LMFDB - Embedded newform 2880.2.t.c.721.3

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			721.3
Root			$-1.39563 + 0.228522i$ of defining polynomial
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.c.2161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{5} +0.690576i q^{7} +O(q^{10})$	$q+(-0.707107 + 0.707107i) q^{5} +0.690576i q^{7} +(-3.06057 + 3.06057i) q^{11} +(-2.33686 - 2.33686i) q^{13} +5.28770 q^{17} +(-5.38887 - 5.38887i) q^{19} -1.60841i q^{23} -1.00000i q^{25} +(-1.70319 - 1.70319i) q^{29} +4.69807 q^{31} +(-0.488311 - 0.488311i) q^{35} +(7.89871 - 7.89871i) q^{37} +5.49891i q^{41} +(0.256166 - 0.256166i) q^{43} -4.60743 q^{47} +6.52310 q^{49} +(4.99318 - 4.99318i) q^{53} -4.32830i q^{55} +(1.46478 - 1.46478i) q^{59} +(9.33004 + 9.33004i) q^{61} +3.30482 q^{65} +(1.94797 + 1.94797i) q^{67} -2.32246i q^{71} -1.29733i q^{73} +(-2.11356 - 2.11356i) q^{77} +5.01968 q^{79} +(7.30477 + 7.30477i) q^{83} +(-3.73897 + 3.73897i) q^{85} -1.81564i q^{89} +(1.61378 - 1.61378i) q^{91} +7.62102 q^{95} +5.27038 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			0.690576i			0.261013i	0.991447	+	0.130507i	$0.0416604\pi$
$7$			0.690576i			0.261013i	−0.991447	+	0.130507i	$0.958340\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.06057	+	3.06057i	−0.922797	+	0.922797i	−0.997226	−	0.0744292i	$-0.976287\pi$
$11$	−3.06057	+	3.06057i	−0.922797	+	0.922797i	0.0744292	+	0.997226i	$0.476287\pi$
$12$		0			0
$13$	−2.33686	−	2.33686i	−0.648128	−	0.648128i	0.304413	−	0.952540i	$-0.401540\pi$
$13$	−2.33686	−	2.33686i	−0.648128	−	0.648128i	−0.952540	+	0.304413i	$0.901540\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.28770			1.28246			0.641228	−	0.767350i	$-0.278425\pi$
$17$	5.28770			1.28246			0.641228	+	0.767350i	$0.278425\pi$
$18$		0			0
$19$	−5.38887	−	5.38887i	−1.23629	−	1.23629i	−0.961505	−	0.274787i	$-0.911393\pi$
$19$	−5.38887	−	5.38887i	−1.23629	−	1.23629i	−0.274787	−	0.961505i	$-0.588607\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.60841i		−	0.335376i	−0.985840	−	0.167688i	$-0.946370\pi$
$23$		−	1.60841i		−	0.335376i	0.985840	−	0.167688i	$-0.0536301\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.70319	−	1.70319i	−0.316274	−	0.316274i	0.531060	−	0.847334i	$-0.321794\pi$
$29$	−1.70319	−	1.70319i	−0.316274	−	0.316274i	−0.847334	+	0.531060i	$0.821794\pi$
$30$		0			0
$31$	4.69807			0.843798			0.421899	−	0.906643i	$-0.361364\pi$
$31$	4.69807			0.843798			0.421899	+	0.906643i	$0.361364\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.488311	−	0.488311i	−0.0825396	−	0.0825396i
$36$		0			0
$37$	7.89871	−	7.89871i	1.29854	−	1.29854i	0.369185	−	0.929356i	$-0.379637\pi$
$37$	7.89871	−	7.89871i	1.29854	−	1.29854i	0.929356	−	0.369185i	$-0.120363\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.49891i			0.858785i	0.903118	+	0.429392i	$0.141272\pi$
$41$			5.49891i			0.858785i	−0.903118	+	0.429392i	$0.858728\pi$
$42$		0			0
$43$	0.256166	−	0.256166i	0.0390650	−	0.0390650i	−0.687304	−	0.726369i	$-0.741206\pi$
$43$	0.256166	−	0.256166i	0.0390650	−	0.0390650i	0.726369	+	0.687304i	$0.241206\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.60743			−0.672063			−0.336032	−	0.941851i	$-0.609085\pi$
$47$	−4.60743			−0.672063			−0.336032	+	0.941851i	$0.609085\pi$
$48$		0			0
$49$	6.52310			0.931872
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.99318	−	4.99318i	0.685866	−	0.685866i	−0.275449	−	0.961316i	$-0.588827\pi$
$53$	4.99318	−	4.99318i	0.685866	−	0.685866i	0.961316	+	0.275449i	$0.0888266\pi$
$54$		0			0
$55$		−	4.32830i		−	0.583628i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.46478	−	1.46478i	0.190698	−	0.190698i	−0.605300	−	0.795998i	$-0.706947\pi$
$59$	1.46478	−	1.46478i	0.190698	−	0.190698i	0.795998	+	0.605300i	$0.206947\pi$
$60$		0			0
$61$	9.33004	+	9.33004i	1.19459	+	1.19459i	0.975764	+	0.218825i	$0.0702224\pi$
$61$	9.33004	+	9.33004i	1.19459	+	1.19459i	0.218825	+	0.975764i	$0.429778\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.30482			0.409912
$66$		0			0
$67$	1.94797	+	1.94797i	0.237982	+	0.237982i	0.816014	−	0.578032i	$-0.196179\pi$
$67$	1.94797	+	1.94797i	0.237982	+	0.237982i	−0.578032	+	0.816014i	$0.696179\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.32246i		−	0.275625i	−0.990458	−	0.137813i	$-0.955993\pi$
$71$		−	2.32246i		−	0.275625i	0.990458	−	0.137813i	$-0.0440072\pi$
$72$		0			0
$73$		−	1.29733i		−	0.151841i	−0.997114	−	0.0759206i	$-0.975810\pi$
$73$		−	1.29733i		−	0.151841i	0.997114	−	0.0759206i	$-0.0241896\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.11356	−	2.11356i	−0.240862	−	0.240862i
$78$		0			0
$79$	5.01968			0.564758			0.282379	−	0.959303i	$-0.408876\pi$
$79$	5.01968			0.564758			0.282379	+	0.959303i	$0.408876\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.30477	+	7.30477i	0.801802	+	0.801802i	0.983377	−	0.181575i	$-0.0581194\pi$
$83$	7.30477	+	7.30477i	0.801802	+	0.801802i	−0.181575	+	0.983377i	$0.558119\pi$
$84$		0			0
$85$	−3.73897	+	3.73897i	−0.405548	+	0.405548i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.81564i		−	0.192458i	−0.995359	−	0.0962290i	$-0.969322\pi$
$89$		−	1.81564i		−	0.192458i	0.995359	−	0.0962290i	$-0.0306781\pi$
$90$		0			0
$91$	1.61378	−	1.61378i	0.169170	−	0.169170i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.62102			0.781900
$96$		0			0
$97$	5.27038			0.535126			0.267563	−	0.963540i	$-0.413782\pi$
$97$	5.27038			0.535126			0.267563	+	0.963540i	$0.413782\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	13.4502	−	13.4502i	1.33834	−	1.33834i	0.440675	−	0.897667i	$-0.354739\pi$
$101$	13.4502	−	13.4502i	1.33834	−	1.33834i	0.897667	−	0.440675i	$-0.145261\pi$
$102$		0			0
$103$			2.64310i			0.260432i	0.991486	+	0.130216i	$0.0415671\pi$
$103$			2.64310i			0.260432i	−0.991486	+	0.130216i	$0.958433\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.28120	+	6.28120i	−0.607227	+	0.607227i	−0.942220	−	0.334994i	$-0.891266\pi$
$107$	−6.28120	+	6.28120i	−0.607227	+	0.607227i	0.334994	+	0.942220i	$0.391266\pi$
$108$		0			0
$109$	−6.89216	−	6.89216i	−0.660149	−	0.660149i	0.295266	−	0.955415i	$-0.404592\pi$
$109$	−6.89216	−	6.89216i	−0.660149	−	0.660149i	−0.955415	+	0.295266i	$0.904592\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.46108			−0.607807			−0.303904	−	0.952703i	$-0.598290\pi$
$113$	−6.46108			−0.607807			−0.303904	+	0.952703i	$0.598290\pi$
$114$		0			0
$115$	1.13732	+	1.13732i	0.106055	+	0.106055i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			3.65156i			0.334738i
$120$		0			0
$121$		−	7.73420i		−	0.703109i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$	16.6123			1.47411			0.737054	−	0.675834i	$-0.236216\pi$
$127$	16.6123			1.47411			0.737054	+	0.675834i	$0.236216\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11.7719	−	11.7719i	−1.02851	−	1.02851i	−0.999581	−	0.0289318i	$-0.990789\pi$
$131$	−11.7719	−	11.7719i	−1.02851	−	1.02851i	−0.0289318	−	0.999581i	$-0.509211\pi$
$132$		0			0
$133$	3.72143	−	3.72143i	0.322689	−	0.322689i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.41495i			0.718937i	0.933157	+	0.359469i	$0.117042\pi$
$137$			8.41495i			0.718937i	−0.933157	+	0.359469i	$0.882958\pi$
$138$		0			0
$139$	1.51845	−	1.51845i	0.128793	−	0.128793i	−0.639772	−	0.768565i	$-0.720971\pi$
$139$	1.51845	−	1.51845i	0.128793	−	0.128793i	0.768565	+	0.639772i	$0.220971\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	14.3042			1.19618
$144$		0			0
$145$	2.40867			0.200029
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.61440	−	2.61440i	0.214180	−	0.214180i	−0.591860	−	0.806040i	$-0.701606\pi$
$149$	2.61440	−	2.61440i	0.214180	−	0.214180i	0.806040	+	0.591860i	$0.201606\pi$
$150$		0			0
$151$		−	12.7143i		−	1.03467i	−0.855782	−	0.517337i	$-0.826923\pi$
$151$		−	12.7143i		−	1.03467i	0.855782	−	0.517337i	$-0.173077\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.32204	+	3.32204i	−0.266832	+	0.266832i
$156$		0			0
$157$	−7.17831	−	7.17831i	−0.572891	−	0.572891i	0.360044	−	0.932935i	$-0.382762\pi$
$157$	−7.17831	−	7.17831i	−0.572891	−	0.572891i	−0.932935	+	0.360044i	$0.882762\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.11073			0.0875376
$162$		0			0
$163$	7.05476	+	7.05476i	0.552572	+	0.552572i	0.927182	−	0.374611i	$-0.122224\pi$
$163$	7.05476	+	7.05476i	0.552572	+	0.552572i	−0.374611	+	0.927182i	$0.622224\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	3.90586i		−	0.302244i	−0.988515	−	0.151122i	$-0.951711\pi$
$167$		−	3.90586i		−	0.302244i	0.988515	−	0.151122i	$-0.0482887\pi$
$168$		0			0
$169$		−	2.07819i		−	0.159861i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.20139	−	8.20139i	−0.623540	−	0.623540i	0.322895	−	0.946435i	$-0.395344\pi$
$173$	−8.20139	−	8.20139i	−0.623540	−	0.623540i	−0.946435	+	0.322895i	$0.895344\pi$
$174$		0			0
$175$	0.690576			0.0522026
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.10363	−	3.10363i	−0.231976	−	0.231976i	0.581541	−	0.813517i	$-0.302450\pi$
$179$	−3.10363	−	3.10363i	−0.231976	−	0.231976i	−0.813517	+	0.581541i	$0.802450\pi$
$180$		0			0
$181$	−1.91041	+	1.91041i	−0.141999	+	0.141999i	−0.774533	−	0.632534i	$-0.782015\pi$
$181$	−1.91041	+	1.91041i	−0.141999	+	0.141999i	0.632534	+	0.774533i	$0.282015\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			11.1705i			0.821269i
$186$		0			0
$187$	−16.1834	+	16.1834i	−1.18345	+	1.18345i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.61041			0.405955			0.202977	−	0.979183i	$-0.434938\pi$
$191$	5.61041			0.405955			0.202977	+	0.979183i	$0.434938\pi$
$192$		0			0
$193$	3.90696			0.281229			0.140615	−	0.990064i	$-0.455092\pi$
$193$	3.90696			0.281229			0.140615	+	0.990064i	$0.455092\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.608436	+	0.608436i	−0.0433493	+	0.0433493i	−0.728449	−	0.685100i	$-0.759759\pi$
$197$	−0.608436	+	0.608436i	−0.0433493	+	0.0433493i	0.685100	+	0.728449i	$0.259759\pi$
$198$		0			0
$199$		−	15.5282i		−	1.10076i	−0.834913	−	0.550382i	$-0.814482\pi$
$199$		−	15.5282i		−	1.10076i	0.834913	−	0.550382i	$-0.185518\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.17618	−	1.17618i	0.0825517	−	0.0825517i
$204$		0			0
$205$	−3.88831	−	3.88831i	−0.271572	−	0.271572i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	32.9861			2.28169
$210$		0			0
$211$	−2.14501	−	2.14501i	−0.147669	−	0.147669i	0.629407	−	0.777076i	$-0.283298\pi$
$211$	−2.14501	−	2.14501i	−0.147669	−	0.147669i	−0.777076	+	0.629407i	$0.783298\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			0.362274i			0.0247069i
$216$		0			0
$217$			3.24437i			0.220242i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.3566	−	12.3566i	−0.831196	−	0.831196i
$222$		0			0
$223$	2.34794			0.157230			0.0786148	−	0.996905i	$-0.474950\pi$
$223$	2.34794			0.157230			0.0786148	+	0.996905i	$0.474950\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.1881	+	13.1881i	0.875325	+	0.875325i	0.993047	−	0.117722i	$-0.0375591\pi$
$227$	13.1881	+	13.1881i	0.875325	+	0.875325i	−0.117722	+	0.993047i	$0.537559\pi$
$228$		0			0
$229$	9.37860	−	9.37860i	0.619755	−	0.619755i	−0.325713	−	0.945469i	$-0.605604\pi$
$229$	9.37860	−	9.37860i	0.619755	−	0.619755i	0.945469	+	0.325713i	$0.105604\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.3435i		−	1.07070i	−0.844630	−	0.535350i	$-0.820180\pi$
$233$		−	16.3435i		−	1.07070i	0.844630	−	0.535350i	$-0.179820\pi$
$234$		0			0
$235$	3.25795	−	3.25795i	0.212525	−	0.212525i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	19.3818			1.25371			0.626854	−	0.779137i	$-0.284342\pi$
$239$	19.3818			1.25371			0.626854	+	0.779137i	$0.284342\pi$
$240$		0			0
$241$	7.15965			0.461193			0.230597	−	0.973049i	$-0.425932\pi$
$241$	7.15965			0.461193			0.230597	+	0.973049i	$0.425932\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.61253	+	4.61253i	−0.294684	+	0.294684i
$246$		0			0
$247$			25.1861i			1.60255i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.4372	−	10.4372i	0.658787	−	0.658787i	−0.296306	−	0.955093i	$-0.595755\pi$
$251$	10.4372	−	10.4372i	0.658787	−	0.658787i	0.955093	+	0.296306i	$0.0957548\pi$
$252$		0			0
$253$	4.92264	+	4.92264i	0.309484	+	0.309484i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.72152			−0.356899			−0.178449	−	0.983949i	$-0.557108\pi$
$257$	−5.72152			−0.356899			−0.178449	+	0.983949i	$0.557108\pi$
$258$		0			0
$259$	5.45466	+	5.45466i	0.338936	+	0.338936i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	27.1378i		−	1.67339i	−0.547669	−	0.836695i	$-0.684485\pi$
$263$		−	27.1378i		−	1.67339i	0.547669	−	0.836695i	$-0.315515\pi$
$264$		0			0
$265$			7.06143i			0.433780i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.0770	−	13.0770i	−0.797320	−	0.797320i	0.185352	−	0.982672i	$-0.440657\pi$
$269$	−13.0770	−	13.0770i	−0.797320	−	0.797320i	−0.982672	+	0.185352i	$0.940657\pi$
$270$		0			0
$271$	−6.55264			−0.398044			−0.199022	−	0.979995i	$-0.563777\pi$
$271$	−6.55264			−0.398044			−0.199022	+	0.979995i	$0.563777\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.06057	+	3.06057i	0.184559	+	0.184559i
$276$		0			0
$277$	−10.2851	+	10.2851i	−0.617973	+	0.617973i	−0.945011	−	0.327038i	$-0.893949\pi$
$277$	−10.2851	+	10.2851i	−0.617973	+	0.617973i	0.327038	+	0.945011i	$0.393949\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			29.9714i			1.78794i	0.448124	+	0.893971i	$0.352092\pi$
$281$			29.9714i			1.78794i	−0.448124	+	0.893971i	$0.647908\pi$
$282$		0			0
$283$	19.1176	−	19.1176i	1.13642	−	1.13642i	0.147334	−	0.989087i	$-0.452931\pi$
$283$	19.1176	−	19.1176i	1.13642	−	1.13642i	0.989087	−	0.147334i	$-0.0470691\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.79741			−0.224154
$288$		0			0
$289$	10.9598			0.644695
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.27952	+	7.27952i	−0.425274	+	0.425274i	−0.887015	−	0.461741i	$-0.847225\pi$
$293$	−7.27952	+	7.27952i	−0.425274	+	0.425274i	0.461741	+	0.887015i	$0.347225\pi$
$294$		0			0
$295$			2.07151i			0.120608i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.75862	+	3.75862i	−0.217366	+	0.217366i
$300$		0			0
$301$	0.176902	+	0.176902i	0.0101965	+	0.0101965i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−13.1947			−0.755525
$306$		0			0
$307$	−7.03304	−	7.03304i	−0.401397	−	0.401397i	0.477328	−	0.878725i	$-0.341605\pi$
$307$	−7.03304	−	7.03304i	−0.401397	−	0.401397i	−0.878725	+	0.477328i	$0.841605\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	14.2833i		−	0.809929i	−0.914332	−	0.404964i	$-0.867284\pi$
$311$		−	14.2833i		−	0.809929i	0.914332	−	0.404964i	$-0.132716\pi$
$312$		0			0
$313$			18.4579i			1.04330i	0.853158	+	0.521652i	$0.174684\pi$
$313$			18.4579i			1.04330i	−0.853158	+	0.521652i	$0.825316\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.21807	+	7.21807i	0.405407	+	0.405407i	0.880133	−	0.474726i	$-0.157453\pi$
$317$	7.21807	+	7.21807i	0.405407	+	0.405407i	−0.474726	+	0.880133i	$0.657453\pi$
$318$		0			0
$319$	10.4255			0.583714
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−28.4948	−	28.4948i	−1.58549	−	1.58549i
$324$		0			0
$325$	−2.33686	+	2.33686i	−0.129626	+	0.129626i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	3.18178i		−	0.175417i
$330$		0			0
$331$	15.4847	−	15.4847i	0.851116	−	0.851116i	−0.139155	−	0.990271i	$-0.544439\pi$
$331$	15.4847	−	15.4847i	0.851116	−	0.851116i	0.990271	+	0.139155i	$0.0444385\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.75484			−0.150513
$336$		0			0
$337$	−26.0210			−1.41746			−0.708728	−	0.705482i	$-0.750731\pi$
$337$	−26.0210			−1.41746			−0.708728	+	0.705482i	$0.750731\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−14.3788	+	14.3788i	−0.778654	+	0.778654i
$342$		0			0
$343$			9.33873i			0.504244i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.8554	+	12.8554i	−0.690115	+	0.690115i	−0.962257	−	0.272142i	$-0.912268\pi$
$347$	−12.8554	+	12.8554i	−0.690115	+	0.690115i	0.272142	+	0.962257i	$0.412268\pi$
$348$		0			0
$349$	−20.0227	−	20.0227i	−1.07179	−	1.07179i	−0.997216	−	0.0745736i	$-0.976240\pi$
$349$	−20.0227	−	20.0227i	−1.07179	−	1.07179i	−0.0745736	−	0.997216i	$-0.523760\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.7062			−0.729510			−0.364755	−	0.931104i	$-0.618847\pi$
$353$	−13.7062			−0.729510			−0.364755	+	0.931104i	$0.618847\pi$
$354$		0			0
$355$	1.64223	+	1.64223i	0.0871603	+	0.0871603i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	32.3506i		−	1.70740i	−0.520764	−	0.853700i	$-0.674353\pi$
$359$		−	32.3506i		−	1.70740i	0.520764	−	0.853700i	$-0.325647\pi$
$360$		0			0
$361$			39.0799i			2.05684i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.917352	+	0.917352i	0.0480164	+	0.0480164i
$366$		0			0
$367$	−16.3714			−0.854582			−0.427291	−	0.904114i	$-0.640532\pi$
$367$	−16.3714			−0.854582			−0.427291	+	0.904114i	$0.640532\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.44817	+	3.44817i	0.179020	+	0.179020i
$372$		0			0
$373$	15.5321	−	15.5321i	0.804222	−	0.804222i	−0.179530	−	0.983752i	$-0.557458\pi$
$373$	15.5321	−	15.5321i	0.804222	−	0.804222i	0.983752	+	0.179530i	$0.0574578\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			7.96022i			0.409972i
$378$		0			0
$379$	−24.9538	+	24.9538i	−1.28179	+	1.28179i	−0.342145	+	0.939647i	$0.611153\pi$
$379$	−24.9538	+	24.9538i	−1.28179	+	1.28179i	−0.939647	+	0.342145i	$0.888847\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.24887			0.319302			0.159651	−	0.987174i	$-0.448963\pi$
$383$	6.24887			0.319302			0.159651	+	0.987174i	$0.448963\pi$
$384$		0			0
$385$	2.98902			0.152335
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.10802	+	2.10802i	−0.106881	+	0.106881i	−0.758525	−	0.651644i	$-0.774080\pi$
$389$	−2.10802	+	2.10802i	−0.106881	+	0.106881i	0.651644	+	0.758525i	$0.274080\pi$
$390$		0			0
$391$		−	8.50478i		−	0.430105i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.54945	+	3.54945i	−0.178592	+	0.178592i
$396$		0			0
$397$	23.4977	+	23.4977i	1.17932	+	1.17932i	0.979919	+	0.199397i	$0.0638983\pi$
$397$	23.4977	+	23.4977i	1.17932	+	1.17932i	0.199397	+	0.979919i	$0.436102\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	20.9893			1.04816			0.524078	−	0.851670i	$-0.324410\pi$
$401$	20.9893			1.04816			0.524078	+	0.851670i	$0.324410\pi$
$402$		0			0
$403$	−10.9787	−	10.9787i	−0.546889	−	0.546889i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			48.3492i			2.39658i
$408$		0			0
$409$		−	18.4025i		−	0.909944i	−0.890506	−	0.454972i	$-0.849649\pi$
$409$		−	18.4025i		−	0.909944i	0.890506	−	0.454972i	$-0.150351\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.01154	+	1.01154i	0.0497746	+	0.0497746i
$414$		0			0
$415$	−10.3305			−0.507104
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.9331	+	14.9331i	0.729530	+	0.729530i	0.970526	−	0.240996i	$-0.0774740\pi$
$419$	14.9331	+	14.9331i	0.729530	+	0.729530i	−0.240996	+	0.970526i	$0.577474\pi$
$420$		0			0
$421$	−16.2680	+	16.2680i	−0.792854	+	0.792854i	−0.981957	−	0.189103i	$-0.939442\pi$
$421$	−16.2680	+	16.2680i	−0.792854	+	0.792854i	0.189103	+	0.981957i	$0.439442\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	5.28770i		−	0.256491i
$426$		0			0
$427$	−6.44310	+	6.44310i	−0.311804	+	0.311804i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.05425			0.339791			0.169896	−	0.985462i	$-0.445657\pi$
$431$	7.05425			0.339791			0.169896	+	0.985462i	$0.445657\pi$
$432$		0			0
$433$	−14.3192			−0.688139			−0.344069	−	0.938944i	$-0.611806\pi$
$433$	−14.3192			−0.688139			−0.344069	+	0.938944i	$0.611806\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.66750	+	8.66750i	−0.414623	+	0.414623i
$438$		0			0
$439$		−	25.9047i		−	1.23637i	−0.786034	−	0.618183i	$-0.787869\pi$
$439$		−	25.9047i		−	1.23637i	0.786034	−	0.618183i	$-0.212131\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−11.1389	+	11.1389i	−0.529224	+	0.529224i	−0.920341	−	0.391117i	$-0.872089\pi$
$443$	−11.1389	+	11.1389i	−0.529224	+	0.529224i	0.391117	+	0.920341i	$0.372089\pi$
$444$		0			0
$445$	1.28385	+	1.28385i	0.0608605	+	0.0608605i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.6659			−0.597740			−0.298870	−	0.954294i	$-0.596610\pi$
$449$	−12.6659			−0.597740			−0.298870	+	0.954294i	$0.596610\pi$
$450$		0			0
$451$	−16.8298	−	16.8298i	−0.792484	−	0.792484i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			2.28223i			0.106992i
$456$		0			0
$457$		−	16.9442i		−	0.792617i	−0.918117	−	0.396308i	$-0.870291\pi$
$457$		−	16.9442i		−	0.792617i	0.918117	−	0.396308i	$-0.129709\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.1888	+	13.1888i	0.614264	+	0.614264i	0.944054	−	0.329790i	$-0.106978\pi$
$461$	13.1888	+	13.1888i	0.614264	+	0.614264i	−0.329790	+	0.944054i	$0.606978\pi$
$462$		0			0
$463$	14.0955			0.655074			0.327537	−	0.944838i	$-0.393781\pi$
$463$	14.0955			0.655074			0.327537	+	0.944838i	$0.393781\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0918	−	12.0918i	−0.559540	−	0.559540i	0.369636	−	0.929177i	$-0.379482\pi$
$467$	−12.0918	−	12.0918i	−0.559540	−	0.559540i	−0.929177	+	0.369636i	$0.879482\pi$
$468$		0			0
$469$	−1.34522	+	1.34522i	−0.0621165	+	0.0621165i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1.56803i			0.0720981i
$474$		0			0
$475$	−5.38887	+	5.38887i	−0.247258	+	0.247258i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14.2523			0.651202			0.325601	−	0.945507i	$-0.394433\pi$
$479$	14.2523			0.651202			0.325601	+	0.945507i	$0.394433\pi$
$480$		0			0
$481$	−36.9163			−1.68324
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.72672	+	3.72672i	−0.169222	+	0.169222i
$486$		0			0
$487$		−	26.0424i		−	1.18010i	−0.807368	−	0.590048i	$-0.799109\pi$
$487$		−	26.0424i		−	1.18010i	0.807368	−	0.590048i	$-0.200891\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.46798	−	3.46798i	0.156508	−	0.156508i	−0.624509	−	0.781017i	$-0.714701\pi$
$491$	3.46798	−	3.46798i	0.156508	−	0.156508i	0.781017	+	0.624509i	$0.214701\pi$
$492$		0			0
$493$	−9.00595	−	9.00595i	−0.405608	−	0.405608i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.60383			0.0719418
$498$		0			0
$499$	5.30274	+	5.30274i	0.237383	+	0.237383i	0.815766	−	0.578383i	$-0.196316\pi$
$499$	5.30274	+	5.30274i	0.237383	+	0.237383i	−0.578383	+	0.815766i	$0.696316\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			28.8492i			1.28632i	0.765731	+	0.643161i	$0.222378\pi$
$503$			28.8492i			1.28632i	−0.765731	+	0.643161i	$0.777622\pi$
$504$		0			0
$505$			19.0214i			0.846442i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.9968	−	12.9968i	−0.576072	−	0.576072i	0.357747	−	0.933819i	$-0.383545\pi$
$509$	−12.9968	−	12.9968i	−0.576072	−	0.576072i	−0.933819	+	0.357747i	$0.883545\pi$
$510$		0			0
$511$	0.895906			0.0396326
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.86895	−	1.86895i	−0.0823558	−	0.0823558i
$516$		0			0
$517$	14.1014	−	14.1014i	0.620178	−	0.620178i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	13.9833i		−	0.612618i	−0.951932	−	0.306309i	$-0.900906\pi$
$521$		−	13.9833i		−	0.612618i	0.951932	−	0.306309i	$-0.0990941\pi$
$522$		0			0
$523$	6.30689	−	6.30689i	0.275781	−	0.275781i	−0.555641	−	0.831422i	$-0.687527\pi$
$523$	6.30689	−	6.30689i	0.275781	−	0.275781i	0.831422	+	0.555641i	$0.187527\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	24.8420			1.08213
$528$		0			0
$529$	20.4130			0.887523
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12.8502	−	12.8502i	0.556602	−	0.556602i
$534$		0			0
$535$		−	8.88296i		−	0.384044i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−19.9644	+	19.9644i	−0.859929	+	0.859929i
$540$		0			0
$541$	3.89317	+	3.89317i	0.167381	+	0.167381i	0.785827	−	0.618446i	$-0.212238\pi$
$541$	3.89317	+	3.89317i	0.167381	+	0.167381i	−0.618446	+	0.785827i	$0.712238\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.74698			0.417515
$546$		0			0
$547$	27.8376	+	27.8376i	1.19025	+	1.19025i	0.976997	+	0.213251i	$0.0684053\pi$
$547$	27.8376	+	27.8376i	1.19025	+	1.19025i	0.213251	+	0.976997i	$0.431595\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			18.3565i			0.782014i
$552$		0			0
$553$			3.46647i			0.147409i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.50454	−	1.50454i	−0.0637492	−	0.0637492i	0.674513	−	0.738263i	$-0.264353\pi$
$557$	−1.50454	−	1.50454i	−0.0637492	−	0.0637492i	−0.738263	+	0.674513i	$0.764353\pi$
$558$		0			0
$559$	−1.19725			−0.0506382
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.66663	−	6.66663i	−0.280965	−	0.280965i	0.552529	−	0.833494i	$-0.313663\pi$
$563$	−6.66663	−	6.66663i	−0.280965	−	0.280965i	−0.833494	+	0.552529i	$0.813663\pi$
$564$		0			0
$565$	4.56867	−	4.56867i	0.192206	−	0.192206i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			8.38187i			0.351386i	0.984445	+	0.175693i	$0.0562167\pi$
$569$			8.38187i			0.351386i	−0.984445	+	0.175693i	$0.943783\pi$
$570$		0			0
$571$	−28.4129	+	28.4129i	−1.18904	+	1.18904i	−0.211708	+	0.977333i	$0.567903\pi$
$571$	−28.4129	+	28.4129i	−1.18904	+	1.18904i	−0.977333	+	0.211708i	$0.932097\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.60841			−0.0670752
$576$		0			0
$577$	23.2045			0.966014			0.483007	−	0.875616i	$-0.339545\pi$
$577$	23.2045			0.966014			0.483007	+	0.875616i	$0.339545\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5.04450	+	5.04450i	−0.209281	+	0.209281i
$582$		0			0
$583$			30.5640i			1.26583i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.0197	+	11.0197i	−0.454832	+	0.454832i	−0.896955	−	0.442123i	$-0.854226\pi$
$587$	−11.0197	+	11.0197i	−0.454832	+	0.454832i	0.442123	+	0.896955i	$0.354226\pi$
$588$		0			0
$589$	−25.3173	−	25.3173i	−1.04318	−	1.04318i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.98847			0.286982			0.143491	−	0.989652i	$-0.454167\pi$
$593$	6.98847			0.286982			0.143491	+	0.989652i	$0.454167\pi$
$594$		0			0
$595$	−2.58204	−	2.58204i	−0.105853	−	0.105853i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	39.9642i		−	1.63289i	−0.577420	−	0.816447i	$-0.695941\pi$
$599$		−	39.9642i		−	1.63289i	0.577420	−	0.816447i	$-0.304059\pi$
$600$		0			0
$601$		−	21.0830i		−	0.859993i	−0.902831	−	0.429997i	$-0.858515\pi$
$601$		−	21.0830i		−	0.859993i	0.902831	−	0.429997i	$-0.141485\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.46890	+	5.46890i	0.222343	+	0.222343i
$606$		0			0
$607$	−22.3189			−0.905897			−0.452949	−	0.891537i	$-0.649628\pi$
$607$	−22.3189			−0.905897			−0.452949	+	0.891537i	$0.649628\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.7669	+	10.7669i	0.435583	+	0.435583i
$612$		0			0
$613$	−10.6045	+	10.6045i	−0.428312	+	0.428312i	−0.888053	−	0.459741i	$-0.847942\pi$
$613$	−10.6045	+	10.6045i	−0.428312	+	0.428312i	0.459741	+	0.888053i	$0.347942\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			33.7636i			1.35927i	0.733550	+	0.679635i	$0.237862\pi$
$617$			33.7636i			1.35927i	−0.733550	+	0.679635i	$0.762138\pi$
$618$		0			0
$619$	−4.86777	+	4.86777i	−0.195652	+	0.195652i	−0.798133	−	0.602481i	$-0.794179\pi$
$619$	−4.86777	+	4.86777i	−0.195652	+	0.195652i	0.602481	+	0.798133i	$0.294179\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.25384			0.0502341
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	41.7661	−	41.7661i	1.66532	−	1.66532i
$630$		0			0
$631$			16.1348i			0.642315i	0.947026	+	0.321157i	$0.104072\pi$
$631$			16.1348i			0.642315i	−0.947026	+	0.321157i	$0.895928\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.7467	+	11.7467i	−0.466154	+	0.466154i
$636$		0			0
$637$	−15.2436	−	15.2436i	−0.603972	−	0.603972i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20.3125			−0.802296			−0.401148	−	0.916013i	$-0.631389\pi$
$641$	−20.3125			−0.802296			−0.401148	+	0.916013i	$0.631389\pi$
$642$		0			0
$643$	−7.78443	−	7.78443i	−0.306988	−	0.306988i	0.536752	−	0.843740i	$-0.319651\pi$
$643$	−7.78443	−	7.78443i	−0.306988	−	0.306988i	−0.843740	+	0.536752i	$0.819651\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			21.7693i			0.855840i	0.903817	+	0.427920i	$0.140754\pi$
$647$			21.7693i			0.855840i	−0.903817	+	0.427920i	$0.859246\pi$
$648$		0			0
$649$			8.96611i			0.351951i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	26.3118	+	26.3118i	1.02966	+	1.02966i	0.999546	+	0.0301152i	$0.00958743\pi$
$653$	26.3118	+	26.3118i	1.02966	+	1.02966i	0.0301152	+	0.999546i	$0.490413\pi$
$654$		0			0
$655$	16.6479			0.650489
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−20.2389	−	20.2389i	−0.788397	−	0.788397i	0.192835	−	0.981231i	$-0.438232\pi$
$659$	−20.2389	−	20.2389i	−0.788397	−	0.788397i	−0.981231	+	0.192835i	$0.938232\pi$
$660$		0			0
$661$	6.81905	−	6.81905i	0.265230	−	0.265230i	−0.561945	−	0.827175i	$-0.689947\pi$
$661$	6.81905	−	6.81905i	0.265230	−	0.265230i	0.827175	+	0.561945i	$0.189947\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			5.26289i			0.204086i
$666$		0			0
$667$	−2.73942	+	2.73942i	−0.106071	+	0.106071i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−57.1105			−2.20473
$672$		0			0
$673$	−8.19512			−0.315899			−0.157949	−	0.987447i	$-0.550488\pi$
$673$	−8.19512			−0.315899			−0.157949	+	0.987447i	$0.550488\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.8834	+	12.8834i	−0.495151	+	0.495151i	−0.909925	−	0.414774i	$-0.863861\pi$
$677$	−12.8834	+	12.8834i	−0.495151	+	0.495151i	0.414774	+	0.909925i	$0.363861\pi$
$678$		0			0
$679$			3.63960i			0.139675i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.0673	−	15.0673i	0.576535	−	0.576535i	−0.357412	−	0.933947i	$-0.616341\pi$
$683$	15.0673	−	15.0673i	0.576535	−	0.576535i	0.933947	+	0.357412i	$0.116341\pi$
$684$		0			0
$685$	−5.95027	−	5.95027i	−0.227348	−	0.227348i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−23.3367			−0.889058
$690$		0			0
$691$	5.23733	+	5.23733i	0.199237	+	0.199237i	0.799673	−	0.600436i	$-0.205006\pi$
$691$	5.23733	+	5.23733i	0.199237	+	0.199237i	−0.600436	+	0.799673i	$0.705006\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			2.14741i			0.0814559i
$696$		0			0
$697$			29.0766i			1.10135i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−21.7664	−	21.7664i	−0.822106	−	0.822106i	0.164303	−	0.986410i	$-0.447462\pi$
$701$	−21.7664	−	21.7664i	−0.822106	−	0.822106i	−0.986410	+	0.164303i	$0.947462\pi$
$702$		0			0
$703$	−85.1304			−3.21075
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.28836	+	9.28836i	0.349325	+	0.349325i
$708$		0			0
$709$	−23.9643	+	23.9643i	−0.899997	+	0.899997i	−0.995435	−	0.0954387i	$-0.969575\pi$
$709$	−23.9643	+	23.9643i	−0.899997	+	0.899997i	0.0954387	+	0.995435i	$0.469575\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	7.55640i		−	0.282990i
$714$		0			0
$715$	−10.1146	+	10.1146i	−0.378266	+	0.378266i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−44.4408			−1.65736			−0.828681	−	0.559721i	$-0.810908\pi$
$719$	−44.4408			−1.65736			−0.828681	+	0.559721i	$0.810908\pi$
$720$		0			0
$721$	−1.82526			−0.0679762
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.70319	+	1.70319i	−0.0632548	+	0.0632548i
$726$		0			0
$727$			46.6543i			1.73031i	0.501504	+	0.865155i	$0.332780\pi$
$727$			46.6543i			1.73031i	−0.501504	+	0.865155i	$0.667220\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.35453	−	1.35453i	0.0500992	−	0.0500992i
$732$		0			0
$733$	19.4202	+	19.4202i	0.717303	+	0.717303i	0.968052	−	0.250749i	$-0.0806770\pi$
$733$	19.4202	+	19.4202i	0.717303	+	0.717303i	−0.250749	+	0.968052i	$0.580677\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.9238			−0.439219
$738$		0			0
$739$	20.5243	+	20.5243i	0.754999	+	0.754999i	0.975408	−	0.220409i	$-0.0707392\pi$
$739$	20.5243	+	20.5243i	0.754999	+	0.754999i	−0.220409	+	0.975408i	$0.570739\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			12.9245i			0.474154i	0.971491	+	0.237077i	$0.0761893\pi$
$743$			12.9245i			0.474154i	−0.971491	+	0.237077i	$0.923811\pi$
$744$		0			0
$745$			3.69732i			0.135459i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.33765	−	4.33765i	−0.158494	−	0.158494i
$750$		0			0
$751$	52.2694			1.90734			0.953668	−	0.300861i	$-0.0972740\pi$
$751$	52.2694			1.90734			0.953668	+	0.300861i	$0.0972740\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.99036	+	8.99036i	0.327193	+	0.327193i
$756$		0			0
$757$	−34.4514	+	34.4514i	−1.25216	+	1.25216i	−0.297407	+	0.954751i	$0.596122\pi$
$757$	−34.4514	+	34.4514i	−1.25216	+	1.25216i	−0.954751	+	0.297407i	$0.903878\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	47.7467i		−	1.73082i	−0.501067	−	0.865408i	$-0.667059\pi$
$761$		−	47.7467i		−	1.73082i	0.501067	−	0.865408i	$-0.332941\pi$
$762$		0			0
$763$	4.75956	−	4.75956i	0.172308	−	0.172308i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.84595			−0.247193
$768$		0			0
$769$	−17.9108			−0.645882			−0.322941	−	0.946419i	$-0.604671\pi$
$769$	−17.9108			−0.645882			−0.322941	+	0.946419i	$0.604671\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3.73170	+	3.73170i	−0.134220	+	0.134220i	−0.771025	−	0.636805i	$-0.780256\pi$
$773$	−3.73170	+	3.73170i	−0.134220	+	0.134220i	0.636805	+	0.771025i	$0.280256\pi$
$774$		0			0
$775$		−	4.69807i		−	0.168760i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	29.6329	−	29.6329i	1.06171	−	1.06171i
$780$		0			0
$781$	7.10805	+	7.10805i	0.254346	+	0.254346i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.1517			0.362328
$786$		0			0
$787$	−2.40160	−	2.40160i	−0.0856076	−	0.0856076i	0.663006	−	0.748614i	$-0.269280\pi$
$787$	−2.40160	−	2.40160i	−0.0856076	−	0.0856076i	−0.748614	+	0.663006i	$0.769280\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	4.46187i		−	0.158646i
$792$		0			0
$793$		−	43.6060i		−	1.54849i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−35.4972	−	35.4972i	−1.25738	−	1.25738i	−0.952341	−	0.305035i	$-0.901332\pi$
$797$	−35.4972	−	35.4972i	−1.25738	−	1.25738i	−0.305035	−	0.952341i	$-0.598668\pi$
$798$		0			0
$799$	−24.3627			−0.861892
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.97058	+	3.97058i	0.140119	+	0.140119i
$804$		0			0
$805$	−0.785403	+	0.785403i	−0.0276818	+	0.0276818i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	11.9182i		−	0.419021i	−0.977806	−	0.209510i	$-0.932813\pi$
$809$		−	11.9182i		−	0.419021i	0.977806	−	0.209510i	$-0.0671870\pi$
$810$		0			0
$811$	22.1494	−	22.1494i	0.777772	−	0.777772i	−0.201680	−	0.979452i	$-0.564640\pi$
$811$	22.1494	−	22.1494i	0.777772	−	0.777772i	0.979452	+	0.201680i	$0.0646400\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.97694			−0.349477
$816$		0			0
$817$	−2.76090			−0.0965915
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.3909	+	13.3909i	−0.467344	+	0.467344i	−0.901053	−	0.433709i	$-0.857205\pi$
$821$	−13.3909	+	13.3909i	−0.467344	+	0.467344i	0.433709	+	0.901053i	$0.357205\pi$
$822$		0			0
$823$			43.9496i			1.53199i	0.642848	+	0.765994i	$0.277753\pi$
$823$			43.9496i			1.53199i	−0.642848	+	0.765994i	$0.722247\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.79096	−	1.79096i	0.0622777	−	0.0622777i	−0.675282	−	0.737560i	$-0.735978\pi$
$827$	1.79096	−	1.79096i	0.0622777	−	0.0622777i	0.737560	+	0.675282i	$0.235978\pi$
$828$		0			0
$829$	13.4979	+	13.4979i	0.468801	+	0.468801i	0.901526	−	0.432725i	$-0.142448\pi$
$829$	13.4979	+	13.4979i	0.468801	+	0.468801i	−0.432725	+	0.901526i	$0.642448\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	34.4923			1.19509
$834$		0			0
$835$	2.76186	+	2.76186i	0.0955780	+	0.0955780i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			14.5332i			0.501741i	0.968021	+	0.250870i	$0.0807168\pi$
$839$			14.5332i			0.501741i	−0.968021	+	0.250870i	$0.919283\pi$
$840$		0			0
$841$		−	23.1983i		−	0.799941i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.46950	+	1.46950i	0.0505524	+	0.0505524i
$846$		0			0
$847$	5.34105			0.183521
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.7043	−	12.7043i	−0.435499	−	0.435499i
$852$		0			0
$853$	11.5836	−	11.5836i	0.396615	−	0.396615i	−0.480423	−	0.877037i	$-0.659517\pi$
$853$	11.5836	−	11.5836i	0.396615	−	0.396615i	0.877037	+	0.480423i	$0.159517\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			15.6443i			0.534399i	0.963641	+	0.267200i	$0.0860983\pi$
$857$			15.6443i			0.534399i	−0.963641	+	0.267200i	$0.913902\pi$
$858$		0			0
$859$	−12.0947	+	12.0947i	−0.412665	+	0.412665i	−0.882666	−	0.470001i	$-0.844254\pi$
$859$	−12.0947	+	12.0947i	−0.412665	+	0.412665i	0.470001	+	0.882666i	$0.344254\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−9.28120			−0.315936			−0.157968	−	0.987444i	$-0.550494\pi$
$863$	−9.28120			−0.315936			−0.157968	+	0.987444i	$0.550494\pi$
$864$		0			0
$865$	11.5985			0.394362
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−15.3631	+	15.3631i	−0.521157	+	0.521157i
$870$		0			0
$871$		−	9.10425i		−	0.308486i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.488311	+	0.488311i	−0.0165079	+	0.0165079i
$876$		0			0
$877$	−2.97610	−	2.97610i	−0.100496	−	0.100496i	0.655071	−	0.755567i	$-0.272639\pi$
$877$	−2.97610	−	2.97610i	−0.100496	−	0.100496i	−0.755567	+	0.655071i	$0.772639\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−29.3318			−0.988214			−0.494107	−	0.869401i	$-0.664505\pi$
$881$	−29.3318			−0.988214			−0.494107	+	0.869401i	$0.664505\pi$
$882$		0			0
$883$	−35.5597	−	35.5597i	−1.19668	−	1.19668i	−0.975155	−	0.221525i	$-0.928897\pi$
$883$	−35.5597	−	35.5597i	−1.19668	−	1.19668i	−0.221525	−	0.975155i	$-0.571103\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	4.51671i		−	0.151656i	−0.997121	−	0.0758282i	$-0.975840\pi$
$887$		−	4.51671i		−	0.151656i	0.997121	−	0.0758282i	$-0.0241601\pi$
$888$		0			0
$889$			11.4721i			0.384762i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	24.8289	+	24.8289i	0.830867	+	0.830867i
$894$		0			0
$895$	4.38920			0.146715
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−8.00169	−	8.00169i	−0.266871	−	0.266871i
$900$		0			0
$901$	26.4025	−	26.4025i	0.879594	−	0.879594i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	2.70172i		−	0.0898083i
$906$		0			0
$907$	5.06769	−	5.06769i	0.168270	−	0.168270i	−0.617949	−	0.786218i	$-0.712036\pi$
$907$	5.06769	−	5.06769i	0.168270	−	0.168270i	0.786218	+	0.617949i	$0.212036\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	36.7140			1.21639			0.608194	−	0.793788i	$-0.291894\pi$
$911$	36.7140			1.21639			0.608194	+	0.793788i	$0.291894\pi$
$912$		0			0
$913$	−44.7135			−1.47980
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.12937	−	8.12937i	0.268456	−	0.268456i
$918$		0			0
$919$		−	21.5651i		−	0.711365i	−0.934607	−	0.355683i	$-0.884248\pi$
$919$		−	21.5651i		−	0.711365i	0.934607	−	0.355683i	$-0.115752\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.42725	+	5.42725i	−0.178640	+	0.178640i
$924$		0			0
$925$	−7.89871	−	7.89871i	−0.259708	−	0.259708i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	45.6603			1.49807			0.749033	−	0.662532i	$-0.230518\pi$
$929$	45.6603			1.49807			0.749033	+	0.662532i	$0.230518\pi$
$930$		0			0
$931$	−35.1522	−	35.1522i	−1.15207	−	1.15207i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	22.8868i		−	0.748478i
$936$		0			0
$937$		−	2.29807i		−	0.0750746i	−0.999295	−	0.0375373i	$-0.988049\pi$
$937$		−	2.29807i		−	0.0750746i	0.999295	−	0.0375373i	$-0.0119513\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	24.1999	+	24.1999i	0.788894	+	0.788894i	0.981313	−	0.192419i	$-0.0616333\pi$
$941$	24.1999	+	24.1999i	0.788894	+	0.788894i	−0.192419	+	0.981313i	$0.561633\pi$
$942$		0			0
$943$	8.84448			0.288016
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24.5182	+	24.5182i	0.796733	+	0.796733i	0.982579	−	0.185846i	$-0.0595025\pi$
$947$	24.5182	+	24.5182i	0.796733	+	0.796733i	−0.185846	+	0.982579i	$0.559502\pi$
$948$		0			0
$949$	−3.03168	+	3.03168i	−0.0984125	+	0.0984125i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	32.3462i		−	1.04780i	−0.851781	−	0.523898i	$-0.824477\pi$
$953$		−	32.3462i		−	1.04780i	0.851781	−	0.523898i	$-0.175523\pi$
$954$		0			0
$955$	−3.96716	+	3.96716i	−0.128374	+	0.128374i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5.81116			−0.187652
$960$		0			0
$961$	−8.92816			−0.288005
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.76264	+	2.76264i	−0.0889326	+	0.0889326i
$966$		0			0
$967$			14.6983i			0.472665i	0.971672	+	0.236333i	$0.0759455\pi$
$967$			14.6983i			0.472665i	−0.971672	+	0.236333i	$0.924054\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	29.1065	−	29.1065i	0.934073	−	0.934073i	−0.0638845	−	0.997957i	$-0.520349\pi$
$971$	29.1065	−	29.1065i	0.934073	−	0.934073i	0.997957	+	0.0638845i	$0.0203489\pi$
$972$		0			0
$973$	1.04860	+	1.04860i	0.0336167	+	0.0336167i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17.3533			−0.555180			−0.277590	−	0.960700i	$-0.589536\pi$
$977$	−17.3533			−0.555180			−0.277590	+	0.960700i	$0.589536\pi$
$978$		0			0
$979$	5.55691	+	5.55691i	0.177600	+	0.177600i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			27.5174i			0.877668i	0.898568	+	0.438834i	$0.144608\pi$
$983$			27.5174i			0.877668i	−0.898568	+	0.438834i	$0.855392\pi$
$984$		0			0
$985$		−	0.860458i		−	0.0274165i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.412020	−	0.412020i	−0.0131015	−	0.0131015i
$990$		0			0
$991$	−6.96363			−0.221207			−0.110604	−	0.993865i	$-0.535278\pi$
$991$	−6.96363			−0.221207			−0.110604	+	0.993865i	$0.535278\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	10.9801	+	10.9801i	0.348092	+	0.348092i
$996$		0			0
$997$	−15.7051	+	15.7051i	−0.497385	+	0.497385i	−0.910623	−	0.413238i	$-0.864398\pi$
$997$	−15.7051	+	15.7051i	−0.497385	+	0.497385i	0.413238	+	0.910623i	$0.364398\pi$
$998$		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.721.3		16
3.2	odd	2		320.2.l.a.81.2		16
4.3	odd	2		720.2.t.c.541.4		16
12.11	even	2		80.2.l.a.61.5	yes	16
15.2	even	4		1600.2.q.g.849.2		16
15.8	even	4		1600.2.q.h.849.7		16
15.14	odd	2		1600.2.l.i.401.7		16
16.5	even	4	inner	2880.2.t.c.2161.2		16
16.11	odd	4		720.2.t.c.181.4		16
24.5	odd	2		640.2.l.a.161.7		16
24.11	even	2		640.2.l.b.161.2		16
48.5	odd	4		320.2.l.a.241.2		16
48.11	even	4		80.2.l.a.21.5	✓	16
48.29	odd	4		640.2.l.a.481.7		16
48.35	even	4		640.2.l.b.481.2		16
60.23	odd	4		400.2.q.g.349.1		16
60.47	odd	4		400.2.q.h.349.8		16
60.59	even	2		400.2.l.h.301.4		16
96.5	odd	8		5120.2.a.t.1.7		8
96.11	even	8		5120.2.a.s.1.7		8
96.53	odd	8		5120.2.a.u.1.2		8
96.59	even	8		5120.2.a.v.1.2		8
240.53	even	4		1600.2.q.g.49.2		16
240.59	even	4		400.2.l.h.101.4		16
240.107	odd	4		400.2.q.g.149.1		16
240.149	odd	4		1600.2.l.i.1201.7		16
240.197	even	4		1600.2.q.h.49.7		16
240.203	odd	4		400.2.q.h.149.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.5	✓	16	48.11	even	4
80.2.l.a.61.5	yes	16	12.11	even	2
320.2.l.a.81.2		16	3.2	odd	2
320.2.l.a.241.2		16	48.5	odd	4
400.2.l.h.101.4		16	240.59	even	4
400.2.l.h.301.4		16	60.59	even	2
400.2.q.g.149.1		16	240.107	odd	4
400.2.q.g.349.1		16	60.23	odd	4
400.2.q.h.149.8		16	240.203	odd	4
400.2.q.h.349.8		16	60.47	odd	4
640.2.l.a.161.7		16	24.5	odd	2
640.2.l.a.481.7		16	48.29	odd	4
640.2.l.b.161.2		16	24.11	even	2
640.2.l.b.481.2		16	48.35	even	4
720.2.t.c.181.4		16	16.11	odd	4
720.2.t.c.541.4		16	4.3	odd	2
1600.2.l.i.401.7		16	15.14	odd	2
1600.2.l.i.1201.7		16	240.149	odd	4
1600.2.q.g.49.2		16	240.53	even	4
1600.2.q.g.849.2		16	15.2	even	4
1600.2.q.h.49.7		16	240.197	even	4
1600.2.q.h.849.7		16	15.8	even	4
2880.2.t.c.721.3		16	1.1	even	1	trivial
2880.2.t.c.2161.2		16	16.5	even	4	inner
5120.2.a.s.1.7		8	96.11	even	8
5120.2.a.t.1.7		8	96.5	odd	8
5120.2.a.u.1.2		8	96.53	odd	8
5120.2.a.v.1.2		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)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{5} +0.690576i q^{7} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{5} +0.690576i q^{7} +(-3.06057 + 3.06057i) q^{11} +(-2.33686 - 2.33686i) q^{13} +5.28770 q^{17} +(-5.38887 - 5.38887i) q^{19} -1.60841i q^{23} -1.00000i q^{25} +(-1.70319 - 1.70319i) q^{29} +4.69807 q^{31} +(-0.488311 - 0.488311i) q^{35} +(7.89871 - 7.89871i) q^{37} +5.49891i q^{41} +(0.256166 - 0.256166i) q^{43} -4.60743 q^{47} +6.52310 q^{49} +(4.99318 - 4.99318i) q^{53} -4.32830i q^{55} +(1.46478 - 1.46478i) q^{59} +(9.33004 + 9.33004i) q^{61} +3.30482 q^{65} +(1.94797 + 1.94797i) q^{67} -2.32246i q^{71} -1.29733i q^{73} +(-2.11356 - 2.11356i) q^{77} +5.01968 q^{79} +(7.30477 + 7.30477i) q^{83} +(-3.73897 + 3.73897i) q^{85} -1.81564i q^{89} +(1.61378 - 1.61378i) q^{91} +7.62102 q^{95} +5.27038 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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

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