LMFDB - Embedded newform 6048.2.c.e.3025.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(3025,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("6048.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.c (of order $2$, degree $1$, 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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$20$
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} + x^{18} + 4x^{16} + 8x^{12} + 4x^{10} + 32x^{8} + 256x^{4} + 256x^{2} + 1024 $ x^20 + x^18 + 4x^16 + 8x^12 + 4x^10 + 32x^8 + 256x^4 + 256x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.6
Root			$-0.725842 - 1.21374i$ of defining polynomial
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.e.3025.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.06888i q^{5} +1.00000 q^{7} +O(q^{10})$	$q-3.06888i q^{5} +1.00000 q^{7} -5.80820i q^{11} +3.52392i q^{13} -6.79736 q^{17} +5.28617i q^{19} +5.65085 q^{23} -4.41801 q^{25} -1.21394i q^{29} -0.107385 q^{31} -3.06888i q^{35} +4.90235i q^{37} -11.6855 q^{41} -1.85684i q^{43} -6.76459 q^{47} +1.00000 q^{49} -11.4861i q^{53} -17.8247 q^{55} -7.85494i q^{59} -12.0556i q^{61} +10.8145 q^{65} -6.66942i q^{67} -1.98478 q^{71} -6.52879 q^{73} -5.80820i q^{77} -2.30741 q^{79} +12.5795i q^{83} +20.8603i q^{85} -7.79151 q^{89} +3.52392i q^{91} +16.2226 q^{95} +4.05981 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{7}+O(q^{10}) $ 20 * q + 20 * q^7	$ 20 q + 20 q^{7} - 28 q^{25} - 36 q^{31} + 20 q^{49} - 48 q^{55} - 64 q^{79} + 56 q^{97}+O(q^{100}) $ 20 * q + 20 * q^7 - 28 * q^25 - 36 * q^31 + 20 * q^49 - 48 * q^55 - 64 * q^79 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 3.06888i	− 1.37244i	−0.727392	−	0.686222i	$-0.759268\pi$
$5$	− 3.06888i	− 1.37244i	0.727392	−	0.686222i	$-0.240732\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.80820i	− 1.75124i	−0.483001	−	0.875620i	$-0.660453\pi$
$11$	− 5.80820i	− 1.75124i	0.483001	−	0.875620i	$-0.339547\pi$
$12$	0	0
$13$	3.52392i	0.977359i	0.872463	+	0.488680i	$0.162521\pi$
$13$	3.52392i	0.977359i	−0.872463	+	0.488680i	$0.837479\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.79736	−1.64860	−0.824301	−	0.566151i	$-0.808432\pi$
$17$	−6.79736	−1.64860	−0.824301	+	0.566151i	$0.808432\pi$
$18$	0	0
$19$	5.28617i	1.21273i	0.795186	+	0.606365i	$0.207373\pi$
$19$	5.28617i	1.21273i	−0.795186	+	0.606365i	$0.792627\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.65085	1.17828	0.589141	−	0.808030i	$-0.299466\pi$
$23$	5.65085	1.17828	0.589141	+	0.808030i	$0.299466\pi$
$24$	0	0
$25$	−4.41801	−0.883601
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1.21394i	− 0.225422i	−0.993628	−	0.112711i	$-0.964047\pi$
$29$	− 1.21394i	− 0.225422i	0.993628	−	0.112711i	$-0.0359534\pi$
$30$	0	0
$31$	−0.107385	−0.0192868	−0.00964342	−	0.999954i	$-0.503070\pi$
$31$	−0.107385	−0.0192868	−0.00964342	+	0.999954i	$0.503070\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.06888i	− 0.518735i
$36$	0	0
$37$	4.90235i	0.805942i	0.915213	+	0.402971i	$0.132022\pi$
$37$	4.90235i	0.805942i	−0.915213	+	0.402971i	$0.867978\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.6855	−1.82497	−0.912485	−	0.409111i	$-0.865839\pi$
$41$	−11.6855	−1.82497	−0.912485	+	0.409111i	$0.865839\pi$
$42$	0	0
$43$	− 1.85684i	− 0.283165i	−0.989926	−	0.141583i	$-0.954781\pi$
$43$	− 1.85684i	− 0.283165i	0.989926	−	0.141583i	$-0.0452191\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.76459	−0.986717	−0.493359	−	0.869826i	$-0.664231\pi$
$47$	−6.76459	−0.986717	−0.493359	+	0.869826i	$0.664231\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.4861i	− 1.57773i	−0.614566	−	0.788865i	$-0.710669\pi$
$53$	− 11.4861i	− 1.57773i	0.614566	−	0.788865i	$-0.289331\pi$
$54$	0	0
$55$	−17.8247	−2.40348
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 7.85494i	− 1.02263i	−0.859394	−	0.511313i	$-0.829159\pi$
$59$	− 7.85494i	− 1.02263i	0.859394	−	0.511313i	$-0.170841\pi$
$60$	0	0
$61$	− 12.0556i	− 1.54356i	−0.635890	−	0.771779i	$-0.719367\pi$
$61$	− 12.0556i	− 1.54356i	0.635890	−	0.771779i	$-0.280633\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.8145	1.34137
$66$	0	0
$67$	− 6.66942i	− 0.814800i	−0.913250	−	0.407400i	$-0.866436\pi$
$67$	− 6.66942i	− 0.814800i	0.913250	−	0.407400i	$-0.133564\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.98478	−0.235550	−0.117775	−	0.993040i	$-0.537576\pi$
$71$	−1.98478	−0.235550	−0.117775	+	0.993040i	$0.537576\pi$
$72$	0	0
$73$	−6.52879	−0.764137	−0.382068	−	0.924134i	$-0.624788\pi$
$73$	−6.52879	−0.764137	−0.382068	+	0.924134i	$0.624788\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.80820i	− 0.661906i
$78$	0	0
$79$	−2.30741	−0.259604	−0.129802	−	0.991540i	$-0.541434\pi$
$79$	−2.30741	−0.259604	−0.129802	+	0.991540i	$0.541434\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.5795i	1.38078i	0.723436	+	0.690391i	$0.242562\pi$
$83$	12.5795i	1.38078i	−0.723436	+	0.690391i	$0.757438\pi$
$84$	0	0
$85$	20.8603i	2.26261i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.79151	−0.825898	−0.412949	−	0.910754i	$-0.635501\pi$
$89$	−7.79151	−0.825898	−0.412949	+	0.910754i	$0.635501\pi$
$90$	0	0
$91$	3.52392i	0.369407i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	16.2226	1.66440
$96$	0	0
$97$	4.05981	0.412212	0.206106	−	0.978530i	$-0.433921\pi$
$97$	4.05981	0.412212	0.206106	+	0.978530i	$0.433921\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.3336i	1.02823i	0.857721	+	0.514116i	$0.171880\pi$
$101$	10.3336i	1.02823i	−0.857721	+	0.514116i	$0.828120\pi$
$102$	0	0
$103$	−8.94679	−0.881554	−0.440777	−	0.897617i	$-0.645297\pi$
$103$	−8.94679	−0.881554	−0.440777	+	0.897617i	$0.645297\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.8845i	1.72896i	0.502669	+	0.864479i	$0.332351\pi$
$107$	17.8845i	1.72896i	−0.502669	+	0.864479i	$0.667649\pi$
$108$	0	0
$109$	11.9610i	1.14566i	0.819676	+	0.572828i	$0.194154\pi$
$109$	11.9610i	1.14566i	−0.819676	+	0.572828i	$0.805846\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.3192	−1.25297	−0.626484	−	0.779434i	$-0.715507\pi$
$113$	−13.3192	−1.25297	−0.626484	+	0.779434i	$0.715507\pi$
$114$	0	0
$115$	− 17.3417i	− 1.61713i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.79736	−0.623113
$120$	0	0
$121$	−22.7352	−2.06684
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.78606i	− 0.159750i
$126$	0	0
$127$	7.42140	0.658543	0.329271	−	0.944235i	$-0.393197\pi$
$127$	7.42140	0.658543	0.329271	+	0.944235i	$0.393197\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.82695i	0.421733i	0.977515	+	0.210866i	$0.0676285\pi$
$131$	4.82695i	0.421733i	−0.977515	+	0.210866i	$0.932372\pi$
$132$	0	0
$133$	5.28617i	0.458369i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.8496	−1.26868	−0.634342	−	0.773052i	$-0.718729\pi$
$137$	−14.8496	−1.26868	−0.634342	+	0.773052i	$0.718729\pi$
$138$	0	0
$139$	− 4.99694i	− 0.423835i	−0.977288	−	0.211918i	$-0.932029\pi$
$139$	− 4.99694i	− 0.423835i	0.977288	−	0.211918i	$-0.0679708\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	20.4676	1.71159
$144$	0	0
$145$	−3.72542	−0.309379
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.54688i	0.782111i	0.920367	+	0.391055i	$0.127890\pi$
$149$	9.54688i	0.782111i	−0.920367	+	0.391055i	$0.872110\pi$
$150$	0	0
$151$	11.4778	0.934052	0.467026	−	0.884244i	$-0.345325\pi$
$151$	11.4778	0.934052	0.467026	+	0.884244i	$0.345325\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.329550i	0.0264701i
$156$	0	0
$157$	− 1.38325i	− 0.110396i	−0.998475	−	0.0551979i	$-0.982421\pi$
$157$	− 1.38325i	− 0.110396i	0.998475	−	0.0551979i	$-0.0175790\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.65085	0.445349
$162$	0	0
$163$	− 10.0101i	− 0.784050i	−0.919955	−	0.392025i	$-0.871775\pi$
$163$	− 10.0101i	− 0.784050i	0.919955	−	0.392025i	$-0.128225\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.63373	−0.126422	−0.0632110	−	0.998000i	$-0.520134\pi$
$167$	−1.63373	−0.126422	−0.0632110	+	0.998000i	$0.520134\pi$
$168$	0	0
$169$	0.581993	0.0447687
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.93513i	− 0.299182i	−0.988748	−	0.149591i	$-0.952204\pi$
$173$	− 3.93513i	− 0.299182i	0.988748	−	0.149591i	$-0.0477957\pi$
$174$	0	0
$175$	−4.41801	−0.333970
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 0.199235i	− 0.0148915i	−0.999972	−	0.00744577i	$-0.997630\pi$
$179$	− 0.199235i	− 0.0148915i	0.999972	−	0.00744577i	$-0.00237008\pi$
$180$	0	0
$181$	− 7.14783i	− 0.531294i	−0.964070	−	0.265647i	$-0.914414\pi$
$181$	− 7.14783i	− 0.531294i	0.964070	−	0.265647i	$-0.0855855\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	15.0447	1.10611
$186$	0	0
$187$	39.4805i	2.88710i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.8761	1.29347	0.646733	−	0.762717i	$-0.276135\pi$
$191$	17.8761	1.29347	0.646733	+	0.762717i	$0.276135\pi$
$192$	0	0
$193$	−14.2993	−1.02928	−0.514642	−	0.857405i	$-0.672075\pi$
$193$	−14.2993	−1.02928	−0.514642	+	0.857405i	$0.672075\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.13436i	0.294561i	0.989095	+	0.147280i	$0.0470520\pi$
$197$	4.13436i	0.294561i	−0.989095	+	0.147280i	$0.952948\pi$
$198$	0	0
$199$	−14.5173	−1.02910	−0.514550	−	0.857460i	$-0.672041\pi$
$199$	−14.5173	−1.02910	−0.514550	+	0.857460i	$0.672041\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.21394i	− 0.0852015i
$204$	0	0
$205$	35.8614i	2.50467i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	30.7031	2.12378
$210$	0	0
$211$	13.9076i	0.957439i	0.877968	+	0.478719i	$0.158899\pi$
$211$	13.9076i	0.957439i	−0.877968	+	0.478719i	$0.841101\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.69841	−0.388628
$216$	0	0
$217$	−0.107385	−0.00728974
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 23.9534i	− 1.61128i
$222$	0	0
$223$	12.1591	0.814231	0.407115	−	0.913377i	$-0.366535\pi$
$223$	12.1591	0.814231	0.407115	+	0.913377i	$0.366535\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.4059i	1.61987i	0.586517	+	0.809937i	$0.300499\pi$
$227$	24.4059i	1.61987i	−0.586517	+	0.809937i	$0.699501\pi$
$228$	0	0
$229$	1.95033i	0.128881i	0.997922	+	0.0644407i	$0.0205263\pi$
$229$	1.95033i	0.128881i	−0.997922	+	0.0644407i	$0.979474\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.3094	1.06847	0.534233	−	0.845337i	$-0.320600\pi$
$233$	16.3094	1.06847	0.534233	+	0.845337i	$0.320600\pi$
$234$	0	0
$235$	20.7597i	1.35421i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.65345	−0.106953	−0.0534764	−	0.998569i	$-0.517030\pi$
$239$	−1.65345	−0.106953	−0.0534764	+	0.998569i	$0.517030\pi$
$240$	0	0
$241$	3.63938	0.234433	0.117217	−	0.993106i	$-0.462603\pi$
$241$	3.63938	0.234433	0.117217	+	0.993106i	$0.462603\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3.06888i	− 0.196063i
$246$	0	0
$247$	−18.6280	−1.18527
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 25.2386i	− 1.59305i	−0.604607	−	0.796524i	$-0.706670\pi$
$251$	− 25.2386i	− 1.59305i	0.604607	−	0.796524i	$-0.293330\pi$
$252$	0	0
$253$	− 32.8213i	− 2.06346i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.2051	−0.886087	−0.443044	−	0.896500i	$-0.646101\pi$
$257$	−14.2051	−0.886087	−0.443044	+	0.896500i	$0.646101\pi$
$258$	0	0
$259$	4.90235i	0.304617i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.3785	1.00994	0.504971	−	0.863137i	$-0.331503\pi$
$263$	16.3785	1.00994	0.504971	+	0.863137i	$0.331503\pi$
$264$	0	0
$265$	−35.2493	−2.16535
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4.73448i	− 0.288666i	−0.989529	−	0.144333i	$-0.953896\pi$
$269$	− 4.73448i	− 0.288666i	0.989529	−	0.144333i	$-0.0461037\pi$
$270$	0	0
$271$	−20.3680	−1.23727	−0.618634	−	0.785679i	$-0.712314\pi$
$271$	−20.3680	−1.23727	−0.618634	+	0.785679i	$0.712314\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	25.6607i	1.54740i
$276$	0	0
$277$	− 29.0135i	− 1.74325i	−0.490170	−	0.871627i	$-0.663065\pi$
$277$	− 29.0135i	− 1.74325i	0.490170	−	0.871627i	$-0.336935\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.5849	−0.989373	−0.494687	−	0.869071i	$-0.664717\pi$
$281$	−16.5849	−0.989373	−0.494687	+	0.869071i	$0.664717\pi$
$282$	0	0
$283$	21.8603i	1.29946i	0.760165	+	0.649730i	$0.225118\pi$
$283$	21.8603i	1.29946i	−0.760165	+	0.649730i	$0.774882\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.6855	−0.689774
$288$	0	0
$289$	29.2041	1.71789
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.59022i	0.209743i	0.994486	+	0.104872i	$0.0334431\pi$
$293$	3.59022i	0.209743i	−0.994486	+	0.104872i	$0.966557\pi$
$294$	0	0
$295$	−24.1059	−1.40350
$296$	0	0
$297$	0	0
$298$	0	0
$299$	19.9131i	1.15161i
$300$	0	0
$301$	− 1.85684i	− 0.107026i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−36.9971	−2.11845
$306$	0	0
$307$	17.1418i	0.978333i	0.872191	+	0.489166i	$0.162699\pi$
$307$	17.1418i	0.978333i	−0.872191	+	0.489166i	$0.837301\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.2814	−1.20676	−0.603379	−	0.797454i	$-0.706180\pi$
$311$	−21.2814	−1.20676	−0.603379	+	0.797454i	$0.706180\pi$
$312$	0	0
$313$	−7.14361	−0.403781	−0.201890	−	0.979408i	$-0.564708\pi$
$313$	−7.14361	−0.403781	−0.201890	+	0.979408i	$0.564708\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.4066i	1.76397i	0.471277	+	0.881986i	$0.343793\pi$
$317$	31.4066i	1.76397i	−0.471277	+	0.881986i	$0.656207\pi$
$318$	0	0
$319$	−7.05078	−0.394768
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 35.9320i	− 1.99931i
$324$	0	0
$325$	− 15.5687i	− 0.863596i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.76459	−0.372944
$330$	0	0
$331$	− 11.6712i	− 0.641506i	−0.947163	−	0.320753i	$-0.896064\pi$
$331$	− 11.6712i	− 0.641506i	0.947163	−	0.320753i	$-0.103936\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−20.4676	−1.11827
$336$	0	0
$337$	20.4008	1.11130	0.555652	−	0.831415i	$-0.312469\pi$
$337$	20.4008	1.11130	0.555652	+	0.831415i	$0.312469\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.623712i	0.0337759i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.5903i	− 0.675881i	−0.941167	−	0.337940i	$-0.890270\pi$
$347$	− 12.5903i	− 0.675881i	0.941167	−	0.337940i	$-0.109730\pi$
$348$	0	0
$349$	12.8022i	0.685286i	0.939466	+	0.342643i	$0.111322\pi$
$349$	12.8022i	0.685286i	−0.939466	+	0.342643i	$0.888678\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	17.2495	0.918099	0.459050	−	0.888411i	$-0.348190\pi$
$353$	17.2495	0.918099	0.459050	+	0.888411i	$0.348190\pi$
$354$	0	0
$355$	6.09103i	0.323278i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.52892	−0.0806935	−0.0403468	−	0.999186i	$-0.512846\pi$
$359$	−1.52892	−0.0806935	−0.0403468	+	0.999186i	$0.512846\pi$
$360$	0	0
$361$	−8.94358	−0.470715
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.0360i	1.04873i
$366$	0	0
$367$	4.04525	0.211160	0.105580	−	0.994411i	$-0.466330\pi$
$367$	4.04525	0.211160	0.105580	+	0.994411i	$0.466330\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 11.4861i	− 0.596326i
$372$	0	0
$373$	25.1416i	1.30178i	0.759171	+	0.650891i	$0.225605\pi$
$373$	25.1416i	1.30178i	−0.759171	+	0.650891i	$0.774395\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.27781	0.220318
$378$	0	0
$379$	− 10.1885i	− 0.523349i	−0.965156	−	0.261675i	$-0.915725\pi$
$379$	− 10.1885i	− 0.523349i	0.965156	−	0.261675i	$-0.0842747\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	35.9423	1.83657	0.918284	−	0.395923i	$-0.129575\pi$
$383$	35.9423	1.83657	0.918284	+	0.395923i	$0.129575\pi$
$384$	0	0
$385$	−17.8247	−0.908429
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.81471i	0.345519i	0.984964	+	0.172760i	$0.0552684\pi$
$389$	6.81471i	0.345519i	−0.984964	+	0.172760i	$0.944732\pi$
$390$	0	0
$391$	−38.4108	−1.94252
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.08116i	0.356292i
$396$	0	0
$397$	− 11.1982i	− 0.562021i	−0.959705	−	0.281010i	$-0.909330\pi$
$397$	− 11.1982i	− 0.562021i	0.959705	−	0.281010i	$-0.0906695\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−29.9404	−1.49515	−0.747577	−	0.664175i	$-0.768783\pi$
$401$	−29.9404	−1.49515	−0.747577	+	0.664175i	$0.768783\pi$
$402$	0	0
$403$	− 0.378415i	− 0.0188502i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	28.4739	1.41140
$408$	0	0
$409$	−11.0213	−0.544968	−0.272484	−	0.962160i	$-0.587845\pi$
$409$	−11.0213	−0.544968	−0.272484	+	0.962160i	$0.587845\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 7.85494i	− 0.386516i
$414$	0	0
$415$	38.6050	1.89505
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.0133i	0.684594i	0.939592	+	0.342297i	$0.111205\pi$
$419$	14.0133i	0.684594i	−0.939592	+	0.342297i	$0.888795\pi$
$420$	0	0
$421$	− 19.4716i	− 0.948988i	−0.880259	−	0.474494i	$-0.842631\pi$
$421$	− 19.4716i	− 0.948988i	0.880259	−	0.474494i	$-0.157369\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	30.0308	1.45671
$426$	0	0
$427$	− 12.0556i	− 0.583410i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−37.2120	−1.79244	−0.896219	−	0.443612i	$-0.853697\pi$
$431$	−37.2120	−1.79244	−0.896219	+	0.443612i	$0.853697\pi$
$432$	0	0
$433$	−23.4042	−1.12474	−0.562368	−	0.826887i	$-0.690109\pi$
$433$	−23.4042	−1.12474	−0.562368	+	0.826887i	$0.690109\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	29.8713i	1.42894i
$438$	0	0
$439$	−21.7533	−1.03823	−0.519115	−	0.854705i	$-0.673738\pi$
$439$	−21.7533	−1.03823	−0.519115	+	0.854705i	$0.673738\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.8369i	0.609901i	0.952368	+	0.304951i	$0.0986400\pi$
$443$	12.8369i	0.609901i	−0.952368	+	0.304951i	$0.901360\pi$
$444$	0	0
$445$	23.9112i	1.13350i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11.7889	0.556353	0.278176	−	0.960530i	$-0.410270\pi$
$449$	11.7889	0.556353	0.278176	+	0.960530i	$0.410270\pi$
$450$	0	0
$451$	67.8718i	3.19596i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.8145	0.506990
$456$	0	0
$457$	−33.5157	−1.56780	−0.783900	−	0.620888i	$-0.786772\pi$
$457$	−33.5157	−1.56780	−0.783900	+	0.620888i	$0.786772\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.0902i	0.469946i	0.972002	+	0.234973i	$0.0755002\pi$
$461$	10.0902i	0.469946i	−0.972002	+	0.234973i	$0.924500\pi$
$462$	0	0
$463$	23.6131	1.09739	0.548697	−	0.836021i	$-0.315124\pi$
$463$	23.6131	1.09739	0.548697	+	0.836021i	$0.315124\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.52944i	− 0.117048i	−0.998286	−	0.0585242i	$-0.981361\pi$
$467$	− 2.52944i	− 0.117048i	0.998286	−	0.0585242i	$-0.0186395\pi$
$468$	0	0
$469$	− 6.66942i	− 0.307965i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.7849	−0.495890
$474$	0	0
$475$	− 23.3543i	− 1.07157i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	15.1106	0.690421	0.345210	−	0.938525i	$-0.387808\pi$
$479$	15.1106	0.690421	0.345210	+	0.938525i	$0.387808\pi$
$480$	0	0
$481$	−17.2755	−0.787695
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 12.4591i	− 0.565737i
$486$	0	0
$487$	−41.1541	−1.86487	−0.932436	−	0.361335i	$-0.882321\pi$
$487$	−41.1541	−1.86487	−0.932436	+	0.361335i	$0.882321\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 36.6868i	− 1.65565i	−0.560984	−	0.827827i	$-0.689577\pi$
$491$	− 36.6868i	− 1.65565i	0.560984	−	0.827827i	$-0.310423\pi$
$492$	0	0
$493$	8.25156i	0.371631i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.98478	−0.0890294
$498$	0	0
$499$	− 20.1824i	− 0.903489i	−0.892147	−	0.451744i	$-0.850802\pi$
$499$	− 20.1824i	− 0.903489i	0.892147	−	0.451744i	$-0.149198\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.5011	−0.468222	−0.234111	−	0.972210i	$-0.575218\pi$
$503$	−10.5011	−0.468222	−0.234111	+	0.972210i	$0.575218\pi$
$504$	0	0
$505$	31.7125	1.41119
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 28.5910i	− 1.26727i	−0.773630	−	0.633637i	$-0.781561\pi$
$509$	− 28.5910i	− 1.26727i	0.773630	−	0.633637i	$-0.218439\pi$
$510$	0	0
$511$	−6.52879	−0.288816
$512$	0	0
$513$	0	0
$514$	0	0
$515$	27.4566i	1.20988i
$516$	0	0
$517$	39.2901i	1.72798i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.8325	0.474578	0.237289	−	0.971439i	$-0.423741\pi$
$521$	10.8325	0.474578	0.237289	+	0.971439i	$0.423741\pi$
$522$	0	0
$523$	− 1.77247i	− 0.0775047i	−0.999249	−	0.0387523i	$-0.987662\pi$
$523$	− 1.77247i	− 0.0775047i	0.999249	−	0.0387523i	$-0.0123383\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.729932	0.0317963
$528$	0	0
$529$	8.93205	0.388350
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 41.1788i	− 1.78365i
$534$	0	0
$535$	54.8853	2.37290
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 5.80820i	− 0.250177i
$540$	0	0
$541$	− 11.0556i	− 0.475316i	−0.971349	−	0.237658i	$-0.923620\pi$
$541$	− 11.0556i	− 0.475316i	0.971349	−	0.237658i	$-0.0763797\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	36.7068	1.57235
$546$	0	0
$547$	− 5.55978i	− 0.237719i	−0.992911	−	0.118859i	$-0.962076\pi$
$547$	− 5.55978i	− 0.237719i	0.992911	−	0.118859i	$-0.0379238\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.41707	0.273376
$552$	0	0
$553$	−2.30741	−0.0981211
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.30560i	− 0.182434i	−0.995831	−	0.0912171i	$-0.970924\pi$
$557$	− 4.30560i	− 0.182434i	0.995831	−	0.0912171i	$-0.0290757\pi$
$558$	0	0
$559$	6.54335	0.276754
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2.43447i	− 0.102601i	−0.998683	−	0.0513003i	$-0.983663\pi$
$563$	− 2.43447i	− 0.102601i	0.998683	−	0.0513003i	$-0.0163366\pi$
$564$	0	0
$565$	40.8751i	1.71963i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.9163	0.457635	0.228818	−	0.973469i	$-0.426514\pi$
$569$	10.9163	0.457635	0.228818	+	0.973469i	$0.426514\pi$
$570$	0	0
$571$	− 11.3442i	− 0.474740i	−0.971419	−	0.237370i	$-0.923715\pi$
$571$	− 11.3442i	− 0.474740i	0.971419	−	0.237370i	$-0.0762854\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−24.9655	−1.04113
$576$	0	0
$577$	7.97962	0.332196	0.166098	−	0.986109i	$-0.446883\pi$
$577$	7.97962	0.332196	0.166098	+	0.986109i	$0.446883\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.5795i	0.521887i
$582$	0	0
$583$	−66.7133	−2.76298
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 41.7410i	− 1.72283i	−0.507898	−	0.861417i	$-0.669577\pi$
$587$	− 41.7410i	− 1.72283i	0.507898	−	0.861417i	$-0.330423\pi$
$588$	0	0
$589$	− 0.567653i	− 0.0233897i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.43527	0.182135	0.0910673	−	0.995845i	$-0.470972\pi$
$593$	4.43527	0.182135	0.0910673	+	0.995845i	$0.470972\pi$
$594$	0	0
$595$	20.8603i	0.855188i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.4875	−0.510224	−0.255112	−	0.966911i	$-0.582112\pi$
$599$	−12.4875	−0.510224	−0.255112	+	0.966911i	$0.582112\pi$
$600$	0	0
$601$	37.1871	1.51690	0.758448	−	0.651734i	$-0.225958\pi$
$601$	37.1871	1.51690	0.758448	+	0.651734i	$0.225958\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	69.7716i	2.83662i
$606$	0	0
$607$	21.8403	0.886470	0.443235	−	0.896405i	$-0.353831\pi$
$607$	21.8403	0.886470	0.443235	+	0.896405i	$0.353831\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 23.8379i	− 0.964377i
$612$	0	0
$613$	− 30.7687i	− 1.24274i	−0.783519	−	0.621368i	$-0.786577\pi$
$613$	− 30.7687i	− 1.24274i	0.783519	−	0.621368i	$-0.213423\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.0402	1.53144	0.765721	−	0.643173i	$-0.222383\pi$
$617$	38.0402	1.53144	0.765721	+	0.643173i	$0.222383\pi$
$618$	0	0
$619$	− 14.9005i	− 0.598903i	−0.954111	−	0.299452i	$-0.903196\pi$
$619$	− 14.9005i	− 0.598903i	0.954111	−	0.299452i	$-0.0968037\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.79151	−0.312160
$624$	0	0
$625$	−27.5712	−1.10285
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 33.3231i	− 1.32868i
$630$	0	0
$631$	26.8787	1.07002	0.535011	−	0.844845i	$-0.320307\pi$
$631$	26.8787	1.07002	0.535011	+	0.844845i	$0.320307\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 22.7754i	− 0.903813i
$636$	0	0
$637$	3.52392i	0.139623i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.4910	−0.493365	−0.246682	−	0.969096i	$-0.579340\pi$
$641$	−12.4910	−0.493365	−0.246682	+	0.969096i	$0.579340\pi$
$642$	0	0
$643$	− 3.53414i	− 0.139373i	−0.997569	−	0.0696864i	$-0.977800\pi$
$643$	− 3.53414i	− 0.139373i	0.997569	−	0.0696864i	$-0.0221999\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.3282	−1.11370	−0.556848	−	0.830614i	$-0.687990\pi$
$647$	−28.3282	−1.11370	−0.556848	+	0.830614i	$0.687990\pi$
$648$	0	0
$649$	−45.6231	−1.79086
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 29.9962i	− 1.17384i	−0.809644	−	0.586921i	$-0.800340\pi$
$653$	− 29.9962i	− 1.17384i	0.809644	−	0.586921i	$-0.199660\pi$
$654$	0	0
$655$	14.8133	0.578804
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.0647i	0.625793i	0.949787	+	0.312896i	$0.101299\pi$
$659$	16.0647i	0.625793i	−0.949787	+	0.312896i	$0.898701\pi$
$660$	0	0
$661$	− 44.3984i	− 1.72690i	−0.504435	−	0.863450i	$-0.668299\pi$
$661$	− 44.3984i	− 1.72690i	0.504435	−	0.863450i	$-0.331701\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.2226	0.629086
$666$	0	0
$667$	− 6.85976i	− 0.265611i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−70.0213	−2.70314
$672$	0	0
$673$	−50.0649	−1.92986	−0.964930	−	0.262508i	$-0.915450\pi$
$673$	−50.0649	−1.92986	−0.964930	+	0.262508i	$0.915450\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.1370i	1.15826i	0.815236	+	0.579129i	$0.196607\pi$
$677$	30.1370i	1.15826i	−0.815236	+	0.579129i	$0.803393\pi$
$678$	0	0
$679$	4.05981	0.155801
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1.49520i	− 0.0572124i	−0.999591	−	0.0286062i	$-0.990893\pi$
$683$	− 1.49520i	− 0.0572124i	0.999591	−	0.0286062i	$-0.00910688\pi$
$684$	0	0
$685$	45.5715i	1.74120i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	40.4759	1.54201
$690$	0	0
$691$	− 0.438347i	− 0.0166755i	−0.999965	−	0.00833774i	$-0.997346\pi$
$691$	− 0.438347i	− 0.0166755i	0.999965	−	0.00833774i	$-0.00265402\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.3350	−0.581690
$696$	0	0
$697$	79.4306	3.00865
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 43.1330i	− 1.62911i	−0.580084	−	0.814556i	$-0.696980\pi$
$701$	− 43.1330i	− 1.62911i	0.580084	−	0.814556i	$-0.303020\pi$
$702$	0	0
$703$	−25.9147	−0.977390
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.3336i	0.388635i
$708$	0	0
$709$	30.8542i	1.15875i	0.815060	+	0.579376i	$0.196704\pi$
$709$	30.8542i	1.15875i	−0.815060	+	0.579376i	$0.803296\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.606814	−0.0227254
$714$	0	0
$715$	− 62.8127i	− 2.34906i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.24406	−0.0836895	−0.0418447	−	0.999124i	$-0.513323\pi$
$719$	−2.24406	−0.0836895	−0.0418447	+	0.999124i	$0.513323\pi$
$720$	0	0
$721$	−8.94679	−0.333196
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.36317i	0.199183i
$726$	0	0
$727$	16.9794	0.629733	0.314866	−	0.949136i	$-0.398040\pi$
$727$	16.9794	0.629733	0.314866	+	0.949136i	$0.398040\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12.6216i	0.466827i
$732$	0	0
$733$	− 34.0255i	− 1.25676i	−0.777906	−	0.628380i	$-0.783718\pi$
$733$	− 34.0255i	− 1.25676i	0.777906	−	0.628380i	$-0.216282\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−38.7374	−1.42691
$738$	0	0
$739$	0.0107994i	0 0.000397263i	1.00000		0.000198632i	$6.32264e-5\pi$
$739$	0.0107994i	0 0.000397263i	−1.00000		0.000198632i	$0.999937\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−45.6360	−1.67422	−0.837112	−	0.547032i	$-0.815758\pi$
$743$	−45.6360	−1.67422	−0.837112	+	0.547032i	$0.815758\pi$
$744$	0	0
$745$	29.2982	1.07340
$746$	0	0
$747$	0	0
$748$	0	0
$749$	17.8845i	0.653485i
$750$	0	0
$751$	3.95000	0.144138	0.0720688	−	0.997400i	$-0.477040\pi$
$751$	3.95000	0.144138	0.0720688	+	0.997400i	$0.477040\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 35.2240i	− 1.28193i
$756$	0	0
$757$	18.6412i	0.677524i	0.940872	+	0.338762i	$0.110008\pi$
$757$	18.6412i	0.677524i	−0.940872	+	0.338762i	$0.889992\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.86842	0.176480	0.0882401	−	0.996099i	$-0.471876\pi$
$761$	4.86842	0.176480	0.0882401	+	0.996099i	$0.471876\pi$
$762$	0	0
$763$	11.9610i	0.433017i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	27.6802	0.999473
$768$	0	0
$769$	−1.48442	−0.0535297	−0.0267649	−	0.999642i	$-0.508521\pi$
$769$	−1.48442	−0.0535297	−0.0267649	+	0.999642i	$0.508521\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.2868i	0.513861i	0.966430	+	0.256931i	$0.0827112\pi$
$773$	14.2868i	0.513861i	−0.966430	+	0.256931i	$0.917289\pi$
$774$	0	0
$775$	0.474426	0.0170419
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 61.7716i	− 2.21320i
$780$	0	0
$781$	11.5280i	0.412504i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.24504	−0.151512
$786$	0	0
$787$	− 22.3799i	− 0.797758i	−0.917004	−	0.398879i	$-0.869399\pi$
$787$	− 22.3799i	− 0.797758i	0.917004	−	0.398879i	$-0.130601\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.3192	−0.473578
$792$	0	0
$793$	42.4829	1.50861
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 23.1872i	− 0.821334i	−0.911785	−	0.410667i	$-0.865296\pi$
$797$	− 23.1872i	− 0.821334i	0.911785	−	0.410667i	$-0.134704\pi$
$798$	0	0
$799$	45.9814	1.62670
$800$	0	0
$801$	0	0
$802$	0	0
$803$	37.9205i	1.33819i
$804$	0	0
$805$	− 17.3417i	− 0.611216i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−44.8263	−1.57601	−0.788004	−	0.615670i	$-0.788886\pi$
$809$	−44.8263	−1.57601	−0.788004	+	0.615670i	$0.788886\pi$
$810$	0	0
$811$	18.4358i	0.647370i	0.946165	+	0.323685i	$0.104922\pi$
$811$	18.4358i	0.647370i	−0.946165	+	0.323685i	$0.895078\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−30.7197	−1.07606
$816$	0	0
$817$	9.81556	0.343403
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 21.0533i	− 0.734764i	−0.930070	−	0.367382i	$-0.880254\pi$
$821$	− 21.0533i	− 0.734764i	0.930070	−	0.367382i	$-0.119746\pi$
$822$	0	0
$823$	32.3206	1.12663	0.563313	−	0.826244i	$-0.309527\pi$
$823$	32.3206	1.12663	0.563313	+	0.826244i	$0.309527\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12.4599i	− 0.433272i	−0.976252	−	0.216636i	$-0.930492\pi$
$827$	− 12.4599i	− 0.433272i	0.976252	−	0.216636i	$-0.0695085\pi$
$828$	0	0
$829$	6.78073i	0.235504i	0.993043	+	0.117752i	$0.0375689\pi$
$829$	6.78073i	0.235504i	−0.993043	+	0.117752i	$0.962431\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.79736	−0.235515
$834$	0	0
$835$	5.01372i	0.173507i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23.6268	−0.815687	−0.407843	−	0.913052i	$-0.633719\pi$
$839$	−23.6268	−0.815687	−0.407843	+	0.913052i	$0.633719\pi$
$840$	0	0
$841$	27.5264	0.949185
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.78606i	− 0.0614425i
$846$	0	0
$847$	−22.7352	−0.781192
$848$	0	0
$849$	0	0
$850$	0	0
$851$	27.7024i	0.949628i
$852$	0	0
$853$	− 17.7202i	− 0.606727i	−0.952875	−	0.303363i	$-0.901890\pi$
$853$	− 17.7202i	− 0.606727i	0.952875	−	0.303363i	$-0.0981096\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.5262	0.462046	0.231023	−	0.972948i	$-0.425793\pi$
$857$	13.5262	0.462046	0.231023	+	0.972948i	$0.425793\pi$
$858$	0	0
$859$	− 27.6249i	− 0.942548i	−0.881987	−	0.471274i	$-0.843794\pi$
$859$	− 27.6249i	− 0.942548i	0.881987	−	0.471274i	$-0.156206\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−39.9545	−1.36007	−0.680034	−	0.733181i	$-0.738035\pi$
$863$	−39.9545	−1.36007	−0.680034	+	0.733181i	$0.738035\pi$
$864$	0	0
$865$	−12.0764	−0.410610
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13.4019i	0.454629i
$870$	0	0
$871$	23.5025	0.796352
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.78606i	− 0.0603800i
$876$	0	0
$877$	11.3922i	0.384687i	0.981328	+	0.192343i	$0.0616087\pi$
$877$	11.3922i	0.384687i	−0.981328	+	0.192343i	$0.938391\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−36.3753	−1.22551	−0.612757	−	0.790271i	$-0.709940\pi$
$881$	−36.3753	−1.22551	−0.612757	+	0.790271i	$0.709940\pi$
$882$	0	0
$883$	− 52.0983i	− 1.75325i	−0.481177	−	0.876624i	$-0.659790\pi$
$883$	− 52.0983i	− 1.75325i	0.481177	−	0.876624i	$-0.340210\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.0095	1.51127	0.755635	−	0.654992i	$-0.227328\pi$
$887$	45.0095	1.51127	0.755635	+	0.654992i	$0.227328\pi$
$888$	0	0
$889$	7.42140	0.248906
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 35.7588i	− 1.19662i
$894$	0	0
$895$	−0.611428	−0.0204378
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.130358i	0.00434768i
$900$	0	0
$901$	78.0748i	2.60105i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.9358	−0.729171
$906$	0	0
$907$	7.96941i	0.264620i	0.991208	+	0.132310i	$0.0422394\pi$
$907$	7.96941i	0.264620i	−0.991208	+	0.132310i	$0.957761\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−47.8457	−1.58520	−0.792600	−	0.609742i	$-0.791273\pi$
$911$	−47.8457	−1.58520	−0.792600	+	0.609742i	$0.791273\pi$
$912$	0	0
$913$	73.0645	2.41808
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.82695i	0.159400i
$918$	0	0
$919$	−36.2279	−1.19505	−0.597525	−	0.801851i	$-0.703849\pi$
$919$	−36.2279	−1.19505	−0.597525	+	0.801851i	$0.703849\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 6.99419i	− 0.230217i
$924$	0	0
$925$	− 21.6586i	− 0.712132i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−9.44479	−0.309874	−0.154937	−	0.987924i	$-0.549517\pi$
$929$	−9.44479	−0.309874	−0.154937	+	0.987924i	$0.549517\pi$
$930$	0	0
$931$	5.28617i	0.173247i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	121.161	3.96238
$936$	0	0
$937$	−15.2959	−0.499695	−0.249847	−	0.968285i	$-0.580380\pi$
$937$	−15.2959	−0.499695	−0.249847	+	0.968285i	$0.580380\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 29.2847i	− 0.954654i	−0.878726	−	0.477327i	$-0.841606\pi$
$941$	− 29.2847i	− 0.954654i	0.878726	−	0.477327i	$-0.158394\pi$
$942$	0	0
$943$	−66.0330	−2.15033
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.0133i	0.747832i	0.927463	+	0.373916i	$0.121985\pi$
$947$	23.0133i	0.747832i	−0.927463	+	0.373916i	$0.878015\pi$
$948$	0	0
$949$	− 23.0069i	− 0.746836i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.48602	0.145316	0.0726582	−	0.997357i	$-0.476852\pi$
$953$	4.48602	0.145316	0.0726582	+	0.997357i	$0.476852\pi$
$954$	0	0
$955$	− 54.8594i	− 1.77521i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14.8496	−0.479518
$960$	0	0
$961$	−30.9885	−0.999628
$962$	0	0
$963$	0	0
$964$	0	0
$965$	43.8827i	1.41263i
$966$	0	0
$967$	18.4520	0.593376	0.296688	−	0.954974i	$-0.404118\pi$
$967$	18.4520	0.593376	0.296688	+	0.954974i	$0.404118\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 21.1370i	− 0.678317i	−0.940729	−	0.339159i	$-0.889858\pi$
$971$	− 21.1370i	− 0.678317i	0.940729	−	0.339159i	$-0.110142\pi$
$972$	0	0
$973$	− 4.99694i	− 0.160195i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.4783	1.67893	0.839465	−	0.543414i	$-0.182869\pi$
$977$	52.4783	1.67893	0.839465	+	0.543414i	$0.182869\pi$
$978$	0	0
$979$	45.2547i	1.44635i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	29.7453	0.948727	0.474364	−	0.880329i	$-0.342678\pi$
$983$	29.7453	0.948727	0.474364	+	0.880329i	$0.342678\pi$
$984$	0	0
$985$	12.6878	0.404268
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 10.4927i	− 0.333649i
$990$	0	0
$991$	−9.11149	−0.289436	−0.144718	−	0.989473i	$-0.546227\pi$
$991$	−9.11149	−0.289436	−0.144718	+	0.989473i	$0.546227\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	44.5517i	1.41238i
$996$	0	0
$997$	− 4.52824i	− 0.143411i	−0.997426	−	0.0717054i	$-0.977156\pi$
$997$	− 4.52824i	− 0.143411i	0.997426	−	0.0717054i	$-0.0228442\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.e.3025.6		20
3.2	odd	2	inner	6048.2.c.e.3025.16		20
4.3	odd	2		1512.2.c.e.757.13	yes	20
8.3	odd	2		1512.2.c.e.757.14	yes	20
8.5	even	2	inner	6048.2.c.e.3025.15		20
12.11	even	2		1512.2.c.e.757.8	yes	20
24.5	odd	2	inner	6048.2.c.e.3025.5		20
24.11	even	2		1512.2.c.e.757.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.e.757.7	✓	20	24.11	even	2
1512.2.c.e.757.8	yes	20	12.11	even	2
1512.2.c.e.757.13	yes	20	4.3	odd	2
1512.2.c.e.757.14	yes	20	8.3	odd	2
6048.2.c.e.3025.5		20	24.5	odd	2	inner
6048.2.c.e.3025.6		20	1.1	even	1	trivial
6048.2.c.e.3025.15		20	8.5	even	2	inner
6048.2.c.e.3025.16		20	3.2	odd	2	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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.06888i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-3.06888i q^{5} +1.00000 q^{7} -5.80820i q^{11} +3.52392i q^{13} -6.79736 q^{17} +5.28617i q^{19} +5.65085 q^{23} -4.41801 q^{25} -1.21394i q^{29} -0.107385 q^{31} -3.06888i q^{35} +4.90235i q^{37} -11.6855 q^{41} -1.85684i q^{43} -6.76459 q^{47} +1.00000 q^{49} -11.4861i q^{53} -17.8247 q^{55} -7.85494i q^{59} -12.0556i q^{61} +10.8145 q^{65} -6.66942i q^{67} -1.98478 q^{71} -6.52879 q^{73} -5.80820i q^{77} -2.30741 q^{79} +12.5795i q^{83} +20.8603i q^{85} -7.79151 q^{89} +3.52392i q^{91} +16.2226 q^{95} +4.05981 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{7}+O(q^{10}) \) 20 * q + 20 * q^7	\( 20 q + 20 q^{7} - 28 q^{25} - 36 q^{31} + 20 q^{49} - 48 q^{55} - 64 q^{79} + 56 q^{97}+O(q^{100}) \) 20 * q + 20 * q^7 - 28 * q^25 - 36 * q^31 + 20 * q^49 - 48 * q^55 - 64 * q^79 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more