LMFDB - Embedded newform 2880.2.bl.b.1871.2

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:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1871.2
Character	$\chi$	$=$	2880.1871
Dual form			2880.2.bl.b.431.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{5} +0.750417 q^{7} +O(q^{10})$	$q+(-0.707107 + 0.707107i) q^{5} +0.750417 q^{7} +(-2.08285 - 2.08285i) q^{11} +(-1.66519 + 1.66519i) q^{13} +1.81077i q^{17} +(1.22715 + 1.22715i) q^{19} -9.36690i q^{23} -1.00000i q^{25} +(4.84824 + 4.84824i) q^{29} +4.95295i q^{31} +(-0.530625 + 0.530625i) q^{35} +(5.94167 + 5.94167i) q^{37} -2.47546 q^{41} +(-5.20107 + 5.20107i) q^{43} -4.43781 q^{47} -6.43687 q^{49} +(-7.54780 + 7.54780i) q^{53} +2.94559 q^{55} +(-0.472792 - 0.472792i) q^{59} +(-4.99164 + 4.99164i) q^{61} -2.35493i q^{65} +(-1.95148 - 1.95148i) q^{67} -0.418759i q^{71} +3.49594i q^{73} +(-1.56300 - 1.56300i) q^{77} +3.40726i q^{79} +(0.548807 - 0.548807i) q^{83} +(-1.28041 - 1.28041i) q^{85} -12.2354 q^{89} +(-1.24958 + 1.24958i) q^{91} -1.73546 q^{95} +6.64896 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 8 q^{7}+O(q^{10}) $ 24 * q + 8 * q^7	$ 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) $ 24 * q + 8 * q^7 + 24 * q^19 + 8 * q^37 - 48 * q^43 + 24 * q^49 - 24 * q^55 + 40 * q^61 + 40 * q^67 + 24 * q^85 - 40 * q^91 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$7$	0.750417			0.283631			0.141815	−	0.989893i	$-0.454706\pi$
$7$	0.750417			0.283631			0.141815	+	0.989893i	$0.454706\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.08285	−	2.08285i	−0.628002	−	0.628002i	0.319563	−	0.947565i	$-0.396464\pi$
$11$	−2.08285	−	2.08285i	−0.628002	−	0.628002i	−0.947565	+	0.319563i	$0.896464\pi$
$12$		0			0
$13$	−1.66519	+	1.66519i	−0.461840	+	0.461840i	−0.899258	−	0.437418i	$-0.855893\pi$
$13$	−1.66519	+	1.66519i	−0.461840	+	0.461840i	0.437418	+	0.899258i	$0.355893\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.81077i			0.439176i	0.975593	+	0.219588i	$0.0704712\pi$
$17$			1.81077i			0.439176i	−0.975593	+	0.219588i	$0.929529\pi$
$18$		0			0
$19$	1.22715	+	1.22715i	0.281529	+	0.281529i	0.833718	−	0.552190i	$-0.186208\pi$
$19$	1.22715	+	1.22715i	0.281529	+	0.281529i	−0.552190	+	0.833718i	$0.686208\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	9.36690i		−	1.95313i	−0.215215	−	0.976567i	$-0.569045\pi$
$23$		−	9.36690i		−	1.95313i	0.215215	−	0.976567i	$-0.430955\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.84824	+	4.84824i	0.900295	+	0.900295i	0.995461	−	0.0951663i	$-0.0303383\pi$
$29$	4.84824	+	4.84824i	0.900295	+	0.900295i	−0.0951663	+	0.995461i	$0.530338\pi$
$30$		0			0
$31$			4.95295i			0.889577i	0.895636	+	0.444788i	$0.146721\pi$
$31$			4.95295i			0.889577i	−0.895636	+	0.444788i	$0.853279\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.530625	+	0.530625i	−0.0896919	+	0.0896919i
$36$		0			0
$37$	5.94167	+	5.94167i	0.976805	+	0.976805i	0.999737	−	0.0229319i	$-0.00730010\pi$
$37$	5.94167	+	5.94167i	0.976805	+	0.976805i	−0.0229319	+	0.999737i	$0.507300\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.47546			−0.386602			−0.193301	−	0.981139i	$-0.561919\pi$
$41$	−2.47546			−0.386602			−0.193301	+	0.981139i	$0.561919\pi$
$42$		0			0
$43$	−5.20107	+	5.20107i	−0.793155	+	0.793155i	−0.982006	−	0.188851i	$-0.939524\pi$
$43$	−5.20107	+	5.20107i	−0.793155	+	0.793155i	0.188851	+	0.982006i	$0.439524\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.43781			−0.647321			−0.323661	−	0.946173i	$-0.604914\pi$
$47$	−4.43781			−0.647321			−0.323661	+	0.946173i	$0.604914\pi$
$48$		0			0
$49$	−6.43687			−0.919554
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.54780	+	7.54780i	−1.03677	+	1.03677i	−0.0374723	+	0.999298i	$0.511931\pi$
$53$	−7.54780	+	7.54780i	−1.03677	+	1.03677i	−0.999298	+	0.0374723i	$0.988069\pi$
$54$		0			0
$55$	2.94559			0.397184
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.472792	−	0.472792i	−0.0615523	−	0.0615523i	0.675661	−	0.737213i	$-0.263859\pi$
$59$	−0.472792	−	0.472792i	−0.0615523	−	0.0615523i	−0.737213	+	0.675661i	$0.763859\pi$
$60$		0			0
$61$	−4.99164	+	4.99164i	−0.639114	+	0.639114i	−0.950337	−	0.311223i	$-0.899261\pi$
$61$	−4.99164	+	4.99164i	−0.639114	+	0.639114i	0.311223	+	0.950337i	$0.399261\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.35493i		−	0.292093i
$66$		0			0
$67$	−1.95148	−	1.95148i	−0.238412	−	0.238412i	0.577781	−	0.816192i	$-0.303919\pi$
$67$	−1.95148	−	1.95148i	−0.238412	−	0.238412i	−0.816192	+	0.577781i	$0.803919\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	0.418759i		−	0.0496975i	−0.999691	−	0.0248488i	$-0.992090\pi$
$71$		−	0.418759i		−	0.0496975i	0.999691	−	0.0248488i	$-0.00791042\pi$
$72$		0			0
$73$			3.49594i			0.409168i	0.978849	+	0.204584i	$0.0655842\pi$
$73$			3.49594i			0.409168i	−0.978849	+	0.204584i	$0.934416\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.56300	−	1.56300i	−0.178121	−	0.178121i
$78$		0			0
$79$			3.40726i			0.383347i	0.981459	+	0.191674i	$0.0613915\pi$
$79$			3.40726i			0.383347i	−0.981459	+	0.191674i	$0.938609\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.548807	−	0.548807i	0.0602394	−	0.0602394i	−0.676345	−	0.736585i	$-0.736437\pi$
$83$	0.548807	−	0.548807i	0.0602394	−	0.0602394i	0.736585	+	0.676345i	$0.236437\pi$
$84$		0			0
$85$	−1.28041	−	1.28041i	−0.138880	−	0.138880i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.2354			−1.29695			−0.648474	−	0.761237i	$-0.724592\pi$
$89$	−12.2354			−1.29695			−0.648474	+	0.761237i	$0.724592\pi$
$90$		0			0
$91$	−1.24958	+	1.24958i	−0.130992	+	0.130992i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.73546			−0.178054
$96$		0			0
$97$	6.64896			0.675100			0.337550	−	0.941308i	$-0.390402\pi$
$97$	6.64896			0.675100			0.337550	+	0.941308i	$0.390402\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.50982	−	8.50982i	0.846759	−	0.846759i	−0.142968	−	0.989727i	$-0.545665\pi$
$101$	8.50982	−	8.50982i	0.846759	−	0.846759i	0.989727	+	0.142968i	$0.0456647\pi$
$102$		0			0
$103$	13.3482			1.31524			0.657621	−	0.753349i	$-0.271563\pi$
$103$	13.3482			1.31524			0.657621	+	0.753349i	$0.271563\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.67726	−	1.67726i	−0.162147	−	0.162147i	0.621370	−	0.783517i	$-0.286576\pi$
$107$	−1.67726	−	1.67726i	−0.162147	−	0.162147i	−0.783517	+	0.621370i	$0.786576\pi$
$108$		0			0
$109$	−10.8094	+	10.8094i	−1.03535	+	1.03535i	−0.0360007	+	0.999352i	$0.511462\pi$
$109$	−10.8094	+	10.8094i	−1.03535	+	1.03535i	−0.999352	+	0.0360007i	$0.988538\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			9.62581i			0.905520i	0.891632	+	0.452760i	$0.149561\pi$
$113$			9.62581i			0.905520i	−0.891632	+	0.452760i	$0.850439\pi$
$114$		0			0
$115$	6.62340	+	6.62340i	0.617635	+	0.617635i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			1.35883i			0.124564i
$120$		0			0
$121$		−	2.32349i		−	0.211226i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$			1.98607i			0.176235i	0.996110	+	0.0881176i	$0.0280851\pi$
$127$			1.98607i			0.176235i	−0.996110	+	0.0881176i	$0.971915\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−14.6078	+	14.6078i	−1.27629	+	1.27629i	−0.333554	+	0.942731i	$0.608248\pi$
$131$	−14.6078	+	14.6078i	−1.27629	+	1.27629i	−0.942731	+	0.333554i	$0.891752\pi$
$132$		0			0
$133$	0.920877	+	0.920877i	0.0798502	+	0.0798502i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−17.5298			−1.49767			−0.748837	−	0.662754i	$-0.769387\pi$
$137$	−17.5298			−1.49767			−0.748837	+	0.662754i	$0.769387\pi$
$138$		0			0
$139$	−9.60358	+	9.60358i	−0.814565	+	0.814565i	−0.985314	−	0.170750i	$-0.945381\pi$
$139$	−9.60358	+	9.60358i	−0.814565	+	0.814565i	0.170750	+	0.985314i	$0.445381\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.93666			0.580073
$144$		0			0
$145$	−6.85644			−0.569397
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.85352	−	3.85352i	0.315693	−	0.315693i	−0.531417	−	0.847110i	$-0.678340\pi$
$149$	3.85352	−	3.85352i	0.315693	−	0.315693i	0.847110	+	0.531417i	$0.178340\pi$
$150$		0			0
$151$	12.1519			0.988912			0.494456	−	0.869203i	$-0.335367\pi$
$151$	12.1519			0.988912			0.494456	+	0.869203i	$0.335367\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.50227	−	3.50227i	−0.281309	−	0.281309i
$156$		0			0
$157$	−3.49520	+	3.49520i	−0.278947	+	0.278947i	−0.832689	−	0.553741i	$-0.813200\pi$
$157$	−3.49520	+	3.49520i	−0.278947	+	0.278947i	0.553741	+	0.832689i	$0.313200\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	7.02908i		−	0.553969i
$162$		0			0
$163$	−6.19907	−	6.19907i	−0.485549	−	0.485549i	0.421349	−	0.906898i	$-0.361557\pi$
$163$	−6.19907	−	6.19907i	−0.485549	−	0.485549i	−0.906898	+	0.421349i	$0.861557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	23.9728i		−	1.85507i	−0.373739	−	0.927534i	$-0.621924\pi$
$167$		−	23.9728i		−	1.85507i	0.373739	−	0.927534i	$-0.378076\pi$
$168$		0			0
$169$			7.45431i			0.573408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.70153	+	4.70153i	0.357450	+	0.357450i	0.862872	−	0.505422i	$-0.168663\pi$
$173$	4.70153	+	4.70153i	0.357450	+	0.357450i	−0.505422	+	0.862872i	$0.668663\pi$
$174$		0			0
$175$		−	0.750417i		−	0.0567262i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.9751	+	10.9751i	−0.820315	+	0.820315i	−0.986153	−	0.165838i	$-0.946967\pi$
$179$	−10.9751	+	10.9751i	−0.820315	+	0.820315i	0.165838	+	0.986153i	$0.446967\pi$
$180$		0			0
$181$	−0.999478	−	0.999478i	−0.0742906	−	0.0742906i	0.668985	−	0.743276i	$-0.266729\pi$
$181$	−0.999478	−	0.999478i	−0.0742906	−	0.0742906i	−0.743276	+	0.668985i	$0.766729\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.40280			−0.617786
$186$		0			0
$187$	3.77155	−	3.77155i	0.275803	−	0.275803i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−20.6166			−1.49177			−0.745884	−	0.666076i	$-0.767973\pi$
$191$	−20.6166			−1.49177			−0.745884	+	0.666076i	$0.767973\pi$
$192$		0			0
$193$	−4.05831			−0.292124			−0.146062	−	0.989275i	$-0.546660\pi$
$193$	−4.05831			−0.292124			−0.146062	+	0.989275i	$0.546660\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.20746	+	4.20746i	−0.299769	+	0.299769i	−0.840923	−	0.541154i	$-0.817987\pi$
$197$	−4.20746	+	4.20746i	−0.299769	+	0.299769i	0.541154	+	0.840923i	$0.317987\pi$
$198$		0			0
$199$	−8.21153			−0.582100			−0.291050	−	0.956708i	$-0.594005\pi$
$199$	−8.21153			−0.582100			−0.291050	+	0.956708i	$0.594005\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.63820	+	3.63820i	0.255351	+	0.255351i
$204$		0			0
$205$	1.75042	−	1.75042i	0.122254	−	0.122254i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	5.11195i		−	0.353601i
$210$		0			0
$211$	−2.03421	−	2.03421i	−0.140041	−	0.140041i	0.633611	−	0.773652i	$-0.281572\pi$
$211$	−2.03421	−	2.03421i	−0.140041	−	0.140041i	−0.773652	+	0.633611i	$0.781572\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	7.35542i		−	0.501635i
$216$		0			0
$217$			3.71678i			0.252311i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.01526	−	3.01526i	−0.202829	−	0.202829i
$222$		0			0
$223$			28.2722i			1.89325i	0.322344	+	0.946623i	$0.395529\pi$
$223$			28.2722i			1.89325i	−0.322344	+	0.946623i	$0.604471\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.39231	+	7.39231i	−0.490645	+	0.490645i	−0.908509	−	0.417864i	$-0.862779\pi$
$227$	−7.39231	+	7.39231i	−0.490645	+	0.490645i	0.417864	+	0.908509i	$0.362779\pi$
$228$		0			0
$229$	14.8710	+	14.8710i	0.982706	+	0.982706i	0.999853	−	0.0171469i	$-0.00545829\pi$
$229$	14.8710	+	14.8710i	0.982706	+	0.982706i	−0.0171469	+	0.999853i	$0.505458\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.2128			−1.25867			−0.629336	−	0.777133i	$-0.716673\pi$
$233$	−19.2128			−1.25867			−0.629336	+	0.777133i	$0.716673\pi$
$234$		0			0
$235$	3.13801	−	3.13801i	0.204701	−	0.204701i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	20.9731			1.35664			0.678318	−	0.734769i	$-0.262709\pi$
$239$	20.9731			1.35664			0.678318	+	0.734769i	$0.262709\pi$
$240$		0			0
$241$	0.400502			0.0257986			0.0128993	−	0.999917i	$-0.495894\pi$
$241$	0.400502			0.0257986			0.0128993	+	0.999917i	$0.495894\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.55156	−	4.55156i	0.290788	−	0.290788i
$246$		0			0
$247$	−4.08688			−0.260042
$248$		0			0
$249$		0			0
$250$		0			0
$251$	21.3187	+	21.3187i	1.34562	+	1.34562i	0.890354	+	0.455269i	$0.150457\pi$
$251$	21.3187	+	21.3187i	1.34562	+	1.34562i	0.455269	+	0.890354i	$0.349543\pi$
$252$		0			0
$253$	−19.5098	+	19.5098i	−1.22657	+	1.22657i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	23.7762i		−	1.48312i	−0.670887	−	0.741560i	$-0.734086\pi$
$257$		−	23.7762i		−	1.48312i	0.670887	−	0.741560i	$-0.265914\pi$
$258$		0			0
$259$	4.45873	+	4.45873i	0.277052	+	0.277052i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4.83991i			0.298441i	0.988804	+	0.149221i	$0.0476765\pi$
$263$			4.83991i			0.298441i	−0.988804	+	0.149221i	$0.952324\pi$
$264$		0			0
$265$		−	10.6742i		−	0.655711i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	20.9861	+	20.9861i	1.27954	+	1.27954i	0.940924	+	0.338619i	$0.109960\pi$
$269$	20.9861	+	20.9861i	1.27954	+	1.27954i	0.338619	+	0.940924i	$0.390040\pi$
$270$		0			0
$271$			19.2132i			1.16712i	0.812071	+	0.583559i	$0.198340\pi$
$271$			19.2132i			1.16712i	−0.812071	+	0.583559i	$0.801660\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.08285	+	2.08285i	−0.125600	+	0.125600i
$276$		0			0
$277$	9.81607	+	9.81607i	0.589790	+	0.589790i	0.937575	−	0.347784i	$-0.113066\pi$
$277$	9.81607	+	9.81607i	0.589790	+	0.589790i	−0.347784	+	0.937575i	$0.613066\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.55344			0.510256			0.255128	−	0.966907i	$-0.417882\pi$
$281$	8.55344			0.510256			0.255128	+	0.966907i	$0.417882\pi$
$282$		0			0
$283$	−0.613273	+	0.613273i	−0.0364553	+	0.0364553i	−0.725099	−	0.688644i	$-0.758206\pi$
$283$	−0.613273	+	0.613273i	−0.0364553	+	0.0364553i	0.688644	+	0.725099i	$0.258206\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.85763			−0.109652
$288$		0			0
$289$	13.7211			0.807125
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.9327	−	15.9327i	0.930801	−	0.930801i	−0.0669549	−	0.997756i	$-0.521328\pi$
$293$	15.9327	−	15.9327i	0.930801	−	0.930801i	0.997756	+	0.0669549i	$0.0213284\pi$
$294$		0			0
$295$	0.668629			0.0389291
$296$		0			0
$297$		0			0
$298$		0			0
$299$	15.5976	+	15.5976i	0.902034	+	0.902034i
$300$		0			0
$301$	−3.90297	+	3.90297i	−0.224963	+	0.224963i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	7.05925i		−	0.404211i
$306$		0			0
$307$	−24.3353	−	24.3353i	−1.38889	−	1.38889i	−0.827657	−	0.561235i	$-0.810327\pi$
$307$	−24.3353	−	24.3353i	−1.38889	−	1.38889i	−0.561235	−	0.827657i	$-0.689673\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			7.54616i			0.427904i	0.976844	+	0.213952i	$0.0686336\pi$
$311$			7.54616i			0.427904i	−0.976844	+	0.213952i	$0.931366\pi$
$312$		0			0
$313$		−	24.9197i		−	1.40855i	−0.709929	−	0.704273i	$-0.751273\pi$
$313$		−	24.9197i		−	1.40855i	0.709929	−	0.704273i	$-0.248727\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.1374	−	12.1374i	−0.681706	−	0.681706i	0.278679	−	0.960384i	$-0.410104\pi$
$317$	−12.1374	−	12.1374i	−0.681706	−	0.681706i	−0.960384	+	0.278679i	$0.910104\pi$
$318$		0			0
$319$		−	20.1963i		−	1.13077i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.22209	+	2.22209i	−0.123640	+	0.123640i
$324$		0			0
$325$	1.66519	+	1.66519i	0.0923679	+	0.0923679i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.33021			−0.183600
$330$		0			0
$331$	−14.1985	+	14.1985i	−0.780418	+	0.780418i	−0.979901	−	0.199483i	$-0.936074\pi$
$331$	−14.1985	+	14.1985i	−0.780418	+	0.780418i	0.199483	+	0.979901i	$0.436074\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.75981			0.150785
$336$		0			0
$337$	32.2946			1.75920			0.879600	−	0.475714i	$-0.157810\pi$
$337$	32.2946			1.75920			0.879600	+	0.475714i	$0.157810\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10.3163	−	10.3163i	0.558656	−	0.558656i
$342$		0			0
$343$	−10.0833			−0.544445
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.0493	−	11.0493i	−0.593159	−	0.593159i	0.345324	−	0.938483i	$-0.387769\pi$
$347$	−11.0493	−	11.0493i	−0.593159	−	0.593159i	−0.938483	+	0.345324i	$0.887769\pi$
$348$		0			0
$349$	25.6937	−	25.6937i	1.37535	−	1.37535i	0.523045	−	0.852305i	$-0.324796\pi$
$349$	25.6937	−	25.6937i	1.37535	−	1.37535i	0.852305	−	0.523045i	$-0.175204\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			14.7815i			0.786740i	0.919380	+	0.393370i	$0.128691\pi$
$353$			14.7815i			0.786740i	−0.919380	+	0.393370i	$0.871309\pi$
$354$		0			0
$355$	0.296107	+	0.296107i	0.0157157	+	0.0157157i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	10.4972i		−	0.554023i	−0.960867	−	0.277012i	$-0.910656\pi$
$359$		−	10.4972i		−	0.554023i	0.960867	−	0.277012i	$-0.0893441\pi$
$360$		0			0
$361$		−	15.9882i		−	0.841483i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.47200	−	2.47200i	−0.129390	−	0.129390i
$366$		0			0
$367$		−	25.0959i		−	1.31000i	−0.755631	−	0.654998i	$-0.772669\pi$
$367$		−	25.0959i		−	1.31000i	0.755631	−	0.654998i	$-0.227331\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.66399	+	5.66399i	−0.294060	+	0.294060i
$372$		0			0
$373$	0.459192	+	0.459192i	0.0237761	+	0.0237761i	0.718895	−	0.695119i	$-0.244648\pi$
$373$	0.459192	+	0.459192i	0.0237761	+	0.0237761i	−0.695119	+	0.718895i	$0.744648\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−16.1464			−0.831584
$378$		0			0
$379$	−11.1426	+	11.1426i	−0.572356	+	0.572356i	−0.932786	−	0.360430i	$-0.882630\pi$
$379$	−11.1426	+	11.1426i	−0.572356	+	0.572356i	0.360430	+	0.932786i	$0.382630\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.51038			−0.485957			−0.242979	−	0.970032i	$-0.578125\pi$
$383$	−9.51038			−0.485957			−0.242979	+	0.970032i	$0.578125\pi$
$384$		0			0
$385$	2.21042			0.112653
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.53161	+	9.53161i	−0.483272	+	0.483272i	−0.906175	−	0.422903i	$-0.861011\pi$
$389$	−9.53161	+	9.53161i	−0.483272	+	0.483272i	0.422903	+	0.906175i	$0.361011\pi$
$390$		0			0
$391$	16.9613			0.857768
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.40930	−	2.40930i	−0.121225	−	0.121225i
$396$		0			0
$397$	12.3827	−	12.3827i	0.621469	−	0.621469i	−0.324438	−	0.945907i	$-0.605175\pi$
$397$	12.3827	−	12.3827i	0.621469	−	0.621469i	0.945907	+	0.324438i	$0.105175\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	20.2893i		−	1.01320i	−0.862182	−	0.506599i	$-0.830902\pi$
$401$		−	20.2893i		−	1.01320i	0.862182	−	0.506599i	$-0.169098\pi$
$402$		0			0
$403$	−8.24759	−	8.24759i	−0.410842	−	0.410842i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	24.7512i		−	1.22687i
$408$		0			0
$409$			26.4946i			1.31008i	0.755596	+	0.655038i	$0.227347\pi$
$409$			26.4946i			1.31008i	−0.755596	+	0.655038i	$0.772653\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.354791	−	0.354791i	−0.0174581	−	0.0174581i
$414$		0			0
$415$			0.776130i			0.0380987i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.89550	+	5.89550i	−0.288014	+	0.288014i	−0.836295	−	0.548280i	$-0.815283\pi$
$419$	−5.89550	+	5.89550i	−0.288014	+	0.288014i	0.548280	+	0.836295i	$0.315283\pi$
$420$		0			0
$421$	−17.4387	−	17.4387i	−0.849912	−	0.849912i	0.140210	−	0.990122i	$-0.455222\pi$
$421$	−17.4387	−	17.4387i	−0.849912	−	0.849912i	−0.990122	+	0.140210i	$0.955222\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.81077			0.0878351
$426$		0			0
$427$	−3.74581	+	3.74581i	−0.181272	+	0.181272i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14.3547			0.691442			0.345721	−	0.938337i	$-0.387634\pi$
$431$	14.3547			0.691442			0.345721	+	0.938337i	$0.387634\pi$
$432$		0			0
$433$	29.6967			1.42713			0.713566	−	0.700588i	$-0.247079\pi$
$433$	29.6967			1.42713			0.713566	+	0.700588i	$0.247079\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	11.4946	−	11.4946i	0.549863	−	0.549863i
$438$		0			0
$439$	−27.7682			−1.32530			−0.662651	−	0.748928i	$-0.730569\pi$
$439$	−27.7682			−1.32530			−0.662651	+	0.748928i	$0.730569\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.9336	+	15.9336i	0.757030	+	0.757030i	0.975781	−	0.218750i	$-0.0701981\pi$
$443$	15.9336	+	15.9336i	0.757030	+	0.757030i	−0.218750	+	0.975781i	$0.570198\pi$
$444$		0			0
$445$	8.65172	−	8.65172i	0.410131	−	0.410131i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			26.5910i			1.25491i	0.778653	+	0.627455i	$0.215903\pi$
$449$			26.5910i			1.25491i	−0.778653	+	0.627455i	$0.784097\pi$
$450$		0			0
$451$	5.15601	+	5.15601i	0.242787	+	0.242787i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	1.76718i		−	0.0828466i
$456$		0			0
$457$		−	37.4320i		−	1.75100i	−0.483221	−	0.875499i	$-0.660533\pi$
$457$		−	37.4320i		−	1.75100i	0.483221	−	0.875499i	$-0.339467\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.6767	+	15.6767i	0.730135	+	0.730135i	0.970646	−	0.240512i	$-0.0773152\pi$
$461$	15.6767	+	15.6767i	0.730135	+	0.730135i	−0.240512	+	0.970646i	$0.577315\pi$
$462$		0			0
$463$		−	8.33480i		−	0.387351i	−0.981066	−	0.193675i	$-0.937959\pi$
$463$		−	8.33480i		−	0.387351i	0.981066	−	0.193675i	$-0.0620409\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14.2346	−	14.2346i	0.658696	−	0.658696i	−0.296375	−	0.955072i	$-0.595778\pi$
$467$	14.2346	−	14.2346i	0.658696	−	0.658696i	0.955072	+	0.296375i	$0.0957779\pi$
$468$		0			0
$469$	−1.46443	−	1.46443i	−0.0676209	−	0.0676209i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21.6661			0.996207
$474$		0			0
$475$	1.22715	−	1.22715i	0.0563057	−	0.0563057i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	25.4854			1.16446			0.582228	−	0.813025i	$-0.302181\pi$
$479$	25.4854			1.16446			0.582228	+	0.813025i	$0.302181\pi$
$480$		0			0
$481$	−19.7880			−0.902254
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.70153	+	4.70153i	−0.213485	+	0.213485i
$486$		0			0
$487$	27.9781			1.26781			0.633905	−	0.773411i	$-0.281451\pi$
$487$	27.9781			1.26781			0.633905	+	0.773411i	$0.281451\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−14.3468	−	14.3468i	−0.647461	−	0.647461i	0.304917	−	0.952379i	$-0.401371\pi$
$491$	−14.3468	−	14.3468i	−0.647461	−	0.647461i	−0.952379	+	0.304917i	$0.901371\pi$
$492$		0			0
$493$	−8.77903	+	8.77903i	−0.395388	+	0.395388i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	0.314244i		−	0.0140958i
$498$		0			0
$499$	−13.7825	−	13.7825i	−0.616992	−	0.616992i	0.327767	−	0.944759i	$-0.393704\pi$
$499$	−13.7825	−	13.7825i	−0.616992	−	0.616992i	−0.944759	+	0.327767i	$0.893704\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			23.6008i			1.05231i	0.850390	+	0.526153i	$0.176366\pi$
$503$			23.6008i			1.05231i	−0.850390	+	0.526153i	$0.823634\pi$
$504$		0			0
$505$			12.0347i			0.535537i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	25.8588	+	25.8588i	1.14617	+	1.14617i	0.987299	+	0.158874i	$0.0507863\pi$
$509$	25.8588	+	25.8588i	1.14617	+	1.14617i	0.158874	+	0.987299i	$0.449214\pi$
$510$		0			0
$511$			2.62341i			0.116053i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.43864	+	9.43864i	−0.415916	+	0.415916i
$516$		0			0
$517$	9.24328	+	9.24328i	0.406519	+	0.406519i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.61959			0.0709555			0.0354778	−	0.999370i	$-0.488705\pi$
$521$	1.61959			0.0709555			0.0354778	+	0.999370i	$0.488705\pi$
$522$		0			0
$523$	12.2302	−	12.2302i	0.534787	−	0.534787i	−0.387206	−	0.921993i	$-0.626560\pi$
$523$	12.2302	−	12.2302i	0.534787	−	0.534787i	0.921993	+	0.387206i	$0.126560\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.96865			−0.390680
$528$		0			0
$529$	−64.7388			−2.81473
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.12211	−	4.12211i	0.178548	−	0.178548i
$534$		0			0
$535$	2.37200			0.102550
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13.4070	+	13.4070i	0.577482	+	0.577482i
$540$		0			0
$541$	−14.5843	+	14.5843i	−0.627026	+	0.627026i	−0.947319	−	0.320293i	$-0.896219\pi$
$541$	−14.5843	+	14.5843i	−0.627026	+	0.627026i	0.320293	+	0.947319i	$0.396219\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	15.2868i		−	0.654814i
$546$		0			0
$547$	5.88016	+	5.88016i	0.251418	+	0.251418i	0.821552	−	0.570134i	$-0.193109\pi$
$547$	5.88016	+	5.88016i	0.251418	+	0.251418i	−0.570134	+	0.821552i	$0.693109\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			11.8991i			0.506918i
$552$		0			0
$553$			2.55687i			0.108729i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.61435	−	4.61435i	−0.195516	−	0.195516i	0.602559	−	0.798075i	$-0.294148\pi$
$557$	−4.61435	−	4.61435i	−0.195516	−	0.195516i	−0.798075	+	0.602559i	$0.794148\pi$
$558$		0			0
$559$		−	17.3215i		−	0.732621i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.9201	−	21.9201i	0.923821	−	0.923821i	−0.0734759	−	0.997297i	$-0.523409\pi$
$563$	21.9201	−	21.9201i	0.923821	−	0.923821i	0.997297	+	0.0734759i	$0.0234092\pi$
$564$		0			0
$565$	−6.80648	−	6.80648i	−0.286351	−	0.286351i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10.4987			0.440128			0.220064	−	0.975485i	$-0.429373\pi$
$569$	10.4987			0.440128			0.220064	+	0.975485i	$0.429373\pi$
$570$		0			0
$571$	27.2368	−	27.2368i	1.13983	−	1.13983i	0.151344	−	0.988481i	$-0.451640\pi$
$571$	27.2368	−	27.2368i	1.13983	−	1.13983i	0.988481	−	0.151344i	$-0.0483603\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−9.36690			−0.390627
$576$		0			0
$577$	5.71248			0.237814			0.118907	−	0.992905i	$-0.462061\pi$
$577$	5.71248			0.237814			0.118907	+	0.992905i	$0.462061\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.411834	−	0.411834i	0.0170857	−	0.0170857i
$582$		0			0
$583$	31.4418			1.30219
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.81040	−	7.81040i	−0.322370	−	0.322370i	0.527306	−	0.849676i	$-0.323202\pi$
$587$	−7.81040	−	7.81040i	−0.322370	−	0.322370i	−0.849676	+	0.527306i	$0.823202\pi$
$588$		0			0
$589$	−6.07804	+	6.07804i	−0.250441	+	0.250441i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			5.56434i			0.228500i	0.993452	+	0.114250i	$0.0364465\pi$
$593$			5.56434i			0.228500i	−0.993452	+	0.114250i	$0.963554\pi$
$594$		0			0
$595$	−0.960838	−	0.960838i	−0.0393905	−	0.0393905i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	20.8570i		−	0.852195i	−0.904677	−	0.426098i	$-0.859888\pi$
$599$		−	20.8570i		−	0.852195i	0.904677	−	0.426098i	$-0.140112\pi$
$600$		0			0
$601$		−	45.8151i		−	1.86884i	−0.356178	−	0.934418i	$-0.615920\pi$
$601$		−	45.8151i		−	1.86884i	0.356178	−	0.934418i	$-0.384080\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.64295	+	1.64295i	0.0667956	+	0.0667956i
$606$		0			0
$607$		−	17.6991i		−	0.718383i	−0.933264	−	0.359191i	$-0.883053\pi$
$607$		−	17.6991i		−	0.718383i	0.933264	−	0.359191i	$-0.116947\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.38978	−	7.38978i	0.298959	−	0.298959i
$612$		0			0
$613$	−22.8661	−	22.8661i	−0.923552	−	0.923552i	0.0737263	−	0.997279i	$-0.476511\pi$
$613$	−22.8661	−	22.8661i	−0.923552	−	0.923552i	−0.997279	+	0.0737263i	$0.976511\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−17.0873			−0.687907			−0.343954	−	0.938987i	$-0.611766\pi$
$617$	−17.0873			−0.687907			−0.343954	+	0.938987i	$0.611766\pi$
$618$		0			0
$619$	−19.5907	+	19.5907i	−0.787419	+	0.787419i	−0.981070	−	0.193652i	$-0.937967\pi$
$619$	−19.5907	+	19.5907i	−0.787419	+	0.787419i	0.193652	+	0.981070i	$0.437967\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−9.18163			−0.367854
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−10.7590	+	10.7590i	−0.428989	+	0.428989i
$630$		0			0
$631$	3.65602			0.145544			0.0727719	−	0.997349i	$-0.476815\pi$
$631$	3.65602			0.145544			0.0727719	+	0.997349i	$0.476815\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.40436	−	1.40436i	−0.0557304	−	0.0557304i
$636$		0			0
$637$	10.7186	−	10.7186i	0.424686	−	0.424686i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			35.6775i			1.40918i	0.709617	+	0.704588i	$0.248868\pi$
$641$			35.6775i			1.40918i	−0.709617	+	0.704588i	$0.751132\pi$
$642$		0			0
$643$	−29.2912	−	29.2912i	−1.15513	−	1.15513i	−0.985509	−	0.169625i	$-0.945744\pi$
$643$	−29.2912	−	29.2912i	−1.15513	−	1.15513i	−0.169625	−	0.985509i	$-0.554256\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			11.4778i			0.451237i	0.974216	+	0.225619i	$0.0724403\pi$
$647$			11.4778i			0.451237i	−0.974216	+	0.225619i	$0.927560\pi$
$648$		0			0
$649$			1.96951i			0.0773100i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.49816	−	1.49816i	−0.0586275	−	0.0586275i	0.677185	−	0.735813i	$-0.263200\pi$
$653$	−1.49816	−	1.49816i	−0.0586275	−	0.0586275i	−0.735813	+	0.677185i	$0.763200\pi$
$654$		0			0
$655$		−	20.6585i		−	0.807194i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.88429	+	8.88429i	−0.346083	+	0.346083i	−0.858648	−	0.512565i	$-0.828695\pi$
$659$	−8.88429	+	8.88429i	−0.346083	+	0.346083i	0.512565	+	0.858648i	$0.328695\pi$
$660$		0			0
$661$	3.63603	+	3.63603i	0.141425	+	0.141425i	0.774275	−	0.632850i	$-0.218115\pi$
$661$	3.63603	+	3.63603i	0.141425	+	0.141425i	−0.632850	+	0.774275i	$0.718115\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.30232			−0.0505017
$666$		0			0
$667$	45.4129	−	45.4129i	1.75840	−	1.75840i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	20.7937			0.802730
$672$		0			0
$673$	21.2098			0.817577			0.408788	−	0.912629i	$-0.365951\pi$
$673$	21.2098			0.817577			0.408788	+	0.912629i	$0.365951\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−11.3920	+	11.3920i	−0.437831	+	0.437831i	−0.891282	−	0.453450i	$-0.850193\pi$
$677$	−11.3920	+	11.3920i	−0.437831	+	0.437831i	0.453450	+	0.891282i	$0.350193\pi$
$678$		0			0
$679$	4.98949			0.191479
$680$		0			0
$681$		0			0
$682$		0			0
$683$	13.9917	+	13.9917i	0.535379	+	0.535379i	0.922168	−	0.386789i	$-0.126416\pi$
$683$	13.9917	+	13.9917i	0.535379	+	0.535379i	−0.386789	+	0.922168i	$0.626416\pi$
$684$		0			0
$685$	12.3955	−	12.3955i	0.473606	−	0.473606i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	25.1370i		−	0.957643i
$690$		0			0
$691$	2.03799	+	2.03799i	0.0775288	+	0.0775288i	0.744808	−	0.667279i	$-0.232541\pi$
$691$	2.03799	+	2.03799i	0.0775288	+	0.0775288i	−0.667279	+	0.744808i	$0.732541\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	13.5815i		−	0.515176i
$696$		0			0
$697$		−	4.48249i		−	0.169786i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.05142	+	3.05142i	0.115251	+	0.115251i	0.762380	−	0.647130i	$-0.224031\pi$
$701$	3.05142	+	3.05142i	0.115251	+	0.115251i	−0.647130	+	0.762380i	$0.724031\pi$
$702$		0			0
$703$			14.5827i			0.549997i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.38591	−	6.38591i	0.240167	−	0.240167i
$708$		0			0
$709$	−5.26339	−	5.26339i	−0.197671	−	0.197671i	0.601330	−	0.799001i	$-0.294638\pi$
$709$	−5.26339	−	5.26339i	−0.197671	−	0.197671i	−0.799001	+	0.601330i	$0.794638\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	46.3938			1.73746
$714$		0			0
$715$	−4.90496	+	4.90496i	−0.183435	+	0.183435i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	47.8137			1.78315			0.891575	−	0.452873i	$-0.149601\pi$
$719$	47.8137			1.78315			0.891575	+	0.452873i	$0.149601\pi$
$720$		0			0
$721$	10.0167			0.373043
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.84824	−	4.84824i	0.180059	−	0.180059i
$726$		0			0
$727$	9.20719			0.341476			0.170738	−	0.985316i	$-0.445385\pi$
$727$	9.20719			0.341476			0.170738	+	0.985316i	$0.445385\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.41792	−	9.41792i	−0.348334	−	0.348334i
$732$		0			0
$733$	−20.8850	+	20.8850i	−0.771405	+	0.771405i	−0.978352	−	0.206947i	$-0.933647\pi$
$733$	−20.8850	+	20.8850i	−0.771405	+	0.771405i	0.206947	+	0.978352i	$0.433647\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			8.12929i			0.299446i
$738$		0			0
$739$	37.2338	+	37.2338i	1.36967	+	1.36967i	0.860910	+	0.508757i	$0.169895\pi$
$739$	37.2338	+	37.2338i	1.36967	+	1.36967i	0.508757	+	0.860910i	$0.330105\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			4.21038i			0.154464i	0.997013	+	0.0772320i	$0.0246082\pi$
$743$			4.21038i			0.154464i	−0.997013	+	0.0772320i	$0.975392\pi$
$744$		0			0
$745$			5.44970i			0.199662i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.25864	−	1.25864i	−0.0459898	−	0.0459898i
$750$		0			0
$751$		−	10.9180i		−	0.398403i	−0.979959	−	0.199202i	$-0.936165\pi$
$751$		−	10.9180i		−	0.398403i	0.979959	−	0.199202i	$-0.0638349\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.59273	+	8.59273i	−0.312721	+	0.312721i
$756$		0			0
$757$	−21.3455	−	21.3455i	−0.775814	−	0.775814i	0.203302	−	0.979116i	$-0.434833\pi$
$757$	−21.3455	−	21.3455i	−0.775814	−	0.775814i	−0.979116	+	0.203302i	$0.934833\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.21797			0.261651			0.130826	−	0.991405i	$-0.458237\pi$
$761$	7.21797			0.261651			0.130826	+	0.991405i	$0.458237\pi$
$762$		0			0
$763$	−8.11155	+	8.11155i	−0.293658	+	0.293658i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.57457			0.0568546
$768$		0			0
$769$	−38.8357			−1.40045			−0.700226	−	0.713921i	$-0.746918\pi$
$769$	−38.8357			−1.40045			−0.700226	+	0.713921i	$0.746918\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.91339	−	4.91339i	0.176722	−	0.176722i	−0.613203	−	0.789925i	$-0.710119\pi$
$773$	4.91339	−	4.91339i	0.176722	−	0.176722i	0.789925	+	0.613203i	$0.210119\pi$
$774$		0			0
$775$	4.95295			0.177915
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.03778	−	3.03778i	−0.108840	−	0.108840i
$780$		0			0
$781$	−0.872211	+	0.872211i	−0.0312102	+	0.0312102i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	4.94296i		−	0.176422i
$786$		0			0
$787$	17.6563	+	17.6563i	0.629379	+	0.629379i	0.947912	−	0.318533i	$-0.103190\pi$
$787$	17.6563	+	17.6563i	0.629379	+	0.629379i	−0.318533	+	0.947912i	$0.603190\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			7.22337i			0.256833i
$792$		0			0
$793$		−	16.6240i		−	0.590336i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−14.3307	−	14.3307i	−0.507618	−	0.507618i	0.406177	−	0.913795i	$-0.366862\pi$
$797$	−14.3307	−	14.3307i	−0.507618	−	0.507618i	−0.913795	+	0.406177i	$0.866862\pi$
$798$		0			0
$799$		−	8.03584i		−	0.284288i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.28150	−	7.28150i	0.256959	−	0.256959i
$804$		0			0
$805$	4.97031	+	4.97031i	0.175180	+	0.175180i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	39.4238			1.38607			0.693034	−	0.720905i	$-0.256274\pi$
$809$	39.4238			1.38607			0.693034	+	0.720905i	$0.256274\pi$
$810$		0			0
$811$	7.90652	−	7.90652i	0.277636	−	0.277636i	−0.554529	−	0.832164i	$-0.687102\pi$
$811$	7.90652	−	7.90652i	0.277636	−	0.277636i	0.832164	+	0.554529i	$0.187102\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.76681			0.307088
$816$		0			0
$817$	−12.7650			−0.446592
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4.52928	+	4.52928i	−0.158073	+	0.158073i	−0.781712	−	0.623639i	$-0.785653\pi$
$821$	−4.52928	+	4.52928i	−0.158073	+	0.158073i	0.623639	+	0.781712i	$0.285653\pi$
$822$		0			0
$823$	−34.2257			−1.19303			−0.596516	−	0.802601i	$-0.703449\pi$
$823$	−34.2257			−1.19303			−0.596516	+	0.802601i	$0.703449\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−30.4656	−	30.4656i	−1.05939	−	1.05939i	−0.998121	−	0.0612712i	$-0.980485\pi$
$827$	−30.4656	−	30.4656i	−1.05939	−	1.05939i	−0.0612712	−	0.998121i	$-0.519515\pi$
$828$		0			0
$829$	16.5777	−	16.5777i	0.575767	−	0.575767i	−0.357967	−	0.933734i	$-0.616530\pi$
$829$	16.5777	−	16.5777i	0.575767	−	0.575767i	0.933734	+	0.357967i	$0.116530\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	11.6557i		−	0.403845i
$834$		0			0
$835$	16.9513	+	16.9513i	0.586624	+	0.586624i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			35.8895i			1.23904i	0.784979	+	0.619522i	$0.212673\pi$
$839$			35.8895i			1.23904i	−0.784979	+	0.619522i	$0.787327\pi$
$840$		0			0
$841$			18.0108i			0.621062i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.27099	−	5.27099i	−0.181328	−	0.181328i
$846$		0			0
$847$		−	1.74358i		−	0.0599102i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	55.6550	−	55.6550i	1.90783	−	1.90783i
$852$		0			0
$853$	24.1542	+	24.1542i	0.827024	+	0.827024i	0.987104	−	0.160080i	$-0.0511752\pi$
$853$	24.1542	+	24.1542i	0.827024	+	0.827024i	−0.160080	+	0.987104i	$0.551175\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5.33820			−0.182349			−0.0911747	−	0.995835i	$-0.529062\pi$
$857$	−5.33820			−0.182349			−0.0911747	+	0.995835i	$0.529062\pi$
$858$		0			0
$859$	22.3520	−	22.3520i	0.762641	−	0.762641i	−0.214158	−	0.976799i	$-0.568701\pi$
$859$	22.3520	−	22.3520i	0.762641	−	0.762641i	0.976799	+	0.214158i	$0.0687007\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−22.5742			−0.768435			−0.384217	−	0.923243i	$-0.625529\pi$
$863$	−22.5742			−0.768435			−0.384217	+	0.923243i	$0.625529\pi$
$864$		0			0
$865$	−6.64896			−0.226072
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.09681	−	7.09681i	0.240743	−	0.240743i
$870$		0			0
$871$	6.49917			0.220216
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.530625	+	0.530625i	0.0179384	+	0.0179384i
$876$		0			0
$877$	26.5772	−	26.5772i	0.897447	−	0.897447i	−0.0977631	−	0.995210i	$-0.531169\pi$
$877$	26.5772	−	26.5772i	0.897447	−	0.897447i	0.995210	+	0.0977631i	$0.0311687\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			49.9704i			1.68355i	0.539831	+	0.841773i	$0.318488\pi$
$881$			49.9704i			1.68355i	−0.539831	+	0.841773i	$0.681512\pi$
$882$		0			0
$883$	−33.9156	−	33.9156i	−1.14135	−	1.14135i	−0.988203	−	0.153147i	$-0.951059\pi$
$883$	−33.9156	−	33.9156i	−1.14135	−	1.14135i	−0.153147	−	0.988203i	$-0.548941\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	44.9975i		−	1.51087i	−0.655226	−	0.755433i	$-0.727426\pi$
$887$		−	44.9975i		−	1.51087i	0.655226	−	0.755433i	$-0.272574\pi$
$888$		0			0
$889$			1.49038i			0.0499857i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5.44588	−	5.44588i	−0.182239	−	0.182239i
$894$		0			0
$895$		−	15.5211i		−	0.518813i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24.0131	+	24.0131i	−0.800882	+	0.800882i
$900$		0			0
$901$	−13.6673	−	13.6673i	−0.455324	−	0.455324i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.41347			0.0469855
$906$		0			0
$907$	5.29252	−	5.29252i	0.175735	−	0.175735i	−0.613759	−	0.789494i	$-0.710343\pi$
$907$	5.29252	−	5.29252i	0.175735	−	0.175735i	0.789494	+	0.613759i	$0.210343\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−26.4816			−0.877373			−0.438687	−	0.898640i	$-0.644556\pi$
$911$	−26.4816			−0.877373			−0.438687	+	0.898640i	$0.644556\pi$
$912$		0			0
$913$	−2.28616			−0.0756609
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.9619	+	10.9619i	−0.361994	+	0.361994i
$918$		0			0
$919$	−46.9764			−1.54961			−0.774805	−	0.632200i	$-0.782152\pi$
$919$	−46.9764			−1.54961			−0.774805	+	0.632200i	$0.782152\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.697311	+	0.697311i	0.0229523	+	0.0229523i
$924$		0			0
$925$	5.94167	−	5.94167i	0.195361	−	0.195361i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	5.52346i		−	0.181219i	−0.995887	−	0.0906095i	$-0.971118\pi$
$929$		−	5.52346i		−	0.181219i	0.995887	−	0.0906095i	$-0.0288815\pi$
$930$		0			0
$931$	−7.89904	−	7.89904i	−0.258881	−	0.258881i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			5.33378i			0.174433i
$936$		0			0
$937$			19.8409i			0.648174i	0.946027	+	0.324087i	$0.105057\pi$
$937$			19.8409i			0.648174i	−0.946027	+	0.324087i	$0.894943\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−32.5548	−	32.5548i	−1.06125	−	1.06125i	−0.997997	−	0.0632574i	$-0.979851\pi$
$941$	−32.5548	−	32.5548i	−1.06125	−	1.06125i	−0.0632574	−	0.997997i	$-0.520149\pi$
$942$		0			0
$943$			23.1874i			0.755086i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33.4414	−	33.4414i	1.08670	−	1.08670i	0.0908334	−	0.995866i	$-0.471047\pi$
$947$	33.4414	−	33.4414i	1.08670	−	1.08670i	0.995866	−	0.0908334i	$-0.0289531\pi$
$948$		0			0
$949$	−5.82139	−	5.82139i	−0.188970	−	0.188970i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.81474			0.253144			0.126572	−	0.991957i	$-0.459602\pi$
$953$	7.81474			0.253144			0.126572	+	0.991957i	$0.459602\pi$
$954$		0			0
$955$	14.5782	−	14.5782i	0.471738	−	0.471738i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−13.1547			−0.424787
$960$		0			0
$961$	6.46825			0.208653
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.86966	−	2.86966i	0.0923776	−	0.0923776i
$966$		0			0
$967$	6.45064			0.207439			0.103719	−	0.994607i	$-0.466926\pi$
$967$	6.45064			0.207439			0.103719	+	0.994607i	$0.466926\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	16.4783	+	16.4783i	0.528813	+	0.528813i	0.920218	−	0.391405i	$-0.128011\pi$
$971$	16.4783	+	16.4783i	0.528813	+	0.528813i	−0.391405	+	0.920218i	$0.628011\pi$
$972$		0			0
$973$	−7.20668	+	7.20668i	−0.231036	+	0.231036i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			6.49980i			0.207947i	0.994580	+	0.103974i	$0.0331557\pi$
$977$			6.49980i			0.207947i	−0.994580	+	0.103974i	$0.966844\pi$
$978$		0			0
$979$	25.4844	+	25.4844i	0.814486	+	0.814486i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			39.3984i			1.25661i	0.777966	+	0.628306i	$0.216251\pi$
$983$			39.3984i			1.25661i	−0.777966	+	0.628306i	$0.783749\pi$
$984$		0			0
$985$		−	5.95025i		−	0.189591i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	48.7179	+	48.7179i	1.54914	+	1.54914i
$990$		0			0
$991$			40.0421i			1.27198i	0.771698	+	0.635989i	$0.219408\pi$
$991$			40.0421i			1.27198i	−0.771698	+	0.635989i	$0.780592\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.80643	−	5.80643i	0.184076	−	0.184076i
$996$		0			0
$997$	−11.8009	−	11.8009i	−0.373737	−	0.373737i	0.495099	−	0.868836i	$-0.335132\pi$
$997$	−11.8009	−	11.8009i	−0.373737	−	0.373737i	−0.868836	+	0.495099i	$0.835132\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bl.b.1871.2		24
3.2	odd	2	inner	2880.2.bl.b.1871.11		24
4.3	odd	2		720.2.bl.b.251.7	yes	24
12.11	even	2		720.2.bl.b.251.6	✓	24
16.3	odd	4	inner	2880.2.bl.b.431.11		24
16.13	even	4		720.2.bl.b.611.6	yes	24
48.29	odd	4		720.2.bl.b.611.7	yes	24
48.35	even	4	inner	2880.2.bl.b.431.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.b.251.6	✓	24	12.11	even	2
720.2.bl.b.251.7	yes	24	4.3	odd	2
720.2.bl.b.611.6	yes	24	16.13	even	4
720.2.bl.b.611.7	yes	24	48.29	odd	4
2880.2.bl.b.431.2		24	48.35	even	4	inner
2880.2.bl.b.431.11		24	16.3	odd	4	inner
2880.2.bl.b.1871.2		24	1.1	even	1	trivial
2880.2.bl.b.1871.11		24	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{5} +0.750417 q^{7} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{5} +0.750417 q^{7} +(-2.08285 - 2.08285i) q^{11} +(-1.66519 + 1.66519i) q^{13} +1.81077i q^{17} +(1.22715 + 1.22715i) q^{19} -9.36690i q^{23} -1.00000i q^{25} +(4.84824 + 4.84824i) q^{29} +4.95295i q^{31} +(-0.530625 + 0.530625i) q^{35} +(5.94167 + 5.94167i) q^{37} -2.47546 q^{41} +(-5.20107 + 5.20107i) q^{43} -4.43781 q^{47} -6.43687 q^{49} +(-7.54780 + 7.54780i) q^{53} +2.94559 q^{55} +(-0.472792 - 0.472792i) q^{59} +(-4.99164 + 4.99164i) q^{61} -2.35493i q^{65} +(-1.95148 - 1.95148i) q^{67} -0.418759i q^{71} +3.49594i q^{73} +(-1.56300 - 1.56300i) q^{77} +3.40726i q^{79} +(0.548807 - 0.548807i) q^{83} +(-1.28041 - 1.28041i) q^{85} -12.2354 q^{89} +(-1.24958 + 1.24958i) q^{91} -1.73546 q^{95} +6.64896 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8 q^{7}+O(q^{10}) \) 24 * q + 8 * q^7	\( 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) \) 24 * q + 8 * q^7 + 24 * q^19 + 8 * q^37 - 48 * q^43 + 24 * q^49 - 24 * q^55 + 40 * q^61 + 40 * q^67 + 24 * q^85 - 40 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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