LMFDB - Embedded newform 6048.2.h.i.2591.3

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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.3
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.i.2591.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.865222i q^{5} -1.00000i q^{7} +O(q^{10})$	$q-0.865222i q^{5} -1.00000i q^{7} -5.82380 q^{11} +0.727273 q^{13} +7.62531i q^{17} +2.29824i q^{19} -7.50524 q^{23} +4.25139 q^{25} +0.231154i q^{29} -10.7347i q^{31} -0.865222 q^{35} +1.64759 q^{37} -0.229547i q^{41} +6.70919i q^{43} +11.6729 q^{47} -1.00000 q^{49} -2.80475i q^{53} +5.03888i q^{55} +11.0135 q^{59} -9.08990 q^{61} -0.629252i q^{65} -11.2616i q^{67} +9.76188 q^{71} +14.5795 q^{73} +5.82380i q^{77} -4.39759i q^{79} +2.47977 q^{83} +6.59759 q^{85} -3.15706i q^{89} -0.727273i q^{91} +1.98848 q^{95} +1.25455 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 8 q^{13} - 24 q^{25} + 48 q^{37} - 24 q^{49} - 48 q^{61} - 32 q^{73} - 80 q^{85} + 64 q^{97}+O(q^{100}) $ 24 * q - 8 * q^13 - 24 * q^25 + 48 * q^37 - 24 * q^49 - 48 * q^61 - 32 * q^73 - 80 * q^85 + 64 * 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
$5$	− 0.865222i	− 0.386939i	−0.981106	−	0.193469i	$-0.938026\pi$
$5$	− 0.865222i	− 0.386939i	0.981106	−	0.193469i	$-0.0619741\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.82380	−1.75594	−0.877971	−	0.478715i	$-0.841103\pi$
$11$	−5.82380	−1.75594	−0.877971	+	0.478715i	$0.841103\pi$
$12$	0	0
$13$	0.727273	0.201709	0.100855	−	0.994901i	$-0.467842\pi$
$13$	0.727273	0.201709	0.100855	+	0.994901i	$0.467842\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.62531i	1.84941i	0.380684	+	0.924705i	$0.375688\pi$
$17$	7.62531i	1.84941i	−0.380684	+	0.924705i	$0.624312\pi$
$18$	0	0
$19$	2.29824i	0.527252i	0.964625	+	0.263626i	$0.0849184\pi$
$19$	2.29824i	0.527252i	−0.964625	+	0.263626i	$0.915082\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.50524	−1.56495	−0.782476	−	0.622681i	$-0.786043\pi$
$23$	−7.50524	−1.56495	−0.782476	+	0.622681i	$0.786043\pi$
$24$	0	0
$25$	4.25139	0.850278
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.231154i	0.0429243i	0.999770	+	0.0214621i	$0.00683214\pi$
$29$	0.231154i	0.0429243i	−0.999770	+	0.0214621i	$0.993168\pi$
$30$	0	0
$31$	− 10.7347i	− 1.92801i	−0.265885	−	0.964005i	$-0.585664\pi$
$31$	− 10.7347i	− 1.92801i	0.265885	−	0.964005i	$-0.414336\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.865222	−0.146249
$36$	0	0
$37$	1.64759	0.270862	0.135431	−	0.990787i	$-0.456758\pi$
$37$	1.64759	0.270862	0.135431	+	0.990787i	$0.456758\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 0.229547i	− 0.0358493i	−0.999839	−	0.0179246i	$-0.994294\pi$
$41$	− 0.229547i	− 0.0358493i	0.999839	−	0.0179246i	$-0.00570590\pi$
$42$	0	0
$43$	6.70919i	1.02314i	0.859241	+	0.511571i	$0.170936\pi$
$43$	6.70919i	1.02314i	−0.859241	+	0.511571i	$0.829064\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.6729	1.70266	0.851332	−	0.524627i	$-0.175795\pi$
$47$	11.6729	1.70266	0.851332	+	0.524627i	$0.175795\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.80475i	− 0.385263i	−0.981271	−	0.192631i	$-0.938298\pi$
$53$	− 2.80475i	− 0.385263i	0.981271	−	0.192631i	$-0.0617021\pi$
$54$	0	0
$55$	5.03888i	0.679442i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.0135	1.43384	0.716920	−	0.697156i	$-0.245551\pi$
$59$	11.0135	1.43384	0.716920	+	0.697156i	$0.245551\pi$
$60$	0	0
$61$	−9.08990	−1.16384	−0.581921	−	0.813245i	$-0.697699\pi$
$61$	−9.08990	−1.16384	−0.581921	+	0.813245i	$0.697699\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.629252i	− 0.0780492i
$66$	0	0
$67$	− 11.2616i	− 1.37582i	−0.725794	−	0.687912i	$-0.758528\pi$
$67$	− 11.2616i	− 1.37582i	0.725794	−	0.687912i	$-0.241472\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.76188	1.15852	0.579261	−	0.815142i	$-0.303341\pi$
$71$	9.76188	1.15852	0.579261	+	0.815142i	$0.303341\pi$
$72$	0	0
$73$	14.5795	1.70640	0.853201	−	0.521582i	$-0.174658\pi$
$73$	14.5795	1.70640	0.853201	+	0.521582i	$0.174658\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.82380i	0.663683i
$78$	0	0
$79$	− 4.39759i	− 0.494767i	−0.968918	−	0.247384i	$-0.920429\pi$
$79$	− 4.39759i	− 0.494767i	0.968918	−	0.247384i	$-0.0795708\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.47977	0.272190	0.136095	−	0.990696i	$-0.456545\pi$
$83$	2.47977	0.272190	0.136095	+	0.990696i	$0.456545\pi$
$84$	0	0
$85$	6.59759	0.715609
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.15706i	− 0.334648i	−0.985902	−	0.167324i	$-0.946487\pi$
$89$	− 3.15706i	− 0.334648i	0.985902	−	0.167324i	$-0.0535126\pi$
$90$	0	0
$91$	− 0.727273i	− 0.0762389i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.98848	0.204014
$96$	0	0
$97$	1.25455	0.127380	0.0636899	−	0.997970i	$-0.479713\pi$
$97$	1.25455	0.127380	0.0636899	+	0.997970i	$0.479713\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.8863i	1.58075i	0.612626	+	0.790373i	$0.290113\pi$
$101$	15.8863i	1.58075i	−0.612626	+	0.790373i	$0.709887\pi$
$102$	0	0
$103$	0.689526i	0.0679410i	0.999423	+	0.0339705i	$0.0108152\pi$
$103$	0.689526i	0.0679410i	−0.999423	+	0.0339705i	$0.989185\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.3665	−1.38887	−0.694433	−	0.719557i	$-0.744345\pi$
$107$	−14.3665	−1.38887	−0.694433	+	0.719557i	$0.744345\pi$
$108$	0	0
$109$	9.69861	0.928958	0.464479	−	0.885584i	$-0.346242\pi$
$109$	9.69861	0.928958	0.464479	+	0.885584i	$0.346242\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 16.4645i	− 1.54885i	−0.632665	−	0.774425i	$-0.718039\pi$
$113$	− 16.4645i	− 1.54885i	0.632665	−	0.774425i	$-0.281961\pi$
$114$	0	0
$115$	6.49370i	0.605541i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.62531	0.699011
$120$	0	0
$121$	22.9166	2.08333
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 8.00451i	− 0.715945i
$126$	0	0
$127$	− 19.0764i	− 1.69275i	−0.532585	−	0.846377i	$-0.678779\pi$
$127$	− 19.0764i	− 1.69275i	0.532585	−	0.846377i	$-0.321221\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.57149	0.661524	0.330762	−	0.943714i	$-0.392694\pi$
$131$	7.57149	0.661524	0.330762	+	0.943714i	$0.392694\pi$
$132$	0	0
$133$	2.29824	0.199282
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.8040i	1.00849i	0.863562	+	0.504243i	$0.168229\pi$
$137$	11.8040i	1.00849i	−0.863562	+	0.504243i	$0.831771\pi$
$138$	0	0
$139$	− 12.7285i	− 1.07962i	−0.841788	−	0.539808i	$-0.818497\pi$
$139$	− 12.7285i	− 1.07962i	0.841788	−	0.539808i	$-0.181503\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.23549	−0.354189
$144$	0	0
$145$	0.200000	0.0166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 19.3820i	− 1.58783i	−0.608028	−	0.793916i	$-0.708039\pi$
$149$	− 19.3820i	− 1.58783i	0.608028	−	0.793916i	$-0.291961\pi$
$150$	0	0
$151$	13.7766i	1.12113i	0.828111	+	0.560564i	$0.189415\pi$
$151$	13.7766i	1.12113i	−0.828111	+	0.560564i	$0.810585\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.28790	−0.746022
$156$	0	0
$157$	−9.10808	−0.726904	−0.363452	−	0.931613i	$-0.618402\pi$
$157$	−9.10808	−0.726904	−0.363452	+	0.931613i	$0.618402\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.50524i	0.591496i
$162$	0	0
$163$	− 17.1137i	− 1.34045i	−0.742157	−	0.670226i	$-0.766197\pi$
$163$	− 17.1137i	− 1.34045i	0.742157	−	0.670226i	$-0.233803\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.37485	0.648065	0.324033	−	0.946046i	$-0.394961\pi$
$167$	8.37485	0.648065	0.324033	+	0.946046i	$0.394961\pi$
$168$	0	0
$169$	−12.4711	−0.959313
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 13.6871i	− 1.04061i	−0.853980	−	0.520306i	$-0.825818\pi$
$173$	− 13.6871i	− 1.04061i	0.853980	−	0.520306i	$-0.174182\pi$
$174$	0	0
$175$	− 4.25139i	− 0.321375i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.23685	−0.690395	−0.345197	−	0.938530i	$-0.612188\pi$
$179$	−9.23685	−0.690395	−0.345197	+	0.938530i	$0.612188\pi$
$180$	0	0
$181$	5.93006	0.440778	0.220389	−	0.975412i	$-0.429267\pi$
$181$	5.93006	0.440778	0.220389	+	0.975412i	$0.429267\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.42553i	− 0.104807i
$186$	0	0
$187$	− 44.4083i	− 3.24746i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.5565	−1.19798	−0.598992	−	0.800755i	$-0.704432\pi$
$191$	−16.5565	−1.19798	−0.598992	+	0.800755i	$0.704432\pi$
$192$	0	0
$193$	−15.1697	−1.09194	−0.545969	−	0.837806i	$-0.683838\pi$
$193$	−15.1697	−1.09194	−0.545969	+	0.837806i	$0.683838\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.02017i	− 0.500166i	−0.968224	−	0.250083i	$-0.919542\pi$
$197$	− 7.02017i	− 0.500166i	0.968224	−	0.250083i	$-0.0804580\pi$
$198$	0	0
$199$	− 7.43368i	− 0.526960i	−0.964665	−	0.263480i	$-0.915130\pi$
$199$	− 7.43368i	− 0.526960i	0.964665	−	0.263480i	$-0.0848703\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.231154	0.0162239
$204$	0	0
$205$	−0.198609	−0.0138715
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 13.3845i	− 0.925823i
$210$	0	0
$211$	25.9303i	1.78511i	0.450935	+	0.892557i	$0.351090\pi$
$211$	25.9303i	1.78511i	−0.450935	+	0.892557i	$0.648910\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.80494	0.395894
$216$	0	0
$217$	−10.7347	−0.728719
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.54568i	0.373043i
$222$	0	0
$223$	− 22.4305i	− 1.50206i	−0.660269	−	0.751030i	$-0.729558\pi$
$223$	− 22.4305i	− 1.50206i	0.660269	−	0.751030i	$-0.270442\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.2642	−1.34498	−0.672491	−	0.740105i	$-0.734776\pi$
$227$	−20.2642	−1.34498	−0.672491	+	0.740105i	$0.734776\pi$
$228$	0	0
$229$	28.4103	1.87741	0.938703	−	0.344726i	$-0.112028\pi$
$229$	28.4103	1.87741	0.938703	+	0.344726i	$0.112028\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.260864i	0.0170898i	0.999963	+	0.00854488i	$0.00271995\pi$
$233$	0.260864i	0.0170898i	−0.999963	+	0.00854488i	$0.997280\pi$
$234$	0	0
$235$	− 10.0996i	− 0.658827i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	18.1228	1.17227	0.586133	−	0.810215i	$-0.300650\pi$
$239$	18.1228	1.17227	0.586133	+	0.810215i	$0.300650\pi$
$240$	0	0
$241$	24.2341	1.56105	0.780526	−	0.625123i	$-0.214951\pi$
$241$	24.2341	1.56105	0.780526	+	0.625123i	$0.214951\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.865222i	0.0552770i
$246$	0	0
$247$	1.67145i	0.106352i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.9471	−1.32217	−0.661083	−	0.750312i	$-0.729903\pi$
$251$	−20.9471	−1.32217	−0.661083	+	0.750312i	$0.729903\pi$
$252$	0	0
$253$	43.7090	2.74796
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.08927i	− 0.255082i	−0.991833	−	0.127541i	$-0.959292\pi$
$257$	− 4.08927i	− 0.255082i	0.991833	−	0.127541i	$-0.0407084\pi$
$258$	0	0
$259$	− 1.64759i	− 0.102376i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	21.5691	1.33001	0.665005	−	0.746839i	$-0.268429\pi$
$263$	21.5691	1.33001	0.665005	+	0.746839i	$0.268429\pi$
$264$	0	0
$265$	−2.42673	−0.149073
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.6825i	1.13909i	0.821959	+	0.569547i	$0.192881\pi$
$269$	18.6825i	1.13909i	−0.821959	+	0.569547i	$0.807119\pi$
$270$	0	0
$271$	4.52107i	0.274636i	0.990527	+	0.137318i	$0.0438482\pi$
$271$	4.52107i	0.274636i	−0.990527	+	0.137318i	$0.956152\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−24.7592	−1.49304
$276$	0	0
$277$	15.0289	0.902997	0.451499	−	0.892272i	$-0.350890\pi$
$277$	15.0289	0.902997	0.451499	+	0.892272i	$0.350890\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.8228i	1.30184i	0.759147	+	0.650919i	$0.225616\pi$
$281$	21.8228i	1.30184i	−0.759147	+	0.650919i	$0.774384\pi$
$282$	0	0
$283$	11.9277i	0.709031i	0.935050	+	0.354515i	$0.115354\pi$
$283$	11.9277i	0.709031i	−0.935050	+	0.354515i	$0.884646\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.229547	−0.0135498
$288$	0	0
$289$	−41.1454	−2.42032
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.35874i	0.137799i	0.997624	+	0.0688995i	$0.0219488\pi$
$293$	2.35874i	0.137799i	−0.997624	+	0.0688995i	$0.978051\pi$
$294$	0	0
$295$	− 9.52915i	− 0.554808i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.45836	−0.315665
$300$	0	0
$301$	6.70919	0.386711
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.86478i	0.450336i
$306$	0	0
$307$	20.0568i	1.14470i	0.820009	+	0.572351i	$0.193968\pi$
$307$	20.0568i	1.14470i	−0.820009	+	0.572351i	$0.806032\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10.4255	0.591173	0.295587	−	0.955316i	$-0.404485\pi$
$311$	10.4255	0.591173	0.295587	+	0.955316i	$0.404485\pi$
$312$	0	0
$313$	16.0767	0.908710	0.454355	−	0.890821i	$-0.349870\pi$
$313$	16.0767	0.908710	0.454355	+	0.890821i	$0.349870\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 5.40845i	− 0.303769i	−0.988398	−	0.151884i	$-0.951466\pi$
$317$	− 5.40845i	− 0.303769i	0.988398	−	0.151884i	$-0.0485341\pi$
$318$	0	0
$319$	− 1.34620i	− 0.0753725i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−17.5248	−0.975105
$324$	0	0
$325$	3.09192	0.171509
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 11.6729i	− 0.643546i
$330$	0	0
$331$	− 12.5645i	− 0.690606i	−0.938491	−	0.345303i	$-0.887776\pi$
$331$	− 12.5645i	− 0.690606i	0.938491	−	0.345303i	$-0.112224\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.74379	−0.532360
$336$	0	0
$337$	29.1719	1.58910	0.794548	−	0.607202i	$-0.207708\pi$
$337$	29.1719	1.58910	0.794548	+	0.607202i	$0.207708\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	62.5167i	3.38547i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.5974	1.53519	0.767595	−	0.640935i	$-0.221453\pi$
$347$	28.5974	1.53519	0.767595	+	0.640935i	$0.221453\pi$
$348$	0	0
$349$	19.8320	1.06158	0.530792	−	0.847502i	$-0.321895\pi$
$349$	19.8320	1.06158	0.530792	+	0.847502i	$0.321895\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 11.8083i	− 0.628494i	−0.949341	−	0.314247i	$-0.898248\pi$
$353$	− 11.8083i	− 0.628494i	0.949341	−	0.314247i	$-0.101752\pi$
$354$	0	0
$355$	− 8.44620i	− 0.448278i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.5304	0.925220	0.462610	−	0.886562i	$-0.346913\pi$
$359$	17.5304	0.925220	0.462610	+	0.886562i	$0.346913\pi$
$360$	0	0
$361$	13.7181	0.722006
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 12.6145i	− 0.660274i
$366$	0	0
$367$	− 2.40621i	− 0.125603i	−0.998026	−	0.0628017i	$-0.979996\pi$
$367$	− 2.40621i	− 0.125603i	0.998026	−	0.0628017i	$-0.0200036\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.80475	−0.145616
$372$	0	0
$373$	−18.3774	−0.951545	−0.475772	−	0.879568i	$-0.657831\pi$
$373$	−18.3774	−0.951545	−0.475772	+	0.879568i	$0.657831\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.168112i	0.00865823i
$378$	0	0
$379$	− 12.6572i	− 0.650155i	−0.945687	−	0.325077i	$-0.894610\pi$
$379$	− 12.6572i	− 0.650155i	0.945687	−	0.325077i	$-0.105390\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.64584	0.135196	0.0675980	−	0.997713i	$-0.478466\pi$
$383$	2.64584	0.135196	0.0675980	+	0.997713i	$0.478466\pi$
$384$	0	0
$385$	5.03888	0.256805
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.84454i	0.245628i	0.992430	+	0.122814i	$0.0391919\pi$
$389$	4.84454i	0.245628i	−0.992430	+	0.122814i	$0.960808\pi$
$390$	0	0
$391$	− 57.2298i	− 2.89424i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.80489	−0.191445
$396$	0	0
$397$	−19.7055	−0.988991	−0.494495	−	0.869180i	$-0.664647\pi$
$397$	−19.7055	−0.988991	−0.494495	+	0.869180i	$0.664647\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.51371i	0.375217i	0.982244	+	0.187608i	$0.0600736\pi$
$401$	7.51371i	0.375217i	−0.982244	+	0.187608i	$0.939926\pi$
$402$	0	0
$403$	− 7.80706i	− 0.388897i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.59522	−0.475617
$408$	0	0
$409$	3.37679	0.166972	0.0834858	−	0.996509i	$-0.473395\pi$
$409$	3.37679	0.166972	0.0834858	+	0.996509i	$0.473395\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 11.0135i	− 0.541940i
$414$	0	0
$415$	− 2.14555i	− 0.105321i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.74114	−0.231620	−0.115810	−	0.993271i	$-0.536946\pi$
$419$	−4.74114	−0.231620	−0.115810	+	0.993271i	$0.536946\pi$
$420$	0	0
$421$	−2.42905	−0.118385	−0.0591924	−	0.998247i	$-0.518853\pi$
$421$	−2.42905	−0.118385	−0.0591924	+	0.998247i	$0.518853\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	32.4182i	1.57251i
$426$	0	0
$427$	9.08990i	0.439891i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0.485660	0.0233934	0.0116967	−	0.999932i	$-0.496277\pi$
$431$	0.485660	0.0233934	0.0116967	+	0.999932i	$0.496277\pi$
$432$	0	0
$433$	6.61560	0.317926	0.158963	−	0.987285i	$-0.449185\pi$
$433$	6.61560	0.317926	0.158963	+	0.987285i	$0.449185\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 17.2488i	− 0.825123i
$438$	0	0
$439$	− 19.2736i	− 0.919877i	−0.887951	−	0.459939i	$-0.847871\pi$
$439$	− 19.2736i	− 0.919877i	0.887951	−	0.459939i	$-0.152129\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.680297	0.0323219	0.0161609	−	0.999869i	$-0.494856\pi$
$443$	0.680297	0.0323219	0.0161609	+	0.999869i	$0.494856\pi$
$444$	0	0
$445$	−2.73156	−0.129488
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 14.2474i	− 0.672378i	−0.941795	−	0.336189i	$-0.890862\pi$
$449$	− 14.2474i	− 0.672378i	0.941795	−	0.336189i	$-0.109138\pi$
$450$	0	0
$451$	1.33684i	0.0629492i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.629252	−0.0294998
$456$	0	0
$457$	17.6350	0.824930	0.412465	−	0.910973i	$-0.364668\pi$
$457$	17.6350	0.824930	0.412465	+	0.910973i	$0.364668\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 15.8412i	− 0.737798i	−0.929469	−	0.368899i	$-0.879735\pi$
$461$	− 15.8412i	− 0.737798i	0.929469	−	0.368899i	$-0.120265\pi$
$462$	0	0
$463$	30.1324i	1.40037i	0.713961	+	0.700186i	$0.246899\pi$
$463$	30.1324i	1.40037i	−0.713961	+	0.700186i	$0.753101\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.2405	−0.982893	−0.491446	−	0.870908i	$-0.663532\pi$
$467$	−21.2405	−0.982893	−0.491446	+	0.870908i	$0.663532\pi$
$468$	0	0
$469$	−11.2616	−0.520013
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 39.0730i	− 1.79658i
$474$	0	0
$475$	9.77070i	0.448311i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16.8779	0.771172	0.385586	−	0.922672i	$-0.373999\pi$
$479$	16.8779	0.771172	0.385586	+	0.922672i	$0.373999\pi$
$480$	0	0
$481$	1.19825	0.0546353
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.08546i	− 0.0492882i
$486$	0	0
$487$	34.5687i	1.56646i	0.621735	+	0.783228i	$0.286428\pi$
$487$	34.5687i	1.56646i	−0.621735	+	0.783228i	$0.713572\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.298579	0.0134747	0.00673735	−	0.999977i	$-0.497855\pi$
$491$	0.298579	0.0134747	0.00673735	+	0.999977i	$0.497855\pi$
$492$	0	0
$493$	−1.76262	−0.0793846
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 9.76188i	− 0.437880i
$498$	0	0
$499$	− 16.8857i	− 0.755908i	−0.925824	−	0.377954i	$-0.876628\pi$
$499$	− 16.8857i	− 0.755908i	0.925824	−	0.377954i	$-0.123372\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.33762	−0.104230	−0.0521148	−	0.998641i	$-0.516596\pi$
$503$	−2.33762	−0.104230	−0.0521148	+	0.998641i	$0.516596\pi$
$504$	0	0
$505$	13.7452	0.611652
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 19.1664i	− 0.849538i	−0.905302	−	0.424769i	$-0.860355\pi$
$509$	− 19.1664i	− 0.849538i	0.905302	−	0.424769i	$-0.139645\pi$
$510$	0	0
$511$	− 14.5795i	− 0.644959i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.596593	0.0262890
$516$	0	0
$517$	−67.9805	−2.98978
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.95754i	0.304815i	0.988318	+	0.152408i	$0.0487027\pi$
$521$	6.95754i	0.304815i	−0.988318	+	0.152408i	$0.951297\pi$
$522$	0	0
$523$	− 6.39416i	− 0.279597i	−0.990180	−	0.139798i	$-0.955355\pi$
$523$	− 6.39416i	− 0.279597i	0.990180	−	0.139798i	$-0.0446455\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	81.8555	3.56568
$528$	0	0
$529$	33.3287	1.44907
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 0.166944i	− 0.00723113i
$534$	0	0
$535$	12.4303i	0.537407i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.82380	0.250849
$540$	0	0
$541$	22.4961	0.967183	0.483591	−	0.875294i	$-0.339332\pi$
$541$	22.4961	0.967183	0.483591	+	0.875294i	$0.339332\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 8.39145i	− 0.359450i
$546$	0	0
$547$	1.17336i	0.0501692i	0.999685	+	0.0250846i	$0.00798552\pi$
$547$	1.17336i	0.0501692i	−0.999685	+	0.0250846i	$0.992014\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.531248	−0.0226319
$552$	0	0
$553$	−4.39759	−0.187004
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 6.87547i	− 0.291323i	−0.989334	−	0.145662i	$-0.953469\pi$
$557$	− 6.87547i	− 0.291323i	0.989334	−	0.145662i	$-0.0465310\pi$
$558$	0	0
$559$	4.87941i	0.206377i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.4969	−0.821696	−0.410848	−	0.911704i	$-0.634767\pi$
$563$	−19.4969	−0.821696	−0.410848	+	0.911704i	$0.634767\pi$
$564$	0	0
$565$	−14.2455	−0.599311
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.1191i	0.466136i	0.972460	+	0.233068i	$0.0748765\pi$
$569$	11.1191i	0.466136i	−0.972460	+	0.233068i	$0.925124\pi$
$570$	0	0
$571$	− 19.8397i	− 0.830267i	−0.909761	−	0.415133i	$-0.863735\pi$
$571$	− 19.8397i	− 0.830267i	0.909761	−	0.415133i	$-0.136265\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−31.9077	−1.33064
$576$	0	0
$577$	19.5159	0.812457	0.406229	−	0.913772i	$-0.366844\pi$
$577$	19.5159	0.812457	0.406229	+	0.913772i	$0.366844\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 2.47977i	− 0.102878i
$582$	0	0
$583$	16.3343i	0.676499i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.10003	−0.169227	−0.0846133	−	0.996414i	$-0.526965\pi$
$587$	−4.10003	−0.169227	−0.0846133	+	0.996414i	$0.526965\pi$
$588$	0	0
$589$	24.6709	1.01655
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 39.1396i	− 1.60727i	−0.595122	−	0.803636i	$-0.702896\pi$
$593$	− 39.1396i	− 1.60727i	0.595122	−	0.803636i	$-0.297104\pi$
$594$	0	0
$595$	− 6.59759i	− 0.270475i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	22.2716	0.909992	0.454996	−	0.890493i	$-0.349641\pi$
$599$	22.2716	0.909992	0.454996	+	0.890493i	$0.349641\pi$
$600$	0	0
$601$	6.57598	0.268240	0.134120	−	0.990965i	$-0.457179\pi$
$601$	6.57598	0.268240	0.134120	+	0.990965i	$0.457179\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 19.8280i	− 0.806121i
$606$	0	0
$607$	29.1564i	1.18342i	0.806150	+	0.591711i	$0.201547\pi$
$607$	29.1564i	1.18342i	−0.806150	+	0.591711i	$0.798453\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.48937	0.343443
$612$	0	0
$613$	24.4722	0.988422	0.494211	−	0.869342i	$-0.335457\pi$
$613$	24.4722	0.988422	0.494211	+	0.869342i	$0.335457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.62276i	0.186105i	0.995661	+	0.0930526i	$0.0296625\pi$
$617$	4.62276i	0.186105i	−0.995661	+	0.0930526i	$0.970338\pi$
$618$	0	0
$619$	28.6712i	1.15239i	0.817312	+	0.576196i	$0.195463\pi$
$619$	28.6712i	1.15239i	−0.817312	+	0.576196i	$0.804537\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.15706	−0.126485
$624$	0	0
$625$	14.3313	0.573251
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.5634i	0.500935i
$630$	0	0
$631$	2.30009i	0.0915653i	0.998951	+	0.0457826i	$0.0145782\pi$
$631$	2.30009i	0.0915653i	−0.998951	+	0.0457826i	$0.985422\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16.5053	−0.654992
$636$	0	0
$637$	−0.727273	−0.0288156
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 27.1632i	− 1.07288i	−0.843938	−	0.536440i	$-0.819769\pi$
$641$	− 27.1632i	− 1.07288i	0.843938	−	0.536440i	$-0.180231\pi$
$642$	0	0
$643$	− 37.0176i	− 1.45983i	−0.683536	−	0.729917i	$-0.739559\pi$
$643$	− 37.0176i	− 1.45983i	0.683536	−	0.729917i	$-0.260441\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.3022	0.758847	0.379423	−	0.925223i	$-0.376122\pi$
$647$	19.3022	0.758847	0.379423	+	0.925223i	$0.376122\pi$
$648$	0	0
$649$	−64.1406	−2.51774
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 8.42983i	− 0.329885i	−0.986303	−	0.164942i	$-0.947256\pi$
$653$	− 8.42983i	− 0.329885i	0.986303	−	0.164942i	$-0.0527438\pi$
$654$	0	0
$655$	− 6.55102i	− 0.255970i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.8983	0.736172	0.368086	−	0.929792i	$-0.380013\pi$
$659$	18.8983	0.736172	0.368086	+	0.929792i	$0.380013\pi$
$660$	0	0
$661$	19.1750	0.745820	0.372910	−	0.927868i	$-0.378360\pi$
$661$	19.1750	0.745820	0.372910	+	0.927868i	$0.378360\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.98848i	− 0.0771101i
$666$	0	0
$667$	− 1.73487i	− 0.0671744i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	52.9377	2.04364
$672$	0	0
$673$	14.3064	0.551471	0.275736	−	0.961234i	$-0.411079\pi$
$673$	14.3064	0.551471	0.275736	+	0.961234i	$0.411079\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.1600i	0.659512i	0.944066	+	0.329756i	$0.106966\pi$
$677$	17.1600i	0.659512i	−0.944066	+	0.329756i	$0.893034\pi$
$678$	0	0
$679$	− 1.25455i	− 0.0481451i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.4867	−0.707375	−0.353687	−	0.935364i	$-0.615072\pi$
$683$	−18.4867	−0.707375	−0.353687	+	0.935364i	$0.615072\pi$
$684$	0	0
$685$	10.2131	0.390223
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 2.03982i	− 0.0777110i
$690$	0	0
$691$	19.4447i	0.739709i	0.929090	+	0.369855i	$0.120593\pi$
$691$	19.4447i	0.739709i	−0.929090	+	0.369855i	$0.879407\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.0130	−0.417746
$696$	0	0
$697$	1.75037	0.0663000
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.3467i	1.07064i	0.844649	+	0.535320i	$0.179809\pi$
$701$	28.3467i	1.07064i	−0.844649	+	0.535320i	$0.820191\pi$
$702$	0	0
$703$	3.78655i	0.142812i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.8863	0.597466
$708$	0	0
$709$	−25.4347	−0.955220	−0.477610	−	0.878572i	$-0.658497\pi$
$709$	−25.4347	−0.955220	−0.477610	+	0.878572i	$0.658497\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	80.5665i	3.01724i
$714$	0	0
$715$	3.66464i	0.137050i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27.3320	−1.01931	−0.509656	−	0.860378i	$-0.670228\pi$
$719$	−27.3320	−1.01931	−0.509656	+	0.860378i	$0.670228\pi$
$720$	0	0
$721$	0.689526	0.0256793
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.982728i	0.0364976i
$726$	0	0
$727$	16.9089i	0.627117i	0.949569	+	0.313559i	$0.101521\pi$
$727$	16.9089i	0.627117i	−0.949569	+	0.313559i	$0.898479\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−51.1597	−1.89221
$732$	0	0
$733$	−32.8540	−1.21349	−0.606745	−	0.794896i	$-0.707525\pi$
$733$	−32.8540	−1.21349	−0.606745	+	0.794896i	$0.707525\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	65.5853i	2.41587i
$738$	0	0
$739$	− 35.6492i	− 1.31138i	−0.755032	−	0.655688i	$-0.772378\pi$
$739$	− 35.6492i	− 1.31138i	0.755032	−	0.655688i	$-0.227622\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.9725	1.31970	0.659852	−	0.751396i	$-0.270619\pi$
$743$	35.9725	1.31970	0.659852	+	0.751396i	$0.270619\pi$
$744$	0	0
$745$	−16.7697	−0.614394
$746$	0	0
$747$	0	0
$748$	0	0
$749$	14.3665i	0.524942i
$750$	0	0
$751$	− 32.6600i	− 1.19178i	−0.803065	−	0.595891i	$-0.796799\pi$
$751$	− 32.6600i	− 1.19178i	0.803065	−	0.595891i	$-0.203201\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.9199	0.433808
$756$	0	0
$757$	1.73551	0.0630783	0.0315391	−	0.999503i	$-0.489959\pi$
$757$	1.73551	0.0630783	0.0315391	+	0.999503i	$0.489959\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.04240i	0.182787i	0.995815	+	0.0913935i	$0.0291321\pi$
$761$	5.04240i	0.182787i	−0.995815	+	0.0913935i	$0.970868\pi$
$762$	0	0
$763$	− 9.69861i	− 0.351113i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00984	0.289219
$768$	0	0
$769$	−15.5543	−0.560902	−0.280451	−	0.959868i	$-0.590484\pi$
$769$	−15.5543	−0.560902	−0.280451	+	0.959868i	$0.590484\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27.4351i	0.986772i	0.869810	+	0.493386i	$0.164241\pi$
$773$	27.4351i	0.986772i	−0.869810	+	0.493386i	$0.835759\pi$
$774$	0	0
$775$	− 45.6374i	− 1.63934i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.527554	0.0189016
$780$	0	0
$781$	−56.8512	−2.03430
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.88051i	0.281267i
$786$	0	0
$787$	− 12.8128i	− 0.456728i	−0.973576	−	0.228364i	$-0.926662\pi$
$787$	− 12.8128i	− 0.456728i	0.973576	−	0.228364i	$-0.0733376\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16.4645	−0.585411
$792$	0	0
$793$	−6.61084	−0.234758
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 21.2364i	− 0.752232i	−0.926573	−	0.376116i	$-0.877259\pi$
$797$	− 21.2364i	− 0.752232i	0.926573	−	0.376116i	$-0.122741\pi$
$798$	0	0
$799$	89.0093i	3.14892i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−84.9081	−2.99634
$804$	0	0
$805$	6.49370	0.228873
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22.0978i	0.776916i	0.921467	+	0.388458i	$0.126992\pi$
$809$	22.0978i	0.776916i	−0.921467	+	0.388458i	$0.873008\pi$
$810$	0	0
$811$	− 7.97299i	− 0.279969i	−0.990154	−	0.139985i	$-0.955295\pi$
$811$	− 7.97299i	− 0.279969i	0.990154	−	0.139985i	$-0.0447054\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14.8072	−0.518673
$816$	0	0
$817$	−15.4193	−0.539454
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.2052i	0.984368i	0.870491	+	0.492184i	$0.163801\pi$
$821$	28.2052i	0.984368i	−0.870491	+	0.492184i	$0.836199\pi$
$822$	0	0
$823$	19.6354i	0.684448i	0.939618	+	0.342224i	$0.111180\pi$
$823$	19.6354i	0.684448i	−0.939618	+	0.342224i	$0.888820\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.34720	0.0468466	0.0234233	−	0.999726i	$-0.492543\pi$
$827$	1.34720	0.0468466	0.0234233	+	0.999726i	$0.492543\pi$
$828$	0	0
$829$	30.0701	1.04438	0.522190	−	0.852829i	$-0.325115\pi$
$829$	30.0701	1.04438	0.522190	+	0.852829i	$0.325115\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 7.62531i	− 0.264201i
$834$	0	0
$835$	− 7.24610i	− 0.250762i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.1288	−0.418731	−0.209366	−	0.977837i	$-0.567140\pi$
$839$	−12.1288	−0.418731	−0.209366	+	0.977837i	$0.567140\pi$
$840$	0	0
$841$	28.9466	0.998158
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.7902i	0.371196i
$846$	0	0
$847$	− 22.9166i	− 0.787424i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−12.3656	−0.423886
$852$	0	0
$853$	−46.5388	−1.59346	−0.796729	−	0.604337i	$-0.793438\pi$
$853$	−46.5388	−1.59346	−0.796729	+	0.604337i	$0.793438\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 43.0088i	− 1.46915i	−0.678526	−	0.734577i	$-0.737381\pi$
$857$	− 43.0088i	− 1.46915i	0.678526	−	0.734577i	$-0.262619\pi$
$858$	0	0
$859$	9.13230i	0.311590i	0.987789	+	0.155795i	$0.0497939\pi$
$859$	9.13230i	0.311590i	−0.987789	+	0.155795i	$0.950206\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14.0928	0.479723	0.239861	−	0.970807i	$-0.422898\pi$
$863$	14.0928	0.479723	0.239861	+	0.970807i	$0.422898\pi$
$864$	0	0
$865$	−11.8424	−0.402654
$866$	0	0
$867$	0	0
$868$	0	0
$869$	25.6107i	0.868782i
$870$	0	0
$871$	− 8.19026i	− 0.277516i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.00451	−0.270602
$876$	0	0
$877$	46.6612	1.57564	0.787818	−	0.615908i	$-0.211211\pi$
$877$	46.6612	1.57564	0.787818	+	0.615908i	$0.211211\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.1212i	0.677901i	0.940804	+	0.338950i	$0.110072\pi$
$881$	20.1212i	0.677901i	−0.940804	+	0.338950i	$0.889928\pi$
$882$	0	0
$883$	47.2857i	1.59129i	0.605763	+	0.795645i	$0.292868\pi$
$883$	47.2857i	1.59129i	−0.605763	+	0.795645i	$0.707132\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.0988	0.372663	0.186331	−	0.982487i	$-0.440340\pi$
$887$	11.0988	0.372663	0.186331	+	0.982487i	$0.440340\pi$
$888$	0	0
$889$	−19.0764	−0.639801
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26.8270i	0.897732i
$894$	0	0
$895$	7.99193i	0.267141i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.48137	0.0827584
$900$	0	0
$901$	21.3871	0.712509
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 5.13081i	− 0.170554i
$906$	0	0
$907$	39.6177i	1.31549i	0.753243	+	0.657743i	$0.228489\pi$
$907$	39.6177i	1.31549i	−0.753243	+	0.657743i	$0.771511\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.1527	1.03213	0.516067	−	0.856548i	$-0.327396\pi$
$911$	31.1527	1.03213	0.516067	+	0.856548i	$0.327396\pi$
$912$	0	0
$913$	−14.4417	−0.477949
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7.57149i	− 0.250033i
$918$	0	0
$919$	− 28.4960i	− 0.939997i	−0.882667	−	0.469999i	$-0.844254\pi$
$919$	− 28.4960i	− 0.939997i	0.882667	−	0.469999i	$-0.155746\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.09955	0.233685
$924$	0	0
$925$	7.00454	0.230308
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 36.4726i	− 1.19663i	−0.801261	−	0.598314i	$-0.795837\pi$
$929$	− 36.4726i	− 1.19663i	0.801261	−	0.598314i	$-0.204163\pi$
$930$	0	0
$931$	− 2.29824i	− 0.0753217i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−38.4230	−1.25657
$936$	0	0
$937$	−26.6030	−0.869081	−0.434541	−	0.900652i	$-0.643089\pi$
$937$	−26.6030	−0.869081	−0.434541	+	0.900652i	$0.643089\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 31.5293i	− 1.02783i	−0.857842	−	0.513913i	$-0.828195\pi$
$941$	− 31.5293i	− 1.02783i	0.857842	−	0.513913i	$-0.171805\pi$
$942$	0	0
$943$	1.72281i	0.0561024i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.7118	1.55042	0.775212	−	0.631701i	$-0.217643\pi$
$947$	47.7118	1.55042	0.775212	+	0.631701i	$0.217643\pi$
$948$	0	0
$949$	10.6033	0.344197
$950$	0	0
$951$	0	0
$952$	0	0
$953$	49.2910i	1.59669i	0.602200	+	0.798345i	$0.294291\pi$
$953$	49.2910i	1.59669i	−0.602200	+	0.798345i	$0.705709\pi$
$954$	0	0
$955$	14.3250i	0.463547i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.8040	0.381172
$960$	0	0
$961$	−84.2338	−2.71722
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.1251i	0.422513i
$966$	0	0
$967$	40.7797i	1.31139i	0.755027	+	0.655693i	$0.227624\pi$
$967$	40.7797i	1.31139i	−0.755027	+	0.655693i	$0.772376\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.18153	0.0379171	0.0189586	−	0.999820i	$-0.493965\pi$
$971$	1.18153	0.0379171	0.0189586	+	0.999820i	$0.493965\pi$
$972$	0	0
$973$	−12.7285	−0.408057
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 48.6889i	− 1.55769i	−0.627214	−	0.778847i	$-0.715805\pi$
$977$	− 48.6889i	− 1.55769i	0.627214	−	0.778847i	$-0.284195\pi$
$978$	0	0
$979$	18.3861i	0.587622i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0.453964	0.0144792	0.00723960	−	0.999974i	$-0.497696\pi$
$983$	0.453964	0.0144792	0.00723960	+	0.999974i	$0.497696\pi$
$984$	0	0
$985$	−6.07401	−0.193534
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 50.3541i	− 1.60117i
$990$	0	0
$991$	5.48824i	0.174340i	0.996193	+	0.0871698i	$0.0277823\pi$
$991$	5.48824i	0.174340i	−0.996193	+	0.0871698i	$0.972218\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−6.43178	−0.203901
$996$	0	0
$997$	17.4879	0.553846	0.276923	−	0.960892i	$-0.410685\pi$
$997$	17.4879	0.553846	0.276923	+	0.960892i	$0.410685\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.i.2591.3	✓	24
3.2	odd	2	inner	6048.2.h.i.2591.22	yes	24
4.3	odd	2	inner	6048.2.h.i.2591.21	yes	24
12.11	even	2	inner	6048.2.h.i.2591.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.i.2591.3	✓	24	1.1	even	1	trivial
6048.2.h.i.2591.4	yes	24	12.11	even	2	inner
6048.2.h.i.2591.21	yes	24	4.3	odd	2	inner
6048.2.h.i.2591.22	yes	24	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.865222i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-0.865222i q^{5} -1.00000i q^{7} -5.82380 q^{11} +0.727273 q^{13} +7.62531i q^{17} +2.29824i q^{19} -7.50524 q^{23} +4.25139 q^{25} +0.231154i q^{29} -10.7347i q^{31} -0.865222 q^{35} +1.64759 q^{37} -0.229547i q^{41} +6.70919i q^{43} +11.6729 q^{47} -1.00000 q^{49} -2.80475i q^{53} +5.03888i q^{55} +11.0135 q^{59} -9.08990 q^{61} -0.629252i q^{65} -11.2616i q^{67} +9.76188 q^{71} +14.5795 q^{73} +5.82380i q^{77} -4.39759i q^{79} +2.47977 q^{83} +6.59759 q^{85} -3.15706i q^{89} -0.727273i q^{91} +1.98848 q^{95} +1.25455 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 8 q^{13} - 24 q^{25} + 48 q^{37} - 24 q^{49} - 48 q^{61} - 32 q^{73} - 80 q^{85} + 64 q^{97}+O(q^{100}) \) 24 * q - 8 * q^13 - 24 * q^25 + 48 * q^37 - 24 * q^49 - 48 * q^61 - 32 * q^73 - 80 * q^85 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more