LMFDB - Embedded newform 6048.2.c.d.3025.12

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			3025.12
Root			$-0.453990 - 0.891007i$ of defining polynomial
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.d.3025.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.60758i q^{5} +1.00000 q^{7} +O(q^{10})$	$q+1.60758i q^{5} +1.00000 q^{7} +0.0549306i q^{11} +3.75621i q^{13} +3.16228 q^{17} -0.726543i q^{19} +7.77604 q^{23} +2.41570 q^{25} -2.75789i q^{29} +2.14475 q^{31} +1.60758i q^{35} +0.600848i q^{37} -4.53658 q^{41} -6.85004i q^{43} +0.744883 q^{47} +1.00000 q^{49} +10.5822i q^{53} -0.0883051 q^{55} +2.58013i q^{59} -9.80423i q^{61} -6.03841 q^{65} -12.2627i q^{67} +9.19025 q^{71} +3.85224 q^{73} +0.0549306i q^{77} -10.9057 q^{79} +13.1624i q^{83} +5.08361i q^{85} +7.90356 q^{89} +3.75621i q^{91} +1.16797 q^{95} +12.1803 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.60758i	0.718930i	0.933158	+	0.359465i	$0.117041\pi$
$5$	1.60758i	0.718930i	−0.933158	+	0.359465i	$0.882959\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.0549306i	0.0165622i	0.999966	+	0.00828109i	$0.00263598\pi$
$11$	0.0549306i	0.0165622i	−0.999966	+	0.00828109i	$0.997364\pi$
$12$	0	0
$13$	3.75621i	1.04179i	0.853622	+	0.520893i	$0.174401\pi$
$13$	3.75621i	1.04179i	−0.853622	+	0.520893i	$0.825599\pi$
$14$	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	$-0.973441\pi$
$19$	− 0.726543i	− 0.166680i	0.996521	−	0.0833401i	$-0.0265588\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.77604	1.62142	0.810708	−	0.585450i	$-0.199082\pi$
$23$	7.77604	1.62142	0.810708	+	0.585450i	$0.199082\pi$
$24$	0	0
$25$	2.41570	0.483139
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.75789i	− 0.512128i	−0.966660	−	0.256064i	$-0.917574\pi$
$29$	− 2.75789i	− 0.512128i	0.966660	−	0.256064i	$-0.0824257\pi$
$30$	0	0
$31$	2.14475	0.385208	0.192604	−	0.981277i	$-0.438307\pi$
$31$	2.14475	0.385208	0.192604	+	0.981277i	$0.438307\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.60758i	0.271730i
$36$	0	0
$37$	0.600848i	0.0987788i	0.998780	+	0.0493894i	$0.0157275\pi$
$37$	0.600848i	0.0987788i	−0.998780	+	0.0493894i	$0.984272\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.53658	−0.708495	−0.354248	−	0.935152i	$-0.615263\pi$
$41$	−4.53658	−0.708495	−0.354248	+	0.935152i	$0.615263\pi$
$42$	0	0
$43$	− 6.85004i	− 1.04462i	−0.852755	−	0.522311i	$-0.825070\pi$
$43$	− 6.85004i	− 1.04462i	0.852755	−	0.522311i	$-0.174930\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.744883	0.108652	0.0543261	−	0.998523i	$-0.482699\pi$
$47$	0.744883	0.108652	0.0543261	+	0.998523i	$0.482699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.5822i	1.45358i	0.686860	+	0.726790i	$0.258989\pi$
$53$	10.5822i	1.45358i	−0.686860	+	0.726790i	$0.741011\pi$
$54$	0	0
$55$	−0.0883051	−0.0119071
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.58013i	0.335905i	0.985795	+	0.167952i	$0.0537155\pi$
$59$	2.58013i	0.335905i	−0.985795	+	0.167952i	$0.946285\pi$
$60$	0	0
$61$	− 9.80423i	− 1.25530i	−0.778495	−	0.627651i	$-0.784016\pi$
$61$	− 9.80423i	− 1.25530i	0.778495	−	0.627651i	$-0.215984\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.03841	−0.748972
$66$	0	0
$67$	− 12.2627i	− 1.49813i	−0.662496	−	0.749065i	$-0.730503\pi$
$67$	− 12.2627i	− 1.49813i	0.662496	−	0.749065i	$-0.269497\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.19025	1.09068	0.545341	−	0.838214i	$-0.316400\pi$
$71$	9.19025	1.09068	0.545341	+	0.838214i	$0.316400\pi$
$72$	0	0
$73$	3.85224	0.450870	0.225435	−	0.974258i	$-0.427620\pi$
$73$	3.85224	0.450870	0.225435	+	0.974258i	$0.427620\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.0549306i	0.00625992i
$78$	0	0
$79$	−10.9057	−1.22698	−0.613491	−	0.789701i	$-0.710235\pi$
$79$	−10.9057	−1.22698	−0.613491	+	0.789701i	$0.710235\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.1624i	1.44476i	0.691499	+	0.722378i	$0.256951\pi$
$83$	13.1624i	1.44476i	−0.691499	+	0.722378i	$0.743049\pi$
$84$	0	0
$85$	5.08361i	0.551394i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.90356	0.837776	0.418888	−	0.908038i	$-0.362420\pi$
$89$	7.90356	0.837776	0.418888	+	0.908038i	$0.362420\pi$
$90$	0	0
$91$	3.75621i	0.393758i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.16797	0.119832
$96$	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	$0.0694788\pi$
$101$	4.35250i	0.433090i	−0.976273	+	0.216545i	$0.930521\pi$
$102$	0	0
$103$	−17.5574	−1.72998	−0.864992	−	0.501785i	$-0.832677\pi$
$103$	−17.5574	−1.72998	−0.864992	+	0.501785i	$0.832677\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6.44944i	− 0.623491i	−0.950166	−	0.311745i	$-0.899086\pi$
$107$	− 6.44944i	− 0.623491i	0.950166	−	0.311745i	$-0.100914\pi$
$108$	0	0
$109$	2.03559i	0.194975i	0.995237	+	0.0974873i	$0.0310805\pi$
$109$	2.03559i	0.194975i	−0.995237	+	0.0974873i	$0.968919\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.61946	−0.246418	−0.123209	−	0.992381i	$-0.539319\pi$
$113$	−2.61946	−0.246418	−0.123209	+	0.992381i	$0.539319\pi$
$114$	0	0
$115$	12.5006i	1.16569i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.16228	0.289886
$120$	0	0
$121$	10.9970	0.999726
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.9213i	1.06627i
$126$	0	0
$127$	13.6733	1.21331	0.606656	−	0.794965i	$-0.292511\pi$
$127$	13.6733	1.21331	0.606656	+	0.794965i	$0.292511\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.2592i	0.983721i	0.870674	+	0.491861i	$0.163683\pi$
$131$	11.2592i	0.983721i	−0.870674	+	0.491861i	$0.836317\pi$
$132$	0	0
$133$	− 0.726543i	− 0.0629992i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.00318	−0.0857076	−0.0428538	−	0.999081i	$-0.513645\pi$
$137$	−1.00318	−0.0857076	−0.0428538	+	0.999081i	$0.513645\pi$
$138$	0	0
$139$	− 2.60253i	− 0.220744i	−0.993890	−	0.110372i	$-0.964796\pi$
$139$	− 2.60253i	− 0.220744i	0.993890	−	0.110372i	$-0.0352042\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.206331	−0.0172543
$144$	0	0
$145$	4.43352	0.368184
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.1514i	1.56895i	0.620163	+	0.784473i	$0.287066\pi$
$149$	19.1514i	1.56895i	−0.620163	+	0.784473i	$0.712934\pi$
$150$	0	0
$151$	−15.8848	−1.29269	−0.646344	−	0.763046i	$-0.723703\pi$
$151$	−15.8848	−1.29269	−0.646344	+	0.763046i	$0.723703\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.44784i	0.276937i
$156$	0	0
$157$	14.7099i	1.17398i	0.809595	+	0.586988i	$0.199687\pi$
$157$	14.7099i	1.17398i	−0.809595	+	0.586988i	$0.800313\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.77604	0.612838
$162$	0	0
$163$	9.69687i	0.759517i	0.925086	+	0.379759i	$0.123993\pi$
$163$	9.69687i	0.759517i	−0.925086	+	0.379759i	$0.876007\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.57013	0.198882	0.0994412	−	0.995043i	$-0.468294\pi$
$167$	2.57013	0.198882	0.0994412	+	0.995043i	$0.468294\pi$
$168$	0	0
$169$	−1.10915	−0.0853193
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.69460i	− 0.128838i	−0.997923	−	0.0644192i	$-0.979481\pi$
$173$	− 1.69460i	− 0.128838i	0.997923	−	0.0644192i	$-0.0205195\pi$
$174$	0	0
$175$	2.41570	0.182609
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 11.1383i	− 0.832518i	−0.909246	−	0.416259i	$-0.863341\pi$
$179$	− 11.1383i	− 0.832518i	0.909246	−	0.416259i	$-0.136659\pi$
$180$	0	0
$181$	− 6.80203i	− 0.505591i	−0.967520	−	0.252796i	$-0.918650\pi$
$181$	− 6.80203i	− 0.505591i	0.967520	−	0.252796i	$-0.0813500\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.965909	−0.0710151
$186$	0	0
$187$	0.173706i	0.0127026i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.6397	1.13165	0.565824	−	0.824526i	$-0.308558\pi$
$191$	15.6397	1.13165	0.565824	+	0.824526i	$0.308558\pi$
$192$	0	0
$193$	19.5893	1.41007	0.705034	−	0.709174i	$-0.250932\pi$
$193$	19.5893	1.41007	0.705034	+	0.709174i	$0.250932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.29916i	− 0.520044i	−0.965603	−	0.260022i	$-0.916270\pi$
$197$	− 7.29916i	− 0.520044i	0.965603	−	0.260022i	$-0.0837298\pi$
$198$	0	0
$199$	−13.8002	−0.978273	−0.489136	−	0.872207i	$-0.662688\pi$
$199$	−13.8002	−0.978273	−0.489136	+	0.872207i	$0.662688\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.75789i	− 0.193566i
$204$	0	0
$205$	− 7.29291i	− 0.509359i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.0399094	0.00276059
$210$	0	0
$211$	− 5.25731i	− 0.361928i	−0.983490	−	0.180964i	$-0.942078\pi$
$211$	− 5.25731i	− 0.361928i	0.983490	−	0.180964i	$-0.0579218\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.0120	0.751011
$216$	0	0
$217$	2.14475	0.145595
$218$	0	0
$219$	0	0
$220$	0	0
$221$	11.8782i	0.799014i
$222$	0	0
$223$	−4.96512	−0.332489	−0.166244	−	0.986085i	$-0.553164\pi$
$223$	−4.96512	−0.332489	−0.166244	+	0.986085i	$0.553164\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.1773i	1.00735i	0.863893	+	0.503676i	$0.168019\pi$
$227$	15.1773i	1.00735i	−0.863893	+	0.503676i	$0.831981\pi$
$228$	0	0
$229$	5.69636i	0.376426i	0.982128	+	0.188213i	$0.0602695\pi$
$229$	5.69636i	0.376426i	−0.982128	+	0.188213i	$0.939730\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.5086	1.08151	0.540756	−	0.841179i	$-0.318138\pi$
$233$	16.5086	1.08151	0.540756	+	0.841179i	$0.318138\pi$
$234$	0	0
$235$	1.19746i	0.0781134i
$236$	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
$241$	−28.5047	−1.83615	−0.918075	−	0.396407i	$-0.870257\pi$
$241$	−28.5047	−1.83615	−0.918075	+	0.396407i	$0.870257\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.60758i	0.102704i
$246$	0	0
$247$	2.72905	0.173645
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.18930i	0.453785i	0.973920	+	0.226892i	$0.0728566\pi$
$251$	7.18930i	0.453785i	−0.973920	+	0.226892i	$0.927143\pi$
$252$	0	0
$253$	0.427142i	0.0268542i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.4469	1.89922	0.949612	−	0.313427i	$-0.101477\pi$
$257$	30.4469	1.89922	0.949612	+	0.313427i	$0.101477\pi$
$258$	0	0
$259$	0.600848i	0.0373349i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.28410	−0.510820	−0.255410	−	0.966833i	$-0.582210\pi$
$263$	−8.28410	−0.510820	−0.255410	+	0.966833i	$0.582210\pi$
$264$	0	0
$265$	−17.0117	−1.04502
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 7.16222i	− 0.436689i	−0.975872	−	0.218344i	$-0.929934\pi$
$269$	− 7.16222i	− 0.436689i	0.975872	−	0.218344i	$-0.0700656\pi$
$270$	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
$275$	0.132696i	0.00800184i
$276$	0	0
$277$	8.91531i	0.535669i	0.963465	+	0.267835i	$0.0863081\pi$
$277$	8.91531i	0.535669i	−0.963465	+	0.267835i	$0.913692\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.3577	0.737198	0.368599	−	0.929589i	$-0.379838\pi$
$281$	12.3577	0.737198	0.368599	+	0.929589i	$0.379838\pi$
$282$	0	0
$283$	− 21.4245i	− 1.27356i	−0.771047	−	0.636778i	$-0.780267\pi$
$283$	− 21.4245i	− 1.27356i	0.771047	−	0.636778i	$-0.219733\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.53658	−0.267786
$288$	0	0
$289$	−7.00000	−0.411765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.0647976i	0.00378552i	0.999998	+	0.00189276i	$0.000602484\pi$
$293$	0.0647976i	0.00378552i	−0.999998	+	0.00189276i	$0.999398\pi$
$294$	0	0
$295$	−4.14776	−0.241492
$296$	0	0
$297$	0	0
$298$	0	0
$299$	29.2085i	1.68917i
$300$	0	0
$301$	− 6.85004i	− 0.394830i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.7611	0.902475
$306$	0	0
$307$	15.8474i	0.904460i	0.891901	+	0.452230i	$0.149371\pi$
$307$	15.8474i	0.904460i	−0.891901	+	0.452230i	$0.850629\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.0254	−0.738602	−0.369301	−	0.929310i	$-0.620403\pi$
$311$	−13.0254	−0.738602	−0.369301	+	0.929310i	$0.620403\pi$
$312$	0	0
$313$	24.0265	1.35806	0.679030	−	0.734110i	$-0.262401\pi$
$313$	24.0265	1.35806	0.679030	+	0.734110i	$0.262401\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.71438i	0.208620i	0.994545	+	0.104310i	$0.0332635\pi$
$317$	3.71438i	0.208620i	−0.994545	+	0.104310i	$0.966737\pi$
$318$	0	0
$319$	0.151493	0.00848195
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2.29753i	− 0.127838i
$324$	0	0
$325$	9.07387i	0.503328i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.744883	0.0410667
$330$	0	0
$331$	3.50059i	0.192410i	0.995362	+	0.0962048i	$0.0306704\pi$
$331$	3.50059i	0.192410i	−0.995362	+	0.0962048i	$0.969330\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	19.7133	1.07705
$336$	0	0
$337$	−16.9917	−0.925595	−0.462797	−	0.886464i	$-0.653154\pi$
$337$	−16.9917	−0.925595	−0.462797	+	0.886464i	$0.653154\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.117812i	0.00637988i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	36.6651i	1.96829i	0.177373	+	0.984144i	$0.443240\pi$
$347$	36.6651i	1.96829i	−0.177373	+	0.984144i	$0.556760\pi$
$348$	0	0
$349$	− 34.7887i	− 1.86220i	−0.364770	−	0.931098i	$-0.618852\pi$
$349$	− 34.7887i	− 1.86220i	0.364770	−	0.931098i	$-0.381148\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.1913	0.648876	0.324438	−	0.945907i	$-0.394825\pi$
$353$	12.1913	0.648876	0.324438	+	0.945907i	$0.394825\pi$
$354$	0	0
$355$	14.7740i	0.784125i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	33.4411	1.76495	0.882477	−	0.470355i	$-0.155874\pi$
$359$	33.4411	1.76495	0.882477	+	0.470355i	$0.155874\pi$
$360$	0	0
$361$	18.4721	0.972218
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.19277i	0.324144i
$366$	0	0
$367$	14.4869	0.756212	0.378106	−	0.925762i	$-0.376575\pi$
$367$	14.4869	0.756212	0.378106	+	0.925762i	$0.376575\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.5822i	0.549401i
$372$	0	0
$373$	28.7612i	1.48920i	0.667512	+	0.744599i	$0.267359\pi$
$373$	28.7612i	1.48920i	−0.667512	+	0.744599i	$0.732641\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.3592	0.533528
$378$	0	0
$379$	− 16.1036i	− 0.827188i	−0.910461	−	0.413594i	$-0.864273\pi$
$379$	− 16.1036i	− 0.827188i	0.910461	−	0.413594i	$-0.135727\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−25.7602	−1.31629	−0.658143	−	0.752893i	$-0.728658\pi$
$383$	−25.7602	−1.31629	−0.658143	+	0.752893i	$0.728658\pi$
$384$	0	0
$385$	−0.0883051	−0.00450045
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9.82322i	− 0.498057i	−0.968496	−	0.249029i	$-0.919889\pi$
$389$	− 9.82322i	− 0.498057i	0.968496	−	0.249029i	$-0.0801113\pi$
$390$	0	0
$391$	24.5900	1.24357
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 17.5317i	− 0.882115i
$396$	0	0
$397$	24.1824i	1.21368i	0.794824	+	0.606840i	$0.207563\pi$
$397$	24.1824i	1.21368i	−0.794824	+	0.606840i	$0.792437\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.4216	−1.16962	−0.584810	−	0.811170i	$-0.698831\pi$
$401$	−23.4216	−1.16962	−0.584810	+	0.811170i	$0.698831\pi$
$402$	0	0
$403$	8.05613i	0.401304i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.0330049	−0.00163599
$408$	0	0
$409$	−32.6813	−1.61599	−0.807994	−	0.589191i	$-0.799447\pi$
$409$	−32.6813	−1.61599	−0.807994	+	0.589191i	$0.799447\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.58013i	0.126960i
$414$	0	0
$415$	−21.1595	−1.03868
$416$	0	0
$417$	0	0
$418$	0	0
$419$	29.3981i	1.43619i	0.695945	+	0.718095i	$0.254986\pi$
$419$	29.3981i	1.43619i	−0.695945	+	0.718095i	$0.745014\pi$
$420$	0	0
$421$	− 1.59493i	− 0.0777319i	−0.999244	−	0.0388660i	$-0.987625\pi$
$421$	− 1.59493i	− 0.0777319i	0.999244	−	0.0388660i	$-0.0123745\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.63910	0.370551
$426$	0	0
$427$	− 9.80423i	− 0.474460i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.7336	−0.517020	−0.258510	−	0.966009i	$-0.583232\pi$
$431$	−10.7336	−0.517020	−0.258510	+	0.966009i	$0.583232\pi$
$432$	0	0
$433$	−8.43352	−0.405289	−0.202645	−	0.979252i	$-0.564954\pi$
$433$	−8.43352	−0.405289	−0.202645	+	0.979252i	$0.564954\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.64962i	− 0.270258i
$438$	0	0
$439$	17.3452	0.827842	0.413921	−	0.910313i	$-0.364159\pi$
$439$	17.3452	0.827842	0.413921	+	0.910313i	$0.364159\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.8824i	1.41976i	0.704325	+	0.709878i	$0.251250\pi$
$443$	29.8824i	1.41976i	−0.704325	+	0.709878i	$0.748750\pi$
$444$	0	0
$445$	12.7056i	0.602302i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.6303	−0.879217	−0.439608	−	0.898190i	$-0.644883\pi$
$449$	−18.6303	−0.879217	−0.439608	+	0.898190i	$0.644883\pi$
$450$	0	0
$451$	− 0.249197i	− 0.0117342i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.03841	−0.283085
$456$	0	0
$457$	−6.63702	−0.310466	−0.155233	−	0.987878i	$-0.549613\pi$
$457$	−6.63702	−0.310466	−0.155233	+	0.987878i	$0.549613\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	42.1500i	1.96312i	0.191155	+	0.981560i	$0.438777\pi$
$461$	42.1500i	1.96312i	−0.191155	+	0.981560i	$0.561223\pi$
$462$	0	0
$463$	27.1209	1.26041	0.630207	−	0.776427i	$-0.282970\pi$
$463$	27.1209	1.26041	0.630207	+	0.776427i	$0.282970\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.06443i	0.0492561i	0.999697	+	0.0246280i	$0.00784014\pi$
$467$	1.06443i	0.0492561i	−0.999697	+	0.0246280i	$0.992160\pi$
$468$	0	0
$469$	− 12.2627i	− 0.566240i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.376277	0.0173012
$474$	0	0
$475$	− 1.75511i	− 0.0805298i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−32.7777	−1.49765	−0.748825	−	0.662767i	$-0.769382\pi$
$479$	−32.7777	−1.49765	−0.748825	+	0.662767i	$0.769382\pi$
$480$	0	0
$481$	−2.25691	−0.102906
$482$	0	0
$483$	0	0
$484$	0	0
$485$	19.5808i	0.889120i
$486$	0	0
$487$	2.44329	0.110716	0.0553580	−	0.998467i	$-0.482370\pi$
$487$	2.44329	0.110716	0.0553580	+	0.998467i	$0.482370\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 25.5978i	− 1.15521i	−0.816315	−	0.577606i	$-0.803987\pi$
$491$	− 25.5978i	− 1.15521i	0.816315	−	0.577606i	$-0.196013\pi$
$492$	0	0
$493$	− 8.72122i	− 0.392784i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.19025	0.412239
$498$	0	0
$499$	− 7.96720i	− 0.356661i	−0.983971	−	0.178330i	$-0.942930\pi$
$499$	− 7.96720i	− 0.356661i	0.983971	−	0.178330i	$-0.0570696\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	31.3158	1.39630	0.698151	−	0.715951i	$-0.254007\pi$
$503$	31.3158	1.39630	0.698151	+	0.715951i	$0.254007\pi$
$504$	0	0
$505$	−6.99698	−0.311362
$506$	0	0
$507$	0	0
$508$	0	0
$509$	40.7362i	1.80560i	0.430061	+	0.902800i	$0.358492\pi$
$509$	40.7362i	1.80560i	−0.430061	+	0.902800i	$0.641508\pi$
$510$	0	0
$511$	3.85224	0.170413
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 28.2249i	− 1.24374i
$516$	0	0
$517$	0.0409168i	0.00179952i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	43.3880	1.90086	0.950432	−	0.310931i	$-0.100641\pi$
$521$	43.3880	1.90086	0.950432	+	0.310931i	$0.100641\pi$
$522$	0	0
$523$	− 17.9726i	− 0.785887i	−0.919563	−	0.392944i	$-0.871457\pi$
$523$	− 17.9726i	− 0.785887i	0.919563	−	0.392944i	$-0.128543\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.78228	0.295441
$528$	0	0
$529$	37.4668	1.62899
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 17.0404i	− 0.738101i
$534$	0	0
$535$	10.3680	0.448246
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.0549306i	0.00236603i
$540$	0	0
$541$	− 11.5151i	− 0.495074i	−0.968878	−	0.247537i	$-0.920379\pi$
$541$	− 11.5151i	− 0.495074i	0.968878	−	0.247537i	$-0.0796212\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.27238	−0.140173
$546$	0	0
$547$	14.5265i	0.621107i	0.950556	+	0.310553i	$0.100514\pi$
$547$	14.5265i	0.621107i	−0.950556	+	0.310553i	$0.899486\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.00373	−0.0853616
$552$	0	0
$553$	−10.9057	−0.463756
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 44.6492i	− 1.89185i	−0.324387	−	0.945924i	$-0.605158\pi$
$557$	− 44.6492i	− 1.89185i	0.324387	−	0.945924i	$-0.394842\pi$
$558$	0	0
$559$	25.7302	1.08827
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 4.26356i	− 0.179688i	−0.995956	−	0.0898438i	$-0.971363\pi$
$563$	− 4.26356i	− 0.179688i	0.995956	−	0.0898438i	$-0.0286368\pi$
$564$	0	0
$565$	− 4.21098i	− 0.177157i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−38.6045	−1.61838	−0.809192	−	0.587544i	$-0.800095\pi$
$569$	−38.6045	−1.61838	−0.809192	+	0.587544i	$0.800095\pi$
$570$	0	0
$571$	− 28.0556i	− 1.17409i	−0.809554	−	0.587045i	$-0.800291\pi$
$571$	− 28.0556i	− 1.17409i	0.809554	−	0.587045i	$-0.199709\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18.7845	0.783370
$576$	0	0
$577$	−2.46468	−0.102606	−0.0513029	−	0.998683i	$-0.516337\pi$
$577$	−2.46468	−0.102606	−0.0513029	+	0.998683i	$0.516337\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.1624i	0.546066i
$582$	0	0
$583$	−0.581287	−0.0240745
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 42.7523i	− 1.76458i	−0.470710	−	0.882288i	$-0.656002\pi$
$587$	− 42.7523i	− 1.76458i	0.470710	−	0.882288i	$-0.343998\pi$
$588$	0	0
$589$	− 1.55825i	− 0.0642065i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.7945	0.730733	0.365367	−	0.930864i	$-0.380944\pi$
$593$	17.7945	0.730733	0.365367	+	0.930864i	$0.380944\pi$
$594$	0	0
$595$	5.08361i	0.208408i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.5626	0.595011	0.297506	−	0.954720i	$-0.403845\pi$
$599$	14.5626	0.595011	0.297506	+	0.954720i	$0.403845\pi$
$600$	0	0
$601$	−27.1497	−1.10746	−0.553730	−	0.832696i	$-0.686796\pi$
$601$	−27.1497	−1.10746	−0.553730	+	0.832696i	$0.686796\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	17.6785i	0.718733i
$606$	0	0
$607$	−10.8708	−0.441231	−0.220616	−	0.975361i	$-0.570807\pi$
$607$	−10.8708	−0.441231	−0.220616	+	0.975361i	$0.570807\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.79794i	0.113192i
$612$	0	0
$613$	− 26.1387i	− 1.05573i	−0.849327	−	0.527866i	$-0.822992\pi$
$613$	− 26.1387i	− 1.05573i	0.849327	−	0.527866i	$-0.177008\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−42.5464	−1.71285	−0.856427	−	0.516269i	$-0.827321\pi$
$617$	−42.5464	−1.71285	−0.856427	+	0.516269i	$0.827321\pi$
$618$	0	0
$619$	41.0012i	1.64798i	0.566606	+	0.823989i	$0.308256\pi$
$619$	41.0012i	1.64798i	−0.566606	+	0.823989i	$0.691744\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.90356	0.316649
$624$	0	0
$625$	−7.08594	−0.283437
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.90005i	0.0757599i
$630$	0	0
$631$	29.8045	1.18650	0.593249	−	0.805019i	$-0.297845\pi$
$631$	29.8045	1.18650	0.593249	+	0.805019i	$0.297845\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	21.9809i	0.872286i
$636$	0	0
$637$	3.75621i	0.148827i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.1768	0.717941	0.358971	−	0.933349i	$-0.383128\pi$
$641$	18.1768	0.717941	0.358971	+	0.933349i	$0.383128\pi$
$642$	0	0
$643$	20.8490i	0.822203i	0.911590	+	0.411101i	$0.134856\pi$
$643$	20.8490i	0.822203i	−0.911590	+	0.411101i	$0.865144\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.7706	0.737948	0.368974	−	0.929440i	$-0.379709\pi$
$647$	18.7706	0.737948	0.368974	+	0.929440i	$0.379709\pi$
$648$	0	0
$649$	−0.141728	−0.00556332
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25.5220i	0.998751i	0.866386	+	0.499376i	$0.166437\pi$
$653$	25.5220i	0.998751i	−0.866386	+	0.499376i	$0.833563\pi$
$654$	0	0
$655$	−18.1000	−0.707227
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 18.8249i	− 0.733314i	−0.930356	−	0.366657i	$-0.880502\pi$
$659$	− 18.8249i	− 0.733314i	0.930356	−	0.366657i	$-0.119498\pi$
$660$	0	0
$661$	− 25.1764i	− 0.979247i	−0.871934	−	0.489623i	$-0.837134\pi$
$661$	− 25.1764i	− 0.979247i	0.871934	−	0.489623i	$-0.162866\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.16797	0.0452921
$666$	0	0
$667$	− 21.4455i	− 0.830372i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.538552	0.0207906
$672$	0	0
$673$	8.29322	0.319680	0.159840	−	0.987143i	$-0.448902\pi$
$673$	8.29322	0.319680	0.159840	+	0.987143i	$0.448902\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 11.6825i	− 0.448993i	−0.974475	−	0.224497i	$-0.927926\pi$
$677$	− 11.6825i	− 0.448993i	0.974475	−	0.224497i	$-0.0720738\pi$
$678$	0	0
$679$	12.1803	0.467439
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.9388i	0.877727i	0.898554	+	0.438864i	$0.144619\pi$
$683$	22.9388i	0.877727i	−0.898554	+	0.438864i	$0.855381\pi$
$684$	0	0
$685$	− 1.61269i	− 0.0616178i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−39.7491	−1.51432
$690$	0	0
$691$	41.2800i	1.57036i	0.619265	+	0.785182i	$0.287431\pi$
$691$	41.2800i	1.57036i	−0.619265	+	0.785182i	$0.712569\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.18377	0.158699
$696$	0	0
$697$	−14.3459	−0.543391
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 22.9596i	− 0.867174i	−0.901112	−	0.433587i	$-0.857248\pi$
$701$	− 22.9596i	− 0.867174i	0.901112	−	0.433587i	$-0.142752\pi$
$702$	0	0
$703$	0.436542	0.0164645
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.35250i	0.163693i
$708$	0	0
$709$	− 22.0831i	− 0.829348i	−0.909970	−	0.414674i	$-0.863896\pi$
$709$	− 22.0831i	− 0.829348i	0.909970	−	0.414674i	$-0.136104\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16.6776	0.624582
$714$	0	0
$715$	− 0.331693i	− 0.0124046i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	29.4594	1.09865	0.549325	−	0.835609i	$-0.314885\pi$
$719$	29.4594	1.09865	0.549325	+	0.835609i	$0.314885\pi$
$720$	0	0
$721$	−17.5574	−0.653873
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 6.66223i	− 0.247429i
$726$	0	0
$727$	25.5893	0.949054	0.474527	−	0.880241i	$-0.342619\pi$
$727$	25.5893	0.949054	0.474527	+	0.880241i	$0.342619\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 21.6617i	− 0.801189i
$732$	0	0
$733$	− 24.3587i	− 0.899711i	−0.893101	−	0.449855i	$-0.851476\pi$
$733$	− 24.3587i	− 0.899711i	0.893101	−	0.449855i	$-0.148524\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.673598	0.0248123
$738$	0	0
$739$	18.8064i	0.691805i	0.938270	+	0.345903i	$0.112427\pi$
$739$	18.8064i	0.691805i	−0.938270	+	0.345903i	$0.887573\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.4039	−1.07873	−0.539363	−	0.842074i	$-0.681335\pi$
$743$	−29.4039	−1.07873	−0.539363	+	0.842074i	$0.681335\pi$
$744$	0	0
$745$	−30.7874	−1.12796
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 6.44944i	− 0.235657i
$750$	0	0
$751$	16.9057	0.616896	0.308448	−	0.951241i	$-0.400190\pi$
$751$	16.9057	0.616896	0.308448	+	0.951241i	$0.400190\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 25.5361i	− 0.929353i
$756$	0	0
$757$	− 30.3657i	− 1.10366i	−0.833957	−	0.551830i	$-0.813930\pi$
$757$	− 30.3657i	− 1.10366i	0.833957	−	0.551830i	$-0.186070\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.0960	−0.655979	−0.327990	−	0.944681i	$-0.606371\pi$
$761$	−18.0960	−0.655979	−0.327990	+	0.944681i	$0.606371\pi$
$762$	0	0
$763$	2.03559i	0.0736935i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.69153	−0.349941
$768$	0	0
$769$	15.4944	0.558743	0.279371	−	0.960183i	$-0.409874\pi$
$769$	15.4944	0.558743	0.279371	+	0.960183i	$0.409874\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 33.1186i	− 1.19119i	−0.803284	−	0.595597i	$-0.796916\pi$
$773$	− 33.1186i	− 1.19119i	0.803284	−	0.595597i	$-0.203084\pi$
$774$	0	0
$775$	5.18105	0.186109
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.29602i	0.118092i
$780$	0	0
$781$	0.504826i	0.0180641i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−23.6473	−0.844008
$786$	0	0
$787$	13.8484i	0.493641i	0.969061	+	0.246820i	$0.0793858\pi$
$787$	13.8484i	0.493641i	−0.969061	+	0.246820i	$0.920614\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.61946	−0.0931372
$792$	0	0
$793$	36.8268	1.30776
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 27.5510i	− 0.975908i	−0.872869	−	0.487954i	$-0.837743\pi$
$797$	− 27.5510i	− 0.975908i	0.872869	−	0.487954i	$-0.162257\pi$
$798$	0	0
$799$	2.35553	0.0833325
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.211606i	0.00746740i
$804$	0	0
$805$	12.5006i	0.440588i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0.181117	0.00636775	0.00318388	−	0.999995i	$-0.498987\pi$
$809$	0.181117	0.00636775	0.00318388	+	0.999995i	$0.498987\pi$
$810$	0	0
$811$	− 23.7508i	− 0.834003i	−0.908906	−	0.417002i	$-0.863081\pi$
$811$	− 23.7508i	− 0.834003i	0.908906	−	0.417002i	$-0.136919\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−15.5885	−0.546040
$816$	0	0
$817$	−4.97685	−0.174118
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 8.16381i	− 0.284919i	−0.989801	−	0.142460i	$-0.954499\pi$
$821$	− 8.16381i	− 0.284919i	0.989801	−	0.142460i	$-0.0455011\pi$
$822$	0	0
$823$	−39.6984	−1.38380	−0.691900	−	0.721993i	$-0.743226\pi$
$823$	−39.6984	−1.38380	−0.691900	+	0.721993i	$0.743226\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 46.0390i	− 1.60093i	−0.599377	−	0.800467i	$-0.704585\pi$
$827$	− 46.0390i	− 1.60093i	0.599377	−	0.800467i	$-0.295415\pi$
$828$	0	0
$829$	− 14.1434i	− 0.491220i	−0.969369	−	0.245610i	$-0.921012\pi$
$829$	− 14.1434i	− 0.491220i	0.969369	−	0.245610i	$-0.0789882\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.16228	0.109566
$834$	0	0
$835$	4.13168i	0.142983i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.0217	1.07099	0.535494	−	0.844539i	$-0.320125\pi$
$839$	31.0217	1.07099	0.535494	+	0.844539i	$0.320125\pi$
$840$	0	0
$841$	21.3940	0.737725
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.78305i	− 0.0613386i
$846$	0	0
$847$	10.9970	0.377861
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.67222i	0.160162i
$852$	0	0
$853$	36.2812i	1.24224i	0.783714	+	0.621121i	$0.213323\pi$
$853$	36.2812i	1.24224i	−0.783714	+	0.621121i	$0.786677\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.6532	−0.944616	−0.472308	−	0.881434i	$-0.656579\pi$
$857$	−27.6532	−0.944616	−0.472308	+	0.881434i	$0.656579\pi$
$858$	0	0
$859$	− 27.0733i	− 0.923729i	−0.886950	−	0.461865i	$-0.847181\pi$
$859$	− 27.0733i	− 0.923729i	0.886950	−	0.461865i	$-0.152819\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.98203	0.305752	0.152876	−	0.988245i	$-0.451147\pi$
$863$	8.98203	0.305752	0.152876	+	0.988245i	$0.451147\pi$
$864$	0	0
$865$	2.72421	0.0926258
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 0.599054i	− 0.0203215i
$870$	0	0
$871$	46.0614	1.56073
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.9213i	0.403014i
$876$	0	0
$877$	27.9552i	0.943980i	0.881604	+	0.471990i	$0.156464\pi$
$877$	27.9552i	0.943980i	−0.881604	+	0.471990i	$0.843536\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	33.0213	1.11252	0.556258	−	0.831010i	$-0.312237\pi$
$881$	33.0213	1.11252	0.556258	+	0.831010i	$0.312237\pi$
$882$	0	0
$883$	19.1451i	0.644283i	0.946692	+	0.322141i	$0.104403\pi$
$883$	19.1451i	0.644283i	−0.946692	+	0.322141i	$0.895597\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.99598	0.302056	0.151028	−	0.988530i	$-0.451742\pi$
$887$	8.99598	0.302056	0.151028	+	0.988530i	$0.451742\pi$
$888$	0	0
$889$	13.6733	0.458588
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 0.541189i	− 0.0181102i
$894$	0	0
$895$	17.9057	0.598522
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 5.91498i	− 0.197276i
$900$	0	0
$901$	33.4639i	1.11484i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.9348	0.363485
$906$	0	0
$907$	− 53.5978i	− 1.77969i	−0.456267	−	0.889843i	$-0.650814\pi$
$907$	− 53.5978i	− 1.77969i	0.456267	−	0.889843i	$-0.349186\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−47.1958	−1.56367	−0.781833	−	0.623487i	$-0.785715\pi$
$911$	−47.1958	−1.56367	−0.781833	+	0.623487i	$0.785715\pi$
$912$	0	0
$913$	−0.723015	−0.0239283
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.2592i	0.371812i
$918$	0	0
$919$	−22.6942	−0.748612	−0.374306	−	0.927305i	$-0.622119\pi$
$919$	−22.6942	−0.748612	−0.374306	+	0.927305i	$0.622119\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	34.5206i	1.13626i
$924$	0	0
$925$	1.45147i	0.0477239i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−31.0001	−1.01708	−0.508541	−	0.861038i	$-0.669815\pi$
$929$	−31.0001	−1.01708	−0.508541	+	0.861038i	$0.669815\pi$
$930$	0	0
$931$	− 0.726543i	− 0.0238115i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.279245	−0.00913230
$936$	0	0
$937$	−49.0011	−1.60080	−0.800398	−	0.599469i	$-0.795378\pi$
$937$	−49.0011	−1.60080	−0.800398	+	0.599469i	$0.795378\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 49.2485i	− 1.60546i	−0.596345	−	0.802728i	$-0.703381\pi$
$941$	− 49.2485i	− 1.60546i	0.596345	−	0.802728i	$-0.296619\pi$
$942$	0	0
$943$	−35.2766	−1.14877
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9.28008i	− 0.301562i	−0.988567	−	0.150781i	$-0.951821\pi$
$947$	− 9.28008i	− 0.301562i	0.988567	−	0.150781i	$-0.0481788\pi$
$948$	0	0
$949$	14.4698i	0.469711i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−50.1855	−1.62567	−0.812834	−	0.582496i	$-0.802076\pi$
$953$	−50.1855	−1.62567	−0.812834	+	0.582496i	$0.802076\pi$
$954$	0	0
$955$	25.1420i	0.813576i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.00318	−0.0323944
$960$	0	0
$961$	−26.4001	−0.851615
$962$	0	0
$963$	0	0
$964$	0	0
$965$	31.4913i	1.01374i
$966$	0	0
$967$	7.81736	0.251389	0.125695	−	0.992069i	$-0.459884\pi$
$967$	7.81736	0.251389	0.125695	+	0.992069i	$0.459884\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.60460i	0.244043i	0.992527	+	0.122022i	$0.0389377\pi$
$971$	7.60460i	0.244043i	−0.992527	+	0.122022i	$0.961062\pi$
$972$	0	0
$973$	− 2.60253i	− 0.0834333i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37.9179	−1.21310	−0.606551	−	0.795044i	$-0.707447\pi$
$977$	−37.9179	−1.21310	−0.606551	+	0.795044i	$0.707447\pi$
$978$	0	0
$979$	0.434147i	0.0138754i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−33.7230	−1.07560	−0.537798	−	0.843074i	$-0.680744\pi$
$983$	−33.7230	−1.07560	−0.537798	+	0.843074i	$0.680744\pi$
$984$	0	0
$985$	11.7340	0.373875
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 53.2662i	− 1.69377i
$990$	0	0
$991$	−46.3137	−1.47120	−0.735602	−	0.677414i	$-0.763101\pi$
$991$	−46.3137	−1.47120	−0.735602	+	0.677414i	$0.763101\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 22.1849i	− 0.703310i
$996$	0	0
$997$	− 5.20068i	− 0.164707i	−0.996603	−	0.0823535i	$-0.973756\pi$
$997$	− 5.20068i	− 0.164707i	0.996603	−	0.0823535i	$-0.0262437\pi$
$998$	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.3025.12		16
3.2	odd	2	inner	6048.2.c.d.3025.6		16
4.3	odd	2		1512.2.c.d.757.16	yes	16
8.3	odd	2		1512.2.c.d.757.13	yes	16
8.5	even	2	inner	6048.2.c.d.3025.5		16
12.11	even	2		1512.2.c.d.757.1	✓	16
24.5	odd	2	inner	6048.2.c.d.3025.11		16
24.11	even	2		1512.2.c.d.757.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.d.757.1	✓	16	12.11	even	2
1512.2.c.d.757.4	yes	16	24.11	even	2
1512.2.c.d.757.13	yes	16	8.3	odd	2
1512.2.c.d.757.16	yes	16	4.3	odd	2
6048.2.c.d.3025.5		16	8.5	even	2	inner
6048.2.c.d.3025.6		16	3.2	odd	2	inner
6048.2.c.d.3025.11		16	24.5	odd	2	inner
6048.2.c.d.3025.12		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.60758i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+1.60758i q^{5} +1.00000 q^{7} +0.0549306i q^{11} +3.75621i q^{13} +3.16228 q^{17} -0.726543i q^{19} +7.77604 q^{23} +2.41570 q^{25} -2.75789i q^{29} +2.14475 q^{31} +1.60758i q^{35} +0.600848i q^{37} -4.53658 q^{41} -6.85004i q^{43} +0.744883 q^{47} +1.00000 q^{49} +10.5822i q^{53} -0.0883051 q^{55} +2.58013i q^{59} -9.80423i q^{61} -6.03841 q^{65} -12.2627i q^{67} +9.19025 q^{71} +3.85224 q^{73} +0.0549306i q^{77} -10.9057 q^{79} +13.1624i q^{83} +5.08361i q^{85} +7.90356 q^{89} +3.75621i q^{91} +1.16797 q^{95} +12.1803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7}+O(q^{10}) \) 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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