LMFDB - Embedded newform 2720.2.l.a.2481.36

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2720,2,Mod(2481,2720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2720.2481"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2720 = 2^{5} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2720.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [36,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$21.7193093498$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 680)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2481.36
Character	$\chi$	$=$	2720.2481
Dual form			2720.2.l.a.2481.35

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.03317 q^{3} -1.00000 q^{5} +1.14609i q^{7} +6.20011 q^{9} +1.02326 q^{11} +5.48640i q^{13} -3.03317 q^{15} +(-3.36364 + 2.38451i) q^{17} -0.131183i q^{19} +3.47629i q^{21} +2.38954i q^{23} +1.00000 q^{25} +9.70646 q^{27} -3.08229 q^{29} -0.752428i q^{31} +3.10373 q^{33} -1.14609i q^{35} +7.71836 q^{37} +16.6412i q^{39} +6.10135i q^{41} +9.64227i q^{43} -6.20011 q^{45} +3.00424 q^{47} +5.68647 q^{49} +(-10.2025 + 7.23263i) q^{51} -13.4640i q^{53} -1.02326 q^{55} -0.397901i q^{57} +10.0052i q^{59} -3.72365 q^{61} +7.10589i q^{63} -5.48640i q^{65} -3.72428i q^{67} +7.24788i q^{69} -6.62521i q^{71} -14.9275i q^{73} +3.03317 q^{75} +1.17275i q^{77} +13.7272i q^{79} +10.8410 q^{81} -3.85242i q^{83} +(3.36364 - 2.38451i) q^{85} -9.34909 q^{87} -3.64410 q^{89} -6.28791 q^{91} -2.28224i q^{93} +0.131183i q^{95} -3.73153i q^{97} +6.34435 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 4 q^{3} - 36 q^{5} + 36 q^{9} - 8 q^{11} + 4 q^{15} + 36 q^{25} - 16 q^{27} - 8 q^{33} - 36 q^{45} - 20 q^{47} - 36 q^{49} + 8 q^{55} - 4 q^{75} + 44 q^{81} - 24 q^{87} - 56 q^{91} - 40 q^{99}+O(q^{100}) $ 36 * q - 4 * q^3 - 36 * q^5 + 36 * q^9 - 8 * q^11 + 4 * q^15 + 36 * q^25 - 16 * q^27 - 8 * q^33 - 36 * q^45 - 20 * q^47 - 36 * q^49 + 8 * q^55 - 4 * q^75 + 44 * q^81 - 24 * q^87 - 56 * q^91 - 40 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2720\mathbb{Z}\right)^\times$.

$n$	$511$	$1601$	$1701$	$2177$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	3.03317			1.75120			0.875600	−	0.483037i	$-0.160466\pi$
$3$	3.03317			1.75120			0.875600	+	0.483037i	$0.160466\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			1.14609i			0.433182i	0.976262	+	0.216591i	$0.0694938\pi$
$7$			1.14609i			0.433182i	−0.976262	+	0.216591i	$0.930506\pi$
$8$		0			0
$9$	6.20011			2.06670
$10$		0			0
$11$	1.02326			0.308526			0.154263	−	0.988030i	$-0.450700\pi$
$11$	1.02326			0.308526			0.154263	+	0.988030i	$0.450700\pi$
$12$		0			0
$13$			5.48640i			1.52165i	0.648956	+	0.760826i	$0.275206\pi$
$13$			5.48640i			1.52165i	−0.648956	+	0.760826i	$0.724794\pi$
$14$		0			0
$15$	−3.03317			−0.783161
$16$		0			0
$17$	−3.36364	+	2.38451i	−0.815803	+	0.578329i
$17$	−3.36364	+	2.38451i	−0.815803	+	0.578329i
$18$		0			0
$19$		−	0.131183i		−	0.0300955i	−0.999887	−	0.0150477i	$-0.995210\pi$
$19$		−	0.131183i		−	0.0300955i	0.999887	−	0.0150477i	$-0.00479003\pi$
$20$		0			0
$21$			3.47629i			0.758588i
$22$		0			0
$23$			2.38954i			0.498254i	0.968471	+	0.249127i	$0.0801436\pi$
$23$			2.38954i			0.498254i	−0.968471	+	0.249127i	$0.919856\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	9.70646			1.86801
$28$		0			0
$29$	−3.08229			−0.572366			−0.286183	−	0.958175i	$-0.592387\pi$
$29$	−3.08229			−0.572366			−0.286183	+	0.958175i	$0.592387\pi$
$30$		0			0
$31$		−	0.752428i		−	0.135140i	−0.997715	−	0.0675700i	$-0.978475\pi$
$31$		−	0.752428i		−	0.135140i	0.997715	−	0.0675700i	$-0.0215246\pi$
$32$		0			0
$33$	3.10373			0.540291
$34$		0			0
$35$		−	1.14609i		−	0.193725i
$36$		0			0
$37$	7.71836			1.26889			0.634445	−	0.772968i	$-0.281229\pi$
$37$	7.71836			1.26889			0.634445	+	0.772968i	$0.281229\pi$
$38$		0			0
$39$			16.6412i			2.66472i
$40$		0			0
$41$			6.10135i			0.952871i	0.879210	+	0.476435i	$0.158071\pi$
$41$			6.10135i			0.952871i	−0.879210	+	0.476435i	$0.841929\pi$
$42$		0			0
$43$			9.64227i			1.47043i	0.677833	+	0.735216i	$0.262919\pi$
$43$			9.64227i			1.47043i	−0.677833	+	0.735216i	$0.737081\pi$
$44$		0			0
$45$	−6.20011			−0.924257
$46$		0			0
$47$	3.00424			0.438214			0.219107	−	0.975701i	$-0.429686\pi$
$47$	3.00424			0.438214			0.219107	+	0.975701i	$0.429686\pi$
$48$		0			0
$49$	5.68647			0.812353
$50$		0			0
$51$	−10.2025	+	7.23263i	−1.42864	+	1.01277i
$52$		0			0
$53$		−	13.4640i		−	1.84942i	−0.380677	−	0.924708i	$-0.624309\pi$
$53$		−	13.4640i		−	1.84942i	0.380677	−	0.924708i	$-0.375691\pi$
$54$		0			0
$55$	−1.02326			−0.137977
$56$		0			0
$57$		−	0.397901i		−	0.0527032i
$58$		0			0
$59$			10.0052i			1.30257i	0.758835	+	0.651283i	$0.225769\pi$
$59$			10.0052i			1.30257i	−0.758835	+	0.651283i	$0.774231\pi$
$60$		0			0
$61$	−3.72365			−0.476765			−0.238383	−	0.971171i	$-0.576617\pi$
$61$	−3.72365			−0.476765			−0.238383	+	0.971171i	$0.576617\pi$
$62$		0			0
$63$			7.10589i			0.895258i
$64$		0			0
$65$		−	5.48640i		−	0.680504i
$66$		0			0
$67$		−	3.72428i		−	0.454993i	−0.973779	−	0.227497i	$-0.926946\pi$
$67$		−	3.72428i		−	0.454993i	0.973779	−	0.227497i	$-0.0730541\pi$
$68$		0			0
$69$			7.24788i			0.872542i
$70$		0			0
$71$		−	6.62521i		−	0.786268i	−0.919481	−	0.393134i	$-0.871391\pi$
$71$		−	6.62521i		−	0.786268i	0.919481	−	0.393134i	$-0.128609\pi$
$72$		0			0
$73$		−	14.9275i		−	1.74713i	−0.486708	−	0.873565i	$-0.661802\pi$
$73$		−	14.9275i		−	1.74713i	0.486708	−	0.873565i	$-0.338198\pi$
$74$		0			0
$75$	3.03317			0.350240
$76$		0			0
$77$			1.17275i			0.133648i
$78$		0			0
$79$			13.7272i			1.54443i	0.635358	+	0.772217i	$0.280852\pi$
$79$			13.7272i			1.54443i	−0.635358	+	0.772217i	$0.719148\pi$
$80$		0			0
$81$	10.8410			1.20456
$82$		0			0
$83$		−	3.85242i		−	0.422858i	−0.977393	−	0.211429i	$-0.932188\pi$
$83$		−	3.85242i		−	0.422858i	0.977393	−	0.211429i	$-0.0678117\pi$
$84$		0			0
$85$	3.36364	−	2.38451i	0.364838	−	0.258637i
$86$		0			0
$87$	−9.34909			−1.00233
$88$		0			0
$89$	−3.64410			−0.386274			−0.193137	−	0.981172i	$-0.561866\pi$
$89$	−3.64410			−0.386274			−0.193137	+	0.981172i	$0.561866\pi$
$90$		0			0
$91$	−6.28791			−0.659152
$92$		0			0
$93$		−	2.28224i		−	0.236657i
$94$		0			0
$95$			0.131183i			0.0134591i
$96$		0			0
$97$		−	3.73153i		−	0.378880i	−0.981892	−	0.189440i	$-0.939333\pi$
$97$		−	3.73153i		−	0.378880i	0.981892	−	0.189440i	$-0.0606672\pi$
$98$		0			0
$99$	6.34435			0.637631
$100$		0			0
$101$		−	12.0231i		−	1.19635i	−0.801366	−	0.598174i	$-0.795893\pi$
$101$		−	12.0231i		−	1.19635i	0.801366	−	0.598174i	$-0.204107\pi$
$102$		0			0
$103$	3.61534			0.356230			0.178115	−	0.984010i	$-0.443000\pi$
$103$	3.61534			0.356230			0.178115	+	0.984010i	$0.443000\pi$
$104$		0			0
$105$		−	3.47629i		−	0.339251i
$106$		0			0
$107$	13.7464			1.32892			0.664459	−	0.747324i	$-0.268662\pi$
$107$	13.7464			1.32892			0.664459	+	0.747324i	$0.268662\pi$
$108$		0			0
$109$	16.8047			1.60960			0.804798	−	0.593549i	$-0.202274\pi$
$109$	16.8047			1.60960			0.804798	+	0.593549i	$0.202274\pi$
$110$		0			0
$111$	23.4111			2.22208
$112$		0			0
$113$		−	2.53084i		−	0.238081i	−0.992889	−	0.119041i	$-0.962018\pi$
$113$		−	2.53084i		−	0.238081i	0.992889	−	0.119041i	$-0.0379818\pi$
$114$		0			0
$115$		−	2.38954i		−	0.222826i
$116$		0			0
$117$			34.0162i			3.14480i
$118$		0			0
$119$	−2.73287	−	3.85504i	−0.250522	−	0.353391i
$120$		0			0
$121$	−9.95293			−0.904812
$122$		0			0
$123$			18.5064i			1.66867i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−9.53470			−0.846068			−0.423034	−	0.906114i	$-0.639035\pi$
$127$	−9.53470			−0.846068			−0.423034	+	0.906114i	$0.639035\pi$
$128$		0			0
$129$			29.2466i			2.57502i
$130$		0			0
$131$	−18.4141			−1.60884			−0.804422	−	0.594059i	$-0.797525\pi$
$131$	−18.4141			−1.60884			−0.804422	+	0.594059i	$0.797525\pi$
$132$		0			0
$133$	0.150348			0.0130368
$134$		0			0
$135$	−9.70646			−0.835399
$136$		0			0
$137$	−13.1861			−1.12656			−0.563282	−	0.826264i	$-0.690462\pi$
$137$	−13.1861			−1.12656			−0.563282	+	0.826264i	$0.690462\pi$
$138$		0			0
$139$	16.9269			1.43572			0.717862	−	0.696185i	$-0.245121\pi$
$139$	16.9269			1.43572			0.717862	+	0.696185i	$0.245121\pi$
$140$		0			0
$141$	9.11237			0.767400
$142$		0			0
$143$			5.61403i			0.469469i
$144$		0			0
$145$	3.08229			0.255970
$146$		0			0
$147$	17.2480			1.42259
$148$		0			0
$149$			11.7779i			0.964881i	0.875929	+	0.482441i	$0.160250\pi$
$149$			11.7779i			0.964881i	−0.875929	+	0.482441i	$0.839750\pi$
$150$		0			0
$151$	−2.90058			−0.236046			−0.118023	−	0.993011i	$-0.537656\pi$
$151$	−2.90058			−0.236046			−0.118023	+	0.993011i	$0.537656\pi$
$152$		0			0
$153$	−20.8549	+	14.7842i	−1.68602	+	1.19523i
$154$		0			0
$155$			0.752428i			0.0604364i
$156$		0			0
$157$		−	9.13785i		−	0.729280i	−0.931149	−	0.364640i	$-0.881192\pi$
$157$		−	9.13785i		−	0.729280i	0.931149	−	0.364640i	$-0.118808\pi$
$158$		0			0
$159$		−	40.8384i		−	3.23870i
$160$		0			0
$161$	−2.73863			−0.215835
$162$		0			0
$163$	22.3320			1.74917			0.874587	−	0.484868i	$-0.161132\pi$
$163$	22.3320			1.74917			0.874587	+	0.484868i	$0.161132\pi$
$164$		0			0
$165$	−3.10373			−0.241625
$166$		0			0
$167$			0.350738i			0.0271409i	0.999908	+	0.0135704i	$0.00431974\pi$
$167$			0.350738i			0.0271409i	−0.999908	+	0.0135704i	$0.995680\pi$
$168$		0			0
$169$	−17.1005			−1.31543
$170$		0			0
$171$		−	0.813350i		−	0.0621984i
$172$		0			0
$173$	21.6007			1.64227			0.821137	−	0.570731i	$-0.193340\pi$
$173$	21.6007			1.64227			0.821137	+	0.570731i	$0.193340\pi$
$174$		0			0
$175$			1.14609i			0.0866364i
$176$		0			0
$177$			30.3474i			2.28105i
$178$		0			0
$179$			20.4689i			1.52991i	0.644082	+	0.764957i	$0.277240\pi$
$179$			20.4689i			1.52991i	−0.644082	+	0.764957i	$0.722760\pi$
$180$		0			0
$181$	10.2177			0.759472			0.379736	−	0.925095i	$-0.376015\pi$
$181$	10.2177			0.759472			0.379736	+	0.925095i	$0.376015\pi$
$182$		0			0
$183$	−11.2945			−0.834911
$184$		0			0
$185$	−7.71836			−0.567465
$186$		0			0
$187$	−3.44190	+	2.43999i	−0.251696	+	0.178430i
$188$		0			0
$189$			11.1245i			0.809188i
$190$		0			0
$191$	10.7086			0.774847			0.387424	−	0.921902i	$-0.373365\pi$
$191$	10.7086			0.774847			0.387424	+	0.921902i	$0.373365\pi$
$192$		0			0
$193$		−	5.68548i		−	0.409250i	−0.978840	−	0.204625i	$-0.934403\pi$
$193$		−	5.68548i		−	0.409250i	0.978840	−	0.204625i	$-0.0655974\pi$
$194$		0			0
$195$		−	16.6412i		−	1.19170i
$196$		0			0
$197$	18.1950			1.29634			0.648170	−	0.761496i	$-0.275535\pi$
$197$	18.1950			1.29634			0.648170	+	0.761496i	$0.275535\pi$
$198$		0			0
$199$		−	6.43510i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$199$		−	6.43510i		−	0.456172i	0.973641	−	0.228086i	$-0.0732468\pi$
$200$		0			0
$201$		−	11.2964i		−	0.796785i
$202$		0			0
$203$		−	3.53258i		−	0.247939i
$204$		0			0
$205$		−	6.10135i		−	0.426137i
$206$		0			0
$207$			14.8154i			1.02974i
$208$		0			0
$209$		−	0.134235i		−	0.00928524i
$210$		0			0
$211$	0.0864097			0.00594869			0.00297434	−	0.999996i	$-0.499053\pi$
$211$	0.0864097			0.00594869			0.00297434	+	0.999996i	$0.499053\pi$
$212$		0			0
$213$		−	20.0954i		−	1.37691i
$214$		0			0
$215$		−	9.64227i		−	0.657597i
$216$		0			0
$217$	0.862351			0.0585402
$218$		0			0
$219$		−	45.2776i		−	3.05957i
$220$		0			0
$221$	−13.0824	−	18.4543i	−0.880016	−	1.24137i
$222$		0			0
$223$	−24.4718			−1.63876			−0.819378	−	0.573254i	$-0.805681\pi$
$223$	−24.4718			−1.63876			−0.819378	+	0.573254i	$0.805681\pi$
$224$		0			0
$225$	6.20011			0.413340
$226$		0			0
$227$	−1.45822			−0.0967854			−0.0483927	−	0.998828i	$-0.515410\pi$
$227$	−1.45822			−0.0967854			−0.0483927	+	0.998828i	$0.515410\pi$
$228$		0			0
$229$		−	15.5745i		−	1.02919i	−0.857434	−	0.514595i	$-0.827942\pi$
$229$		−	15.5745i		−	1.02919i	0.857434	−	0.514595i	$-0.172058\pi$
$230$		0			0
$231$			3.55716i			0.234044i
$232$		0			0
$233$		−	27.4971i		−	1.80140i	−0.434445	−	0.900698i	$-0.643055\pi$
$233$		−	27.4971i		−	1.80140i	0.434445	−	0.900698i	$-0.356945\pi$
$234$		0			0
$235$	−3.00424			−0.195975
$236$		0			0
$237$			41.6370i			2.70461i
$238$		0			0
$239$	−24.6460			−1.59422			−0.797109	−	0.603836i	$-0.793638\pi$
$239$	−24.6460			−1.59422			−0.797109	+	0.603836i	$0.793638\pi$
$240$		0			0
$241$			4.94166i			0.318320i	0.987253	+	0.159160i	$0.0508786\pi$
$241$			4.94166i			0.318320i	−0.987253	+	0.159160i	$0.949121\pi$
$242$		0			0
$243$	3.76319			0.241409
$244$		0			0
$245$	−5.68647			−0.363295
$246$		0			0
$247$	0.719723			0.0457949
$248$		0			0
$249$		−	11.6850i		−	0.740509i
$250$		0			0
$251$		−	14.0131i		−	0.884498i	−0.896892	−	0.442249i	$-0.854181\pi$
$251$		−	14.0131i		−	0.884498i	0.896892	−	0.442249i	$-0.145819\pi$
$252$		0			0
$253$			2.44513i			0.153724i
$254$		0			0
$255$	10.2025	−	7.23263i	0.638905	−	0.452925i
$256$		0			0
$257$	4.66350			0.290901			0.145451	−	0.989366i	$-0.453537\pi$
$257$	4.66350			0.290901			0.145451	+	0.989366i	$0.453537\pi$
$258$		0			0
$259$			8.84595i			0.549661i
$260$		0			0
$261$	−19.1105			−1.18291
$262$		0			0
$263$	−14.1307			−0.871339			−0.435669	−	0.900107i	$-0.643488\pi$
$263$	−14.1307			−0.871339			−0.435669	+	0.900107i	$0.643488\pi$
$264$		0			0
$265$			13.4640i			0.827084i
$266$		0			0
$267$	−11.0532			−0.676443
$268$		0			0
$269$	−8.14611			−0.496677			−0.248339	−	0.968673i	$-0.579885\pi$
$269$	−8.14611			−0.496677			−0.248339	+	0.968673i	$0.579885\pi$
$270$		0			0
$271$	15.1993			0.923289			0.461644	−	0.887065i	$-0.347260\pi$
$271$	15.1993			0.923289			0.461644	+	0.887065i	$0.347260\pi$
$272$		0			0
$273$	−19.0723			−1.15431
$274$		0			0
$275$	1.02326			0.0617052
$276$		0			0
$277$	8.47445			0.509180			0.254590	−	0.967049i	$-0.418059\pi$
$277$	8.47445			0.509180			0.254590	+	0.967049i	$0.418059\pi$
$278$		0			0
$279$		−	4.66513i		−	0.279294i
$280$		0			0
$281$	−18.0326			−1.07574			−0.537868	−	0.843029i	$-0.680770\pi$
$281$	−18.0326			−1.07574			−0.537868	+	0.843029i	$0.680770\pi$
$282$		0			0
$283$	−19.9346			−1.18499			−0.592493	−	0.805576i	$-0.701856\pi$
$283$	−19.9346			−1.18499			−0.592493	+	0.805576i	$0.701856\pi$
$284$		0			0
$285$			0.397901i			0.0235696i
$286$		0			0
$287$	−6.99271			−0.412766
$288$		0			0
$289$	5.62819	−	16.0413i	0.331070	−	0.943606i
$290$		0			0
$291$		−	11.3184i		−	0.663494i
$292$		0			0
$293$		−	7.57931i		−	0.442788i	−0.975184	−	0.221394i	$-0.928939\pi$
$293$		−	7.57931i		−	0.442788i	0.975184	−	0.221394i	$-0.0710607\pi$
$294$		0			0
$295$		−	10.0052i		−	0.582525i
$296$		0			0
$297$	9.93227			0.576329
$298$		0			0
$299$	−13.1100			−0.758169
$300$		0			0
$301$	−11.0509			−0.636965
$302$		0			0
$303$		−	36.4682i		−	2.09504i
$304$		0			0
$305$	3.72365			0.213216
$306$		0			0
$307$			19.8260i			1.13153i	0.824567	+	0.565765i	$0.191419\pi$
$307$			19.8260i			1.13153i	−0.824567	+	0.565765i	$0.808581\pi$
$308$		0			0
$309$	10.9659			0.623830
$310$		0			0
$311$		−	16.7810i		−	0.951565i	−0.879563	−	0.475782i	$-0.842165\pi$
$311$		−	16.7810i		−	0.951565i	0.879563	−	0.475782i	$-0.157835\pi$
$312$		0			0
$313$		−	10.1370i		−	0.572979i	−0.958083	−	0.286490i	$-0.907512\pi$
$313$		−	10.1370i		−	0.572979i	0.958083	−	0.286490i	$-0.0924884\pi$
$314$		0			0
$315$		−	7.10589i		−	0.400372i
$316$		0			0
$317$	−16.3655			−0.919177			−0.459589	−	0.888132i	$-0.652003\pi$
$317$	−16.3655			−0.919177			−0.459589	+	0.888132i	$0.652003\pi$
$318$		0			0
$319$	−3.15400			−0.176590
$320$		0			0
$321$	41.6953			2.32720
$322$		0			0
$323$	0.312808	+	0.441254i	0.0174051	+	0.0245520i
$324$		0			0
$325$			5.48640i			0.304330i
$326$		0			0
$327$	50.9714			2.81872
$328$		0			0
$329$			3.44314i			0.189826i
$330$		0			0
$331$		−	25.3181i		−	1.39161i	−0.718232	−	0.695804i	$-0.755048\pi$
$331$		−	25.3181i		−	1.39161i	0.718232	−	0.695804i	$-0.244952\pi$
$332$		0			0
$333$	47.8547			2.62242
$334$		0			0
$335$			3.72428i			0.203479i
$336$		0			0
$337$			13.6016i			0.740928i	0.928847	+	0.370464i	$0.120801\pi$
$337$			13.6016i			0.740928i	−0.928847	+	0.370464i	$0.879199\pi$
$338$		0			0
$339$		−	7.67645i		−	0.416928i
$340$		0			0
$341$		−	0.769932i		−	0.0416942i
$342$		0			0
$343$			14.5399i			0.785079i
$344$		0			0
$345$		−	7.24788i		−	0.390213i
$346$		0			0
$347$	2.58624			0.138837			0.0694183	−	0.997588i	$-0.477886\pi$
$347$	2.58624			0.138837			0.0694183	+	0.997588i	$0.477886\pi$
$348$		0			0
$349$			14.5552i			0.779123i	0.921000	+	0.389561i	$0.127373\pi$
$349$			14.5552i			0.779123i	−0.921000	+	0.389561i	$0.872627\pi$
$350$		0			0
$351$			53.2535i			2.84246i
$352$		0			0
$353$	12.1646			0.647456			0.323728	−	0.946150i	$-0.395064\pi$
$353$	12.1646			0.647456			0.323728	+	0.946150i	$0.395064\pi$
$354$		0			0
$355$			6.62521i			0.351630i
$356$		0			0
$357$	−8.28925	−	11.6930i	−0.438714	−	0.618859i
$358$		0			0
$359$	−2.79598			−0.147566			−0.0737830	−	0.997274i	$-0.523507\pi$
$359$	−2.79598			−0.147566			−0.0737830	+	0.997274i	$0.523507\pi$
$360$		0			0
$361$	18.9828			0.999094
$362$		0			0
$363$	−30.1889			−1.58451
$364$		0			0
$365$			14.9275i			0.781340i
$366$		0			0
$367$		−	8.02582i		−	0.418944i	−0.977815	−	0.209472i	$-0.932825\pi$
$367$		−	8.02582i		−	0.418944i	0.977815	−	0.209472i	$-0.0671746\pi$
$368$		0			0
$369$			37.8290i			1.96930i
$370$		0			0
$371$	15.4309			0.801134
$372$		0			0
$373$		−	7.11793i		−	0.368553i	−0.982874	−	0.184276i	$-0.941006\pi$
$373$		−	7.11793i		−	0.368553i	0.982874	−	0.184276i	$-0.0589941\pi$
$374$		0			0
$375$	−3.03317			−0.156632
$376$		0			0
$377$		−	16.9106i		−	0.870943i
$378$		0			0
$379$	6.32540			0.324914			0.162457	−	0.986716i	$-0.448058\pi$
$379$	6.32540			0.324914			0.162457	+	0.986716i	$0.448058\pi$
$380$		0			0
$381$	−28.9203			−1.48163
$382$		0			0
$383$	−2.87869			−0.147094			−0.0735472	−	0.997292i	$-0.523432\pi$
$383$	−2.87869			−0.147094			−0.0735472	+	0.997292i	$0.523432\pi$
$384$		0			0
$385$		−	1.17275i		−	0.0597691i
$386$		0			0
$387$			59.7831i			3.03895i
$388$		0			0
$389$			0.0104623i			0.000530459i	1.00000		0.000265230i	$8.44253e-5\pi$
$389$			0.0104623i			0.000530459i	−1.00000		0.000265230i	$0.999916\pi$
$390$		0			0
$391$	−5.69789	−	8.03757i	−0.288155	−	0.406477i
$392$		0			0
$393$	−55.8529			−2.81741
$394$		0			0
$395$		−	13.7272i		−	0.690692i
$396$		0			0
$397$	10.7463			0.539343			0.269672	−	0.962952i	$-0.413085\pi$
$397$	10.7463			0.539343			0.269672	+	0.962952i	$0.413085\pi$
$398$		0			0
$399$	0.456031			0.0228301
$400$		0			0
$401$		−	17.1586i		−	0.856858i	−0.903575	−	0.428429i	$-0.859067\pi$
$401$		−	17.1586i		−	0.856858i	0.903575	−	0.428429i	$-0.140933\pi$
$402$		0			0
$403$	4.12811			0.205636
$404$		0			0
$405$	−10.8410			−0.538694
$406$		0			0
$407$	7.89792			0.391486
$408$		0			0
$409$	−24.4746			−1.21019			−0.605096	−	0.796153i	$-0.706865\pi$
$409$	−24.4746			−1.21019			−0.605096	+	0.796153i	$0.706865\pi$
$410$		0			0
$411$	−39.9957			−1.97284
$412$		0			0
$413$	−11.4669			−0.564248
$414$		0			0
$415$			3.85242i			0.189108i
$416$		0			0
$417$	51.3422			2.51424
$418$		0			0
$419$	31.5503			1.54133			0.770667	−	0.637238i	$-0.219923\pi$
$419$	31.5503			1.54133			0.770667	+	0.637238i	$0.219923\pi$
$420$		0			0
$421$			34.3300i			1.67314i	0.547858	+	0.836571i	$0.315443\pi$
$421$			34.3300i			1.67314i	−0.547858	+	0.836571i	$0.684557\pi$
$422$		0			0
$423$	18.6266			0.905657
$424$		0			0
$425$	−3.36364	+	2.38451i	−0.163161	+	0.115666i
$426$		0			0
$427$		−	4.26765i		−	0.206526i
$428$		0			0
$429$			17.0283i			0.822134i
$430$		0			0
$431$			23.6707i			1.14018i	0.821584	+	0.570088i	$0.193091\pi$
$431$			23.6707i			1.14018i	−0.821584	+	0.570088i	$0.806909\pi$
$432$		0			0
$433$	−35.0893			−1.68628			−0.843142	−	0.537691i	$-0.819297\pi$
$433$	−35.0893			−1.68628			−0.843142	+	0.537691i	$0.819297\pi$
$434$		0			0
$435$	9.34909			0.448255
$436$		0			0
$437$	0.313468			0.0149952
$438$		0			0
$439$			23.9757i			1.14430i	0.820150	+	0.572149i	$0.193890\pi$
$439$			23.9757i			1.14430i	−0.820150	+	0.572149i	$0.806110\pi$
$440$		0			0
$441$	35.2567			1.67889
$442$		0			0
$443$			2.98605i			0.141872i	0.997481	+	0.0709358i	$0.0225985\pi$
$443$			2.98605i			0.141872i	−0.997481	+	0.0709358i	$0.977401\pi$
$444$		0			0
$445$	3.64410			0.172747
$446$		0			0
$447$			35.7243i			1.68970i
$448$		0			0
$449$		−	23.9517i		−	1.13035i	−0.824970	−	0.565176i	$-0.808808\pi$
$449$		−	23.9517i		−	1.13035i	0.824970	−	0.565176i	$-0.191192\pi$
$450$		0			0
$451$			6.24329i			0.293985i
$452$		0			0
$453$	−8.79794			−0.413363
$454$		0			0
$455$	6.28791			0.294782
$456$		0			0
$457$	−18.6341			−0.871666			−0.435833	−	0.900027i	$-0.643546\pi$
$457$	−18.6341			−0.871666			−0.435833	+	0.900027i	$0.643546\pi$
$458$		0			0
$459$	−32.6491	+	23.1452i	−1.52393	+	1.08032i
$460$		0			0
$461$		−	13.9508i		−	0.649753i	−0.945756	−	0.324877i	$-0.894677\pi$
$461$		−	13.9508i		−	0.649753i	0.945756	−	0.324877i	$-0.105323\pi$
$462$		0			0
$463$	20.8212			0.967644			0.483822	−	0.875166i	$-0.339248\pi$
$463$	20.8212			0.967644			0.483822	+	0.875166i	$0.339248\pi$
$464$		0			0
$465$			2.28224i			0.105836i
$466$		0			0
$467$		−	16.4551i		−	0.761453i	−0.924688	−	0.380726i	$-0.875674\pi$
$467$		−	16.4551i		−	0.761453i	0.924688	−	0.380726i	$-0.124326\pi$
$468$		0			0
$469$	4.26837			0.197095
$470$		0			0
$471$		−	27.7166i		−	1.27712i
$472$		0			0
$473$			9.86659i			0.453666i
$474$		0			0
$475$		−	0.131183i		−	0.00601910i
$476$		0			0
$477$		−	83.4780i		−	3.82219i
$478$		0			0
$479$			4.95366i			0.226338i	0.993576	+	0.113169i	$0.0361002\pi$
$479$			4.95366i			0.226338i	−0.993576	+	0.113169i	$0.963900\pi$
$480$		0			0
$481$			42.3460i			1.93081i
$482$		0			0
$483$	−8.30674			−0.377970
$484$		0			0
$485$			3.73153i			0.169440i
$486$		0			0
$487$			38.6316i			1.75057i	0.483611	+	0.875283i	$0.339325\pi$
$487$			38.6316i			1.75057i	−0.483611	+	0.875283i	$0.660675\pi$
$488$		0			0
$489$	67.7366			3.06315
$490$		0			0
$491$			34.3423i			1.54985i	0.632054	+	0.774924i	$0.282212\pi$
$491$			34.3423i			1.54985i	−0.632054	+	0.774924i	$0.717788\pi$
$492$		0			0
$493$	10.3677	−	7.34975i	0.466938	−	0.331016i
$494$		0			0
$495$	−6.34435			−0.285157
$496$		0			0
$497$	7.59310			0.340597
$498$		0			0
$499$	12.3584			0.553240			0.276620	−	0.960979i	$-0.410786\pi$
$499$	12.3584			0.553240			0.276620	+	0.960979i	$0.410786\pi$
$500$		0			0
$501$			1.06385i			0.0475291i
$502$		0			0
$503$		−	41.3000i		−	1.84147i	−0.390183	−	0.920737i	$-0.627588\pi$
$503$		−	41.3000i		−	1.84147i	0.390183	−	0.920737i	$-0.372412\pi$
$504$		0			0
$505$			12.0231i			0.535023i
$506$		0			0
$507$	−51.8688			−2.30357
$508$		0			0
$509$		−	27.2031i		−	1.20576i	−0.797833	−	0.602878i	$-0.794020\pi$
$509$		−	27.2031i		−	1.20576i	0.797833	−	0.602878i	$-0.205980\pi$
$510$		0			0
$511$	17.1083			0.756825
$512$		0			0
$513$		−	1.27332i		−	0.0562186i
$514$		0			0
$515$	−3.61534			−0.159311
$516$		0			0
$517$	3.07413			0.135200
$518$		0			0
$519$	65.5187			2.87595
$520$		0			0
$521$		−	28.1207i		−	1.23199i	−0.787750	−	0.615995i	$-0.788754\pi$
$521$		−	28.1207i		−	1.23199i	0.787750	−	0.615995i	$-0.211246\pi$
$522$		0			0
$523$		−	39.9777i		−	1.74810i	−0.485834	−	0.874051i	$-0.661484\pi$
$523$		−	39.9777i		−	1.74810i	0.485834	−	0.874051i	$-0.338516\pi$
$524$		0			0
$525$			3.47629i			0.151718i
$526$		0			0
$527$	1.79417	+	2.53090i	0.0781554	+	0.110248i
$528$		0			0
$529$	17.2901			0.751743
$530$		0			0
$531$			62.0333i			2.69202i
$532$		0			0
$533$	−33.4744			−1.44994
$534$		0			0
$535$	−13.7464			−0.594311
$536$		0			0
$537$			62.0855i			2.67919i
$538$		0			0
$539$	5.81877			0.250632
$540$		0			0
$541$	−26.3948			−1.13480			−0.567401	−	0.823442i	$-0.692051\pi$
$541$	−26.3948			−1.13480			−0.567401	+	0.823442i	$0.692051\pi$
$542$		0			0
$543$	30.9919			1.32999
$544$		0			0
$545$	−16.8047			−0.719833
$546$		0			0
$547$	−3.38133			−0.144575			−0.0722877	−	0.997384i	$-0.523030\pi$
$547$	−3.38133			−0.144575			−0.0722877	+	0.997384i	$0.523030\pi$
$548$		0			0
$549$	−23.0871			−0.985331
$550$		0			0
$551$			0.404344i			0.0172256i
$552$		0			0
$553$	−15.7327			−0.669021
$554$		0			0
$555$	−23.4111			−0.993745
$556$		0			0
$557$			6.09342i			0.258187i	0.991632	+	0.129093i	$0.0412067\pi$
$557$			6.09342i			0.258187i	−0.991632	+	0.129093i	$0.958793\pi$
$558$		0			0
$559$	−52.9013			−2.23749
$560$		0			0
$561$	−10.4399	+	7.40089i	−0.440771	+	0.312466i
$562$		0			0
$563$		−	41.4270i		−	1.74594i	−0.487774	−	0.872970i	$-0.662191\pi$
$563$		−	41.4270i		−	1.74594i	0.487774	−	0.872970i	$-0.337809\pi$
$564$		0			0
$565$			2.53084i			0.106473i
$566$		0			0
$567$			12.4248i			0.521792i
$568$		0			0
$569$	−1.99334			−0.0835653			−0.0417827	−	0.999127i	$-0.513304\pi$
$569$	−1.99334			−0.0835653			−0.0417827	+	0.999127i	$0.513304\pi$
$570$		0			0
$571$	−12.1058			−0.506614			−0.253307	−	0.967386i	$-0.581518\pi$
$571$	−12.1058			−0.506614			−0.253307	+	0.967386i	$0.581518\pi$
$572$		0			0
$573$	32.4810			1.35691
$574$		0			0
$575$			2.38954i			0.0996508i
$576$		0			0
$577$	19.1678			0.797967			0.398984	−	0.916958i	$-0.369363\pi$
$577$	19.1678			0.797967			0.398984	+	0.916958i	$0.369363\pi$
$578$		0			0
$579$		−	17.2450i		−	0.716678i
$580$		0			0
$581$	4.41523			0.183174
$582$		0			0
$583$		−	13.7772i		−	0.570593i
$584$		0			0
$585$		−	34.0162i		−	1.40640i
$586$		0			0
$587$		−	16.6238i		−	0.686137i	−0.939310	−	0.343068i	$-0.888534\pi$
$587$		−	16.6238i		−	0.686137i	0.939310	−	0.343068i	$-0.111466\pi$
$588$		0			0
$589$	−0.0987059			−0.00406710
$590$		0			0
$591$	55.1885			2.27015
$592$		0			0
$593$	24.6362			1.01169			0.505844	−	0.862625i	$-0.331181\pi$
$593$	24.6362			1.01169			0.505844	+	0.862625i	$0.331181\pi$
$594$		0			0
$595$	2.73287	+	3.85504i	0.112037	+	0.158041i
$596$		0			0
$597$		−	19.5188i		−	0.798849i
$598$		0			0
$599$	−0.908614			−0.0371249			−0.0185625	−	0.999828i	$-0.505909\pi$
$599$	−0.908614			−0.0371249			−0.0185625	+	0.999828i	$0.505909\pi$
$600$		0			0
$601$		−	25.0001i		−	1.01977i	−0.860241	−	0.509887i	$-0.829687\pi$
$601$		−	25.0001i		−	1.01977i	0.860241	−	0.509887i	$-0.170313\pi$
$602$		0			0
$603$		−	23.0909i		−	0.940336i
$604$		0			0
$605$	9.95293			0.404644
$606$		0			0
$607$		−	36.9035i		−	1.49787i	−0.662645	−	0.748933i	$-0.730566\pi$
$607$		−	36.9035i		−	1.49787i	0.662645	−	0.748933i	$-0.269434\pi$
$608$		0			0
$609$		−	10.7149i		−	0.434190i
$610$		0			0
$611$			16.4825i			0.666809i
$612$		0			0
$613$			6.42995i			0.259703i	0.991533	+	0.129852i	$0.0414501\pi$
$613$			6.42995i			0.259703i	−0.991533	+	0.129852i	$0.958550\pi$
$614$		0			0
$615$		−	18.5064i		−	0.746251i
$616$		0			0
$617$		−	14.0811i		−	0.566883i	−0.958989	−	0.283442i	$-0.908524\pi$
$617$		−	14.0811i		−	0.566883i	0.958989	−	0.283442i	$-0.0914762\pi$
$618$		0			0
$619$	−0.313074			−0.0125835			−0.00629176	−	0.999980i	$-0.502003\pi$
$619$	−0.313074			−0.0125835			−0.00629176	+	0.999980i	$0.502003\pi$
$620$		0			0
$621$			23.1940i			0.930743i
$622$		0			0
$623$		−	4.17647i		−	0.167327i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	0.407158i		−	0.0162603i
$628$		0			0
$629$	−25.9618	+	18.4045i	−1.03517	+	0.733837i
$630$		0			0
$631$	−47.2286			−1.88014			−0.940070	−	0.340982i	$-0.889241\pi$
$631$	−47.2286			−1.88014			−0.940070	+	0.340982i	$0.889241\pi$
$632$		0			0
$633$	0.262095			0.0104173
$634$		0			0
$635$	9.53470			0.378373
$636$		0			0
$637$			31.1982i			1.23612i
$638$		0			0
$639$		−	41.0770i		−	1.62498i
$640$		0			0
$641$			33.8478i			1.33691i	0.743754	+	0.668453i	$0.233043\pi$
$641$			33.8478i			1.33691i	−0.743754	+	0.668453i	$0.766957\pi$
$642$		0			0
$643$	12.8818			0.508008			0.254004	−	0.967203i	$-0.418252\pi$
$643$	12.8818			0.508008			0.254004	+	0.967203i	$0.418252\pi$
$644$		0			0
$645$		−	29.2466i		−	1.15158i
$646$		0			0
$647$	12.8495			0.505166			0.252583	−	0.967575i	$-0.418720\pi$
$647$	12.8495			0.505166			0.252583	+	0.967575i	$0.418720\pi$
$648$		0			0
$649$			10.2380i			0.401875i
$650$		0			0
$651$	2.61565			0.102516
$652$		0			0
$653$	−35.7866			−1.40044			−0.700219	−	0.713928i	$-0.746914\pi$
$653$	−35.7866			−1.40044			−0.700219	+	0.713928i	$0.746914\pi$
$654$		0			0
$655$	18.4141			0.719497
$656$		0			0
$657$		−	92.5520i		−	3.61080i
$658$		0			0
$659$		−	31.8870i		−	1.24214i	−0.783754	−	0.621071i	$-0.786698\pi$
$659$		−	31.8870i		−	1.24214i	0.783754	−	0.621071i	$-0.213302\pi$
$660$		0			0
$661$			4.27000i			0.166084i	0.996546	+	0.0830420i	$0.0264636\pi$
$661$			4.27000i			0.166084i	−0.996546	+	0.0830420i	$0.973536\pi$
$662$		0			0
$663$	−39.6811	−	55.9749i	−1.54108	−	2.17389i
$664$		0			0
$665$	−0.150348			−0.00583024
$666$		0			0
$667$		−	7.36525i		−	0.285184i
$668$		0			0
$669$	−74.2272			−2.86979
$670$		0			0
$671$	−3.81028			−0.147094
$672$		0			0
$673$			48.6589i			1.87566i	0.347089	+	0.937832i	$0.387170\pi$
$673$			48.6589i			1.87566i	−0.347089	+	0.937832i	$0.612830\pi$
$674$		0			0
$675$	9.70646			0.373602
$676$		0			0
$677$	−13.6262			−0.523698			−0.261849	−	0.965109i	$-0.584332\pi$
$677$	−13.6262			−0.523698			−0.261849	+	0.965109i	$0.584332\pi$
$678$		0			0
$679$	4.27668			0.164124
$680$		0			0
$681$	−4.42302			−0.169491
$682$		0			0
$683$	−4.24075			−0.162268			−0.0811339	−	0.996703i	$-0.525854\pi$
$683$	−4.24075			−0.162268			−0.0811339	+	0.996703i	$0.525854\pi$
$684$		0			0
$685$	13.1861			0.503815
$686$		0			0
$687$		−	47.2400i		−	1.80232i
$688$		0			0
$689$	73.8686			2.81417
$690$		0			0
$691$	37.5781			1.42954			0.714769	−	0.699361i	$-0.246532\pi$
$691$	37.5781			1.42954			0.714769	+	0.699361i	$0.246532\pi$
$692$		0			0
$693$			7.27120i			0.276210i
$694$		0			0
$695$	−16.9269			−0.642075
$696$		0			0
$697$	−14.5487	−	20.5228i	−0.551073	−	0.777355i
$698$		0			0
$699$		−	83.4034i		−	3.15461i
$700$		0			0
$701$		−	24.2590i		−	0.916249i	−0.888888	−	0.458124i	$-0.848521\pi$
$701$		−	24.2590i		−	0.916249i	0.888888	−	0.458124i	$-0.151479\pi$
$702$		0			0
$703$		−	1.01252i		−	0.0381879i
$704$		0			0
$705$	−9.11237			−0.343192
$706$		0			0
$707$	13.7796			0.518236
$708$		0			0
$709$	21.6792			0.814181			0.407090	−	0.913388i	$-0.366543\pi$
$709$	21.6792			0.814181			0.407090	+	0.913388i	$0.366543\pi$
$710$		0			0
$711$			85.1103i			3.19189i
$712$		0			0
$713$	1.79796			0.0673340
$714$		0			0
$715$		−	5.61403i		−	0.209953i
$716$		0			0
$717$	−74.7555			−2.79179
$718$		0			0
$719$			23.9297i			0.892427i	0.894926	+	0.446214i	$0.147228\pi$
$719$			23.9297i			0.892427i	−0.894926	+	0.446214i	$0.852772\pi$
$720$		0			0
$721$			4.14351i			0.154312i
$722$		0			0
$723$			14.9889i			0.557442i
$724$		0			0
$725$	−3.08229			−0.114473
$726$		0			0
$727$	5.88741			0.218352			0.109176	−	0.994022i	$-0.465179\pi$
$727$	5.88741			0.218352			0.109176	+	0.994022i	$0.465179\pi$
$728$		0			0
$729$	−21.1086			−0.781800
$730$		0			0
$731$	−22.9921	−	32.4332i	−0.850394	−	1.19958i
$732$		0			0
$733$		−	17.0070i		−	0.628167i	−0.949395	−	0.314084i	$-0.898303\pi$
$733$		−	17.0070i		−	0.628167i	0.949395	−	0.314084i	$-0.101697\pi$
$734$		0			0
$735$	−17.2480			−0.636203
$736$		0			0
$737$		−	3.81093i		−	0.140377i
$738$		0			0
$739$			26.7139i			0.982687i	0.870966	+	0.491344i	$0.163494\pi$
$739$			26.7139i			0.982687i	−0.870966	+	0.491344i	$0.836506\pi$
$740$		0			0
$741$	2.18304			0.0801960
$742$		0			0
$743$			49.2505i			1.80683i	0.428771	+	0.903413i	$0.358947\pi$
$743$			49.2505i			1.80683i	−0.428771	+	0.903413i	$0.641053\pi$
$744$		0			0
$745$		−	11.7779i		−	0.431508i
$746$		0			0
$747$		−	23.8854i		−	0.873921i
$748$		0			0
$749$			15.7547i			0.575664i
$750$		0			0
$751$		−	9.44928i		−	0.344809i	−0.985026	−	0.172405i	$-0.944846\pi$
$751$		−	9.44928i		−	0.344809i	0.985026	−	0.172405i	$-0.0551537\pi$
$752$		0			0
$753$		−	42.5040i		−	1.54893i
$754$		0			0
$755$	2.90058			0.105563
$756$		0			0
$757$		−	1.63895i		−	0.0595685i	−0.999556	−	0.0297843i	$-0.990518\pi$
$757$		−	1.63895i		−	0.0595685i	0.999556	−	0.0297843i	$-0.00948203\pi$
$758$		0			0
$759$			7.41650i			0.269202i
$760$		0			0
$761$	28.2428			1.02380			0.511901	−	0.859044i	$-0.328941\pi$
$761$	28.2428			1.02380			0.511901	+	0.859044i	$0.328941\pi$
$762$		0			0
$763$			19.2597i			0.697248i
$764$		0			0
$765$	20.8549	−	14.7842i	0.754012	−	0.534525i
$766$		0			0
$767$	−54.8925			−1.98205
$768$		0			0
$769$	5.94202			0.214275			0.107137	−	0.994244i	$-0.465832\pi$
$769$	5.94202			0.214275			0.107137	+	0.994244i	$0.465832\pi$
$770$		0			0
$771$	14.1452			0.509427
$772$		0			0
$773$		−	20.5173i		−	0.737957i	−0.929438	−	0.368978i	$-0.879708\pi$
$773$		−	20.5173i		−	0.737957i	0.929438	−	0.368978i	$-0.120292\pi$
$774$		0			0
$775$		−	0.752428i		−	0.0270280i
$776$		0			0
$777$			26.8312i			0.962566i
$778$		0			0
$779$	0.800395			0.0286771
$780$		0			0
$781$		−	6.77935i		−	0.242584i
$782$		0			0
$783$	−29.9181			−1.06919
$784$		0			0
$785$			9.13785i			0.326144i
$786$		0			0
$787$	30.1991			1.07648			0.538241	−	0.842791i	$-0.319089\pi$
$787$	30.1991			1.07648			0.538241	+	0.842791i	$0.319089\pi$
$788$		0			0
$789$	−42.8609			−1.52589
$790$		0			0
$791$	2.90057			0.103132
$792$		0			0
$793$		−	20.4294i		−	0.725471i
$794$		0			0
$795$			40.8384i			1.44839i
$796$		0			0
$797$		−	24.2150i		−	0.857738i	−0.903367	−	0.428869i	$-0.858912\pi$
$797$		−	24.2150i		−	0.857738i	0.903367	−	0.428869i	$-0.141088\pi$
$798$		0			0
$799$	−10.1052	+	7.16365i	−0.357496	+	0.253432i
$800$		0			0
$801$	−22.5938			−0.798313
$802$		0			0
$803$		−	15.2748i		−	0.539035i
$804$		0			0
$805$	2.73863			0.0965242
$806$		0			0
$807$	−24.7085			−0.869781
$808$		0			0
$809$		−	42.8270i		−	1.50572i	−0.658183	−	0.752858i	$-0.728675\pi$
$809$		−	42.8270i		−	1.50572i	0.658183	−	0.752858i	$-0.271325\pi$
$810$		0			0
$811$	35.3651			1.24184			0.620919	−	0.783875i	$-0.286760\pi$
$811$	35.3651			1.24184			0.620919	+	0.783875i	$0.286760\pi$
$812$		0			0
$813$	46.1019			1.61686
$814$		0			0
$815$	−22.3320			−0.782255
$816$		0			0
$817$	1.26490			0.0442534
$818$		0			0
$819$	−38.9857			−1.36227
$820$		0			0
$821$	40.2892			1.40610			0.703051	−	0.711139i	$-0.251821\pi$
$821$	40.2892			1.40610			0.703051	+	0.711139i	$0.251821\pi$
$822$		0			0
$823$		−	15.8735i		−	0.553315i	−0.960969	−	0.276658i	$-0.910773\pi$
$823$		−	15.8735i		−	0.553315i	0.960969	−	0.276658i	$-0.0892268\pi$
$824$		0			0
$825$	3.10373			0.108058
$826$		0			0
$827$	−50.7719			−1.76551			−0.882756	−	0.469833i	$-0.844314\pi$
$827$	−50.7719			−1.76551			−0.882756	+	0.469833i	$0.844314\pi$
$828$		0			0
$829$			17.3126i			0.601290i	0.953736	+	0.300645i	$0.0972019\pi$
$829$			17.3126i			0.601290i	−0.953736	+	0.300645i	$0.902798\pi$
$830$		0			0
$831$	25.7044			0.891677
$832$		0			0
$833$	−19.1273	+	13.5595i	−0.662721	+	0.469808i
$834$		0			0
$835$		−	0.350738i		−	0.0121378i
$836$		0			0
$837$		−	7.30341i		−	0.252443i
$838$		0			0
$839$			37.2470i			1.28591i	0.765904	+	0.642955i	$0.222292\pi$
$839$			37.2470i			1.28591i	−0.765904	+	0.642955i	$0.777708\pi$
$840$		0			0
$841$	−19.4995			−0.672397
$842$		0			0
$843$	−54.6960			−1.88383
$844$		0			0
$845$	17.1005			0.588276
$846$		0			0
$847$		−	11.4070i		−	0.391948i
$848$		0			0
$849$	−60.4648			−2.07515
$850$		0			0
$851$			18.4433i			0.632230i
$852$		0			0
$853$	−23.8943			−0.818125			−0.409062	−	0.912506i	$-0.634144\pi$
$853$	−23.8943			−0.818125			−0.409062	+	0.912506i	$0.634144\pi$
$854$		0			0
$855$			0.813350i			0.0278160i
$856$		0			0
$857$		−	22.7277i		−	0.776363i	−0.921583	−	0.388181i	$-0.873103\pi$
$857$		−	22.7277i		−	0.776363i	0.921583	−	0.388181i	$-0.126897\pi$
$858$		0			0
$859$			48.5724i			1.65727i	0.559790	+	0.828635i	$0.310882\pi$
$859$			48.5724i			1.65727i	−0.559790	+	0.828635i	$0.689118\pi$
$860$		0			0
$861$	−21.2100			−0.722836
$862$		0			0
$863$	−32.4151			−1.10342			−0.551711	−	0.834035i	$-0.686025\pi$
$863$	−32.4151			−1.10342			−0.551711	+	0.834035i	$0.686025\pi$
$864$		0			0
$865$	−21.6007			−0.734447
$866$		0			0
$867$	17.0713	−	48.6560i	0.579770	−	1.65244i
$868$		0			0
$869$			14.0466i			0.476498i
$870$		0			0
$871$	20.4329			0.692342
$872$		0			0
$873$		−	23.1359i		−	0.783032i
$874$		0			0
$875$		−	1.14609i		−	0.0387450i
$876$		0			0
$877$	48.4637			1.63650			0.818252	−	0.574860i	$-0.194944\pi$
$877$	48.4637			1.63650			0.818252	+	0.574860i	$0.194944\pi$
$878$		0			0
$879$		−	22.9893i		−	0.775410i
$880$		0			0
$881$		−	19.3636i		−	0.652377i	−0.945305	−	0.326188i	$-0.894236\pi$
$881$		−	19.3636i		−	0.652377i	0.945305	−	0.326188i	$-0.105764\pi$
$882$		0			0
$883$			48.0780i			1.61795i	0.587841	+	0.808977i	$0.299978\pi$
$883$			48.0780i			1.61795i	−0.587841	+	0.808977i	$0.700022\pi$
$884$		0			0
$885$		−	30.3474i		−	1.02012i
$886$		0			0
$887$		−	57.0934i		−	1.91701i	−0.285078	−	0.958504i	$-0.592019\pi$
$887$		−	57.0934i		−	1.91701i	0.285078	−	0.958504i	$-0.407981\pi$
$888$		0			0
$889$		−	10.9276i		−	0.366501i
$890$		0			0
$891$	11.0932			0.371636
$892$		0			0
$893$		−	0.394106i		−	0.0131883i
$894$		0			0
$895$		−	20.4689i		−	0.684198i
$896$		0			0
$897$	−39.7647			−1.32771
$898$		0			0
$899$			2.31920i			0.0773496i
$900$		0			0
$901$	32.1050	+	45.2879i	1.06957	+	1.50876i
$902$		0			0
$903$	−33.5193			−1.11545
$904$		0			0
$905$	−10.2177			−0.339646
$906$		0			0
$907$	2.35855			0.0783144			0.0391572	−	0.999233i	$-0.487533\pi$
$907$	2.35855			0.0783144			0.0391572	+	0.999233i	$0.487533\pi$
$908$		0			0
$909$		−	74.5448i		−	2.47249i
$910$		0			0
$911$			36.5217i			1.21002i	0.796218	+	0.605009i	$0.206831\pi$
$911$			36.5217i			1.21002i	−0.796218	+	0.605009i	$0.793169\pi$
$912$		0			0
$913$		−	3.94204i		−	0.130463i
$914$		0			0
$915$	11.2945			0.373384
$916$		0			0
$917$		−	21.1042i		−	0.696922i
$918$		0			0
$919$	−28.1923			−0.929978			−0.464989	−	0.885317i	$-0.653942\pi$
$919$	−28.1923			−0.929978			−0.464989	+	0.885317i	$0.653942\pi$
$920$		0			0
$921$			60.1356i			1.98153i
$922$		0			0
$923$	36.3485			1.19643
$924$		0			0
$925$	7.71836			0.253778
$926$		0			0
$927$	22.4155			0.736222
$928$		0			0
$929$			13.5645i			0.445038i	0.974928	+	0.222519i	$0.0714280\pi$
$929$			13.5645i			0.445038i	−0.974928	+	0.222519i	$0.928572\pi$
$930$		0			0
$931$		−	0.745970i		−	0.0244482i
$932$		0			0
$933$		−	50.8997i		−	1.66638i
$934$		0			0
$935$	3.44190	−	2.43999i	0.112562	−	0.0797961i
$936$		0			0
$937$	−36.5498			−1.19403			−0.597015	−	0.802230i	$-0.703647\pi$
$937$	−36.5498			−1.19403			−0.597015	+	0.802230i	$0.703647\pi$
$938$		0			0
$939$		−	30.7473i		−	1.00340i
$940$		0			0
$941$	12.6597			0.412695			0.206347	−	0.978479i	$-0.433842\pi$
$941$	12.6597			0.412695			0.206347	+	0.978479i	$0.433842\pi$
$942$		0			0
$943$	−14.5794			−0.474772
$944$		0			0
$945$		−	11.1245i		−	0.361880i
$946$		0			0
$947$	−15.4599			−0.502378			−0.251189	−	0.967938i	$-0.580822\pi$
$947$	−15.4599			−0.502378			−0.251189	+	0.967938i	$0.580822\pi$
$948$		0			0
$949$	81.8981			2.65852
$950$		0			0
$951$	−49.6393			−1.60966
$952$		0			0
$953$	36.0503			1.16778			0.583891	−	0.811832i	$-0.301529\pi$
$953$	36.0503			1.16778			0.583891	+	0.811832i	$0.301529\pi$
$954$		0			0
$955$	−10.7086			−0.346522
$956$		0			0
$957$	−9.56660			−0.309244
$958$		0			0
$959$		−	15.1125i		−	0.488007i
$960$		0			0
$961$	30.4339			0.981737
$962$		0			0
$963$	85.2294			2.74648
$964$		0			0
$965$			5.68548i			0.183022i
$966$		0			0
$967$	41.1709			1.32397			0.661984	−	0.749518i	$-0.269715\pi$
$967$	41.1709			1.32397			0.661984	+	0.749518i	$0.269715\pi$
$968$		0			0
$969$	0.948799	+	1.33840i	0.0304798	+	0.0429955i
$970$		0			0
$971$			28.5053i			0.914777i	0.889267	+	0.457389i	$0.151215\pi$
$971$			28.5053i			0.914777i	−0.889267	+	0.457389i	$0.848785\pi$
$972$		0			0
$973$			19.3998i			0.621930i
$974$		0			0
$975$			16.6412i			0.532944i
$976$		0			0
$977$	−15.7285			−0.503198			−0.251599	−	0.967832i	$-0.580956\pi$
$977$	−15.7285			−0.503198			−0.251599	+	0.967832i	$0.580956\pi$
$978$		0			0
$979$	−3.72888			−0.119175
$980$		0			0
$981$	104.191			3.32655
$982$		0			0
$983$			24.3987i			0.778196i	0.921196	+	0.389098i	$0.127213\pi$
$983$			24.3987i			0.778196i	−0.921196	+	0.389098i	$0.872787\pi$
$984$		0			0
$985$	−18.1950			−0.579741
$986$		0			0
$987$			10.4436i			0.332424i
$988$		0			0
$989$	−23.0406			−0.732649
$990$		0			0
$991$		−	6.73788i		−	0.214036i	−0.994257	−	0.107018i	$-0.965870\pi$
$991$		−	6.73788i		−	0.214036i	0.994257	−	0.107018i	$-0.0341302\pi$
$992$		0			0
$993$		−	76.7940i		−	2.43698i
$994$		0			0
$995$			6.43510i			0.204007i
$996$		0			0
$997$	−49.0657			−1.55393			−0.776964	−	0.629545i	$-0.783241\pi$
$997$	−49.0657			−1.55393			−0.776964	+	0.629545i	$0.783241\pi$
$998$		0			0
$999$	74.9180			2.37030

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2720.2.l.a.2481.36		36
4.3	odd	2		680.2.l.b.101.6	yes	36
8.3	odd	2		680.2.l.a.101.5	✓	36
8.5	even	2		2720.2.l.b.2481.2		36
17.16	even	2		2720.2.l.b.2481.1		36
68.67	odd	2		680.2.l.a.101.6	yes	36
136.67	odd	2		680.2.l.b.101.5	yes	36
136.101	even	2	inner	2720.2.l.a.2481.35		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.l.a.101.5	✓	36	8.3	odd	2
680.2.l.a.101.6	yes	36	68.67	odd	2
680.2.l.b.101.5	yes	36	136.67	odd	2
680.2.l.b.101.6	yes	36	4.3	odd	2
2720.2.l.a.2481.35		36	136.101	even	2	inner
2720.2.l.a.2481.36		36	1.1	even	1	trivial
2720.2.l.b.2481.1		36	17.16	even	2
2720.2.l.b.2481.2		36	8.5	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2720 = 2^{5} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2720.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.03317 q^{3} -1.00000 q^{5} +1.14609i q^{7} +6.20011 q^{9} +1.02326 q^{11} +5.48640i q^{13} -3.03317 q^{15} +(-3.36364 + 2.38451i) q^{17} -0.131183i q^{19} +3.47629i q^{21} +2.38954i q^{23} +1.00000 q^{25} +9.70646 q^{27} -3.08229 q^{29} -0.752428i q^{31} +3.10373 q^{33} -1.14609i q^{35} +7.71836 q^{37} +16.6412i q^{39} +6.10135i q^{41} +9.64227i q^{43} -6.20011 q^{45} +3.00424 q^{47} +5.68647 q^{49} +(-10.2025 + 7.23263i) q^{51} -13.4640i q^{53} -1.02326 q^{55} -0.397901i q^{57} +10.0052i q^{59} -3.72365 q^{61} +7.10589i q^{63} -5.48640i q^{65} -3.72428i q^{67} +7.24788i q^{69} -6.62521i q^{71} -14.9275i q^{73} +3.03317 q^{75} +1.17275i q^{77} +13.7272i q^{79} +10.8410 q^{81} -3.85242i q^{83} +(3.36364 - 2.38451i) q^{85} -9.34909 q^{87} -3.64410 q^{89} -6.28791 q^{91} -2.28224i q^{93} +0.131183i q^{95} -3.73153i q^{97} +6.34435 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} - 36 q^{5} + 36 q^{9} - 8 q^{11} + 4 q^{15} + 36 q^{25} - 16 q^{27} - 8 q^{33} - 36 q^{45} - 20 q^{47} - 36 q^{49} + 8 q^{55} - 4 q^{75} + 44 q^{81} - 24 q^{87} - 56 q^{91} - 40 q^{99}+O(q^{100}) \) 36 * q - 4 * q^3 - 36 * q^5 + 36 * q^9 - 8 * q^11 + 4 * q^15 + 36 * q^25 - 16 * q^27 - 8 * q^33 - 36 * q^45 - 20 * q^47 - 36 * q^49 + 8 * q^55 - 4 * q^75 + 44 * q^81 - 24 * q^87 - 56 * q^91 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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