	0.524564i	$-0.824228\pi$
$941$	−10.0250	+	10.0250i	−0.326807	+	0.326807i	0.524564	+	0.851371i	$0.324228\pi$
$942$		0			0
$943$			0.669121i			0.0217896i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−12.0109	−	12.0109i	−0.390303	−	0.390303i	0.484492	−	0.874796i	$-0.339004\pi$
$947$	−12.0109	−	12.0109i	−0.390303	−	0.390303i	−0.874796	+	0.484492i	$0.839004\pi$
$948$		0			0
$949$	−68.0724	+	68.0724i	−2.20972	+	2.20972i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20.1336			0.652192			0.326096	−	0.945337i	$-0.394267\pi$
$953$	20.1336			0.652192			0.326096	+	0.945337i	$0.394267\pi$
$954$		0			0
$955$	10.4958	+	10.4958i	0.339635	+	0.339635i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−34.7041			−1.12065
$960$		0			0
$961$	30.0073			0.967978
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.80997	−	4.80997i	−0.154838	−	0.154838i
$966$		0			0
$967$	−32.5320			−1.04616			−0.523079	−	0.852284i	$-0.675217\pi$
$967$	−32.5320			−1.04616			−0.523079	+	0.852284i	$0.675217\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.355869	−	0.355869i	0.0114204	−	0.0114204i	−0.701374	−	0.712794i	$-0.747429\pi$
$971$	0.355869	−	0.355869i	0.0114204	−	0.0114204i	0.712794	+	0.701374i	$0.247429\pi$
$972$		0			0
$973$	39.4965	+	39.4965i	1.26620	+	1.26620i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	16.5742i		−	0.530256i	−0.964213	−	0.265128i	$-0.914586\pi$
$977$		−	16.5742i		−	0.530256i	0.964213	−	0.265128i	$-0.0854142\pi$
$978$		0			0
$979$	−16.6281	+	16.6281i	−0.531436	+	0.531436i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	7.38385i		−	0.235508i	−0.993043	−	0.117754i	$-0.962431\pi$
$983$		−	7.38385i		−	0.235508i	0.993043	−	0.117754i	$-0.0375695\pi$
$984$		0			0
$985$			25.5379i			0.813706i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.76499	+	1.76499i	−0.0561233	+	0.0561233i
$990$		0			0
$991$		−	28.8218i		−	0.915554i	−0.889067	−	0.457777i	$-0.848646\pi$
$991$		−	28.8218i		−	0.915554i	0.889067	−	0.457777i	$-0.151354\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9.84528	−	9.84528i	−0.312116	−	0.312116i
$996$		0			0
$997$	17.3238	−	17.3238i	0.548649	−	0.548649i	−0.377401	−	0.926050i	$-0.623182\pi$
$997$	17.3238	−	17.3238i	0.548649	−	0.548649i	0.926050	+	0.377401i	$0.123182\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bl.c.431.9		32
3.2	odd	2	inner	2880.2.bl.c.431.7		32
4.3	odd	2		720.2.bl.c.611.10	yes	32
12.11	even	2		720.2.bl.c.611.7	yes	32
16.5	even	4		720.2.bl.c.251.7	✓	32
16.11	odd	4	inner	2880.2.bl.c.1871.7		32
48.5	odd	4		720.2.bl.c.251.10	yes	32
48.11	even	4	inner	2880.2.bl.c.1871.9		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.c.251.7	✓	32	16.5	even	4
720.2.bl.c.251.10	yes	32	48.5	odd	4
720.2.bl.c.611.7	yes	32	12.11	even	2
720.2.bl.c.611.10	yes	32	4.3	odd	2
2880.2.bl.c.431.7		32	3.2	odd	2	inner
2880.2.bl.c.431.9		32	1.1	even	1	trivial
2880.2.bl.c.1871.7		32	16.11	odd	4	inner
2880.2.bl.c.1871.9		32	48.11	even	4	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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +3.46834 q^{7} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{5} +3.46834 q^{7} +(2.99622 - 2.99622i) q^{11} +(4.77434 + 4.77434i) q^{13} -2.39050i q^{17} +(3.53066 - 3.53066i) q^{19} -0.278836i q^{23} +1.00000i q^{25} +(3.01517 - 3.01517i) q^{29} -0.996340i q^{31} +(2.45249 + 2.45249i) q^{35} +(-7.22519 + 7.22519i) q^{37} -2.39969 q^{41} +(-6.32983 - 6.32983i) q^{43} +4.37750 q^{47} +5.02938 q^{49} +(-7.49769 - 7.49769i) q^{53} +4.23729 q^{55} +(7.26952 - 7.26952i) q^{59} +(-0.978845 - 0.978845i) q^{61} +6.75194i q^{65} +(-4.87547 + 4.87547i) q^{67} +13.1542i q^{71} +14.2580i q^{73} +(10.3919 - 10.3919i) q^{77} -2.98407i q^{79} +(-11.2372 - 11.2372i) q^{83} +(1.69034 - 1.69034i) q^{85} -5.54970 q^{89} +(16.5590 + 16.5590i) q^{91} +4.99310 q^{95} +13.0596 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 8 q^{19} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 16 q^{67} + 16 q^{85} + 16 q^{91}+O(q^{100}) \) 32 * q + 8 * q^19 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 16 * q^67 + 16 * q^85 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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