LMFDB - Embedded newform 6048.2.c.f.3025.21

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:	$24$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.21
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.f.3025.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.04340i q^{5} -1.00000 q^{7} +O(q^{10})$	$q+3.04340i q^{5} -1.00000 q^{7} -0.128573i q^{11} -6.30135i q^{13} +5.32168 q^{17} -6.68215i q^{19} -5.18212 q^{23} -4.26229 q^{25} +9.96821i q^{29} -3.27122 q^{31} -3.04340i q^{35} +0.796970i q^{37} -2.96782 q^{41} +6.99100i q^{43} -4.76595 q^{47} +1.00000 q^{49} -1.14264i q^{53} +0.391299 q^{55} +11.0999i q^{59} +14.6662i q^{61} +19.1775 q^{65} -1.05725i q^{67} -1.10582 q^{71} -12.1019 q^{73} +0.128573i q^{77} -2.62426 q^{79} -6.46403i q^{83} +16.1960i q^{85} -2.23283 q^{89} +6.30135i q^{91} +20.3365 q^{95} -7.88097 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{7}+O(q^{10}) $ 24 * q - 24 * q^7	$ 24 q - 24 q^{7} - 24 q^{25} - 8 q^{31} + 24 q^{49} - 16 q^{55} - 8 q^{79}+O(q^{100}) $ 24 * q - 24 * q^7 - 24 * q^25 - 8 * q^31 + 24 * q^49 - 16 * q^55 - 8 * q^79

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
$5$	3.04340i	1.36105i	0.732725	+	0.680525i	$0.238248\pi$
$5$	3.04340i	1.36105i	−0.732725	+	0.680525i	$0.761752\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 0.128573i	− 0.0387662i	−0.999812	−	0.0193831i	$-0.993830\pi$
$11$	− 0.128573i	− 0.0387662i	0.999812	−	0.0193831i	$-0.00617022\pi$
$12$	0	0
$13$	− 6.30135i	− 1.74768i	−0.486214	−	0.873840i	$-0.661622\pi$
$13$	− 6.30135i	− 1.74768i	0.486214	−	0.873840i	$-0.338378\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.32168	1.29070	0.645348	−	0.763889i	$-0.276712\pi$
$17$	5.32168	1.29070	0.645348	+	0.763889i	$0.276712\pi$
$18$	0	0
$19$	− 6.68215i	− 1.53299i	−0.642250	−	0.766495i	$-0.721999\pi$
$19$	− 6.68215i	− 1.53299i	0.642250	−	0.766495i	$-0.278001\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.18212	−1.08055	−0.540273	−	0.841490i	$-0.681679\pi$
$23$	−5.18212	−1.08055	−0.540273	+	0.841490i	$0.681679\pi$
$24$	0	0
$25$	−4.26229	−0.852459
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.96821i	1.85105i	0.378686	+	0.925525i	$0.376376\pi$
$29$	9.96821i	1.85105i	−0.378686	+	0.925525i	$0.623624\pi$
$30$	0	0
$31$	−3.27122	−0.587528	−0.293764	−	0.955878i	$-0.594908\pi$
$31$	−3.27122	−0.587528	−0.293764	+	0.955878i	$0.594908\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.04340i	− 0.514429i
$36$	0	0
$37$	0.796970i	0.131021i	0.997852	+	0.0655106i	$0.0208676\pi$
$37$	0.796970i	0.131021i	−0.997852	+	0.0655106i	$0.979132\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.96782	−0.463496	−0.231748	−	0.972776i	$-0.574444\pi$
$41$	−2.96782	−0.463496	−0.231748	+	0.972776i	$0.574444\pi$
$42$	0	0
$43$	6.99100i	1.06612i	0.846078	+	0.533059i	$0.178958\pi$
$43$	6.99100i	1.06612i	−0.846078	+	0.533059i	$0.821042\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.76595	−0.695186	−0.347593	−	0.937646i	$-0.613001\pi$
$47$	−4.76595	−0.695186	−0.347593	+	0.937646i	$0.613001\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.14264i	− 0.156954i	−0.996916	−	0.0784772i	$-0.974994\pi$
$53$	− 1.14264i	− 0.156954i	0.996916	−	0.0784772i	$-0.0250058\pi$
$54$	0	0
$55$	0.391299	0.0527627
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.0999i	1.44508i	0.691329	+	0.722540i	$0.257025\pi$
$59$	11.0999i	1.44508i	−0.691329	+	0.722540i	$0.742975\pi$
$60$	0	0
$61$	14.6662i	1.87782i	0.344165	+	0.938909i	$0.388162\pi$
$61$	14.6662i	1.87782i	−0.344165	+	0.938909i	$0.611838\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	19.1775	2.37868
$66$	0	0
$67$	− 1.05725i	− 0.129163i	−0.997912	−	0.0645816i	$-0.979429\pi$
$67$	− 1.05725i	− 0.129163i	0.997912	−	0.0645816i	$-0.0205713\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.10582	−0.131237	−0.0656185	−	0.997845i	$-0.520902\pi$
$71$	−1.10582	−0.131237	−0.0656185	+	0.997845i	$0.520902\pi$
$72$	0	0
$73$	−12.1019	−1.41642	−0.708212	−	0.706000i	$-0.750498\pi$
$73$	−12.1019	−1.41642	−0.708212	+	0.706000i	$0.750498\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.128573i	0.0146522i
$78$	0	0
$79$	−2.62426	−0.295253	−0.147626	−	0.989043i	$-0.547163\pi$
$79$	−2.62426	−0.295253	−0.147626	+	0.989043i	$0.547163\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.46403i	− 0.709520i	−0.934957	−	0.354760i	$-0.884563\pi$
$83$	− 6.46403i	− 0.709520i	0.934957	−	0.354760i	$-0.115437\pi$
$84$	0	0
$85$	16.1960i	1.75670i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.23283	−0.236680	−0.118340	−	0.992973i	$-0.537757\pi$
$89$	−2.23283	−0.236680	−0.118340	+	0.992973i	$0.537757\pi$
$90$	0	0
$91$	6.30135i	0.660561i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	20.3365	2.08648
$96$	0	0
$97$	−7.88097	−0.800192	−0.400096	−	0.916473i	$-0.631023\pi$
$97$	−7.88097	−0.800192	−0.400096	+	0.916473i	$0.631023\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.9693i	1.68851i	0.535945	+	0.844253i	$0.319955\pi$
$101$	16.9693i	1.68851i	−0.535945	+	0.844253i	$0.680045\pi$
$102$	0	0
$103$	10.7049	1.05479	0.527393	−	0.849621i	$-0.323170\pi$
$103$	10.7049	1.05479	0.527393	+	0.849621i	$0.323170\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.31462i	− 0.127089i	−0.997979	−	0.0635445i	$-0.979760\pi$
$107$	− 1.31462i	− 0.127089i	0.997979	−	0.0635445i	$-0.0202405\pi$
$108$	0	0
$109$	10.2677i	0.983465i	0.870746	+	0.491732i	$0.163636\pi$
$109$	10.2677i	0.983465i	−0.870746	+	0.491732i	$0.836364\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.1796	−1.23983	−0.619916	−	0.784668i	$-0.712834\pi$
$113$	−13.1796	−1.23983	−0.619916	+	0.784668i	$0.712834\pi$
$114$	0	0
$115$	− 15.7713i	− 1.47068i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.32168	−0.487837
$120$	0	0
$121$	10.9835	0.998497
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.24514i	0.200811i
$126$	0	0
$127$	0.840201	0.0745558	0.0372779	−	0.999305i	$-0.488131\pi$
$127$	0.840201	0.0745558	0.0372779	+	0.999305i	$0.488131\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 7.39979i	− 0.646523i	−0.946310	−	0.323261i	$-0.895221\pi$
$131$	− 7.39979i	− 0.646523i	0.946310	−	0.323261i	$-0.104779\pi$
$132$	0	0
$133$	6.68215i	0.579416i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.61682	0.479877	0.239939	−	0.970788i	$-0.422873\pi$
$137$	5.61682	0.479877	0.239939	+	0.970788i	$0.422873\pi$
$138$	0	0
$139$	− 5.19553i	− 0.440679i	−0.975423	−	0.220339i	$-0.929284\pi$
$139$	− 5.19553i	− 0.440679i	0.975423	−	0.220339i	$-0.0707165\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.810183	−0.0677509
$144$	0	0
$145$	−30.3373	−2.51937
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.6207i	0.952009i	0.879443	+	0.476004i	$0.157915\pi$
$149$	11.6207i	0.952009i	−0.879443	+	0.476004i	$0.842085\pi$
$150$	0	0
$151$	−21.5154	−1.75090	−0.875448	−	0.483313i	$-0.839433\pi$
$151$	−21.5154	−1.75090	−0.875448	+	0.483313i	$0.839433\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 9.95563i	− 0.799655i
$156$	0	0
$157$	− 18.3001i	− 1.46050i	−0.683178	−	0.730252i	$-0.739403\pi$
$157$	− 18.3001i	− 1.46050i	0.683178	−	0.730252i	$-0.260597\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.18212	0.408408
$162$	0	0
$163$	18.8550i	1.47684i	0.674344	+	0.738418i	$0.264427\pi$
$163$	18.8550i	1.47684i	−0.674344	+	0.738418i	$0.735573\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.8283	−1.61174	−0.805872	−	0.592089i	$-0.798303\pi$
$167$	−20.8283	−1.61174	−0.805872	+	0.592089i	$0.798303\pi$
$168$	0	0
$169$	−26.7070	−2.05438
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.49174i	0.417529i	0.977966	+	0.208765i	$0.0669443\pi$
$173$	5.49174i	0.417529i	−0.977966	+	0.208765i	$0.933056\pi$
$174$	0	0
$175$	4.26229	0.322199
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 7.62941i	− 0.570249i	−0.958491	−	0.285124i	$-0.907965\pi$
$179$	− 7.62941i	− 0.570249i	0.958491	−	0.285124i	$-0.0920349\pi$
$180$	0	0
$181$	− 10.0176i	− 0.744600i	−0.928113	−	0.372300i	$-0.878569\pi$
$181$	− 10.0176i	− 0.744600i	0.928113	−	0.372300i	$-0.121431\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.42550	−0.178326
$186$	0	0
$187$	− 0.684223i	− 0.0500354i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.25601	0.452669	0.226335	−	0.974050i	$-0.427326\pi$
$191$	6.25601	0.452669	0.226335	+	0.974050i	$0.427326\pi$
$192$	0	0
$193$	2.46446	0.177396	0.0886980	−	0.996059i	$-0.471729\pi$
$193$	2.46446	0.177396	0.0886980	+	0.996059i	$0.471729\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 23.8105i	− 1.69643i	−0.529654	−	0.848214i	$-0.677678\pi$
$197$	− 23.8105i	− 1.69643i	0.529654	−	0.848214i	$-0.322322\pi$
$198$	0	0
$199$	−24.1296	−1.71050	−0.855252	−	0.518212i	$-0.826598\pi$
$199$	−24.1296	−1.71050	−0.855252	+	0.518212i	$0.826598\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 9.96821i	− 0.699631i
$204$	0	0
$205$	− 9.03227i	− 0.630841i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.859144	−0.0594282
$210$	0	0
$211$	14.9668i	1.03036i	0.857083	+	0.515179i	$0.172274\pi$
$211$	14.9668i	1.03036i	−0.857083	+	0.515179i	$0.827726\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−21.2764	−1.45104
$216$	0	0
$217$	3.27122	0.222065
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 33.5337i	− 2.25572i
$222$	0	0
$223$	−11.6108	−0.777518	−0.388759	−	0.921340i	$-0.627096\pi$
$223$	−11.6108	−0.777518	−0.388759	+	0.921340i	$0.627096\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 0.939445i	− 0.0623531i	−0.999514	−	0.0311766i	$-0.990075\pi$
$227$	− 0.939445i	− 0.0623531i	0.999514	−	0.0311766i	$-0.00992542\pi$
$228$	0	0
$229$	− 0.0137124i	0 0.000906143i	−1.00000		0.000453071i	$-0.999856\pi$
$229$	− 0.0137124i	0 0.000906143i	1.00000		0.000453071i	$-0.000144217\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.2195	−1.19360	−0.596800	−	0.802390i	$-0.703561\pi$
$233$	−18.2195	−1.19360	−0.596800	+	0.802390i	$0.703561\pi$
$234$	0	0
$235$	− 14.5047i	− 0.946183i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.37153	−0.0887169	−0.0443585	−	0.999016i	$-0.514124\pi$
$239$	−1.37153	−0.0887169	−0.0443585	+	0.999016i	$0.514124\pi$
$240$	0	0
$241$	−10.5054	−0.676711	−0.338355	−	0.941018i	$-0.609871\pi$
$241$	−10.5054	−0.676711	−0.338355	+	0.941018i	$0.609871\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.04340i	0.194436i
$246$	0	0
$247$	−42.1066	−2.67918
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 20.0556i	− 1.26590i	−0.774192	−	0.632950i	$-0.781844\pi$
$251$	− 20.0556i	− 1.26590i	0.774192	−	0.632950i	$-0.218156\pi$
$252$	0	0
$253$	0.666280i	0.0418887i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.6631	1.53844	0.769220	−	0.638984i	$-0.220645\pi$
$257$	24.6631	1.53844	0.769220	+	0.638984i	$0.220645\pi$
$258$	0	0
$259$	− 0.796970i	− 0.0495213i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.0542	−0.866616	−0.433308	−	0.901246i	$-0.642654\pi$
$263$	−14.0542	−0.866616	−0.433308	+	0.901246i	$0.642654\pi$
$264$	0	0
$265$	3.47753	0.213623
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.1276i	0.800404i	0.916427	+	0.400202i	$0.131060\pi$
$269$	13.1276i	0.800404i	−0.916427	+	0.400202i	$0.868940\pi$
$270$	0	0
$271$	15.4626	0.939285	0.469642	−	0.882857i	$-0.344383\pi$
$271$	15.4626	0.939285	0.469642	+	0.882857i	$0.344383\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.548015i	0.0330466i
$276$	0	0
$277$	− 11.1880i	− 0.672223i	−0.941822	−	0.336112i	$-0.890888\pi$
$277$	− 11.1880i	− 0.672223i	0.941822	−	0.336112i	$-0.109112\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−0.179766	−0.0107240	−0.00536198	−	0.999986i	$-0.501707\pi$
$281$	−0.179766	−0.0107240	−0.00536198	+	0.999986i	$0.501707\pi$
$282$	0	0
$283$	29.0480i	1.72673i	0.504583	+	0.863363i	$0.331646\pi$
$283$	29.0480i	1.72673i	−0.504583	+	0.863363i	$0.668354\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.96782	0.175185
$288$	0	0
$289$	11.3202	0.665897
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.42498i	0.258510i	0.991611	+	0.129255i	$0.0412586\pi$
$293$	4.42498i	0.258510i	−0.991611	+	0.129255i	$0.958741\pi$
$294$	0	0
$295$	−33.7813	−1.96683
$296$	0	0
$297$	0	0
$298$	0	0
$299$	32.6543i	1.88845i
$300$	0	0
$301$	− 6.99100i	− 0.402955i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−44.6352	−2.55581
$306$	0	0
$307$	− 8.36831i	− 0.477605i	−0.971068	−	0.238802i	$-0.923245\pi$
$307$	− 8.36831i	− 0.477605i	0.971068	−	0.238802i	$-0.0767548\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.1701	1.59738	0.798690	−	0.601743i	$-0.205527\pi$
$311$	28.1701	1.59738	0.798690	+	0.601743i	$0.205527\pi$
$312$	0	0
$313$	14.0081	0.791784	0.395892	−	0.918297i	$-0.370435\pi$
$313$	14.0081	0.791784	0.395892	+	0.918297i	$0.370435\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.4390i	0.810974i	0.914101	+	0.405487i	$0.132898\pi$
$317$	14.4390i	0.810974i	−0.914101	+	0.405487i	$0.867102\pi$
$318$	0	0
$319$	1.28164	0.0717582
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 35.5603i	− 1.97863i
$324$	0	0
$325$	26.8582i	1.48982i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.76595	0.262756
$330$	0	0
$331$	32.5738i	1.79042i	0.445648	+	0.895208i	$0.352973\pi$
$331$	32.5738i	1.79042i	−0.445648	+	0.895208i	$0.647027\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.21762	0.175798
$336$	0	0
$337$	−2.17181	−0.118306	−0.0591529	−	0.998249i	$-0.518840\pi$
$337$	−2.17181	−0.118306	−0.0591529	+	0.998249i	$0.518840\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.420590i	0.0227762i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.79477i	0.150031i	0.997182	+	0.0750157i	$0.0239007\pi$
$347$	2.79477i	0.150031i	−0.997182	+	0.0750157i	$0.976099\pi$
$348$	0	0
$349$	− 3.19085i	− 0.170802i	−0.996347	−	0.0854011i	$-0.972783\pi$
$349$	− 3.19085i	− 0.170802i	0.996347	−	0.0854011i	$-0.0272172\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.10245	0.431250	0.215625	−	0.976476i	$-0.430821\pi$
$353$	8.10245	0.431250	0.215625	+	0.976476i	$0.430821\pi$
$354$	0	0
$355$	− 3.36546i	− 0.178620i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.2635	0.911134	0.455567	−	0.890202i	$-0.349437\pi$
$359$	17.2635	0.911134	0.455567	+	0.890202i	$0.349437\pi$
$360$	0	0
$361$	−25.6511	−1.35006
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 36.8310i	− 1.92782i
$366$	0	0
$367$	13.5508	0.707348	0.353674	−	0.935369i	$-0.384932\pi$
$367$	13.5508	0.707348	0.353674	+	0.935369i	$0.384932\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.14264i	0.0593232i
$372$	0	0
$373$	− 16.3399i	− 0.846048i	−0.906118	−	0.423024i	$-0.860969\pi$
$373$	− 16.3399i	− 0.846048i	0.906118	−	0.423024i	$-0.139031\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	62.8132	3.23504
$378$	0	0
$379$	− 2.37754i	− 0.122126i	−0.998134	−	0.0610630i	$-0.980551\pi$
$379$	− 2.37754i	− 0.122126i	0.998134	−	0.0610630i	$-0.0194491\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	17.6210	0.900389	0.450194	−	0.892931i	$-0.351355\pi$
$383$	17.6210	0.900389	0.450194	+	0.892931i	$0.351355\pi$
$384$	0	0
$385$	−0.391299	−0.0199424
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 12.6831i	− 0.643058i	−0.946900	−	0.321529i	$-0.895803\pi$
$389$	− 12.6831i	− 0.643058i	0.946900	−	0.321529i	$-0.104197\pi$
$390$	0	0
$391$	−27.5776	−1.39466
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 7.98668i	− 0.401854i
$396$	0	0
$397$	34.3856i	1.72576i	0.505405	+	0.862882i	$0.331343\pi$
$397$	34.3856i	1.72576i	−0.505405	+	0.862882i	$0.668657\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.75947	−0.187739	−0.0938696	−	0.995585i	$-0.529924\pi$
$401$	−3.75947	−0.187739	−0.0938696	+	0.995585i	$0.529924\pi$
$402$	0	0
$403$	20.6131i	1.02681i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.102469	0.00507919
$408$	0	0
$409$	13.0996	0.647731	0.323866	−	0.946103i	$-0.395017\pi$
$409$	13.0996	0.647731	0.323866	+	0.946103i	$0.395017\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 11.0999i	− 0.546189i
$414$	0	0
$415$	19.6726	0.965692
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 24.4344i	− 1.19370i	−0.802353	−	0.596849i	$-0.796419\pi$
$419$	− 24.4344i	− 1.19370i	0.802353	−	0.596849i	$-0.203581\pi$
$420$	0	0
$421$	− 26.3907i	− 1.28620i	−0.765781	−	0.643101i	$-0.777647\pi$
$421$	− 26.3907i	− 1.28620i	0.765781	−	0.643101i	$-0.222353\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−22.6825	−1.10027
$426$	0	0
$427$	− 14.6662i	− 0.709749i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.7161	−1.57588	−0.787940	−	0.615751i	$-0.788853\pi$
$431$	−32.7161	−1.57588	−0.787940	+	0.615751i	$0.788853\pi$
$432$	0	0
$433$	12.3235	0.592228	0.296114	−	0.955153i	$-0.404309\pi$
$433$	12.3235	0.592228	0.296114	+	0.955153i	$0.404309\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.6277i	1.65647i
$438$	0	0
$439$	−4.67878	−0.223306	−0.111653	−	0.993747i	$-0.535615\pi$
$439$	−4.67878	−0.223306	−0.111653	+	0.993747i	$0.535615\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.6471i	1.69364i	0.531877	+	0.846821i	$0.321487\pi$
$443$	35.6471i	1.69364i	−0.531877	+	0.846821i	$0.678513\pi$
$444$	0	0
$445$	− 6.79540i	− 0.322133i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.02786	0.284472	0.142236	−	0.989833i	$-0.454571\pi$
$449$	6.02786	0.284472	0.142236	+	0.989833i	$0.454571\pi$
$450$	0	0
$451$	0.381581i	0.0179680i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−19.1775	−0.899057
$456$	0	0
$457$	4.88761	0.228633	0.114316	−	0.993444i	$-0.463532\pi$
$457$	4.88761	0.228633	0.114316	+	0.993444i	$0.463532\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.16306i	0.426766i	0.976969	+	0.213383i	$0.0684483\pi$
$461$	9.16306i	0.426766i	−0.976969	+	0.213383i	$0.931552\pi$
$462$	0	0
$463$	−32.8412	−1.52626	−0.763130	−	0.646245i	$-0.776338\pi$
$463$	−32.8412	−1.52626	−0.763130	+	0.646245i	$0.776338\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.40559i	0.296415i	0.988956	+	0.148208i	$0.0473504\pi$
$467$	6.40559i	0.296415i	−0.988956	+	0.148208i	$0.952650\pi$
$468$	0	0
$469$	1.05725i	0.0488191i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.898854	0.0413293
$474$	0	0
$475$	28.4813i	1.30681i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−27.3218	−1.24837	−0.624184	−	0.781278i	$-0.714568\pi$
$479$	−27.3218	−1.24837	−0.624184	+	0.781278i	$0.714568\pi$
$480$	0	0
$481$	5.02199	0.228983
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 23.9850i	− 1.08910i
$486$	0	0
$487$	−9.05264	−0.410214	−0.205107	−	0.978740i	$-0.565754\pi$
$487$	−9.05264	−0.410214	−0.205107	+	0.978740i	$0.565754\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.1183i	0.456632i	0.973587	+	0.228316i	$0.0733219\pi$
$491$	10.1183i	0.456632i	−0.973587	+	0.228316i	$0.926678\pi$
$492$	0	0
$493$	53.0476i	2.38914i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.10582	0.0496029
$498$	0	0
$499$	19.9396i	0.892621i	0.894878	+	0.446310i	$0.147262\pi$
$499$	19.9396i	0.892621i	−0.894878	+	0.446310i	$0.852738\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.3936	−1.08766	−0.543828	−	0.839197i	$-0.683026\pi$
$503$	−24.3936	−1.08766	−0.543828	+	0.839197i	$0.683026\pi$
$504$	0	0
$505$	−51.6443	−2.29814
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 7.20394i	− 0.319309i	−0.987173	−	0.159655i	$-0.948962\pi$
$509$	− 7.20394i	− 0.319309i	0.987173	−	0.159655i	$-0.0510381\pi$
$510$	0	0
$511$	12.1019	0.535358
$512$	0	0
$513$	0	0
$514$	0	0
$515$	32.5793i	1.43562i
$516$	0	0
$517$	0.612773i	0.0269497i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.8648	0.651240	0.325620	−	0.945501i	$-0.394427\pi$
$521$	14.8648	0.651240	0.325620	+	0.945501i	$0.394427\pi$
$522$	0	0
$523$	− 32.1286i	− 1.40489i	−0.711740	−	0.702443i	$-0.752093\pi$
$523$	− 32.1286i	− 1.40489i	0.711740	−	0.702443i	$-0.247907\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.4084	−0.758320
$528$	0	0
$529$	3.85438	0.167582
$530$	0	0
$531$	0	0
$532$	0	0
$533$	18.7013i	0.810042i
$534$	0	0
$535$	4.00091	0.172975
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 0.128573i	− 0.00553803i
$540$	0	0
$541$	− 30.6123i	− 1.31612i	−0.752964	−	0.658062i	$-0.771376\pi$
$541$	− 30.6123i	− 1.31612i	0.752964	−	0.658062i	$-0.228624\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−31.2487	−1.33855
$546$	0	0
$547$	− 7.64518i	− 0.326884i	−0.986553	−	0.163442i	$-0.947740\pi$
$547$	− 7.64518i	− 0.326884i	0.986553	−	0.163442i	$-0.0522597\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	66.6091	2.83764
$552$	0	0
$553$	2.62426	0.111595
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.9682i	− 1.05794i	−0.848641	−	0.528969i	$-0.822579\pi$
$557$	− 24.9682i	− 1.05794i	0.848641	−	0.528969i	$-0.177421\pi$
$558$	0	0
$559$	44.0527	1.86323
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.1836i	1.39852i	0.714865	+	0.699262i	$0.246488\pi$
$563$	33.1836i	1.39852i	−0.714865	+	0.699262i	$0.753512\pi$
$564$	0	0
$565$	− 40.1108i	− 1.68748i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.0679	−0.715523	−0.357761	−	0.933813i	$-0.616460\pi$
$569$	−17.0679	−0.715523	−0.357761	+	0.933813i	$0.616460\pi$
$570$	0	0
$571$	− 5.13058i	− 0.214708i	−0.994221	−	0.107354i	$-0.965762\pi$
$571$	− 5.13058i	− 0.214708i	0.994221	−	0.107354i	$-0.0342378\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	22.0877	0.921121
$576$	0	0
$577$	−47.5819	−1.98086	−0.990429	−	0.138021i	$-0.955926\pi$
$577$	−47.5819	−1.98086	−0.990429	+	0.138021i	$0.955926\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.46403i	0.268173i
$582$	0	0
$583$	−0.146913	−0.00608452
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.6803i	0.977390i	0.872455	+	0.488695i	$0.162527\pi$
$587$	23.6803i	0.977390i	−0.872455	+	0.488695i	$0.837473\pi$
$588$	0	0
$589$	21.8588i	0.900675i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−17.8919	−0.734734	−0.367367	−	0.930076i	$-0.619741\pi$
$593$	−17.8919	−0.734734	−0.367367	+	0.930076i	$0.619741\pi$
$594$	0	0
$595$	− 16.1960i	− 0.663971i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.0167	1.30817	0.654083	−	0.756423i	$-0.273055\pi$
$599$	32.0167	1.30817	0.654083	+	0.756423i	$0.273055\pi$
$600$	0	0
$601$	23.7187	0.967507	0.483754	−	0.875204i	$-0.339273\pi$
$601$	23.7187	0.967507	0.483754	+	0.875204i	$0.339273\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	33.4271i	1.35901i
$606$	0	0
$607$	14.4444	0.586281	0.293141	−	0.956069i	$-0.405300\pi$
$607$	14.4444	0.586281	0.293141	+	0.956069i	$0.405300\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.0319i	1.21496i
$612$	0	0
$613$	16.2817i	0.657610i	0.944398	+	0.328805i	$0.106646\pi$
$613$	16.2817i	0.657610i	−0.944398	+	0.328805i	$0.893354\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28.7753	−1.15845	−0.579225	−	0.815167i	$-0.696645\pi$
$617$	−28.7753	−1.15845	−0.579225	+	0.815167i	$0.696645\pi$
$618$	0	0
$619$	− 36.0871i	− 1.45046i	−0.688506	−	0.725231i	$-0.741733\pi$
$619$	− 36.0871i	− 1.45046i	0.688506	−	0.725231i	$-0.258267\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.23283	0.0894564
$624$	0	0
$625$	−28.1443	−1.12577
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.24122i	0.169108i
$630$	0	0
$631$	−18.0509	−0.718595	−0.359297	−	0.933223i	$-0.616984\pi$
$631$	−18.0509	−0.718595	−0.359297	+	0.933223i	$0.616984\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.55707i	0.101474i
$636$	0	0
$637$	− 6.30135i	− 0.249668i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.0253	−0.474971	−0.237486	−	0.971391i	$-0.576323\pi$
$641$	−12.0253	−0.474971	−0.237486	+	0.971391i	$0.576323\pi$
$642$	0	0
$643$	9.54316i	0.376345i	0.982136	+	0.188173i	$0.0602565\pi$
$643$	9.54316i	0.376345i	−0.982136	+	0.188173i	$0.939744\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.2201	1.26670	0.633351	−	0.773865i	$-0.281679\pi$
$647$	32.2201	1.26670	0.633351	+	0.773865i	$0.281679\pi$
$648$	0	0
$649$	1.42714	0.0560202
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.5245i	0.568388i	0.958767	+	0.284194i	$0.0917259\pi$
$653$	14.5245i	0.568388i	−0.958767	+	0.284194i	$0.908274\pi$
$654$	0	0
$655$	22.5205	0.879950
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.5726i	0.411849i	0.978568	+	0.205924i	$0.0660201\pi$
$659$	10.5726i	0.411849i	−0.978568	+	0.205924i	$0.933980\pi$
$660$	0	0
$661$	27.0965i	1.05393i	0.849887	+	0.526965i	$0.176670\pi$
$661$	27.0965i	1.05393i	−0.849887	+	0.526965i	$0.823330\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−20.3365	−0.788614
$666$	0	0
$667$	− 51.6565i	− 2.00015i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.88568	0.0727959
$672$	0	0
$673$	38.2417	1.47411	0.737055	−	0.675833i	$-0.236216\pi$
$673$	38.2417	1.47411	0.737055	+	0.675833i	$0.236216\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 14.2289i	− 0.546860i	−0.961892	−	0.273430i	$-0.911842\pi$
$677$	− 14.2289i	− 0.546860i	0.961892	−	0.273430i	$-0.0881582\pi$
$678$	0	0
$679$	7.88097	0.302444
$680$	0	0
$681$	0	0
$682$	0	0
$683$	39.0552i	1.49441i	0.664596	+	0.747203i	$0.268603\pi$
$683$	39.0552i	1.49441i	−0.664596	+	0.747203i	$0.731397\pi$
$684$	0	0
$685$	17.0942i	0.653137i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7.20020	−0.274306
$690$	0	0
$691$	20.1602i	0.766929i	0.923556	+	0.383464i	$0.125269\pi$
$691$	20.1602i	0.766929i	−0.923556	+	0.383464i	$0.874731\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.8121	0.599786
$696$	0	0
$697$	−15.7938	−0.598232
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.4760i	0.848908i	0.905449	+	0.424454i	$0.139534\pi$
$701$	22.4760i	0.848908i	−0.905449	+	0.424454i	$0.860466\pi$
$702$	0	0
$703$	5.32548	0.200854
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 16.9693i	− 0.638195i
$708$	0	0
$709$	20.2609i	0.760915i	0.924798	+	0.380458i	$0.124233\pi$
$709$	20.2609i	0.760915i	−0.924798	+	0.380458i	$0.875767\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16.9518	0.634852
$714$	0	0
$715$	− 2.46571i	− 0.0922124i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.2440	0.792269	0.396134	−	0.918193i	$-0.370351\pi$
$719$	21.2440	0.792269	0.396134	+	0.918193i	$0.370351\pi$
$720$	0	0
$721$	−10.7049	−0.398672
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 42.4874i	− 1.57794i
$726$	0	0
$727$	32.9048	1.22037	0.610185	−	0.792259i	$-0.291095\pi$
$727$	32.9048	1.22037	0.610185	+	0.792259i	$0.291095\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	37.2039i	1.37603i
$732$	0	0
$733$	− 32.0598i	− 1.18415i	−0.805881	−	0.592077i	$-0.798308\pi$
$733$	− 32.0598i	− 1.18415i	0.805881	−	0.592077i	$-0.201692\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.135933	−0.00500716
$738$	0	0
$739$	46.3826i	1.70621i	0.521738	+	0.853106i	$0.325284\pi$
$739$	46.3826i	1.70621i	−0.521738	+	0.853106i	$0.674716\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.1782	−0.446777	−0.223388	−	0.974730i	$-0.571712\pi$
$743$	−12.1782	−0.446777	−0.223388	+	0.974730i	$0.571712\pi$
$744$	0	0
$745$	−35.3666	−1.29573
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.31462i	0.0480351i
$750$	0	0
$751$	−24.4371	−0.891723	−0.445862	−	0.895102i	$-0.647103\pi$
$751$	−24.4371	−0.891723	−0.445862	+	0.895102i	$0.647103\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 65.4799i	− 2.38306i
$756$	0	0
$757$	− 7.15224i	− 0.259953i	−0.991517	−	0.129976i	$-0.958510\pi$
$757$	− 7.15224i	− 0.259953i	0.991517	−	0.129976i	$-0.0414902\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.7809	0.535806	0.267903	−	0.963446i	$-0.413669\pi$
$761$	14.7809	0.535806	0.267903	+	0.963446i	$0.413669\pi$
$762$	0	0
$763$	− 10.2677i	− 0.371715i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	69.9441	2.52554
$768$	0	0
$769$	−1.90576	−0.0687235	−0.0343618	−	0.999409i	$-0.510940\pi$
$769$	−1.90576	−0.0687235	−0.0343618	+	0.999409i	$0.510940\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 6.63435i	− 0.238621i	−0.992857	−	0.119310i	$-0.961932\pi$
$773$	− 6.63435i	− 0.238621i	0.992857	−	0.119310i	$-0.0380684\pi$
$774$	0	0
$775$	13.9429	0.500843
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.8314i	0.710534i
$780$	0	0
$781$	0.142179i	0.00508756i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	55.6944	1.98782
$786$	0	0
$787$	− 2.91096i	− 0.103765i	−0.998653	−	0.0518823i	$-0.983478\pi$
$787$	− 2.91096i	− 0.103765i	0.998653	−	0.0518823i	$-0.0165221\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13.1796	0.468613
$792$	0	0
$793$	92.4170	3.28182
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6.96910i	− 0.246858i	−0.992353	−	0.123429i	$-0.960611\pi$
$797$	− 6.96910i	− 0.246858i	0.992353	−	0.123429i	$-0.0393891\pi$
$798$	0	0
$799$	−25.3629	−0.897274
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.55598i	0.0549094i
$804$	0	0
$805$	15.7713i	0.555864i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21.3012	−0.748909	−0.374454	−	0.927245i	$-0.622170\pi$
$809$	−21.3012	−0.748909	−0.374454	+	0.927245i	$0.622170\pi$
$810$	0	0
$811$	18.1357i	0.636832i	0.947951	+	0.318416i	$0.103151\pi$
$811$	18.1357i	0.636832i	−0.947951	+	0.318416i	$0.896849\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−57.3832	−2.01005
$816$	0	0
$817$	46.7149	1.63435
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 24.3030i	− 0.848180i	−0.905620	−	0.424090i	$-0.860594\pi$
$821$	− 24.3030i	− 0.848180i	0.905620	−	0.424090i	$-0.139406\pi$
$822$	0	0
$823$	−1.29306	−0.0450734	−0.0225367	−	0.999746i	$-0.507174\pi$
$823$	−1.29306	−0.0450734	−0.0225367	+	0.999746i	$0.507174\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.6268i	0.821585i	0.911729	+	0.410792i	$0.134748\pi$
$827$	23.6268i	0.821585i	−0.911729	+	0.410792i	$0.865252\pi$
$828$	0	0
$829$	− 5.66492i	− 0.196751i	−0.995149	−	0.0983754i	$-0.968635\pi$
$829$	− 5.66492i	− 0.196751i	0.995149	−	0.0983754i	$-0.0313646\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.32168	0.184385
$834$	0	0
$835$	− 63.3890i	− 2.19367i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21.1554	0.730366	0.365183	−	0.930936i	$-0.381006\pi$
$839$	21.1554	0.730366	0.365183	+	0.930936i	$0.381006\pi$
$840$	0	0
$841$	−70.3652	−2.42639
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 81.2801i	− 2.79612i
$846$	0	0
$847$	−10.9835	−0.377396
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 4.13000i	− 0.141574i
$852$	0	0
$853$	− 4.41255i	− 0.151083i	−0.997143	−	0.0755413i	$-0.975932\pi$
$853$	− 4.41255i	− 0.151083i	0.997143	−	0.0755413i	$-0.0240685\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.9783	0.716606	0.358303	−	0.933605i	$-0.383356\pi$
$857$	20.9783	0.716606	0.358303	+	0.933605i	$0.383356\pi$
$858$	0	0
$859$	− 15.6318i	− 0.533351i	−0.963786	−	0.266676i	$-0.914075\pi$
$859$	− 15.6318i	− 0.533351i	0.963786	−	0.266676i	$-0.0859253\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.1880	−0.551047	−0.275523	−	0.961294i	$-0.588851\pi$
$863$	−16.1880	−0.551047	−0.275523	+	0.961294i	$0.588851\pi$
$864$	0	0
$865$	−16.7136	−0.568278
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.337409i	0.0114458i
$870$	0	0
$871$	−6.66208	−0.225736
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 2.24514i	− 0.0758995i
$876$	0	0
$877$	− 38.1001i	− 1.28655i	−0.765636	−	0.643274i	$-0.777576\pi$
$877$	− 38.1001i	− 1.28655i	0.765636	−	0.643274i	$-0.222424\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−46.4289	−1.56423	−0.782114	−	0.623135i	$-0.785859\pi$
$881$	−46.4289	−1.56423	−0.782114	+	0.623135i	$0.785859\pi$
$882$	0	0
$883$	− 17.8873i	− 0.601956i	−0.953631	−	0.300978i	$-0.902687\pi$
$883$	− 17.8873i	− 0.601956i	0.953631	−	0.300978i	$-0.0973131\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35.5228	−1.19274	−0.596369	−	0.802711i	$-0.703390\pi$
$887$	−35.5228	−1.19274	−0.596369	+	0.802711i	$0.703390\pi$
$888$	0	0
$889$	−0.840201	−0.0281795
$890$	0	0
$891$	0	0
$892$	0	0
$893$	31.8468i	1.06571i
$894$	0	0
$895$	23.2194	0.776137
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 32.6082i	− 1.08754i
$900$	0	0
$901$	− 6.08079i	− 0.202580i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	30.4875	1.01344
$906$	0	0
$907$	− 41.8996i	− 1.39125i	−0.718404	−	0.695627i	$-0.755127\pi$
$907$	− 41.8996i	− 1.39125i	0.718404	−	0.695627i	$-0.244873\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.35427	−0.177395	−0.0886974	−	0.996059i	$-0.528270\pi$
$911$	−5.35427	−0.177395	−0.0886974	+	0.996059i	$0.528270\pi$
$912$	0	0
$913$	−0.831099	−0.0275054
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.39979i	0.244363i
$918$	0	0
$919$	54.8843	1.81047	0.905234	−	0.424914i	$-0.139696\pi$
$919$	54.8843	1.81047	0.905234	+	0.424914i	$0.139696\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.96817i	0.229360i
$924$	0	0
$925$	− 3.39692i	− 0.111690i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.4091	−0.341512	−0.170756	−	0.985313i	$-0.554621\pi$
$929$	−10.4091	−0.341512	−0.170756	+	0.985313i	$0.554621\pi$
$930$	0	0
$931$	− 6.68215i	− 0.218999i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.08237	0.0681007
$936$	0	0
$937$	49.8177	1.62747	0.813737	−	0.581233i	$-0.197430\pi$
$937$	49.8177	1.62747	0.813737	+	0.581233i	$0.197430\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 5.61503i	− 0.183045i	−0.995803	−	0.0915224i	$-0.970827\pi$
$941$	− 5.61503i	− 0.183045i	0.995803	−	0.0915224i	$-0.0291733\pi$
$942$	0	0
$943$	15.3796	0.500829
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 22.3495i	− 0.726262i	−0.931738	−	0.363131i	$-0.881708\pi$
$947$	− 22.3495i	− 0.726262i	0.931738	−	0.363131i	$-0.118292\pi$
$948$	0	0
$949$	76.2585i	2.47545i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.9449	−0.646080	−0.323040	−	0.946385i	$-0.604705\pi$
$953$	−19.9449	−0.646080	−0.323040	+	0.946385i	$0.604705\pi$
$954$	0	0
$955$	19.0396i	0.616106i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.61682	−0.181376
$960$	0	0
$961$	−20.2991	−0.654811
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.50035i	0.241445i
$966$	0	0
$967$	−32.6951	−1.05140	−0.525702	−	0.850669i	$-0.676197\pi$
$967$	−32.6951	−1.05140	−0.525702	+	0.850669i	$0.676197\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.7391i	1.08274i	0.840785	+	0.541369i	$0.182094\pi$
$971$	33.7391i	1.08274i	−0.840785	+	0.541369i	$0.817906\pi$
$972$	0	0
$973$	5.19553i	0.166561i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−45.8374	−1.46647	−0.733235	−	0.679976i	$-0.761990\pi$
$977$	−45.8374	−1.46647	−0.733235	+	0.679976i	$0.761990\pi$
$978$	0	0
$979$	0.287081i	0.00917516i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−14.0557	−0.448307	−0.224154	−	0.974554i	$-0.571962\pi$
$983$	−14.0557	−0.448307	−0.224154	+	0.974554i	$0.571962\pi$
$984$	0	0
$985$	72.4649	2.30892
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 36.2282i	− 1.15199i
$990$	0	0
$991$	−18.3875	−0.584099	−0.292050	−	0.956403i	$-0.594337\pi$
$991$	−18.3875	−0.584099	−0.292050	+	0.956403i	$0.594337\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 73.4362i	− 2.32808i
$996$	0	0
$997$	− 20.8011i	− 0.658776i	−0.944195	−	0.329388i	$-0.893158\pi$
$997$	− 20.8011i	− 0.658776i	0.944195	−	0.329388i	$-0.106842\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.f.3025.21		24
3.2	odd	2	inner	6048.2.c.f.3025.3		24
4.3	odd	2		1512.2.c.g.757.12	yes	24
8.3	odd	2		1512.2.c.g.757.11	✓	24
8.5	even	2	inner	6048.2.c.f.3025.4		24
12.11	even	2		1512.2.c.g.757.13	yes	24
24.5	odd	2	inner	6048.2.c.f.3025.22		24
24.11	even	2		1512.2.c.g.757.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.g.757.11	✓	24	8.3	odd	2
1512.2.c.g.757.12	yes	24	4.3	odd	2
1512.2.c.g.757.13	yes	24	12.11	even	2
1512.2.c.g.757.14	yes	24	24.11	even	2
6048.2.c.f.3025.3		24	3.2	odd	2	inner
6048.2.c.f.3025.4		24	8.5	even	2	inner
6048.2.c.f.3025.21		24	1.1	even	1	trivial
6048.2.c.f.3025.22		24	24.5	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)

\(f(q)\)	\(=\)	\(q+3.04340i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+3.04340i q^{5} -1.00000 q^{7} -0.128573i q^{11} -6.30135i q^{13} +5.32168 q^{17} -6.68215i q^{19} -5.18212 q^{23} -4.26229 q^{25} +9.96821i q^{29} -3.27122 q^{31} -3.04340i q^{35} +0.796970i q^{37} -2.96782 q^{41} +6.99100i q^{43} -4.76595 q^{47} +1.00000 q^{49} -1.14264i q^{53} +0.391299 q^{55} +11.0999i q^{59} +14.6662i q^{61} +19.1775 q^{65} -1.05725i q^{67} -1.10582 q^{71} -12.1019 q^{73} +0.128573i q^{77} -2.62426 q^{79} -6.46403i q^{83} +16.1960i q^{85} -2.23283 q^{89} +6.30135i q^{91} +20.3365 q^{95} -7.88097 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{7}+O(q^{10}) \) 24 * q - 24 * q^7	\( 24 q - 24 q^{7} - 24 q^{25} - 8 q^{31} + 24 q^{49} - 16 q^{55} - 8 q^{79}+O(q^{100}) \) 24 * q - 24 * q^7 - 24 * q^25 - 8 * q^31 + 24 * q^49 - 16 * q^55 - 8 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more