LMFDB - Embedded newform 2880.2.t.c.721.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(721,2880)]

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([0, 3, 0, 0]))

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

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

chi := DirichletCharacter("2880.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.t (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:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 $ x^16 - 4x^15 + 4x^14 + 7x^12 - 8x^11 - 28x^10 + 28x^9 + 17x^8 + 56x^7 - 112x^6 - 64x^5 + 112x^4 + 256x^2 - 512*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			721.2
Root			$1.38652 - 0.278517i$ of defining polynomial
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.c.2161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{5} -0.982011i q^{7} +O(q^{10})$	$q+(-0.707107 + 0.707107i) q^{5} -0.982011i q^{7} +(-1.62645 + 1.62645i) q^{11} +(-0.690562 - 0.690562i) q^{13} +2.19577 q^{17} +(-1.92659 - 1.92659i) q^{19} -2.01442i q^{23} -1.00000i q^{25} +(5.27182 + 5.27182i) q^{29} -0.435286 q^{31} +(0.694387 + 0.694387i) q^{35} +(-5.79805 + 5.79805i) q^{37} +3.93139i q^{41} +(0.507592 - 0.507592i) q^{43} -9.21960 q^{47} +6.03565 q^{49} +(-6.29357 + 6.29357i) q^{53} -2.30015i q^{55} +(-5.67778 + 5.67778i) q^{59} +(-3.60301 - 3.60301i) q^{61} +0.976603 q^{65} +(-4.53563 - 4.53563i) q^{67} -10.3984i q^{71} +9.24439i q^{73} +(1.59719 + 1.59719i) q^{77} -15.4493 q^{79} +(-0.683244 - 0.683244i) q^{83} +(-1.55264 + 1.55264i) q^{85} +5.44401i q^{89} +(-0.678140 + 0.678140i) q^{91} +2.72461 q^{95} +5.54540 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) $ 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

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
$7$		−	0.982011i		−	0.371165i	−0.982629	−	0.185583i	$-0.940583\pi$
$7$		−	0.982011i		−	0.371165i	0.982629	−	0.185583i	$-0.0594172\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.62645	+	1.62645i	−0.490393	+	0.490393i	−0.908430	−	0.418037i	$-0.862718\pi$
$11$	−1.62645	+	1.62645i	−0.490393	+	0.490393i	0.418037	+	0.908430i	$0.362718\pi$
$12$		0			0
$13$	−0.690562	−	0.690562i	−0.191528	−	0.191528i	0.604828	−	0.796356i	$-0.293242\pi$
$13$	−0.690562	−	0.690562i	−0.191528	−	0.191528i	−0.796356	+	0.604828i	$0.793242\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.19577			0.532552			0.266276	−	0.963897i	$-0.414207\pi$
$17$	2.19577			0.532552			0.266276	+	0.963897i	$0.414207\pi$
$18$		0			0
$19$	−1.92659	−	1.92659i	−0.441991	−	0.441991i	0.450690	−	0.892681i	$-0.351178\pi$
$19$	−1.92659	−	1.92659i	−0.441991	−	0.441991i	−0.892681	+	0.450690i	$0.851178\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.01442i		−	0.420035i	−0.977698	−	0.210018i	$-0.932648\pi$
$23$		−	2.01442i		−	0.420035i	0.977698	−	0.210018i	$-0.0673522\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.27182	+	5.27182i	0.978952	+	0.978952i	0.999783	−	0.0208314i	$-0.00663132\pi$
$29$	5.27182	+	5.27182i	0.978952	+	0.978952i	−0.0208314	+	0.999783i	$0.506631\pi$
$30$		0			0
$31$	−0.435286			−0.0781797			−0.0390898	−	0.999236i	$-0.512446\pi$
$31$	−0.435286			−0.0781797			−0.0390898	+	0.999236i	$0.512446\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.694387	+	0.694387i	0.117373	+	0.117373i
$36$		0			0
$37$	−5.79805	+	5.79805i	−0.953194	+	0.953194i	−0.998953	−	0.0457583i	$-0.985430\pi$
$37$	−5.79805	+	5.79805i	−0.953194	+	0.953194i	0.0457583	+	0.998953i	$0.485430\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.93139i			0.613980i	0.951713	+	0.306990i	$0.0993218\pi$
$41$			3.93139i			0.613980i	−0.951713	+	0.306990i	$0.900678\pi$
$42$		0			0
$43$	0.507592	−	0.507592i	0.0774071	−	0.0774071i	−0.667343	−	0.744750i	$-0.732569\pi$
$43$	0.507592	−	0.507592i	0.0774071	−	0.0774071i	0.744750	+	0.667343i	$0.232569\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.21960			−1.34482			−0.672409	−	0.740180i	$-0.734740\pi$
$47$	−9.21960			−1.34482			−0.672409	+	0.740180i	$0.734740\pi$
$48$		0			0
$49$	6.03565			0.862236
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.29357	+	6.29357i	−0.864488	+	0.864488i	−0.991856	−	0.127367i	$-0.959347\pi$
$53$	−6.29357	+	6.29357i	−0.864488	+	0.864488i	0.127367	+	0.991856i	$0.459347\pi$
$54$		0			0
$55$		−	2.30015i		−	0.310152i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.67778	+	5.67778i	−0.739183	+	0.739183i	−0.972420	−	0.233237i	$-0.925068\pi$
$59$	−5.67778	+	5.67778i	−0.739183	+	0.739183i	0.233237	+	0.972420i	$0.425068\pi$
$60$		0			0
$61$	−3.60301	−	3.60301i	−0.461318	−	0.461318i	0.437770	−	0.899087i	$-0.355769\pi$
$61$	−3.60301	−	3.60301i	−0.461318	−	0.461318i	−0.899087	+	0.437770i	$0.855769\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.976603			0.121133
$66$		0			0
$67$	−4.53563	−	4.53563i	−0.554116	−	0.554116i	0.373510	−	0.927626i	$-0.378154\pi$
$67$	−4.53563	−	4.53563i	−0.554116	−	0.554116i	−0.927626	+	0.373510i	$0.878154\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3984i		−	1.23407i	−0.786937	−	0.617033i	$-0.788335\pi$
$71$		−	10.3984i		−	1.23407i	0.786937	−	0.617033i	$-0.211665\pi$
$72$		0			0
$73$			9.24439i			1.08197i	0.841031	+	0.540987i	$0.181949\pi$
$73$			9.24439i			1.08197i	−0.841031	+	0.540987i	$0.818051\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.59719	+	1.59719i	0.182017	+	0.182017i
$78$		0			0
$79$	−15.4493			−1.73818			−0.869091	−	0.494653i	$-0.835295\pi$
$79$	−15.4493			−1.73818			−0.869091	+	0.494653i	$0.835295\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.683244	−	0.683244i	−0.0749957	−	0.0749957i	0.668614	−	0.743610i	$-0.266888\pi$
$83$	−0.683244	−	0.683244i	−0.0749957	−	0.0749957i	−0.743610	+	0.668614i	$0.766888\pi$
$84$		0			0
$85$	−1.55264	+	1.55264i	−0.168408	+	0.168408i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			5.44401i			0.577064i	0.957470	+	0.288532i	$0.0931672\pi$
$89$			5.44401i			0.577064i	−0.957470	+	0.288532i	$0.906833\pi$
$90$		0			0
$91$	−0.678140	+	0.678140i	−0.0710884	+	0.0710884i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.72461			0.279540
$96$		0			0
$97$	5.54540			0.563050			0.281525	−	0.959554i	$-0.409160\pi$
$97$	5.54540			0.563050			0.281525	+	0.959554i	$0.409160\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.291294	−	0.291294i	0.0289848	−	0.0289848i	−0.692466	−	0.721451i	$-0.743476\pi$
$101$	0.291294	−	0.291294i	0.0289848	−	0.0289848i	0.721451	+	0.692466i	$0.243476\pi$
$102$		0			0
$103$			4.50219i			0.443614i	0.975091	+	0.221807i	$0.0711955\pi$
$103$			4.50219i			0.443614i	−0.975091	+	0.221807i	$0.928805\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.49890	+	6.49890i	−0.628272	+	0.628272i	−0.947633	−	0.319361i	$-0.896532\pi$
$107$	−6.49890	+	6.49890i	−0.628272	+	0.628272i	0.319361	+	0.947633i	$0.396532\pi$
$108$		0			0
$109$	−2.51950	−	2.51950i	−0.241324	−	0.241324i	0.576074	−	0.817398i	$-0.304584\pi$
$109$	−2.51950	−	2.51950i	−0.241324	−	0.241324i	−0.817398	+	0.576074i	$0.804584\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.38101			−0.506203			−0.253102	−	0.967440i	$-0.581451\pi$
$113$	−5.38101			−0.506203			−0.253102	+	0.967440i	$0.581451\pi$
$114$		0			0
$115$	1.42441	+	1.42441i	0.132827	+	0.132827i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	2.15627i		−	0.197665i
$120$		0			0
$121$			5.70933i			0.519030i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$	−4.86578			−0.431768			−0.215884	−	0.976419i	$-0.569263\pi$
$127$	−4.86578			−0.431768			−0.215884	+	0.976419i	$0.569263\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.00581	+	8.00581i	0.699471	+	0.699471i	0.964296	−	0.264825i	$-0.0853143\pi$
$131$	8.00581	+	8.00581i	0.699471	+	0.699471i	−0.264825	+	0.964296i	$0.585314\pi$
$132$		0			0
$133$	−1.89194	+	1.89194i	−0.164052	+	0.164052i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	13.5567i		−	1.15822i	−0.815248	−	0.579112i	$-0.803399\pi$
$137$		−	13.5567i		−	1.15822i	0.815248	−	0.579112i	$-0.196601\pi$
$138$		0			0
$139$	−8.22645	+	8.22645i	−0.697758	+	0.697758i	−0.963927	−	0.266168i	$-0.914242\pi$
$139$	−8.22645	+	8.22645i	−0.697758	+	0.697758i	0.266168	+	0.963927i	$0.414242\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.24633			0.187847
$144$		0			0
$145$	−7.45547			−0.619143
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.6363	+	12.6363i	−1.03521	+	1.03521i	−0.0358519	+	0.999357i	$0.511414\pi$
$149$	−12.6363	+	12.6363i	−1.03521	+	1.03521i	−0.999357	+	0.0358519i	$0.988586\pi$
$150$		0			0
$151$			15.1562i			1.23339i	0.787201	+	0.616696i	$0.211529\pi$
$151$			15.1562i			1.23339i	−0.787201	+	0.616696i	$0.788471\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.307794	−	0.307794i	0.0247226	−	0.0247226i
$156$		0			0
$157$	−1.75816	−	1.75816i	−0.140316	−	0.140316i	0.633460	−	0.773776i	$-0.281634\pi$
$157$	−1.75816	−	1.75816i	−0.140316	−	0.140316i	−0.773776	+	0.633460i	$0.781634\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.97818			−0.155903
$162$		0			0
$163$	13.9102	+	13.9102i	1.08953	+	1.08953i	0.995576	+	0.0939562i	$0.0299514\pi$
$163$	13.9102	+	13.9102i	1.08953	+	1.08953i	0.0939562	+	0.995576i	$0.470049\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18.8620i		−	1.45958i	−0.683669	−	0.729792i	$-0.739617\pi$
$167$		−	18.8620i		−	1.45958i	0.683669	−	0.729792i	$-0.260383\pi$
$168$		0			0
$169$		−	12.0462i		−	0.926634i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−16.0724	−	16.0724i	−1.22196	−	1.22196i	−0.966933	−	0.255031i	$-0.917914\pi$
$173$	−16.0724	−	16.0724i	−1.22196	−	1.22196i	−0.255031	−	0.966933i	$-0.582086\pi$
$174$		0			0
$175$	−0.982011			−0.0742331
$176$		0			0
$177$		0			0
$178$		0			0
$179$	16.4341	+	16.4341i	1.22834	+	1.22834i	0.964591	+	0.263749i	$0.0849591\pi$
$179$	16.4341	+	16.4341i	1.22834	+	1.22834i	0.263749	+	0.964591i	$0.415041\pi$
$180$		0			0
$181$	−15.4539	+	15.4539i	−1.14868	+	1.14868i	−0.161870	+	0.986812i	$0.551753\pi$
$181$	−15.4539	+	15.4539i	−1.14868	+	1.14868i	−0.986812	+	0.161870i	$0.948247\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	8.19969i		−	0.602853i
$186$		0			0
$187$	−3.57130	+	3.57130i	−0.261160	+	0.261160i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−14.7872			−1.06997			−0.534983	−	0.844863i	$-0.679682\pi$
$191$	−14.7872			−1.06997			−0.534983	+	0.844863i	$0.679682\pi$
$192$		0			0
$193$	−11.2912			−0.812758			−0.406379	−	0.913705i	$-0.633209\pi$
$193$	−11.2912			−0.812758			−0.406379	+	0.913705i	$0.633209\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.6152	−	10.6152i	0.756302	−	0.756302i	−0.219345	−	0.975647i	$-0.570392\pi$
$197$	10.6152	−	10.6152i	0.756302	−	0.756302i	0.975647	+	0.219345i	$0.0703920\pi$
$198$		0			0
$199$		−	4.68789i		−	0.332316i	−0.986099	−	0.166158i	$-0.946864\pi$
$199$		−	4.68789i		−	0.332316i	0.986099	−	0.166158i	$-0.0531361\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.17698	−	5.17698i	0.363353	−	0.363353i
$204$		0			0
$205$	−2.77991	−	2.77991i	−0.194157	−	0.194157i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.26701			0.433498
$210$		0			0
$211$	2.63215	+	2.63215i	0.181205	+	0.181205i	0.791881	−	0.610676i	$-0.209102\pi$
$211$	2.63215	+	2.63215i	0.181205	+	0.181205i	−0.610676	+	0.791881i	$0.709102\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			0.717844i			0.0489565i
$216$		0			0
$217$			0.427456i			0.0290176i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.51632	−	1.51632i	−0.101998	−	0.101998i
$222$		0			0
$223$	−3.45644			−0.231461			−0.115730	−	0.993281i	$-0.536921\pi$
$223$	−3.45644			−0.231461			−0.115730	+	0.993281i	$0.536921\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.74550	−	4.74550i	−0.314970	−	0.314970i	0.531862	−	0.846831i	$-0.321493\pi$
$227$	−4.74550	−	4.74550i	−0.314970	−	0.314970i	−0.846831	+	0.531862i	$0.821493\pi$
$228$		0			0
$229$	−13.3576	+	13.3576i	−0.882697	+	0.882697i	−0.993808	−	0.111111i	$-0.964559\pi$
$229$	−13.3576	+	13.3576i	−0.882697	+	0.882697i	0.111111	+	0.993808i	$0.464559\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4.82691i		−	0.316222i	−0.987421	−	0.158111i	$-0.949460\pi$
$233$		−	4.82691i		−	0.316222i	0.987421	−	0.158111i	$-0.0505403\pi$
$234$		0			0
$235$	6.51924	−	6.51924i	0.425269	−	0.425269i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.82497			−0.570840			−0.285420	−	0.958403i	$-0.592133\pi$
$239$	−8.82497			−0.570840			−0.285420	+	0.958403i	$0.592133\pi$
$240$		0			0
$241$	−3.74147			−0.241009			−0.120504	−	0.992713i	$-0.538451\pi$
$241$	−3.74147			−0.241009			−0.120504	+	0.992713i	$0.538451\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.26785	+	4.26785i	−0.272663	+	0.272663i
$246$		0			0
$247$			2.66087i			0.169307i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.99322	+	5.99322i	−0.378289	+	0.378289i	−0.870484	−	0.492196i	$-0.836194\pi$
$251$	−5.99322	+	5.99322i	−0.378289	+	0.378289i	0.492196	+	0.870484i	$0.336194\pi$
$252$		0			0
$253$	3.27635	+	3.27635i	0.205982	+	0.205982i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.7662			0.921091			0.460545	−	0.887636i	$-0.347654\pi$
$257$	14.7662			0.921091			0.460545	+	0.887636i	$0.347654\pi$
$258$		0			0
$259$	5.69375	+	5.69375i	0.353793	+	0.353793i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			6.79486i			0.418989i	0.977810	+	0.209494i	$0.0671818\pi$
$263$			6.79486i			0.418989i	−0.977810	+	0.209494i	$0.932818\pi$
$264$		0			0
$265$		−	8.90045i		−	0.546750i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.03990	+	6.03990i	0.368259	+	0.368259i	0.866842	−	0.498583i	$-0.166146\pi$
$269$	6.03990	+	6.03990i	0.368259	+	0.368259i	−0.498583	+	0.866842i	$0.666146\pi$
$270$		0			0
$271$	24.6221			1.49568			0.747842	−	0.663877i	$-0.231090\pi$
$271$	24.6221			1.49568			0.747842	+	0.663877i	$0.231090\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.62645	+	1.62645i	0.0980785	+	0.0980785i
$276$		0			0
$277$	9.98018	−	9.98018i	0.599651	−	0.599651i	−0.340569	−	0.940220i	$-0.610620\pi$
$277$	9.98018	−	9.98018i	0.599651	−	0.599651i	0.940220	+	0.340569i	$0.110620\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	14.4611i		−	0.862675i	−0.902191	−	0.431337i	$-0.858042\pi$
$281$		−	14.4611i		−	0.862675i	0.902191	−	0.431337i	$-0.141958\pi$
$282$		0			0
$283$	−20.0783	+	20.0783i	−1.19353	+	1.19353i	−0.217462	+	0.976069i	$0.569778\pi$
$283$	−20.0783	+	20.0783i	−1.19353	+	1.19353i	−0.976069	+	0.217462i	$0.930222\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.86067			0.227888
$288$		0			0
$289$	−12.1786			−0.716388
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.4038	+	15.4038i	−0.899899	+	0.899899i	−0.995427	−	0.0955279i	$-0.969546\pi$
$293$	−15.4038	+	15.4038i	−0.899899	+	0.899899i	0.0955279	+	0.995427i	$0.469546\pi$
$294$		0			0
$295$		−	8.02959i		−	0.467501i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.39108	+	1.39108i	−0.0804484	+	0.0804484i
$300$		0			0
$301$	−0.498461	−	0.498461i	−0.0287308	−	0.0287308i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.09542			0.291763
$306$		0			0
$307$	−9.12398	−	9.12398i	−0.520733	−	0.520733i	0.397060	−	0.917793i	$-0.370031\pi$
$307$	−9.12398	−	9.12398i	−0.520733	−	0.520733i	−0.917793	+	0.397060i	$0.870031\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			0.642911i			0.0364561i	0.999834	+	0.0182281i	$0.00580249\pi$
$311$			0.642911i			0.0364561i	−0.999834	+	0.0182281i	$0.994198\pi$
$312$		0			0
$313$			21.3775i			1.20833i	0.796860	+	0.604164i	$0.206493\pi$
$313$			21.3775i			1.20833i	−0.796860	+	0.604164i	$0.793507\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.66200	−	8.66200i	−0.486507	−	0.486507i	0.420695	−	0.907202i	$-0.361786\pi$
$317$	−8.66200	−	8.66200i	−0.486507	−	0.486507i	−0.907202	+	0.420695i	$0.861786\pi$
$318$		0			0
$319$	−17.1487			−0.960141
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.23035	−	4.23035i	−0.235383	−	0.235383i
$324$		0			0
$325$	−0.690562	+	0.690562i	−0.0383055	+	0.0383055i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			9.05375i			0.499150i
$330$		0			0
$331$	8.43941	−	8.43941i	0.463872	−	0.463872i	−0.436050	−	0.899922i	$-0.643623\pi$
$331$	8.43941	−	8.43941i	0.463872	−	0.463872i	0.899922	+	0.436050i	$0.143623\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.41435			0.350454
$336$		0			0
$337$	30.7047			1.67259			0.836295	−	0.548280i	$-0.184717\pi$
$337$	30.7047			1.67259			0.836295	+	0.548280i	$0.184717\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.707970	−	0.707970i	0.0383387	−	0.0383387i
$342$		0			0
$343$		−	12.8012i		−	0.691197i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	13.6418	−	13.6418i	0.732329	−	0.732329i	−0.238752	−	0.971081i	$-0.576738\pi$
$347$	13.6418	−	13.6418i	0.732329	−	0.732329i	0.971081	+	0.238752i	$0.0767383\pi$
$348$		0			0
$349$	9.97321	+	9.97321i	0.533854	+	0.533854i	0.921717	−	0.387863i	$-0.126787\pi$
$349$	9.97321	+	9.97321i	0.533854	+	0.533854i	−0.387863	+	0.921717i	$0.626787\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	26.7843			1.42559			0.712793	−	0.701374i	$-0.247430\pi$
$353$	26.7843			1.42559			0.712793	+	0.701374i	$0.247430\pi$
$354$		0			0
$355$	7.35280	+	7.35280i	0.390246	+	0.390246i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			19.1190i			1.00906i	0.863393	+	0.504532i	$0.168335\pi$
$359$			19.1190i			1.00906i	−0.863393	+	0.504532i	$0.831665\pi$
$360$		0			0
$361$		−	11.5765i		−	0.609288i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.53677	−	6.53677i	−0.342150	−	0.342150i
$366$		0			0
$367$	−4.24385			−0.221527			−0.110764	−	0.993847i	$-0.535330\pi$
$367$	−4.24385			−0.221527			−0.110764	+	0.993847i	$0.535330\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.18035	+	6.18035i	0.320868	+	0.320868i
$372$		0			0
$373$	23.9514	−	23.9514i	1.24016	−	1.24016i	0.280221	−	0.959935i	$-0.409592\pi$
$373$	23.9514	−	23.9514i	1.24016	−	1.24016i	0.959935	−	0.280221i	$-0.0904078\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	7.28104i		−	0.374992i
$378$		0			0
$379$	7.45685	−	7.45685i	0.383033	−	0.383033i	−0.489161	−	0.872194i	$-0.662697\pi$
$379$	7.45685	−	7.45685i	0.383033	−	0.383033i	0.872194	+	0.489161i	$0.162697\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.19667			0.265538			0.132769	−	0.991147i	$-0.457613\pi$
$383$	5.19667			0.265538			0.132769	+	0.991147i	$0.457613\pi$
$384$		0			0
$385$	−2.25877			−0.115117
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.3846	+	10.3846i	−0.526522	+	0.526522i	−0.919534	−	0.393011i	$-0.871433\pi$
$389$	−10.3846	+	10.3846i	−0.526522	+	0.526522i	0.393011	+	0.919534i	$0.371433\pi$
$390$		0			0
$391$		−	4.42320i		−	0.223691i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.9243	−	10.9243i	0.549661	−	0.549661i
$396$		0			0
$397$	9.93104	+	9.93104i	0.498425	+	0.498425i	0.910947	−	0.412523i	$-0.135352\pi$
$397$	9.93104	+	9.93104i	0.498425	+	0.498425i	−0.412523	+	0.910947i	$0.635352\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.51392			−0.475102			−0.237551	−	0.971375i	$-0.576345\pi$
$401$	−9.51392			−0.475102			−0.237551	+	0.971375i	$0.576345\pi$
$402$		0			0
$403$	0.300592	+	0.300592i	0.0149736	+	0.0149736i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	18.8605i		−	0.934879i
$408$		0			0
$409$		−	4.81799i		−	0.238234i	−0.992880	−	0.119117i	$-0.961994\pi$
$409$		−	4.81799i		−	0.238234i	0.992880	−	0.119117i	$-0.0380064\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.57564	+	5.57564i	0.274359	+	0.274359i
$414$		0			0
$415$	0.966253			0.0474315
$416$		0			0
$417$		0			0
$418$		0			0
$419$	21.4380	+	21.4380i	1.04731	+	1.04731i	0.998824	+	0.0484914i	$0.0154413\pi$
$419$	21.4380	+	21.4380i	1.04731	+	1.04731i	0.0484914	+	0.998824i	$0.484559\pi$
$420$		0			0
$421$	−4.80145	+	4.80145i	−0.234008	+	0.234008i	−0.814363	−	0.580355i	$-0.802914\pi$
$421$	−4.80145	+	4.80145i	−0.234008	+	0.234008i	0.580355	+	0.814363i	$0.302914\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	2.19577i		−	0.106510i
$426$		0			0
$427$	−3.53819	+	3.53819i	−0.171225	+	0.171225i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13.2369			0.637597			0.318799	−	0.947822i	$-0.396721\pi$
$431$	13.2369			0.637597			0.318799	+	0.947822i	$0.396721\pi$
$432$		0			0
$433$	−1.50709			−0.0724259			−0.0362129	−	0.999344i	$-0.511529\pi$
$433$	−1.50709			−0.0724259			−0.0362129	+	0.999344i	$0.511529\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.88097	+	3.88097i	−0.185652	+	0.185652i
$438$		0			0
$439$		−	10.3092i		−	0.492033i	−0.969266	−	0.246016i	$-0.920878\pi$
$439$		−	10.3092i		−	0.492033i	0.969266	−	0.246016i	$-0.0791217\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.2651	+	14.2651i	−0.677755	+	0.677755i	−0.959492	−	0.281736i	$-0.909090\pi$
$443$	−14.2651	+	14.2651i	−0.677755	+	0.677755i	0.281736	+	0.959492i	$0.409090\pi$
$444$		0			0
$445$	−3.84950	−	3.84950i	−0.182484	−	0.182484i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	19.5711			0.923618			0.461809	−	0.886979i	$-0.347201\pi$
$449$	19.5711			0.923618			0.461809	+	0.886979i	$0.347201\pi$
$450$		0			0
$451$	−6.39420	−	6.39420i	−0.301091	−	0.301091i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	0.959035i		−	0.0449602i
$456$		0			0
$457$		−	39.0185i		−	1.82521i	−0.408845	−	0.912604i	$-0.634068\pi$
$457$		−	39.0185i		−	1.82521i	0.408845	−	0.912604i	$-0.365932\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.6941	−	19.6941i	−0.917245	−	0.917245i	0.0795833	−	0.996828i	$-0.474641\pi$
$461$	−19.6941	−	19.6941i	−0.917245	−	0.917245i	−0.996828	+	0.0795833i	$0.974641\pi$
$462$		0			0
$463$	14.9979			0.697009			0.348505	−	0.937307i	$-0.386690\pi$
$463$	14.9979			0.697009			0.348505	+	0.937307i	$0.386690\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.88870	+	4.88870i	0.226222	+	0.226222i	0.811112	−	0.584890i	$-0.198862\pi$
$467$	4.88870	+	4.88870i	0.226222	+	0.226222i	−0.584890	+	0.811112i	$0.698862\pi$
$468$		0			0
$469$	−4.45404	+	4.45404i	−0.205669	+	0.205669i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1.65114i			0.0759197i
$474$		0			0
$475$	−1.92659	+	1.92659i	−0.0883982	+	0.0883982i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−27.3381			−1.24911			−0.624555	−	0.780981i	$-0.714720\pi$
$479$	−27.3381			−1.24911			−0.624555	+	0.780981i	$0.714720\pi$
$480$		0			0
$481$	8.00784			0.365126
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.92119	+	3.92119i	−0.178052	+	0.178052i
$486$		0			0
$487$		−	35.4769i		−	1.60761i	−0.594892	−	0.803806i	$-0.702805\pi$
$487$		−	35.4769i		−	1.60761i	0.594892	−	0.803806i	$-0.297195\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.55614	−	3.55614i	0.160486	−	0.160486i	−0.622296	−	0.782782i	$-0.713800\pi$
$491$	3.55614	−	3.55614i	0.160486	−	0.160486i	0.782782	+	0.622296i	$0.213800\pi$
$492$		0			0
$493$	11.5757	+	11.5757i	0.521343	+	0.521343i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10.2114			−0.458042
$498$		0			0
$499$	17.6521	+	17.6521i	0.790218	+	0.790218i	0.981529	−	0.191312i	$-0.0612742\pi$
$499$	17.6521	+	17.6521i	0.790218	+	0.790218i	−0.191312	+	0.981529i	$0.561274\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			31.8567i			1.42042i	0.703990	+	0.710210i	$0.251400\pi$
$503$			31.8567i			1.42042i	−0.703990	+	0.710210i	$0.748600\pi$
$504$		0			0
$505$			0.411952i			0.0183316i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.61054	+	5.61054i	0.248683	+	0.248683i	0.820430	−	0.571747i	$-0.193734\pi$
$509$	5.61054	+	5.61054i	0.248683	+	0.248683i	−0.571747	+	0.820430i	$0.693734\pi$
$510$		0			0
$511$	9.07810			0.401591
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.18353	−	3.18353i	−0.140283	−	0.140283i
$516$		0			0
$517$	14.9952	−	14.9952i	0.659489	−	0.659489i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			33.1977i			1.45442i	0.686417	+	0.727208i	$0.259182\pi$
$521$			33.1977i			1.45442i	−0.686417	+	0.727208i	$0.740818\pi$
$522$		0			0
$523$	2.60707	−	2.60707i	0.113999	−	0.113999i	−0.647806	−	0.761805i	$-0.724313\pi$
$523$	2.60707	−	2.60707i	0.113999	−	0.113999i	0.761805	+	0.647806i	$0.224313\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.955787			−0.0416347
$528$		0			0
$529$	18.9421			0.823570
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.71487	−	2.71487i	0.117594	−	0.117594i
$534$		0			0
$535$		−	9.19083i		−	0.397354i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9.81668	+	9.81668i	−0.422834	+	0.422834i
$540$		0			0
$541$	−22.6839	−	22.6839i	−0.975257	−	0.975257i	0.0244439	−	0.999701i	$-0.492218\pi$
$541$	−22.6839	−	22.6839i	−0.975257	−	0.975257i	−0.999701	+	0.0244439i	$0.992218\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.56311			0.152627
$546$		0			0
$547$	3.02284	+	3.02284i	0.129248	+	0.129248i	0.768771	−	0.639524i	$-0.220868\pi$
$547$	3.02284	+	3.02284i	0.129248	+	0.129248i	−0.639524	+	0.768771i	$0.720868\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	20.3133i		−	0.865375i
$552$		0			0
$553$			15.1714i			0.645153i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.27495	−	9.27495i	−0.392992	−	0.392992i	0.482760	−	0.875753i	$-0.339634\pi$
$557$	−9.27495	−	9.27495i	−0.392992	−	0.392992i	−0.875753	+	0.482760i	$0.839634\pi$
$558$		0			0
$559$	−0.701048			−0.0296512
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.3025	−	20.3025i	−0.855649	−	0.855649i	0.135173	−	0.990822i	$-0.456841\pi$
$563$	−20.3025	−	20.3025i	−0.855649	−	0.855649i	−0.990822	+	0.135173i	$0.956841\pi$
$564$		0			0
$565$	3.80495	−	3.80495i	0.160076	−	0.160076i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	14.3362i		−	0.601005i	−0.953781	−	0.300503i	$-0.902846\pi$
$569$		−	14.3362i		−	0.601005i	0.953781	−	0.300503i	$-0.0971544\pi$
$570$		0			0
$571$	−8.54368	+	8.54368i	−0.357542	+	0.357542i	−0.862906	−	0.505364i	$-0.831358\pi$
$571$	−8.54368	+	8.54368i	−0.357542	+	0.357542i	0.505364	+	0.862906i	$0.331358\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.01442			−0.0840071
$576$		0			0
$577$	−8.68179			−0.361428			−0.180714	−	0.983536i	$-0.557841\pi$
$577$	−8.68179			−0.361428			−0.180714	+	0.983536i	$0.557841\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.670953	+	0.670953i	−0.0278358	+	0.0278358i
$582$		0			0
$583$		−	20.4723i		−	0.847877i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21.9042	−	21.9042i	0.904082	−	0.904082i	−0.0917043	−	0.995786i	$-0.529231\pi$
$587$	21.9042	−	21.9042i	0.904082	−	0.904082i	0.995786	+	0.0917043i	$0.0292315\pi$
$588$		0			0
$589$	0.838619	+	0.838619i	0.0345547	+	0.0345547i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	17.5142			0.719222			0.359611	−	0.933102i	$-0.382909\pi$
$593$	17.5142			0.719222			0.359611	+	0.933102i	$0.382909\pi$
$594$		0			0
$595$	1.52471	+	1.52471i	0.0625071	+	0.0625071i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			19.0276i			0.777447i	0.921354	+	0.388724i	$0.127084\pi$
$599$			19.0276i			0.777447i	−0.921354	+	0.388724i	$0.872916\pi$
$600$		0			0
$601$		−	5.52545i		−	0.225388i	−0.993630	−	0.112694i	$-0.964052\pi$
$601$		−	5.52545i		−	0.225388i	0.993630	−	0.112694i	$-0.0359479\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.03711	−	4.03711i	−0.164132	−	0.164132i
$606$		0			0
$607$	12.1064			0.491384			0.245692	−	0.969348i	$-0.420985\pi$
$607$	12.1064			0.491384			0.245692	+	0.969348i	$0.420985\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.36671	+	6.36671i	0.257570	+	0.257570i
$612$		0			0
$613$	17.8073	−	17.8073i	0.719230	−	0.719230i	−0.249218	−	0.968448i	$-0.580173\pi$
$613$	17.8073	−	17.8073i	0.719230	−	0.719230i	0.968448	+	0.249218i	$0.0801734\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.10944i		−	0.0446642i	−0.999751	−	0.0223321i	$-0.992891\pi$
$617$		−	1.10944i		−	0.0446642i	0.999751	−	0.0223321i	$-0.00710912\pi$
$618$		0			0
$619$	31.8702	−	31.8702i	1.28097	−	1.28097i	0.340859	−	0.940115i	$-0.389282\pi$
$619$	31.8702	−	31.8702i	1.28097	−	1.28097i	0.940115	−	0.340859i	$-0.110718\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.34608			0.214186
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.7312	+	12.7312i	−0.507626	+	0.507626i
$630$		0			0
$631$			6.80064i			0.270729i	0.990796	+	0.135365i	$0.0432206\pi$
$631$			6.80064i			0.270729i	−0.990796	+	0.135365i	$0.956779\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.44063	−	3.44063i	0.136537	−	0.136537i
$636$		0			0
$637$	−4.16800	−	4.16800i	−0.165142	−	0.165142i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.2566			−0.563100			−0.281550	−	0.959547i	$-0.590849\pi$
$641$	−14.2566			−0.563100			−0.281550	+	0.959547i	$0.590849\pi$
$642$		0			0
$643$	−14.4137	−	14.4137i	−0.568422	−	0.568422i	0.363264	−	0.931686i	$-0.381662\pi$
$643$	−14.4137	−	14.4137i	−0.568422	−	0.568422i	−0.931686	+	0.363264i	$0.881662\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	20.5723i		−	0.808782i	−0.914586	−	0.404391i	$-0.867483\pi$
$647$		−	20.5723i		−	0.808782i	0.914586	−	0.404391i	$-0.132517\pi$
$648$		0			0
$649$		−	18.4692i		−	0.724980i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.79946	+	9.79946i	0.383482	+	0.383482i	0.872355	−	0.488873i	$-0.162592\pi$
$653$	9.79946	+	9.79946i	0.383482	+	0.383482i	−0.488873	+	0.872355i	$0.662592\pi$
$654$		0			0
$655$	−11.3219			−0.442384
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.70669	−	8.70669i	−0.339165	−	0.339165i	0.516888	−	0.856053i	$-0.327090\pi$
$659$	−8.70669	−	8.70669i	−0.339165	−	0.339165i	−0.856053	+	0.516888i	$0.827090\pi$
$660$		0			0
$661$	19.7899	−	19.7899i	0.769737	−	0.769737i	−0.208323	−	0.978060i	$-0.566801\pi$
$661$	19.7899	−	19.7899i	0.769737	−	0.769737i	0.978060	+	0.208323i	$0.0668006\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	2.67560i		−	0.103755i
$666$		0			0
$667$	10.6196	−	10.6196i	0.411194	−	0.411194i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	11.7202			0.452454
$672$		0			0
$673$	−14.0829			−0.542857			−0.271429	−	0.962459i	$-0.587496\pi$
$673$	−14.0829			−0.542857			−0.271429	+	0.962459i	$0.587496\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	29.8166	−	29.8166i	1.14594	−	1.14594i	0.158601	−	0.987343i	$-0.449302\pi$
$677$	29.8166	−	29.8166i	1.14594	−	1.14594i	0.987343	−	0.158601i	$-0.0506984\pi$
$678$		0			0
$679$		−	5.44564i		−	0.208984i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0646	+	12.0646i	−0.461641	+	0.461641i	−0.899193	−	0.437552i	$-0.855845\pi$
$683$	−12.0646	+	12.0646i	−0.461641	+	0.461641i	0.437552	+	0.899193i	$0.355845\pi$
$684$		0			0
$685$	9.58601	+	9.58601i	0.366263	+	0.366263i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8.69221			0.331147
$690$		0			0
$691$	−2.58867	−	2.58867i	−0.0984776	−	0.0984776i	0.656152	−	0.754629i	$-0.272183\pi$
$691$	−2.58867	−	2.58867i	−0.0984776	−	0.0984776i	−0.754629	+	0.656152i	$0.772183\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	11.6340i		−	0.441301i
$696$		0			0
$697$			8.63242i			0.326976i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26.9943	+	26.9943i	1.01956	+	1.01956i	0.999805	+	0.0197572i	$0.00628933\pi$
$701$	26.9943	+	26.9943i	1.01956	+	1.01956i	0.0197572	+	0.999805i	$0.493711\pi$
$702$		0			0
$703$	22.3410			0.842606
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.286054	−	0.286054i	−0.0107582	−	0.0107582i
$708$		0			0
$709$	−35.0639	+	35.0639i	−1.31685	+	1.31685i	−0.400598	+	0.916254i	$0.631198\pi$
$709$	−35.0639	+	35.0639i	−1.31685	+	1.31685i	−0.916254	+	0.400598i	$0.868802\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			0.876848i			0.0328382i
$714$		0			0
$715$	−1.58839	+	1.58839i	−0.0594026	+	0.0594026i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−0.436840			−0.0162914			−0.00814568	−	0.999967i	$-0.502593\pi$
$719$	−0.436840			−0.0162914			−0.00814568	+	0.999967i	$0.502593\pi$
$720$		0			0
$721$	4.42120			0.164654
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.27182	−	5.27182i	0.195790	−	0.195790i
$726$		0			0
$727$			38.8072i			1.43928i	0.694348	+	0.719640i	$0.255693\pi$
$727$			38.8072i			1.43928i	−0.694348	+	0.719640i	$0.744307\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.11455	−	1.11455i	0.0412233	−	0.0412233i
$732$		0			0
$733$	−24.3059	−	24.3059i	−0.897758	−	0.897758i	0.0974793	−	0.995238i	$-0.468922\pi$
$733$	−24.3059	−	24.3059i	−0.897758	−	0.897758i	−0.995238	+	0.0974793i	$0.968922\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.7539			0.543469
$738$		0			0
$739$	27.0262	+	27.0262i	0.994174	+	0.994174i	0.999983	−	0.00580951i	$-0.00184923\pi$
$739$	27.0262	+	27.0262i	0.994174	+	0.994174i	−0.00580951	+	0.999983i	$0.501849\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	12.1663i		−	0.446337i	−0.974780	−	0.223169i	$-0.928360\pi$
$743$		−	12.1663i		−	0.446337i	0.974780	−	0.223169i	$-0.0716401\pi$
$744$		0			0
$745$		−	17.8705i		−	0.654724i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.38199	+	6.38199i	0.233193	+	0.233193i
$750$		0			0
$751$	40.8606			1.49102			0.745512	−	0.666492i	$-0.232205\pi$
$751$	40.8606			1.49102			0.745512	+	0.666492i	$0.232205\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.7170	−	10.7170i	−0.390033	−	0.390033i
$756$		0			0
$757$	−0.00399171	+	0.00399171i	−0.000145081	+	0.000145081i	−0.707179	−	0.707034i	$-0.750033\pi$
$757$	−0.00399171	+	0.00399171i	−0.000145081	+	0.000145081i	0.707034	+	0.707179i	$0.250033\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			0.751325i			0.0272355i	0.999907	+	0.0136178i	$0.00433480\pi$
$761$			0.751325i			0.0272355i	−0.999907	+	0.0136178i	$0.995665\pi$
$762$		0			0
$763$	−2.47418	+	2.47418i	−0.0895712	+	0.0895712i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7.84172			0.283148
$768$		0			0
$769$	35.4522			1.27844			0.639219	−	0.769025i	$-0.279258\pi$
$769$	35.4522			1.27844			0.639219	+	0.769025i	$0.279258\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.50186	−	5.50186i	0.197888	−	0.197888i	−0.601206	−	0.799094i	$-0.705313\pi$
$773$	5.50186	−	5.50186i	0.197888	−	0.197888i	0.799094	+	0.601206i	$0.205313\pi$
$774$		0			0
$775$			0.435286i			0.0156359i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.57419	−	7.57419i	0.271373	−	0.271373i
$780$		0			0
$781$	16.9125	+	16.9125i	0.605177	+	0.605177i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.48641			0.0887438
$786$		0			0
$787$	−28.6944	−	28.6944i	−1.02284	−	1.02284i	−0.999733	−	0.0231107i	$-0.992643\pi$
$787$	−28.6944	−	28.6944i	−1.02284	−	1.02284i	−0.0231107	−	0.999733i	$-0.507357\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			5.28422i			0.187885i
$792$		0			0
$793$			4.97620i			0.176710i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.29277	+	6.29277i	0.222901	+	0.222901i	0.809719	−	0.586818i	$-0.199619\pi$
$797$	6.29277	+	6.29277i	0.222901	+	0.222901i	−0.586818	+	0.809719i	$0.699619\pi$
$798$		0			0
$799$	−20.2441			−0.716185
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−15.0355	−	15.0355i	−0.530592	−	0.530592i
$804$		0			0
$805$	1.39879	−	1.39879i	0.0493007	−	0.0493007i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			27.0850i			0.952257i	0.879376	+	0.476128i	$0.157960\pi$
$809$			27.0850i			0.952257i	−0.879376	+	0.476128i	$0.842040\pi$
$810$		0			0
$811$	14.6690	−	14.6690i	0.515098	−	0.515098i	−0.400986	−	0.916084i	$-0.631332\pi$
$811$	14.6690	−	14.6690i	0.515098	−	0.515098i	0.916084	+	0.400986i	$0.131332\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.6720			−0.689081
$816$		0			0
$817$	−1.95585			−0.0684264
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.4717	−	15.4717i	0.539965	−	0.539965i	−0.383553	−	0.923519i	$-0.625300\pi$
$821$	15.4717	−	15.4717i	0.539965	−	0.539965i	0.923519	+	0.383553i	$0.125300\pi$
$822$		0			0
$823$			7.64319i			0.266425i	0.991088	+	0.133212i	$0.0425292\pi$
$823$			7.64319i			0.266425i	−0.991088	+	0.133212i	$0.957471\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.781185	+	0.781185i	−0.0271645	+	0.0271645i	−0.720559	−	0.693394i	$-0.756115\pi$
$827$	−0.781185	+	0.781185i	−0.0271645	+	0.0271645i	0.693394	+	0.720559i	$0.256115\pi$
$828$		0			0
$829$	28.9122	+	28.9122i	1.00416	+	1.00416i	0.999991	+	0.00417165i	$0.00132788\pi$
$829$	28.9122	+	28.9122i	1.00416	+	1.00416i	0.00417165	+	0.999991i	$0.498672\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	13.2529			0.459186
$834$		0			0
$835$	13.3374	+	13.3374i	0.461561	+	0.461561i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	35.9665i		−	1.24170i	−0.783928	−	0.620851i	$-0.786787\pi$
$839$		−	35.9665i		−	1.24170i	0.783928	−	0.620851i	$-0.213213\pi$
$840$		0			0
$841$			26.5841i			0.916692i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.51798	+	8.51798i	0.293028	+	0.293028i
$846$		0			0
$847$	5.60663			0.192646
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11.6797	+	11.6797i	0.400375	+	0.400375i
$852$		0			0
$853$	8.53167	−	8.53167i	0.292119	−	0.292119i	−0.545798	−	0.837917i	$-0.683773\pi$
$853$	8.53167	−	8.53167i	0.292119	−	0.292119i	0.837917	+	0.545798i	$0.183773\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	20.6681i		−	0.706010i	−0.935621	−	0.353005i	$-0.885160\pi$
$857$		−	20.6681i		−	0.706010i	0.935621	−	0.353005i	$-0.114840\pi$
$858$		0			0
$859$	−26.6003	+	26.6003i	−0.907590	+	0.907590i	−0.996077	−	0.0884877i	$-0.971797\pi$
$859$	−26.6003	+	26.6003i	−0.907590	+	0.907590i	0.0884877	+	0.996077i	$0.471797\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.2911			0.826880			0.413440	−	0.910531i	$-0.364327\pi$
$863$	24.2911			0.826880			0.413440	+	0.910531i	$0.364327\pi$
$864$		0			0
$865$	22.7298			0.772838
$866$		0			0
$867$		0			0
$868$		0			0
$869$	25.1275	−	25.1275i	0.852391	−	0.852391i
$870$		0			0
$871$			6.26428i			0.212257i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.694387	−	0.694387i	0.0234746	−	0.0234746i
$876$		0			0
$877$	−17.5305	−	17.5305i	−0.591963	−	0.591963i	0.346198	−	0.938161i	$-0.387472\pi$
$877$	−17.5305	−	17.5305i	−0.591963	−	0.591963i	−0.938161	+	0.346198i	$0.887472\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−35.1334			−1.18367			−0.591837	−	0.806058i	$-0.701597\pi$
$881$	−35.1334			−1.18367			−0.591837	+	0.806058i	$0.701597\pi$
$882$		0			0
$883$	−18.0965	−	18.0965i	−0.608997	−	0.608997i	0.333687	−	0.942684i	$-0.391707\pi$
$883$	−18.0965	−	18.0965i	−0.608997	−	0.608997i	−0.942684	+	0.333687i	$0.891707\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	14.6666i		−	0.492455i	−0.969212	−	0.246228i	$-0.920809\pi$
$887$		−	14.6666i		−	0.492455i	0.969212	−	0.246228i	$-0.0791911\pi$
$888$		0			0
$889$			4.77825i			0.160257i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	17.7624	+	17.7624i	0.594397	+	0.594397i
$894$		0			0
$895$	−23.2413			−0.776871
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.29475	−	2.29475i	−0.0765341	−	0.0765341i
$900$		0			0
$901$	−13.8192	+	13.8192i	−0.460385	+	0.460385i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	21.8552i		−	0.726490i
$906$		0			0
$907$	25.4429	−	25.4429i	0.844817	−	0.844817i	−0.144664	−	0.989481i	$-0.546210\pi$
$907$	25.4429	−	25.4429i	0.844817	−	0.844817i	0.989481	+	0.144664i	$0.0462100\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	14.6852			0.486542			0.243271	−	0.969958i	$-0.421780\pi$
$911$	14.6852			0.486542			0.243271	+	0.969958i	$0.421780\pi$
$912$		0			0
$913$	2.22252			0.0735547
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.86179	−	7.86179i	0.259619	−	0.259619i
$918$		0			0
$919$			46.2157i			1.52451i	0.647274	+	0.762257i	$0.275909\pi$
$919$			46.2157i			1.52451i	−0.647274	+	0.762257i	$0.724091\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7.18076	+	7.18076i	−0.236358	+	0.236358i
$924$		0			0
$925$	5.79805	+	5.79805i	0.190639	+	0.190639i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−52.0543			−1.70785			−0.853923	−	0.520400i	$-0.825783\pi$
$929$	−52.0543			−1.70785			−0.853923	+	0.520400i	$0.825783\pi$
$930$		0			0
$931$	−11.6283	−	11.6283i	−0.381101	−	0.381101i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	5.05059i		−	0.165172i
$936$		0			0
$937$			33.7454i			1.10241i	0.834368	+	0.551207i	$0.185833\pi$
$937$			33.7454i			1.10241i	−0.834368	+	0.551207i	$0.814167\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	14.5814	+	14.5814i	0.475341	+	0.475341i	0.903638	−	0.428297i	$-0.140886\pi$
$941$	14.5814	+	14.5814i	0.475341	+	0.475341i	−0.428297	+	0.903638i	$0.640886\pi$
$942$		0			0
$943$	7.91946			0.257893
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−37.7582	−	37.7582i	−1.22698	−	1.22698i	−0.965102	−	0.261876i	$-0.915659\pi$
$947$	−37.7582	−	37.7582i	−1.22698	−	1.22698i	−0.261876	−	0.965102i	$-0.584341\pi$
$948$		0			0
$949$	6.38383	−	6.38383i	0.207228	−	0.207228i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			48.6441i			1.57574i	0.615844	+	0.787868i	$0.288815\pi$
$953$			48.6441i			1.57574i	−0.615844	+	0.787868i	$0.711185\pi$
$954$		0			0
$955$	10.4562	−	10.4562i	0.338353	−	0.338353i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−13.3128			−0.429893
$960$		0			0
$961$	−30.8105			−0.993888
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7.98408	−	7.98408i	0.257017	−	0.257017i
$966$		0			0
$967$		−	12.4521i		−	0.400433i	−0.979752	−	0.200216i	$-0.935835\pi$
$967$		−	12.4521i		−	0.400433i	0.979752	−	0.200216i	$-0.0641645\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	14.1931	−	14.1931i	0.455478	−	0.455478i	−0.441690	−	0.897168i	$-0.645621\pi$
$971$	14.1931	−	14.1931i	0.455478	−	0.455478i	0.897168	+	0.441690i	$0.145621\pi$
$972$		0			0
$973$	8.07846	+	8.07846i	0.258984	+	0.258984i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.3144			−0.585929			−0.292965	−	0.956123i	$-0.594642\pi$
$977$	−18.3144			−0.585929			−0.292965	+	0.956123i	$0.594642\pi$
$978$		0			0
$979$	−8.85441	−	8.85441i	−0.282988	−	0.282988i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			27.0583i			0.863027i	0.902107	+	0.431513i	$0.142020\pi$
$983$			27.0583i			0.863027i	−0.902107	+	0.431513i	$0.857980\pi$
$984$		0			0
$985$			15.0122i			0.478328i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.02250	−	1.02250i	−0.0325137	−	0.0325137i
$990$		0			0
$991$	−25.7759			−0.818799			−0.409400	−	0.912355i	$-0.634262\pi$
$991$	−25.7759			−0.818799			−0.409400	+	0.912355i	$0.634262\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.31484	+	3.31484i	0.105087	+	0.105087i
$996$		0			0
$997$	11.1158	−	11.1158i	0.352041	−	0.352041i	−0.508828	−	0.860868i	$-0.669921\pi$
$997$	11.1158	−	11.1158i	0.352041	−	0.352041i	0.860868	+	0.508828i	$0.169921\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.c.721.2		16
3.2	odd	2		320.2.l.a.81.8		16
4.3	odd	2		720.2.t.c.541.1		16
12.11	even	2		80.2.l.a.61.8	yes	16
15.2	even	4		1600.2.q.g.849.8		16
15.8	even	4		1600.2.q.h.849.1		16
15.14	odd	2		1600.2.l.i.401.1		16
16.5	even	4	inner	2880.2.t.c.2161.3		16
16.11	odd	4		720.2.t.c.181.1		16
24.5	odd	2		640.2.l.a.161.1		16
24.11	even	2		640.2.l.b.161.8		16
48.5	odd	4		320.2.l.a.241.8		16
48.11	even	4		80.2.l.a.21.8	✓	16
48.29	odd	4		640.2.l.a.481.1		16
48.35	even	4		640.2.l.b.481.8		16
60.23	odd	4		400.2.q.g.349.6		16
60.47	odd	4		400.2.q.h.349.3		16
60.59	even	2		400.2.l.h.301.1		16
96.5	odd	8		5120.2.a.t.1.1		8
96.11	even	8		5120.2.a.s.1.1		8
96.53	odd	8		5120.2.a.u.1.8		8
96.59	even	8		5120.2.a.v.1.8		8
240.53	even	4		1600.2.q.g.49.8		16
240.59	even	4		400.2.l.h.101.1		16
240.107	odd	4		400.2.q.g.149.6		16
240.149	odd	4		1600.2.l.i.1201.1		16
240.197	even	4		1600.2.q.h.49.1		16
240.203	odd	4		400.2.q.h.149.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.8	✓	16	48.11	even	4
80.2.l.a.61.8	yes	16	12.11	even	2
320.2.l.a.81.8		16	3.2	odd	2
320.2.l.a.241.8		16	48.5	odd	4
400.2.l.h.101.1		16	240.59	even	4
400.2.l.h.301.1		16	60.59	even	2
400.2.q.g.149.6		16	240.107	odd	4
400.2.q.g.349.6		16	60.23	odd	4
400.2.q.h.149.3		16	240.203	odd	4
400.2.q.h.349.3		16	60.47	odd	4
640.2.l.a.161.1		16	24.5	odd	2
640.2.l.a.481.1		16	48.29	odd	4
640.2.l.b.161.8		16	24.11	even	2
640.2.l.b.481.8		16	48.35	even	4
720.2.t.c.181.1		16	16.11	odd	4
720.2.t.c.541.1		16	4.3	odd	2
1600.2.l.i.401.1		16	15.14	odd	2
1600.2.l.i.1201.1		16	240.149	odd	4
1600.2.q.g.49.8		16	240.53	even	4
1600.2.q.g.849.8		16	15.2	even	4
1600.2.q.h.49.1		16	240.197	even	4
1600.2.q.h.849.1		16	15.8	even	4
2880.2.t.c.721.2		16	1.1	even	1	trivial
2880.2.t.c.2161.3		16	16.5	even	4	inner
5120.2.a.s.1.1		8	96.11	even	8
5120.2.a.t.1.1		8	96.5	odd	8
5120.2.a.u.1.8		8	96.53	odd	8
5120.2.a.v.1.8		8	96.59	even	8

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.t (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{5} -0.982011i q^{7} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{5} -0.982011i q^{7} +(-1.62645 + 1.62645i) q^{11} +(-0.690562 - 0.690562i) q^{13} +2.19577 q^{17} +(-1.92659 - 1.92659i) q^{19} -2.01442i q^{23} -1.00000i q^{25} +(5.27182 + 5.27182i) q^{29} -0.435286 q^{31} +(0.694387 + 0.694387i) q^{35} +(-5.79805 + 5.79805i) q^{37} +3.93139i q^{41} +(0.507592 - 0.507592i) q^{43} -9.21960 q^{47} +6.03565 q^{49} +(-6.29357 + 6.29357i) q^{53} -2.30015i q^{55} +(-5.67778 + 5.67778i) q^{59} +(-3.60301 - 3.60301i) q^{61} +0.976603 q^{65} +(-4.53563 - 4.53563i) q^{67} -10.3984i q^{71} +9.24439i q^{73} +(1.59719 + 1.59719i) q^{77} -15.4493 q^{79} +(-0.683244 - 0.683244i) q^{83} +(-1.55264 + 1.55264i) q^{85} +5.44401i q^{89} +(-0.678140 + 0.678140i) q^{91} +2.72461 q^{95} +5.54540 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) \) 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

Introduction

L-functions

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