LMFDB - Embedded newform 6048.2.c.e.3025.7

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + x^{18} + 4x^{16} + 8x^{12} + 4x^{10} + 32x^{8} + 256x^{4} + 256x^{2} + 1024 $ x^20 + x^18 + 4x^16 + 8x^12 + 4x^10 + 32x^8 + 256x^4 + 256x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.7
Root			$0.328272 - 1.37559i$ of defining polynomial
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.e.3025.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.512447i q^{5} +1.00000 q^{7} +O(q^{10})$	$q-0.512447i q^{5} +1.00000 q^{7} -1.82890i q^{11} -1.80627i q^{13} -8.11456 q^{17} +3.43946i q^{19} -3.65626 q^{23} +4.73740 q^{25} +7.98990i q^{29} +1.56895 q^{31} -0.512447i q^{35} +8.44370i q^{37} +2.30828 q^{41} -10.7778i q^{43} +11.3833 q^{47} +1.00000 q^{49} +8.87795i q^{53} -0.937212 q^{55} -2.50234i q^{59} +5.31310i q^{61} -0.925616 q^{65} +6.44648i q^{67} +9.10975 q^{71} +9.24419 q^{73} -1.82890i q^{77} -3.64300 q^{79} -4.53105i q^{83} +4.15828i q^{85} +11.7359 q^{89} -1.80627i q^{91} +1.76254 q^{95} +15.2323 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{7}+O(q^{10}) $ 20 * q + 20 * q^7	$ 20 q + 20 q^{7} - 28 q^{25} - 36 q^{31} + 20 q^{49} - 48 q^{55} - 64 q^{79} + 56 q^{97}+O(q^{100}) $ 20 * q + 20 * q^7 - 28 * q^25 - 36 * q^31 + 20 * q^49 - 48 * q^55 - 64 * q^79 + 56 * 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
$5$	− 0.512447i	− 0.229173i	−0.993413	−	0.114587i	$-0.963446\pi$
$5$	− 0.512447i	− 0.229173i	0.993413	−	0.114587i	$-0.0365543\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.82890i	− 0.551433i	−0.961239	−	0.275716i	$-0.911085\pi$
$11$	− 1.82890i	− 0.551433i	0.961239	−	0.275716i	$-0.0889151\pi$
$12$	0	0
$13$	− 1.80627i	− 0.500968i	−0.968121	−	0.250484i	$-0.919410\pi$
$13$	− 1.80627i	− 0.500968i	0.968121	−	0.250484i	$-0.0805898\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−8.11456	−1.96807	−0.984035	−	0.177977i	$-0.943045\pi$
$17$	−8.11456	−1.96807	−0.984035	+	0.177977i	$0.943045\pi$
$18$	0	0
$19$	3.43946i	0.789066i	0.918882	+	0.394533i	$0.129094\pi$
$19$	3.43946i	0.789066i	−0.918882	+	0.394533i	$0.870906\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.65626	−0.762384	−0.381192	−	0.924496i	$-0.624486\pi$
$23$	−3.65626	−0.762384	−0.381192	+	0.924496i	$0.624486\pi$
$24$	0	0
$25$	4.73740	0.947480
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.98990i	1.48369i	0.670573	+	0.741843i	$0.266048\pi$
$29$	7.98990i	1.48369i	−0.670573	+	0.741843i	$0.733952\pi$
$30$	0	0
$31$	1.56895	0.281792	0.140896	−	0.990024i	$-0.455002\pi$
$31$	1.56895	0.281792	0.140896	+	0.990024i	$0.455002\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.512447i	− 0.0866193i
$36$	0	0
$37$	8.44370i	1.38814i	0.719909	+	0.694068i	$0.244183\pi$
$37$	8.44370i	1.38814i	−0.719909	+	0.694068i	$0.755817\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.30828	0.360493	0.180247	−	0.983621i	$-0.442310\pi$
$41$	2.30828	0.360493	0.180247	+	0.983621i	$0.442310\pi$
$42$	0	0
$43$	− 10.7778i	− 1.64360i	−0.569773	−	0.821802i	$-0.692969\pi$
$43$	− 10.7778i	− 1.64360i	0.569773	−	0.821802i	$-0.307031\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.3833	1.66043	0.830216	−	0.557442i	$-0.188217\pi$
$47$	11.3833	1.66043	0.830216	+	0.557442i	$0.188217\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.87795i	1.21948i	0.792601	+	0.609740i	$0.208726\pi$
$53$	8.87795i	1.21948i	−0.792601	+	0.609740i	$0.791274\pi$
$54$	0	0
$55$	−0.937212	−0.126374
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2.50234i	− 0.325778i	−0.986644	−	0.162889i	$-0.947919\pi$
$59$	− 2.50234i	− 0.325778i	0.986644	−	0.162889i	$-0.0520812\pi$
$60$	0	0
$61$	5.31310i	0.680272i	0.940376	+	0.340136i	$0.110473\pi$
$61$	5.31310i	0.680272i	−0.940376	+	0.340136i	$0.889527\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.925616	−0.114809
$66$	0	0
$67$	6.44648i	0.787562i	0.919204	+	0.393781i	$0.128833\pi$
$67$	6.44648i	0.787562i	−0.919204	+	0.393781i	$0.871167\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.10975	1.08113	0.540564	−	0.841303i	$-0.318211\pi$
$71$	9.10975	1.08113	0.540564	+	0.841303i	$0.318211\pi$
$72$	0	0
$73$	9.24419	1.08195	0.540975	−	0.841039i	$-0.318055\pi$
$73$	9.24419	1.08195	0.540975	+	0.841039i	$0.318055\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.82890i	− 0.208422i
$78$	0	0
$79$	−3.64300	−0.409870	−0.204935	−	0.978776i	$-0.565698\pi$
$79$	−3.64300	−0.409870	−0.204935	+	0.978776i	$0.565698\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.53105i	− 0.497348i	−0.968587	−	0.248674i	$-0.920005\pi$
$83$	− 4.53105i	− 0.497348i	0.968587	−	0.248674i	$-0.0799948\pi$
$84$	0	0
$85$	4.15828i	0.451029i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.7359	1.24401	0.622003	−	0.783015i	$-0.286319\pi$
$89$	11.7359	1.24401	0.622003	+	0.783015i	$0.286319\pi$
$90$	0	0
$91$	− 1.80627i	− 0.189348i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.76254	0.180833
$96$	0	0
$97$	15.2323	1.54661	0.773303	−	0.634037i	$-0.218603\pi$
$97$	15.2323	1.54661	0.773303	+	0.634037i	$0.218603\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.13524i	0.809487i	0.914430	+	0.404744i	$0.132639\pi$
$101$	8.13524i	0.809487i	−0.914430	+	0.404744i	$0.867361\pi$
$102$	0	0
$103$	15.9816	1.57471	0.787356	−	0.616498i	$-0.211449\pi$
$103$	15.9816	1.57471	0.787356	+	0.616498i	$0.211449\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.1695i	− 1.17647i	−0.808690	−	0.588235i	$-0.799823\pi$
$107$	− 12.1695i	− 1.17647i	0.808690	−	0.588235i	$-0.200177\pi$
$108$	0	0
$109$	− 10.8452i	− 1.03878i	−0.854537	−	0.519391i	$-0.826159\pi$
$109$	− 10.8452i	− 1.03878i	0.854537	−	0.519391i	$-0.173841\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.07563	−0.289331	−0.144665	−	0.989481i	$-0.546211\pi$
$113$	−3.07563	−0.289331	−0.144665	+	0.989481i	$0.546211\pi$
$114$	0	0
$115$	1.87364i	0.174718i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.11456	−0.743860
$120$	0	0
$121$	7.65514	0.695922
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 4.98990i	− 0.446310i
$126$	0	0
$127$	−6.67524	−0.592332	−0.296166	−	0.955137i	$-0.595708\pi$
$127$	−6.67524	−0.592332	−0.296166	+	0.955137i	$0.595708\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.26995i	− 0.110956i	−0.998460	−	0.0554778i	$-0.982332\pi$
$131$	− 1.26995i	− 0.110956i	0.998460	−	0.0554778i	$-0.0176682\pi$
$132$	0	0
$133$	3.43946i	0.298239i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.8507	−1.69596	−0.847980	−	0.530029i	$-0.822181\pi$
$137$	−19.8507	−1.69596	−0.847980	+	0.530029i	$0.822181\pi$
$138$	0	0
$139$	− 13.9758i	− 1.18541i	−0.805419	−	0.592706i	$-0.798059\pi$
$139$	− 13.9758i	− 1.18541i	0.805419	−	0.592706i	$-0.201941\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.30348	−0.276251
$144$	0	0
$145$	4.09440	0.340021
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 11.8561i	− 0.971286i	−0.874157	−	0.485643i	$-0.838585\pi$
$149$	− 11.8561i	− 0.971286i	0.874157	−	0.485643i	$-0.161415\pi$
$150$	0	0
$151$	13.4949	1.09820	0.549100	−	0.835757i	$-0.314971\pi$
$151$	13.4949	1.09820	0.549100	+	0.835757i	$0.314971\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.804003i	− 0.0645791i
$156$	0	0
$157$	9.88593i	0.788983i	0.918900	+	0.394492i	$0.129079\pi$
$157$	9.88593i	0.788983i	−0.918900	+	0.394492i	$0.870921\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.65626	−0.288154
$162$	0	0
$163$	− 4.43706i	− 0.347537i	−0.984786	−	0.173769i	$-0.944405\pi$
$163$	− 4.43706i	− 0.347537i	0.984786	−	0.173769i	$-0.0555945\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.38391	−0.416619	−0.208310	−	0.978063i	$-0.566796\pi$
$167$	−5.38391	−0.416619	−0.208310	+	0.978063i	$0.566796\pi$
$168$	0	0
$169$	9.73740	0.749031
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.9611i	1.36556i	0.730624	+	0.682780i	$0.239229\pi$
$173$	17.9611i	1.36556i	−0.730624	+	0.682780i	$0.760771\pi$
$174$	0	0
$175$	4.73740	0.358114
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 16.0482i	− 1.19950i	−0.800188	−	0.599749i	$-0.795267\pi$
$179$	− 16.0482i	− 1.19950i	0.800188	−	0.599749i	$-0.204733\pi$
$180$	0	0
$181$	5.91861i	0.439927i	0.975508	+	0.219964i	$0.0705938\pi$
$181$	5.91861i	0.439927i	−0.975508	+	0.219964i	$0.929406\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.32695	0.318123
$186$	0	0
$187$	14.8407i	1.08526i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.0697837	0.00504937	0.00252469	−	0.999997i	$-0.499196\pi$
$191$	0.0697837	0.00504937	0.00252469	+	0.999997i	$0.499196\pi$
$192$	0	0
$193$	−8.24356	−0.593384	−0.296692	−	0.954973i	$-0.595884\pi$
$193$	−8.24356	−0.593384	−0.296692	+	0.954973i	$0.595884\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.91295i	− 0.136292i	−0.997675	−	0.0681459i	$-0.978292\pi$
$197$	− 1.91295i	− 0.136292i	0.997675	−	0.0681459i	$-0.0217083\pi$
$198$	0	0
$199$	3.70579	0.262696	0.131348	−	0.991336i	$-0.458069\pi$
$199$	3.70579	0.262696	0.131348	+	0.991336i	$0.458069\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.98990i	0.560781i
$204$	0	0
$205$	− 1.18287i	− 0.0826153i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.29041	0.435117
$210$	0	0
$211$	25.6006i	1.76242i	0.472724	+	0.881210i	$0.343271\pi$
$211$	25.6006i	1.76242i	−0.472724	+	0.881210i	$0.656729\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.52307	−0.376670
$216$	0	0
$217$	1.56895	0.106507
$218$	0	0
$219$	0	0
$220$	0	0
$221$	14.6571i	0.985941i
$222$	0	0
$223$	14.2639	0.955182	0.477591	−	0.878582i	$-0.341510\pi$
$223$	14.2639	0.955182	0.477591	+	0.878582i	$0.341510\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.58553i	0.171608i	0.996312	+	0.0858039i	$0.0273458\pi$
$227$	2.58553i	0.171608i	−0.996312	+	0.0858039i	$0.972654\pi$
$228$	0	0
$229$	− 27.5800i	− 1.82254i	−0.411813	−	0.911268i	$-0.635104\pi$
$229$	− 27.5800i	− 1.82254i	0.411813	−	0.911268i	$-0.364896\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.61196	−0.367652	−0.183826	−	0.982959i	$-0.558848\pi$
$233$	−5.61196	−0.367652	−0.183826	+	0.982959i	$0.558848\pi$
$234$	0	0
$235$	− 5.83336i	− 0.380526i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.69275	0.109495	0.0547476	−	0.998500i	$-0.482565\pi$
$239$	1.69275	0.109495	0.0547476	+	0.998500i	$0.482565\pi$
$240$	0	0
$241$	−22.6246	−1.45738	−0.728689	−	0.684845i	$-0.759870\pi$
$241$	−22.6246	−1.45738	−0.728689	+	0.684845i	$0.759870\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 0.512447i	− 0.0327390i
$246$	0	0
$247$	6.21258	0.395297
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.1391i	1.14493i	0.819930	+	0.572464i	$0.194012\pi$
$251$	18.1391i	1.14493i	−0.819930	+	0.572464i	$0.805988\pi$
$252$	0	0
$253$	6.68693i	0.420403i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.62562	0.538051	0.269026	−	0.963133i	$-0.413298\pi$
$257$	8.62562	0.538051	0.269026	+	0.963133i	$0.413298\pi$
$258$	0	0
$259$	8.44370i	0.524666i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.79262	0.172201	0.0861003	−	0.996286i	$-0.472559\pi$
$263$	2.79262	0.172201	0.0861003	+	0.996286i	$0.472559\pi$
$264$	0	0
$265$	4.54948	0.279472
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 17.8391i	− 1.08767i	−0.839192	−	0.543836i	$-0.816971\pi$
$269$	− 17.8391i	− 1.08767i	0.839192	−	0.543836i	$-0.183029\pi$
$270$	0	0
$271$	22.5304	1.36863	0.684313	−	0.729188i	$-0.260102\pi$
$271$	22.5304	1.36863	0.684313	+	0.729188i	$0.260102\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 8.66421i	− 0.522472i
$276$	0	0
$277$	2.18249i	0.131133i	0.997848	+	0.0655666i	$0.0208855\pi$
$277$	2.18249i	0.131133i	−0.997848	+	0.0655666i	$0.979115\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.54152	−0.449890	−0.224945	−	0.974372i	$-0.572220\pi$
$281$	−7.54152	−0.449890	−0.224945	+	0.974372i	$0.572220\pi$
$282$	0	0
$283$	11.5743i	0.688021i	0.938966	+	0.344011i	$0.111786\pi$
$283$	11.5743i	0.688021i	−0.938966	+	0.344011i	$0.888214\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.30828	0.136254
$288$	0	0
$289$	48.8460	2.87330
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 8.12045i	− 0.474402i	−0.971461	−	0.237201i	$-0.923770\pi$
$293$	− 8.12045i	− 0.474402i	0.971461	−	0.237201i	$-0.0762300\pi$
$294$	0	0
$295$	−1.28232	−0.0746595
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.60419i	0.381930i
$300$	0	0
$301$	− 10.7778i	− 0.621224i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.72268	0.155900
$306$	0	0
$307$	2.73852i	0.156295i	0.996942	+	0.0781477i	$0.0249006\pi$
$307$	2.73852i	0.156295i	−0.996942	+	0.0781477i	$0.975099\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	17.9488	1.01778	0.508891	−	0.860831i	$-0.330055\pi$
$311$	17.9488	1.01778	0.508891	+	0.860831i	$0.330055\pi$
$312$	0	0
$313$	5.95819	0.336777	0.168388	−	0.985721i	$-0.446144\pi$
$313$	5.95819	0.336777	0.168388	+	0.985721i	$0.446144\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.6033i	1.21336i	0.794945	+	0.606682i	$0.207500\pi$
$317$	21.6033i	1.21336i	−0.794945	+	0.606682i	$0.792500\pi$
$318$	0	0
$319$	14.6127	0.818154
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 27.9097i	− 1.55294i
$324$	0	0
$325$	− 8.55701i	− 0.474657i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.3833	0.627584
$330$	0	0
$331$	12.6066i	0.692920i	0.938065	+	0.346460i	$0.112616\pi$
$331$	12.6066i	0.692920i	−0.938065	+	0.346460i	$0.887384\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.30348	0.180488
$336$	0	0
$337$	−28.9818	−1.57874	−0.789370	−	0.613917i	$-0.789593\pi$
$337$	−28.9818	−1.57874	−0.789370	+	0.613917i	$0.789593\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.86945i	− 0.155389i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.2675i	− 0.658553i	−0.944234	−	0.329277i	$-0.893195\pi$
$347$	− 12.2675i	− 0.658553i	0.944234	−	0.329277i	$-0.106805\pi$
$348$	0	0
$349$	28.3289i	1.51641i	0.652016	+	0.758205i	$0.273923\pi$
$349$	28.3289i	1.51641i	−0.652016	+	0.758205i	$0.726077\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.40996	0.0750444	0.0375222	−	0.999296i	$-0.488054\pi$
$353$	1.40996	0.0750444	0.0375222	+	0.999296i	$0.488054\pi$
$354$	0	0
$355$	− 4.66826i	− 0.247766i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.0581	0.900290	0.450145	−	0.892955i	$-0.351372\pi$
$359$	17.0581	0.900290	0.450145	+	0.892955i	$0.351372\pi$
$360$	0	0
$361$	7.17014	0.377376
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 4.73715i	− 0.247954i
$366$	0	0
$367$	25.4558	1.32878	0.664390	−	0.747386i	$-0.268692\pi$
$367$	25.4558	1.32878	0.664390	+	0.747386i	$0.268692\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.87795i	0.460920i
$372$	0	0
$373$	20.3176i	1.05201i	0.850483	+	0.526003i	$0.176310\pi$
$373$	20.3176i	1.05201i	−0.850483	+	0.526003i	$0.823690\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.4319	0.743280
$378$	0	0
$379$	− 11.8832i	− 0.610397i	−0.952289	−	0.305198i	$-0.901277\pi$
$379$	− 11.8832i	− 0.610397i	0.952289	−	0.305198i	$-0.0987228\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−18.6261	−0.951749	−0.475874	−	0.879513i	$-0.657868\pi$
$383$	−18.6261	−0.951749	−0.475874	+	0.879513i	$0.657868\pi$
$384$	0	0
$385$	−0.937212	−0.0477647
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.7636i	1.05276i	0.850251	+	0.526378i	$0.176450\pi$
$389$	20.7636i	1.05276i	−0.850251	+	0.526378i	$0.823550\pi$
$390$	0	0
$391$	29.6690	1.50042
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.86684i	0.0939311i
$396$	0	0
$397$	20.9726i	1.05259i	0.850303	+	0.526293i	$0.176418\pi$
$397$	20.9726i	1.05259i	−0.850303	+	0.526293i	$0.823582\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.476166	0.0237786	0.0118893	−	0.999929i	$-0.496215\pi$
$401$	0.476166	0.0237786	0.0118893	+	0.999929i	$0.496215\pi$
$402$	0	0
$403$	− 2.83394i	− 0.141169i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.4427	0.765464
$408$	0	0
$409$	−2.08700	−0.103196	−0.0515978	−	0.998668i	$-0.516431\pi$
$409$	−2.08700	−0.103196	−0.0515978	+	0.998668i	$0.516431\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2.50234i	− 0.123132i
$414$	0	0
$415$	−2.32192	−0.113979
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15.8104i	0.772388i	0.922418	+	0.386194i	$0.126210\pi$
$419$	15.8104i	0.772388i	−0.922418	+	0.386194i	$0.873790\pi$
$420$	0	0
$421$	− 21.8824i	− 1.06648i	−0.845963	−	0.533241i	$-0.820974\pi$
$421$	− 21.8824i	− 1.06648i	0.845963	−	0.533241i	$-0.179026\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−38.4419	−1.86471
$426$	0	0
$427$	5.31310i	0.257119i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.3231	1.21977	0.609885	−	0.792490i	$-0.291216\pi$
$431$	25.3231	1.21977	0.609885	+	0.792490i	$0.291216\pi$
$432$	0	0
$433$	30.9197	1.48590	0.742952	−	0.669344i	$-0.233425\pi$
$433$	30.9197	1.48590	0.742952	+	0.669344i	$0.233425\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 12.5756i	− 0.601571i
$438$	0	0
$439$	8.75669	0.417934	0.208967	−	0.977923i	$-0.432990\pi$
$439$	8.75669	0.417934	0.208967	+	0.977923i	$0.432990\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	27.6190i	1.31222i	0.754666	+	0.656109i	$0.227799\pi$
$443$	27.6190i	1.31222i	−0.754666	+	0.656109i	$0.772201\pi$
$444$	0	0
$445$	− 6.01404i	− 0.285093i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13.6994	−0.646516	−0.323258	−	0.946311i	$-0.604778\pi$
$449$	−13.6994	−0.646516	−0.323258	+	0.946311i	$0.604778\pi$
$450$	0	0
$451$	− 4.22161i	− 0.198788i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.925616	−0.0433935
$456$	0	0
$457$	5.85452	0.273863	0.136931	−	0.990581i	$-0.456276\pi$
$457$	5.85452	0.273863	0.136931	+	0.990581i	$0.456276\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.42544i	0.112964i	0.998404	+	0.0564821i	$0.0179884\pi$
$461$	2.42544i	0.112964i	−0.998404	+	0.0564821i	$0.982012\pi$
$462$	0	0
$463$	1.26366	0.0587273	0.0293636	−	0.999569i	$-0.490652\pi$
$463$	1.26366	0.0587273	0.0293636	+	0.999569i	$0.490652\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 27.2456i	− 1.26078i	−0.776280	−	0.630388i	$-0.782896\pi$
$467$	− 27.2456i	− 1.26078i	0.776280	−	0.630388i	$-0.217104\pi$
$468$	0	0
$469$	6.44648i	0.297671i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−19.7115	−0.906338
$474$	0	0
$475$	16.2941i	0.747624i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−27.4035	−1.25210	−0.626050	−	0.779783i	$-0.715329\pi$
$479$	−27.4035	−1.25210	−0.626050	+	0.779783i	$0.715329\pi$
$480$	0	0
$481$	15.2516	0.695412
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 7.80574i	− 0.354440i
$486$	0	0
$487$	−27.0530	−1.22589	−0.612944	−	0.790126i	$-0.710015\pi$
$487$	−27.0530	−1.22589	−0.612944	+	0.790126i	$0.710015\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	29.2739i	1.32111i	0.750776	+	0.660556i	$0.229680\pi$
$491$	29.2739i	1.32111i	−0.750776	+	0.660556i	$0.770320\pi$
$492$	0	0
$493$	− 64.8345i	− 2.92000i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.10975	0.408628
$498$	0	0
$499$	− 39.8348i	− 1.78325i	−0.452775	−	0.891625i	$-0.649566\pi$
$499$	− 39.8348i	− 1.78325i	0.452775	−	0.891625i	$-0.350434\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−36.3079	−1.61889	−0.809444	−	0.587197i	$-0.800231\pi$
$503$	−36.3079	−1.61889	−0.809444	+	0.587197i	$0.800231\pi$
$504$	0	0
$505$	4.16888	0.185513
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 22.2853i	− 0.987778i	−0.869525	−	0.493889i	$-0.835575\pi$
$509$	− 22.2853i	− 0.987778i	0.869525	−	0.493889i	$-0.164425\pi$
$510$	0	0
$511$	9.24419	0.408939
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 8.18971i	− 0.360882i
$516$	0	0
$517$	− 20.8190i	− 0.915617i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	28.8909	1.26573	0.632866	−	0.774261i	$-0.281878\pi$
$521$	28.8909	1.26573	0.632866	+	0.774261i	$0.281878\pi$
$522$	0	0
$523$	22.7284i	0.993842i	0.867796	+	0.496921i	$0.165536\pi$
$523$	22.7284i	0.993842i	−0.867796	+	0.496921i	$0.834464\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.7313	−0.554585
$528$	0	0
$529$	−9.63174	−0.418771
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.16938i	− 0.180596i
$534$	0	0
$535$	−6.23622	−0.269615
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.82890i	− 0.0787761i
$540$	0	0
$541$	12.7291i	0.547268i	0.961834	+	0.273634i	$0.0882257\pi$
$541$	12.7291i	0.547268i	−0.961834	+	0.273634i	$0.911774\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.55759	−0.238061
$546$	0	0
$547$	− 28.7154i	− 1.22778i	−0.789391	−	0.613890i	$-0.789604\pi$
$547$	− 28.7154i	− 1.22778i	0.789391	−	0.613890i	$-0.210396\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−27.4809	−1.17073
$552$	0	0
$553$	−3.64300	−0.154916
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.36295i	− 0.311978i	−0.987759	−	0.155989i	$-0.950143\pi$
$557$	− 7.36295i	− 0.311978i	0.987759	−	0.155989i	$-0.0498565\pi$
$558$	0	0
$559$	−19.4676	−0.823394
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 15.9713i	− 0.673110i	−0.941664	−	0.336555i	$-0.890738\pi$
$563$	− 15.9713i	− 0.673110i	0.941664	−	0.336555i	$-0.109262\pi$
$564$	0	0
$565$	1.57610i	0.0663068i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.9769	1.13093	0.565465	−	0.824772i	$-0.308697\pi$
$569$	26.9769	1.13093	0.565465	+	0.824772i	$0.308697\pi$
$570$	0	0
$571$	35.5632i	1.48827i	0.668027	+	0.744137i	$0.267139\pi$
$571$	35.5632i	1.48827i	−0.668027	+	0.744137i	$0.732861\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−17.3212	−0.722343
$576$	0	0
$577$	−23.4330	−0.975528	−0.487764	−	0.872976i	$-0.662187\pi$
$577$	−23.4330	−0.975528	−0.487764	+	0.872976i	$0.662187\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4.53105i	− 0.187980i
$582$	0	0
$583$	16.2369	0.672462
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.1113i	1.53175i	0.642991	+	0.765874i	$0.277693\pi$
$587$	37.1113i	1.53175i	−0.642991	+	0.765874i	$0.722307\pi$
$588$	0	0
$589$	5.39633i	0.222352i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−23.8317	−0.978650	−0.489325	−	0.872102i	$-0.662757\pi$
$593$	−23.8317	−0.978650	−0.489325	+	0.872102i	$0.662757\pi$
$594$	0	0
$595$	4.15828i	0.170473i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.0955	0.494210	0.247105	−	0.968989i	$-0.420521\pi$
$599$	12.0955	0.494210	0.247105	+	0.968989i	$0.420521\pi$
$600$	0	0
$601$	20.4752	0.835202	0.417601	−	0.908631i	$-0.362871\pi$
$601$	20.4752	0.835202	0.417601	+	0.908631i	$0.362871\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3.92285i	− 0.159487i
$606$	0	0
$607$	24.0329	0.975466	0.487733	−	0.872993i	$-0.337824\pi$
$607$	24.0329	0.975466	0.487733	+	0.872993i	$0.337824\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 20.5614i	− 0.831824i
$612$	0	0
$613$	45.2862i	1.82909i	0.404480	+	0.914547i	$0.367452\pi$
$613$	45.2862i	1.82909i	−0.404480	+	0.914547i	$0.632548\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.4393	0.822855	0.411428	−	0.911442i	$-0.365030\pi$
$617$	20.4393	0.822855	0.411428	+	0.911442i	$0.365030\pi$
$618$	0	0
$619$	15.3328i	0.616277i	0.951342	+	0.308138i	$0.0997060\pi$
$619$	15.3328i	0.616277i	−0.951342	+	0.308138i	$0.900294\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.7359	0.470190
$624$	0	0
$625$	21.1299	0.845197
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 68.5169i	− 2.73195i
$630$	0	0
$631$	−20.4869	−0.815572	−0.407786	−	0.913078i	$-0.633699\pi$
$631$	−20.4869	−0.815572	−0.407786	+	0.913078i	$0.633699\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.42070i	0.135747i
$636$	0	0
$637$	− 1.80627i	− 0.0715669i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.6868	1.68603	0.843013	−	0.537893i	$-0.180780\pi$
$641$	42.6868	1.68603	0.843013	+	0.537893i	$0.180780\pi$
$642$	0	0
$643$	29.7804i	1.17442i	0.809434	+	0.587211i	$0.199774\pi$
$643$	29.7804i	1.17442i	−0.809434	+	0.587211i	$0.800226\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.63482	0.260842	0.130421	−	0.991459i	$-0.458367\pi$
$647$	6.63482	0.260842	0.130421	+	0.991459i	$0.458367\pi$
$648$	0	0
$649$	−4.57653	−0.179644
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 4.48147i	− 0.175373i	−0.996148	−	0.0876867i	$-0.972053\pi$
$653$	− 4.48147i	− 0.175373i	0.996148	−	0.0876867i	$-0.0279474\pi$
$654$	0	0
$655$	−0.650779	−0.0254281
$656$	0	0
$657$	0	0
$658$	0	0
$659$	49.4245i	1.92530i	0.270741	+	0.962652i	$0.412731\pi$
$659$	49.4245i	1.92530i	−0.270741	+	0.962652i	$0.587269\pi$
$660$	0	0
$661$	12.5279i	0.487278i	0.969866	+	0.243639i	$0.0783413\pi$
$661$	12.5279i	0.487278i	−0.969866	+	0.243639i	$0.921659\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.76254	0.0683483
$666$	0	0
$667$	− 29.2132i	− 1.13114i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.71710	0.375125
$672$	0	0
$673$	−26.3343	−1.01511	−0.507556	−	0.861619i	$-0.669451\pi$
$673$	−26.3343	−1.01511	−0.507556	+	0.861619i	$0.669451\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.4868i	0.556774i	0.960469	+	0.278387i	$0.0897998\pi$
$677$	14.4868i	0.556774i	−0.960469	+	0.278387i	$0.910200\pi$
$678$	0	0
$679$	15.2323	0.584562
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.0232069i	0 0.000887989i	1.00000		0.000443994i	$0.000141328\pi$
$683$	0.0232069i	0 0.000887989i	−1.00000		0.000443994i	$0.999859\pi$
$684$	0	0
$685$	10.1724i	0.388668i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16.0360	0.610921
$690$	0	0
$691$	− 17.2719i	− 0.657054i	−0.944495	−	0.328527i	$-0.893448\pi$
$691$	− 17.2719i	− 0.657054i	0.944495	−	0.328527i	$-0.106552\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.16186	−0.271665
$696$	0	0
$697$	−18.7307	−0.709476
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 23.9519i	− 0.904651i	−0.891853	−	0.452326i	$-0.850595\pi$
$701$	− 23.9519i	− 0.904651i	0.891853	−	0.452326i	$-0.149405\pi$
$702$	0	0
$703$	−29.0417	−1.09533
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.13524i	0.305957i
$708$	0	0
$709$	32.1099i	1.20591i	0.797774	+	0.602956i	$0.206011\pi$
$709$	32.1099i	1.20591i	−0.797774	+	0.602956i	$0.793989\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.73649	−0.214833
$714$	0	0
$715$	1.69286i	0.0633092i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19.4708	0.726139	0.363070	−	0.931762i	$-0.381729\pi$
$719$	19.4708	0.726139	0.363070	+	0.931762i	$0.381729\pi$
$720$	0	0
$721$	15.9816	0.595185
$722$	0	0
$723$	0	0
$724$	0	0
$725$	37.8513i	1.40576i
$726$	0	0
$727$	−18.3066	−0.678954	−0.339477	−	0.940614i	$-0.610250\pi$
$727$	−18.3066	−0.678954	−0.339477	+	0.940614i	$0.610250\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	87.4573i	3.23473i
$732$	0	0
$733$	36.3167i	1.34139i	0.741735	+	0.670693i	$0.234003\pi$
$733$	36.3167i	1.34139i	−0.741735	+	0.670693i	$0.765997\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.7899	0.434288
$738$	0	0
$739$	− 15.6764i	− 0.576665i	−0.957530	−	0.288332i	$-0.906899\pi$
$739$	− 15.6764i	− 0.576665i	0.957530	−	0.288332i	$-0.0931008\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−49.3578	−1.81076	−0.905381	−	0.424599i	$-0.860415\pi$
$743$	−49.3578	−1.81076	−0.905381	+	0.424599i	$0.860415\pi$
$744$	0	0
$745$	−6.07560	−0.222593
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 12.1695i	− 0.444664i
$750$	0	0
$751$	−29.7930	−1.08716	−0.543582	−	0.839356i	$-0.682932\pi$
$751$	−29.7930	−1.08716	−0.543582	+	0.839356i	$0.682932\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 6.91542i	− 0.251678i
$756$	0	0
$757$	− 13.6736i	− 0.496974i	−0.968635	−	0.248487i	$-0.920067\pi$
$757$	− 13.6736i	− 0.496974i	0.968635	−	0.248487i	$-0.0799333\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.34618	−0.121299	−0.0606494	−	0.998159i	$-0.519317\pi$
$761$	−3.34618	−0.121299	−0.0606494	+	0.998159i	$0.519317\pi$
$762$	0	0
$763$	− 10.8452i	− 0.392623i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.51990	−0.163204
$768$	0	0
$769$	10.2544	0.369783	0.184891	−	0.982759i	$-0.440807\pi$
$769$	10.2544	0.369783	0.184891	+	0.982759i	$0.440807\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 27.9261i	− 1.00443i	−0.864742	−	0.502217i	$-0.832518\pi$
$773$	− 27.9261i	− 1.00443i	0.864742	−	0.502217i	$-0.167482\pi$
$774$	0	0
$775$	7.43274	0.266992
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.93924i	0.284453i
$780$	0	0
$781$	− 16.6608i	− 0.596170i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.06601	0.180814
$786$	0	0
$787$	− 19.9536i	− 0.711268i	−0.934625	−	0.355634i	$-0.884265\pi$
$787$	− 19.9536i	− 0.711268i	0.934625	−	0.355634i	$-0.115735\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.07563	−0.109357
$792$	0	0
$793$	9.59687	0.340795
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.3652i	0.792217i	0.918204	+	0.396108i	$0.129640\pi$
$797$	22.3652i	0.792217i	−0.918204	+	0.396108i	$0.870360\pi$
$798$	0	0
$799$	−92.3708	−3.26785
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 16.9067i	− 0.596623i
$804$	0	0
$805$	1.87364i	0.0660371i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.28121	−0.291152	−0.145576	−	0.989347i	$-0.546504\pi$
$809$	−8.28121	−0.291152	−0.145576	+	0.989347i	$0.546504\pi$
$810$	0	0
$811$	− 20.5177i	− 0.720474i	−0.932861	−	0.360237i	$-0.882696\pi$
$811$	− 20.5177i	− 0.720474i	0.932861	−	0.360237i	$-0.117304\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.27376	−0.0796462
$816$	0	0
$817$	37.0699	1.29691
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.88906i	0.240430i	0.992748	+	0.120215i	$0.0383584\pi$
$821$	6.88906i	0.240430i	−0.992748	+	0.120215i	$0.961642\pi$
$822$	0	0
$823$	6.14442	0.214181	0.107091	−	0.994249i	$-0.465847\pi$
$823$	6.14442	0.214181	0.107091	+	0.994249i	$0.465847\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.6562i	1.03125i	0.856815	+	0.515623i	$0.172440\pi$
$827$	29.6562i	1.03125i	−0.856815	+	0.515623i	$0.827560\pi$
$828$	0	0
$829$	7.16328i	0.248791i	0.992233	+	0.124396i	$0.0396992\pi$
$829$	7.16328i	0.248791i	−0.992233	+	0.124396i	$0.960301\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.11456	−0.281153
$834$	0	0
$835$	2.75897i	0.0954780i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−15.6136	−0.539041	−0.269520	−	0.962995i	$-0.586865\pi$
$839$	−15.6136	−0.539041	−0.269520	+	0.962995i	$0.586865\pi$
$840$	0	0
$841$	−34.8385	−1.20133
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 4.98990i	− 0.171658i
$846$	0	0
$847$	7.65514	0.263034
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 30.8724i	− 1.05829i
$852$	0	0
$853$	− 0.960301i	− 0.0328801i	−0.999865	−	0.0164400i	$-0.994767\pi$
$853$	− 0.960301i	− 0.0328801i	0.999865	−	0.0164400i	$-0.00523327\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.3062	−0.591168	−0.295584	−	0.955317i	$-0.595514\pi$
$857$	−17.3062	−0.591168	−0.295584	+	0.955317i	$0.595514\pi$
$858$	0	0
$859$	− 15.5416i	− 0.530273i	−0.964211	−	0.265137i	$-0.914583\pi$
$859$	− 15.5416i	− 0.530273i	0.964211	−	0.265137i	$-0.0854171\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	30.9082	1.05213	0.526063	−	0.850445i	$-0.323667\pi$
$863$	30.9082	1.05213	0.526063	+	0.850445i	$0.323667\pi$
$864$	0	0
$865$	9.20413	0.312950
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.66267i	0.226016i
$870$	0	0
$871$	11.6441	0.394544
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 4.98990i	− 0.168689i
$876$	0	0
$877$	− 30.0443i	− 1.01452i	−0.861792	−	0.507262i	$-0.830658\pi$
$877$	− 30.0443i	− 1.01452i	0.861792	−	0.507262i	$-0.169342\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	43.7362	1.47351	0.736754	−	0.676160i	$-0.236357\pi$
$881$	43.7362	1.47351	0.736754	+	0.676160i	$0.236357\pi$
$882$	0	0
$883$	− 11.9694i	− 0.402804i	−0.979509	−	0.201402i	$-0.935450\pi$
$883$	− 11.9694i	− 0.402804i	0.979509	−	0.201402i	$-0.0645497\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	50.6980	1.70227	0.851136	−	0.524945i	$-0.175914\pi$
$887$	50.6980	1.70227	0.851136	+	0.524945i	$0.175914\pi$
$888$	0	0
$889$	−6.67524	−0.223880
$890$	0	0
$891$	0	0
$892$	0	0
$893$	39.1525i	1.31019i
$894$	0	0
$895$	−8.22384	−0.274893
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.5357i	0.418090i
$900$	0	0
$901$	− 72.0406i	− 2.40002i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.03297	0.100819
$906$	0	0
$907$	− 5.94868i	− 0.197523i	−0.995111	−	0.0987614i	$-0.968512\pi$
$907$	− 5.94868i	− 0.197523i	0.995111	−	0.0987614i	$-0.0314881\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−49.6827	−1.64606	−0.823031	−	0.567996i	$-0.807719\pi$
$911$	−49.6827	−1.64606	−0.823031	+	0.567996i	$0.807719\pi$
$912$	0	0
$913$	−8.28683	−0.274254
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.26995i	− 0.0419373i
$918$	0	0
$919$	−16.5497	−0.545923	−0.272961	−	0.962025i	$-0.588003\pi$
$919$	−16.5497	−0.545923	−0.272961	+	0.962025i	$0.588003\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 16.4546i	− 0.541611i
$924$	0	0
$925$	40.0012i	1.31523i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	15.8292	0.519339	0.259669	−	0.965698i	$-0.416386\pi$
$929$	15.8292	0.519339	0.259669	+	0.965698i	$0.416386\pi$
$930$	0	0
$931$	3.43946i	0.112724i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.60506	0.248712
$936$	0	0
$937$	−14.1814	−0.463286	−0.231643	−	0.972801i	$-0.574410\pi$
$937$	−14.1814	−0.463286	−0.231643	+	0.972801i	$0.574410\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 59.4006i	− 1.93641i	−0.250166	−	0.968203i	$-0.580485\pi$
$941$	− 59.4006i	− 1.93641i	0.250166	−	0.968203i	$-0.419515\pi$
$942$	0	0
$943$	−8.43969	−0.274834
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.81038i	0.221308i	0.993859	+	0.110654i	$0.0352945\pi$
$947$	6.81038i	0.221308i	−0.993859	+	0.110654i	$0.964706\pi$
$948$	0	0
$949$	− 16.6975i	− 0.542023i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.4827	0.825466	0.412733	−	0.910852i	$-0.364574\pi$
$953$	25.4827	0.825466	0.412733	+	0.910852i	$0.364574\pi$
$954$	0	0
$955$	− 0.0357604i	− 0.00115718i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19.8507	−0.641012
$960$	0	0
$961$	−28.5384	−0.920593
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.22439i	0.135988i
$966$	0	0
$967$	−44.8473	−1.44219	−0.721096	−	0.692835i	$-0.756361\pi$
$967$	−44.8473	−1.44219	−0.721096	+	0.692835i	$0.756361\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 23.4868i	− 0.753728i	−0.926269	−	0.376864i	$-0.877002\pi$
$971$	− 23.4868i	− 0.753728i	0.926269	−	0.376864i	$-0.122998\pi$
$972$	0	0
$973$	− 13.9758i	− 0.448044i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−54.0970	−1.73072	−0.865359	−	0.501153i	$-0.832909\pi$
$977$	−54.0970	−1.73072	−0.865359	+	0.501153i	$0.832909\pi$
$978$	0	0
$979$	− 21.4638i	− 0.685986i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15.0476	0.479943	0.239972	−	0.970780i	$-0.422862\pi$
$983$	15.0476	0.479943	0.239972	+	0.970780i	$0.422862\pi$
$984$	0	0
$985$	−0.980283	−0.0312344
$986$	0	0
$987$	0	0
$988$	0	0
$989$	39.4066i	1.25306i
$990$	0	0
$991$	−24.0651	−0.764455	−0.382227	−	0.924068i	$-0.624843\pi$
$991$	−24.0651	−0.764455	−0.382227	+	0.924068i	$0.624843\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.89902i	− 0.0602030i
$996$	0	0
$997$	46.1183i	1.46058i	0.683136	+	0.730291i	$0.260616\pi$
$997$	46.1183i	1.46058i	−0.683136	+	0.730291i	$0.739384\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.e.3025.7		20
3.2	odd	2	inner	6048.2.c.e.3025.13		20
4.3	odd	2		1512.2.c.e.757.9	✓	20
8.3	odd	2		1512.2.c.e.757.10	yes	20
8.5	even	2	inner	6048.2.c.e.3025.14		20
12.11	even	2		1512.2.c.e.757.12	yes	20
24.5	odd	2	inner	6048.2.c.e.3025.8		20
24.11	even	2		1512.2.c.e.757.11	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.e.757.9	✓	20	4.3	odd	2
1512.2.c.e.757.10	yes	20	8.3	odd	2
1512.2.c.e.757.11	yes	20	24.11	even	2
1512.2.c.e.757.12	yes	20	12.11	even	2
6048.2.c.e.3025.7		20	1.1	even	1	trivial
6048.2.c.e.3025.8		20	24.5	odd	2	inner
6048.2.c.e.3025.13		20	3.2	odd	2	inner
6048.2.c.e.3025.14		20	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.512447i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-0.512447i q^{5} +1.00000 q^{7} -1.82890i q^{11} -1.80627i q^{13} -8.11456 q^{17} +3.43946i q^{19} -3.65626 q^{23} +4.73740 q^{25} +7.98990i q^{29} +1.56895 q^{31} -0.512447i q^{35} +8.44370i q^{37} +2.30828 q^{41} -10.7778i q^{43} +11.3833 q^{47} +1.00000 q^{49} +8.87795i q^{53} -0.937212 q^{55} -2.50234i q^{59} +5.31310i q^{61} -0.925616 q^{65} +6.44648i q^{67} +9.10975 q^{71} +9.24419 q^{73} -1.82890i q^{77} -3.64300 q^{79} -4.53105i q^{83} +4.15828i q^{85} +11.7359 q^{89} -1.80627i q^{91} +1.76254 q^{95} +15.2323 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{7}+O(q^{10}) \) 20 * q + 20 * q^7	\( 20 q + 20 q^{7} - 28 q^{25} - 36 q^{31} + 20 q^{49} - 48 q^{55} - 64 q^{79} + 56 q^{97}+O(q^{100}) \) 20 * q + 20 * q^7 - 28 * q^25 - 36 * q^31 + 20 * q^49 - 48 * q^55 - 64 * q^79 + 56 * q^97

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