LMFDB - Embedded newform 6048.2.c.d.3025.13

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:	$16$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.13
Root			$0.891007 + 0.453990i$ of defining polynomial
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.d.3025.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.63506i q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.63506i q^{5} +1.00000 q^{7} -4.29757i q^{11} -0.0480111i q^{13} +3.16228 q^{17} +0.726543i q^{19} -5.28146 q^{23} -1.94356 q^{25} +8.00208i q^{29} -4.90868 q^{31} +2.63506i q^{35} +9.10736i q^{37} -1.45412 q^{41} -8.85816i q^{43} -9.23016 q^{47} +1.00000 q^{49} +3.14726i q^{53} +11.3244 q^{55} +5.90515i q^{59} -2.19577i q^{61} +0.126512 q^{65} +8.55452i q^{67} -3.86725 q^{71} -7.56044 q^{73} -4.29757i q^{77} +14.6139 q^{79} +9.05240i q^{83} +8.33280i q^{85} -5.15394 q^{89} -0.0480111i q^{91} -1.91449 q^{95} +12.1803 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{7}+O(q^{10}) $ 16 * q + 16 * q^7	$ 16 q + 16 q^{7} - 32 q^{25} - 40 q^{31} + 16 q^{49} + 72 q^{55} + 24 q^{73} - 24 q^{79} + 16 q^{97}+O(q^{100}) $ 16 * q + 16 * q^7 - 32 * q^25 - 40 * q^31 + 16 * q^49 + 72 * q^55 + 24 * q^73 - 24 * q^79 + 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
$5$	2.63506i	1.17844i	0.807974	+	0.589218i	$0.200564\pi$
$5$	2.63506i	1.17844i	−0.807974	+	0.589218i	$0.799436\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.29757i	− 1.29577i	−0.761740	−	0.647883i	$-0.775654\pi$
$11$	− 4.29757i	− 1.29577i	0.761740	−	0.647883i	$-0.224346\pi$
$12$	0	0
$13$	− 0.0480111i	− 0.0133159i	−0.999978	−	0.00665794i	$-0.997881\pi$
$13$	− 0.0480111i	− 0.0133159i	0.999978	−	0.00665794i	$-0.00211930\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.16228	0.766965	0.383482	−	0.923548i	$-0.374725\pi$
$17$	3.16228	0.766965	0.383482	+	0.923548i	$0.374725\pi$
$18$	0	0
$19$	0.726543i	0.166680i	0.996521	+	0.0833401i	$0.0265588\pi$
$19$	0.726543i	0.166680i	−0.996521	+	0.0833401i	$0.973441\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.28146	−1.10126	−0.550631	−	0.834749i	$-0.685613\pi$
$23$	−5.28146	−1.10126	−0.550631	+	0.834749i	$0.685613\pi$
$24$	0	0
$25$	−1.94356	−0.388712
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.00208i	1.48595i	0.669319	+	0.742975i	$0.266586\pi$
$29$	8.00208i	1.48595i	−0.669319	+	0.742975i	$0.733414\pi$
$30$	0	0
$31$	−4.90868	−0.881625	−0.440812	−	0.897599i	$-0.645310\pi$
$31$	−4.90868	−0.881625	−0.440812	+	0.897599i	$0.645310\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.63506i	0.445407i
$36$	0	0
$37$	9.10736i	1.49724i	0.662999	+	0.748620i	$0.269283\pi$
$37$	9.10736i	1.49724i	−0.662999	+	0.748620i	$0.730717\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.45412	−0.227096	−0.113548	−	0.993533i	$-0.536222\pi$
$41$	−1.45412	−0.227096	−0.113548	+	0.993533i	$0.536222\pi$
$42$	0	0
$43$	− 8.85816i	− 1.35086i	−0.737426	−	0.675428i	$-0.763959\pi$
$43$	− 8.85816i	− 1.35086i	0.737426	−	0.675428i	$-0.236041\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.23016	−1.34636	−0.673179	−	0.739480i	$-0.735072\pi$
$47$	−9.23016	−1.34636	−0.673179	+	0.739480i	$0.735072\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.14726i	0.432309i	0.976359	+	0.216154i	$0.0693515\pi$
$53$	3.14726i	0.432309i	−0.976359	+	0.216154i	$0.930649\pi$
$54$	0	0
$55$	11.3244	1.52698
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.90515i	0.768785i	0.923170	+	0.384392i	$0.125589\pi$
$59$	5.90515i	0.768785i	−0.923170	+	0.384392i	$0.874411\pi$
$60$	0	0
$61$	− 2.19577i	− 0.281140i	−0.990071	−	0.140570i	$-0.955106\pi$
$61$	− 2.19577i	− 0.281140i	0.990071	−	0.140570i	$-0.0448935\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.126512	0.0156919
$66$	0	0
$67$	8.55452i	1.04510i	0.852608	+	0.522550i	$0.175019\pi$
$67$	8.55452i	1.04510i	−0.852608	+	0.522550i	$0.824981\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.86725	−0.458958	−0.229479	−	0.973314i	$-0.573702\pi$
$71$	−3.86725	−0.458958	−0.229479	+	0.973314i	$0.573702\pi$
$72$	0	0
$73$	−7.56044	−0.884883	−0.442441	−	0.896797i	$-0.645888\pi$
$73$	−7.56044	−0.884883	−0.442441	+	0.896797i	$0.645888\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.29757i	− 0.489754i
$78$	0	0
$79$	14.6139	1.64419	0.822094	−	0.569352i	$-0.192806\pi$
$79$	14.6139	1.64419	0.822094	+	0.569352i	$0.192806\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.05240i	0.993631i	0.867856	+	0.496815i	$0.165497\pi$
$83$	9.05240i	0.993631i	−0.867856	+	0.496815i	$0.834503\pi$
$84$	0	0
$85$	8.33280i	0.903819i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.15394	−0.546317	−0.273159	−	0.961969i	$-0.588068\pi$
$89$	−5.15394	−0.546317	−0.273159	+	0.961969i	$0.588068\pi$
$90$	0	0
$91$	− 0.0480111i	− 0.00503293i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.91449	−0.196422
$96$	0	0
$97$	12.1803	1.23673	0.618363	−	0.785893i	$-0.287796\pi$
$97$	12.1803	1.23673	0.618363	+	0.785893i	$0.287796\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 4.35250i	− 0.433090i	−0.976273	−	0.216545i	$-0.930521\pi$
$101$	− 4.35250i	− 0.433090i	0.976273	−	0.216545i	$-0.0694788\pi$
$102$	0	0
$103$	12.3214	1.21406	0.607030	−	0.794679i	$-0.292361\pi$
$103$	12.3214	1.21406	0.607030	+	0.794679i	$0.292361\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.69053i	0.936819i	0.883512	+	0.468409i	$0.155173\pi$
$107$	9.69053i	0.936819i	−0.883512	+	0.468409i	$0.844827\pi$
$108$	0	0
$109$	9.08902i	0.870570i	0.900293	+	0.435285i	$0.143352\pi$
$109$	9.08902i	0.870570i	−0.900293	+	0.435285i	$0.856648\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.5205	1.27190	0.635951	−	0.771729i	$-0.280608\pi$
$113$	13.5205	1.27190	0.635951	+	0.771729i	$0.280608\pi$
$114$	0	0
$115$	− 13.9170i	− 1.29777i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.16228	0.289886
$120$	0	0
$121$	−7.46912	−0.679011
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.05391i	0.720364i
$126$	0	0
$127$	10.9792	0.974242	0.487121	−	0.873334i	$-0.338047\pi$
$127$	10.9792	0.974242	0.487121	+	0.873334i	$0.338047\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 0.770817i	− 0.0673466i	−0.999433	−	0.0336733i	$-0.989279\pi$
$131$	− 0.770817i	− 0.0673466i	0.999433	−	0.0336733i	$-0.0107206\pi$
$132$	0	0
$133$	0.726543i	0.0629992i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.9782	−0.937933	−0.468967	−	0.883216i	$-0.655374\pi$
$137$	−10.9782	−0.937933	−0.468967	+	0.883216i	$0.655374\pi$
$138$	0	0
$139$	22.0189i	1.86762i	0.357767	+	0.933811i	$0.383538\pi$
$139$	22.0189i	1.86762i	−0.357767	+	0.933811i	$0.616462\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.206331	−0.0172543
$144$	0	0
$145$	−21.0860	−1.75110
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.06643i	0.415058i	0.978229	+	0.207529i	$0.0665422\pi$
$149$	5.06643i	0.415058i	−0.978229	+	0.207529i	$0.933458\pi$
$150$	0	0
$151$	6.94054	0.564813	0.282407	−	0.959295i	$-0.408867\pi$
$151$	6.94054	0.564813	0.282407	+	0.959295i	$0.408867\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 12.9347i	− 1.03894i
$156$	0	0
$157$	− 18.4181i	− 1.46992i	−0.678109	−	0.734962i	$-0.737200\pi$
$157$	− 18.4181i	− 1.46992i	0.678109	−	0.734962i	$-0.262800\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.28146	−0.416238
$162$	0	0
$163$	10.5949i	0.829859i	0.909854	+	0.414929i	$0.136194\pi$
$163$	10.5949i	0.829859i	−0.909854	+	0.414929i	$0.863806\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.3800	−1.34490	−0.672451	−	0.740142i	$-0.734758\pi$
$167$	−17.3800	−1.34490	−0.672451	+	0.740142i	$0.734758\pi$
$168$	0	0
$169$	12.9977	0.999823
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.9078i	1.74165i	0.491595	+	0.870824i	$0.336414\pi$
$173$	22.9078i	1.74165i	−0.491595	+	0.870824i	$0.663586\pi$
$174$	0	0
$175$	−1.94356	−0.146919
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.6267i	1.61646i	0.588869	+	0.808228i	$0.299573\pi$
$179$	21.6267i	1.61646i	−0.588869	+	0.808228i	$0.700427\pi$
$180$	0	0
$181$	− 12.6144i	− 0.937619i	−0.883299	−	0.468810i	$-0.844683\pi$
$181$	− 12.6144i	− 0.937619i	0.883299	−	0.468810i	$-0.155317\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−23.9985	−1.76440
$186$	0	0
$187$	− 13.5901i	− 0.993808i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.5578	−0.981006	−0.490503	−	0.871439i	$-0.663187\pi$
$191$	−13.5578	−0.981006	−0.490503	+	0.871439i	$0.663187\pi$
$192$	0	0
$193$	−26.0614	−1.87594	−0.937971	−	0.346713i	$-0.887298\pi$
$193$	−26.0614	−1.87594	−0.937971	+	0.346713i	$0.887298\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.18922i	− 0.227222i	−0.993525	−	0.113611i	$-0.963758\pi$
$197$	− 3.18922i	− 0.227222i	0.993525	−	0.113611i	$-0.0362418\pi$
$198$	0	0
$199$	−25.2129	−1.78730	−0.893648	−	0.448768i	$-0.851863\pi$
$199$	−25.2129	−1.78730	−0.893648	+	0.448768i	$0.851863\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.00208i	0.561636i
$204$	0	0
$205$	− 3.83171i	− 0.267618i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.12237	0.215979
$210$	0	0
$211$	5.25731i	0.361928i	0.983490	+	0.180964i	$0.0579218\pi$
$211$	5.25731i	0.361928i	−0.983490	+	0.180964i	$0.942078\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	23.3418	1.59190
$216$	0	0
$217$	−4.90868	−0.333223
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 0.151824i	− 0.0102128i
$222$	0	0
$223$	−2.27095	−0.152074	−0.0760370	−	0.997105i	$-0.524227\pi$
$223$	−2.27095	−0.152074	−0.0760370	+	0.997105i	$0.524227\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.44780i	− 0.0960940i	−0.998845	−	0.0480470i	$-0.984700\pi$
$227$	− 1.44780i	− 0.0960940i	0.998845	−	0.0480470i	$-0.0152997\pi$
$228$	0	0
$229$	− 13.1128i	− 0.866516i	−0.901270	−	0.433258i	$-0.857364\pi$
$229$	− 13.1128i	− 0.866516i	0.901270	−	0.433258i	$-0.142636\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.6984	0.831903	0.415951	−	0.909387i	$-0.363449\pi$
$233$	12.6984	0.831903	0.415951	+	0.909387i	$0.363449\pi$
$234$	0	0
$235$	− 24.3221i	− 1.58660i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.1437	−0.979564	−0.489782	−	0.871845i	$-0.662924\pi$
$239$	−15.1437	−0.979564	−0.489782	+	0.871845i	$0.662924\pi$
$240$	0	0
$241$	−17.0920	−1.10099	−0.550497	−	0.834837i	$-0.685562\pi$
$241$	−17.0920	−1.10099	−0.550497	+	0.834837i	$0.685562\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.63506i	0.168348i
$246$	0	0
$247$	0.0348821	0.00221949
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.7844i	0.743822i	0.928268	+	0.371911i	$0.121297\pi$
$251$	11.7844i	0.743822i	−0.928268	+	0.371911i	$0.878703\pi$
$252$	0	0
$253$	22.6975i	1.42698i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.8905	−0.928847	−0.464423	−	0.885613i	$-0.653738\pi$
$257$	−14.8905	−0.928847	−0.464423	+	0.885613i	$0.653738\pi$
$258$	0	0
$259$	9.10736i	0.565904i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.1032	1.05463	0.527315	−	0.849670i	$-0.323199\pi$
$263$	17.1032	1.05463	0.527315	+	0.849670i	$0.323199\pi$
$264$	0	0
$265$	−8.29322	−0.509448
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2.32461i	− 0.141734i	−0.997486	−	0.0708670i	$-0.977423\pi$
$269$	− 2.32461i	− 0.141734i	0.997486	−	0.0708670i	$-0.0225766\pi$
$270$	0	0
$271$	−12.4721	−0.757628	−0.378814	−	0.925473i	$-0.623668\pi$
$271$	−12.4721	−0.757628	−0.378814	+	0.925473i	$0.623668\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.35259i	0.503680i
$276$	0	0
$277$	15.6257i	0.938858i	0.882970	+	0.469429i	$0.155540\pi$
$277$	15.6257i	0.938858i	−0.882970	+	0.469429i	$0.844460\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.0872	−1.55623	−0.778115	−	0.628122i	$-0.783824\pi$
$281$	−26.0872	−1.55623	−0.778115	+	0.628122i	$0.783824\pi$
$282$	0	0
$283$	− 17.4083i	− 1.03482i	−0.855739	−	0.517408i	$-0.826897\pi$
$283$	− 17.4083i	− 1.03482i	0.855739	−	0.517408i	$-0.173103\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.45412	−0.0858342
$288$	0	0
$289$	−7.00000	−0.411765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	13.6647i	0.798299i	0.916886	+	0.399149i	$0.130695\pi$
$293$	13.6647i	0.798299i	−0.916886	+	0.399149i	$0.869305\pi$
$294$	0	0
$295$	−15.5604	−0.905964
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.253569i	0.0146643i
$300$	0	0
$301$	− 8.85816i	− 0.510576i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.78600	0.331306
$306$	0	0
$307$	− 8.43102i	− 0.481184i	−0.970626	−	0.240592i	$-0.922659\pi$
$307$	− 8.43102i	− 0.481184i	0.970626	−	0.240592i	$-0.0773415\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	25.4195	1.44141	0.720703	−	0.693244i	$-0.243819\pi$
$311$	25.4195	1.44141	0.720703	+	0.693244i	$0.243819\pi$
$312$	0	0
$313$	−24.3183	−1.37455	−0.687277	−	0.726396i	$-0.741194\pi$
$313$	−24.3183	−1.37455	−0.687277	+	0.726396i	$0.741194\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.2593i	0.857047i	0.903531	+	0.428523i	$0.140966\pi$
$317$	15.2593i	0.857047i	−0.903531	+	0.428523i	$0.859034\pi$
$318$	0	0
$319$	34.3895	1.92544
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.29753i	0.127838i
$324$	0	0
$325$	0.0933124i	0.00517604i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.23016	−0.508875
$330$	0	0
$331$	− 22.9170i	− 1.25963i	−0.776744	−	0.629816i	$-0.783130\pi$
$331$	− 22.9170i	− 1.25963i	0.776744	−	0.629816i	$-0.216870\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−22.5417	−1.23158
$336$	0	0
$337$	34.0474	1.85468	0.927340	−	0.374221i	$-0.122090\pi$
$337$	34.0474	1.85468	0.927340	+	0.374221i	$0.122090\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.0954i	1.14238i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.72245i	− 0.0924662i	−0.998931	−	0.0462331i	$-0.985278\pi$
$347$	− 1.72245i	− 0.0924662i	0.998931	−	0.0462331i	$-0.0147217\pi$
$348$	0	0
$349$	7.95586i	0.425868i	0.977067	+	0.212934i	$0.0683019\pi$
$349$	7.95586i	0.425868i	−0.977067	+	0.212934i	$0.931698\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.1711	−1.23327	−0.616637	−	0.787247i	$-0.711505\pi$
$353$	−23.1711	−1.23327	−0.616637	+	0.787247i	$0.711505\pi$
$354$	0	0
$355$	− 10.1905i	− 0.540853i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.7889	−0.991640	−0.495820	−	0.868425i	$-0.665133\pi$
$359$	−18.7889	−0.991640	−0.495820	+	0.868425i	$0.665133\pi$
$360$	0	0
$361$	18.4721	0.972218
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 19.9222i	− 1.04278i
$366$	0	0
$367$	−25.1394	−1.31227	−0.656134	−	0.754645i	$-0.727809\pi$
$367$	−25.1394	−1.31227	−0.656134	+	0.754645i	$0.727809\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.14726i	0.163397i
$372$	0	0
$373$	− 25.0530i	− 1.29719i	−0.761132	−	0.648597i	$-0.775356\pi$
$373$	− 25.0530i	− 1.29719i	0.761132	−	0.648597i	$-0.224644\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.384189	0.0197867
$378$	0	0
$379$	15.2282i	0.782222i	0.920344	+	0.391111i	$0.127909\pi$
$379$	15.2282i	0.782222i	−0.920344	+	0.391111i	$0.872091\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−29.5703	−1.51097	−0.755487	−	0.655164i	$-0.772600\pi$
$383$	−29.5703	−1.51097	−0.755487	+	0.655164i	$0.772600\pi$
$384$	0	0
$385$	11.3244	0.577143
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1.90315i	− 0.0964934i	−0.998835	−	0.0482467i	$-0.984637\pi$
$389$	− 1.90315i	− 0.0964934i	0.998835	−	0.0482467i	$-0.0153634\pi$
$390$	0	0
$391$	−16.7015	−0.844629
$392$	0	0
$393$	0	0
$394$	0	0
$395$	38.5085i	1.93757i
$396$	0	0
$397$	− 16.7660i	− 0.841462i	−0.907185	−	0.420731i	$-0.861774\pi$
$397$	− 16.7660i	− 0.841462i	0.907185	−	0.420731i	$-0.138226\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.2567	−0.861759	−0.430880	−	0.902409i	$-0.641797\pi$
$401$	−17.2567	−0.861759	−0.430880	+	0.902409i	$0.641797\pi$
$402$	0	0
$403$	0.235671i	0.0117396i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	39.1395	1.94007
$408$	0	0
$409$	1.55671	0.0769744	0.0384872	−	0.999259i	$-0.487746\pi$
$409$	1.55671	0.0769744	0.0384872	+	0.999259i	$0.487746\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.90515i	0.290573i
$414$	0	0
$415$	−23.8537	−1.17093
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 26.1570i	− 1.27785i	−0.769268	−	0.638926i	$-0.779379\pi$
$419$	− 26.1570i	− 1.27785i	0.769268	−	0.638926i	$-0.220621\pi$
$420$	0	0
$421$	− 0.696870i	− 0.0339634i	−0.999856	−	0.0169817i	$-0.994594\pi$
$421$	− 0.696870i	− 0.0339634i	0.999856	−	0.0169817i	$-0.00540570\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.14608	−0.298128
$426$	0	0
$427$	− 2.19577i	− 0.106261i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.65116	−0.368543	−0.184272	−	0.982875i	$-0.558993\pi$
$431$	−7.65116	−0.368543	−0.184272	+	0.982875i	$0.558993\pi$
$432$	0	0
$433$	17.0860	0.821101	0.410550	−	0.911838i	$-0.365337\pi$
$433$	17.0860	0.821101	0.410550	+	0.911838i	$0.365337\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.83721i	− 0.183559i
$438$	0	0
$439$	3.23837	0.154559	0.0772796	−	0.997009i	$-0.475377\pi$
$439$	3.23837	0.154559	0.0772796	+	0.997009i	$0.475377\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 3.42501i	− 0.162727i	−0.996684	−	0.0813636i	$-0.974073\pi$
$443$	− 3.42501i	− 0.162727i	0.996684	−	0.0813636i	$-0.0259275\pi$
$444$	0	0
$445$	− 13.5810i	− 0.643800i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	39.7647	1.87661	0.938305	−	0.345808i	$-0.112395\pi$
$449$	39.7647	1.87661	0.938305	+	0.345808i	$0.112395\pi$
$450$	0	0
$451$	6.24920i	0.294263i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.126512	0.00593098
$456$	0	0
$457$	7.46983	0.349424	0.174712	−	0.984620i	$-0.444101\pi$
$457$	7.46983	0.349424	0.174712	+	0.984620i	$0.444101\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.0075i	0.698967i	0.936942	+	0.349484i	$0.113643\pi$
$461$	15.0075i	0.698967i	−0.936942	+	0.349484i	$0.886357\pi$
$462$	0	0
$463$	4.29553	0.199630	0.0998150	−	0.995006i	$-0.468175\pi$
$463$	4.29553	0.199630	0.0998150	+	0.995006i	$0.468175\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.3945i	1.22139i	0.791865	+	0.610696i	$0.209110\pi$
$467$	26.3945i	1.22139i	−0.791865	+	0.610696i	$0.790890\pi$
$468$	0	0
$469$	8.55452i	0.395011i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−38.0686	−1.75039
$474$	0	0
$475$	− 1.41208i	− 0.0647906i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.66267	−0.304425	−0.152213	−	0.988348i	$-0.548640\pi$
$479$	−6.66267	−0.304425	−0.152213	+	0.988348i	$0.548640\pi$
$480$	0	0
$481$	0.437254	0.0199371
$482$	0	0
$483$	0	0
$484$	0	0
$485$	32.0960i	1.45740i
$486$	0	0
$487$	36.6813	1.66219	0.831095	−	0.556131i	$-0.187715\pi$
$487$	36.6813	1.66219	0.831095	+	0.556131i	$0.187715\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.38460i	0.197874i	0.995094	+	0.0989371i	$0.0315443\pi$
$491$	4.38460i	0.197874i	−0.995094	+	0.0989371i	$0.968456\pi$
$492$	0	0
$493$	25.3048i	1.13967i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.86725	−0.173470
$498$	0	0
$499$	35.6754i	1.59705i	0.601961	+	0.798525i	$0.294386\pi$
$499$	35.6754i	1.59705i	−0.601961	+	0.798525i	$0.705614\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.3657	0.506770	0.253385	−	0.967365i	$-0.418456\pi$
$503$	11.3657	0.506770	0.253385	+	0.967365i	$0.418456\pi$
$504$	0	0
$505$	11.4691	0.510369
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 2.78884i	− 0.123613i	−0.998088	−	0.0618066i	$-0.980314\pi$
$509$	− 2.78884i	− 0.123613i	0.998088	−	0.0618066i	$-0.0196862\pi$
$510$	0	0
$511$	−7.56044	−0.334454
$512$	0	0
$513$	0	0
$514$	0	0
$515$	32.4676i	1.43069i
$516$	0	0
$517$	39.6673i	1.74457i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	40.3056	1.76582	0.882910	−	0.469542i	$-0.155581\pi$
$521$	40.3056	1.76582	0.882910	+	0.469542i	$0.155581\pi$
$522$	0	0
$523$	35.9726i	1.57297i	0.617608	+	0.786486i	$0.288102\pi$
$523$	35.9726i	1.57297i	−0.617608	+	0.786486i	$0.711898\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.5226	−0.676175
$528$	0	0
$529$	4.89387	0.212777
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.0698140i	0.00302398i
$534$	0	0
$535$	−25.5352	−1.10398
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.29757i	− 0.185109i
$540$	0	0
$541$	− 10.1931i	− 0.438234i	−0.975699	−	0.219117i	$-0.929682\pi$
$541$	− 10.1931i	− 0.438234i	0.975699	−	0.219117i	$-0.0703177\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−23.9501	−1.02591
$546$	0	0
$547$	12.3063i	0.526182i	0.964771	+	0.263091i	$0.0847419\pi$
$547$	12.3063i	0.526182i	−0.964771	+	0.263091i	$0.915258\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.81385	−0.247679
$552$	0	0
$553$	14.6139	0.621445
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 27.2392i	− 1.15416i	−0.816687	−	0.577082i	$-0.804191\pi$
$557$	− 27.2392i	− 1.15416i	0.816687	−	0.577082i	$-0.195809\pi$
$558$	0	0
$559$	−0.425290	−0.0179878
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 32.9187i	− 1.38736i	−0.720285	−	0.693678i	$-0.755989\pi$
$563$	− 32.9187i	− 1.38736i	0.720285	−	0.693678i	$-0.244011\pi$
$564$	0	0
$565$	35.6274i	1.49886i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.9554	1.08811	0.544053	−	0.839051i	$-0.316889\pi$
$569$	25.9554	1.08811	0.544053	+	0.839051i	$0.316889\pi$
$570$	0	0
$571$	− 0.527977i	− 0.0220951i	−0.999939	−	0.0110476i	$-0.996483\pi$
$571$	− 0.527977i	− 0.0220951i	0.999939	−	0.0110476i	$-0.00351662\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10.2648	0.428073
$576$	0	0
$577$	43.1860	1.79786	0.898929	−	0.438094i	$-0.144346\pi$
$577$	43.1860	1.79786	0.898929	+	0.438094i	$0.144346\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.05240i	0.375557i
$582$	0	0
$583$	13.5256	0.560171
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.5375i	0.847675i	0.905738	+	0.423837i	$0.139317\pi$
$587$	20.5375i	0.847675i	−0.905738	+	0.423837i	$0.860683\pi$
$588$	0	0
$589$	− 3.56636i	− 0.146949i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−47.4930	−1.95030	−0.975152	−	0.221537i	$-0.928893\pi$
$593$	−47.4930	−1.95030	−0.975152	+	0.221537i	$0.928893\pi$
$594$	0	0
$595$	8.33280i	0.341612i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.31521	0.217174	0.108587	−	0.994087i	$-0.465367\pi$
$599$	5.31521	0.217174	0.108587	+	0.994087i	$0.465367\pi$
$600$	0	0
$601$	29.9137	1.22020	0.610102	−	0.792323i	$-0.291129\pi$
$601$	29.9137	1.22020	0.610102	+	0.792323i	$0.291129\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 19.6816i	− 0.800171i
$606$	0	0
$607$	17.3429	0.703927	0.351964	−	0.936014i	$-0.385514\pi$
$607$	17.3429	0.703927	0.351964	+	0.936014i	$0.385514\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.443150i	0.0179279i
$612$	0	0
$613$	17.8469i	0.720830i	0.932792	+	0.360415i	$0.117365\pi$
$613$	17.8469i	0.720830i	−0.932792	+	0.360415i	$0.882635\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.51870	0.141658	0.0708289	−	0.997488i	$-0.477436\pi$
$617$	3.51870	0.141658	0.0708289	+	0.997488i	$0.477436\pi$
$618$	0	0
$619$	− 5.33557i	− 0.214455i	−0.994235	−	0.107227i	$-0.965803\pi$
$619$	− 5.33557i	− 0.214455i	0.994235	−	0.107227i	$-0.0341973\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.15394	−0.206488
$624$	0	0
$625$	−30.9404	−1.23761
$626$	0	0
$627$	0	0
$628$	0	0
$629$	28.8000i	1.14833i
$630$	0	0
$631$	−13.1520	−0.523574	−0.261787	−	0.965126i	$-0.584312\pi$
$631$	−13.1520	−0.523574	−0.261787	+	0.965126i	$0.584312\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	28.9308i	1.14808i
$636$	0	0
$637$	− 0.0480111i	− 0.00190227i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.20177	0.323950	0.161975	−	0.986795i	$-0.448214\pi$
$641$	8.20177	0.323950	0.161975	+	0.986795i	$0.448214\pi$
$642$	0	0
$643$	32.8167i	1.29416i	0.762421	+	0.647082i	$0.224011\pi$
$643$	32.8167i	1.29416i	−0.762421	+	0.647082i	$0.775989\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.7207	1.52227	0.761134	−	0.648595i	$-0.224643\pi$
$647$	38.7207	1.52227	0.761134	+	0.648595i	$0.224643\pi$
$648$	0	0
$649$	25.3778	0.996166
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 28.7630i	− 1.12558i	−0.826598	−	0.562792i	$-0.809727\pi$
$653$	− 28.7630i	− 1.12558i	0.826598	−	0.562792i	$-0.190273\pi$
$654$	0	0
$655$	2.03115	0.0793637
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 30.6124i	− 1.19249i	−0.802803	−	0.596244i	$-0.796659\pi$
$659$	− 30.6124i	− 1.19249i	0.802803	−	0.596244i	$-0.203341\pi$
$660$	0	0
$661$	− 44.1975i	− 1.71908i	−0.511066	−	0.859541i	$-0.670749\pi$
$661$	− 44.1975i	− 1.71908i	0.511066	−	0.859541i	$-0.329251\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.91449	−0.0742406
$666$	0	0
$667$	− 42.2627i	− 1.63642i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.43649	−0.364292
$672$	0	0
$673$	17.0117	0.655754	0.327877	−	0.944720i	$-0.393667\pi$
$673$	17.0117	0.655754	0.327877	+	0.944720i	$0.393667\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 45.4750i	− 1.74775i	−0.486155	−	0.873873i	$-0.661601\pi$
$677$	− 45.4750i	− 1.74775i	0.486155	−	0.873873i	$-0.338399\pi$
$678$	0	0
$679$	12.1803	0.467439
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 34.4287i	− 1.31738i	−0.752416	−	0.658689i	$-0.771111\pi$
$683$	− 34.4287i	− 1.31738i	0.752416	−	0.658689i	$-0.228889\pi$
$684$	0	0
$685$	− 28.9283i	− 1.10529i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.151103	0.00575657
$690$	0	0
$691$	28.9692i	1.10204i	0.834491	+	0.551021i	$0.185762\pi$
$691$	28.9692i	1.10204i	−0.834491	+	0.551021i	$0.814238\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−58.0213	−2.20087
$696$	0	0
$697$	−4.59834	−0.174175
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 12.9846i	− 0.490421i	−0.969470	−	0.245211i	$-0.921143\pi$
$701$	− 12.9846i	− 0.490421i	0.969470	−	0.245211i	$-0.0788571\pi$
$702$	0	0
$703$	−6.61688	−0.249560
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.35250i	− 0.163693i
$708$	0	0
$709$	− 5.62511i	− 0.211255i	−0.994406	−	0.105628i	$-0.966315\pi$
$709$	− 5.62511i	− 0.211255i	0.994406	−	0.105628i	$-0.0336852\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	25.9250	0.970899
$714$	0	0
$715$	− 0.543695i	− 0.0203330i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	13.3194	0.496732	0.248366	−	0.968666i	$-0.420106\pi$
$719$	13.3194	0.496732	0.248366	+	0.968666i	$0.420106\pi$
$720$	0	0
$721$	12.3214	0.458871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 15.5525i	− 0.577606i
$726$	0	0
$727$	−20.0614	−0.744037	−0.372019	−	0.928225i	$-0.621334\pi$
$727$	−20.0614	−0.744037	−0.372019	+	0.928225i	$0.621334\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 28.0120i	− 1.03606i
$732$	0	0
$733$	4.06695i	0.150216i	0.997175	+	0.0751081i	$0.0239302\pi$
$733$	4.06695i	0.150216i	−0.997175	+	0.0751081i	$0.976070\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.7637	1.35421
$738$	0	0
$739$	− 2.22283i	− 0.0817680i	−0.999164	−	0.0408840i	$-0.986983\pi$
$739$	− 2.22283i	− 0.0817680i	0.999164	−	0.0408840i	$-0.0130174\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.6763	−1.05203	−0.526015	−	0.850475i	$-0.676314\pi$
$743$	−28.6763	−1.05203	−0.526015	+	0.850475i	$0.676314\pi$
$744$	0	0
$745$	−13.3504	−0.489120
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.69053i	0.354084i
$750$	0	0
$751$	−8.61386	−0.314324	−0.157162	−	0.987573i	$-0.550235\pi$
$751$	−8.61386	−0.314324	−0.157162	+	0.987573i	$0.550235\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.2888i	0.665596i
$756$	0	0
$757$	− 27.8835i	− 1.01344i	−0.862109	−	0.506722i	$-0.830857\pi$
$757$	− 27.8835i	− 1.01344i	0.862109	−	0.506722i	$-0.169143\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.20889	0.152572	0.0762861	−	0.997086i	$-0.475694\pi$
$761$	4.20889	0.152572	0.0762861	+	0.997086i	$0.475694\pi$
$762$	0	0
$763$	9.08902i	0.329045i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.283512	0.0102370
$768$	0	0
$769$	21.5187	0.775986	0.387993	−	0.921662i	$-0.373168\pi$
$769$	21.5187	0.775986	0.387993	+	0.921662i	$0.373168\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	30.1139i	1.08312i	0.840661	+	0.541562i	$0.182167\pi$
$773$	30.1139i	1.08312i	−0.840661	+	0.541562i	$0.817833\pi$
$774$	0	0
$775$	9.54031	0.342698
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.05648i	− 0.0378524i
$780$	0	0
$781$	16.6198i	0.594703i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	48.5328	1.73221
$786$	0	0
$787$	48.9845i	1.74611i	0.487624	+	0.873054i	$0.337864\pi$
$787$	48.9845i	1.74611i	−0.487624	+	0.873054i	$0.662136\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13.5205	0.480734
$792$	0	0
$793$	−0.105421	−0.00374363
$794$	0	0
$795$	0	0
$796$	0	0
$797$	47.5262i	1.68347i	0.539894	+	0.841733i	$0.318464\pi$
$797$	47.5262i	1.68347i	−0.539894	+	0.841733i	$0.681536\pi$
$798$	0	0
$799$	−29.1883	−1.03261
$800$	0	0
$801$	0	0
$802$	0	0
$803$	32.4915i	1.14660i
$804$	0	0
$805$	− 13.9170i	− 0.490510i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.1063	1.05848	0.529240	−	0.848472i	$-0.322477\pi$
$809$	30.1063	1.05848	0.529240	+	0.848472i	$0.322477\pi$
$810$	0	0
$811$	− 30.2492i	− 1.06219i	−0.847311	−	0.531096i	$-0.821780\pi$
$811$	− 30.2492i	− 1.06219i	0.847311	−	0.531096i	$-0.178220\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−27.9183	−0.977935
$816$	0	0
$817$	6.43583	0.225161
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.4111i	0.537851i	0.963161	+	0.268926i	$0.0866686\pi$
$821$	15.4111i	0.537851i	−0.963161	+	0.268926i	$0.913331\pi$
$822$	0	0
$823$	20.0591	0.699217	0.349608	−	0.936896i	$-0.386315\pi$
$823$	20.0591	0.699217	0.349608	+	0.936896i	$0.386315\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 35.3363i	− 1.22876i	−0.789009	−	0.614382i	$-0.789406\pi$
$827$	− 35.3363i	− 1.22876i	0.789009	−	0.614382i	$-0.210594\pi$
$828$	0	0
$829$	− 52.3976i	− 1.81985i	−0.414778	−	0.909923i	$-0.636141\pi$
$829$	− 52.3976i	− 1.81985i	0.414778	−	0.909923i	$-0.363859\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.16228	0.109566
$834$	0	0
$835$	− 45.7973i	− 1.58488i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−43.5132	−1.50224	−0.751121	−	0.660164i	$-0.770487\pi$
$839$	−43.5132	−1.50224	−0.751121	+	0.660164i	$0.770487\pi$
$840$	0	0
$841$	−35.0334	−1.20805
$842$	0	0
$843$	0	0
$844$	0	0
$845$	34.2498i	1.17823i
$846$	0	0
$847$	−7.46912	−0.256642
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 48.1002i	− 1.64885i
$852$	0	0
$853$	− 28.8647i	− 0.988310i	−0.869374	−	0.494155i	$-0.835478\pi$
$853$	− 28.8647i	− 0.988310i	0.869374	−	0.494155i	$-0.164522\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.7856	−0.368429	−0.184214	−	0.982886i	$-0.558974\pi$
$857$	−10.7856	−0.368429	−0.184214	+	0.982886i	$0.558974\pi$
$858$	0	0
$859$	− 47.7595i	− 1.62953i	−0.579789	−	0.814767i	$-0.696865\pi$
$859$	− 47.7595i	− 1.62953i	0.579789	−	0.814767i	$-0.303135\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.2620	1.40457	0.702287	−	0.711894i	$-0.252162\pi$
$863$	41.2620	1.40457	0.702287	+	0.711894i	$0.252162\pi$
$864$	0	0
$865$	−60.3635	−2.05242
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 62.8041i	− 2.13048i
$870$	0	0
$871$	0.410712	0.0139164
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.05391i	0.272272i
$876$	0	0
$877$	34.8776i	1.17773i	0.808230	+	0.588867i	$0.200426\pi$
$877$	34.8776i	1.17773i	−0.808230	+	0.588867i	$0.799574\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−6.15123	−0.207240	−0.103620	−	0.994617i	$-0.533043\pi$
$881$	−6.15123	−0.207240	−0.103620	+	0.994617i	$0.533043\pi$
$882$	0	0
$883$	24.2713i	0.816796i	0.912804	+	0.408398i	$0.133912\pi$
$883$	24.2713i	0.816796i	−0.912804	+	0.408398i	$0.866088\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.8061	0.429987	0.214994	−	0.976615i	$-0.431027\pi$
$887$	12.8061	0.429987	0.214994	+	0.976615i	$0.431027\pi$
$888$	0	0
$889$	10.9792	0.368229
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 6.70611i	− 0.224411i
$894$	0	0
$895$	−56.9878	−1.90489
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 39.2797i	− 1.31005i
$900$	0	0
$901$	9.95250i	0.331566i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33.2397	1.10492
$906$	0	0
$907$	26.7650i	0.888717i	0.895849	+	0.444358i	$0.146568\pi$
$907$	26.7650i	0.888717i	−0.895849	+	0.444358i	$0.853432\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34.9594	1.15826	0.579128	−	0.815237i	$-0.303393\pi$
$911$	34.9594	1.15826	0.579128	+	0.815237i	$0.303393\pi$
$912$	0	0
$913$	38.9034	1.28751
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 0.770817i	− 0.0254546i
$918$	0	0
$919$	−17.3058	−0.570867	−0.285433	−	0.958399i	$-0.592138\pi$
$919$	−17.3058	−0.570867	−0.285433	+	0.958399i	$0.592138\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.185671i	0.00611143i
$924$	0	0
$925$	− 17.7007i	− 0.581995i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8.69526	−0.285282	−0.142641	−	0.989774i	$-0.545559\pi$
$929$	−8.69526	−0.285282	−0.142641	+	0.989774i	$0.545559\pi$
$930$	0	0
$931$	0.726543i	0.0238115i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	35.8108	1.17114
$936$	0	0
$937$	24.8633	0.812249	0.406125	−	0.913818i	$-0.366880\pi$
$937$	24.8633	0.812249	0.406125	+	0.913818i	$0.366880\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 15.1562i	− 0.494078i	−0.969006	−	0.247039i	$-0.920542\pi$
$941$	− 15.1562i	− 0.494078i	0.969006	−	0.247039i	$-0.0794575\pi$
$942$	0	0
$943$	7.67990	0.250092
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 25.6626i	− 0.833922i	−0.908924	−	0.416961i	$-0.863095\pi$
$947$	− 25.6626i	− 0.833922i	0.908924	−	0.416961i	$-0.136905\pi$
$948$	0	0
$949$	0.362985i	0.0117830i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	9.66478	0.313073	0.156536	−	0.987672i	$-0.449967\pi$
$953$	9.66478	0.313073	0.156536	+	0.987672i	$0.449967\pi$
$954$	0	0
$955$	− 35.7256i	− 1.15605i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.9782	−0.354505
$960$	0	0
$961$	−6.90488	−0.222738
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 68.6735i	− 2.21068i
$966$	0	0
$967$	−6.28949	−0.202256	−0.101128	−	0.994873i	$-0.532245\pi$
$967$	−6.28949	−0.202256	−0.101128	+	0.994873i	$0.532245\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.3396i	0.909462i	0.890629	+	0.454731i	$0.150265\pi$
$971$	28.3396i	0.909462i	−0.890629	+	0.454731i	$0.849735\pi$
$972$	0	0
$973$	22.0189i	0.705895i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.0973	0.898911	0.449455	−	0.893303i	$-0.351618\pi$
$977$	28.0973	0.898911	0.449455	+	0.893303i	$0.351618\pi$
$978$	0	0
$979$	22.1494i	0.707899i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	38.4571	1.22659	0.613296	−	0.789853i	$-0.289843\pi$
$983$	38.4571	1.22659	0.613296	+	0.789853i	$0.289843\pi$
$984$	0	0
$985$	8.40380	0.267767
$986$	0	0
$987$	0	0
$988$	0	0
$989$	46.7841i	1.48765i
$990$	0	0
$991$	30.2448	0.960759	0.480380	−	0.877061i	$-0.340499\pi$
$991$	30.2448	0.960759	0.480380	+	0.877061i	$0.340499\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 66.4376i	− 2.10621i
$996$	0	0
$997$	17.2007i	0.544751i	0.962191	+	0.272375i	$0.0878093\pi$
$997$	17.2007i	0.544751i	−0.962191	+	0.272375i	$0.912191\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.d.3025.13		16
3.2	odd	2	inner	6048.2.c.d.3025.3		16
4.3	odd	2		1512.2.c.d.757.14	yes	16
8.3	odd	2		1512.2.c.d.757.15	yes	16
8.5	even	2	inner	6048.2.c.d.3025.4		16
12.11	even	2		1512.2.c.d.757.3	yes	16
24.5	odd	2	inner	6048.2.c.d.3025.14		16
24.11	even	2		1512.2.c.d.757.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.d.757.2	✓	16	24.11	even	2
1512.2.c.d.757.3	yes	16	12.11	even	2
1512.2.c.d.757.14	yes	16	4.3	odd	2
1512.2.c.d.757.15	yes	16	8.3	odd	2
6048.2.c.d.3025.3		16	3.2	odd	2	inner
6048.2.c.d.3025.4		16	8.5	even	2	inner
6048.2.c.d.3025.13		16	1.1	even	1	trivial
6048.2.c.d.3025.14		16	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)

\(f(q)\)	\(=\)	\(q+2.63506i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.63506i q^{5} +1.00000 q^{7} -4.29757i q^{11} -0.0480111i q^{13} +3.16228 q^{17} +0.726543i q^{19} -5.28146 q^{23} -1.94356 q^{25} +8.00208i q^{29} -4.90868 q^{31} +2.63506i q^{35} +9.10736i q^{37} -1.45412 q^{41} -8.85816i q^{43} -9.23016 q^{47} +1.00000 q^{49} +3.14726i q^{53} +11.3244 q^{55} +5.90515i q^{59} -2.19577i q^{61} +0.126512 q^{65} +8.55452i q^{67} -3.86725 q^{71} -7.56044 q^{73} -4.29757i q^{77} +14.6139 q^{79} +9.05240i q^{83} +8.33280i q^{85} -5.15394 q^{89} -0.0480111i q^{91} -1.91449 q^{95} +12.1803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7}+O(q^{10}) \) 16 * q + 16 * q^7	\( 16 q + 16 q^{7} - 32 q^{25} - 40 q^{31} + 16 q^{49} + 72 q^{55} + 24 q^{73} - 24 q^{79} + 16 q^{97}+O(q^{100}) \) 16 * q + 16 * q^7 - 32 * q^25 - 40 * q^31 + 16 * q^49 + 72 * q^55 + 24 * q^73 - 24 * q^79 + 16 * q^97

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