LMFDB - Embedded newform 6048.2.j.b.5615.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(5615,6048)]

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([1, 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("6048.5615");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.j (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:	$8$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.3
Root			$0.500000 + 1.56488i$ of defining polynomial
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.b.5615.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.12976 q^{5} -1.00000i q^{7} +O(q^{10})$	$q-1.12976 q^{5} -1.00000i q^{7} +3.86182i q^{11} +5.08658i q^{13} -2.00000i q^{17} -7.99158 q^{19} -7.68044 q^{23} -3.72363 q^{25} +4.73205 q^{29} +4.08658i q^{31} +1.12976i q^{35} -8.28273i q^{37} -8.41249i q^{41} +10.0148 q^{43} -1.93662 q^{47} -1.00000 q^{49} +7.72363 q^{53} -4.36294i q^{55} -10.9916i q^{59} +11.4641i q^{61} -5.74663i q^{65} -5.08658 q^{67} +13.6804 q^{71} -2.63706 q^{73} +3.86182 q^{77} -7.17295i q^{79} -14.9052i q^{83} +2.25953i q^{85} -12.6889i q^{89} +5.08658 q^{91} +9.02861 q^{95} +4.17315 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} - 16 q^{19} + 16 q^{23} + 32 q^{25} + 24 q^{29} - 8 q^{43} - 8 q^{47} - 8 q^{49} - 8 q^{67} + 32 q^{71} + 8 q^{73} + 8 q^{91} + 112 q^{95} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 - 16 * q^19 + 16 * q^23 + 32 * q^25 + 24 * q^29 - 8 * q^43 - 8 * q^47 - 8 * q^49 - 8 * q^67 + 32 * q^71 + 8 * q^73 + 8 * q^91 + 112 * q^95 - 32 * 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
$5$	−1.12976	−0.505246	−0.252623	−	0.967565i	$-0.581293\pi$
$5$	−1.12976	−0.505246	−0.252623	+	0.967565i	$0.581293\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.86182i	1.16438i	0.813052	+	0.582191i	$0.197804\pi$
$11$	3.86182i	1.16438i	−0.813052	+	0.582191i	$0.802196\pi$
$12$	0	0
$13$	5.08658i	1.41076i	0.708828	+	0.705381i	$0.249224\pi$
$13$	5.08658i	1.41076i	−0.708828	+	0.705381i	$0.750776\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	−7.99158	−1.83339	−0.916697	−	0.399583i	$-0.869155\pi$
$19$	−7.99158	−1.83339	−0.916697	+	0.399583i	$0.869155\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.68044	−1.60148	−0.800741	−	0.599010i	$-0.795561\pi$
$23$	−7.68044	−1.60148	−0.800741	+	0.599010i	$0.795561\pi$
$24$	0	0
$25$	−3.72363	−0.744726
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.73205	0.878720	0.439360	−	0.898311i	$-0.355205\pi$
$29$	4.73205	0.878720	0.439360	+	0.898311i	$0.355205\pi$
$30$	0	0
$31$	4.08658i	0.733971i	0.930227	+	0.366985i	$0.119610\pi$
$31$	4.08658i	0.733971i	−0.930227	+	0.366985i	$0.880390\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.12976i	0.190965i
$36$	0	0
$37$	− 8.28273i	− 1.36167i	−0.732436	−	0.680836i	$-0.761617\pi$
$37$	− 8.28273i	− 1.36167i	0.732436	−	0.680836i	$-0.238383\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.41249i	− 1.31381i	−0.753973	−	0.656905i	$-0.771865\pi$
$41$	− 8.41249i	− 1.31381i	0.753973	−	0.656905i	$-0.228135\pi$
$42$	0	0
$43$	10.0148	1.52724	0.763620	−	0.645666i	$-0.223420\pi$
$43$	10.0148	1.52724	0.763620	+	0.645666i	$0.223420\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.93662	−0.282485	−0.141243	−	0.989975i	$-0.545110\pi$
$47$	−1.93662	−0.282485	−0.141243	+	0.989975i	$0.545110\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.72363	1.06092	0.530461	−	0.847709i	$-0.322019\pi$
$53$	7.72363	1.06092	0.530461	+	0.847709i	$0.322019\pi$
$54$	0	0
$55$	− 4.36294i	− 0.588299i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 10.9916i	− 1.43098i	−0.698622	−	0.715491i	$-0.746203\pi$
$59$	− 10.9916i	− 1.43098i	0.698622	−	0.715491i	$-0.253797\pi$
$60$	0	0
$61$	11.4641i	1.46783i	0.679243	+	0.733914i	$0.262308\pi$
$61$	11.4641i	1.46783i	−0.679243	+	0.733914i	$0.737692\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 5.74663i	− 0.712782i
$66$	0	0
$67$	−5.08658	−0.621424	−0.310712	−	0.950504i	$-0.600567\pi$
$67$	−5.08658	−0.621424	−0.310712	+	0.950504i	$0.600567\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.6804	1.62357	0.811785	−	0.583957i	$-0.198496\pi$
$71$	13.6804	1.62357	0.811785	+	0.583957i	$0.198496\pi$
$72$	0	0
$73$	−2.63706	−0.308644	−0.154322	−	0.988021i	$-0.549319\pi$
$73$	−2.63706	−0.308644	−0.154322	+	0.988021i	$0.549319\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.86182	0.440095
$78$	0	0
$79$	− 7.17295i	− 0.807020i	−0.914975	−	0.403510i	$-0.867790\pi$
$79$	− 7.17295i	− 0.807020i	0.914975	−	0.403510i	$-0.132210\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 14.9052i	− 1.63606i	−0.575177	−	0.818029i	$-0.695067\pi$
$83$	− 14.9052i	− 1.63606i	0.575177	−	0.818029i	$-0.304933\pi$
$84$	0	0
$85$	2.25953i	0.245080i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.6889i	− 1.34502i	−0.740090	−	0.672508i	$-0.765217\pi$
$89$	− 12.6889i	− 1.34502i	0.740090	−	0.672508i	$-0.234783\pi$
$90$	0	0
$91$	5.08658	0.533218
$92$	0	0
$93$	0	0
$94$	0	0
$95$	9.02861	0.926316
$96$	0	0
$97$	4.17315	0.423719	0.211860	−	0.977300i	$-0.432048\pi$
$97$	4.17315	0.423719	0.211860	+	0.977300i	$0.432048\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	− 5.37753i	− 0.529863i	−0.964267	−	0.264932i	$-0.914651\pi$
$103$	− 5.37753i	− 0.529863i	0.964267	−	0.264932i	$-0.0853494\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.8418i	1.24147i	0.784022	+	0.620733i	$0.213165\pi$
$107$	12.8418i	1.24147i	−0.784022	+	0.620733i	$0.786835\pi$
$108$	0	0
$109$	13.0782i	1.25266i	0.779558	+	0.626330i	$0.215444\pi$
$109$	13.0782i	1.25266i	−0.779558	+	0.626330i	$0.784556\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.05496i	0.663675i	0.943337	+	0.331837i	$0.107669\pi$
$113$	7.05496i	0.663675i	−0.943337	+	0.331837i	$0.892331\pi$
$114$	0	0
$115$	8.67709	0.809143
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−3.91362	−0.355784
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.85565	0.881516
$126$	0	0
$127$	4.63706i	0.411472i	0.978608	+	0.205736i	$0.0659589\pi$
$127$	4.63706i	0.411472i	−0.978608	+	0.205736i	$0.934041\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.36930i	0.731229i	0.930766	+	0.365615i	$0.119141\pi$
$131$	8.36930i	0.731229i	−0.930766	+	0.365615i	$0.880859\pi$
$132$	0	0
$133$	7.99158i	0.692958i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13.9366i	− 1.19069i	−0.803472	−	0.595343i	$-0.797016\pi$
$137$	− 13.9366i	− 1.19069i	0.803472	−	0.595343i	$-0.202984\pi$
$138$	0	0
$139$	13.9832	1.18604	0.593018	−	0.805189i	$-0.297936\pi$
$139$	13.9832	1.18604	0.593018	+	0.805189i	$0.297936\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−19.6434	−1.64266
$144$	0	0
$145$	−5.34610	−0.443970
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.7700	−1.78347	−0.891735	−	0.452558i	$-0.850512\pi$
$149$	−21.7700	−1.78347	−0.891735	+	0.452558i	$0.850512\pi$
$150$	0	0
$151$	− 16.0464i	− 1.30584i	−0.757428	−	0.652919i	$-0.773544\pi$
$151$	− 16.0464i	− 1.30584i	0.757428	−	0.652919i	$-0.226456\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 4.61687i	− 0.370836i
$156$	0	0
$157$	5.49572i	0.438606i	0.975657	+	0.219303i	$0.0703783\pi$
$157$	5.49572i	0.438606i	−0.975657	+	0.219303i	$0.929622\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.68044i	0.605304i
$162$	0	0
$163$	−1.62247	−0.127082	−0.0635410	−	0.997979i	$-0.520239\pi$
$163$	−1.62247	−0.127082	−0.0635410	+	0.997979i	$0.520239\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.70905	0.673926	0.336963	−	0.941518i	$-0.390600\pi$
$167$	8.70905	0.673926	0.336963	+	0.941518i	$0.390600\pi$
$168$	0	0
$169$	−12.8732	−0.990250
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.53891	0.117001	0.0585005	−	0.998287i	$-0.481368\pi$
$173$	1.53891	0.117001	0.0585005	+	0.998287i	$0.481368\pi$
$174$	0	0
$175$	3.72363i	0.281480i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 26.7386i	− 1.99854i	−0.0382400	−	0.999269i	$-0.512175\pi$
$179$	− 26.7386i	− 1.99854i	0.0382400	−	0.999269i	$-0.487825\pi$
$180$	0	0
$181$	− 3.10135i	− 0.230522i	−0.993335	−	0.115261i	$-0.963230\pi$
$181$	− 3.10135i	− 0.230522i	0.993335	−	0.115261i	$-0.0367704\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.35753i	0.687980i
$186$	0	0
$187$	7.72363	0.564808
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.8132	0.999489	0.499745	−	0.866173i	$-0.333427\pi$
$191$	13.8132	0.999489	0.499745	+	0.866173i	$0.333427\pi$
$192$	0	0
$193$	7.05496	0.507827	0.253913	−	0.967227i	$-0.418282\pi$
$193$	7.05496	0.507827	0.253913	+	0.967227i	$0.418282\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.362748	0.0258448	0.0129224	−	0.999917i	$-0.495887\pi$
$197$	0.362748	0.0258448	0.0129224	+	0.999917i	$0.495887\pi$
$198$	0	0
$199$	7.70029i	0.545859i	0.962034	+	0.272930i	$0.0879926\pi$
$199$	7.70029i	0.545859i	−0.962034	+	0.272930i	$0.912007\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.73205i	− 0.332125i
$204$	0	0
$205$	9.50414i	0.663798i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 30.8620i	− 2.13477i
$210$	0	0
$211$	14.5191	0.999533	0.499767	−	0.866160i	$-0.333419\pi$
$211$	14.5191	0.999533	0.499767	+	0.866160i	$0.333419\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−11.3143	−0.771632
$216$	0	0
$217$	4.08658	0.277415
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.1732	0.684320
$222$	0	0
$223$	− 19.2743i	− 1.29070i	−0.763886	−	0.645352i	$-0.776711\pi$
$223$	− 19.2743i	− 1.29070i	0.763886	−	0.645352i	$-0.223289\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 23.2341i	− 1.54210i	−0.636773	−	0.771052i	$-0.719731\pi$
$227$	− 23.2341i	− 1.54210i	0.636773	−	0.771052i	$-0.280269\pi$
$228$	0	0
$229$	13.4641i	0.889733i	0.895597	+	0.444866i	$0.146749\pi$
$229$	13.4641i	0.889733i	−0.895597	+	0.444866i	$0.853251\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 17.4641i	− 1.14411i	−0.820215	−	0.572056i	$-0.806146\pi$
$233$	− 17.4641i	− 1.14411i	0.820215	−	0.572056i	$-0.193854\pi$
$234$	0	0
$235$	2.18793	0.142725
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.82724	−0.118194	−0.0590972	−	0.998252i	$-0.518822\pi$
$239$	−1.82724	−0.118194	−0.0590972	+	0.998252i	$0.518822\pi$
$240$	0	0
$241$	−21.5193	−1.38618	−0.693089	−	0.720852i	$-0.743751\pi$
$241$	−21.5193	−1.38618	−0.693089	+	0.720852i	$0.743751\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.12976	0.0721780
$246$	0	0
$247$	− 40.6498i	− 2.58648i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 1.15818i	− 0.0731035i	−0.999332	−	0.0365517i	$-0.988363\pi$
$251$	− 1.15818i	− 0.0731035i	0.999332	−	0.0365517i	$-0.0116374\pi$
$252$	0	0
$253$	− 29.6604i	− 1.86474i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.0812i	0.691224i	0.938377	+	0.345612i	$0.112329\pi$
$257$	11.0812i	0.691224i	−0.938377	+	0.345612i	$0.887671\pi$
$258$	0	0
$259$	−8.28273	−0.514664
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.7354	−0.661973	−0.330987	−	0.943635i	$-0.607381\pi$
$263$	−10.7354	−0.661973	−0.330987	+	0.943635i	$0.607381\pi$
$264$	0	0
$265$	−8.72589	−0.536027
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.75204	0.350708	0.175354	−	0.984505i	$-0.443893\pi$
$269$	5.75204	0.350708	0.175354	+	0.984505i	$0.443893\pi$
$270$	0	0
$271$	1.33735i	0.0812380i	0.999175	+	0.0406190i	$0.0129330\pi$
$271$	1.33735i	0.0812380i	−0.999175	+	0.0406190i	$0.987067\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 14.3800i	− 0.867145i
$276$	0	0
$277$	− 3.13537i	− 0.188386i	−0.995554	−	0.0941930i	$-0.969973\pi$
$277$	− 3.13537i	− 0.188386i	0.995554	−	0.0941930i	$-0.0300271\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 13.3839i	− 0.798416i	−0.916860	−	0.399208i	$-0.869285\pi$
$281$	− 13.3839i	− 0.798416i	0.916860	−	0.399208i	$-0.130715\pi$
$282$	0	0
$283$	−21.8104	−1.29649	−0.648247	−	0.761430i	$-0.724498\pi$
$283$	−21.8104	−1.29649	−0.648247	+	0.761430i	$0.724498\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.41249	−0.496574
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−30.6982	−1.79341	−0.896705	−	0.442629i	$-0.854046\pi$
$293$	−30.6982	−1.79341	−0.896705	+	0.442629i	$0.854046\pi$
$294$	0	0
$295$	12.4179i	0.722998i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 39.0671i	− 2.25931i
$300$	0	0
$301$	− 10.0148i	− 0.577242i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 12.9517i	− 0.741614i
$306$	0	0
$307$	−9.10977	−0.519922	−0.259961	−	0.965619i	$-0.583710\pi$
$307$	−9.10977	−0.519922	−0.259961	+	0.965619i	$0.583710\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.5275	1.56094	0.780470	−	0.625193i	$-0.214980\pi$
$311$	27.5275	1.56094	0.780470	+	0.625193i	$0.214980\pi$
$312$	0	0
$313$	5.36294	0.303131	0.151566	−	0.988447i	$-0.451568\pi$
$313$	5.36294	0.303131	0.151566	+	0.988447i	$0.451568\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.30832	0.410476	0.205238	−	0.978712i	$-0.434203\pi$
$317$	7.30832	0.410476	0.205238	+	0.978712i	$0.434203\pi$
$318$	0	0
$319$	18.2743i	1.02316i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	15.9832i	0.889327i
$324$	0	0
$325$	− 18.9405i	− 1.05063i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.93662i	0.106769i
$330$	0	0
$331$	−10.5823	−0.581655	−0.290828	−	0.956775i	$-0.593931\pi$
$331$	−10.5823	−0.581655	−0.290828	+	0.956775i	$0.593931\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.74663	0.313972
$336$	0	0
$337$	−4.29095	−0.233743	−0.116872	−	0.993147i	$-0.537287\pi$
$337$	−4.29095	−0.233743	−0.116872	+	0.993147i	$0.537287\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−15.7816	−0.854622
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 10.3405i	− 0.555107i	−0.960710	−	0.277554i	$-0.910476\pi$
$347$	− 10.3405i	− 0.555107i	0.960710	−	0.277554i	$-0.0895236\pi$
$348$	0	0
$349$	− 11.1182i	− 0.595143i	−0.954700	−	0.297572i	$-0.903823\pi$
$349$	− 11.1182i	− 0.595143i	0.954700	−	0.297572i	$-0.0961767\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 23.3407i	− 1.24230i	−0.783692	−	0.621150i	$-0.786666\pi$
$353$	− 23.3407i	− 1.24230i	0.783692	−	0.621150i	$-0.213334\pi$
$354$	0	0
$355$	−15.4557	−0.820302
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.3923	1.18182	0.590910	−	0.806737i	$-0.298769\pi$
$359$	22.3923	1.18182	0.590910	+	0.806737i	$0.298769\pi$
$360$	0	0
$361$	44.8654	2.36133
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.97925	0.155941
$366$	0	0
$367$	− 1.93696i	− 0.101109i	−0.998721	−	0.0505543i	$-0.983901\pi$
$367$	− 1.93696i	− 0.101109i	0.998721	−	0.0505543i	$-0.0160988\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 7.72363i	− 0.400991i
$372$	0	0
$373$	15.1246i	0.783120i	0.920153	+	0.391560i	$0.128064\pi$
$373$	15.1246i	0.783120i	−0.920153	+	0.391560i	$0.871936\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0699i	1.23966i
$378$	0	0
$379$	−6.15837	−0.316334	−0.158167	−	0.987412i	$-0.550558\pi$
$379$	−6.15837	−0.316334	−0.158167	+	0.987412i	$0.550558\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.8822	0.760445	0.380222	−	0.924895i	$-0.375847\pi$
$383$	14.8822	0.760445	0.380222	+	0.924895i	$0.375847\pi$
$384$	0	0
$385$	−4.36294	−0.222356
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.82685	0.0926250	0.0463125	−	0.998927i	$-0.485253\pi$
$389$	1.82685	0.0926250	0.0463125	+	0.998927i	$0.485253\pi$
$390$	0	0
$391$	15.3609i	0.776833i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.10375i	0.407744i
$396$	0	0
$397$	2.18999i	0.109912i	0.998489	+	0.0549562i	$0.0175019\pi$
$397$	2.18999i	0.109912i	−0.998489	+	0.0549562i	$0.982498\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 3.59071i	− 0.179312i	−0.995973	−	0.0896558i	$-0.971423\pi$
$401$	− 3.59071i	− 0.179312i	0.995973	−	0.0896558i	$-0.0285767\pi$
$402$	0	0
$403$	−20.7867	−1.03546
$404$	0	0
$405$	0	0
$406$	0	0
$407$	31.9864	1.58551
$408$	0	0
$409$	−35.3588	−1.74838	−0.874191	−	0.485583i	$-0.838607\pi$
$409$	−35.3588	−1.74838	−0.874191	+	0.485583i	$0.838607\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.9916	−0.540860
$414$	0	0
$415$	16.8394i	0.826612i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2.44952i	− 0.119667i	−0.998208	−	0.0598334i	$-0.980943\pi$
$419$	− 2.44952i	− 0.119667i	0.998208	−	0.0598334i	$-0.0190569\pi$
$420$	0	0
$421$	6.26589i	0.305381i	0.988274	+	0.152690i	$0.0487937\pi$
$421$	6.26589i	0.305381i	−0.988274	+	0.152690i	$0.951206\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.44726i	0.361245i
$426$	0	0
$427$	11.4641	0.554787
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.01177	0.145072	0.0725359	−	0.997366i	$-0.476891\pi$
$431$	3.01177	0.145072	0.0725359	+	0.997366i	$0.476891\pi$
$432$	0	0
$433$	−7.00020	−0.336408	−0.168204	−	0.985752i	$-0.553797\pi$
$433$	−7.00020	−0.336408	−0.168204	+	0.985752i	$0.553797\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	61.3789	2.93615
$438$	0	0
$439$	− 1.33735i	− 0.0638281i	−0.999491	−	0.0319140i	$-0.989840\pi$
$439$	− 1.33735i	− 0.0638281i	0.999491	−	0.0319140i	$-0.0101603\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.90821i	0.0906618i	0.998972	+	0.0453309i	$0.0144342\pi$
$443$	1.90821i	0.0906618i	−0.998972	+	0.0453309i	$0.985566\pi$
$444$	0	0
$445$	14.3354i	0.679565i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 10.0695i	− 0.475211i	−0.971362	−	0.237606i	$-0.923637\pi$
$449$	− 10.0695i	− 0.475211i	0.971362	−	0.237606i	$-0.0763625\pi$
$450$	0	0
$451$	32.4875	1.52978
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.74663	−0.269406
$456$	0	0
$457$	10.3205	0.482773	0.241387	−	0.970429i	$-0.422398\pi$
$457$	10.3205	0.482773	0.241387	+	0.970429i	$0.422398\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17.9420	−0.835644	−0.417822	−	0.908529i	$-0.637206\pi$
$461$	−17.9420	−0.835644	−0.417822	+	0.908529i	$0.637206\pi$
$462$	0	0
$463$	21.6664i	1.00692i	0.864017	+	0.503462i	$0.167941\pi$
$463$	21.6664i	1.00692i	−0.864017	+	0.503462i	$0.832059\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 18.3525i	− 0.849251i	−0.905369	−	0.424625i	$-0.860406\pi$
$467$	− 18.3525i	− 0.849251i	0.905369	−	0.424625i	$-0.139594\pi$
$468$	0	0
$469$	5.08658i	0.234876i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	38.6752i	1.77829i
$474$	0	0
$475$	29.7577	1.36538
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.6518	0.669459	0.334730	−	0.942314i	$-0.391355\pi$
$479$	14.6518	0.669459	0.334730	+	0.942314i	$0.391355\pi$
$480$	0	0
$481$	42.1307	1.92100
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.71468	−0.214083
$486$	0	0
$487$	6.60789i	0.299432i	0.988729	+	0.149716i	$0.0478360\pi$
$487$	6.60789i	0.299432i	−0.988729	+	0.149716i	$0.952164\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.08323i	0.319662i	0.987144	+	0.159831i	$0.0510949\pi$
$491$	7.08323i	0.319662i	−0.987144	+	0.159831i	$0.948905\pi$
$492$	0	0
$493$	− 9.46410i	− 0.426242i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 13.6804i	− 0.613652i
$498$	0	0
$499$	7.13297	0.319316	0.159658	−	0.987172i	$-0.448961\pi$
$499$	7.13297	0.319316	0.159658	+	0.987172i	$0.448961\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14.7321	−0.656870	−0.328435	−	0.944527i	$-0.606521\pi$
$503$	−14.7321	−0.656870	−0.328435	+	0.944527i	$0.606521\pi$
$504$	0	0
$505$	−2.25953	−0.100548
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6.21955	−0.275677	−0.137838	−	0.990455i	$-0.544015\pi$
$509$	−6.21955	−0.275677	−0.137838	+	0.990455i	$0.544015\pi$
$510$	0	0
$511$	2.63706i	0.116657i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.07534i	0.267711i
$516$	0	0
$517$	− 7.47888i	− 0.328921i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.0897i	0.880147i	0.897962	+	0.440073i	$0.145048\pi$
$521$	20.0897i	0.880147i	−0.897962	+	0.440073i	$0.854952\pi$
$522$	0	0
$523$	−25.5885	−1.11891	−0.559453	−	0.828862i	$-0.688989\pi$
$523$	−25.5885	−1.11891	−0.559453	+	0.828862i	$0.688989\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.17315	0.356028
$528$	0	0
$529$	35.9892	1.56475
$530$	0	0
$531$	0	0
$532$	0	0
$533$	42.7908	1.85347
$534$	0	0
$535$	− 14.5082i	− 0.627246i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 3.86182i	− 0.166340i
$540$	0	0
$541$	− 25.7932i	− 1.10894i	−0.832205	−	0.554469i	$-0.812922\pi$
$541$	− 25.7932i	− 1.10894i	0.832205	−	0.554469i	$-0.187078\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 14.7752i	− 0.632902i
$546$	0	0
$547$	1.41771	0.0606167	0.0303084	−	0.999541i	$-0.490351\pi$
$547$	1.41771	0.0606167	0.0303084	+	0.999541i	$0.490351\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−37.8166	−1.61104
$552$	0	0
$553$	−7.17295	−0.305025
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.36275	0.184856	0.0924278	−	0.995719i	$-0.470537\pi$
$557$	4.36275	0.184856	0.0924278	+	0.995719i	$0.470537\pi$
$558$	0	0
$559$	50.9409i	2.15457i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.4007i	1.07051i	0.844690	+	0.535256i	$0.179785\pi$
$563$	25.4007i	1.07051i	−0.844690	+	0.535256i	$0.820215\pi$
$564$	0	0
$565$	− 7.97044i	− 0.335319i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 24.9746i	− 1.04699i	−0.852029	−	0.523495i	$-0.824628\pi$
$569$	− 24.9746i	− 1.04699i	0.852029	−	0.523495i	$-0.175372\pi$
$570$	0	0
$571$	8.02956	0.336026	0.168013	−	0.985785i	$-0.446265\pi$
$571$	8.02956	0.336026	0.168013	+	0.985785i	$0.446265\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	28.5991	1.19267
$576$	0	0
$577$	−21.7808	−0.906748	−0.453374	−	0.891320i	$-0.649780\pi$
$577$	−21.7808	−0.906748	−0.453374	+	0.891320i	$0.649780\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14.9052	−0.618372
$582$	0	0
$583$	29.8272i	1.23532i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.2913i	0.548592i	0.961645	+	0.274296i	$0.0884449\pi$
$587$	13.2913i	0.548592i	−0.961645	+	0.274296i	$0.911555\pi$
$588$	0	0
$589$	− 32.6582i	− 1.34566i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 0.746075i	− 0.0306376i	−0.999883	−	0.0153188i	$-0.995124\pi$
$593$	− 0.746075i	− 0.0306376i	0.999883	−	0.0153188i	$-0.00487632\pi$
$594$	0	0
$595$	2.25953	0.0926317
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−42.1595	−1.72259	−0.861296	−	0.508104i	$-0.830346\pi$
$599$	−42.1595	−1.72259	−0.861296	+	0.508104i	$0.830346\pi$
$600$	0	0
$601$	−12.6371	−0.515476	−0.257738	−	0.966215i	$-0.582977\pi$
$601$	−12.6371	−0.515476	−0.257738	+	0.966215i	$0.582977\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.42147	0.179758
$606$	0	0
$607$	− 44.5950i	− 1.81006i	−0.425352	−	0.905028i	$-0.639850\pi$
$607$	− 44.5950i	− 1.81006i	0.425352	−	0.905028i	$-0.360150\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 9.85078i	− 0.398520i
$612$	0	0
$613$	− 16.6859i	− 0.673935i	−0.941516	−	0.336968i	$-0.890599\pi$
$613$	− 16.6859i	− 0.673935i	0.941516	−	0.336968i	$-0.109401\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.1225i	− 1.21269i	−0.795203	−	0.606343i	$-0.792636\pi$
$617$	− 30.1225i	− 1.21269i	0.795203	−	0.606343i	$-0.207364\pi$
$618$	0	0
$619$	−32.5002	−1.30629	−0.653147	−	0.757231i	$-0.726552\pi$
$619$	−32.5002	−1.30629	−0.653147	+	0.757231i	$0.726552\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.6889	−0.508368
$624$	0	0
$625$	7.48359	0.299343
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.5655	−0.660508
$630$	0	0
$631$	− 18.8734i	− 0.751340i	−0.926754	−	0.375670i	$-0.877413\pi$
$631$	− 18.8734i	− 0.751340i	0.926754	−	0.375670i	$-0.122587\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 5.23878i	− 0.207895i
$636$	0	0
$637$	− 5.08658i	− 0.201537i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13.1417i	0.519067i	0.965734	+	0.259534i	$0.0835688\pi$
$641$	13.1417i	0.519067i	−0.965734	+	0.259534i	$0.916431\pi$
$642$	0	0
$643$	−11.3121	−0.446105	−0.223053	−	0.974806i	$-0.571602\pi$
$643$	−11.3121	−0.446105	−0.223053	+	0.974806i	$0.571602\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.1038	0.397219	0.198610	−	0.980079i	$-0.436357\pi$
$647$	10.1038	0.397219	0.198610	+	0.980079i	$0.436357\pi$
$648$	0	0
$649$	42.4475	1.66621
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.2661	−0.714807	−0.357404	−	0.933950i	$-0.616338\pi$
$653$	−18.2661	−0.714807	−0.357404	+	0.933950i	$0.616338\pi$
$654$	0	0
$655$	− 9.45534i	− 0.369451i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 37.9781i	− 1.47942i	−0.672927	−	0.739709i	$-0.734963\pi$
$659$	− 37.9781i	− 1.47942i	0.672927	−	0.739709i	$-0.265037\pi$
$660$	0	0
$661$	8.98899i	0.349631i	0.984601	+	0.174816i	$0.0559329\pi$
$661$	8.98899i	0.349631i	−0.984601	+	0.174816i	$0.944067\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 9.02861i	− 0.350114i
$666$	0	0
$667$	−36.3442	−1.40725
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−44.2722	−1.70911
$672$	0	0
$673$	21.4937	0.828520	0.414260	−	0.910159i	$-0.364040\pi$
$673$	21.4937	0.828520	0.414260	+	0.910159i	$0.364040\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.5920	0.522383	0.261192	−	0.965287i	$-0.415885\pi$
$677$	13.5920	0.522383	0.261192	+	0.965287i	$0.415885\pi$
$678$	0	0
$679$	− 4.17315i	− 0.160151i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 13.0200i	− 0.498196i	−0.968478	−	0.249098i	$-0.919866\pi$
$683$	− 13.0200i	− 0.498196i	0.968478	−	0.249098i	$-0.0801342\pi$
$684$	0	0
$685$	15.7451i	0.601590i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	39.2868i	1.49671i
$690$	0	0
$691$	8.07556	0.307209	0.153604	−	0.988132i	$-0.450912\pi$
$691$	8.07556	0.307209	0.153604	+	0.988132i	$0.450912\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.7977	−0.599240
$696$	0	0
$697$	−16.8250	−0.637292
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.91978	0.374665	0.187333	−	0.982297i	$-0.440016\pi$
$701$	9.91978	0.374665	0.187333	+	0.982297i	$0.440016\pi$
$702$	0	0
$703$	66.1921i	2.49648i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.00000i	− 0.0752177i
$708$	0	0
$709$	33.4709i	1.25702i	0.777800	+	0.628512i	$0.216336\pi$
$709$	33.4709i	1.25702i	−0.777800	+	0.628512i	$0.783664\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 31.3867i	− 1.17544i
$714$	0	0
$715$	22.1924	0.829950
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−29.1075	−1.08553	−0.542764	−	0.839886i	$-0.682622\pi$
$719$	−29.1075	−1.08553	−0.542764	+	0.839886i	$0.682622\pi$
$720$	0	0
$721$	−5.37753	−0.200270
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−17.6204	−0.654406
$726$	0	0
$727$	− 2.39230i	− 0.0887257i	−0.999015	−	0.0443628i	$-0.985874\pi$
$727$	− 2.39230i	− 0.0887257i	0.999015	−	0.0443628i	$-0.0141258\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 20.0296i	− 0.740820i
$732$	0	0
$733$	− 20.3315i	− 0.750962i	−0.926830	−	0.375481i	$-0.877478\pi$
$733$	− 20.3315i	− 0.750962i	0.926830	−	0.375481i	$-0.122522\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 19.6434i	− 0.723574i
$738$	0	0
$739$	−36.5823	−1.34570	−0.672851	−	0.739778i	$-0.734930\pi$
$739$	−36.5823	−1.34570	−0.672851	+	0.739778i	$0.734930\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.9864	−0.513110	−0.256555	−	0.966530i	$-0.582587\pi$
$743$	−13.9864	−0.513110	−0.256555	+	0.966530i	$0.582587\pi$
$744$	0	0
$745$	24.5950	0.901091
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.8418	0.469230
$750$	0	0
$751$	47.5657i	1.73570i	0.496830	+	0.867848i	$0.334497\pi$
$751$	47.5657i	1.73570i	−0.496830	+	0.867848i	$0.665503\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.1287i	0.659769i
$756$	0	0
$757$	41.3844i	1.50414i	0.659082	+	0.752071i	$0.270945\pi$
$757$	41.3844i	1.50414i	−0.659082	+	0.752071i	$0.729055\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 19.8335i	− 0.718966i	−0.933152	−	0.359483i	$-0.882953\pi$
$761$	− 19.8335i	− 0.718966i	0.933152	−	0.359483i	$-0.117047\pi$
$762$	0	0
$763$	13.0782	0.473461
$764$	0	0
$765$	0	0
$766$	0	0
$767$	55.9095	2.01878
$768$	0	0
$769$	−20.2739	−0.731096	−0.365548	−	0.930792i	$-0.619118\pi$
$769$	−20.2739	−0.731096	−0.365548	+	0.930792i	$0.619118\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−0.271102	−0.00975088	−0.00487544	−	0.999988i	$-0.501552\pi$
$773$	−0.271102	−0.00975088	−0.00487544	+	0.999988i	$0.501552\pi$
$774$	0	0
$775$	− 15.2169i	− 0.546607i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	67.2291i	2.40873i
$780$	0	0
$781$	52.8313i	1.89045i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 6.20887i	− 0.221604i
$786$	0	0
$787$	50.0760	1.78501	0.892507	−	0.451033i	$-0.148944\pi$
$787$	50.0760	1.78501	0.892507	+	0.451033i	$0.148944\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.05496	0.250845
$792$	0	0
$793$	−58.3130	−2.07076
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.1762	−0.962629	−0.481314	−	0.876548i	$-0.659840\pi$
$797$	−27.1762	−0.962629	−0.481314	+	0.876548i	$0.659840\pi$
$798$	0	0
$799$	3.87325i	0.137026i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 10.1838i	− 0.359379i
$804$	0	0
$805$	− 8.67709i	− 0.305827i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	33.5566i	1.17979i	0.807480	+	0.589894i	$0.200831\pi$
$809$	33.5566i	1.17979i	−0.807480	+	0.589894i	$0.799169\pi$
$810$	0	0
$811$	3.54206	0.124379	0.0621893	−	0.998064i	$-0.480192\pi$
$811$	3.54206	0.124379	0.0621893	+	0.998064i	$0.480192\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.83301	0.0642077
$816$	0	0
$817$	−80.0339	−2.80003
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−31.0847	−1.08486	−0.542431	−	0.840100i	$-0.682496\pi$
$821$	−31.0847	−1.08486	−0.542431	+	0.840100i	$0.682496\pi$
$822$	0	0
$823$	− 36.2323i	− 1.26298i	−0.775385	−	0.631489i	$-0.782444\pi$
$823$	− 36.2323i	− 1.26298i	0.775385	−	0.631489i	$-0.217556\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	53.4991i	1.86034i	0.367123	+	0.930172i	$0.380343\pi$
$827$	53.4991i	1.86034i	−0.367123	+	0.930172i	$0.619657\pi$
$828$	0	0
$829$	− 46.2003i	− 1.60460i	−0.596920	−	0.802301i	$-0.703609\pi$
$829$	− 46.2003i	− 1.60460i	0.596920	−	0.802301i	$-0.296391\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.00000i	0.0692959i
$834$	0	0
$835$	−9.83918	−0.340499
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.0211	1.00192	0.500960	−	0.865470i	$-0.332980\pi$
$839$	29.0211	1.00192	0.500960	+	0.865470i	$0.332980\pi$
$840$	0	0
$841$	−6.60770	−0.227852
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.5437	0.500320
$846$	0	0
$847$	3.91362i	0.134474i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	63.6150i	2.18069i
$852$	0	0
$853$	− 50.8777i	− 1.74202i	−0.491266	−	0.871009i	$-0.663466\pi$
$853$	− 50.8777i	− 1.74202i	0.491266	−	0.871009i	$-0.336534\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 22.3553i	− 0.763642i	−0.924236	−	0.381821i	$-0.875297\pi$
$857$	− 22.3553i	− 0.763642i	0.924236	−	0.381821i	$-0.124703\pi$
$858$	0	0
$859$	−44.4134	−1.51537	−0.757684	−	0.652622i	$-0.773669\pi$
$859$	−44.4134	−1.51537	−0.757684	+	0.652622i	$0.773669\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8.80145	−0.299605	−0.149802	−	0.988716i	$-0.547864\pi$
$863$	−8.80145	−0.299605	−0.149802	+	0.988716i	$0.547864\pi$
$864$	0	0
$865$	−1.73860	−0.0591143
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27.7006	0.939679
$870$	0	0
$871$	− 25.8732i	− 0.876681i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 9.85565i	− 0.333182i
$876$	0	0
$877$	− 25.5280i	− 0.862020i	−0.902347	−	0.431010i	$-0.858157\pi$
$877$	− 25.5280i	− 0.862020i	0.902347	−	0.431010i	$-0.141843\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41.0812i	1.38406i	0.721869	+	0.692030i	$0.243283\pi$
$881$	41.0812i	1.38406i	−0.721869	+	0.692030i	$0.756717\pi$
$882$	0	0
$883$	−2.80145	−0.0942763	−0.0471381	−	0.998888i	$-0.515010\pi$
$883$	−2.80145	−0.0942763	−0.0471381	+	0.998888i	$0.515010\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.4366	−1.52561	−0.762806	−	0.646628i	$-0.776179\pi$
$887$	−45.4366	−1.52561	−0.762806	+	0.646628i	$0.776179\pi$
$888$	0	0
$889$	4.63706	0.155522
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.4767	0.517907
$894$	0	0
$895$	30.2083i	1.00975i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.3379i	0.644954i
$900$	0	0
$901$	− 15.4473i	− 0.514623i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.50380i	0.116470i
$906$	0	0
$907$	−46.5318	−1.54506	−0.772531	−	0.634977i	$-0.781010\pi$
$907$	−46.5318	−1.54506	−0.772531	+	0.634977i	$0.781010\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.9174	0.825550	0.412775	−	0.910833i	$-0.364560\pi$
$911$	24.9174	0.825550	0.412775	+	0.910833i	$0.364560\pi$
$912$	0	0
$913$	57.5611	1.90500
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.36930	0.276379
$918$	0	0
$919$	− 8.77228i	− 0.289371i	−0.989478	−	0.144685i	$-0.953783\pi$
$919$	− 8.77228i	− 0.289371i	0.989478	−	0.144685i	$-0.0462170\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	69.5866i	2.29047i
$924$	0	0
$925$	30.8418i	1.01407i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.96002i	0.129924i	0.997888	+	0.0649620i	$0.0206926\pi$
$929$	3.96002i	0.129924i	−0.997888	+	0.0649620i	$0.979307\pi$
$930$	0	0
$931$	7.99158	0.261913
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.72589	−0.285367
$936$	0	0
$937$	−11.6077	−0.379207	−0.189603	−	0.981861i	$-0.560720\pi$
$937$	−11.6077	−0.379207	−0.189603	+	0.981861i	$0.560720\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.05758	0.0670751	0.0335376	−	0.999437i	$-0.489323\pi$
$941$	2.05758	0.0670751	0.0335376	+	0.999437i	$0.489323\pi$
$942$	0	0
$943$	64.6117i	2.10404i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 53.3723i	− 1.73437i	−0.497989	−	0.867184i	$-0.665928\pi$
$947$	− 53.3723i	− 1.73437i	0.497989	−	0.867184i	$-0.334072\pi$
$948$	0	0
$949$	− 13.4136i	− 0.435423i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	45.9433i	1.48825i	0.668040	+	0.744125i	$0.267133\pi$
$953$	45.9433i	1.48825i	−0.668040	+	0.744125i	$0.732867\pi$
$954$	0	0
$955$	−15.6057	−0.504988
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.9366	−0.450037
$960$	0	0
$961$	14.2999	0.461287
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.97044	−0.256578
$966$	0	0
$967$	− 1.43455i	− 0.0461319i	−0.999734	−	0.0230659i	$-0.992657\pi$
$967$	− 1.43455i	− 0.0461319i	0.999734	−	0.0230659i	$-0.00734276\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.6030i	0.436542i	0.975888	+	0.218271i	$0.0700417\pi$
$971$	13.6030i	0.436542i	−0.975888	+	0.218271i	$0.929958\pi$
$972$	0	0
$973$	− 13.9832i	− 0.448280i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 10.3860i	− 0.332278i	−0.986102	−	0.166139i	$-0.946870\pi$
$977$	− 10.3860i	− 0.332278i	0.986102	−	0.166139i	$-0.0531300\pi$
$978$	0	0
$979$	49.0020	1.56611
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−20.0062	−0.638098	−0.319049	−	0.947738i	$-0.603363\pi$
$983$	−20.0062	−0.638098	−0.319049	+	0.947738i	$0.603363\pi$
$984$	0	0
$985$	−0.409820	−0.0130580
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−76.9179	−2.44585
$990$	0	0
$991$	− 45.7220i	− 1.45241i	−0.687480	−	0.726203i	$-0.741283\pi$
$991$	− 45.7220i	− 1.45241i	0.687480	−	0.726203i	$-0.258717\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 8.69952i	− 0.275793i
$996$	0	0
$997$	− 41.6964i	− 1.32054i	−0.751030	−	0.660269i	$-0.770442\pi$
$997$	− 41.6964i	− 1.32054i	0.751030	−	0.660269i	$-0.229558\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.b.5615.3		8
3.2	odd	2		6048.2.j.a.5615.5		8
4.3	odd	2		1512.2.j.b.323.5	yes	8
8.3	odd	2		6048.2.j.a.5615.6		8
8.5	even	2		1512.2.j.a.323.2	✓	8
12.11	even	2		1512.2.j.a.323.4	yes	8
24.5	odd	2		1512.2.j.b.323.7	yes	8
24.11	even	2	inner	6048.2.j.b.5615.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.a.323.2	✓	8	8.5	even	2
1512.2.j.a.323.4	yes	8	12.11	even	2
1512.2.j.b.323.5	yes	8	4.3	odd	2
1512.2.j.b.323.7	yes	8	24.5	odd	2
6048.2.j.a.5615.5		8	3.2	odd	2
6048.2.j.a.5615.6		8	8.3	odd	2
6048.2.j.b.5615.3		8	1.1	even	1	trivial
6048.2.j.b.5615.4		8	24.11	even	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.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.12976 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-1.12976 q^{5} -1.00000i q^{7} +3.86182i q^{11} +5.08658i q^{13} -2.00000i q^{17} -7.99158 q^{19} -7.68044 q^{23} -3.72363 q^{25} +4.73205 q^{29} +4.08658i q^{31} +1.12976i q^{35} -8.28273i q^{37} -8.41249i q^{41} +10.0148 q^{43} -1.93662 q^{47} -1.00000 q^{49} +7.72363 q^{53} -4.36294i q^{55} -10.9916i q^{59} +11.4641i q^{61} -5.74663i q^{65} -5.08658 q^{67} +13.6804 q^{71} -2.63706 q^{73} +3.86182 q^{77} -7.17295i q^{79} -14.9052i q^{83} +2.25953i q^{85} -12.6889i q^{89} +5.08658 q^{91} +9.02861 q^{95} +4.17315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} - 16 q^{19} + 16 q^{23} + 32 q^{25} + 24 q^{29} - 8 q^{43} - 8 q^{47} - 8 q^{49} - 8 q^{67} + 32 q^{71} + 8 q^{73} + 8 q^{91} + 112 q^{95} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 - 16 * q^19 + 16 * q^23 + 32 * q^25 + 24 * q^29 - 8 * q^43 - 8 * q^47 - 8 * q^49 - 8 * q^67 + 32 * q^71 + 8 * q^73 + 8 * q^91 + 112 * q^95 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more