LMFDB - Embedded newform 2880.2.bl.a.431.4

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:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.3317760000.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 17x^{4} + 256 $ x^8 + 17*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			431.4
Root			$-1.72286 + 1.01575i$ of defining polynomial
Character	$\chi$	$=$	2880.431
Dual form			2880.2.bl.a.1871.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +2.87298 q^{7} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{5} +2.87298 q^{7} +(-2.03151 + 2.03151i) q^{11} +(3.87298 + 3.87298i) q^{13} +1.41421i q^{17} +(-5.87298 + 5.87298i) q^{19} -5.47723i q^{23} +1.00000i q^{25} +(-2.82843 + 2.82843i) q^{29} +4.00000i q^{31} +(2.03151 + 2.03151i) q^{35} +(-1.00000 + 1.00000i) q^{37} -11.1341 q^{41} +(6.00000 + 6.00000i) q^{43} -1.41421 q^{47} +1.25403 q^{49} +(1.41421 + 1.41421i) q^{53} -2.87298 q^{55} +(9.10257 - 9.10257i) q^{59} +(-10.7460 - 10.7460i) q^{61} +5.47723i q^{65} +(-3.12702 + 3.12702i) q^{67} +9.71987i q^{71} -5.74597i q^{73} +(-5.83648 + 5.83648i) q^{77} -3.74597i q^{79} +(-1.41421 - 1.41421i) q^{83} +(-1.00000 + 1.00000i) q^{85} -5.83648 q^{89} +(11.1270 + 11.1270i) q^{91} -8.30565 q^{95} +17.4919 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} - 16 q^{19} - 8 q^{37} + 48 q^{43} + 72 q^{49} + 8 q^{55} - 24 q^{61} - 56 q^{67} - 8 q^{85} + 120 q^{91} + 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 - 16 * q^19 - 8 * q^37 + 48 * q^43 + 72 * q^49 + 8 * q^55 - 24 * q^61 - 56 * q^67 - 8 * q^85 + 120 * q^91 + 16 * q^97

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
$7$	2.87298			1.08589			0.542943	−	0.839770i	$-0.317310\pi$
$7$	2.87298			1.08589			0.542943	+	0.839770i	$0.317310\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.03151	+	2.03151i	−0.612522	+	0.612522i	−0.943603	−	0.331080i	$-0.892587\pi$
$11$	−2.03151	+	2.03151i	−0.612522	+	0.612522i	0.331080	+	0.943603i	$0.392587\pi$
$12$		0			0
$13$	3.87298	+	3.87298i	1.07417	+	1.07417i	0.997019	+	0.0771531i	$0.0245830\pi$
$13$	3.87298	+	3.87298i	1.07417	+	1.07417i	0.0771531	+	0.997019i	$0.475417\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.41421i			0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
$17$			1.41421i			0.342997i	−0.985184	+	0.171499i	$0.945139\pi$
$18$		0			0
$19$	−5.87298	+	5.87298i	−1.34735	+	1.34735i	−0.458831	+	0.888523i	$0.651732\pi$
$19$	−5.87298	+	5.87298i	−1.34735	+	1.34735i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	5.47723i		−	1.14208i	−0.820922	−	0.571040i	$-0.806540\pi$
$23$		−	5.47723i		−	1.14208i	0.820922	−	0.571040i	$-0.193460\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.82843	+	2.82843i	−0.525226	+	0.525226i	−0.919145	−	0.393919i	$-0.871119\pi$
$29$	−2.82843	+	2.82843i	−0.525226	+	0.525226i	0.393919	+	0.919145i	$0.371119\pi$
$30$		0			0
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.03151	+	2.03151i	0.343387	+	0.343387i
$36$		0			0
$37$	−1.00000	+	1.00000i	−0.164399	+	0.164399i	−0.784512	−	0.620113i	$-0.787087\pi$
$37$	−1.00000	+	1.00000i	−0.164399	+	0.164399i	0.620113	+	0.784512i	$0.287087\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.1341			−1.73885			−0.869426	−	0.494064i	$-0.835511\pi$
$41$	−11.1341			−1.73885			−0.869426	+	0.494064i	$0.835511\pi$
$42$		0			0
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	0.996660	−	0.0816682i	$-0.0260248\pi$
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	−0.0816682	+	0.996660i	$0.526025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.41421			−0.206284			−0.103142	−	0.994667i	$-0.532890\pi$
$47$	−1.41421			−0.206284			−0.103142	+	0.994667i	$0.532890\pi$
$48$		0			0
$49$	1.25403			0.179148
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.41421	+	1.41421i	0.194257	+	0.194257i	0.797533	−	0.603276i	$-0.206138\pi$
$53$	1.41421	+	1.41421i	0.194257	+	0.194257i	−0.603276	+	0.797533i	$0.706138\pi$
$54$		0			0
$55$	−2.87298			−0.387393
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.10257	−	9.10257i	1.18505	−	1.18505i	0.206636	−	0.978418i	$-0.433748\pi$
$59$	9.10257	−	9.10257i	1.18505	−	1.18505i	0.978418	−	0.206636i	$-0.0662517\pi$
$60$		0			0
$61$	−10.7460	−	10.7460i	−1.37588	−	1.37588i	−0.851459	−	0.524421i	$-0.824282\pi$
$61$	−10.7460	−	10.7460i	−1.37588	−	1.37588i	−0.524421	−	0.851459i	$-0.675718\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			5.47723i			0.679366i
$66$		0			0
$67$	−3.12702	+	3.12702i	−0.382026	+	0.382026i	−0.871832	−	0.489806i	$-0.837068\pi$
$67$	−3.12702	+	3.12702i	−0.382026	+	0.382026i	0.489806	+	0.871832i	$0.337068\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			9.71987i			1.15354i	0.816908	+	0.576768i	$0.195686\pi$
$71$			9.71987i			1.15354i	−0.816908	+	0.576768i	$0.804314\pi$
$72$		0			0
$73$		−	5.74597i		−	0.672515i	−0.941770	−	0.336257i	$-0.890839\pi$
$73$		−	5.74597i		−	0.672515i	0.941770	−	0.336257i	$-0.109161\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.83648	+	5.83648i	−0.665129	+	0.665129i
$78$		0			0
$79$		−	3.74597i		−	0.421454i	−0.977545	−	0.210727i	$-0.932417\pi$
$79$		−	3.74597i		−	0.421454i	0.977545	−	0.210727i	$-0.0675831\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.41421	−	1.41421i	−0.155230	−	0.155230i	0.625219	−	0.780449i	$-0.285010\pi$
$83$	−1.41421	−	1.41421i	−0.155230	−	0.155230i	−0.780449	+	0.625219i	$0.785010\pi$
$84$		0			0
$85$	−1.00000	+	1.00000i	−0.108465	+	0.108465i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.83648			−0.618666			−0.309333	−	0.950954i	$-0.600106\pi$
$89$	−5.83648			−0.618666			−0.309333	+	0.950954i	$0.600106\pi$
$90$		0			0
$91$	11.1270	+	11.1270i	1.16643	+	1.16643i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.30565			−0.852142
$96$		0			0
$97$	17.4919			1.77604			0.888018	−	0.459808i	$-0.152082\pi$
$97$	17.4919			1.77604			0.888018	+	0.459808i	$0.152082\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.23458	+	1.23458i	0.122846	+	0.122846i	0.765857	−	0.643011i	$-0.222315\pi$
$101$	1.23458	+	1.23458i	0.122846	+	0.122846i	−0.643011	+	0.765857i	$0.722315\pi$
$102$		0			0
$103$	−0.872983			−0.0860176			−0.0430088	−	0.999075i	$-0.513694\pi$
$103$	−0.872983			−0.0860176			−0.0430088	+	0.999075i	$0.513694\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.47723	−	5.47723i	0.529503	−	0.529503i	−0.390921	−	0.920424i	$-0.627843\pi$
$107$	5.47723	−	5.47723i	0.529503	−	0.529503i	0.920424	+	0.390921i	$0.127843\pi$
$108$		0			0
$109$	12.7460	+	12.7460i	1.22084	+	1.22084i	0.967333	+	0.253509i	$0.0815848\pi$
$109$	12.7460	+	12.7460i	1.22084	+	1.22084i	0.253509	+	0.967333i	$0.418415\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.3687i		−	1.16355i	−0.813351	−	0.581773i	$-0.802359\pi$
$113$		−	12.3687i		−	1.16355i	0.813351	−	0.581773i	$-0.197641\pi$
$114$		0			0
$115$	3.87298	−	3.87298i	0.361158	−	0.361158i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			4.06301i			0.372456i
$120$		0			0
$121$			2.74597i			0.249633i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$			6.61895i			0.587337i	0.955907	+	0.293668i	$0.0948762\pi$
$127$			6.61895i			0.587337i	−0.955907	+	0.293668i	$0.905124\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.45378	+	6.45378i	0.563869	+	0.563869i	0.930404	−	0.366535i	$-0.119456\pi$
$131$	6.45378	+	6.45378i	0.563869	+	0.563869i	−0.366535	+	0.930404i	$0.619456\pi$
$132$		0			0
$133$	−16.8730	+	16.8730i	−1.46307	+	1.46307i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−15.5563			−1.32907			−0.664534	−	0.747258i	$-0.731370\pi$
$137$	−15.5563			−1.32907			−0.664534	+	0.747258i	$0.731370\pi$
$138$		0			0
$139$	8.74597	+	8.74597i	0.741823	+	0.741823i	0.972929	−	0.231105i	$-0.0742343\pi$
$139$	8.74597	+	8.74597i	0.741823	+	0.741823i	−0.231105	+	0.972929i	$0.574234\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−15.7360			−1.31591
$144$		0			0
$145$	−4.00000			−0.332182
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00806	+	3.00806i	0.246430	+	0.246430i	0.819504	−	0.573074i	$-0.194249\pi$
$149$	3.00806	+	3.00806i	0.246430	+	0.246430i	−0.573074	+	0.819504i	$0.694249\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.82843	+	2.82843i	−0.227185	+	0.227185i
$156$		0			0
$157$	−2.74597	−	2.74597i	−0.219152	−	0.219152i	0.588989	−	0.808141i	$-0.299526\pi$
$157$	−2.74597	−	2.74597i	−0.219152	−	0.219152i	−0.808141	+	0.588989i	$0.799526\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	15.7360i		−	1.24017i
$162$		0			0
$163$	8.87298	−	8.87298i	0.694986	−	0.694986i	−0.268339	−	0.963325i	$-0.586475\pi$
$163$	8.87298	−	8.87298i	0.694986	−	0.694986i	0.963325	+	0.268339i	$0.0864747\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			8.66491i			0.670511i	0.942127	+	0.335255i	$0.108823\pi$
$167$			8.66491i			0.670511i	−0.942127	+	0.335255i	$0.891177\pi$
$168$		0			0
$169$			17.0000i			1.30769i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.3687	+	12.3687i	−0.940372	+	0.940372i	−0.998320	−	0.0579475i	$-0.981544\pi$
$173$	−12.3687	+	12.3687i	−0.940372	+	0.940372i	0.0579475	+	0.998320i	$0.481544\pi$
$174$		0			0
$175$			2.87298i			0.217177i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−11.7514	−	11.7514i	−0.878339	−	0.878339i	0.115024	−	0.993363i	$-0.463306\pi$
$179$	−11.7514	−	11.7514i	−0.878339	−	0.878339i	−0.993363	+	0.115024i	$0.963306\pi$
$180$		0			0
$181$	−9.00000	+	9.00000i	−0.668965	+	0.668965i	−0.957476	−	0.288512i	$-0.906840\pi$
$181$	−9.00000	+	9.00000i	−0.668965	+	0.668965i	0.288512	+	0.957476i	$0.406840\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.41421			−0.103975
$186$		0			0
$187$	−2.87298	−	2.87298i	−0.210093	−	0.210093i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.42227			−0.319984			−0.159992	−	0.987118i	$-0.551147\pi$
$191$	−4.42227			−0.319984			−0.159992	+	0.987118i	$0.551147\pi$
$192$		0			0
$193$	6.25403			0.450175			0.225088	−	0.974339i	$-0.427733\pi$
$193$	6.25403			0.450175			0.225088	+	0.974339i	$0.427733\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.00806	+	3.00806i	0.214315	+	0.214315i	0.806098	−	0.591783i	$-0.201576\pi$
$197$	3.00806	+	3.00806i	0.214315	+	0.214315i	−0.591783	+	0.806098i	$0.701576\pi$
$198$		0			0
$199$	15.7460			1.11620			0.558101	−	0.829773i	$-0.311530\pi$
$199$	15.7460			1.11620			0.558101	+	0.829773i	$0.311530\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.12602	+	8.12602i	−0.570335	+	0.570335i
$204$		0			0
$205$	−7.87298	−	7.87298i	−0.549873	−	0.549873i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	23.8620i		−	1.65057i
$210$		0			0
$211$	−10.7460	+	10.7460i	−0.739783	+	0.739783i	−0.972536	−	0.232753i	$-0.925227\pi$
$211$	−10.7460	+	10.7460i	−0.739783	+	0.739783i	0.232753	+	0.972536i	$0.425227\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			8.48528i			0.578691i
$216$		0			0
$217$			11.4919i			0.780123i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.47723	+	5.47723i	−0.368438	+	0.368438i
$222$		0			0
$223$		−	9.12702i		−	0.611190i	−0.952162	−	0.305595i	$-0.901145\pi$
$223$		−	9.12702i		−	0.611190i	0.952162	−	0.305595i	$-0.0988554\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.9625	+	13.9625i	0.926724	+	0.926724i	0.997493	−	0.0707686i	$-0.0225452\pi$
$227$	13.9625	+	13.9625i	0.926724	+	0.926724i	−0.0707686	+	0.997493i	$0.522545\pi$
$228$		0			0
$229$	18.4919	−	18.4919i	1.22198	−	1.22198i	0.255055	−	0.966927i	$-0.417907\pi$
$229$	18.4919	−	18.4919i	1.22198	−	1.22198i	0.966927	−	0.255055i	$-0.0820935\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	23.6824			1.55148			0.775742	−	0.631050i	$-0.217376\pi$
$233$	23.6824			1.55148			0.775742	+	0.631050i	$0.217376\pi$
$234$		0			0
$235$	−1.00000	−	1.00000i	−0.0652328	−	0.0652328i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.9076			−0.834920			−0.417460	−	0.908695i	$-0.637080\pi$
$239$	−12.9076			−0.834920			−0.417460	+	0.908695i	$0.637080\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.886735	+	0.886735i	0.0566515	+	0.0566515i
$246$		0			0
$247$	−45.4919			−2.89458
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.617292	−	0.617292i	0.0389632	−	0.0389632i	−0.687357	−	0.726320i	$-0.741229\pi$
$251$	0.617292	−	0.617292i	0.0389632	−	0.0389632i	0.726320	+	0.687357i	$0.241229\pi$
$252$		0			0
$253$	11.1270	+	11.1270i	0.699550	+	0.699550i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			20.8539i			1.30083i	0.759578	+	0.650417i	$0.225406\pi$
$257$			20.8539i			1.30083i	−0.759578	+	0.650417i	$0.774594\pi$
$258$		0			0
$259$	−2.87298	+	2.87298i	−0.178518	+	0.178518i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2.28954i			0.141179i	0.997505	+	0.0705896i	$0.0224881\pi$
$263$			2.28954i			0.141179i	−0.997505	+	0.0705896i	$0.977512\pi$
$264$		0			0
$265$			2.00000i			0.122859i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.25070	−	7.25070i	0.442083	−	0.442083i	−0.450629	−	0.892711i	$-0.648800\pi$
$269$	7.25070	−	7.25070i	0.442083	−	0.442083i	0.892711	+	0.450629i	$0.148800\pi$
$270$		0			0
$271$		−	3.74597i		−	0.227551i	−0.993506	−	0.113776i	$-0.963705\pi$
$271$		−	3.74597i		−	0.227551i	0.993506	−	0.113776i	$-0.0362945\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.03151	−	2.03151i	−0.122504	−	0.122504i
$276$		0			0
$277$	−12.7460	+	12.7460i	−0.765831	+	0.765831i	−0.977370	−	0.211539i	$-0.932153\pi$
$277$	−12.7460	+	12.7460i	−0.765831	+	0.765831i	0.211539	+	0.977370i	$0.432153\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	18.0255			1.07531			0.537656	−	0.843164i	$-0.319310\pi$
$281$	18.0255			1.07531			0.537656	+	0.843164i	$0.319310\pi$
$282$		0			0
$283$	12.8730	+	12.8730i	0.765219	+	0.765219i	0.977261	−	0.212041i	$-0.0680111\pi$
$283$	12.8730	+	12.8730i	0.765219	+	0.765219i	−0.212041	+	0.977261i	$0.568011\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−31.9880			−1.88819
$288$		0			0
$289$	15.0000			0.882353
$290$		0			0
$291$		0			0
$292$		0			0
$293$	19.4397	+	19.4397i	1.13568	+	1.13568i	0.989216	+	0.146466i	$0.0467897\pi$
$293$	19.4397	+	19.4397i	1.13568	+	1.13568i	0.146466	+	0.989216i	$0.453210\pi$
$294$		0			0
$295$	12.8730			0.749494
$296$		0			0
$297$		0			0
$298$		0			0
$299$	21.2132	−	21.2132i	1.22679	−	1.22679i
$300$		0			0
$301$	17.2379	+	17.2379i	0.993576	+	0.993576i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	15.1971i		−	0.870183i
$306$		0			0
$307$	1.74597	−	1.74597i	0.0996476	−	0.0996476i	−0.655525	−	0.755173i	$-0.727553\pi$
$307$	1.74597	−	1.74597i	0.0996476	−	0.0996476i	0.755173	+	0.655525i	$0.227553\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			10.0791i			0.571535i	0.958299	+	0.285767i	$0.0922485\pi$
$311$			10.0791i			0.571535i	−0.958299	+	0.285767i	$0.907752\pi$
$312$		0			0
$313$			5.74597i			0.324781i	0.986727	+	0.162391i	$0.0519205\pi$
$313$			5.74597i			0.324781i	−0.986727	+	0.162391i	$0.948080\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.4933	+	11.4933i	−0.645530	+	0.645530i	−0.951909	−	0.306379i	$-0.900882\pi$
$317$	−11.4933	+	11.4933i	−0.645530	+	0.645530i	0.306379	+	0.951909i	$0.400882\pi$
$318$		0			0
$319$		−	11.4919i		−	0.643425i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.30565	−	8.30565i	−0.462139	−	0.462139i
$324$		0			0
$325$	−3.87298	+	3.87298i	−0.214834	+	0.214834i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.06301			−0.224001
$330$		0			0
$331$	9.87298	+	9.87298i	0.542668	+	0.542668i	0.924310	−	0.381642i	$-0.124641\pi$
$331$	9.87298	+	9.87298i	0.542668	+	0.542668i	−0.381642	+	0.924310i	$0.624641\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.42227			−0.241614
$336$		0			0
$337$	0.508067			0.0276762			0.0138381	−	0.999904i	$-0.495595\pi$
$337$	0.508067			0.0276762			0.0138381	+	0.999904i	$0.495595\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.12602	−	8.12602i	−0.440049	−	0.440049i
$342$		0			0
$343$	−16.5081			−0.891352
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.89144	+	6.89144i	−0.369952	+	0.369952i	−0.867460	−	0.497508i	$-0.834249\pi$
$347$	−6.89144	+	6.89144i	−0.369952	+	0.369952i	0.497508	+	0.867460i	$0.334249\pi$
$348$		0			0
$349$	−3.00000	−	3.00000i	−0.160586	−	0.160586i	0.622240	−	0.782826i	$-0.286223\pi$
$349$	−3.00000	−	3.00000i	−0.160586	−	0.160586i	−0.782826	+	0.622240i	$0.786223\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			1.05496i			0.0561497i	0.999606	+	0.0280748i	$0.00893767\pi$
$353$			1.05496i			0.0561497i	−0.999606	+	0.0280748i	$0.991062\pi$
$354$		0			0
$355$	−6.87298	+	6.87298i	−0.364780	+	0.364780i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.7829i		−	0.727432i	−0.931510	−	0.363716i	$-0.881508\pi$
$359$		−	13.7829i		−	0.727432i	0.931510	−	0.363716i	$-0.118492\pi$
$360$		0			0
$361$		−	49.9839i		−	2.63073i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.06301	−	4.06301i	0.212668	−	0.212668i
$366$		0			0
$367$		−	19.1270i		−	0.998422i	−0.866480	−	0.499211i	$-0.833623\pi$
$367$		−	19.1270i		−	0.998422i	0.866480	−	0.499211i	$-0.166377\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.06301	+	4.06301i	0.210941	+	0.210941i
$372$		0			0
$373$	4.74597	−	4.74597i	0.245737	−	0.245737i	−0.573482	−	0.819218i	$-0.694408\pi$
$373$	4.74597	−	4.74597i	0.245737	−	0.245737i	0.819218	+	0.573482i	$0.194408\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−21.9089			−1.12837
$378$		0			0
$379$	−10.7460	−	10.7460i	−0.551983	−	0.551983i	0.375030	−	0.927013i	$-0.377633\pi$
$379$	−10.7460	−	10.7460i	−0.551983	−	0.551983i	−0.927013	+	0.375030i	$0.877633\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	26.5108			1.35464			0.677319	−	0.735689i	$-0.263142\pi$
$383$	26.5108			1.35464			0.677319	+	0.735689i	$0.263142\pi$
$384$		0			0
$385$	−8.25403			−0.420664
$386$		0			0
$387$		0			0
$388$		0			0
$389$	22.0885	+	22.0885i	1.11993	+	1.11993i	0.991751	+	0.128182i	$0.0409143\pi$
$389$	22.0885	+	22.0885i	1.11993	+	1.11993i	0.128182	+	0.991751i	$0.459086\pi$
$390$		0			0
$391$	7.74597			0.391730
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.64880	−	2.64880i	0.133276	−	0.133276i
$396$		0			0
$397$	−13.6190	−	13.6190i	−0.683516	−	0.683516i	0.277275	−	0.960791i	$-0.410569\pi$
$397$	−13.6190	−	13.6190i	−0.683516	−	0.683516i	−0.960791	+	0.277275i	$0.910569\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	19.2601i		−	0.961804i	−0.876774	−	0.480902i	$-0.840309\pi$
$401$		−	19.2601i		−	0.961804i	0.876774	−	0.480902i	$-0.159691\pi$
$402$		0			0
$403$	−15.4919	+	15.4919i	−0.771708	+	0.771708i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4.06301i		−	0.201396i
$408$		0			0
$409$			0.508067i			0.0251223i	0.999921	+	0.0125611i	$0.00399844\pi$
$409$			0.508067i			0.0251223i	−0.999921	+	0.0125611i	$0.996002\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	26.1515	−	26.1515i	1.28683	−	1.28683i
$414$		0			0
$415$		−	2.00000i		−	0.0981761i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.4002	+	14.4002i	0.703494	+	0.703494i	0.965159	−	0.261665i	$-0.0842714\pi$
$419$	14.4002	+	14.4002i	0.703494	+	0.703494i	−0.261665	+	0.965159i	$0.584271\pi$
$420$		0			0
$421$	−9.00000	+	9.00000i	−0.438633	+	0.438633i	−0.891552	−	0.452919i	$-0.850383\pi$
$421$	−9.00000	+	9.00000i	−0.438633	+	0.438633i	0.452919	+	0.891552i	$0.350383\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.41421			−0.0685994
$426$		0			0
$427$	−30.8730	−	30.8730i	−1.49405	−	1.49405i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−22.2682			−1.07262			−0.536310	−	0.844021i	$-0.680182\pi$
$431$	−22.2682			−1.07262			−0.536310	+	0.844021i	$0.680182\pi$
$432$		0			0
$433$	39.4919			1.89786			0.948931	−	0.315485i	$-0.102167\pi$
$433$	39.4919			1.89786			0.948931	+	0.315485i	$0.102167\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	32.1677	+	32.1677i	1.53879	+	1.53879i
$438$		0			0
$439$	16.2540			0.775763			0.387881	−	0.921709i	$-0.373207\pi$
$439$	16.2540			0.775763			0.387881	+	0.921709i	$0.373207\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	26.1515	−	26.1515i	1.24250	−	1.24250i	0.283536	−	0.958962i	$-0.408493\pi$
$443$	26.1515	−	26.1515i	1.24250	−	1.24250i	0.958962	−	0.283536i	$-0.0915074\pi$
$444$		0			0
$445$	−4.12702	−	4.12702i	−0.195639	−	0.195639i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	32.5269i		−	1.53504i	−0.641025	−	0.767520i	$-0.721491\pi$
$449$		−	32.5269i		−	1.53504i	0.641025	−	0.767520i	$-0.278509\pi$
$450$		0			0
$451$	22.6190	−	22.6190i	1.06508	−	1.06508i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			15.7360i			0.737714i
$456$		0			0
$457$		−	27.2379i		−	1.27414i	−0.770808	−	0.637068i	$-0.780147\pi$
$457$		−	27.2379i		−	1.27414i	0.770808	−	0.637068i	$-0.219853\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	22.0885	−	22.0885i	1.02877	−	1.02877i	0.0291916	−	0.999574i	$-0.490707\pi$
$461$	22.0885	−	22.0885i	1.02877	−	1.02877i	0.999574	−	0.0291916i	$-0.00929330\pi$
$462$		0			0
$463$		−	28.3649i		−	1.31823i	−0.752042	−	0.659115i	$-0.770931\pi$
$463$		−	28.3649i		−	1.31823i	0.752042	−	0.659115i	$-0.229069\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.46917	+	2.46917i	0.114260	+	0.114260i	0.761925	−	0.647665i	$-0.224255\pi$
$467$	2.46917	+	2.46917i	0.114260	+	0.114260i	−0.647665	+	0.761925i	$0.724255\pi$
$468$		0			0
$469$	−8.98387	+	8.98387i	−0.414836	+	0.414836i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−24.3781			−1.12090
$474$		0			0
$475$	−5.87298	−	5.87298i	−0.269471	−	0.269471i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.42227			0.202059			0.101029	−	0.994883i	$-0.467786\pi$
$479$	4.42227			0.202059			0.101029	+	0.994883i	$0.467786\pi$
$480$		0			0
$481$	−7.74597			−0.353186
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.3687	+	12.3687i	0.561632	+	0.561632i
$486$		0			0
$487$	−34.6190			−1.56873			−0.784367	−	0.620297i	$-0.787012\pi$
$487$	−34.6190			−1.56873			−0.784367	+	0.620297i	$0.787012\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.68030	−	4.68030i	0.211219	−	0.211219i	−0.593566	−	0.804785i	$-0.702280\pi$
$491$	4.68030	−	4.68030i	0.211219	−	0.211219i	0.804785	+	0.593566i	$0.202280\pi$
$492$		0			0
$493$	−4.00000	−	4.00000i	−0.180151	−	0.180151i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			27.9250i			1.25261i
$498$		0			0
$499$	−0.381050	+	0.381050i	−0.0170581	+	0.0170581i	−0.715584	−	0.698526i	$-0.753840\pi$
$499$	−0.381050	+	0.381050i	−0.0170581	+	0.0170581i	0.698526	+	0.715584i	$0.253840\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	37.8245i		−	1.68651i	−0.537513	−	0.843256i	$-0.680636\pi$
$503$		−	37.8245i		−	1.68651i	0.537513	−	0.843256i	$-0.319364\pi$
$504$		0			0
$505$			1.74597i			0.0776945i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.1421	−	14.1421i	0.626839	−	0.626839i	−0.320432	−	0.947271i	$-0.603828\pi$
$509$	14.1421	−	14.1421i	0.626839	−	0.626839i	0.947271	+	0.320432i	$0.103828\pi$
$510$		0			0
$511$		−	16.5081i		−	0.730274i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.617292	−	0.617292i	−0.0272012	−	0.0272012i
$516$		0			0
$517$	2.87298	−	2.87298i	0.126354	−	0.126354i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	15.5563			0.681536			0.340768	−	0.940147i	$-0.389313\pi$
$521$	15.5563			0.681536			0.340768	+	0.940147i	$0.389313\pi$
$522$		0			0
$523$	−4.61895	−	4.61895i	−0.201973	−	0.201973i	0.598872	−	0.800845i	$-0.295616\pi$
$523$	−4.61895	−	4.61895i	−0.201973	−	0.201973i	−0.800845	+	0.598872i	$0.795616\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.65685			−0.246416
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−43.1221	−	43.1221i	−1.86783	−	1.86783i
$534$		0			0
$535$	7.74597			0.334887
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.54758	+	2.54758i	−0.109732	+	0.109732i
$540$		0			0
$541$	−10.7460	−	10.7460i	−0.462005	−	0.462005i	0.437307	−	0.899312i	$-0.355932\pi$
$541$	−10.7460	−	10.7460i	−0.462005	−	0.462005i	−0.899312	+	0.437307i	$0.855932\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			18.0255i			0.772128i
$546$		0			0
$547$	−1.74597	+	1.74597i	−0.0746521	+	0.0746521i	−0.743447	−	0.668795i	$-0.766810\pi$
$547$	−1.74597	+	1.74597i	−0.0746521	+	0.0746521i	0.668795	+	0.743447i	$0.266810\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	33.2226i		−	1.41533i
$552$		0			0
$553$		−	10.7621i		−	0.457651i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−26.1515	+	26.1515i	−1.10808	+	1.10808i	−0.114673	+	0.993403i	$0.536582\pi$
$557$	−26.1515	+	26.1515i	−1.10808	+	1.10808i	−0.993403	+	0.114673i	$0.963418\pi$
$558$		0			0
$559$			46.4758i			1.96572i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.2051	+	18.2051i	0.767255	+	0.767255i	0.977622	−	0.210367i	$-0.0674659\pi$
$563$	18.2051	+	18.2051i	0.767255	+	0.767255i	−0.210367	+	0.977622i	$0.567466\pi$
$564$		0			0
$565$	8.74597	−	8.74597i	0.367946	−	0.367946i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.47723			−0.229617			−0.114809	−	0.993388i	$-0.536625\pi$
$569$	−5.47723			−0.229617			−0.114809	+	0.993388i	$0.536625\pi$
$570$		0			0
$571$	31.3649	+	31.3649i	1.31258	+	1.31258i	0.919507	+	0.393074i	$0.128588\pi$
$571$	31.3649	+	31.3649i	1.31258	+	1.31258i	0.393074	+	0.919507i	$0.371412\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.47723			0.228416
$576$		0			0
$577$	−16.2540			−0.676664			−0.338332	−	0.941027i	$-0.609863\pi$
$577$	−16.2540			−0.676664			−0.338332	+	0.941027i	$0.609863\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.06301	−	4.06301i	−0.168562	−	0.168562i
$582$		0			0
$583$	−5.74597			−0.237974
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.0885	+	22.0885i	−0.911691	+	0.911691i	−0.996405	−	0.0847141i	$-0.973002\pi$
$587$	−22.0885	+	22.0885i	−0.911691	+	0.911691i	0.0847141	+	0.996405i	$0.473002\pi$
$588$		0			0
$589$	−23.4919	−	23.4919i	−0.967968	−	0.967968i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			9.18098i			0.377018i	0.982071	+	0.188509i	$0.0603654\pi$
$593$			9.18098i			0.377018i	−0.982071	+	0.188509i	$0.939635\pi$
$594$		0			0
$595$	−2.87298	+	2.87298i	−0.117781	+	0.117781i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			27.5658i			1.12631i	0.826353	+	0.563153i	$0.190412\pi$
$599$			27.5658i			1.12631i	−0.826353	+	0.563153i	$0.809588\pi$
$600$		0			0
$601$			16.9839i			0.692786i	0.938089	+	0.346393i	$0.112594\pi$
$601$			16.9839i			0.692786i	−0.938089	+	0.346393i	$0.887406\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.94169	+	1.94169i	−0.0789410	+	0.0789410i
$606$		0			0
$607$		−	42.3649i		−	1.71954i	−0.510682	−	0.859769i	$-0.670607\pi$
$607$		−	42.3649i		−	1.71954i	0.510682	−	0.859769i	$-0.329393\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.47723	−	5.47723i	−0.221585	−	0.221585i
$612$		0			0
$613$	5.61895	−	5.61895i	0.226947	−	0.226947i	−0.584469	−	0.811416i	$-0.698697\pi$
$613$	5.61895	−	5.61895i	0.226947	−	0.226947i	0.811416	+	0.584469i	$0.198697\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	42.4036			1.70710			0.853552	−	0.521007i	$-0.174444\pi$
$617$	42.4036			1.70710			0.853552	+	0.521007i	$0.174444\pi$
$618$		0			0
$619$	−26.7460	−	26.7460i	−1.07501	−	1.07501i	−0.996948	−	0.0780627i	$-0.975127\pi$
$619$	−26.7460	−	26.7460i	−1.07501	−	1.07501i	−0.0780627	−	0.996948i	$-0.524873\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−16.7681			−0.671800
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.41421	−	1.41421i	−0.0563884	−	0.0563884i
$630$		0			0
$631$	−22.0000			−0.875806			−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000			−0.875806			−0.437903	+	0.899022i	$0.644279\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.68030	+	4.68030i	−0.185732	+	0.185732i
$636$		0			0
$637$	4.85685	+	4.85685i	0.192435	+	0.192435i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	12.3687i		−	0.488533i	−0.969708	−	0.244266i	$-0.921453\pi$
$641$		−	12.3687i		−	0.488533i	0.969708	−	0.244266i	$-0.0785471\pi$
$642$		0			0
$643$	3.12702	−	3.12702i	0.123317	−	0.123317i	−0.642755	−	0.766072i	$-0.722209\pi$
$643$	3.12702	−	3.12702i	0.123317	−	0.123317i	0.766072	+	0.642755i	$0.222209\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			42.7628i			1.68118i	0.541671	+	0.840591i	$0.317792\pi$
$647$			42.7628i			1.68118i	−0.541671	+	0.840591i	$0.682208\pi$
$648$		0			0
$649$			36.9839i			1.45174i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.5483	+	12.5483i	−0.491053	+	0.491053i	−0.908638	−	0.417585i	$-0.862877\pi$
$653$	−12.5483	+	12.5483i	−0.491053	+	0.491053i	0.417585	+	0.908638i	$0.362877\pi$
$654$		0			0
$655$			9.12702i			0.356622i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−27.3077	−	27.3077i	−1.06376	−	1.06376i	−0.997824	−	0.0659338i	$-0.978997\pi$
$659$	−27.3077	−	27.3077i	−1.06376	−	1.06376i	−0.0659338	−	0.997824i	$-0.521003\pi$
$660$		0			0
$661$	−7.00000	+	7.00000i	−0.272268	+	0.272268i	−0.830013	−	0.557744i	$-0.811667\pi$
$661$	−7.00000	+	7.00000i	−0.272268	+	0.272268i	0.557744	+	0.830013i	$0.311667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−23.8620			−0.925329
$666$		0			0
$667$	15.4919	+	15.4919i	0.599850	+	0.599850i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	43.6610			1.68551
$672$		0			0
$673$	−15.4919			−0.597170			−0.298585	−	0.954383i	$-0.596515\pi$
$673$	−15.4919			−0.597170			−0.298585	+	0.954383i	$0.596515\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−26.3312	−	26.3312i	−1.01199	−	1.01199i	−0.999927	−	0.0120619i	$-0.996160\pi$
$677$	−26.3312	−	26.3312i	−1.01199	−	1.01199i	−0.0120619	−	0.999927i	$-0.503840\pi$
$678$		0			0
$679$	50.2540			1.92857
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.71987	−	9.71987i	0.371920	−	0.371920i	−0.496256	−	0.868176i	$-0.665292\pi$
$683$	9.71987	−	9.71987i	0.371920	−	0.371920i	0.868176	+	0.496256i	$0.165292\pi$
$684$		0			0
$685$	−11.0000	−	11.0000i	−0.420288	−	0.420288i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			10.9545i			0.417331i
$690$		0			0
$691$	29.3649	−	29.3649i	1.11709	−	1.11709i	0.124928	−	0.992166i	$-0.460130\pi$
$691$	29.3649	−	29.3649i	1.11709	−	1.11709i	0.992166	−	0.124928i	$-0.0398701\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			12.3687i			0.469170i
$696$		0			0
$697$		−	15.7460i		−	0.596421i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.53218	+	6.53218i	−0.246717	+	0.246717i	−0.819622	−	0.572905i	$-0.805816\pi$
$701$	−6.53218	+	6.53218i	−0.246717	+	0.246717i	0.572905	+	0.819622i	$0.305816\pi$
$702$		0			0
$703$		−	11.7460i		−	0.443008i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.54694	+	3.54694i	0.133396	+	0.133396i
$708$		0			0
$709$	32.7460	−	32.7460i	1.22980	−	1.22980i	0.265762	−	0.964039i	$-0.414377\pi$
$709$	32.7460	−	32.7460i	1.22980	−	1.22980i	0.964039	−	0.265762i	$-0.0856235\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21.9089			0.820495
$714$		0			0
$715$	−11.1270	−	11.1270i	−0.416127	−	0.416127i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	49.8339			1.85849			0.929246	−	0.369462i	$-0.120458\pi$
$719$	49.8339			1.85849			0.929246	+	0.369462i	$0.120458\pi$
$720$		0			0
$721$	−2.50807			−0.0934053
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.82843	−	2.82843i	−0.105045	−	0.105045i
$726$		0			0
$727$	41.6028			1.54296			0.771482	−	0.636252i	$-0.219516\pi$
$727$	41.6028			1.54296			0.771482	+	0.636252i	$0.219516\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.48528	+	8.48528i	−0.313839	+	0.313839i
$732$		0			0
$733$	−18.7460	−	18.7460i	−0.692398	−	0.692398i	0.270361	−	0.962759i	$-0.412857\pi$
$733$	−18.7460	−	18.7460i	−0.692398	−	0.692398i	−0.962759	+	0.270361i	$0.912857\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	12.7051i		−	0.467999i
$738$		0			0
$739$	0.381050	−	0.381050i	0.0140171	−	0.0140171i	−0.700063	−	0.714081i	$-0.746845\pi$
$739$	0.381050	−	0.381050i	0.0140171	−	0.0140171i	0.714081	+	0.700063i	$0.246845\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1.41421i			0.0518825i	0.999663	+	0.0259412i	$0.00825828\pi$
$743$			1.41421i			0.0518825i	−0.999663	+	0.0259412i	$0.991742\pi$
$744$		0			0
$745$			4.25403i			0.155856i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15.7360	−	15.7360i	0.574980	−	0.574980i
$750$		0			0
$751$			17.7460i			0.647560i	0.946132	+	0.323780i	$0.104954\pi$
$751$			17.7460i			0.647560i	−0.946132	+	0.323780i	$0.895046\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.65685	+	5.65685i	0.205874	+	0.205874i
$756$		0			0
$757$	2.49193	−	2.49193i	0.0905709	−	0.0905709i	−0.660370	−	0.750941i	$-0.729600\pi$
$757$	2.49193	−	2.49193i	0.0905709	−	0.0905709i	0.750941	+	0.660370i	$0.229600\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	39.9344			1.44762			0.723811	−	0.689998i	$-0.242389\pi$
$761$	39.9344			1.44762			0.723811	+	0.689998i	$0.242389\pi$
$762$		0			0
$763$	36.6190	+	36.6190i	1.32570	+	1.32570i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	70.5082			2.54590
$768$		0			0
$769$	7.23790			0.261005			0.130503	−	0.991448i	$-0.458341\pi$
$769$	7.23790			0.261005			0.130503	+	0.991448i	$0.458341\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−30.9331	−	30.9331i	−1.11259	−	1.11259i	−0.992800	−	0.119785i	$-0.961779\pi$
$773$	−30.9331	−	30.9331i	−1.11259	−	1.11259i	−0.119785	−	0.992800i	$-0.538221\pi$
$774$		0			0
$775$	−4.00000			−0.143684
$776$		0			0
$777$		0			0
$778$		0			0
$779$	65.3903	−	65.3903i	2.34285	−	2.34285i
$780$		0			0
$781$	−19.7460	−	19.7460i	−0.706566	−	0.706566i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	3.88338i		−	0.138604i
$786$		0			0
$787$	−26.6190	+	26.6190i	−0.948863	+	0.948863i	−0.998755	−	0.0498915i	$-0.984112\pi$
$787$	−26.6190	+	26.6190i	−0.948863	+	0.948863i	0.0498915	+	0.998755i	$0.484112\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	35.5350i		−	1.26348i
$792$		0			0
$793$		−	83.2379i		−	2.95586i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−25.0966	+	25.0966i	−0.888967	+	0.888967i	−0.994424	−	0.105457i	$-0.966369\pi$
$797$	−25.0966	+	25.0966i	−0.888967	+	0.888967i	0.105457	+	0.994424i	$0.466369\pi$
$798$		0			0
$799$		−	2.00000i		−	0.0707549i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11.6730	+	11.6730i	0.411930	+	0.411930i
$804$		0			0
$805$	11.1270	−	11.1270i	0.392176	−	0.392176i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−26.8701			−0.944701			−0.472350	−	0.881411i	$-0.656594\pi$
$809$	−26.8701			−0.944701			−0.472350	+	0.881411i	$0.656594\pi$
$810$		0			0
$811$	−24.4919	−	24.4919i	−0.860028	−	0.860028i	0.131313	−	0.991341i	$-0.458081\pi$
$811$	−24.4919	−	24.4919i	−0.860028	−	0.860028i	−0.991341	+	0.131313i	$0.958081\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.5483			0.439548
$816$		0			0
$817$	−70.4758			−2.46564
$818$		0			0
$819$		0			0
$820$		0			0
$821$	10.9545	+	10.9545i	0.382313	+	0.382313i	0.871935	−	0.489622i	$-0.162865\pi$
$821$	10.9545	+	10.9545i	0.382313	+	0.382313i	−0.489622	+	0.871935i	$0.662865\pi$
$822$		0			0
$823$	−14.8730			−0.518440			−0.259220	−	0.965818i	$-0.583465\pi$
$823$	−14.8730			−0.518440			−0.259220	+	0.965818i	$0.583465\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.93834	−	4.93834i	0.171723	−	0.171723i	−0.616013	−	0.787736i	$-0.711253\pi$
$827$	4.93834	−	4.93834i	0.171723	−	0.171723i	0.787736	+	0.616013i	$0.211253\pi$
$828$		0			0
$829$	11.2540	+	11.2540i	0.390869	+	0.390869i	0.874997	−	0.484128i	$-0.160863\pi$
$829$	11.2540	+	11.2540i	0.390869	+	0.390869i	−0.484128	+	0.874997i	$0.660863\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1.77347i			0.0614471i
$834$		0			0
$835$	−6.12702	+	6.12702i	−0.212034	+	0.212034i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	44.1771i		−	1.52516i	−0.646893	−	0.762581i	$-0.723932\pi$
$839$		−	44.1771i		−	1.52516i	0.646893	−	0.762581i	$-0.276068\pi$
$840$		0			0
$841$			13.0000i			0.448276i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.0208	+	12.0208i	−0.413529	+	0.413529i
$846$		0			0
$847$			7.88912i			0.271073i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.47723	+	5.47723i	0.187757	+	0.187757i
$852$		0			0
$853$	−19.6190	+	19.6190i	−0.671740	+	0.671740i	−0.958117	−	0.286377i	$-0.907549\pi$
$853$	−19.6190	+	19.6190i	−0.671740	+	0.671740i	0.286377	+	0.958117i	$0.407549\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.88338			−0.132654			−0.0663269	−	0.997798i	$-0.521128\pi$
$857$	−3.88338			−0.132654			−0.0663269	+	0.997798i	$0.521128\pi$
$858$		0			0
$859$	−22.4919	−	22.4919i	−0.767415	−	0.767415i	0.210236	−	0.977651i	$-0.432577\pi$
$859$	−22.4919	−	22.4919i	−0.767415	−	0.767415i	−0.977651	+	0.210236i	$0.932577\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	50.3728			1.71471			0.857355	−	0.514725i	$-0.172106\pi$
$863$	50.3728			1.71471			0.857355	+	0.514725i	$0.172106\pi$
$864$		0			0
$865$	−17.4919			−0.594744
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.60995	+	7.60995i	0.258150	+	0.258150i
$870$		0			0
$871$	−24.2218			−0.820723
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.03151	+	2.03151i	−0.0686774	+	0.0686774i
$876$		0			0
$877$	3.00000	+	3.00000i	0.101303	+	0.101303i	0.755942	−	0.654639i	$-0.227179\pi$
$877$	3.00000	+	3.00000i	0.101303	+	0.101303i	−0.654639	+	0.755942i	$0.727179\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			41.5283i			1.39912i	0.714573	+	0.699561i	$0.246621\pi$
$881$			41.5283i			1.39912i	−0.714573	+	0.699561i	$0.753379\pi$
$882$		0			0
$883$	22.3649	−	22.3649i	0.752639	−	0.752639i	−0.222332	−	0.974971i	$-0.571367\pi$
$883$	22.3649	−	22.3649i	0.752639	−	0.752639i	0.974971	+	0.222332i	$0.0713668\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			8.30565i			0.278877i	0.990231	+	0.139438i	$0.0445297\pi$
$887$			8.30565i			0.278877i	−0.990231	+	0.139438i	$0.955470\pi$
$888$		0			0
$889$			19.0161i			0.637781i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.30565	−	8.30565i	0.277938	−	0.277938i
$894$		0			0
$895$		−	16.6190i		−	0.555510i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.3137	−	11.3137i	−0.377333	−	0.377333i
$900$		0			0
$901$	−2.00000	+	2.00000i	−0.0666297	+	0.0666297i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.7279			−0.423090
$906$		0			0
$907$	−18.0000	−	18.0000i	−0.597680	−	0.597680i	0.342014	−	0.939695i	$-0.388891\pi$
$907$	−18.0000	−	18.0000i	−0.597680	−	0.597680i	−0.939695	+	0.342014i	$0.888891\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−22.6274			−0.749680			−0.374840	−	0.927090i	$-0.622302\pi$
$911$	−22.6274			−0.749680			−0.374840	+	0.927090i	$0.622302\pi$
$912$		0			0
$913$	5.74597			0.190164
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.5416	+	18.5416i	0.612297	+	0.612297i
$918$		0			0
$919$	−35.4919			−1.17077			−0.585385	−	0.810755i	$-0.699057\pi$
$919$	−35.4919			−1.17077			−0.585385	+	0.810755i	$0.699057\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−37.6449	+	37.6449i	−1.23910	+	1.23910i
$924$		0			0
$925$	−1.00000	−	1.00000i	−0.0328798	−	0.0328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	31.4491i		−	1.03181i	−0.856645	−	0.515906i	$-0.827455\pi$
$929$		−	31.4491i		−	1.03181i	0.856645	−	0.515906i	$-0.172545\pi$
$930$		0			0
$931$	−7.36492	+	7.36492i	−0.241375	+	0.241375i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	4.06301i		−	0.132875i
$936$		0			0
$937$		−	32.2540i		−	1.05369i	−0.849960	−	0.526847i	$-0.823374\pi$
$937$		−	32.2540i		−	1.05369i	0.849960	−	0.526847i	$-0.176626\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15.1971	−	15.1971i	0.495411	−	0.495411i	−0.414595	−	0.910006i	$-0.636077\pi$
$941$	15.1971	−	15.1971i	0.495411	−	0.495411i	0.910006	+	0.414595i	$0.136077\pi$
$942$		0			0
$943$			60.9839i			1.98591i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−31.8084	−	31.8084i	−1.03363	−	1.03363i	−0.999414	−	0.0342197i	$-0.989105\pi$
$947$	−31.8084	−	31.8084i	−1.03363	−	1.03363i	−0.0342197	−	0.999414i	$-0.510895\pi$
$948$		0			0
$949$	22.2540	−	22.2540i	0.722396	−	0.722396i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29.3392			0.950391			0.475195	−	0.879880i	$-0.342377\pi$
$953$	29.3392			0.950391			0.475195	+	0.879880i	$0.342377\pi$
$954$		0			0
$955$	−3.12702	−	3.12702i	−0.101188	−	0.101188i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−44.6931			−1.44322
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.42227	+	4.42227i	0.142358	+	0.142358i
$966$		0			0
$967$	8.61895			0.277167			0.138583	−	0.990351i	$-0.455745\pi$
$967$	8.61895			0.277167			0.138583	+	0.990351i	$0.455745\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9.97790	+	9.97790i	−0.320206	+	0.320206i	−0.848846	−	0.528640i	$-0.822702\pi$
$971$	−9.97790	+	9.97790i	−0.320206	+	0.320206i	0.528640	+	0.848846i	$0.322702\pi$
$972$		0			0
$973$	25.1270	+	25.1270i	0.805535	+	0.805535i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			32.5269i			1.04063i	0.853975	+	0.520314i	$0.174185\pi$
$977$			32.5269i			1.04063i	−0.853975	+	0.520314i	$0.825815\pi$
$978$		0			0
$979$	11.8569	−	11.8569i	0.378947	−	0.378947i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	7.78958i		−	0.248449i	−0.992254	−	0.124225i	$-0.960356\pi$
$983$		−	7.78958i		−	0.248449i	0.992254	−	0.124225i	$-0.0396443\pi$
$984$		0			0
$985$			4.25403i			0.135545i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	32.8634	−	32.8634i	1.04499	−	1.04499i
$990$		0			0
$991$		−	15.2379i		−	0.484048i	−0.970270	−	0.242024i	$-0.922189\pi$
$991$		−	15.2379i		−	0.484048i	0.970270	−	0.242024i	$-0.0778112\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	11.1341	+	11.1341i	0.352974	+	0.352974i
$996$		0			0
$997$	−1.87298	+	1.87298i	−0.0593180	+	0.0593180i	−0.736143	−	0.676826i	$-0.763355\pi$
$997$	−1.87298	+	1.87298i	−0.0593180	+	0.0593180i	0.676826	+	0.736143i	$0.263355\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bl.a.431.4		8
3.2	odd	2	inner	2880.2.bl.a.431.2		8
4.3	odd	2		720.2.bl.a.611.3	yes	8
12.11	even	2		720.2.bl.a.611.1	yes	8
16.5	even	4		720.2.bl.a.251.1	✓	8
16.11	odd	4	inner	2880.2.bl.a.1871.2		8
48.5	odd	4		720.2.bl.a.251.3	yes	8
48.11	even	4	inner	2880.2.bl.a.1871.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.a.251.1	✓	8	16.5	even	4
720.2.bl.a.251.3	yes	8	48.5	odd	4
720.2.bl.a.611.1	yes	8	12.11	even	2
720.2.bl.a.611.3	yes	8	4.3	odd	2
2880.2.bl.a.431.2		8	3.2	odd	2	inner
2880.2.bl.a.431.4		8	1.1	even	1	trivial
2880.2.bl.a.1871.2		8	16.11	odd	4	inner
2880.2.bl.a.1871.4		8	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)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +2.87298 q^{7} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{5} +2.87298 q^{7} +(-2.03151 + 2.03151i) q^{11} +(3.87298 + 3.87298i) q^{13} +1.41421i q^{17} +(-5.87298 + 5.87298i) q^{19} -5.47723i q^{23} +1.00000i q^{25} +(-2.82843 + 2.82843i) q^{29} +4.00000i q^{31} +(2.03151 + 2.03151i) q^{35} +(-1.00000 + 1.00000i) q^{37} -11.1341 q^{41} +(6.00000 + 6.00000i) q^{43} -1.41421 q^{47} +1.25403 q^{49} +(1.41421 + 1.41421i) q^{53} -2.87298 q^{55} +(9.10257 - 9.10257i) q^{59} +(-10.7460 - 10.7460i) q^{61} +5.47723i q^{65} +(-3.12702 + 3.12702i) q^{67} +9.71987i q^{71} -5.74597i q^{73} +(-5.83648 + 5.83648i) q^{77} -3.74597i q^{79} +(-1.41421 - 1.41421i) q^{83} +(-1.00000 + 1.00000i) q^{85} -5.83648 q^{89} +(11.1270 + 11.1270i) q^{91} -8.30565 q^{95} +17.4919 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} - 16 q^{19} - 8 q^{37} + 48 q^{43} + 72 q^{49} + 8 q^{55} - 24 q^{61} - 56 q^{67} - 8 q^{85} + 120 q^{91} + 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 - 16 * q^19 - 8 * q^37 + 48 * q^43 + 72 * q^49 + 8 * q^55 - 24 * q^61 - 56 * q^67 - 8 * q^85 + 120 * q^91 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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