LMFDB - Embedded newform 2240.2.l.c.1569.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1569,2240)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))

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

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

chi := DirichletCharacter("2240.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.l (of order $2$, degree $1$, 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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.8
Character	$\chi$	$=$	2240.1569
Dual form			2240.2.l.c.1569.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.29255 q^{3} +(0.437032 + 2.19294i) q^{5} +1.00000i q^{7} -1.32931 q^{9} +O(q^{10})$	$q-1.29255 q^{3} +(0.437032 + 2.19294i) q^{5} +1.00000i q^{7} -1.32931 q^{9} +0.711996i q^{11} +1.85608 q^{13} +(-0.564886 - 2.83449i) q^{15} -3.75616i q^{17} -7.57843i q^{19} -1.29255i q^{21} -8.45250i q^{23} +(-4.61801 + 1.91677i) q^{25} +5.59586 q^{27} +5.24684i q^{29} -0.349014 q^{31} -0.920291i q^{33} +(-2.19294 + 0.437032i) q^{35} +7.50603 q^{37} -2.39908 q^{39} +2.74755 q^{41} -2.07092 q^{43} +(-0.580952 - 2.91511i) q^{45} -4.69919i q^{47} -1.00000 q^{49} +4.85503i q^{51} +9.81202 q^{53} +(-1.56137 + 0.311165i) q^{55} +9.79551i q^{57} +6.54305i q^{59} +6.50330i q^{61} -1.32931i q^{63} +(0.811166 + 4.07028i) q^{65} +10.4937 q^{67} +10.9253i q^{69} -5.53965 q^{71} +9.10513i q^{73} +(5.96901 - 2.47753i) q^{75} -0.711996 q^{77} +4.54792 q^{79} -3.24499 q^{81} +4.92352 q^{83} +(8.23706 - 1.64156i) q^{85} -6.78181i q^{87} -5.58518 q^{89} +1.85608i q^{91} +0.451118 q^{93} +(16.6191 - 3.31202i) q^{95} +14.1673i q^{97} -0.946466i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{5} + 12 q^{9}+O(q^{10}) $ 24 * q - 4 * q^5 + 12 * q^9	$ 24 q - 4 q^{5} + 12 q^{9} - 4 q^{13} - 24 q^{37} - 48 q^{45} - 24 q^{49} + 88 q^{53} + 36 q^{65} + 20 q^{77} + 16 q^{81} - 56 q^{85} - 40 q^{89} + 24 q^{93}+O(q^{100}) $ 24 * q - 4 * q^5 + 12 * q^9 - 4 * q^13 - 24 * q^37 - 48 * q^45 - 24 * q^49 + 88 * q^53 + 36 * q^65 + 20 * q^77 + 16 * q^81 - 56 * q^85 - 40 * q^89 + 24 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−1.29255			−0.746254			−0.373127	−	0.927780i	$-0.621715\pi$
$3$	−1.29255			−0.746254			−0.373127	+	0.927780i	$0.621715\pi$
$4$		0			0
$5$	0.437032	+	2.19294i	0.195447	+	0.980714i
$5$	0.437032	+	2.19294i	0.195447	+	0.980714i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.32931			−0.443104
$10$		0			0
$11$			0.711996i			0.214675i	0.994223	+	0.107337i	$0.0342325\pi$
$11$			0.711996i			0.214675i	−0.994223	+	0.107337i	$0.965767\pi$
$12$		0			0
$13$	1.85608			0.514784			0.257392	−	0.966307i	$-0.417137\pi$
$13$	1.85608			0.514784			0.257392	+	0.966307i	$0.417137\pi$
$14$		0			0
$15$	−0.564886	−	2.83449i	−0.145853	−	0.731862i
$16$		0			0
$17$		−	3.75616i		−	0.911004i	−0.890235	−	0.455502i	$-0.849460\pi$
$17$		−	3.75616i		−	0.911004i	0.890235	−	0.455502i	$-0.150540\pi$
$18$		0			0
$19$		−	7.57843i		−	1.73861i	−0.494274	−	0.869306i	$-0.664566\pi$
$19$		−	7.57843i		−	1.73861i	0.494274	−	0.869306i	$-0.335434\pi$
$20$		0			0
$21$		−	1.29255i		−	0.282058i
$22$		0			0
$23$		−	8.45250i		−	1.76247i	−0.472681	−	0.881234i	$-0.656714\pi$
$23$		−	8.45250i		−	1.76247i	0.472681	−	0.881234i	$-0.343286\pi$
$24$		0			0
$25$	−4.61801	+	1.91677i	−0.923601	+	0.383355i
$26$		0			0
$27$	5.59586			1.07692
$28$		0			0
$29$			5.24684i			0.974314i	0.873314	+	0.487157i	$0.161966\pi$
$29$			5.24684i			0.974314i	−0.873314	+	0.487157i	$0.838034\pi$
$30$		0			0
$31$	−0.349014			−0.0626847			−0.0313424	−	0.999509i	$-0.509978\pi$
$31$	−0.349014			−0.0626847			−0.0313424	+	0.999509i	$0.509978\pi$
$32$		0			0
$33$		−	0.920291i		−	0.160202i
$34$		0			0
$35$	−2.19294	+	0.437032i	−0.370675	+	0.0738719i
$36$		0			0
$37$	7.50603			1.23398			0.616992	−	0.786969i	$-0.288351\pi$
$37$	7.50603			1.23398			0.616992	+	0.786969i	$0.288351\pi$
$38$		0			0
$39$	−2.39908			−0.384160
$40$		0			0
$41$	2.74755			0.429095			0.214547	−	0.976714i	$-0.431172\pi$
$41$	2.74755			0.429095			0.214547	+	0.976714i	$0.431172\pi$
$42$		0			0
$43$	−2.07092			−0.315813			−0.157906	−	0.987454i	$-0.550474\pi$
$43$	−2.07092			−0.315813			−0.157906	+	0.987454i	$0.550474\pi$
$44$		0			0
$45$	−0.580952	−	2.91511i	−0.0866032	−	0.434559i
$46$		0			0
$47$		−	4.69919i		−	0.685448i	−0.939436	−	0.342724i	$-0.888651\pi$
$47$		−	4.69919i		−	0.685448i	0.939436	−	0.342724i	$-0.111349\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$			4.85503i			0.679841i
$52$		0			0
$53$	9.81202			1.34778			0.673892	−	0.738830i	$-0.264621\pi$
$53$	9.81202			1.34778			0.673892	+	0.738830i	$0.264621\pi$
$54$		0			0
$55$	−1.56137	+	0.311165i	−0.210535	+	0.0419575i
$56$		0			0
$57$			9.79551i			1.29745i
$58$		0			0
$59$			6.54305i			0.851832i	0.904763	+	0.425916i	$0.140048\pi$
$59$			6.54305i			0.851832i	−0.904763	+	0.425916i	$0.859952\pi$
$60$		0			0
$61$			6.50330i			0.832663i	0.909213	+	0.416331i	$0.136684\pi$
$61$			6.50330i			0.832663i	−0.909213	+	0.416331i	$0.863316\pi$
$62$		0			0
$63$		−	1.32931i		−	0.167478i
$64$		0			0
$65$	0.811166	+	4.07028i	0.100613	+	0.504856i
$66$		0			0
$67$	10.4937			1.28201			0.641004	−	0.767538i	$-0.278518\pi$
$67$	10.4937			1.28201			0.641004	+	0.767538i	$0.278518\pi$
$68$		0			0
$69$			10.9253i			1.31525i
$70$		0			0
$71$	−5.53965			−0.657435			−0.328718	−	0.944428i	$-0.606616\pi$
$71$	−5.53965			−0.657435			−0.328718	+	0.944428i	$0.606616\pi$
$72$		0			0
$73$			9.10513i			1.06567i	0.846218	+	0.532837i	$0.178874\pi$
$73$			9.10513i			1.06567i	−0.846218	+	0.532837i	$0.821126\pi$
$74$		0			0
$75$	5.96901	−	2.47753i	0.689242	−	0.286080i
$76$		0			0
$77$	−0.711996			−0.0811395
$78$		0			0
$79$	4.54792			0.511681			0.255841	−	0.966719i	$-0.417648\pi$
$79$	4.54792			0.511681			0.255841	+	0.966719i	$0.417648\pi$
$80$		0			0
$81$	−3.24499			−0.360554
$82$		0			0
$83$	4.92352			0.540427			0.270213	−	0.962800i	$-0.412906\pi$
$83$	4.92352			0.540427			0.270213	+	0.962800i	$0.412906\pi$
$84$		0			0
$85$	8.23706	−	1.64156i	0.893434	−	0.178053i
$86$		0			0
$87$		−	6.78181i		−	0.727086i
$88$		0			0
$89$	−5.58518			−0.592028			−0.296014	−	0.955184i	$-0.595657\pi$
$89$	−5.58518			−0.592028			−0.296014	+	0.955184i	$0.595657\pi$
$90$		0			0
$91$			1.85608i			0.194570i
$92$		0			0
$93$	0.451118			0.0467787
$94$		0			0
$95$	16.6191	−	3.31202i	1.70508	−	0.339806i
$96$		0			0
$97$			14.1673i			1.43847i	0.694768	+	0.719234i	$0.255507\pi$
$97$			14.1673i			1.43847i	−0.694768	+	0.719234i	$0.744493\pi$
$98$		0			0
$99$		−	0.946466i		−	0.0951234i
$100$		0			0
$101$			1.69475i			0.168634i	0.996439	+	0.0843168i	$0.0268708\pi$
$101$			1.69475i			0.168634i	−0.996439	+	0.0843168i	$0.973129\pi$
$102$		0			0
$103$		−	11.3218i		−	1.11557i	−0.829984	−	0.557787i	$-0.811650\pi$
$103$		−	11.3218i		−	1.11557i	0.829984	−	0.557787i	$-0.188350\pi$
$104$		0			0
$105$	2.83449	−	0.564886i	0.276618	−	0.0551272i
$106$		0			0
$107$	−2.25954			−0.218438			−0.109219	−	0.994018i	$-0.534835\pi$
$107$	−2.25954			−0.218438			−0.109219	+	0.994018i	$0.534835\pi$
$108$		0			0
$109$		−	18.5031i		−	1.77228i	−0.463422	−	0.886138i	$-0.653379\pi$
$109$		−	18.5031i		−	1.77228i	0.463422	−	0.886138i	$-0.346621\pi$
$110$		0			0
$111$	−9.70193			−0.920866
$112$		0			0
$113$		−	19.7639i		−	1.85923i	−0.368535	−	0.929614i	$-0.620140\pi$
$113$		−	19.7639i		−	1.85923i	0.368535	−	0.929614i	$-0.379860\pi$
$114$		0			0
$115$	18.5359	−	3.69401i	1.72848	−	0.344468i
$116$		0			0
$117$	−2.46731			−0.228103
$118$		0			0
$119$	3.75616			0.344327
$120$		0			0
$121$	10.4931			0.953915
$122$		0			0
$123$	−3.55134			−0.320214
$124$		0			0
$125$	−6.22159	−	9.28934i	−0.556476	−	0.830864i
$126$		0			0
$127$		−	8.43468i		−	0.748456i	−0.927337	−	0.374228i	$-0.877908\pi$
$127$		−	8.43468i		−	0.748456i	0.927337	−	0.374228i	$-0.122092\pi$
$128$		0			0
$129$	2.67677			0.235677
$130$		0			0
$131$			1.23358i			0.107779i	0.998547	+	0.0538894i	$0.0171618\pi$
$131$			1.23358i			0.107779i	−0.998547	+	0.0538894i	$0.982838\pi$
$132$		0			0
$133$	7.57843			0.657133
$134$		0			0
$135$	2.44557	+	12.2714i	0.210481	+	1.05615i
$136$		0			0
$137$			16.5866i			1.41709i	0.705665	+	0.708546i	$0.250648\pi$
$137$			16.5866i			1.41709i	−0.705665	+	0.708546i	$0.749352\pi$
$138$		0			0
$139$			18.3119i			1.55319i	0.629998	+	0.776597i	$0.283056\pi$
$139$			18.3119i			1.55319i	−0.629998	+	0.776597i	$0.716944\pi$
$140$		0			0
$141$			6.07394i			0.511518i
$142$		0			0
$143$			1.32152i			0.110511i
$144$		0			0
$145$	−11.5060	+	2.29304i	−0.955524	+	0.190426i
$146$		0			0
$147$	1.29255			0.106608
$148$		0			0
$149$		−	11.2743i		−	0.923626i	−0.886977	−	0.461813i	$-0.847199\pi$
$149$		−	11.2743i		−	0.923626i	0.886977	−	0.461813i	$-0.152801\pi$
$150$		0			0
$151$	14.5681			1.18554			0.592769	−	0.805372i	$-0.298035\pi$
$151$	14.5681			1.18554			0.592769	+	0.805372i	$0.298035\pi$
$152$		0			0
$153$			4.99312i			0.403670i
$154$		0			0
$155$	−0.152530	−	0.765367i	−0.0122515	−	0.0614758i
$156$		0			0
$157$	10.1294			0.808415			0.404207	−	0.914667i	$-0.367547\pi$
$157$	10.1294			0.808415			0.404207	+	0.914667i	$0.367547\pi$
$158$		0			0
$159$	−12.6825			−1.00579
$160$		0			0
$161$	8.45250			0.666150
$162$		0			0
$163$	−6.96429			−0.545486			−0.272743	−	0.962087i	$-0.587931\pi$
$163$	−6.96429			−0.545486			−0.272743	+	0.962087i	$0.587931\pi$
$164$		0			0
$165$	2.01815	−	0.402197i	0.157113	−	0.0313110i
$166$		0			0
$167$			4.55615i			0.352566i	0.984340	+	0.176283i	$0.0564074\pi$
$167$			4.55615i			0.352566i	−0.984340	+	0.176283i	$0.943593\pi$
$168$		0			0
$169$	−9.55497			−0.734998
$170$		0			0
$171$			10.0741i			0.770386i
$172$		0			0
$173$	21.1315			1.60660			0.803300	−	0.595574i	$-0.203075\pi$
$173$	21.1315			1.60660			0.803300	+	0.595574i	$0.203075\pi$
$174$		0			0
$175$	−1.91677	−	4.61801i	−0.144894	−	0.349088i
$176$		0			0
$177$		−	8.45722i		−	0.635684i
$178$		0			0
$179$		−	21.6347i		−	1.61705i	−0.588458	−	0.808527i	$-0.700265\pi$
$179$		−	21.6347i		−	1.61705i	0.588458	−	0.808527i	$-0.299735\pi$
$180$		0			0
$181$			9.01846i			0.670337i	0.942158	+	0.335168i	$0.108793\pi$
$181$			9.01846i			0.670337i	−0.942158	+	0.335168i	$0.891207\pi$
$182$		0			0
$183$		−	8.40585i		−	0.621378i
$184$		0			0
$185$	3.28038	+	16.4603i	0.241178	+	1.21019i
$186$		0			0
$187$	2.67438			0.195570
$188$		0			0
$189$			5.59586i			0.407039i
$190$		0			0
$191$	12.1324			0.877872			0.438936	−	0.898518i	$-0.355355\pi$
$191$	12.1324			0.877872			0.438936	+	0.898518i	$0.355355\pi$
$192$		0			0
$193$		−	10.9798i		−	0.790342i	−0.918608	−	0.395171i	$-0.870685\pi$
$193$		−	10.9798i		−	0.790342i	0.918608	−	0.395171i	$-0.129315\pi$
$194$		0			0
$195$	−1.04847	−	5.26104i	−0.0750828	−	0.376751i
$196$		0			0
$197$	−0.349165			−0.0248770			−0.0124385	−	0.999923i	$-0.503959\pi$
$197$	−0.349165			−0.0248770			−0.0124385	+	0.999923i	$0.503959\pi$
$198$		0			0
$199$	21.8923			1.55190			0.775951	−	0.630793i	$-0.217270\pi$
$199$	21.8923			1.55190			0.775951	+	0.630793i	$0.217270\pi$
$200$		0			0
$201$	−13.5636			−0.956704
$202$		0			0
$203$	−5.24684			−0.368256
$204$		0			0
$205$	1.20077	+	6.02521i	0.0838651	+	0.420819i
$206$		0			0
$207$			11.2360i			0.780957i
$208$		0			0
$209$	5.39582			0.373236
$210$		0			0
$211$			14.6498i			1.00853i	0.863548	+	0.504267i	$0.168237\pi$
$211$			14.6498i			1.00853i	−0.863548	+	0.504267i	$0.831763\pi$
$212$		0			0
$213$	7.16028			0.490614
$214$		0			0
$215$	−0.905060	−	4.54142i	−0.0617246	−	0.309722i
$216$		0			0
$217$		−	0.349014i		−	0.0236926i
$218$		0			0
$219$		−	11.7688i		−	0.795265i
$220$		0			0
$221$		−	6.97174i		−	0.468970i
$222$		0			0
$223$		−	18.9248i		−	1.26730i	−0.773619	−	0.633651i	$-0.781556\pi$
$223$		−	18.9248i		−	1.26730i	0.773619	−	0.633651i	$-0.218444\pi$
$224$		0			0
$225$	6.13877	−	2.54799i	0.409252	−	0.169866i
$226$		0			0
$227$	−18.2624			−1.21212			−0.606058	−	0.795420i	$-0.707250\pi$
$227$	−18.2624			−1.21212			−0.606058	+	0.795420i	$0.707250\pi$
$228$		0			0
$229$		−	7.23742i		−	0.478262i	−0.970987	−	0.239131i	$-0.923137\pi$
$229$		−	7.23742i		−	0.478262i	0.970987	−	0.239131i	$-0.0768626\pi$
$230$		0			0
$231$	0.920291			0.0605507
$232$		0			0
$233$		−	18.6650i		−	1.22278i	−0.791329	−	0.611391i	$-0.790610\pi$
$233$		−	18.6650i		−	1.22278i	0.791329	−	0.611391i	$-0.209390\pi$
$234$		0			0
$235$	10.3051	−	2.05370i	0.672228	−	0.133968i
$236$		0			0
$237$	−5.87842			−0.381845
$238$		0			0
$239$	−0.993443			−0.0642605			−0.0321302	−	0.999484i	$-0.510229\pi$
$239$	−0.993443			−0.0642605			−0.0321302	+	0.999484i	$0.510229\pi$
$240$		0			0
$241$	21.2196			1.36687			0.683436	−	0.730010i	$-0.260485\pi$
$241$	21.2196			1.36687			0.683436	+	0.730010i	$0.260485\pi$
$242$		0			0
$243$	−12.5933			−0.807858
$244$		0			0
$245$	−0.437032	−	2.19294i	−0.0279210	−	0.140102i
$246$		0			0
$247$		−	14.0662i		−	0.895009i
$248$		0			0
$249$	−6.36390			−0.403296
$250$		0			0
$251$			15.2669i			0.963641i	0.876270	+	0.481820i	$0.160024\pi$
$251$			15.2669i			0.963641i	−0.876270	+	0.481820i	$0.839976\pi$
$252$		0			0
$253$	6.01815			0.378358
$254$		0			0
$255$	−10.6468	+	2.12181i	−0.666729	+	0.132873i
$256$		0			0
$257$		−	28.5310i		−	1.77971i	−0.456240	−	0.889857i	$-0.650804\pi$
$257$		−	28.5310i		−	1.77971i	0.456240	−	0.889857i	$-0.349196\pi$
$258$		0			0
$259$			7.50603i			0.466402i
$260$		0			0
$261$		−	6.97469i		−	0.431723i
$262$		0			0
$263$			5.51697i			0.340191i	0.985428	+	0.170095i	$0.0544076\pi$
$263$			5.51697i			0.340191i	−0.985428	+	0.170095i	$0.945592\pi$
$264$		0			0
$265$	4.28817	+	21.5172i	0.263420	+	1.32179i
$266$		0			0
$267$	7.21912			0.441803
$268$		0			0
$269$			8.79256i			0.536092i	0.963406	+	0.268046i	$0.0863779\pi$
$269$			8.79256i			0.536092i	−0.963406	+	0.268046i	$0.913622\pi$
$270$		0			0
$271$	1.95225			0.118591			0.0592954	−	0.998240i	$-0.481115\pi$
$271$	1.95225			0.118591			0.0592954	+	0.998240i	$0.481115\pi$
$272$		0			0
$273$		−	2.39908i		−	0.145199i
$274$		0			0
$275$	−1.36474	−	3.28800i	−0.0822967	−	0.198274i
$276$		0			0
$277$	−17.1637			−1.03127			−0.515635	−	0.856808i	$-0.672444\pi$
$277$	−17.1637			−1.03127			−0.515635	+	0.856808i	$0.672444\pi$
$278$		0			0
$279$	0.463948			0.0277759
$280$		0			0
$281$	24.2087			1.44417			0.722085	−	0.691805i	$-0.243184\pi$
$281$	24.2087			1.44417			0.722085	+	0.691805i	$0.243184\pi$
$282$		0			0
$283$	−11.8658			−0.705350			−0.352675	−	0.935746i	$-0.614728\pi$
$283$	−11.8658			−0.705350			−0.352675	+	0.935746i	$0.614728\pi$
$284$		0			0
$285$	−21.4810	+	4.28095i	−1.27242	+	0.253582i
$286$		0			0
$287$			2.74755i			0.162183i
$288$		0			0
$289$	2.89123			0.170072
$290$		0			0
$291$		−	18.3119i		−	1.07346i
$292$		0			0
$293$	−11.5627			−0.675502			−0.337751	−	0.941235i	$-0.609666\pi$
$293$	−11.5627			−0.675502			−0.337751	+	0.941235i	$0.609666\pi$
$294$		0			0
$295$	−14.3485	+	2.85952i	−0.835404	+	0.166488i
$296$		0			0
$297$			3.98423i			0.231188i
$298$		0			0
$299$		−	15.6885i		−	0.907290i
$300$		0			0
$301$		−	2.07092i		−	0.119366i
$302$		0			0
$303$		−	2.19055i		−	0.125844i
$304$		0			0
$305$	−14.2614	+	2.84215i	−0.816604	+	0.162741i
$306$		0			0
$307$	30.2849			1.72845			0.864225	−	0.503106i	$-0.167809\pi$
$307$	30.2849			1.72845			0.864225	+	0.503106i	$0.167809\pi$
$308$		0			0
$309$			14.6341i			0.832503i
$310$		0			0
$311$	−30.8962			−1.75196			−0.875981	−	0.482346i	$-0.839785\pi$
$311$	−30.8962			−1.75196			−0.875981	+	0.482346i	$0.839785\pi$
$312$		0			0
$313$		−	7.39647i		−	0.418073i	−0.977908	−	0.209037i	$-0.932967\pi$
$313$		−	7.39647i		−	0.418073i	0.977908	−	0.209037i	$-0.0670328\pi$
$314$		0			0
$315$	2.91511	−	0.580952i	0.164248	−	0.0327330i
$316$		0			0
$317$	25.4720			1.43065			0.715326	−	0.698791i	$-0.246278\pi$
$317$	25.4720			1.43065			0.715326	+	0.698791i	$0.246278\pi$
$318$		0			0
$319$	−3.73573			−0.209161
$320$		0			0
$321$	2.92058			0.163011
$322$		0			0
$323$	−28.4658			−1.58388
$324$		0			0
$325$	−8.57139	+	3.55768i	−0.475455	+	0.197345i
$326$		0			0
$327$			23.9162i			1.32257i
$328$		0			0
$329$	4.69919			0.259075
$330$		0			0
$331$		−	9.33587i		−	0.513146i	−0.966525	−	0.256573i	$-0.917407\pi$
$331$		−	9.33587i		−	0.513146i	0.966525	−	0.256573i	$-0.0825934\pi$
$332$		0			0
$333$	−9.97786			−0.546784
$334$		0			0
$335$	4.58608	+	23.0121i	0.250564	+	1.25728i
$336$		0			0
$337$			5.92273i			0.322632i	0.986903	+	0.161316i	$0.0515738\pi$
$337$			5.92273i			0.322632i	−0.986903	+	0.161316i	$0.948426\pi$
$338$		0			0
$339$			25.5458i			1.38746i
$340$		0			0
$341$		−	0.248496i		−	0.0134568i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−23.9585	+	4.77470i	−1.28988	+	0.257061i
$346$		0			0
$347$	−0.107524			−0.00577220			−0.00288610	−	0.999996i	$-0.500919\pi$
$347$	−0.107524			−0.00577220			−0.00288610	+	0.999996i	$0.500919\pi$
$348$		0			0
$349$			4.23635i			0.226766i	0.993551	+	0.113383i	$0.0361688\pi$
$349$			4.23635i			0.226766i	−0.993551	+	0.113383i	$0.963831\pi$
$350$		0			0
$351$	10.3864			0.554383
$352$		0			0
$353$		−	23.0592i		−	1.22732i	−0.789571	−	0.613659i	$-0.789697\pi$
$353$		−	23.0592i		−	1.22732i	0.789571	−	0.613659i	$-0.210303\pi$
$354$		0			0
$355$	−2.42100	−	12.1481i	−0.128494	−	0.644756i
$356$		0			0
$357$	−4.85503			−0.256956
$358$		0			0
$359$	−1.94196			−0.102493			−0.0512464	−	0.998686i	$-0.516319\pi$
$359$	−1.94196			−0.102493			−0.0512464	+	0.998686i	$0.516319\pi$
$360$		0			0
$361$	−38.4326			−2.02277
$362$		0			0
$363$	−13.5628			−0.711863
$364$		0			0
$365$	−19.9670	+	3.97923i	−1.04512	+	0.208283i
$366$		0			0
$367$			16.2406i			0.847753i	0.905720	+	0.423876i	$0.139331\pi$
$367$			16.2406i			0.847753i	−0.905720	+	0.423876i	$0.860669\pi$
$368$		0			0
$369$	−3.65235			−0.190134
$370$		0			0
$371$			9.81202i			0.509415i
$372$		0			0
$373$	0.818104			0.0423598			0.0211799	−	0.999776i	$-0.493258\pi$
$373$	0.818104			0.0423598			0.0211799	+	0.999776i	$0.493258\pi$
$374$		0			0
$375$	8.04172	+	12.0069i	0.415273	+	0.620036i
$376$		0			0
$377$			9.73856i			0.501561i
$378$		0			0
$379$		−	1.90152i		−	0.0976745i	−0.998807	−	0.0488373i	$-0.984448\pi$
$379$		−	1.90152i		−	0.0976745i	0.998807	−	0.0488373i	$-0.0155516\pi$
$380$		0			0
$381$			10.9022i			0.558539i
$382$		0			0
$383$			6.24670i			0.319191i	0.987182	+	0.159596i	$0.0510190\pi$
$383$			6.24670i			0.319191i	−0.987182	+	0.159596i	$0.948981\pi$
$384$		0			0
$385$	−0.311165	−	1.56137i	−0.0158584	−	0.0795747i
$386$		0			0
$387$	2.75290			0.139938
$388$		0			0
$389$		−	18.6095i		−	0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$389$		−	18.6095i		−	0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$390$		0			0
$391$	−31.7490			−1.60561
$392$		0			0
$393$		−	1.59447i		−	0.0804304i
$394$		0			0
$395$	1.98759	+	9.97334i	0.100006	+	0.501813i
$396$		0			0
$397$	−3.57650			−0.179499			−0.0897497	−	0.995964i	$-0.528607\pi$
$397$	−3.57650			−0.179499			−0.0897497	+	0.995964i	$0.528607\pi$
$398$		0			0
$399$	−9.79551			−0.490389
$400$		0			0
$401$	−25.9125			−1.29401			−0.647004	−	0.762487i	$-0.723978\pi$
$401$	−25.9125			−1.29401			−0.647004	+	0.762487i	$0.723978\pi$
$402$		0			0
$403$	−0.647797			−0.0322691
$404$		0			0
$405$	−1.41816	−	7.11608i	−0.0704692	−	0.353601i
$406$		0			0
$407$			5.34427i			0.264905i
$408$		0			0
$409$	20.1037			0.994062			0.497031	−	0.867733i	$-0.334424\pi$
$409$	20.1037			0.994062			0.497031	+	0.867733i	$0.334424\pi$
$410$		0			0
$411$		−	21.4391i		−	1.05751i
$412$		0			0
$413$	−6.54305			−0.321962
$414$		0			0
$415$	2.15174	+	10.7970i	0.105625	+	0.530004i
$416$		0			0
$417$		−	23.6690i		−	1.15908i
$418$		0			0
$419$		−	0.413171i		−	0.0201847i	−0.999949	−	0.0100924i	$-0.996787\pi$
$419$		−	0.413171i		−	0.0201847i	0.999949	−	0.0100924i	$-0.00321256\pi$
$420$		0			0
$421$		−	5.85147i		−	0.285183i	−0.989782	−	0.142591i	$-0.954456\pi$
$421$		−	5.85147i		−	0.285183i	0.989782	−	0.142591i	$-0.0455435\pi$
$422$		0			0
$423$			6.24670i			0.303725i
$424$		0			0
$425$	7.19972	+	17.3460i	0.349238	+	0.841404i
$426$		0			0
$427$	−6.50330			−0.314717
$428$		0			0
$429$		−	1.70813i		−	0.0824695i
$430$		0			0
$431$	−5.55603			−0.267625			−0.133812	−	0.991007i	$-0.542722\pi$
$431$	−5.55603			−0.267625			−0.133812	+	0.991007i	$0.542722\pi$
$432$		0			0
$433$		−	24.1942i		−	1.16270i	−0.813654	−	0.581350i	$-0.802525\pi$
$433$		−	24.1942i		−	1.16270i	0.813654	−	0.581350i	$-0.197475\pi$
$434$		0			0
$435$	14.8721	−	2.96387i	0.713064	−	0.142107i
$436$		0			0
$437$	−64.0567			−3.06425
$438$		0			0
$439$	27.9665			1.33477			0.667385	−	0.744713i	$-0.267414\pi$
$439$	27.9665			1.33477			0.667385	+	0.744713i	$0.267414\pi$
$440$		0			0
$441$	1.32931			0.0633006
$442$		0			0
$443$	−18.5932			−0.883391			−0.441696	−	0.897165i	$-0.645623\pi$
$443$	−18.5932			−0.883391			−0.441696	+	0.897165i	$0.645623\pi$
$444$		0			0
$445$	−2.44090	−	12.2480i	−0.115710	−	0.580610i
$446$		0			0
$447$			14.5726i			0.689260i
$448$		0			0
$449$	23.3079			1.09997			0.549985	−	0.835174i	$-0.314633\pi$
$449$	23.3079			1.09997			0.549985	+	0.835174i	$0.314633\pi$
$450$		0			0
$451$			1.95624i			0.0921159i
$452$		0			0
$453$	−18.8301			−0.884714
$454$		0			0
$455$	−4.07028	+	0.811166i	−0.190818	+	0.0380281i
$456$		0			0
$457$			30.1043i			1.40822i	0.710092	+	0.704109i	$0.248653\pi$
$457$			30.1043i			1.40822i	−0.710092	+	0.704109i	$0.751347\pi$
$458$		0			0
$459$		−	21.0190i		−	0.981081i
$460$		0			0
$461$			30.9144i			1.43983i	0.694063	+	0.719914i	$0.255819\pi$
$461$			30.9144i			1.43983i	−0.694063	+	0.719914i	$0.744181\pi$
$462$		0			0
$463$		−	5.26313i		−	0.244599i	−0.992493	−	0.122299i	$-0.960973\pi$
$463$		−	5.26313i		−	0.244599i	0.992493	−	0.122299i	$-0.0390268\pi$
$464$		0			0
$465$	0.197153	+	0.989276i	0.00914275	+	0.0458766i
$466$		0			0
$467$	4.20172			0.194433			0.0972163	−	0.995263i	$-0.469006\pi$
$467$	4.20172			0.194433			0.0972163	+	0.995263i	$0.469006\pi$
$468$		0			0
$469$			10.4937i			0.484553i
$470$		0			0
$471$	−13.0928			−0.603283
$472$		0			0
$473$		−	1.47449i		−	0.0677971i
$474$		0			0
$475$	14.5261	+	34.9972i	0.666505	+	1.60578i
$476$		0			0
$477$	−13.0432			−0.597209
$478$		0			0
$479$	−28.2017			−1.28857			−0.644283	−	0.764787i	$-0.722844\pi$
$479$	−28.2017			−1.28857			−0.644283	+	0.764787i	$0.722844\pi$
$480$		0			0
$481$	13.9318			0.635235
$482$		0			0
$483$	−10.9253			−0.497117
$484$		0			0
$485$	−31.0680	+	6.19155i	−1.41073	+	0.281144i
$486$		0			0
$487$			11.8032i			0.534852i	0.963578	+	0.267426i	$0.0861731\pi$
$487$			11.8032i			0.534852i	−0.963578	+	0.267426i	$0.913827\pi$
$488$		0			0
$489$	9.00170			0.407071
$490$		0			0
$491$			0.305569i			0.0137901i	0.999976	+	0.00689507i	$0.00219479\pi$
$491$			0.305569i			0.0137901i	−0.999976	+	0.00689507i	$0.997805\pi$
$492$		0			0
$493$	19.7080			0.887604
$494$		0			0
$495$	2.07555	−	0.413636i	0.0932889	−	0.0185916i
$496$		0			0
$497$		−	5.53965i		−	0.248487i
$498$		0			0
$499$			5.15472i			0.230757i	0.993322	+	0.115378i	$0.0368081\pi$
$499$			5.15472i			0.230757i	−0.993322	+	0.115378i	$0.963192\pi$
$500$		0			0
$501$		−	5.88906i		−	0.263104i
$502$		0			0
$503$			21.5322i			0.960076i	0.877248	+	0.480038i	$0.159377\pi$
$503$			21.5322i			0.960076i	−0.877248	+	0.480038i	$0.840623\pi$
$504$		0			0
$505$	−3.71649	+	0.740659i	−0.165381	+	0.0329589i
$506$		0			0
$507$	12.3503			0.548495
$508$		0			0
$509$		−	16.1494i		−	0.715810i	−0.933758	−	0.357905i	$-0.883491\pi$
$509$		−	16.1494i		−	0.715810i	0.933758	−	0.357905i	$-0.116509\pi$
$510$		0			0
$511$	−9.10513			−0.402787
$512$		0			0
$513$		−	42.4078i		−	1.87235i
$514$		0			0
$515$	24.8282	−	4.94801i	1.09406	−	0.218035i
$516$		0			0
$517$	3.34581			0.147148
$518$		0			0
$519$	−27.3136			−1.19893
$520$		0			0
$521$	−7.44181			−0.326032			−0.163016	−	0.986623i	$-0.552122\pi$
$521$	−7.44181			−0.326032			−0.163016	+	0.986623i	$0.552122\pi$
$522$		0			0
$523$	−18.1132			−0.792036			−0.396018	−	0.918243i	$-0.629608\pi$
$523$	−18.1132			−0.792036			−0.396018	+	0.918243i	$0.629608\pi$
$524$		0			0
$525$	2.47753	+	5.96901i	0.108128	+	0.260509i
$526$		0			0
$527$			1.31095i			0.0571060i
$528$		0			0
$529$	−48.4447			−2.10629
$530$		0			0
$531$		−	8.69775i		−	0.377450i
$532$		0			0
$533$	5.09967			0.220891
$534$		0			0
$535$	−0.987493	−	4.95505i	−0.0426931	−	0.214226i
$536$		0			0
$537$			27.9640i			1.20673i
$538$		0			0
$539$		−	0.711996i		−	0.0306679i
$540$		0			0
$541$		−	22.3864i		−	0.962466i	−0.876593	−	0.481233i	$-0.840189\pi$
$541$		−	22.3864i		−	0.962466i	0.876593	−	0.481233i	$-0.159811\pi$
$542$		0			0
$543$		−	11.6568i		−	0.500242i
$544$		0			0
$545$	40.5763	−	8.08645i	1.73810	−	0.346385i
$546$		0			0
$547$	−32.7568			−1.40058			−0.700290	−	0.713858i	$-0.746946\pi$
$547$	−32.7568			−1.40058			−0.700290	+	0.713858i	$0.746946\pi$
$548$		0			0
$549$		−	8.64492i		−	0.368956i
$550$		0			0
$551$	39.7628			1.69395
$552$		0			0
$553$			4.54792i			0.193397i
$554$		0			0
$555$	−4.24005	−	21.2758i	−0.179980	−	0.903107i
$556$		0			0
$557$	3.01493			0.127747			0.0638734	−	0.997958i	$-0.479655\pi$
$557$	3.01493			0.127747			0.0638734	+	0.997958i	$0.479655\pi$
$558$		0			0
$559$	−3.84380			−0.162575
$560$		0			0
$561$	−3.45677			−0.145945
$562$		0			0
$563$	2.92403			0.123233			0.0616166	−	0.998100i	$-0.480374\pi$
$563$	2.92403			0.123233			0.0616166	+	0.998100i	$0.480374\pi$
$564$		0			0
$565$	43.3410	−	8.63744i	1.82337	−	0.363380i
$566$		0			0
$567$		−	3.24499i		−	0.136277i
$568$		0			0
$569$	−22.9064			−0.960285			−0.480143	−	0.877190i	$-0.659415\pi$
$569$	−22.9064			−0.960285			−0.480143	+	0.877190i	$0.659415\pi$
$570$		0			0
$571$		−	39.5106i		−	1.65347i	−0.562593	−	0.826734i	$-0.690196\pi$
$571$		−	39.5106i		−	1.65347i	0.562593	−	0.826734i	$-0.309804\pi$
$572$		0			0
$573$	−15.6818			−0.655116
$574$		0			0
$575$	16.2015	+	39.0337i	0.675650	+	1.62782i
$576$		0			0
$577$			19.6976i			0.820023i	0.912080	+	0.410012i	$0.134475\pi$
$577$			19.6976i			0.820023i	−0.912080	+	0.410012i	$0.865525\pi$
$578$		0			0
$579$			14.1919i			0.589796i
$580$		0			0
$581$			4.92352i			0.204262i
$582$		0			0
$583$			6.98612i			0.289336i
$584$		0			0
$585$	−1.07829	−	5.41067i	−0.0445820	−	0.223704i
$586$		0			0
$587$	−31.1220			−1.28454			−0.642270	−	0.766478i	$-0.722007\pi$
$587$	−31.1220			−1.28454			−0.642270	+	0.766478i	$0.722007\pi$
$588$		0			0
$589$			2.64498i			0.108984i
$590$		0			0
$591$	0.451314			0.0185646
$592$		0			0
$593$		−	31.6520i		−	1.29979i	−0.760023	−	0.649896i	$-0.774812\pi$
$593$		−	31.6520i		−	1.29979i	0.760023	−	0.649896i	$-0.225188\pi$
$594$		0			0
$595$	1.64156	+	8.23706i	0.0672976	+	0.337686i
$596$		0			0
$597$	−28.2969			−1.15811
$598$		0			0
$599$	46.9208			1.91713			0.958566	−	0.284870i	$-0.0919504\pi$
$599$	46.9208			1.91713			0.958566	+	0.284870i	$0.0919504\pi$
$600$		0			0
$601$	20.7934			0.848179			0.424089	−	0.905620i	$-0.360594\pi$
$601$	20.7934			0.848179			0.424089	+	0.905620i	$0.360594\pi$
$602$		0			0
$603$	−13.9494			−0.568063
$604$		0			0
$605$	4.58580	+	23.0107i	0.186439	+	0.935518i
$606$		0			0
$607$			2.52360i			0.102430i	0.998688	+	0.0512148i	$0.0163093\pi$
$607$			2.52360i			0.102430i	−0.998688	+	0.0512148i	$0.983691\pi$
$608$		0			0
$609$	6.78181			0.274813
$610$		0			0
$611$		−	8.72207i		−	0.352857i
$612$		0			0
$613$	41.0993			1.65999			0.829993	−	0.557774i	$-0.188344\pi$
$613$	41.0993			1.65999			0.829993	+	0.557774i	$0.188344\pi$
$614$		0			0
$615$	−1.55205	−	7.78790i	−0.0625847	−	0.314038i
$616$		0			0
$617$		−	14.4277i		−	0.580839i	−0.956899	−	0.290420i	$-0.906205\pi$
$617$		−	14.4277i		−	0.580839i	0.956899	−	0.290420i	$-0.0937949\pi$
$618$		0			0
$619$			1.03872i			0.0417497i	0.999782	+	0.0208749i	$0.00664516\pi$
$619$			1.03872i			0.0417497i	−0.999782	+	0.0208749i	$0.993355\pi$
$620$		0			0
$621$		−	47.2990i		−	1.89804i
$622$		0			0
$623$		−	5.58518i		−	0.223765i
$624$		0			0
$625$	17.6520	−	17.7033i	0.706078	−	0.708134i
$626$		0			0
$627$	−6.97437			−0.278529
$628$		0			0
$629$		−	28.1939i		−	1.12416i
$630$		0			0
$631$	23.4094			0.931912			0.465956	−	0.884808i	$-0.345710\pi$
$631$	23.4094			0.931912			0.465956	+	0.884808i	$0.345710\pi$
$632$		0			0
$633$		−	18.9356i		−	0.752623i
$634$		0			0
$635$	18.4968	−	3.68622i	0.734022	−	0.146283i
$636$		0			0
$637$	−1.85608			−0.0735406
$638$		0			0
$639$	7.36392			0.291312
$640$		0			0
$641$	−15.1178			−0.597115			−0.298558	−	0.954392i	$-0.596506\pi$
$641$	−15.1178			−0.597115			−0.298558	+	0.954392i	$0.596506\pi$
$642$		0			0
$643$	−45.9767			−1.81314			−0.906572	−	0.422051i	$-0.861310\pi$
$643$	−45.9767			−1.81314			−0.906572	+	0.422051i	$0.861310\pi$
$644$		0			0
$645$	1.16984	+	5.87001i	0.0460622	+	0.231132i
$646$		0			0
$647$			30.0253i			1.18042i	0.807251	+	0.590208i	$0.200954\pi$
$647$			30.0253i			1.18042i	−0.807251	+	0.590208i	$0.799046\pi$
$648$		0			0
$649$	−4.65863			−0.182867
$650$		0			0
$651$			0.451118i			0.0176807i
$652$		0			0
$653$	−24.8944			−0.974192			−0.487096	−	0.873348i	$-0.661944\pi$
$653$	−24.8944			−0.974192			−0.487096	+	0.873348i	$0.661944\pi$
$654$		0			0
$655$	−2.70518	+	0.539116i	−0.105700	+	0.0210650i
$656$		0			0
$657$		−	12.1036i		−	0.472205i
$658$		0			0
$659$			36.8754i			1.43646i	0.695804	+	0.718231i	$0.255048\pi$
$659$			36.8754i			1.43646i	−0.695804	+	0.718231i	$0.744952\pi$
$660$		0			0
$661$			15.5668i			0.605477i	0.953074	+	0.302739i	$0.0979010\pi$
$661$			15.5668i			0.605477i	−0.953074	+	0.302739i	$0.902099\pi$
$662$		0			0
$663$			9.01133i			0.349971i
$664$		0			0
$665$	3.31202	+	16.6191i	0.128435	+	0.644460i
$666$		0			0
$667$	44.3489			1.71720
$668$		0			0
$669$			24.4613i			0.945729i
$670$		0			0
$671$	−4.63033			−0.178752
$672$		0			0
$673$		−	8.30821i		−	0.320258i	−0.987096	−	0.160129i	$-0.948809\pi$
$673$		−	8.30821i		−	0.320258i	0.987096	−	0.160129i	$-0.0511910\pi$
$674$		0			0
$675$	−25.8417	+	10.7260i	−0.994647	+	0.412844i
$676$		0			0
$677$	−23.8042			−0.914872			−0.457436	−	0.889243i	$-0.651232\pi$
$677$	−23.8042			−0.914872			−0.457436	+	0.889243i	$0.651232\pi$
$678$		0			0
$679$	−14.1673			−0.543690
$680$		0			0
$681$	23.6050			0.904547
$682$		0			0
$683$	−9.33136			−0.357055			−0.178527	−	0.983935i	$-0.557133\pi$
$683$	−9.33136			−0.357055			−0.178527	+	0.983935i	$0.557133\pi$
$684$		0			0
$685$	−36.3735	+	7.24889i	−1.38976	+	0.276966i
$686$		0			0
$687$			9.35473i			0.356905i
$688$		0			0
$689$	18.2119			0.693818
$690$		0			0
$691$			21.3925i			0.813810i	0.913471	+	0.406905i	$0.133392\pi$
$691$			21.3925i			0.813810i	−0.913471	+	0.406905i	$0.866608\pi$
$692$		0			0
$693$	0.946466			0.0359533
$694$		0			0
$695$	−40.1569	+	8.00288i	−1.52324	+	0.303567i
$696$		0			0
$697$		−	10.3202i		−	0.390907i
$698$		0			0
$699$			24.1254i			0.912507i
$700$		0			0
$701$		−	47.0880i		−	1.77849i	−0.457431	−	0.889245i	$-0.651230\pi$
$701$		−	47.0880i		−	1.77849i	0.457431	−	0.889245i	$-0.348770\pi$
$702$		0			0
$703$		−	56.8840i		−	2.14542i
$704$		0			0
$705$	−13.3198	+	2.65451i	−0.501653	+	0.0999746i
$706$		0			0
$707$	−1.69475			−0.0637375
$708$		0			0
$709$			33.9549i			1.27520i	0.770366	+	0.637602i	$0.220073\pi$
$709$			33.9549i			1.27520i	−0.770366	+	0.637602i	$0.779927\pi$
$710$		0			0
$711$	−6.04561			−0.226728
$712$		0			0
$713$			2.95004i			0.110480i
$714$		0			0
$715$	−2.89802	+	0.577548i	−0.108380	+	0.0215991i
$716$		0			0
$717$	1.28407			0.0479547
$718$		0			0
$719$	−1.89494			−0.0706692			−0.0353346	−	0.999376i	$-0.511250\pi$
$719$	−1.89494			−0.0706692			−0.0353346	+	0.999376i	$0.511250\pi$
$720$		0			0
$721$	11.3218			0.421648
$722$		0			0
$723$	−27.4274			−1.02003
$724$		0			0
$725$	−10.0570	−	24.2300i	−0.373508	−	0.899878i
$726$		0			0
$727$		−	33.9082i		−	1.25759i	−0.777573	−	0.628793i	$-0.783549\pi$
$727$		−	33.9082i		−	1.25759i	0.777573	−	0.628793i	$-0.216451\pi$
$728$		0			0
$729$	26.0124			0.963422
$730$		0			0
$731$			7.77873i			0.287707i
$732$		0			0
$733$	31.2130			1.15288			0.576439	−	0.817140i	$-0.304442\pi$
$733$	31.2130			1.15288			0.576439	+	0.817140i	$0.304442\pi$
$734$		0			0
$735$	0.564886	+	2.83449i	0.0208361	+	0.104552i
$736$		0			0
$737$			7.47147i			0.275215i
$738$		0			0
$739$		−	38.8444i		−	1.42891i	−0.699680	−	0.714456i	$-0.746674\pi$
$739$		−	38.8444i		−	1.42891i	0.699680	−	0.714456i	$-0.253326\pi$
$740$		0			0
$741$			18.1812i			0.667905i
$742$		0			0
$743$		−	5.04781i		−	0.185186i	−0.995704	−	0.0925931i	$-0.970484\pi$
$743$		−	5.04781i		−	0.185186i	0.995704	−	0.0925931i	$-0.0295156\pi$
$744$		0			0
$745$	24.7239	−	4.92723i	0.905813	−	0.180520i
$746$		0			0
$747$	−6.54490			−0.239465
$748$		0			0
$749$		−	2.25954i		−	0.0825620i
$750$		0			0
$751$	50.8436			1.85531			0.927655	−	0.373437i	$-0.121821\pi$
$751$	50.8436			1.85531			0.927655	+	0.373437i	$0.121821\pi$
$752$		0			0
$753$		−	19.7333i		−	0.719121i
$754$		0			0
$755$	6.36675	+	31.9471i	0.231710	+	1.16267i
$756$		0			0
$757$	19.8379			0.721021			0.360511	−	0.932755i	$-0.382602\pi$
$757$	19.8379			0.721021			0.360511	+	0.932755i	$0.382602\pi$
$758$		0			0
$759$	−7.77876			−0.282351
$760$		0			0
$761$	−33.7095			−1.22197			−0.610985	−	0.791642i	$-0.709226\pi$
$761$	−33.7095			−1.22197			−0.610985	+	0.791642i	$0.709226\pi$
$762$		0			0
$763$	18.5031			0.669857
$764$		0			0
$765$	−10.9496	+	2.18215i	−0.395885	+	0.0788959i
$766$		0			0
$767$			12.1444i			0.438509i
$768$		0			0
$769$	34.8641			1.25723			0.628616	−	0.777716i	$-0.283622\pi$
$769$	34.8641			1.25723			0.628616	+	0.777716i	$0.283622\pi$
$770$		0			0
$771$			36.8777i			1.32812i
$772$		0			0
$773$	−1.74412			−0.0627315			−0.0313657	−	0.999508i	$-0.509986\pi$
$773$	−1.74412			−0.0627315			−0.0313657	+	0.999508i	$0.509986\pi$
$774$		0			0
$775$	1.61175	−	0.668980i	0.0578957	−	0.0240305i
$776$		0			0
$777$		−	9.70193i		−	0.348055i
$778$		0			0
$779$		−	20.8221i		−	0.746029i
$780$		0			0
$781$		−	3.94421i		−	0.141135i
$782$		0			0
$783$			29.3606i			1.04926i
$784$		0			0
$785$	4.42688	+	22.2132i	0.158002	+	0.792824i
$786$		0			0
$787$	26.1967			0.933810			0.466905	−	0.884308i	$-0.345369\pi$
$787$	26.1967			0.933810			0.466905	+	0.884308i	$0.345369\pi$
$788$		0			0
$789$		−	7.13096i		−	0.253869i
$790$		0			0
$791$	19.7639			0.702722
$792$		0			0
$793$			12.0706i			0.428641i
$794$		0			0
$795$	−5.54267	−	27.8121i	−0.196578	−	0.986393i
$796$		0			0
$797$	24.7303			0.875992			0.437996	−	0.898977i	$-0.355688\pi$
$797$	24.7303			0.875992			0.437996	+	0.898977i	$0.355688\pi$
$798$		0			0
$799$	−17.6509			−0.624445
$800$		0			0
$801$	7.42445			0.262330
$802$		0			0
$803$	−6.48282			−0.228774
$804$		0			0
$805$	3.69401	+	18.5359i	0.130197	+	0.653303i
$806$		0			0
$807$		−	11.3648i		−	0.400061i
$808$		0			0
$809$	−41.4914			−1.45876			−0.729379	−	0.684110i	$-0.760191\pi$
$809$	−41.4914			−1.45876			−0.729379	+	0.684110i	$0.760191\pi$
$810$		0			0
$811$			12.1293i			0.425919i	0.977061	+	0.212960i	$0.0683103\pi$
$811$			12.1293i			0.425919i	−0.977061	+	0.212960i	$0.931690\pi$
$812$		0			0
$813$	−2.52338			−0.0884988
$814$		0			0
$815$	−3.04362	−	15.2723i	−0.106613	−	0.534966i
$816$		0			0
$817$			15.6944i			0.549076i
$818$		0			0
$819$		−	2.46731i		−	0.0862148i
$820$		0			0
$821$		−	6.87577i		−	0.239966i	−0.992776	−	0.119983i	$-0.961716\pi$
$821$		−	6.87577i		−	0.239966i	0.992776	−	0.119983i	$-0.0382840\pi$
$822$		0			0
$823$		−	36.0346i		−	1.25609i	−0.778178	−	0.628044i	$-0.783856\pi$
$823$		−	36.0346i		−	1.25609i	0.778178	−	0.628044i	$-0.216144\pi$
$824$		0			0
$825$	1.76399	+	4.24991i	0.0614143	+	0.147963i
$826$		0			0
$827$	18.6307			0.647853			0.323927	−	0.946082i	$-0.394997\pi$
$827$	18.6307			0.647853			0.323927	+	0.946082i	$0.394997\pi$
$828$		0			0
$829$		−	14.9525i		−	0.519321i	−0.965700	−	0.259661i	$-0.916389\pi$
$829$		−	14.9525i		−	0.519321i	0.965700	−	0.259661i	$-0.0836107\pi$
$830$		0			0
$831$	22.1850			0.769590
$832$		0			0
$833$			3.75616i			0.130143i
$834$		0			0
$835$	−9.99139	+	1.99119i	−0.345766	+	0.0689078i
$836$		0			0
$837$	−1.95303			−0.0675066
$838$		0			0
$839$	−13.4613			−0.464734			−0.232367	−	0.972628i	$-0.574647\pi$
$839$	−13.4613			−0.464734			−0.232367	+	0.972628i	$0.574647\pi$
$840$		0			0
$841$	1.47064			0.0507119
$842$		0			0
$843$	−31.2910			−1.07772
$844$		0			0
$845$	−4.17583	−	20.9535i	−0.143653	−	0.720823i
$846$		0			0
$847$			10.4931i			0.360546i
$848$		0			0
$849$	15.3372			0.526371
$850$		0			0
$851$		−	63.4447i		−	2.17486i
$852$		0			0
$853$	48.6691			1.66640			0.833200	−	0.552972i	$-0.186507\pi$
$853$	48.6691			1.66640			0.833200	+	0.552972i	$0.186507\pi$
$854$		0			0
$855$	−22.0920	+	4.40271i	−0.755529	+	0.150569i
$856$		0			0
$857$		−	0.947295i		−	0.0323590i	−0.999869	−	0.0161795i	$-0.994850\pi$
$857$		−	0.947295i		−	0.0323590i	0.999869	−	0.0161795i	$-0.00515032\pi$
$858$		0			0
$859$		−	0.917522i		−	0.0313055i	−0.999877	−	0.0156527i	$-0.995017\pi$
$859$		−	0.917522i		−	0.0313055i	0.999877	−	0.0156527i	$-0.00498262\pi$
$860$		0			0
$861$		−	3.55134i		−	0.121029i
$862$		0			0
$863$			20.8807i			0.710789i	0.934716	+	0.355394i	$0.115653\pi$
$863$			20.8807i			0.710789i	−0.934716	+	0.355394i	$0.884347\pi$
$864$		0			0
$865$	9.23516	+	46.3403i	0.314005	+	1.57562i
$866$		0			0
$867$	−3.73706			−0.126917
$868$		0			0
$869$			3.23811i			0.109845i
$870$		0			0
$871$	19.4771			0.659957
$872$		0			0
$873$		−	18.8327i		−	0.637391i
$874$		0			0
$875$	9.28934	−	6.22159i	0.314037	−	0.210328i
$876$		0			0
$877$	48.7337			1.64562			0.822810	−	0.568316i	$-0.192405\pi$
$877$	48.7337			1.64562			0.822810	+	0.568316i	$0.192405\pi$
$878$		0			0
$879$	14.9454			0.504097
$880$		0			0
$881$	−32.1537			−1.08329			−0.541643	−	0.840609i	$-0.682198\pi$
$881$	−32.1537			−1.08329			−0.541643	+	0.840609i	$0.682198\pi$
$882$		0			0
$883$	51.4051			1.72992			0.864959	−	0.501842i	$-0.167344\pi$
$883$	51.4051			1.72992			0.864959	+	0.501842i	$0.167344\pi$
$884$		0			0
$885$	18.5462	−	3.69608i	0.623424	−	0.124242i
$886$		0			0
$887$			38.2250i			1.28347i	0.766926	+	0.641735i	$0.221785\pi$
$887$			38.2250i			1.28347i	−0.766926	+	0.641735i	$0.778215\pi$
$888$		0			0
$889$	8.43468			0.282890
$890$		0			0
$891$		−	2.31042i		−	0.0774020i
$892$		0			0
$893$	−35.6125			−1.19173
$894$		0			0
$895$	47.4437	−	9.45507i	1.58587	−	0.316048i
$896$		0			0
$897$			20.2782i			0.677069i
$898$		0			0
$899$		−	1.83122i		−	0.0610746i
$900$		0			0
$901$		−	36.8556i		−	1.22784i
$902$		0			0
$903$			2.67677i			0.0890774i
$904$		0			0
$905$	−19.7770	+	3.94135i	−0.657409	+	0.131015i
$906$		0			0
$907$	17.6808			0.587081			0.293540	−	0.955947i	$-0.405166\pi$
$907$	17.6808			0.587081			0.293540	+	0.955947i	$0.405166\pi$
$908$		0			0
$909$		−	2.25285i		−	0.0747223i
$910$		0			0
$911$	1.52564			0.0505467			0.0252733	−	0.999681i	$-0.491954\pi$
$911$	1.52564			0.0505467			0.0252733	+	0.999681i	$0.491954\pi$
$912$		0			0
$913$			3.50553i			0.116016i
$914$		0			0
$915$	18.4336	−	3.67362i	0.609394	−	0.121446i
$916$		0			0
$917$	−1.23358			−0.0407365
$918$		0			0
$919$	−29.3054			−0.966697			−0.483348	−	0.875428i	$-0.660580\pi$
$919$	−29.3054			−0.966697			−0.483348	+	0.875428i	$0.660580\pi$
$920$		0			0
$921$	−39.1447			−1.28986
$922$		0			0
$923$	−10.2820			−0.338437
$924$		0			0
$925$	−34.6629	+	14.3874i	−1.13971	+	0.473054i
$926$		0			0
$927$			15.0503i			0.494316i
$928$		0			0
$929$	−29.2981			−0.961241			−0.480620	−	0.876929i	$-0.659588\pi$
$929$	−29.2981			−0.961241			−0.480620	+	0.876929i	$0.659588\pi$
$930$		0			0
$931$			7.57843i			0.248373i
$932$		0			0
$933$	39.9349			1.30741
$934$		0			0
$935$	1.16879	+	5.86476i	0.0382235	+	0.191798i
$936$		0			0
$937$			20.4748i			0.668884i	0.942416	+	0.334442i	$0.108548\pi$
$937$			20.4748i			0.668884i	−0.942416	+	0.334442i	$0.891452\pi$
$938$		0			0
$939$			9.56031i			0.311989i
$940$		0			0
$941$			45.1365i			1.47141i	0.677303	+	0.735704i	$0.263149\pi$
$941$			45.1365i			1.47141i	−0.677303	+	0.735704i	$0.736851\pi$
$942$		0			0
$943$		−	23.2236i		−	0.756265i
$944$		0			0
$945$	−12.2714	+	2.44557i	−0.399189	+	0.0795543i
$946$		0			0
$947$	60.0236			1.95050			0.975252	−	0.221097i	$-0.0709637\pi$
$947$	60.0236			1.95050			0.975252	+	0.221097i	$0.0709637\pi$
$948$		0			0
$949$			16.8998i			0.548592i
$950$		0			0
$951$	−32.9239			−1.06763
$952$		0			0
$953$			42.6147i			1.38043i	0.723606	+	0.690213i	$0.242483\pi$
$953$			42.6147i			1.38043i	−0.723606	+	0.690213i	$0.757517\pi$
$954$		0			0
$955$	5.30226	+	26.6058i	0.171577	+	0.860942i
$956$		0			0
$957$	4.82862			0.156087
$958$		0			0
$959$	−16.5866			−0.535610
$960$		0			0
$961$	−30.8782			−0.996071
$962$		0			0
$963$	3.00364			0.0967910
$964$		0			0
$965$	24.0780	−	4.79852i	0.775100	−	0.154470i
$966$		0			0
$967$		−	32.3051i		−	1.03886i	−0.854513	−	0.519431i	$-0.826144\pi$
$967$		−	32.3051i		−	1.03886i	0.854513	−	0.519431i	$-0.173856\pi$
$968$		0			0
$969$	36.7935			1.18198
$970$		0			0
$971$		−	30.5761i		−	0.981235i	−0.871375	−	0.490617i	$-0.836771\pi$
$971$		−	30.5761i		−	0.981235i	0.871375	−	0.490617i	$-0.163229\pi$
$972$		0			0
$973$	−18.3119			−0.587052
$974$		0			0
$975$	11.0790	−	4.59849i	0.354810	−	0.147269i
$976$		0			0
$977$		−	33.2108i		−	1.06251i	−0.847212	−	0.531254i	$-0.821721\pi$
$977$		−	33.2108i		−	1.06251i	0.847212	−	0.531254i	$-0.178279\pi$
$978$		0			0
$979$		−	3.97663i		−	0.127094i
$980$		0			0
$981$			24.5964i			0.785303i
$982$		0			0
$983$		−	24.5681i		−	0.783601i	−0.920050	−	0.391800i	$-0.871852\pi$
$983$		−	24.5681i		−	0.783601i	0.920050	−	0.391800i	$-0.128148\pi$
$984$		0			0
$985$	−0.152596	−	0.765700i	−0.00486212	−	0.0243972i
$986$		0			0
$987$	−6.07394			−0.193336
$988$		0			0
$989$			17.5045i			0.556610i
$990$		0			0
$991$	−7.13397			−0.226618			−0.113309	−	0.993560i	$-0.536145\pi$
$991$	−7.13397			−0.226618			−0.113309	+	0.993560i	$0.536145\pi$
$992$		0			0
$993$			12.0671i			0.382937i
$994$		0			0
$995$	9.56763	+	48.0085i	0.303314	+	1.52197i
$996$		0			0
$997$	−13.2715			−0.420312			−0.210156	−	0.977668i	$-0.567397\pi$
$997$	−13.2715			−0.420312			−0.210156	+	0.977668i	$0.567397\pi$
$998$		0			0
$999$	42.0027			1.32891

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.l.c.1569.8	yes	24
4.3	odd	2	inner	2240.2.l.c.1569.17	yes	24
5.4	even	2		2240.2.l.d.1569.17	yes	24
8.3	odd	2		2240.2.l.d.1569.7	yes	24
8.5	even	2		2240.2.l.d.1569.18	yes	24
20.19	odd	2		2240.2.l.d.1569.8	yes	24
40.19	odd	2	inner	2240.2.l.c.1569.18	yes	24
40.29	even	2	inner	2240.2.l.c.1569.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.l.c.1569.7	✓	24	40.29	even	2	inner
2240.2.l.c.1569.8	yes	24	1.1	even	1	trivial
2240.2.l.c.1569.17	yes	24	4.3	odd	2	inner
2240.2.l.c.1569.18	yes	24	40.19	odd	2	inner
2240.2.l.d.1569.7	yes	24	8.3	odd	2
2240.2.l.d.1569.8	yes	24	20.19	odd	2
2240.2.l.d.1569.17	yes	24	5.4	even	2
2240.2.l.d.1569.18	yes	24	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.29255 q^{3} +(0.437032 + 2.19294i) q^{5} +1.00000i q^{7} -1.32931 q^{9} +O(q^{10})\)	\(q-1.29255 q^{3} +(0.437032 + 2.19294i) q^{5} +1.00000i q^{7} -1.32931 q^{9} +0.711996i q^{11} +1.85608 q^{13} +(-0.564886 - 2.83449i) q^{15} -3.75616i q^{17} -7.57843i q^{19} -1.29255i q^{21} -8.45250i q^{23} +(-4.61801 + 1.91677i) q^{25} +5.59586 q^{27} +5.24684i q^{29} -0.349014 q^{31} -0.920291i q^{33} +(-2.19294 + 0.437032i) q^{35} +7.50603 q^{37} -2.39908 q^{39} +2.74755 q^{41} -2.07092 q^{43} +(-0.580952 - 2.91511i) q^{45} -4.69919i q^{47} -1.00000 q^{49} +4.85503i q^{51} +9.81202 q^{53} +(-1.56137 + 0.311165i) q^{55} +9.79551i q^{57} +6.54305i q^{59} +6.50330i q^{61} -1.32931i q^{63} +(0.811166 + 4.07028i) q^{65} +10.4937 q^{67} +10.9253i q^{69} -5.53965 q^{71} +9.10513i q^{73} +(5.96901 - 2.47753i) q^{75} -0.711996 q^{77} +4.54792 q^{79} -3.24499 q^{81} +4.92352 q^{83} +(8.23706 - 1.64156i) q^{85} -6.78181i q^{87} -5.58518 q^{89} +1.85608i q^{91} +0.451118 q^{93} +(16.6191 - 3.31202i) q^{95} +14.1673i q^{97} -0.946466i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} + 12 q^{9}+O(q^{10}) \) 24 * q - 4 * q^5 + 12 * q^9	\( 24 q - 4 q^{5} + 12 q^{9} - 4 q^{13} - 24 q^{37} - 48 q^{45} - 24 q^{49} + 88 q^{53} + 36 q^{65} + 20 q^{77} + 16 q^{81} - 56 q^{85} - 40 q^{89} + 24 q^{93}+O(q^{100}) \) 24 * q - 4 * q^5 + 12 * q^9 - 4 * q^13 - 24 * q^37 - 48 * q^45 - 24 * q^49 + 88 * q^53 + 36 * q^65 + 20 * q^77 + 16 * q^81 - 56 * q^85 - 40 * q^89 + 24 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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