LMFDB - Embedded newform 2880.2.t.d.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:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + \cdots + 1024 $ x^20 - 2x^18 - 4x^17 + 7x^16 + 16x^15 + 6x^14 - 36x^13 - 42x^12 + 40x^11 + 136x^10 + 80x^9 - 168x^8 - 288x^7 + 96x^6 + 512x^5 + 448x^4 - 512x^3 - 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			721.3
Root			$-0.720859 - 1.21670i$ of defining polynomial
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.d.2161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{5} -0.0588949i q^{7} +O(q^{10})$	$q+(-0.707107 + 0.707107i) q^{5} -0.0588949i q^{7} +(2.23289 - 2.23289i) q^{11} +(2.84870 + 2.84870i) q^{13} -5.98228 q^{17} +(0.617238 + 0.617238i) q^{19} -0.746698i q^{23} -1.00000i q^{25} +(1.13987 + 1.13987i) q^{29} +8.55143 q^{31} +(0.0416450 + 0.0416450i) q^{35} +(-2.01811 + 2.01811i) q^{37} +7.71113i q^{41} +(2.94233 - 2.94233i) q^{43} -0.789616 q^{47} +6.99653 q^{49} +(6.80791 - 6.80791i) q^{53} +3.15778i q^{55} +(-9.36045 + 9.36045i) q^{59} +(-0.814225 - 0.814225i) q^{61} -4.02867 q^{65} +(-5.46701 - 5.46701i) q^{67} -7.40423i q^{71} +11.6114i q^{73} +(-0.131506 - 0.131506i) q^{77} +17.4027 q^{79} +(-7.55090 - 7.55090i) q^{83} +(4.23011 - 4.23011i) q^{85} +16.3007i q^{89} +(0.167774 - 0.167774i) q^{91} -0.872906 q^{95} +12.3159 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q + 8 q^{11} + 24 q^{17} + 4 q^{19} - 16 q^{29} + 16 q^{37} + 8 q^{43} - 52 q^{49} + 16 q^{53} - 16 q^{59} - 4 q^{61} + 8 q^{67} + 40 q^{77} - 56 q^{79} - 48 q^{83} + 4 q^{85} + 8 q^{91} + 56 q^{97}+O(q^{100}) $ 20 * q + 8 * q^11 + 24 * q^17 + 4 * q^19 - 16 * q^29 + 16 * q^37 + 8 * q^43 - 52 * q^49 + 16 * q^53 - 16 * q^59 - 4 * q^61 + 8 * q^67 + 40 * q^77 - 56 * q^79 - 48 * q^83 + 4 * q^85 + 8 * q^91 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.707107	+	0.707107i	−0.316228	+	0.316228i
$5$	−0.707107	+	0.707107i	−0.316228	+	0.316228i
$6$		0			0
$7$		−	0.0588949i		−	0.0222602i	−0.999938	−	0.0111301i	$-0.996457\pi$
$7$		−	0.0588949i		−	0.0222602i	0.999938	−	0.0111301i	$-0.00354289\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.23289	−	2.23289i	0.673242	−	0.673242i	−0.285220	−	0.958462i	$-0.592067\pi$
$11$	2.23289	−	2.23289i	0.673242	−	0.673242i	0.958462	+	0.285220i	$0.0920668\pi$
$12$		0			0
$13$	2.84870	+	2.84870i	0.790087	+	0.790087i	0.981508	−	0.191421i	$-0.0613095\pi$
$13$	2.84870	+	2.84870i	0.790087	+	0.790087i	−0.191421	+	0.981508i	$0.561310\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.98228			−1.45092			−0.725458	−	0.688267i	$-0.758372\pi$
$17$	−5.98228			−1.45092			−0.725458	+	0.688267i	$0.758372\pi$
$18$		0			0
$19$	0.617238	+	0.617238i	0.141604	+	0.141604i	0.774355	−	0.632751i	$-0.218074\pi$
$19$	0.617238	+	0.617238i	0.141604	+	0.141604i	−0.632751	+	0.774355i	$0.718074\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.746698i		−	0.155697i	−0.996965	−	0.0778486i	$-0.975195\pi$
$23$		−	0.746698i		−	0.155697i	0.996965	−	0.0778486i	$-0.0248051\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.13987	+	1.13987i	0.211668	+	0.211668i	0.804976	−	0.593308i	$-0.202178\pi$
$29$	1.13987	+	1.13987i	0.211668	+	0.211668i	−0.593308	+	0.804976i	$0.702178\pi$
$30$		0			0
$31$	8.55143			1.53588			0.767941	−	0.640520i	$-0.221281\pi$
$31$	8.55143			1.53588			0.767941	+	0.640520i	$0.221281\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.0416450	+	0.0416450i	0.00703929	+	0.00703929i
$36$		0			0
$37$	−2.01811	+	2.01811i	−0.331774	+	0.331774i	−0.853260	−	0.521486i	$-0.825378\pi$
$37$	−2.01811	+	2.01811i	−0.331774	+	0.331774i	0.521486	+	0.853260i	$0.325378\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.71113i			1.20428i	0.798392	+	0.602138i	$0.205684\pi$
$41$			7.71113i			1.20428i	−0.798392	+	0.602138i	$0.794316\pi$
$42$		0			0
$43$	2.94233	−	2.94233i	0.448701	−	0.448701i	−0.446221	−	0.894923i	$-0.647231\pi$
$43$	2.94233	−	2.94233i	0.448701	−	0.448701i	0.894923	+	0.446221i	$0.147231\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.789616			−0.115177			−0.0575887	−	0.998340i	$-0.518341\pi$
$47$	−0.789616			−0.115177			−0.0575887	+	0.998340i	$0.518341\pi$
$48$		0			0
$49$	6.99653			0.999504
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.80791	−	6.80791i	0.935138	−	0.935138i	−0.0628826	−	0.998021i	$-0.520029\pi$
$53$	6.80791	−	6.80791i	0.935138	−	0.935138i	0.998021	+	0.0628826i	$0.0200294\pi$
$54$		0			0
$55$			3.15778i			0.425795i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.36045	+	9.36045i	−1.21863	+	1.21863i	−0.250514	+	0.968113i	$0.580599\pi$
$59$	−9.36045	+	9.36045i	−1.21863	+	1.21863i	−0.968113	+	0.250514i	$0.919401\pi$
$60$		0			0
$61$	−0.814225	−	0.814225i	−0.104251	−	0.104251i	0.653057	−	0.757308i	$-0.273486\pi$
$61$	−0.814225	−	0.814225i	−0.104251	−	0.104251i	−0.757308	+	0.653057i	$0.773486\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.02867			−0.499695
$66$		0			0
$67$	−5.46701	−	5.46701i	−0.667901	−	0.667901i	0.289329	−	0.957230i	$-0.406568\pi$
$67$	−5.46701	−	5.46701i	−0.667901	−	0.667901i	−0.957230	+	0.289329i	$0.906568\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.40423i		−	0.878720i	−0.898311	−	0.439360i	$-0.855205\pi$
$71$		−	7.40423i		−	0.878720i	0.898311	−	0.439360i	$-0.144795\pi$
$72$		0			0
$73$			11.6114i			1.35901i	0.733670	+	0.679506i	$0.237806\pi$
$73$			11.6114i			1.35901i	−0.733670	+	0.679506i	$0.762194\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.131506	−	0.131506i	−0.0149865	−	0.0149865i
$78$		0			0
$79$	17.4027			1.95796			0.978978	−	0.203964i	$-0.0653827\pi$
$79$	17.4027			1.95796			0.978978	+	0.203964i	$0.0653827\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.55090	−	7.55090i	−0.828819	−	0.828819i	0.158534	−	0.987354i	$-0.449323\pi$
$83$	−7.55090	−	7.55090i	−0.828819	−	0.828819i	−0.987354	+	0.158534i	$0.949323\pi$
$84$		0			0
$85$	4.23011	−	4.23011i	0.458820	−	0.458820i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			16.3007i			1.72788i	0.503599	+	0.863938i	$0.332009\pi$
$89$			16.3007i			1.72788i	−0.503599	+	0.863938i	$0.667991\pi$
$90$		0			0
$91$	0.167774	−	0.167774i	0.0175875	−	0.0175875i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.872906			−0.0895583
$96$		0			0
$97$	12.3159			1.25049			0.625247	−	0.780427i	$-0.284998\pi$
$97$	12.3159			1.25049			0.625247	+	0.780427i	$0.284998\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.663582	−	0.663582i	0.0660289	−	0.0660289i	−0.673321	−	0.739350i	$-0.735133\pi$
$101$	0.663582	−	0.663582i	0.0660289	−	0.0660289i	0.739350	+	0.673321i	$0.235133\pi$
$102$		0			0
$103$			14.9036i			1.46850i	0.678880	+	0.734249i	$0.262466\pi$
$103$			14.9036i			1.46850i	−0.678880	+	0.734249i	$0.737534\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.43861	−	3.43861i	0.332423	−	0.332423i	−0.521083	−	0.853506i	$-0.674472\pi$
$107$	3.43861	−	3.43861i	0.332423	−	0.332423i	0.853506	+	0.521083i	$0.174472\pi$
$108$		0			0
$109$	0.0571202	+	0.0571202i	0.00547113	+	0.00547113i	0.709837	−	0.704366i	$-0.248769\pi$
$109$	0.0571202	+	0.0571202i	0.00547113	+	0.00547113i	−0.704366	+	0.709837i	$0.748769\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.9834			1.40952			0.704759	−	0.709447i	$-0.251055\pi$
$113$	14.9834			1.40952			0.704759	+	0.709447i	$0.251055\pi$
$114$		0			0
$115$	0.527995	+	0.527995i	0.0492358	+	0.0492358i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			0.352326i			0.0322977i
$120$		0			0
$121$			1.02840i			0.0934912i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$	−6.74629			−0.598636			−0.299318	−	0.954153i	$-0.596759\pi$
$127$	−6.74629			−0.598636			−0.299318	+	0.954153i	$0.596759\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.2459	+	10.2459i	0.895185	+	0.895185i	0.995005	−	0.0998208i	$-0.0318269\pi$
$131$	10.2459	+	10.2459i	0.895185	+	0.895185i	−0.0998208	+	0.995005i	$0.531827\pi$
$132$		0			0
$133$	0.0363522	−	0.0363522i	0.00315214	−	0.00315214i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			19.4514i			1.66185i	0.556388	+	0.830923i	$0.312187\pi$
$137$			19.4514i			1.66185i	−0.556388	+	0.830923i	$0.687813\pi$
$138$		0			0
$139$	1.09587	−	1.09587i	0.0929501	−	0.0929501i	−0.659103	−	0.752053i	$-0.729064\pi$
$139$	1.09587	−	1.09587i	0.0929501	−	0.0929501i	0.752053	+	0.659103i	$0.229064\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.7217			1.06384
$144$		0			0
$145$	−1.61202			−0.133871
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.6510	−	13.6510i	1.11834	−	1.11834i	0.126349	−	0.991986i	$-0.459674\pi$
$149$	13.6510	−	13.6510i	1.11834	−	1.11834i	0.991986	−	0.126349i	$-0.0403260\pi$
$150$		0			0
$151$			13.4811i			1.09708i	0.836125	+	0.548538i	$0.184816\pi$
$151$			13.4811i			1.09708i	−0.836125	+	0.548538i	$0.815184\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.04678	+	6.04678i	−0.485689	+	0.485689i
$156$		0			0
$157$	11.1090	+	11.1090i	0.886593	+	0.886593i	0.994194	−	0.107602i	$-0.0343171\pi$
$157$	11.1090	+	11.1090i	0.886593	+	0.886593i	−0.107602	+	0.994194i	$0.534317\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.0439767			−0.00346585
$162$		0			0
$163$	1.97598	+	1.97598i	0.154771	+	0.154771i	0.780245	−	0.625474i	$-0.215094\pi$
$163$	1.97598	+	1.97598i	0.154771	+	0.154771i	−0.625474	+	0.780245i	$0.715094\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2.12777i		−	0.164652i	−0.996605	−	0.0823259i	$-0.973765\pi$
$167$		−	2.12777i		−	0.164652i	0.996605	−	0.0823259i	$-0.0262348\pi$
$168$		0			0
$169$			3.23018i			0.248476i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.21877	+	9.21877i	0.700890	+	0.700890i	0.964602	−	0.263711i	$-0.0849466\pi$
$173$	9.21877	+	9.21877i	0.700890	+	0.700890i	−0.263711	+	0.964602i	$0.584947\pi$
$174$		0			0
$175$	−0.0588949			−0.00445204
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.02407	+	5.02407i	0.375516	+	0.375516i	0.869482	−	0.493965i	$-0.164453\pi$
$179$	5.02407	+	5.02407i	0.375516	+	0.375516i	−0.493965	+	0.869482i	$0.664453\pi$
$180$		0			0
$181$	−15.1363	+	15.1363i	−1.12507	+	1.12507i	−0.134102	+	0.990967i	$0.542815\pi$
$181$	−15.1363	+	15.1363i	−1.12507	+	1.12507i	−0.990967	+	0.134102i	$0.957185\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	2.85403i		−	0.209833i
$186$		0			0
$187$	−13.3578	+	13.3578i	−0.976817	+	0.976817i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.4425			0.900310			0.450155	−	0.892950i	$-0.351369\pi$
$191$	12.4425			0.900310			0.450155	+	0.892950i	$0.351369\pi$
$192$		0			0
$193$	0.241933			0.0174147			0.00870734	−	0.999962i	$-0.497228\pi$
$193$	0.241933			0.0174147			0.00870734	+	0.999962i	$0.497228\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.37260	−	2.37260i	0.169041	−	0.169041i	−0.617517	−	0.786558i	$-0.711861\pi$
$197$	2.37260	−	2.37260i	0.169041	−	0.169041i	0.786558	+	0.617517i	$0.211861\pi$
$198$		0			0
$199$		−	19.8275i		−	1.40553i	−0.711420	−	0.702767i	$-0.751948\pi$
$199$		−	19.8275i		−	1.40553i	0.711420	−	0.702767i	$-0.248052\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.0671324	−	0.0671324i	0.00471177	−	0.00471177i
$204$		0			0
$205$	−5.45259	−	5.45259i	−0.380826	−	0.380826i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.75645			0.190668
$210$		0			0
$211$	−3.54907	−	3.54907i	−0.244328	−	0.244328i	0.574310	−	0.818638i	$-0.305270\pi$
$211$	−3.54907	−	3.54907i	−0.244328	−	0.244328i	−0.818638	+	0.574310i	$0.805270\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			4.16109i			0.283784i
$216$		0			0
$217$		−	0.503636i		−	0.0341890i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−17.0417	−	17.0417i	−1.14635	−	1.14635i
$222$		0			0
$223$	−21.6789			−1.45173			−0.725864	−	0.687838i	$-0.758560\pi$
$223$	−21.6789			−1.45173			−0.725864	+	0.687838i	$0.758560\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.2117	−	10.2117i	−0.677775	−	0.677775i	0.281721	−	0.959496i	$-0.409095\pi$
$227$	−10.2117	−	10.2117i	−0.677775	−	0.677775i	−0.959496	+	0.281721i	$0.909095\pi$
$228$		0			0
$229$	17.0933	−	17.0933i	1.12956	−	1.12956i	0.139312	−	0.990249i	$-0.455511\pi$
$229$	17.0933	−	17.0933i	1.12956	−	1.12956i	0.990249	−	0.139312i	$-0.0444890\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	24.2409i		−	1.58807i	−0.607871	−	0.794036i	$-0.707976\pi$
$233$		−	24.2409i		−	1.58807i	0.607871	−	0.794036i	$-0.292024\pi$
$234$		0			0
$235$	0.558343	−	0.558343i	0.0364223	−	0.0364223i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	21.0658			1.36263			0.681317	−	0.731989i	$-0.261408\pi$
$239$	21.0658			1.36263			0.681317	+	0.731989i	$0.261408\pi$
$240$		0			0
$241$	−22.0578			−1.42087			−0.710434	−	0.703764i	$-0.751501\pi$
$241$	−22.0578			−1.42087			−0.710434	+	0.703764i	$0.751501\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.94729	+	4.94729i	−0.316071	+	0.316071i
$246$		0			0
$247$			3.51665i			0.223759i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.22942	−	8.22942i	0.519436	−	0.519436i	−0.397964	−	0.917401i	$-0.630283\pi$
$251$	8.22942	−	8.22942i	0.519436	−	0.519436i	0.917401	+	0.397964i	$0.130283\pi$
$252$		0			0
$253$	−1.66729	−	1.66729i	−0.104822	−	0.104822i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.32164			−0.456712			−0.228356	−	0.973578i	$-0.573335\pi$
$257$	−7.32164			−0.456712			−0.228356	+	0.973578i	$0.573335\pi$
$258$		0			0
$259$	0.118856	+	0.118856i	0.00738536	+	0.00738536i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			18.6430i			1.14958i	0.818302	+	0.574788i	$0.194915\pi$
$263$			18.6430i			1.14958i	−0.818302	+	0.574788i	$0.805085\pi$
$264$		0			0
$265$			9.62784i			0.591433i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.22889	+	3.22889i	0.196869	+	0.196869i	0.798656	−	0.601787i	$-0.205544\pi$
$269$	3.22889	+	3.22889i	0.196869	+	0.196869i	−0.601787	+	0.798656i	$0.705544\pi$
$270$		0			0
$271$	27.9381			1.69712			0.848560	−	0.529099i	$-0.177470\pi$
$271$	27.9381			1.69712			0.848560	+	0.529099i	$0.177470\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.23289	−	2.23289i	−0.134648	−	0.134648i
$276$		0			0
$277$	−1.58682	+	1.58682i	−0.0953431	+	0.0953431i	−0.753170	−	0.657826i	$-0.771476\pi$
$277$	−1.58682	+	1.58682i	−0.0953431	+	0.0953431i	0.657826	+	0.753170i	$0.271476\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			6.36028i			0.379423i	0.981840	+	0.189711i	$0.0607552\pi$
$281$			6.36028i			0.379423i	−0.981840	+	0.189711i	$0.939245\pi$
$282$		0			0
$283$	−3.64115	+	3.64115i	−0.216444	+	0.216444i	−0.806998	−	0.590554i	$-0.798909\pi$
$283$	−3.64115	+	3.64115i	−0.216444	+	0.216444i	0.590554	+	0.806998i	$0.298909\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.454146			0.0268074
$288$		0			0
$289$	18.7876			1.10516
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.09157	−	4.09157i	0.239032	−	0.239032i	−0.577417	−	0.816449i	$-0.695939\pi$
$293$	4.09157	−	4.09157i	0.239032	−	0.239032i	0.816449	+	0.577417i	$0.195939\pi$
$294$		0			0
$295$		−	13.2377i		−	0.770727i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.12712	−	2.12712i	0.123014	−	0.123014i
$300$		0			0
$301$	−0.173288	−	0.173288i	−0.00998818	−	0.00998818i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.15149			0.0659341
$306$		0			0
$307$	−0.832070	−	0.832070i	−0.0474887	−	0.0474887i	0.682964	−	0.730452i	$-0.260691\pi$
$307$	−0.832070	−	0.832070i	−0.0474887	−	0.0474887i	−0.730452	+	0.682964i	$0.760691\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			13.8376i			0.784657i	0.919825	+	0.392329i	$0.128330\pi$
$311$			13.8376i			0.784657i	−0.919825	+	0.392329i	$0.871670\pi$
$312$		0			0
$313$			5.09179i			0.287805i	0.989592	+	0.143902i	$0.0459651\pi$
$313$			5.09179i			0.287805i	−0.989592	+	0.143902i	$0.954035\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.3055	−	16.3055i	−0.915811	−	0.915811i	0.0809105	−	0.996721i	$-0.474217\pi$
$317$	−16.3055	−	16.3055i	−0.915811	−	0.915811i	−0.996721	+	0.0809105i	$0.974217\pi$
$318$		0			0
$319$	5.09040			0.285007
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.69249	−	3.69249i	−0.205456	−	0.205456i
$324$		0			0
$325$	2.84870	−	2.84870i	0.158017	−	0.158017i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			0.0465044i			0.00256387i
$330$		0			0
$331$	−5.33950	+	5.33950i	−0.293485	+	0.293485i	−0.838455	−	0.544970i	$-0.816541\pi$
$331$	−5.33950	+	5.33950i	−0.293485	+	0.293485i	0.544970	+	0.838455i	$0.316541\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.73152			0.422418
$336$		0			0
$337$	−10.9232			−0.595023			−0.297512	−	0.954718i	$-0.596157\pi$
$337$	−10.9232			−0.595023			−0.297512	+	0.954718i	$0.596157\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	19.0944	−	19.0944i	1.03402	−	1.03402i
$342$		0			0
$343$		−	0.824325i		−	0.0445094i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.5294	+	19.5294i	−1.04839	+	1.04839i	−0.0496243	+	0.998768i	$0.515802\pi$
$347$	−19.5294	+	19.5294i	−1.04839	+	1.04839i	−0.998768	+	0.0496243i	$0.984198\pi$
$348$		0			0
$349$	−9.27622	−	9.27622i	−0.496545	−	0.496545i	0.413816	−	0.910361i	$-0.364196\pi$
$349$	−9.27622	−	9.27622i	−0.496545	−	0.496545i	−0.910361	+	0.413816i	$0.864196\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.31510			−0.176445			−0.0882225	−	0.996101i	$-0.528119\pi$
$353$	−3.31510			−0.176445			−0.0882225	+	0.996101i	$0.528119\pi$
$354$		0			0
$355$	5.23558	+	5.23558i	0.277876	+	0.277876i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.61854i			0.243757i	0.992545	+	0.121879i	$0.0388919\pi$
$359$			4.61854i			0.243757i	−0.992545	+	0.121879i	$0.961108\pi$
$360$		0			0
$361$		−	18.2380i		−	0.959897i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.21050	−	8.21050i	−0.429757	−	0.429757i
$366$		0			0
$367$	3.80336			0.198534			0.0992668	−	0.995061i	$-0.468350\pi$
$367$	3.80336			0.198534			0.0992668	+	0.995061i	$0.468350\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.400951	−	0.400951i	−0.0208164	−	0.0208164i
$372$		0			0
$373$	18.4506	−	18.4506i	0.955338	−	0.955338i	−0.0437062	−	0.999044i	$-0.513917\pi$
$373$	18.4506	−	18.4506i	0.955338	−	0.955338i	0.999044	+	0.0437062i	$0.0139165\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6.49428i			0.334472i
$378$		0			0
$379$	−2.95913	+	2.95913i	−0.152000	+	0.152000i	−0.779011	−	0.627010i	$-0.784278\pi$
$379$	−2.95913	+	2.95913i	−0.152000	+	0.152000i	0.627010	+	0.779011i	$0.284278\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	37.0073			1.89099			0.945493	−	0.325642i	$-0.105580\pi$
$383$	37.0073			1.89099			0.945493	+	0.325642i	$0.105580\pi$
$384$		0			0
$385$	0.185977			0.00947829
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.0915	+	18.0915i	−0.917276	+	0.917276i	−0.996831	−	0.0795542i	$-0.974650\pi$
$389$	−18.0915	+	18.0915i	−0.917276	+	0.917276i	0.0795542	+	0.996831i	$0.474650\pi$
$390$		0			0
$391$			4.46695i			0.225904i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12.3056	+	12.3056i	−0.619160	+	0.619160i
$396$		0			0
$397$	−17.0980	−	17.0980i	−0.858125	−	0.858125i	0.132992	−	0.991117i	$-0.457542\pi$
$397$	−17.0980	−	17.0980i	−0.858125	−	0.858125i	−0.991117	+	0.132992i	$0.957542\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.8173			−0.540189			−0.270094	−	0.962834i	$-0.587055\pi$
$401$	−10.8173			−0.540189			−0.270094	+	0.962834i	$0.587055\pi$
$402$		0			0
$403$	24.3605	+	24.3605i	1.21348	+	1.21348i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			9.01241i			0.446729i
$408$		0			0
$409$			4.30551i			0.212894i	0.994318	+	0.106447i	$0.0339474\pi$
$409$			4.30551i			0.212894i	−0.994318	+	0.106447i	$0.966053\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.551283	+	0.551283i	0.0271269	+	0.0271269i
$414$		0			0
$415$	10.6786			0.524191
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16.3215	−	16.3215i	−0.797357	−	0.797357i	0.185321	−	0.982678i	$-0.440667\pi$
$419$	−16.3215	−	16.3215i	−0.797357	−	0.797357i	−0.982678	+	0.185321i	$0.940667\pi$
$420$		0			0
$421$	7.16177	−	7.16177i	0.349043	−	0.349043i	−0.510710	−	0.859753i	$-0.670617\pi$
$421$	7.16177	−	7.16177i	0.349043	−	0.349043i	0.859753	+	0.510710i	$0.170617\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			5.98228i			0.290183i
$426$		0			0
$427$	−0.0479538	+	0.0479538i	−0.00232064	+	0.00232064i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−14.4255			−0.694851			−0.347426	−	0.937708i	$-0.612944\pi$
$431$	−14.4255			−0.694851			−0.347426	+	0.937708i	$0.612944\pi$
$432$		0			0
$433$	−31.3821			−1.50813			−0.754063	−	0.656802i	$-0.771909\pi$
$433$	−31.3821			−1.50813			−0.754063	+	0.656802i	$0.771909\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.460890	−	0.460890i	0.0220474	−	0.0220474i
$438$		0			0
$439$		−	7.91484i		−	0.377755i	−0.982001	−	0.188877i	$-0.939515\pi$
$439$		−	7.91484i		−	0.377755i	0.982001	−	0.188877i	$-0.0604849\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−11.2877	+	11.2877i	−0.536294	+	0.536294i	−0.922438	−	0.386144i	$-0.873807\pi$
$443$	−11.2877	+	11.2877i	−0.536294	+	0.536294i	0.386144	+	0.922438i	$0.373807\pi$
$444$		0			0
$445$	−11.5264	−	11.5264i	−0.546402	−	0.546402i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	16.7071			0.788458			0.394229	−	0.919012i	$-0.371012\pi$
$449$	16.7071			0.788458			0.394229	+	0.919012i	$0.371012\pi$
$450$		0			0
$451$	17.2181	+	17.2181i	0.810769	+	0.810769i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			0.237268i			0.0111233i
$456$		0			0
$457$			25.5371i			1.19458i	0.802027	+	0.597288i	$0.203755\pi$
$457$			25.5371i			1.19458i	−0.802027	+	0.597288i	$0.796245\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.05719	+	1.05719i	0.0492382	+	0.0492382i	0.731297	−	0.682059i	$-0.238915\pi$
$461$	1.05719	+	1.05719i	0.0492382	+	0.0492382i	−0.682059	+	0.731297i	$0.738915\pi$
$462$		0			0
$463$	−33.8953			−1.57525			−0.787623	−	0.616157i	$-0.788688\pi$
$463$	−33.8953			−1.57525			−0.787623	+	0.616157i	$0.788688\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−15.8661	−	15.8661i	−0.734194	−	0.734194i	0.237253	−	0.971448i	$-0.423753\pi$
$467$	−15.8661	−	15.8661i	−0.734194	−	0.734194i	−0.971448	+	0.237253i	$0.923753\pi$
$468$		0			0
$469$	−0.321979	+	0.321979i	−0.0148676	+	0.0148676i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	13.1398i		−	0.604169i
$474$		0			0
$475$	0.617238	−	0.617238i	0.0283208	−	0.0283208i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.85047			−0.0845500			−0.0422750	−	0.999106i	$-0.513461\pi$
$479$	−1.85047			−0.0845500			−0.0422750	+	0.999106i	$0.513461\pi$
$480$		0			0
$481$	−11.4980			−0.524261
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.70869	+	8.70869i	−0.395441	+	0.395441i
$486$		0			0
$487$		−	7.72194i		−	0.349915i	−0.984576	−	0.174957i	$-0.944021\pi$
$487$		−	7.72194i		−	0.349915i	0.984576	−	0.174957i	$-0.0559787\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	16.2289	−	16.2289i	0.732399	−	0.732399i	−0.238696	−	0.971094i	$-0.576720\pi$
$491$	16.2289	−	16.2289i	0.732399	−	0.732399i	0.971094	+	0.238696i	$0.0767199\pi$
$492$		0			0
$493$	−6.81900	−	6.81900i	−0.307112	−	0.307112i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.436071			−0.0195605
$498$		0			0
$499$	−20.2479	−	20.2479i	−0.906418	−	0.906418i	0.0895629	−	0.995981i	$-0.471453\pi$
$499$	−20.2479	−	20.2479i	−0.906418	−	0.906418i	−0.995981	+	0.0895629i	$0.971453\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	18.2912i		−	0.815566i	−0.913079	−	0.407783i	$-0.866302\pi$
$503$		−	18.2912i		−	0.815566i	0.913079	−	0.407783i	$-0.133698\pi$
$504$		0			0
$505$			0.938447i			0.0417603i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.0470	−	16.0470i	−0.711272	−	0.711272i	0.255529	−	0.966801i	$-0.417750\pi$
$509$	−16.0470	−	16.0470i	−0.711272	−	0.711272i	−0.966801	+	0.255529i	$0.917750\pi$
$510$		0			0
$511$	0.683853			0.0302519
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−10.5385	−	10.5385i	−0.464380	−	0.464380i
$516$		0			0
$517$	−1.76313	+	1.76313i	−0.0775422	+	0.0775422i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	8.02188i		−	0.351445i	−0.984440	−	0.175722i	$-0.943774\pi$
$521$		−	8.02188i		−	0.351445i	0.984440	−	0.175722i	$-0.0562261\pi$
$522$		0			0
$523$	31.8339	−	31.8339i	1.39200	−	1.39200i	0.571158	−	0.820840i	$-0.306494\pi$
$523$	31.8339	−	31.8339i	1.39200	−	1.39200i	0.820840	−	0.571158i	$-0.193506\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−51.1570			−2.22844
$528$		0			0
$529$	22.4424			0.975758
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−21.9667	+	21.9667i	−0.951483	+	0.951483i
$534$		0			0
$535$			4.86293i			0.210243i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	15.6225	−	15.6225i	0.672908	−	0.672908i
$540$		0			0
$541$	16.7467	+	16.7467i	0.719996	+	0.719996i	0.968604	−	0.248608i	$-0.0799730\pi$
$541$	16.7467	+	16.7467i	0.719996	+	0.719996i	−0.248608	+	0.968604i	$0.579973\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.0807802			−0.00346024
$546$		0			0
$547$	−24.8600	−	24.8600i	−1.06294	−	1.06294i	−0.997882	−	0.0650571i	$-0.979277\pi$
$547$	−24.8600	−	24.8600i	−1.06294	−	1.06294i	−0.0650571	−	0.997882i	$-0.520723\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			1.40714i			0.0599461i
$552$		0			0
$553$		−	1.02493i		−	0.0435845i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.0423	−	12.0423i	−0.510248	−	0.510248i	0.404354	−	0.914603i	$-0.367496\pi$
$557$	−12.0423	−	12.0423i	−0.510248	−	0.510248i	−0.914603	+	0.404354i	$0.867496\pi$
$558$		0			0
$559$	16.7636			0.709026
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−15.2672	−	15.2672i	−0.643435	−	0.643435i	0.307963	−	0.951398i	$-0.400353\pi$
$563$	−15.2672	−	15.2672i	−0.643435	−	0.643435i	−0.951398	+	0.307963i	$0.900353\pi$
$564$		0			0
$565$	−10.5948	+	10.5948i	−0.445729	+	0.445729i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			33.2847i			1.39537i	0.716406	+	0.697683i	$0.245786\pi$
$569$			33.2847i			1.39537i	−0.716406	+	0.697683i	$0.754214\pi$
$570$		0			0
$571$	10.4781	−	10.4781i	0.438494	−	0.438494i	−0.453011	−	0.891505i	$-0.649650\pi$
$571$	10.4781	−	10.4781i	0.438494	−	0.438494i	0.891505	+	0.453011i	$0.149650\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.746698			−0.0311394
$576$		0			0
$577$	−38.6407			−1.60863			−0.804316	−	0.594202i	$-0.797468\pi$
$577$	−38.6407			−1.60863			−0.804316	+	0.594202i	$0.797468\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.444710	+	0.444710i	−0.0184497	+	0.0184497i
$582$		0			0
$583$		−	30.4026i		−	1.25915i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−21.1392	+	21.1392i	−0.872507	+	0.872507i	−0.992745	−	0.120238i	$-0.961634\pi$
$587$	−21.1392	+	21.1392i	−0.872507	+	0.872507i	0.120238	+	0.992745i	$0.461634\pi$
$588$		0			0
$589$	5.27827	+	5.27827i	0.217487	+	0.217487i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4.03330			0.165628			0.0828138	−	0.996565i	$-0.473609\pi$
$593$	4.03330			0.165628			0.0828138	+	0.996565i	$0.473609\pi$
$594$		0			0
$595$	−0.249132	−	0.249132i	−0.0102134	−	0.0102134i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			6.44275i			0.263244i	0.991300	+	0.131622i	$0.0420184\pi$
$599$			6.44275i			0.263244i	−0.991300	+	0.131622i	$0.957982\pi$
$600$		0			0
$601$		−	36.3011i		−	1.48075i	−0.672193	−	0.740376i	$-0.734648\pi$
$601$		−	36.3011i		−	1.48075i	0.672193	−	0.740376i	$-0.265352\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.727191	−	0.727191i	−0.0295645	−	0.0295645i
$606$		0			0
$607$	−19.5463			−0.793360			−0.396680	−	0.917957i	$-0.629838\pi$
$607$	−19.5463			−0.793360			−0.396680	+	0.917957i	$0.629838\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.24938	−	2.24938i	−0.0910002	−	0.0910002i
$612$		0			0
$613$	18.5711	−	18.5711i	0.750080	−	0.750080i	−0.224414	−	0.974494i	$-0.572047\pi$
$613$	18.5711	−	18.5711i	0.750080	−	0.750080i	0.974494	+	0.224414i	$0.0720467\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	19.7311i		−	0.794342i	−0.917745	−	0.397171i	$-0.869992\pi$
$617$		−	19.7311i		−	0.794342i	0.917745	−	0.397171i	$-0.130008\pi$
$618$		0			0
$619$	−22.1910	+	22.1910i	−0.891930	+	0.891930i	−0.994705	−	0.102775i	$-0.967228\pi$
$619$	−22.1910	+	22.1910i	−0.891930	+	0.891930i	0.102775	+	0.994705i	$0.467228\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.960031			0.0384628
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.0729	−	12.0729i	0.481377	−	0.481377i
$630$		0			0
$631$			13.1148i			0.522093i	0.965326	+	0.261046i	$0.0840676\pi$
$631$			13.1148i			0.522093i	−0.965326	+	0.261046i	$0.915932\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.77035	−	4.77035i	0.189305	−	0.189305i
$636$		0			0
$637$	19.9310	+	19.9310i	0.789696	+	0.789696i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−30.5395			−1.20624			−0.603119	−	0.797652i	$-0.706075\pi$
$641$	−30.5395			−1.20624			−0.603119	+	0.797652i	$0.706075\pi$
$642$		0			0
$643$	4.01306	+	4.01306i	0.158260	+	0.158260i	0.781795	−	0.623535i	$-0.214304\pi$
$643$	4.01306	+	4.01306i	0.158260	+	0.158260i	−0.623535	+	0.781795i	$0.714304\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	9.01207i		−	0.354301i	−0.984184	−	0.177151i	$-0.943312\pi$
$647$		−	9.01207i		−	0.354301i	0.984184	−	0.177151i	$-0.0566880\pi$
$648$		0			0
$649$			41.8017i			1.64086i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.2241	+	25.2241i	0.987096	+	0.987096i	0.999918	−	0.0128219i	$-0.00408146\pi$
$653$	25.2241	+	25.2241i	0.987096	+	0.987096i	−0.0128219	+	0.999918i	$0.504081\pi$
$654$		0			0
$655$	−14.4898			−0.566165
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15.8387	+	15.8387i	0.616987	+	0.616987i	0.944757	−	0.327770i	$-0.106297\pi$
$659$	15.8387	+	15.8387i	0.616987	+	0.616987i	−0.327770	+	0.944757i	$0.606297\pi$
$660$		0			0
$661$	8.25357	−	8.25357i	0.321027	−	0.321027i	−0.528134	−	0.849161i	$-0.677108\pi$
$661$	8.25357	−	8.25357i	0.321027	−	0.321027i	0.849161	+	0.528134i	$0.177108\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			0.0514098i			0.00199359i
$666$		0			0
$667$	0.851136	−	0.851136i	0.0329561	−	0.0329561i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.63615			−0.140372
$672$		0			0
$673$	29.8226			1.14958			0.574789	−	0.818302i	$-0.305084\pi$
$673$	29.8226			1.14958			0.574789	+	0.818302i	$0.305084\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−11.3009	+	11.3009i	−0.434328	+	0.434328i	−0.890098	−	0.455770i	$-0.849364\pi$
$677$	−11.3009	+	11.3009i	−0.434328	+	0.434328i	0.455770	+	0.890098i	$0.349364\pi$
$678$		0			0
$679$		−	0.725347i		−	0.0278363i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	24.7739	−	24.7739i	0.947949	−	0.947949i	−0.0507621	−	0.998711i	$-0.516165\pi$
$683$	24.7739	−	24.7739i	0.947949	−	0.947949i	0.998711	+	0.0507621i	$0.0161650\pi$
$684$		0			0
$685$	−13.7542	−	13.7542i	−0.525522	−	0.525522i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	38.7874			1.47768
$690$		0			0
$691$	−30.4140	−	30.4140i	−1.15700	−	1.15700i	−0.985117	−	0.171888i	$-0.945013\pi$
$691$	−30.4140	−	30.4140i	−1.15700	−	1.15700i	−0.171888	−	0.985117i	$-0.554987\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			1.54979i			0.0587868i
$696$		0			0
$697$		−	46.1301i		−	1.74730i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	9.35541	+	9.35541i	0.353349	+	0.353349i	0.861354	−	0.508005i	$-0.169617\pi$
$701$	9.35541	+	9.35541i	0.353349	+	0.353349i	−0.508005	+	0.861354i	$0.669617\pi$
$702$		0			0
$703$	−2.49130			−0.0939612
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.0390816	−	0.0390816i	−0.00146982	−	0.00146982i
$708$		0			0
$709$	15.6531	−	15.6531i	0.587866	−	0.587866i	−0.349187	−	0.937053i	$-0.613542\pi$
$709$	15.6531	−	15.6531i	0.587866	−	0.587866i	0.937053	+	0.349187i	$0.113542\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	6.38533i		−	0.239133i
$714$		0			0
$715$	−8.99558	+	8.99558i	−0.336416	+	0.336416i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	51.4373			1.91829			0.959144	−	0.282919i	$-0.0913028\pi$
$719$	51.4373			1.91829			0.959144	+	0.282919i	$0.0913028\pi$
$720$		0			0
$721$	0.877748			0.0326891
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.13987	−	1.13987i	0.0423336	−	0.0423336i
$726$		0			0
$727$		−	15.7975i		−	0.585898i	−0.956128	−	0.292949i	$-0.905363\pi$
$727$		−	15.7975i		−	0.585898i	0.956128	−	0.292949i	$-0.0946366\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−17.6018	+	17.6018i	−0.651028	+	0.651028i
$732$		0			0
$733$	28.7672	+	28.7672i	1.06254	+	1.06254i	0.997909	+	0.0646310i	$0.0205870\pi$
$733$	28.7672	+	28.7672i	1.06254	+	1.06254i	0.0646310	+	0.997909i	$0.479413\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.4145			−0.899318
$738$		0			0
$739$	−6.12767	−	6.12767i	−0.225410	−	0.225410i	0.585362	−	0.810772i	$-0.300952\pi$
$739$	−6.12767	−	6.12767i	−0.225410	−	0.225410i	−0.810772	+	0.585362i	$0.800952\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	22.0466i		−	0.808810i	−0.914580	−	0.404405i	$-0.867479\pi$
$743$		−	22.0466i		−	0.808810i	0.914580	−	0.404405i	$-0.132521\pi$
$744$		0			0
$745$			19.3055i			0.707297i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.202517	−	0.202517i	−0.00739980	−	0.00739980i
$750$		0			0
$751$	−40.6203			−1.48226			−0.741128	−	0.671364i	$-0.765709\pi$
$751$	−40.6203			−1.48226			−0.741128	+	0.671364i	$0.765709\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9.53258	−	9.53258i	−0.346926	−	0.346926i
$756$		0			0
$757$	−12.9575	+	12.9575i	−0.470949	+	0.470949i	−0.902222	−	0.431272i	$-0.858065\pi$
$757$	−12.9575	+	12.9575i	−0.470949	+	0.470949i	0.431272	+	0.902222i	$0.358065\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			33.9075i			1.22915i	0.788860	+	0.614573i	$0.210672\pi$
$761$			33.9075i			1.22915i	−0.788860	+	0.614573i	$0.789328\pi$
$762$		0			0
$763$	0.00336409	−	0.00336409i	0.000121788	−	0.000121788i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−53.3302			−1.92564
$768$		0			0
$769$	8.30816			0.299600			0.149800	−	0.988716i	$-0.452137\pi$
$769$	8.30816			0.299600			0.149800	+	0.988716i	$0.452137\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.90838	+	9.90838i	−0.356380	+	0.356380i	−0.862477	−	0.506097i	$-0.831088\pi$
$773$	−9.90838	+	9.90838i	−0.356380	+	0.356380i	0.506097	+	0.862477i	$0.331088\pi$
$774$		0			0
$775$		−	8.55143i		−	0.307176i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.75960	+	4.75960i	−0.170530	+	0.170530i
$780$		0			0
$781$	−16.5328	−	16.5328i	−0.591591	−	0.591591i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.7105			−0.560730
$786$		0			0
$787$	−32.2948	−	32.2948i	−1.15119	−	1.15119i	−0.986315	−	0.164872i	$-0.947279\pi$
$787$	−32.2948	−	32.2948i	−1.15119	−	1.15119i	−0.164872	−	0.986315i	$-0.552721\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	0.882445i		−	0.0313761i
$792$		0			0
$793$		−	4.63897i		−	0.164735i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.3637	+	21.3637i	0.756740	+	0.756740i	0.975728	−	0.218987i	$-0.0702754\pi$
$797$	21.3637	+	21.3637i	0.756740	+	0.756740i	−0.218987	+	0.975728i	$0.570275\pi$
$798$		0			0
$799$	4.72370			0.167113
$800$		0			0
$801$		0			0
$802$		0			0
$803$	25.9270	+	25.9270i	0.914944	+	0.914944i
$804$		0			0
$805$	0.0310962	−	0.0310962i	0.00109600	−	0.00109600i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	37.9895i		−	1.33564i	−0.744323	−	0.667820i	$-0.767228\pi$
$809$		−	37.9895i		−	1.33564i	0.744323	−	0.667820i	$-0.232772\pi$
$810$		0			0
$811$	−6.45375	+	6.45375i	−0.226622	+	0.226622i	−0.811280	−	0.584658i	$-0.801229\pi$
$811$	−6.45375	+	6.45375i	−0.226622	+	0.226622i	0.584658	+	0.811280i	$0.301229\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.79446			−0.0978857
$816$		0			0
$817$	3.63224			0.127076
$818$		0			0
$819$		0			0
$820$		0			0
$821$	10.5389	−	10.5389i	0.367809	−	0.367809i	−0.498869	−	0.866677i	$-0.666251\pi$
$821$	10.5389	−	10.5389i	0.367809	−	0.367809i	0.866677	+	0.498869i	$0.166251\pi$
$822$		0			0
$823$			35.0064i			1.22025i	0.792306	+	0.610124i	$0.208880\pi$
$823$			35.0064i			1.22025i	−0.792306	+	0.610124i	$0.791120\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	13.0789	−	13.0789i	0.454799	−	0.454799i	−0.442145	−	0.896944i	$-0.645782\pi$
$827$	13.0789	−	13.0789i	0.454799	−	0.454799i	0.896944	+	0.442145i	$0.145782\pi$
$828$		0			0
$829$	−28.3502	−	28.3502i	−0.984642	−	0.984642i	0.0152419	−	0.999884i	$-0.495148\pi$
$829$	−28.3502	−	28.3502i	−0.984642	−	0.984642i	−0.999884	+	0.0152419i	$0.995148\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−41.8552			−1.45020
$834$		0			0
$835$	1.50456	+	1.50456i	0.0520674	+	0.0520674i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	52.5560i		−	1.81444i	−0.420661	−	0.907218i	$-0.638202\pi$
$839$		−	52.5560i		−	1.81444i	0.420661	−	0.907218i	$-0.361798\pi$
$840$		0			0
$841$		−	26.4014i		−	0.910393i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.28408	−	2.28408i	−0.0785749	−	0.0785749i
$846$		0			0
$847$	0.0605677			0.00208113
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.50691	+	1.50691i	0.0516564	+	0.0516564i
$852$		0			0
$853$	−33.3957	+	33.3957i	−1.14345	+	1.14345i	−0.155632	+	0.987815i	$0.549741\pi$
$853$	−33.3957	+	33.3957i	−1.14345	+	1.14345i	−0.987815	+	0.155632i	$0.950259\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			6.36548i			0.217441i	0.994072	+	0.108720i	$0.0346753\pi$
$857$			6.36548i			0.217441i	−0.994072	+	0.108720i	$0.965325\pi$
$858$		0			0
$859$	−30.2446	+	30.2446i	−1.03193	+	1.03193i	−0.0324590	+	0.999473i	$0.510334\pi$
$859$	−30.2446	+	30.2446i	−1.03193	+	1.03193i	−0.999473	+	0.0324590i	$0.989666\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−16.2067			−0.551683			−0.275841	−	0.961203i	$-0.588956\pi$
$863$	−16.2067			−0.551683			−0.275841	+	0.961203i	$0.588956\pi$
$864$		0			0
$865$	−13.0373			−0.443282
$866$		0			0
$867$		0			0
$868$		0			0
$869$	38.8583	−	38.8583i	1.31818	−	1.31818i
$870$		0			0
$871$		−	31.1477i		−	1.05540i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.0416450	−	0.0416450i	0.00140786	−	0.00140786i
$876$		0			0
$877$	−7.08871	−	7.08871i	−0.239369	−	0.239369i	0.577220	−	0.816589i	$-0.304137\pi$
$877$	−7.08871	−	7.08871i	−0.239369	−	0.239369i	−0.816589	+	0.577220i	$0.804137\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	46.7030			1.57346			0.786731	−	0.617296i	$-0.211772\pi$
$881$	46.7030			1.57346			0.786731	+	0.617296i	$0.211772\pi$
$882$		0			0
$883$	−35.6005	−	35.6005i	−1.19805	−	1.19805i	−0.974749	−	0.223303i	$-0.928316\pi$
$883$	−35.6005	−	35.6005i	−1.19805	−	1.19805i	−0.223303	−	0.974749i	$-0.571684\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			2.62187i			0.0880338i	0.999031	+	0.0440169i	$0.0140156\pi$
$887$			2.62187i			0.0880338i	−0.999031	+	0.0440169i	$0.985984\pi$
$888$		0			0
$889$			0.397322i			0.0133258i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.487381	−	0.487381i	−0.0163096	−	0.0163096i
$894$		0			0
$895$	−7.10510			−0.237497
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.74749	+	9.74749i	0.325097	+	0.325097i
$900$		0			0
$901$	−40.7268	+	40.7268i	−1.35681	+	1.35681i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	21.4059i		−	0.711557i
$906$		0			0
$907$	11.1287	−	11.1287i	0.369522	−	0.369522i	−0.497781	−	0.867303i	$-0.665852\pi$
$907$	11.1287	−	11.1287i	0.369522	−	0.369522i	0.867303	+	0.497781i	$0.165852\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40.1961			−1.33176			−0.665878	−	0.746061i	$-0.731943\pi$
$911$	−40.1961			−1.33176			−0.665878	+	0.746061i	$0.731943\pi$
$912$		0			0
$913$	−33.7207			−1.11599
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.603429	−	0.603429i	0.0199270	−	0.0199270i
$918$		0			0
$919$		−	0.413348i		−	0.0136351i	−0.999977	−	0.00681755i	$-0.997830\pi$
$919$		−	0.413348i		−	0.0136351i	0.999977	−	0.00681755i	$-0.00217011\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	21.0924	−	21.0924i	0.694266	−	0.694266i
$924$		0			0
$925$	2.01811	+	2.01811i	0.0663549	+	0.0663549i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	28.8881			0.947787			0.473893	−	0.880582i	$-0.342848\pi$
$929$	28.8881			0.947787			0.473893	+	0.880582i	$0.342848\pi$
$930$		0			0
$931$	4.31853	+	4.31853i	0.141534	+	0.141534i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	18.8907i		−	0.617793i
$936$		0			0
$937$		−	5.60125i		−	0.182985i	−0.995806	−	0.0914925i	$-0.970836\pi$
$937$		−	5.60125i		−	0.182985i	0.995806	−	0.0914925i	$-0.0291637\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15.8730	+	15.8730i	0.517446	+	0.517446i	0.916798	−	0.399352i	$-0.130765\pi$
$941$	15.8730	+	15.8730i	0.517446	+	0.517446i	−0.399352	+	0.916798i	$0.630765\pi$
$942$		0			0
$943$	5.75788			0.187502
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.0697	+	12.0697i	0.392214	+	0.392214i	0.875476	−	0.483262i	$-0.160548\pi$
$947$	12.0697	+	12.0697i	0.392214	+	0.392214i	−0.483262	+	0.875476i	$0.660548\pi$
$948$		0			0
$949$	−33.0774	+	33.0774i	−1.07374	+	1.07374i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	5.02008i		−	0.162616i	−0.996689	−	0.0813081i	$-0.974090\pi$
$953$		−	5.02008i		−	0.162616i	0.996689	−	0.0813081i	$-0.0259098\pi$
$954$		0			0
$955$	−8.79820	+	8.79820i	−0.284703	+	0.284703i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.14559			0.0369930
$960$		0			0
$961$	42.1270			1.35893
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.171072	+	0.171072i	−0.00550701	+	0.00550701i
$966$		0			0
$967$			33.9334i			1.09123i	0.838038	+	0.545613i	$0.183703\pi$
$967$			33.9334i			1.09123i	−0.838038	+	0.545613i	$0.816297\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.06974	−	1.06974i	0.0343297	−	0.0343297i	−0.689734	−	0.724063i	$-0.742272\pi$
$971$	1.06974	−	1.06974i	0.0343297	−	0.0343297i	0.724063	+	0.689734i	$0.242272\pi$
$972$		0			0
$973$	−0.0645409	−	0.0645409i	−0.00206909	−	0.00206909i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.01239			0.0323891			0.0161946	−	0.999869i	$-0.494845\pi$
$977$	1.01239			0.0323891			0.0161946	+	0.999869i	$0.494845\pi$
$978$		0			0
$979$	36.3978	+	36.3978i	1.16328	+	1.16328i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			27.2297i			0.868494i	0.900794	+	0.434247i	$0.142985\pi$
$983$			27.2297i			0.868494i	−0.900794	+	0.434247i	$0.857015\pi$
$984$		0			0
$985$			3.35537i			0.106911i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.19703	−	2.19703i	−0.0698616	−	0.0698616i
$990$		0			0
$991$	−2.24865			−0.0714308			−0.0357154	−	0.999362i	$-0.511371\pi$
$991$	−2.24865			−0.0714308			−0.0357154	+	0.999362i	$0.511371\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14.0202	+	14.0202i	0.444469	+	0.444469i
$996$		0			0
$997$	−11.6266	+	11.6266i	−0.368220	+	0.368220i	−0.866828	−	0.498608i	$-0.833845\pi$
$997$	−11.6266	+	11.6266i	−0.368220	+	0.368220i	0.498608	+	0.866828i	$0.333845\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.d.721.3		20
3.2	odd	2		960.2.s.c.721.3		20
4.3	odd	2		720.2.t.d.541.7		20
12.11	even	2		240.2.s.c.61.4	✓	20
16.5	even	4	inner	2880.2.t.d.2161.3		20
16.11	odd	4		720.2.t.d.181.7		20
24.5	odd	2		1920.2.s.f.1441.8		20
24.11	even	2		1920.2.s.e.1441.3		20
48.5	odd	4		960.2.s.c.241.3		20
48.11	even	4		240.2.s.c.181.4	yes	20
48.29	odd	4		1920.2.s.f.481.8		20
48.35	even	4		1920.2.s.e.481.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.s.c.61.4	✓	20	12.11	even	2
240.2.s.c.181.4	yes	20	48.11	even	4
720.2.t.d.181.7		20	16.11	odd	4
720.2.t.d.541.7		20	4.3	odd	2
960.2.s.c.241.3		20	48.5	odd	4
960.2.s.c.721.3		20	3.2	odd	2
1920.2.s.e.481.3		20	48.35	even	4
1920.2.s.e.1441.3		20	24.11	even	2
1920.2.s.f.481.8		20	48.29	odd	4
1920.2.s.f.1441.8		20	24.5	odd	2
2880.2.t.d.721.3		20	1.1	even	1	trivial
2880.2.t.d.2161.3		20	16.5	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{5} -0.0588949i q^{7} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{5} -0.0588949i q^{7} +(2.23289 - 2.23289i) q^{11} +(2.84870 + 2.84870i) q^{13} -5.98228 q^{17} +(0.617238 + 0.617238i) q^{19} -0.746698i q^{23} -1.00000i q^{25} +(1.13987 + 1.13987i) q^{29} +8.55143 q^{31} +(0.0416450 + 0.0416450i) q^{35} +(-2.01811 + 2.01811i) q^{37} +7.71113i q^{41} +(2.94233 - 2.94233i) q^{43} -0.789616 q^{47} +6.99653 q^{49} +(6.80791 - 6.80791i) q^{53} +3.15778i q^{55} +(-9.36045 + 9.36045i) q^{59} +(-0.814225 - 0.814225i) q^{61} -4.02867 q^{65} +(-5.46701 - 5.46701i) q^{67} -7.40423i q^{71} +11.6114i q^{73} +(-0.131506 - 0.131506i) q^{77} +17.4027 q^{79} +(-7.55090 - 7.55090i) q^{83} +(4.23011 - 4.23011i) q^{85} +16.3007i q^{89} +(0.167774 - 0.167774i) q^{91} -0.872906 q^{95} +12.3159 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q + 8 q^{11} + 24 q^{17} + 4 q^{19} - 16 q^{29} + 16 q^{37} + 8 q^{43} - 52 q^{49} + 16 q^{53} - 16 q^{59} - 4 q^{61} + 8 q^{67} + 40 q^{77} - 56 q^{79} - 48 q^{83} + 4 q^{85} + 8 q^{91} + 56 q^{97}+O(q^{100}) \) 20 * q + 8 * q^11 + 24 * q^17 + 4 * q^19 - 16 * q^29 + 16 * q^37 + 8 * q^43 - 52 * q^49 + 16 * q^53 - 16 * q^59 - 4 * q^61 + 8 * q^67 + 40 * q^77 - 56 * q^79 - 48 * q^83 + 4 * q^85 + 8 * q^91 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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