LMFDB - Embedded newform 2880.2.bl.a.431.3

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			431.3
Root			$1.01575 - 1.72286i$ of defining polynomial
Character	$\chi$	$=$	2880.431
Dual form			2880.2.bl.a.1871.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} -4.87298 q^{7} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{5} -4.87298 q^{7} +(3.44572 - 3.44572i) q^{11} +(-3.87298 - 3.87298i) q^{13} +1.41421i q^{17} +(1.87298 - 1.87298i) q^{19} +5.47723i q^{23} +1.00000i q^{25} +(-2.82843 + 2.82843i) q^{29} +4.00000i q^{31} +(-3.44572 - 3.44572i) q^{35} +(-1.00000 + 1.00000i) q^{37} -0.179629 q^{41} +(6.00000 + 6.00000i) q^{43} -1.41421 q^{47} +16.7460 q^{49} +(1.41421 + 1.41421i) q^{53} +4.87298 q^{55} +(3.62535 - 3.62535i) q^{59} +(4.74597 + 4.74597i) q^{61} -5.47723i q^{65} +(-10.8730 + 10.8730i) q^{67} -1.23458i q^{71} +9.74597i q^{73} +(-16.7909 + 16.7909i) q^{77} +11.7460i q^{79} +(-1.41421 - 1.41421i) q^{83} +(-1.00000 + 1.00000i) q^{85} -16.7909 q^{89} +(18.8730 + 18.8730i) q^{91} +2.64880 q^{95} -13.4919 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−4.87298			−1.84181			−0.920907	−	0.389782i	$-0.872550\pi$
$7$	−4.87298			−1.84181			−0.920907	+	0.389782i	$0.872550\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.44572	−	3.44572i	1.03892	−	1.03892i	0.0397124	−	0.999211i	$-0.487356\pi$
$11$	3.44572	−	3.44572i	1.03892	−	1.03892i	0.999211	−	0.0397124i	$-0.0126442\pi$
$12$		0			0
$13$	−3.87298	−	3.87298i	−1.07417	−	1.07417i	−0.997019	−	0.0771531i	$-0.975417\pi$
$13$	−3.87298	−	3.87298i	−1.07417	−	1.07417i	−0.0771531	−	0.997019i	$-0.524583\pi$
$14$		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
$19$	1.87298	−	1.87298i	0.429692	−	0.429692i	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	1.87298	−	1.87298i	0.429692	−	0.429692i	0.888523	+	0.458831i	$0.151732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.47723i			1.14208i	0.820922	+	0.571040i	$0.193460\pi$
$23$			5.47723i			1.14208i	−0.820922	+	0.571040i	$0.806540\pi$
$24$		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
$35$	−3.44572	−	3.44572i	−0.582433	−	0.582433i
$36$		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
$41$	−0.179629			−0.0280533			−0.0140266	−	0.999902i	$-0.504465\pi$
$41$	−0.179629			−0.0280533			−0.0140266	+	0.999902i	$0.504465\pi$
$42$		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
$49$	16.7460			2.39228
$50$		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
$55$	4.87298			0.657073
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.62535	−	3.62535i	0.471980	−	0.471980i	−0.430575	−	0.902555i	$-0.641689\pi$
$59$	3.62535	−	3.62535i	0.471980	−	0.471980i	0.902555	+	0.430575i	$0.141689\pi$
$60$		0			0
$61$	4.74597	+	4.74597i	0.607659	+	0.607659i	0.942334	−	0.334675i	$-0.108626\pi$
$61$	4.74597	+	4.74597i	0.607659	+	0.607659i	−0.334675	+	0.942334i	$0.608626\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	5.47723i		−	0.679366i
$66$		0			0
$67$	−10.8730	+	10.8730i	−1.32835	+	1.32835i	−0.421534	+	0.906813i	$0.638508\pi$
$67$	−10.8730	+	10.8730i	−1.32835	+	1.32835i	−0.906813	+	0.421534i	$0.861492\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.23458i		−	0.146518i	−0.997313	−	0.0732591i	$-0.976660\pi$
$71$		−	1.23458i		−	0.146518i	0.997313	−	0.0732591i	$-0.0233400\pi$
$72$		0			0
$73$			9.74597i			1.14068i	0.821409	+	0.570340i	$0.193188\pi$
$73$			9.74597i			1.14068i	−0.821409	+	0.570340i	$0.806812\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−16.7909	+	16.7909i	−1.91350	+	1.91350i
$78$		0			0
$79$			11.7460i			1.32152i	0.750595	+	0.660762i	$0.229767\pi$
$79$			11.7460i			1.32152i	−0.750595	+	0.660762i	$0.770233\pi$
$80$		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
$89$	−16.7909			−1.77984			−0.889918	−	0.456121i	$-0.849238\pi$
$89$	−16.7909			−1.77984			−0.889918	+	0.456121i	$0.849238\pi$
$90$		0			0
$91$	18.8730	+	18.8730i	1.97843	+	1.97843i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.64880			0.271761
$96$		0			0
$97$	−13.4919			−1.36990			−0.684949	−	0.728591i	$-0.740176\pi$
$97$	−13.4919			−1.36990			−0.684949	+	0.728591i	$0.740176\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.71987	−	9.71987i	−0.967163	−	0.967163i	0.0323149	−	0.999478i	$-0.489712\pi$
$101$	−9.71987	−	9.71987i	−0.967163	−	0.967163i	−0.999478	+	0.0323149i	$0.989712\pi$
$102$		0			0
$103$	6.87298			0.677215			0.338608	−	0.940928i	$-0.390044\pi$
$103$	6.87298			0.677215			0.338608	+	0.940928i	$0.390044\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.47723	+	5.47723i	−0.529503	+	0.529503i	−0.920424	−	0.390921i	$-0.872157\pi$
$107$	−5.47723	+	5.47723i	−0.529503	+	0.529503i	0.390921	+	0.920424i	$0.372157\pi$
$108$		0			0
$109$	−2.74597	−	2.74597i	−0.263016	−	0.263016i	0.563262	−	0.826278i	$-0.309546\pi$
$109$	−2.74597	−	2.74597i	−0.263016	−	0.263016i	−0.826278	+	0.563262i	$0.809546\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			9.54024i			0.897470i	0.893665	+	0.448735i	$0.148125\pi$
$113$			9.54024i			0.897470i	−0.893665	+	0.448735i	$0.851875\pi$
$114$		0			0
$115$	−3.87298	+	3.87298i	−0.361158	+	0.361158i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	6.89144i		−	0.631737i
$120$		0			0
$121$		−	12.7460i		−	1.15872i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$		−	16.6190i		−	1.47469i	−0.675515	−	0.737347i	$-0.736078\pi$
$127$		−	16.6190i		−	1.47469i	0.675515	−	0.737347i	$-0.263922\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	11.9310	+	11.9310i	1.04242	+	1.04242i	0.999060	+	0.0433567i	$0.0138052\pi$
$131$	11.9310	+	11.9310i	1.04242	+	1.04242i	0.0433567	+	0.999060i	$0.486195\pi$
$132$		0			0
$133$	−9.12702	+	9.12702i	−0.791413	+	0.791413i
$134$		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
$139$	−6.74597	−	6.74597i	−0.572185	−	0.572185i	0.360553	−	0.932739i	$-0.382588\pi$
$139$	−6.74597	−	6.74597i	−0.572185	−	0.572185i	−0.932739	+	0.360553i	$0.882588\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−26.6904			−2.23197
$144$		0			0
$145$	−4.00000			−0.332182
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.9625	+	13.9625i	1.14385	+	1.14385i	0.987739	+	0.156114i	$0.0498968\pi$
$149$	13.9625	+	13.9625i	1.14385	+	1.14385i	0.156114	+	0.987739i	$0.450103\pi$
$150$		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
$157$	12.7460	+	12.7460i	1.01724	+	1.01724i	0.999849	+	0.0173901i	$0.00553572\pi$
$157$	12.7460	+	12.7460i	1.01724	+	1.01724i	0.0173901	+	0.999849i	$0.494464\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	26.6904i		−	2.10350i
$162$		0			0
$163$	1.12702	−	1.12702i	0.0882748	−	0.0882748i	−0.661591	−	0.749865i	$-0.730118\pi$
$163$	1.12702	−	1.12702i	0.0882748	−	0.0882748i	0.749865	+	0.661591i	$0.230118\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			19.6194i			1.51819i	0.650979	+	0.759096i	$0.274359\pi$
$167$			19.6194i			1.51819i	−0.650979	+	0.759096i	$0.725641\pi$
$168$		0			0
$169$			17.0000i			1.30769i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.54024	−	9.54024i	0.725331	−	0.725331i	−0.244355	−	0.969686i	$-0.578576\pi$
$173$	9.54024	−	9.54024i	0.725331	−	0.725331i	0.969686	+	0.244355i	$0.0785762\pi$
$174$		0			0
$175$		−	4.87298i		−	0.368363i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.68030	+	4.68030i	0.349822	+	0.349822i	0.860043	−	0.510221i	$-0.170436\pi$
$179$	4.68030	+	4.68030i	0.349822	+	0.349822i	−0.510221	+	0.860043i	$0.670436\pi$
$180$		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
$187$	4.87298	+	4.87298i	0.356348	+	0.356348i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−15.3767			−1.11262			−0.556310	−	0.830975i	$-0.687783\pi$
$191$	−15.3767			−1.11262			−0.556310	+	0.830975i	$0.687783\pi$
$192$		0			0
$193$	21.7460			1.56531			0.782654	−	0.622456i	$-0.213865\pi$
$193$	21.7460			1.56531			0.782654	+	0.622456i	$0.213865\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	13.9625	+	13.9625i	0.994787	+	0.994787i	0.999986	−	0.00519905i	$-0.00165492\pi$
$197$	13.9625	+	13.9625i	0.994787	+	0.994787i	−0.00519905	+	0.999986i	$0.501655\pi$
$198$		0			0
$199$	0.254033			0.0180079			0.00900397	−	0.999959i	$-0.497134\pi$
$199$	0.254033			0.0180079			0.00900397	+	0.999959i	$0.497134\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.7829	−	13.7829i	0.967368	−	0.967368i
$204$		0			0
$205$	−0.127017	−	0.127017i	−0.00887123	−	0.00887123i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	12.9076i		−	0.892834i
$210$		0			0
$211$	4.74597	−	4.74597i	0.326726	−	0.326726i	−0.524614	−	0.851340i	$-0.675790\pi$
$211$	4.74597	−	4.74597i	0.326726	−	0.326726i	0.851340	+	0.524614i	$0.175790\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			8.48528i			0.578691i
$216$		0			0
$217$		−	19.4919i		−	1.32320i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.47723	−	5.47723i	0.368438	−	0.368438i
$222$		0			0
$223$		−	16.8730i		−	1.12990i	−0.825126	−	0.564949i	$-0.808896\pi$
$223$		−	16.8730i		−	1.12990i	0.825126	−	0.564949i	$-0.191104\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.00806	+	3.00806i	0.199652	+	0.199652i	0.799851	−	0.600199i	$-0.204912\pi$
$227$	3.00806	+	3.00806i	0.199652	+	0.199652i	−0.600199	+	0.799851i	$0.704912\pi$
$228$		0			0
$229$	−12.4919	+	12.4919i	−0.825490	+	0.825490i	−0.986889	−	0.161399i	$-0.948399\pi$
$229$	−12.4919	+	12.4919i	−0.825490	+	0.825490i	0.161399	+	0.986889i	$0.448399\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.77347			0.116184			0.0580920	−	0.998311i	$-0.481498\pi$
$233$	1.77347			0.116184			0.0580920	+	0.998311i	$0.481498\pi$
$234$		0			0
$235$	−1.00000	−	1.00000i	−0.0652328	−	0.0652328i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−23.8620			−1.54350			−0.771752	−	0.635923i	$-0.780619\pi$
$239$	−23.8620			−1.54350			−0.771752	+	0.635923i	$0.780619\pi$
$240$		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
$245$	11.8412	+	11.8412i	0.756506	+	0.756506i
$246$		0			0
$247$	−14.5081			−0.923126
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.85993	+	4.85993i	−0.306756	+	0.306756i	−0.843650	−	0.536894i	$-0.819598\pi$
$251$	−4.85993	+	4.85993i	−0.306756	+	0.306756i	0.536894	+	0.843650i	$0.319598\pi$
$252$		0			0
$253$	18.8730	+	18.8730i	1.18653	+	1.18653i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	1.05496i		−	0.0658064i	−0.999459	−	0.0329032i	$-0.989525\pi$
$257$		−	1.05496i		−	0.0658064i	0.999459	−	0.0329032i	$-0.0104753\pi$
$258$		0			0
$259$	4.87298	−	4.87298i	0.302792	−	0.302792i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	30.5738i		−	1.88526i	−0.333836	−	0.942631i	$-0.608343\pi$
$263$		−	30.5738i		−	1.88526i	0.333836	−	0.942631i	$-0.391657\pi$
$264$		0			0
$265$			2.00000i			0.122859i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.2051	−	18.2051i	1.10999	−	1.10999i	0.116836	−	0.993151i	$-0.462725\pi$
$269$	18.2051	−	18.2051i	1.10999	−	1.10999i	0.993151	−	0.116836i	$-0.0372753\pi$
$270$		0			0
$271$			11.7460i			0.713517i	0.934197	+	0.356758i	$0.116118\pi$
$271$			11.7460i			0.713517i	−0.934197	+	0.356758i	$0.883882\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.44572	+	3.44572i	0.207785	+	0.207785i
$276$		0			0
$277$	2.74597	−	2.74597i	0.164989	−	0.164989i	−0.619784	−	0.784773i	$-0.712780\pi$
$277$	2.74597	−	2.74597i	0.164989	−	0.164989i	0.784773	+	0.619784i	$0.212780\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.88338			−0.231663			−0.115832	−	0.993269i	$-0.536953\pi$
$281$	−3.88338			−0.231663			−0.115832	+	0.993269i	$0.536953\pi$
$282$		0			0
$283$	5.12702	+	5.12702i	0.304770	+	0.304770i	0.842877	−	0.538107i	$-0.180860\pi$
$283$	5.12702	+	5.12702i	0.304770	+	0.304770i	−0.538107	+	0.842877i	$0.680860\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.875328			0.0516690
$288$		0			0
$289$	15.0000			0.882353
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.46917	−	2.46917i	−0.144250	−	0.144250i	0.631294	−	0.775544i	$-0.282524\pi$
$293$	−2.46917	−	2.46917i	−0.144250	−	0.144250i	−0.775544	+	0.631294i	$0.782524\pi$
$294$		0			0
$295$	5.12702			0.298506
$296$		0			0
$297$		0			0
$298$		0			0
$299$	21.2132	−	21.2132i	1.22679	−	1.22679i
$300$		0			0
$301$	−29.2379	−	29.2379i	−1.68524	−	1.68524i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			6.71181i			0.384317i
$306$		0			0
$307$	−13.7460	+	13.7460i	−0.784524	+	0.784524i	−0.980591	−	0.196067i	$-0.937183\pi$
$307$	−13.7460	+	13.7460i	−0.784524	+	0.784524i	0.196067	+	0.980591i	$0.437183\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			21.0336i			1.19270i	0.802723	+	0.596352i	$0.203384\pi$
$311$			21.0336i			1.19270i	−0.802723	+	0.596352i	$0.796616\pi$
$312$		0			0
$313$		−	9.74597i		−	0.550875i	−0.961319	−	0.275437i	$-0.911177\pi$
$313$		−	9.74597i		−	0.550875i	0.961319	−	0.275437i	$-0.0888226\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−22.4478	+	22.4478i	−1.26079	+	1.26079i	−0.310084	+	0.950709i	$0.600357\pi$
$317$	−22.4478	+	22.4478i	−1.26079	+	1.26079i	−0.950709	+	0.310084i	$0.899643\pi$
$318$		0			0
$319$			19.4919i			1.09134i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.64880	+	2.64880i	0.147383	+	0.147383i
$324$		0			0
$325$	3.87298	−	3.87298i	0.214834	−	0.214834i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.89144			0.379937
$330$		0			0
$331$	2.12702	+	2.12702i	0.116911	+	0.116911i	0.763142	−	0.646231i	$-0.223656\pi$
$331$	2.12702	+	2.12702i	0.116911	+	0.116911i	−0.646231	+	0.763142i	$0.723656\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.3767			−0.840120
$336$		0			0
$337$	31.4919			1.71547			0.857737	−	0.514088i	$-0.171870\pi$
$337$	31.4919			1.71547			0.857737	+	0.514088i	$0.171870\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	13.7829	+	13.7829i	0.746385	+	0.746385i
$342$		0			0
$343$	−47.4919			−2.56432
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.06301	−	4.06301i	0.218114	−	0.218114i	−0.589589	−	0.807703i	$-0.700710\pi$
$347$	4.06301	−	4.06301i	0.218114	−	0.218114i	0.807703	+	0.589589i	$0.200710\pi$
$348$		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
$353$		−	20.8539i		−	1.10994i	−0.831869	−	0.554972i	$-0.812729\pi$
$353$		−	20.8539i		−	1.10994i	0.831869	−	0.554972i	$-0.187271\pi$
$354$		0			0
$355$	0.872983	−	0.872983i	0.0463331	−	0.0463331i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			8.12602i			0.428875i	0.976738	+	0.214438i	$0.0687919\pi$
$359$			8.12602i			0.428875i	−0.976738	+	0.214438i	$0.931208\pi$
$360$		0			0
$361$			11.9839i			0.630730i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.89144	+	6.89144i	−0.360714	+	0.360714i
$366$		0			0
$367$		−	26.8730i		−	1.40276i	−0.712788	−	0.701379i	$-0.752568\pi$
$367$		−	26.8730i		−	1.40276i	0.712788	−	0.701379i	$-0.247432\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.89144	−	6.89144i	−0.357786	−	0.357786i
$372$		0			0
$373$	−10.7460	+	10.7460i	−0.556405	+	0.556405i	−0.928282	−	0.371877i	$-0.878714\pi$
$373$	−10.7460	+	10.7460i	−0.556405	+	0.556405i	0.371877	+	0.928282i	$0.378714\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21.9089			1.12837
$378$		0			0
$379$	4.74597	+	4.74597i	0.243784	+	0.243784i	0.818414	−	0.574630i	$-0.194854\pi$
$379$	4.74597	+	4.74597i	0.243784	+	0.243784i	−0.574630	+	0.818414i	$0.694854\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.60190			0.235146			0.117573	−	0.993064i	$-0.462489\pi$
$383$	4.60190			0.235146			0.117573	+	0.993064i	$0.462489\pi$
$384$		0			0
$385$	−23.7460			−1.21021
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.7748	−	10.7748i	−0.546305	−	0.546305i	0.379065	−	0.925370i	$-0.376246\pi$
$389$	−10.7748	−	10.7748i	−0.546305	−	0.546305i	−0.925370	+	0.379065i	$0.876246\pi$
$390$		0			0
$391$	−7.74597			−0.391730
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.30565	+	8.30565i	−0.417903	+	0.417903i
$396$		0			0
$397$	9.61895	+	9.61895i	0.482761	+	0.482761i	0.906012	−	0.423251i	$-0.139111\pi$
$397$	9.61895	+	9.61895i	0.482761	+	0.482761i	−0.423251	+	0.906012i	$0.639111\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			13.6032i			0.679314i	0.940549	+	0.339657i	$0.110311\pi$
$401$			13.6032i			0.679314i	−0.940549	+	0.339657i	$0.889689\pi$
$402$		0			0
$403$	15.4919	−	15.4919i	0.771708	−	0.771708i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6.89144i			0.341596i
$408$		0			0
$409$			31.4919i			1.55718i	0.627536	+	0.778588i	$0.284064\pi$
$409$			31.4919i			1.55718i	−0.627536	+	0.778588i	$0.715936\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−17.6663	+	17.6663i	−0.869300	+	0.869300i
$414$		0			0
$415$		−	2.00000i		−	0.0981761i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.9860	−	12.9860i	−0.634406	−	0.634406i	0.314764	−	0.949170i	$-0.398075\pi$
$419$	−12.9860	−	12.9860i	−0.634406	−	0.634406i	−0.949170	+	0.314764i	$0.898075\pi$
$420$		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
$427$	−23.1270	−	23.1270i	−1.11919	−	1.11919i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.359257			−0.0173048			−0.00865241	−	0.999963i	$-0.502754\pi$
$431$	−0.359257			−0.0173048			−0.00865241	+	0.999963i	$0.502754\pi$
$432$		0			0
$433$	8.50807			0.408872			0.204436	−	0.978880i	$-0.434464\pi$
$433$	8.50807			0.408872			0.204436	+	0.978880i	$0.434464\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10.2588	+	10.2588i	0.490743	+	0.490743i
$438$		0			0
$439$	31.7460			1.51515			0.757576	−	0.652747i	$-0.226383\pi$
$439$	31.7460			1.51515			0.757576	+	0.652747i	$0.226383\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.6663	+	17.6663i	−0.839349	+	0.839349i	−0.988773	−	0.149424i	$-0.952258\pi$
$443$	−17.6663	+	17.6663i	−0.839349	+	0.839349i	0.149424	+	0.988773i	$0.452258\pi$
$444$		0			0
$445$	−11.8730	−	11.8730i	−0.562833	−	0.562833i
$446$		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
$451$	−0.618950	+	0.618950i	−0.0291452	+	0.0291452i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			26.6904i			1.25127i
$456$		0			0
$457$			19.2379i			0.899911i	0.893051	+	0.449956i	$0.148560\pi$
$457$			19.2379i			0.899911i	−0.893051	+	0.449956i	$0.851440\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.7748	+	10.7748i	−0.501834	+	0.501834i	−0.912007	−	0.410174i	$-0.865468\pi$
$461$	−10.7748	+	10.7748i	−0.501834	+	0.501834i	0.410174	+	0.912007i	$0.365468\pi$
$462$		0			0
$463$			10.3649i			0.481699i	0.970563	+	0.240849i	$0.0774259\pi$
$463$			10.3649i			0.481699i	−0.970563	+	0.240849i	$0.922574\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.4397	−	19.4397i	−0.899564	−	0.899564i	0.0958338	−	0.995397i	$-0.469448\pi$
$467$	−19.4397	−	19.4397i	−0.899564	−	0.899564i	−0.995397	+	0.0958338i	$0.969448\pi$
$468$		0			0
$469$	52.9839	−	52.9839i	2.44657	−	2.44657i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	41.3486			1.90121
$474$		0			0
$475$	1.87298	+	1.87298i	0.0859384	+	0.0859384i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.3767			0.702580			0.351290	−	0.936267i	$-0.385743\pi$
$479$	15.3767			0.702580			0.351290	+	0.936267i	$0.385743\pi$
$480$		0			0
$481$	7.74597			0.353186
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.54024	−	9.54024i	−0.433200	−	0.433200i
$486$		0			0
$487$	−11.3810			−0.515725			−0.257862	−	0.966182i	$-0.583018\pi$
$487$	−11.3810			−0.515725			−0.257862	+	0.966182i	$0.583018\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11.7514	+	11.7514i	−0.530332	+	0.530332i	−0.920671	−	0.390339i	$-0.872358\pi$
$491$	−11.7514	+	11.7514i	−0.530332	+	0.530332i	0.390339	+	0.920671i	$0.372358\pi$
$492$		0			0
$493$	−4.00000	−	4.00000i	−0.180151	−	0.180151i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			6.01611i			0.269859i
$498$		0			0
$499$	−23.6190	+	23.6190i	−1.05733	+	1.05733i	−0.0590759	+	0.998253i	$0.518815\pi$
$499$	−23.6190	+	23.6190i	−1.05733	+	1.05733i	−0.998253	+	0.0590759i	$0.981185\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	15.9156i		−	0.709642i	−0.934934	−	0.354821i	$-0.884542\pi$
$503$		−	15.9156i		−	0.709642i	0.934934	−	0.354821i	$-0.115458\pi$
$504$		0			0
$505$		−	13.7460i		−	0.611687i
$506$		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
$511$		−	47.4919i		−	2.10092i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.85993	+	4.85993i	0.214154	+	0.214154i
$516$		0			0
$517$	−4.87298	+	4.87298i	−0.214314	+	0.214314i
$518$		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
$523$	18.6190	+	18.6190i	0.814150	+	0.814150i	0.985253	−	0.171103i	$-0.0547332\pi$
$523$	18.6190	+	18.6190i	0.814150	+	0.814150i	−0.171103	+	0.985253i	$0.554733\pi$
$524$		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
$533$	0.695699	+	0.695699i	0.0301341	+	0.0301341i
$534$		0			0
$535$	−7.74597			−0.334887
$536$		0			0
$537$		0			0
$538$		0			0
$539$	57.7019	−	57.7019i	2.48540	−	2.48540i
$540$		0			0
$541$	4.74597	+	4.74597i	0.204045	+	0.204045i	0.801731	−	0.597686i	$-0.203913\pi$
$541$	4.74597	+	4.74597i	0.204045	+	0.204045i	−0.597686	+	0.801731i	$0.703913\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	3.88338i		−	0.166346i
$546$		0			0
$547$	13.7460	−	13.7460i	0.587735	−	0.587735i	−0.349282	−	0.937018i	$-0.613575\pi$
$547$	13.7460	−	13.7460i	0.587735	−	0.587735i	0.937018	+	0.349282i	$0.113575\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			10.5952i			0.451370i
$552$		0			0
$553$		−	57.2379i		−	2.43400i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	17.6663	−	17.6663i	0.748544	−	0.748544i	−0.225662	−	0.974206i	$-0.572455\pi$
$557$	17.6663	−	17.6663i	0.748544	−	0.748544i	0.974206	+	0.225662i	$0.0724545\pi$
$558$		0			0
$559$		−	46.4758i		−	1.96572i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.25070	+	7.25070i	0.305580	+	0.305580i	0.843192	−	0.537612i	$-0.180673\pi$
$563$	7.25070	+	7.25070i	0.305580	+	0.305580i	−0.537612	+	0.843192i	$0.680673\pi$
$564$		0			0
$565$	−6.74597	+	6.74597i	−0.283805	+	0.283805i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.47723			0.229617			0.114809	−	0.993388i	$-0.463375\pi$
$569$	5.47723			0.229617			0.114809	+	0.993388i	$0.463375\pi$
$570$		0			0
$571$	−7.36492	−	7.36492i	−0.308212	−	0.308212i	0.536004	−	0.844216i	$-0.319933\pi$
$571$	−7.36492	−	7.36492i	−0.308212	−	0.308212i	−0.844216	+	0.536004i	$0.819933\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.47723			−0.228416
$576$		0			0
$577$	−31.7460			−1.32160			−0.660801	−	0.750561i	$-0.729783\pi$
$577$	−31.7460			−1.32160			−0.660801	+	0.750561i	$0.729783\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6.89144	+	6.89144i	0.285905	+	0.285905i
$582$		0			0
$583$	9.74597			0.403637
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.7748	−	10.7748i	0.444725	−	0.444725i	−0.448872	−	0.893596i	$-0.648174\pi$
$587$	10.7748	−	10.7748i	0.444725	−	0.444725i	0.893596	+	0.448872i	$0.148174\pi$
$588$		0			0
$589$	7.49193	+	7.49193i	0.308700	+	0.308700i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	34.6368i		−	1.42236i	−0.703008	−	0.711182i	$-0.748160\pi$
$593$		−	34.6368i		−	1.42236i	0.703008	−	0.711182i	$-0.251840\pi$
$594$		0			0
$595$	4.87298	−	4.87298i	0.199773	−	0.199773i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	16.2520i		−	0.664041i	−0.943272	−	0.332020i	$-0.892270\pi$
$599$		−	16.2520i		−	0.664041i	0.943272	−	0.332020i	$-0.107730\pi$
$600$		0			0
$601$		−	44.9839i		−	1.83493i	−0.397816	−	0.917465i	$-0.630232\pi$
$601$		−	44.9839i		−	1.83493i	0.397816	−	0.917465i	$-0.369768\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.01276	−	9.01276i	0.366421	−	0.366421i
$606$		0			0
$607$		−	3.63508i		−	0.147543i	−0.997275	−	0.0737717i	$-0.976496\pi$
$607$		−	3.63508i		−	0.147543i	0.997275	−	0.0737717i	$-0.0235036\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.47723	+	5.47723i	0.221585	+	0.221585i
$612$		0			0
$613$	−17.6190	+	17.6190i	−0.711623	+	0.711623i	−0.966875	−	0.255252i	$-0.917842\pi$
$613$	−17.6190	+	17.6190i	−0.711623	+	0.711623i	0.255252	+	0.966875i	$0.417842\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−45.2320			−1.82097			−0.910486	−	0.413539i	$-0.864292\pi$
$617$	−45.2320			−1.82097			−0.910486	+	0.413539i	$0.864292\pi$
$618$		0			0
$619$	−11.2540	−	11.2540i	−0.452338	−	0.452338i	0.443792	−	0.896130i	$-0.353633\pi$
$619$	−11.2540	−	11.2540i	−0.452338	−	0.452338i	−0.896130	+	0.443792i	$0.853633\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	81.8219			3.27813
$624$		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
$635$	11.7514	−	11.7514i	0.466339	−	0.466339i
$636$		0			0
$637$	−64.8569	−	64.8569i	−2.56972	−	2.56972i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			9.54024i			0.376817i	0.982091	+	0.188408i	$0.0603328\pi$
$641$			9.54024i			0.376817i	−0.982091	+	0.188408i	$0.939667\pi$
$642$		0			0
$643$	10.8730	−	10.8730i	0.428789	−	0.428789i	−0.459427	−	0.888216i	$-0.651945\pi$
$643$	10.8730	−	10.8730i	0.428789	−	0.428789i	0.888216	+	0.459427i	$0.151945\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	22.9639i		−	0.902802i	−0.892321	−	0.451401i	$-0.850924\pi$
$647$		−	22.9639i		−	0.902802i	0.892321	−	0.451401i	$-0.149076\pi$
$648$		0			0
$649$		−	24.9839i		−	0.980703i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.59384	+	1.59384i	−0.0623719	+	0.0623719i	−0.737605	−	0.675233i	$-0.764043\pi$
$653$	−1.59384	+	1.59384i	−0.0623719	+	0.0623719i	0.675233	+	0.737605i	$0.264043\pi$
$654$		0			0
$655$			16.8730i			0.659282i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.8760	−	10.8760i	−0.423671	−	0.423671i	0.462795	−	0.886465i	$-0.346847\pi$
$659$	−10.8760	−	10.8760i	−0.423671	−	0.423671i	−0.886465	+	0.462795i	$0.846847\pi$
$660$		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
$665$	−12.9076			−0.500533
$666$		0			0
$667$	−15.4919	−	15.4919i	−0.599850	−	0.599850i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	32.7065			1.26262
$672$		0			0
$673$	15.4919			0.597170			0.298585	−	0.954383i	$-0.403485\pi$
$673$	15.4919			0.597170			0.298585	+	0.954383i	$0.403485\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.53218	+	6.53218i	0.251052	+	0.251052i	0.821402	−	0.570350i	$-0.193192\pi$
$677$	6.53218	+	6.53218i	0.251052	+	0.251052i	−0.570350	+	0.821402i	$0.693192\pi$
$678$		0			0
$679$	65.7460			2.52310
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.23458	+	1.23458i	−0.0472401	+	0.0472401i	−0.730332	−	0.683092i	$-0.760635\pi$
$683$	−1.23458	+	1.23458i	−0.0472401	+	0.0472401i	0.683092	+	0.730332i	$0.260635\pi$
$684$		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
$691$	−9.36492	+	9.36492i	−0.356258	+	0.356258i	−0.862432	−	0.506173i	$-0.831060\pi$
$691$	−9.36492	+	9.36492i	−0.356258	+	0.356258i	0.506173	+	0.862432i	$0.331060\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	9.54024i		−	0.361882i
$696$		0			0
$697$		−	0.254033i		−	0.00962220i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26.3312	−	26.3312i	0.994515	−	0.994515i	−0.00547041	−	0.999985i	$-0.501741\pi$
$701$	26.3312	−	26.3312i	0.994515	−	0.994515i	0.999985	+	0.00547041i	$0.00174129\pi$
$702$		0			0
$703$			3.74597i			0.141282i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	47.3647	+	47.3647i	1.78133	+	1.78133i
$708$		0			0
$709$	17.2540	−	17.2540i	0.647989	−	0.647989i	−0.304518	−	0.952507i	$-0.598495\pi$
$709$	17.2540	−	17.2540i	0.647989	−	0.647989i	0.952507	+	0.304518i	$0.0984953\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−21.9089			−0.820495
$714$		0			0
$715$	−18.8730	−	18.8730i	−0.705810	−	0.705810i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.8928			−0.592701			−0.296351	−	0.955079i	$-0.595770\pi$
$719$	−15.8928			−0.592701			−0.296351	+	0.955079i	$0.595770\pi$
$720$		0			0
$721$	−33.4919			−1.24730
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.82843	−	2.82843i	−0.105045	−	0.105045i
$726$		0			0
$727$	−43.6028			−1.61714			−0.808569	−	0.588401i	$-0.799758\pi$
$727$	−43.6028			−1.61714			−0.808569	+	0.588401i	$0.799758\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.48528	+	8.48528i	−0.313839	+	0.313839i
$732$		0			0
$733$	−3.25403	−	3.25403i	−0.120190	−	0.120190i	0.644453	−	0.764644i	$-0.277085\pi$
$733$	−3.25403	−	3.25403i	−0.120190	−	0.120190i	−0.764644	+	0.644453i	$0.777085\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			74.9305i			2.76010i
$738$		0			0
$739$	23.6190	−	23.6190i	0.868837	−	0.868837i	−0.123507	−	0.992344i	$-0.539414\pi$
$739$	23.6190	−	23.6190i	0.868837	−	0.868837i	0.992344	+	0.123507i	$0.0394140\pi$
$740$		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
$745$			19.7460i			0.723436i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	26.6904	−	26.6904i	0.975247	−	0.975247i
$750$		0			0
$751$			2.25403i			0.0822508i	0.999154	+	0.0411254i	$0.0130943\pi$
$751$			2.25403i			0.0822508i	−0.999154	+	0.0411254i	$0.986906\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.65685	+	5.65685i	0.205874	+	0.205874i
$756$		0			0
$757$	−28.4919	+	28.4919i	−1.03556	+	1.03556i	−0.0362128	+	0.999344i	$0.511529\pi$
$757$	−28.4919	+	28.4919i	−1.03556	+	1.03556i	−0.999344	+	0.0362128i	$0.988471\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.7923			−0.934970			−0.467485	−	0.884001i	$-0.654840\pi$
$761$	−25.7923			−0.934970			−0.467485	+	0.884001i	$0.654840\pi$
$762$		0			0
$763$	13.3810	+	13.3810i	0.484427	+	0.484427i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−28.0818			−1.01398
$768$		0			0
$769$	−39.2379			−1.41495			−0.707477	−	0.706736i	$-0.750167\pi$
$769$	−39.2379			−1.41495			−0.707477	+	0.706736i	$0.750167\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.9786	−	19.9786i	−0.718581	−	0.718581i	0.249734	−	0.968315i	$-0.419657\pi$
$773$	−19.9786	−	19.9786i	−0.718581	−	0.718581i	−0.968315	+	0.249734i	$0.919657\pi$
$774$		0			0
$775$	−4.00000			−0.143684
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.336442	+	0.336442i	−0.0120543	+	0.0120543i
$780$		0			0
$781$	−4.25403	−	4.25403i	−0.152221	−	0.152221i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			18.0255i			0.643358i
$786$		0			0
$787$	−3.38105	+	3.38105i	−0.120521	+	0.120521i	−0.764795	−	0.644274i	$-0.777160\pi$
$787$	−3.38105	+	3.38105i	−0.120521	+	0.120521i	0.644274	+	0.764795i	$0.277160\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	46.4894i		−	1.65297i
$792$		0			0
$793$		−	36.7621i		−	1.30546i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.18768	+	3.18768i	−0.112914	+	0.112914i	−0.761306	−	0.648393i	$-0.775442\pi$
$797$	−3.18768	+	3.18768i	−0.112914	+	0.112914i	0.648393	+	0.761306i	$0.275442\pi$
$798$		0			0
$799$		−	2.00000i		−	0.0707549i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	33.5819	+	33.5819i	1.18508	+	1.18508i
$804$		0			0
$805$	18.8730	−	18.8730i	0.665185	−	0.665185i
$806$		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
$811$	6.49193	+	6.49193i	0.227963	+	0.227963i	0.811841	−	0.583878i	$-0.198466\pi$
$811$	6.49193	+	6.49193i	0.227963	+	0.227963i	−0.583878	+	0.811841i	$0.698466\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.59384			0.0558299
$816$		0			0
$817$	22.4758			0.786329
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.9545	−	10.9545i	−0.382313	−	0.382313i	0.489622	−	0.871935i	$-0.337135\pi$
$821$	−10.9545	−	10.9545i	−0.382313	−	0.382313i	−0.871935	+	0.489622i	$0.837135\pi$
$822$		0			0
$823$	−7.12702			−0.248432			−0.124216	−	0.992255i	$-0.539642\pi$
$823$	−7.12702			−0.248432			−0.124216	+	0.992255i	$0.539642\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−38.8795	+	38.8795i	−1.35197	+	1.35197i	−0.468517	+	0.883454i	$0.655212\pi$
$827$	−38.8795	+	38.8795i	−1.35197	+	1.35197i	−0.883454	+	0.468517i	$0.844788\pi$
$828$		0			0
$829$	26.7460	+	26.7460i	0.928926	+	0.928926i	0.997637	−	0.0687108i	$-0.0218886\pi$
$829$	26.7460	+	26.7460i	0.928926	+	0.928926i	−0.0687108	+	0.997637i	$0.521889\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			23.6824i			0.820546i
$834$		0			0
$835$	−13.8730	+	13.8730i	−0.480094	+	0.480094i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			21.5496i			0.743976i	0.928237	+	0.371988i	$0.121324\pi$
$839$			21.5496i			0.743976i	−0.928237	+	0.371988i	$0.878676\pi$
$840$		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
$847$			62.1109i			2.13416i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.47723	−	5.47723i	−0.187757	−	0.187757i
$852$		0			0
$853$	3.61895	−	3.61895i	0.123910	−	0.123910i	−0.642432	−	0.766343i	$-0.722075\pi$
$853$	3.61895	−	3.61895i	0.123910	−	0.123910i	0.766343	+	0.642432i	$0.222075\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.0255			0.615740			0.307870	−	0.951428i	$-0.400384\pi$
$857$	18.0255			0.615740			0.307870	+	0.951428i	$0.400384\pi$
$858$		0			0
$859$	8.49193	+	8.49193i	0.289741	+	0.289741i	0.836978	−	0.547237i	$-0.184320\pi$
$859$	8.49193	+	8.49193i	0.289741	+	0.289741i	−0.547237	+	0.836978i	$0.684320\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	17.5094			0.596029			0.298014	−	0.954561i	$-0.403676\pi$
$863$	17.5094			0.596029			0.298014	+	0.954561i	$0.403676\pi$
$864$		0			0
$865$	13.4919			0.458739
$866$		0			0
$867$		0			0
$868$		0			0
$869$	40.4733	+	40.4733i	1.37296	+	1.37296i
$870$		0			0
$871$	84.2218			2.85375
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.44572	−	3.44572i	0.116487	−	0.116487i
$876$		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
$881$		−	13.2440i		−	0.446201i	−0.974795	−	0.223101i	$-0.928382\pi$
$881$		−	13.2440i		−	0.446201i	0.974795	−	0.223101i	$-0.0716179\pi$
$882$		0			0
$883$	−16.3649	+	16.3649i	−0.550723	+	0.550723i	−0.926650	−	0.375926i	$-0.877325\pi$
$883$	−16.3649	+	16.3649i	−0.550723	+	0.550723i	0.375926	+	0.926650i	$0.377325\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	2.64880i		−	0.0889379i	−0.999011	−	0.0444690i	$-0.985840\pi$
$887$		−	2.64880i		−	0.0889379i	0.999011	−	0.0444690i	$-0.0141596\pi$
$888$		0			0
$889$			80.9839i			2.71611i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.64880	+	2.64880i	−0.0886387	+	0.0886387i
$894$		0			0
$895$			6.61895i			0.221247i
$896$		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
$913$	−9.74597			−0.322544
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−58.1396	−	58.1396i	−1.91994	−	1.91994i
$918$		0			0
$919$	−4.50807			−0.148707			−0.0743537	−	0.997232i	$-0.523689\pi$
$919$	−4.50807			−0.148707			−0.0743537	+	0.997232i	$0.523689\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.78153	+	4.78153i	−0.157386	+	0.157386i
$924$		0			0
$925$	−1.00000	−	1.00000i	−0.0328798	−	0.0328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			34.2776i			1.12461i	0.826930	+	0.562305i	$0.190085\pi$
$929$			34.2776i			1.12461i	−0.826930	+	0.562305i	$0.809915\pi$
$930$		0			0
$931$	31.3649	−	31.3649i	1.02794	−	1.02794i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			6.89144i			0.225374i
$936$		0			0
$937$		−	47.7460i		−	1.55979i	−0.625909	−	0.779896i	$-0.715272\pi$
$937$		−	47.7460i		−	1.55979i	0.625909	−	0.779896i	$-0.284728\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.71181	+	6.71181i	−0.218799	+	0.218799i	−0.807992	−	0.589193i	$-0.799446\pi$
$941$	−6.71181	+	6.71181i	−0.218799	+	0.218799i	0.589193	+	0.807992i	$0.299446\pi$
$942$		0			0
$943$		−	0.983867i		−	0.0320391i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.0094	+	12.0094i	0.390253	+	0.390253i	0.874778	−	0.484524i	$-0.161007\pi$
$947$	12.0094	+	12.0094i	0.390253	+	0.390253i	−0.484524	+	0.874778i	$0.661007\pi$
$948$		0			0
$949$	37.7460	−	37.7460i	1.22529	−	1.22529i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.43033			0.240692			0.120346	−	0.992732i	$-0.461600\pi$
$953$	7.43033			0.240692			0.120346	+	0.992732i	$0.461600\pi$
$954$		0			0
$955$	−10.8730	−	10.8730i	−0.351841	−	0.351841i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	75.8058			2.44790
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	15.3767	+	15.3767i	0.494994	+	0.494994i
$966$		0			0
$967$	−14.6190			−0.470114			−0.235057	−	0.971982i	$-0.575528\pi$
$967$	−14.6190			−0.470114			−0.235057	+	0.971982i	$0.575528\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	28.3627	−	28.3627i	0.910202	−	0.910202i	−0.0860861	−	0.996288i	$-0.527436\pi$
$971$	28.3627	−	28.3627i	0.910202	−	0.910202i	0.996288	+	0.0860861i	$0.0274360\pi$
$972$		0			0
$973$	32.8730	+	32.8730i	1.05386	+	1.05386i
$974$		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
$979$	−57.8569	+	57.8569i	−1.84911	+	1.84911i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	51.6074i		−	1.64602i	−0.568027	−	0.823010i	$-0.692293\pi$
$983$		−	51.6074i		−	1.64602i	0.568027	−	0.823010i	$-0.307707\pi$
$984$		0			0
$985$			19.7460i			0.629159i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32.8634	+	32.8634i	−1.04499	+	1.04499i
$990$		0			0
$991$			31.2379i			0.992305i	0.868236	+	0.496152i	$0.165254\pi$
$991$			31.2379i			0.992305i	−0.868236	+	0.496152i	$0.834746\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.179629	+	0.179629i	0.00569461	+	0.00569461i
$996$		0			0
$997$	5.87298	−	5.87298i	0.185999	−	0.185999i	−0.607965	−	0.793964i	$-0.708014\pi$
$997$	5.87298	−	5.87298i	0.185999	−	0.185999i	0.793964	+	0.607965i	$0.208014\pi$
$998$		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.431.3		8
3.2	odd	2	inner	2880.2.bl.a.431.1		8
4.3	odd	2		720.2.bl.a.611.4	yes	8
12.11	even	2		720.2.bl.a.611.2	yes	8
16.5	even	4		720.2.bl.a.251.2	✓	8
16.11	odd	4	inner	2880.2.bl.a.1871.1		8
48.5	odd	4		720.2.bl.a.251.4	yes	8
48.11	even	4	inner	2880.2.bl.a.1871.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.a.251.2	✓	8	16.5	even	4
720.2.bl.a.251.4	yes	8	48.5	odd	4
720.2.bl.a.611.2	yes	8	12.11	even	2
720.2.bl.a.611.4	yes	8	4.3	odd	2
2880.2.bl.a.431.1		8	3.2	odd	2	inner
2880.2.bl.a.431.3		8	1.1	even	1	trivial
2880.2.bl.a.1871.1		8	16.11	odd	4	inner
2880.2.bl.a.1871.3		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)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} -4.87298 q^{7} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{5} -4.87298 q^{7} +(3.44572 - 3.44572i) q^{11} +(-3.87298 - 3.87298i) q^{13} +1.41421i q^{17} +(1.87298 - 1.87298i) q^{19} +5.47723i q^{23} +1.00000i q^{25} +(-2.82843 + 2.82843i) q^{29} +4.00000i q^{31} +(-3.44572 - 3.44572i) q^{35} +(-1.00000 + 1.00000i) q^{37} -0.179629 q^{41} +(6.00000 + 6.00000i) q^{43} -1.41421 q^{47} +16.7460 q^{49} +(1.41421 + 1.41421i) q^{53} +4.87298 q^{55} +(3.62535 - 3.62535i) q^{59} +(4.74597 + 4.74597i) q^{61} -5.47723i q^{65} +(-10.8730 + 10.8730i) q^{67} -1.23458i q^{71} +9.74597i q^{73} +(-16.7909 + 16.7909i) q^{77} +11.7460i q^{79} +(-1.41421 - 1.41421i) q^{83} +(-1.00000 + 1.00000i) q^{85} -16.7909 q^{89} +(18.8730 + 18.8730i) q^{91} +2.64880 q^{95} -13.4919 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 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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