LMFDB - Embedded newform 6048.2.h.c.2591.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(2591,6048)]

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, 0, 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("6048.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.8
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.c.2591.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.14626i q^{5} +1.00000i q^{7} +O(q^{10})$	$q+3.14626i q^{5} +1.00000i q^{7} +1.41421 q^{11} +0.449490 q^{13} -2.51059i q^{17} -2.44949i q^{19} -5.65685 q^{23} -4.89898 q^{25} -8.34242i q^{29} -1.55051i q^{31} -3.14626 q^{35} +9.00000 q^{37} -10.0745i q^{41} -8.34847i q^{43} -0.460702 q^{47} -1.00000 q^{49} -5.02118i q^{53} +4.44949i q^{55} -11.4887 q^{59} -5.55051 q^{61} +1.41421i q^{65} -13.7980i q^{67} +5.65685 q^{71} +9.79796 q^{73} +1.41421i q^{77} -0.348469i q^{79} -2.36773 q^{83} +7.89898 q^{85} +1.55708i q^{89} +0.449490i q^{91} +7.70674 q^{95} +10.2474 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{13} + 72 q^{37} - 8 q^{49} - 64 q^{61} + 24 q^{85} - 16 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 + 72 * q^37 - 8 * q^49 - 64 * q^61 + 24 * q^85 - 16 * 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
$5$	3.14626i	1.40705i	0.710669	+	0.703526i	$0.248392\pi$
$5$	3.14626i	1.40705i	−0.710669	+	0.703526i	$0.751608\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	0.449490	0.124666	0.0623330	−	0.998055i	$-0.480146\pi$
$13$	0.449490	0.124666	0.0623330	+	0.998055i	$0.480146\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.51059i	− 0.608907i	−0.952527	−	0.304454i	$-0.901526\pi$
$17$	− 2.51059i	− 0.608907i	0.952527	−	0.304454i	$-0.0984739\pi$
$18$	0	0
$19$	− 2.44949i	− 0.561951i	−0.959715	−	0.280976i	$-0.909342\pi$
$19$	− 2.44949i	− 0.561951i	0.959715	−	0.280976i	$-0.0906580\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	−4.89898	−0.979796
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.34242i	− 1.54915i	−0.632483	−	0.774574i	$-0.717964\pi$
$29$	− 8.34242i	− 1.54915i	0.632483	−	0.774574i	$-0.282036\pi$
$30$	0	0
$31$	− 1.55051i	− 0.278480i	−0.990259	−	0.139240i	$-0.955534\pi$
$31$	− 1.55051i	− 0.278480i	0.990259	−	0.139240i	$-0.0444659\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.14626	−0.531816
$36$	0	0
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.0745i	− 1.57337i	−0.617356	−	0.786684i	$-0.711796\pi$
$41$	− 10.0745i	− 1.57337i	0.617356	−	0.786684i	$-0.288204\pi$
$42$	0	0
$43$	− 8.34847i	− 1.27313i	−0.771223	−	0.636565i	$-0.780355\pi$
$43$	− 8.34847i	− 1.27313i	0.771223	−	0.636565i	$-0.219645\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.460702	−0.0672003	−0.0336001	−	0.999435i	$-0.510697\pi$
$47$	−0.460702	−0.0672003	−0.0336001	+	0.999435i	$0.510697\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 5.02118i	− 0.689712i	−0.938656	−	0.344856i	$-0.887928\pi$
$53$	− 5.02118i	− 0.689712i	0.938656	−	0.344856i	$-0.112072\pi$
$54$	0	0
$55$	4.44949i	0.599969i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.4887	−1.49570	−0.747849	−	0.663868i	$-0.768914\pi$
$59$	−11.4887	−1.49570	−0.747849	+	0.663868i	$0.768914\pi$
$60$	0	0
$61$	−5.55051	−0.710670	−0.355335	−	0.934739i	$-0.615633\pi$
$61$	−5.55051	−0.710670	−0.355335	+	0.934739i	$0.615633\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.41421i	0.175412i
$66$	0	0
$67$	− 13.7980i	− 1.68569i	−0.538157	−	0.842844i	$-0.680879\pi$
$67$	− 13.7980i	− 1.68569i	0.538157	−	0.842844i	$-0.319121\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	9.79796	1.14676	0.573382	−	0.819288i	$-0.305631\pi$
$73$	9.79796	1.14676	0.573382	+	0.819288i	$0.305631\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.41421i	0.161165i
$78$	0	0
$79$	− 0.348469i	− 0.0392059i	−0.999808	−	0.0196029i	$-0.993760\pi$
$79$	− 0.348469i	− 0.0392059i	0.999808	−	0.0196029i	$-0.00624020\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.36773	−0.259892	−0.129946	−	0.991521i	$-0.541480\pi$
$83$	−2.36773	−0.259892	−0.129946	+	0.991521i	$0.541480\pi$
$84$	0	0
$85$	7.89898	0.856765
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.55708i	0.165050i	0.996589	+	0.0825250i	$0.0262984\pi$
$89$	1.55708i	0.165050i	−0.996589	+	0.0825250i	$0.973702\pi$
$90$	0	0
$91$	0.449490i	0.0471193i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.70674	0.790695
$96$	0	0
$97$	10.2474	1.04047	0.520235	−	0.854023i	$-0.325844\pi$
$97$	10.2474	1.04047	0.520235	+	0.854023i	$0.325844\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.65685i	0.562878i	0.959579	+	0.281439i	$0.0908117\pi$
$101$	5.65685i	0.562878i	−0.959579	+	0.281439i	$0.909188\pi$
$102$	0	0
$103$	− 5.79796i	− 0.571290i	−0.958336	−	0.285645i	$-0.907792\pi$
$103$	− 5.79796i	− 0.571290i	0.958336	−	0.285645i	$-0.0922078\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.8564	−1.33955	−0.669775	−	0.742564i	$-0.733609\pi$
$107$	−13.8564	−1.33955	−0.669775	+	0.742564i	$0.733609\pi$
$108$	0	0
$109$	0.797959	0.0764306	0.0382153	−	0.999270i	$-0.487833\pi$
$109$	0.797959	0.0764306	0.0382153	+	0.999270i	$0.487833\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 10.5352i	− 0.991065i	−0.868589	−	0.495533i	$-0.834973\pi$
$113$	− 10.5352i	− 0.991065i	0.868589	−	0.495533i	$-0.165027\pi$
$114$	0	0
$115$	− 17.7980i	− 1.65967i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.51059	0.230145
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0.317837i	0.0284282i
$126$	0	0
$127$	12.5505i	1.11368i	0.830621	+	0.556839i	$0.187986\pi$
$127$	12.5505i	1.11368i	−0.830621	+	0.556839i	$0.812014\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.84961	−0.685823	−0.342912	−	0.939368i	$-0.611413\pi$
$131$	−7.84961	−0.685823	−0.342912	+	0.939368i	$0.611413\pi$
$132$	0	0
$133$	2.44949	0.212398
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.4422i	1.06301i	0.847056	+	0.531504i	$0.178373\pi$
$137$	12.4422i	1.06301i	−0.847056	+	0.531504i	$0.821627\pi$
$138$	0	0
$139$	0.449490i	0.0381252i	0.999818	+	0.0190626i	$0.00606819\pi$
$139$	0.449490i	0.0381252i	−0.999818	+	0.0190626i	$0.993932\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.635674	0.0531578
$144$	0	0
$145$	26.2474	2.17973
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.4634i	1.43066i	0.698789	+	0.715328i	$0.253723\pi$
$149$	17.4634i	1.43066i	−0.698789	+	0.715328i	$0.746277\pi$
$150$	0	0
$151$	8.34847i	0.679389i	0.940536	+	0.339694i	$0.110324\pi$
$151$	8.34847i	0.679389i	−0.940536	+	0.339694i	$0.889676\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.87832	0.391836
$156$	0	0
$157$	−10.2474	−0.817835	−0.408918	−	0.912571i	$-0.634094\pi$
$157$	−10.2474	−0.817835	−0.408918	+	0.912571i	$0.634094\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 5.65685i	− 0.445823i
$162$	0	0
$163$	0.550510i	0.0431193i	0.999768	+	0.0215596i	$0.00686318\pi$
$163$	0.550510i	0.0431193i	−0.999768	+	0.0215596i	$0.993137\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.58166	−0.741451	−0.370725	−	0.928742i	$-0.620891\pi$
$167$	−9.58166	−0.741451	−0.370725	+	0.928742i	$0.620891\pi$
$168$	0	0
$169$	−12.7980	−0.984458
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 7.84961i	− 0.596795i	−0.954442	−	0.298397i	$-0.903548\pi$
$173$	− 7.84961i	− 0.596795i	0.954442	−	0.298397i	$-0.0964520\pi$
$174$	0	0
$175$	− 4.89898i	− 0.370328i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.1708	0.834948	0.417474	−	0.908689i	$-0.362915\pi$
$179$	11.1708	0.834948	0.417474	+	0.908689i	$0.362915\pi$
$180$	0	0
$181$	3.79796	0.282300	0.141150	−	0.989988i	$-0.454920\pi$
$181$	3.79796	0.282300	0.141150	+	0.989988i	$0.454920\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	28.3164i	2.08186i
$186$	0	0
$187$	− 3.55051i	− 0.259639i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.84961	0.567978	0.283989	−	0.958828i	$-0.408342\pi$
$191$	7.84961	0.567978	0.283989	+	0.958828i	$0.408342\pi$
$192$	0	0
$193$	20.5959	1.48253	0.741263	−	0.671214i	$-0.234227\pi$
$193$	20.5959	1.48253	0.741263	+	0.671214i	$0.234227\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 9.89949i	− 0.705310i	−0.935753	−	0.352655i	$-0.885279\pi$
$197$	− 9.89949i	− 0.705310i	0.935753	−	0.352655i	$-0.114721\pi$
$198$	0	0
$199$	− 12.6969i	− 0.900062i	−0.893013	−	0.450031i	$-0.851413\pi$
$199$	− 12.6969i	− 0.900062i	0.893013	−	0.450031i	$-0.148587\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.34242	0.585523
$204$	0	0
$205$	31.6969	2.21381
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 3.46410i	− 0.239617i
$210$	0	0
$211$	− 17.5959i	− 1.21135i	−0.795711	−	0.605676i	$-0.792903\pi$
$211$	− 17.5959i	− 1.21135i	0.795711	−	0.605676i	$-0.207097\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	26.2665	1.79136
$216$	0	0
$217$	1.55051	0.105255
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.12848i	− 0.0759101i
$222$	0	0
$223$	− 28.2474i	− 1.89159i	−0.324766	−	0.945795i	$-0.605285\pi$
$223$	− 28.2474i	− 1.89159i	0.324766	−	0.945795i	$-0.394715\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.6848	−1.10741	−0.553706	−	0.832712i	$-0.686787\pi$
$227$	−16.6848	−1.10741	−0.553706	+	0.832712i	$0.686787\pi$
$228$	0	0
$229$	23.7980	1.57261	0.786307	−	0.617836i	$-0.211991\pi$
$229$	23.7980	1.57261	0.786307	+	0.617836i	$0.211991\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 7.07107i	− 0.463241i	−0.972806	−	0.231621i	$-0.925597\pi$
$233$	− 7.07107i	− 0.463241i	0.972806	−	0.231621i	$-0.0744028\pi$
$234$	0	0
$235$	− 1.44949i	− 0.0945543i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.3704	1.25297	0.626483	−	0.779435i	$-0.284494\pi$
$239$	19.3704	1.25297	0.626483	+	0.779435i	$0.284494\pi$
$240$	0	0
$241$	9.34847	0.602188	0.301094	−	0.953594i	$-0.402648\pi$
$241$	9.34847	0.602188	0.301094	+	0.953594i	$0.402648\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3.14626i	− 0.201007i
$246$	0	0
$247$	− 1.10102i	− 0.0700563i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	13.6814	0.863564	0.431782	−	0.901978i	$-0.357885\pi$
$251$	13.6814	0.863564	0.431782	+	0.901978i	$0.357885\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.1421i	0.882162i	0.897467	+	0.441081i	$0.145405\pi$
$257$	14.1421i	0.882162i	−0.897467	+	0.441081i	$0.854595\pi$
$258$	0	0
$259$	9.00000i	0.559233i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.7347	1.15523	0.577616	−	0.816308i	$-0.303983\pi$
$263$	18.7347	1.15523	0.577616	+	0.816308i	$0.303983\pi$
$264$	0	0
$265$	15.7980	0.970461
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.9453i	1.39900i	0.714634	+	0.699498i	$0.246593\pi$
$269$	22.9453i	1.39900i	−0.714634	+	0.699498i	$0.753407\pi$
$270$	0	0
$271$	8.69694i	0.528301i	0.964481	+	0.264151i	$0.0850916\pi$
$271$	8.69694i	0.528301i	−0.964481	+	0.264151i	$0.914908\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.92820	−0.417786
$276$	0	0
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.38551i	0.261617i	0.991408	+	0.130809i	$0.0417574\pi$
$281$	4.38551i	0.261617i	−0.991408	+	0.130809i	$0.958243\pi$
$282$	0	0
$283$	2.65153i	0.157617i	0.996890	+	0.0788086i	$0.0251116\pi$
$283$	2.65153i	0.157617i	−0.996890	+	0.0788086i	$0.974888\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.0745	0.594677
$288$	0	0
$289$	10.6969	0.629232
$290$	0	0
$291$	0	0
$292$	0	0
$293$	11.6315i	0.679522i	0.940512	+	0.339761i	$0.110346\pi$
$293$	11.6315i	0.679522i	−0.940512	+	0.339761i	$0.889654\pi$
$294$	0	0
$295$	− 36.1464i	− 2.10453i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.54270	−0.147048
$300$	0	0
$301$	8.34847	0.481198
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 17.4634i	− 0.999950i
$306$	0	0
$307$	− 2.20204i	− 0.125677i	−0.998024	−	0.0628386i	$-0.979985\pi$
$307$	− 2.20204i	− 0.125677i	0.998024	−	0.0628386i	$-0.0200153\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.8954	−1.18487	−0.592434	−	0.805619i	$-0.701833\pi$
$311$	−20.8954	−1.18487	−0.592434	+	0.805619i	$0.701833\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.5851i	− 0.706847i	−0.935463	−	0.353424i	$-0.885017\pi$
$317$	− 12.5851i	− 0.706847i	0.935463	−	0.353424i	$-0.114983\pi$
$318$	0	0
$319$	− 11.7980i	− 0.660559i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.14966	−0.342176
$324$	0	0
$325$	−2.20204	−0.122147
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 0.460702i	− 0.0253993i
$330$	0	0
$331$	− 28.1464i	− 1.54707i	−0.633755	−	0.773534i	$-0.718487\pi$
$331$	− 28.1464i	− 1.54707i	0.633755	−	0.773534i	$-0.281513\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	43.4120	2.37185
$336$	0	0
$337$	−20.5959	−1.12193	−0.560966	−	0.827839i	$-0.689570\pi$
$337$	−20.5959	−1.12193	−0.560966	+	0.827839i	$0.689570\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.19275i	− 0.118744i
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−32.5911	−1.74958	−0.874792	−	0.484499i	$-0.839002\pi$
$347$	−32.5911	−1.74958	−0.874792	+	0.484499i	$0.839002\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.6595i	1.20604i	0.797724	+	0.603022i	$0.206037\pi$
$353$	22.6595i	1.20604i	−0.797724	+	0.603022i	$0.793963\pi$
$354$	0	0
$355$	17.7980i	0.944618i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.3923	0.548485	0.274242	−	0.961661i	$-0.411573\pi$
$359$	10.3923	0.548485	0.274242	+	0.961661i	$0.411573\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	30.8270i	1.61356i
$366$	0	0
$367$	17.1010i	0.892666i	0.894867	+	0.446333i	$0.147270\pi$
$367$	17.1010i	0.892666i	−0.894867	+	0.446333i	$0.852730\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.02118	0.260687
$372$	0	0
$373$	−12.1010	−0.626567	−0.313284	−	0.949660i	$-0.601429\pi$
$373$	−12.1010	−0.626567	−0.313284	+	0.949660i	$0.601429\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.74983i	− 0.193126i
$378$	0	0
$379$	− 11.0454i	− 0.567364i	−0.958918	−	0.283682i	$-0.908444\pi$
$379$	− 11.0454i	− 0.567364i	0.958918	−	0.283682i	$-0.0915561\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.02458	−0.410037	−0.205018	−	0.978758i	$-0.565725\pi$
$383$	−8.02458	−0.410037	−0.205018	+	0.978758i	$0.565725\pi$
$384$	0	0
$385$	−4.44949	−0.226767
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 25.9487i	− 1.31565i	−0.753171	−	0.657824i	$-0.771477\pi$
$389$	− 25.9487i	− 1.31565i	0.753171	−	0.657824i	$-0.228523\pi$
$390$	0	0
$391$	14.2020i	0.718228i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.09638	0.0551647
$396$	0	0
$397$	21.7980	1.09401	0.547004	−	0.837130i	$-0.315768\pi$
$397$	21.7980	1.09401	0.547004	+	0.837130i	$0.315768\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.47848i	0.123769i	0.998083	+	0.0618847i	$0.0197111\pi$
$401$	2.47848i	0.123769i	−0.998083	+	0.0618847i	$0.980289\pi$
$402$	0	0
$403$	− 0.696938i	− 0.0347170i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.7279	0.630900
$408$	0	0
$409$	5.79796	0.286691	0.143345	−	0.989673i	$-0.454214\pi$
$409$	5.79796	0.286691	0.143345	+	0.989673i	$0.454214\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 11.4887i	− 0.565321i
$414$	0	0
$415$	− 7.44949i	− 0.365681i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.36773	−0.115671	−0.0578355	−	0.998326i	$-0.518420\pi$
$419$	−2.36773	−0.115671	−0.0578355	+	0.998326i	$0.518420\pi$
$420$	0	0
$421$	32.8990	1.60340	0.801699	−	0.597728i	$-0.203930\pi$
$421$	32.8990	1.60340	0.801699	+	0.597728i	$0.203930\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.2993i	0.596605i
$426$	0	0
$427$	− 5.55051i	− 0.268608i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.92820	−0.333720	−0.166860	−	0.985981i	$-0.553363\pi$
$431$	−6.92820	−0.333720	−0.166860	+	0.985981i	$0.553363\pi$
$432$	0	0
$433$	26.4495	1.27108	0.635541	−	0.772067i	$-0.280777\pi$
$433$	26.4495	1.27108	0.635541	+	0.772067i	$0.280777\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.8564i	0.662842i
$438$	0	0
$439$	5.10102i	0.243458i	0.992563	+	0.121729i	$0.0388439\pi$
$439$	5.10102i	0.243458i	−0.992563	+	0.121729i	$0.961156\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−31.8912	−1.51520	−0.757599	−	0.652720i	$-0.773628\pi$
$443$	−31.8912	−1.51520	−0.757599	+	0.652720i	$0.773628\pi$
$444$	0	0
$445$	−4.89898	−0.232234
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 26.5843i	− 1.25459i	−0.778781	−	0.627296i	$-0.784162\pi$
$449$	− 26.5843i	− 1.25459i	0.778781	−	0.627296i	$-0.215838\pi$
$450$	0	0
$451$	− 14.2474i	− 0.670886i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.41421	−0.0662994
$456$	0	0
$457$	11.5959	0.542434	0.271217	−	0.962518i	$-0.412574\pi$
$457$	11.5959	0.542434	0.271217	+	0.962518i	$0.412574\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 30.5091i	− 1.42095i	−0.703721	−	0.710476i	$-0.748480\pi$
$461$	− 30.5091i	− 1.42095i	0.703721	−	0.710476i	$-0.251520\pi$
$462$	0	0
$463$	− 31.2474i	− 1.45219i	−0.687593	−	0.726096i	$-0.741333\pi$
$463$	− 31.2474i	− 1.45219i	0.687593	−	0.726096i	$-0.258667\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.2275	−0.889744	−0.444872	−	0.895594i	$-0.646751\pi$
$467$	−19.2275	−0.889744	−0.444872	+	0.895594i	$0.646751\pi$
$468$	0	0
$469$	13.7980	0.637131
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 11.8065i	− 0.542864i
$474$	0	0
$475$	12.0000i	0.550598i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14.6028	−0.667221	−0.333610	−	0.942711i	$-0.608267\pi$
$479$	−14.6028	−0.667221	−0.333610	+	0.942711i	$0.608267\pi$
$480$	0	0
$481$	4.04541	0.184455
$482$	0	0
$483$	0	0
$484$	0	0
$485$	32.2412i	1.46400i
$486$	0	0
$487$	− 19.5959i	− 0.887976i	−0.896033	−	0.443988i	$-0.853563\pi$
$487$	− 19.5959i	− 0.887976i	0.896033	−	0.443988i	$-0.146437\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.4634	−0.788111	−0.394055	−	0.919087i	$-0.628928\pi$
$491$	−17.4634	−0.788111	−0.394055	+	0.919087i	$0.628928\pi$
$492$	0	0
$493$	−20.9444	−0.943288
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.65685i	0.253745i
$498$	0	0
$499$	25.2474i	1.13023i	0.825012	+	0.565116i	$0.191168\pi$
$499$	25.2474i	1.13023i	−0.825012	+	0.565116i	$0.808832\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.4169	0.821168	0.410584	−	0.911823i	$-0.365325\pi$
$503$	18.4169	0.821168	0.410584	+	0.911823i	$0.365325\pi$
$504$	0	0
$505$	−17.7980	−0.791999
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.87492i	0.0831042i	0.999136	+	0.0415521i	$0.0132302\pi$
$509$	1.87492i	0.0831042i	−0.999136	+	0.0415521i	$0.986770\pi$
$510$	0	0
$511$	9.79796i	0.433436i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18.2419	0.803835
$516$	0	0
$517$	−0.651531	−0.0286543
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.68896i	0.249238i	0.992205	+	0.124619i	$0.0397709\pi$
$521$	5.68896i	0.249238i	−0.992205	+	0.124619i	$0.960229\pi$
$522$	0	0
$523$	− 12.0454i	− 0.526709i	−0.964699	−	0.263354i	$-0.915171\pi$
$523$	− 12.0454i	− 0.526709i	0.964699	−	0.263354i	$-0.0848289\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.89270	−0.169568
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.52837i	− 0.196145i
$534$	0	0
$535$	− 43.5959i	− 1.88482i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.41421	−0.0609145
$540$	0	0
$541$	−29.0000	−1.24681	−0.623404	−	0.781900i	$-0.714251\pi$
$541$	−29.0000	−1.24681	−0.623404	+	0.781900i	$0.714251\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.51059i	0.107542i
$546$	0	0
$547$	0.146428i	0.00626082i	0.999995	+	0.00313041i	$0.000996442\pi$
$547$	0.146428i	0.00626082i	−0.999995	+	0.00313041i	$0.999004\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−20.4347	−0.870546
$552$	0	0
$553$	0.348469	0.0148184
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 27.7128i	− 1.17423i	−0.809504	−	0.587115i	$-0.800264\pi$
$557$	− 27.7128i	− 1.17423i	0.809504	−	0.587115i	$-0.199736\pi$
$558$	0	0
$559$	− 3.75255i	− 0.158716i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−32.6698	−1.37687	−0.688433	−	0.725300i	$-0.741701\pi$
$563$	−32.6698	−1.37687	−0.688433	+	0.725300i	$0.741701\pi$
$564$	0	0
$565$	33.1464	1.39448
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 4.38551i	− 0.183850i	−0.995766	−	0.0919250i	$-0.970698\pi$
$569$	− 4.38551i	− 0.183850i	0.995766	−	0.0919250i	$-0.0293020\pi$
$570$	0	0
$571$	− 14.3485i	− 0.600465i	−0.953866	−	0.300232i	$-0.902936\pi$
$571$	− 14.3485i	− 0.600465i	0.953866	−	0.300232i	$-0.0970642\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	27.7128	1.15570
$576$	0	0
$577$	−1.59592	−0.0664389	−0.0332195	−	0.999448i	$-0.510576\pi$
$577$	−1.59592	−0.0664389	−0.0332195	+	0.999448i	$0.510576\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 2.36773i	− 0.0982298i
$582$	0	0
$583$	− 7.10102i	− 0.294094i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.8844	−1.02709	−0.513544	−	0.858063i	$-0.671668\pi$
$587$	−24.8844	−1.02709	−0.513544	+	0.858063i	$0.671668\pi$
$588$	0	0
$589$	−3.79796	−0.156492
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 0.603566i	− 0.0247855i	−0.999923	−	0.0123928i	$-0.996055\pi$
$593$	− 0.603566i	− 0.0247855i	0.999923	−	0.0123928i	$-0.00394484\pi$
$594$	0	0
$595$	7.89898i	0.323827i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	29.1270	1.19010	0.595049	−	0.803689i	$-0.297133\pi$
$599$	29.1270	1.19010	0.595049	+	0.803689i	$0.297133\pi$
$600$	0	0
$601$	46.7423	1.90666	0.953330	−	0.301930i	$-0.0976310\pi$
$601$	46.7423	1.90666	0.953330	+	0.301930i	$0.0976310\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 28.3164i	− 1.15122i
$606$	0	0
$607$	− 1.75255i	− 0.0711339i	−0.999367	−	0.0355669i	$-0.988676\pi$
$607$	− 1.75255i	− 0.0711339i	0.999367	−	0.0355669i	$-0.0113237\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.207081	−0.00837759
$612$	0	0
$613$	−41.3939	−1.67188	−0.835941	−	0.548819i	$-0.815078\pi$
$613$	−41.3939	−1.67188	−0.835941	+	0.548819i	$0.815078\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.38551i	0.176554i	0.996096	+	0.0882769i	$0.0281360\pi$
$617$	4.38551i	0.176554i	−0.996096	+	0.0882769i	$0.971864\pi$
$618$	0	0
$619$	47.8434i	1.92299i	0.274827	+	0.961494i	$0.411379\pi$
$619$	47.8434i	1.92299i	−0.274827	+	0.961494i	$0.588621\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.55708	−0.0623830
$624$	0	0
$625$	−25.4949	−1.01980
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 22.5953i	− 0.900934i
$630$	0	0
$631$	− 33.7423i	− 1.34326i	−0.740886	−	0.671631i	$-0.765594\pi$
$631$	− 33.7423i	− 1.34326i	0.740886	−	0.671631i	$-0.234406\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−39.4872	−1.56700
$636$	0	0
$637$	−0.449490	−0.0178094
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 25.9487i	− 1.02491i	−0.858714	−	0.512455i	$-0.828736\pi$
$641$	− 25.9487i	− 1.02491i	0.858714	−	0.512455i	$-0.171264\pi$
$642$	0	0
$643$	38.4949i	1.51809i	0.651038	+	0.759045i	$0.274334\pi$
$643$	38.4949i	1.51809i	−0.651038	+	0.759045i	$0.725666\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	35.4982	1.39558	0.697789	−	0.716303i	$-0.254167\pi$
$647$	35.4982	1.39558	0.697789	+	0.716303i	$0.254167\pi$
$648$	0	0
$649$	−16.2474	−0.637768
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.43539i	0.251836i	0.992041	+	0.125918i	$0.0401877\pi$
$653$	6.43539i	0.251836i	−0.992041	+	0.125918i	$0.959812\pi$
$654$	0	0
$655$	− 24.6969i	− 0.964989i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.2351	0.476612	0.238306	−	0.971190i	$-0.423408\pi$
$659$	12.2351	0.476612	0.238306	+	0.971190i	$0.423408\pi$
$660$	0	0
$661$	42.0454	1.63538	0.817688	−	0.575661i	$-0.195255\pi$
$661$	42.0454	1.63538	0.817688	+	0.575661i	$0.195255\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.70674i	0.298855i
$666$	0	0
$667$	47.1918i	1.82728i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.84961	−0.303031
$672$	0	0
$673$	−12.8990	−0.497219	−0.248610	−	0.968604i	$-0.579974\pi$
$673$	−12.8990	−0.497219	−0.248610	+	0.968604i	$0.579974\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.285729i	0.0109815i	0.999985	+	0.00549073i	$0.00174776\pi$
$677$	0.285729i	0.0109815i	−0.999985	+	0.00549073i	$0.998252\pi$
$678$	0	0
$679$	10.2474i	0.393261i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	33.1626	1.26893	0.634466	−	0.772951i	$-0.281220\pi$
$683$	33.1626	1.26893	0.634466	+	0.772951i	$0.281220\pi$
$684$	0	0
$685$	−39.1464	−1.49571
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 2.25697i	− 0.0859837i
$690$	0	0
$691$	40.9444i	1.55760i	0.627274	+	0.778799i	$0.284171\pi$
$691$	40.9444i	1.55760i	−0.627274	+	0.778799i	$0.715829\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.41421	−0.0536442
$696$	0	0
$697$	−25.2929	−0.958035
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 43.2049i	− 1.63183i	−0.578173	−	0.815914i	$-0.696234\pi$
$701$	− 43.2049i	− 1.63183i	0.578173	−	0.815914i	$-0.303766\pi$
$702$	0	0
$703$	− 22.0454i	− 0.831458i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.65685	−0.212748
$708$	0	0
$709$	52.3939	1.96769	0.983847	−	0.179013i	$-0.0572905\pi$
$709$	52.3939	1.96769	0.983847	+	0.179013i	$0.0572905\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.77101i	0.328477i
$714$	0	0
$715$	2.00000i	0.0747958i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32.5590	−1.21425	−0.607123	−	0.794608i	$-0.707677\pi$
$719$	−32.5590	−1.21425	−0.607123	+	0.794608i	$0.707677\pi$
$720$	0	0
$721$	5.79796	0.215927
$722$	0	0
$723$	0	0
$724$	0	0
$725$	40.8693i	1.51785i
$726$	0	0
$727$	26.0454i	0.965971i	0.875628	+	0.482985i	$0.160448\pi$
$727$	26.0454i	0.965971i	−0.875628	+	0.482985i	$0.839552\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−20.9596	−0.775218
$732$	0	0
$733$	17.5959	0.649920	0.324960	−	0.945728i	$-0.394649\pi$
$733$	17.5959	0.649920	0.324960	+	0.945728i	$0.394649\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 19.5133i	− 0.718780i
$738$	0	0
$739$	− 20.0000i	− 0.735712i	−0.929883	−	0.367856i	$-0.880092\pi$
$739$	− 20.0000i	− 0.735712i	0.929883	−	0.367856i	$-0.119908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−32.8769	−1.20613	−0.603067	−	0.797690i	$-0.706055\pi$
$743$	−32.8769	−1.20613	−0.603067	+	0.797690i	$0.706055\pi$
$744$	0	0
$745$	−54.9444	−2.01301
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 13.8564i	− 0.506302i
$750$	0	0
$751$	25.7980i	0.941381i	0.882298	+	0.470690i	$0.155995\pi$
$751$	25.7980i	0.941381i	−0.882298	+	0.470690i	$0.844005\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−26.2665	−0.955935
$756$	0	0
$757$	−33.6969	−1.22474	−0.612368	−	0.790573i	$-0.709783\pi$
$757$	−33.6969	−1.22474	−0.612368	+	0.790573i	$0.709783\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	19.1954i	0.695834i	0.937525	+	0.347917i	$0.113111\pi$
$761$	19.1954i	0.695834i	−0.937525	+	0.347917i	$0.886889\pi$
$762$	0	0
$763$	0.797959i	0.0288881i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.16404	−0.186463
$768$	0	0
$769$	35.7980	1.29091	0.645454	−	0.763799i	$-0.276668\pi$
$769$	35.7980	1.29091	0.645454	+	0.763799i	$0.276668\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	12.5529i	0.451498i	0.974185	+	0.225749i	$0.0724830\pi$
$773$	12.5529i	0.451498i	−0.974185	+	0.225749i	$0.927517\pi$
$774$	0	0
$775$	7.59592i	0.272853i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.6773	−0.884156
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 32.2412i	− 1.15074i
$786$	0	0
$787$	− 37.3939i	− 1.33295i	−0.745528	−	0.666474i	$-0.767803\pi$
$787$	− 37.3939i	− 1.33295i	0.745528	−	0.666474i	$-0.232197\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.5352	0.374588
$792$	0	0
$793$	−2.49490	−0.0885964
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.5836i	1.43754i	0.695245	+	0.718772i	$0.255296\pi$
$797$	40.5836i	1.43754i	−0.695245	+	0.718772i	$0.744704\pi$
$798$	0	0
$799$	1.15663i	0.0409187i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.8564	0.488982
$804$	0	0
$805$	17.7980	0.627296
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 8.13534i	− 0.286023i	−0.989721	−	0.143012i	$-0.954321\pi$
$809$	− 8.13534i	− 0.286023i	0.989721	−	0.143012i	$-0.0456786\pi$
$810$	0	0
$811$	− 16.0454i	− 0.563430i	−0.959498	−	0.281715i	$-0.909097\pi$
$811$	− 16.0454i	− 0.563430i	0.959498	−	0.281715i	$-0.0909033\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.73205	−0.0606711
$816$	0	0
$817$	−20.4495	−0.715437
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35.9124i	1.25335i	0.779281	+	0.626675i	$0.215585\pi$
$821$	35.9124i	1.25335i	−0.779281	+	0.626675i	$0.784415\pi$
$822$	0	0
$823$	42.8434i	1.49343i	0.665146	+	0.746713i	$0.268369\pi$
$823$	42.8434i	1.49343i	−0.665146	+	0.746713i	$0.731631\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.1708	−0.388448	−0.194224	−	0.980957i	$-0.562219\pi$
$827$	−11.1708	−0.388448	−0.194224	+	0.980957i	$0.562219\pi$
$828$	0	0
$829$	−52.0454	−1.80761	−0.903806	−	0.427943i	$-0.859238\pi$
$829$	−52.0454	−1.80761	−0.903806	+	0.427943i	$0.859238\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.51059i	0.0869868i
$834$	0	0
$835$	− 30.1464i	− 1.04326i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.56048	0.157445	0.0787226	−	0.996897i	$-0.474916\pi$
$839$	4.56048	0.157445	0.0787226	+	0.996897i	$0.474916\pi$
$840$	0	0
$841$	−40.5959	−1.39986
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 40.2658i	− 1.38518i
$846$	0	0
$847$	− 9.00000i	− 0.309244i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−50.9117	−1.74523
$852$	0	0
$853$	24.2020	0.828662	0.414331	−	0.910126i	$-0.364016\pi$
$853$	24.2020	0.828662	0.414331	+	0.910126i	$0.364016\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 25.1380i	− 0.858698i	−0.903139	−	0.429349i	$-0.858743\pi$
$857$	− 25.1380i	− 0.858698i	0.903139	−	0.429349i	$-0.141257\pi$
$858$	0	0
$859$	− 29.1010i	− 0.992914i	−0.868061	−	0.496457i	$-0.834634\pi$
$859$	− 29.1010i	− 0.992914i	0.868061	−	0.496457i	$-0.165366\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.4347	0.695604	0.347802	−	0.937568i	$-0.386928\pi$
$863$	20.4347	0.695604	0.347802	+	0.937568i	$0.386928\pi$
$864$	0	0
$865$	24.6969	0.839721
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 0.492810i	− 0.0167174i
$870$	0	0
$871$	− 6.20204i	− 0.210148i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.317837	−0.0107449
$876$	0	0
$877$	38.5959	1.30329	0.651646	−	0.758523i	$-0.274079\pi$
$877$	38.5959	1.30329	0.651646	+	0.758523i	$0.274079\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 48.4974i	− 1.63392i	−0.576695	−	0.816960i	$-0.695658\pi$
$881$	− 48.4974i	− 1.63392i	0.576695	−	0.816960i	$-0.304342\pi$
$882$	0	0
$883$	− 7.65153i	− 0.257495i	−0.991677	−	0.128747i	$-0.958904\pi$
$883$	− 7.65153i	− 0.257495i	0.991677	−	0.128747i	$-0.0410956\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.8304	1.13591	0.567956	−	0.823059i	$-0.307734\pi$
$887$	33.8304	1.13591	0.567956	+	0.823059i	$0.307734\pi$
$888$	0	0
$889$	−12.5505	−0.420931
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.12848i	0.0377633i
$894$	0	0
$895$	35.1464i	1.17482i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.9350	−0.431406
$900$	0	0
$901$	−12.6061	−0.419971
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.9494i	0.397211i
$906$	0	0
$907$	23.2474i	0.771919i	0.922516	+	0.385959i	$0.126130\pi$
$907$	23.2474i	0.771919i	−0.922516	+	0.385959i	$0.873870\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	49.2117	1.63046	0.815229	−	0.579139i	$-0.196611\pi$
$911$	49.2117	1.63046	0.815229	+	0.579139i	$0.196611\pi$
$912$	0	0
$913$	−3.34847	−0.110818
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7.84961i	− 0.259217i
$918$	0	0
$919$	28.8434i	0.951455i	0.879593	+	0.475727i	$0.157815\pi$
$919$	28.8434i	0.951455i	−0.879593	+	0.475727i	$0.842185\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.54270	0.0836939
$924$	0	0
$925$	−44.0908	−1.44970
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 3.14626i	− 0.103226i	−0.998667	−	0.0516128i	$-0.983564\pi$
$929$	− 3.14626i	− 0.103226i	0.998667	−	0.0516128i	$-0.0164362\pi$
$930$	0	0
$931$	2.44949i	0.0802788i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.1708	0.365326
$936$	0	0
$937$	54.7423	1.78835	0.894177	−	0.447713i	$-0.147761\pi$
$937$	54.7423	1.78835	0.894177	+	0.447713i	$0.147761\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 58.8576i	− 1.91870i	−0.282213	−	0.959352i	$-0.591068\pi$
$941$	− 58.8576i	− 1.91870i	0.282213	−	0.959352i	$-0.408932\pi$
$942$	0	0
$943$	56.9898i	1.85584i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43.4120	1.41070	0.705351	−	0.708859i	$-0.250790\pi$
$947$	43.4120	1.41070	0.705351	+	0.708859i	$0.250790\pi$
$948$	0	0
$949$	4.40408	0.142963
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.81405i	0.123549i	0.998090	+	0.0617745i	$0.0196760\pi$
$953$	3.81405i	0.123549i	−0.998090	+	0.0617745i	$0.980324\pi$
$954$	0	0
$955$	24.6969i	0.799174i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.4422	−0.401779
$960$	0	0
$961$	28.5959	0.922449
$962$	0	0
$963$	0	0
$964$	0	0
$965$	64.8002i	2.08599i
$966$	0	0
$967$	12.8990i	0.414803i	0.978256	+	0.207402i	$0.0665007\pi$
$967$	12.8990i	0.414803i	−0.978256	+	0.207402i	$0.933499\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	39.2015	1.25804	0.629018	−	0.777391i	$-0.283457\pi$
$971$	39.2015	1.25804	0.629018	+	0.777391i	$0.283457\pi$
$972$	0	0
$973$	−0.449490	−0.0144100
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.4626i	1.00658i	0.864118	+	0.503290i	$0.167877\pi$
$977$	31.4626i	1.00658i	−0.864118	+	0.503290i	$0.832123\pi$
$978$	0	0
$979$	2.20204i	0.0703775i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−9.58166	−0.305607	−0.152804	−	0.988257i	$-0.548830\pi$
$983$	−9.58166	−0.305607	−0.152804	+	0.988257i	$0.548830\pi$
$984$	0	0
$985$	31.1464	0.992408
$986$	0	0
$987$	0	0
$988$	0	0
$989$	47.2261i	1.50170i
$990$	0	0
$991$	7.65153i	0.243059i	0.992588	+	0.121529i	$0.0387799\pi$
$991$	7.65153i	0.243059i	−0.992588	+	0.121529i	$0.961220\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	39.9479	1.26643
$996$	0	0
$997$	−41.8434	−1.32519	−0.662596	−	0.748977i	$-0.730545\pi$
$997$	−41.8434	−1.32519	−0.662596	+	0.748977i	$0.730545\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.c.2591.8	yes	8
3.2	odd	2	inner	6048.2.h.c.2591.2	yes	8
4.3	odd	2	inner	6048.2.h.c.2591.7	yes	8
12.11	even	2	inner	6048.2.h.c.2591.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.c.2591.1	✓	8	12.11	even	2	inner
6048.2.h.c.2591.2	yes	8	3.2	odd	2	inner
6048.2.h.c.2591.7	yes	8	4.3	odd	2	inner
6048.2.h.c.2591.8	yes	8	1.1	even	1	trivial

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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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.14626i q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+3.14626i q^{5} +1.00000i q^{7} +1.41421 q^{11} +0.449490 q^{13} -2.51059i q^{17} -2.44949i q^{19} -5.65685 q^{23} -4.89898 q^{25} -8.34242i q^{29} -1.55051i q^{31} -3.14626 q^{35} +9.00000 q^{37} -10.0745i q^{41} -8.34847i q^{43} -0.460702 q^{47} -1.00000 q^{49} -5.02118i q^{53} +4.44949i q^{55} -11.4887 q^{59} -5.55051 q^{61} +1.41421i q^{65} -13.7980i q^{67} +5.65685 q^{71} +9.79796 q^{73} +1.41421i q^{77} -0.348469i q^{79} -2.36773 q^{83} +7.89898 q^{85} +1.55708i q^{89} +0.449490i q^{91} +7.70674 q^{95} +10.2474 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{13} + 72 q^{37} - 8 q^{49} - 64 q^{61} + 24 q^{85} - 16 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 + 72 * q^37 - 8 * q^49 - 64 * q^61 + 24 * q^85 - 16 * q^97

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