LMFDB - Embedded newform 6048.2.c.g.3025.14

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:	$24$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.14
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.g.3025.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.26947i q^{5} -1.00000 q^{7} +O(q^{10})$	$q+1.26947i q^{5} -1.00000 q^{7} +2.02805i q^{11} +4.05339i q^{13} -1.33658 q^{17} -3.28346i q^{19} -2.25680 q^{23} +3.38845 q^{25} -3.58798i q^{29} +0.464312 q^{31} -1.26947i q^{35} +11.8743i q^{37} -1.73758 q^{41} -1.70380i q^{43} +8.45083 q^{47} +1.00000 q^{49} +11.7544i q^{53} -2.57454 q^{55} -5.49489i q^{59} +6.02150i q^{61} -5.14565 q^{65} -0.381481i q^{67} -9.47622 q^{71} -10.0336 q^{73} -2.02805i q^{77} -4.90442 q^{79} +6.95800i q^{83} -1.69674i q^{85} -17.4322 q^{89} -4.05339i q^{91} +4.16824 q^{95} +7.62794 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{7}+O(q^{10}) $ 24 * q - 24 * q^7	$ 24 q - 24 q^{7} - 24 q^{25} + 16 q^{31} + 24 q^{49} + 8 q^{55} - 32 q^{79}+O(q^{100}) $ 24 * q - 24 * q^7 - 24 * q^25 + 16 * q^31 + 24 * q^49 + 8 * q^55 - 32 * q^79

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
$5$	1.26947i	0.567723i	0.958865	+	0.283862i	$0.0916156\pi$
$5$	1.26947i	0.567723i	−0.958865	+	0.283862i	$0.908384\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.02805i	0.611480i	0.952115	+	0.305740i	$0.0989039\pi$
$11$	2.02805i	0.611480i	−0.952115	+	0.305740i	$0.901096\pi$
$12$	0	0
$13$	4.05339i	1.12421i	0.827066	+	0.562104i	$0.190008\pi$
$13$	4.05339i	1.12421i	−0.827066	+	0.562104i	$0.809992\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.33658	−0.324167	−0.162084	−	0.986777i	$-0.551821\pi$
$17$	−1.33658	−0.324167	−0.162084	+	0.986777i	$0.551821\pi$
$18$	0	0
$19$	− 3.28346i	− 0.753277i	−0.926360	−	0.376638i	$-0.877080\pi$
$19$	− 3.28346i	− 0.753277i	0.926360	−	0.376638i	$-0.122920\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.25680	−0.470576	−0.235288	−	0.971926i	$-0.575603\pi$
$23$	−2.25680	−0.470576	−0.235288	+	0.971926i	$0.575603\pi$
$24$	0	0
$25$	3.38845	0.677690
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.58798i	− 0.666272i	−0.942879	−	0.333136i	$-0.891893\pi$
$29$	− 3.58798i	− 0.666272i	0.942879	−	0.333136i	$-0.108107\pi$
$30$	0	0
$31$	0.464312	0.0833930	0.0416965	−	0.999130i	$-0.486724\pi$
$31$	0.464312	0.0833930	0.0416965	+	0.999130i	$0.486724\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.26947i	− 0.214579i
$36$	0	0
$37$	11.8743i	1.95212i	0.217507	+	0.976059i	$0.430207\pi$
$37$	11.8743i	1.95212i	−0.217507	+	0.976059i	$0.569793\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.73758	−0.271365	−0.135683	−	0.990752i	$-0.543323\pi$
$41$	−1.73758	−0.271365	−0.135683	+	0.990752i	$0.543323\pi$
$42$	0	0
$43$	− 1.70380i	− 0.259827i	−0.991525	−	0.129913i	$-0.958530\pi$
$43$	− 1.70380i	− 0.259827i	0.991525	−	0.129913i	$-0.0414699\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.45083	1.23268	0.616340	−	0.787480i	$-0.288615\pi$
$47$	8.45083	1.23268	0.616340	+	0.787480i	$0.288615\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.7544i	1.61459i	0.590146	+	0.807296i	$0.299070\pi$
$53$	11.7544i	1.61459i	−0.590146	+	0.807296i	$0.700930\pi$
$54$	0	0
$55$	−2.57454	−0.347152
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 5.49489i	− 0.715374i	−0.933842	−	0.357687i	$-0.883565\pi$
$59$	− 5.49489i	− 0.715374i	0.933842	−	0.357687i	$-0.116435\pi$
$60$	0	0
$61$	6.02150i	0.770975i	0.922713	+	0.385487i	$0.125967\pi$
$61$	6.02150i	0.770975i	−0.922713	+	0.385487i	$0.874033\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.14565	−0.638239
$66$	0	0
$67$	− 0.381481i	− 0.0466054i	−0.999728	−	0.0233027i	$-0.992582\pi$
$67$	− 0.381481i	− 0.0466054i	0.999728	−	0.0233027i	$-0.00741815\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.47622	−1.12462	−0.562310	−	0.826926i	$-0.690087\pi$
$71$	−9.47622	−1.12462	−0.562310	+	0.826926i	$0.690087\pi$
$72$	0	0
$73$	−10.0336	−1.17434	−0.587171	−	0.809463i	$-0.699759\pi$
$73$	−10.0336	−1.17434	−0.587171	+	0.809463i	$0.699759\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.02805i	− 0.231118i
$78$	0	0
$79$	−4.90442	−0.551791	−0.275895	−	0.961188i	$-0.588974\pi$
$79$	−4.90442	−0.551791	−0.275895	+	0.961188i	$0.588974\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.95800i	0.763740i	0.924216	+	0.381870i	$0.124720\pi$
$83$	6.95800i	0.763740i	−0.924216	+	0.381870i	$0.875280\pi$
$84$	0	0
$85$	− 1.69674i	− 0.184037i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17.4322	−1.84781	−0.923903	−	0.382628i	$-0.875019\pi$
$89$	−17.4322	−1.84781	−0.923903	+	0.382628i	$0.875019\pi$
$90$	0	0
$91$	− 4.05339i	− 0.424911i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.16824	0.427653
$96$	0	0
$97$	7.62794	0.774500	0.387250	−	0.921975i	$-0.373425\pi$
$97$	7.62794	0.774500	0.387250	+	0.921975i	$0.373425\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 5.27674i	− 0.525055i	−0.964924	−	0.262528i	$-0.915444\pi$
$101$	− 5.27674i	− 0.525055i	0.964924	−	0.262528i	$-0.0845561\pi$
$102$	0	0
$103$	5.07253	0.499812	0.249906	−	0.968270i	$-0.419600\pi$
$103$	5.07253	0.499812	0.249906	+	0.968270i	$0.419600\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.22269i	− 0.504896i	−0.967610	−	0.252448i	$-0.918764\pi$
$107$	− 5.22269i	− 0.504896i	0.967610	−	0.252448i	$-0.0812357\pi$
$108$	0	0
$109$	− 10.8354i	− 1.03784i	−0.854822	−	0.518922i	$-0.826333\pi$
$109$	− 10.8354i	− 1.03784i	0.854822	−	0.518922i	$-0.173667\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.8783	−1.02334	−0.511672	−	0.859181i	$-0.670974\pi$
$113$	−10.8783	−1.02334	−0.511672	+	0.859181i	$0.670974\pi$
$114$	0	0
$115$	− 2.86494i	− 0.267157i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.33658	0.122524
$120$	0	0
$121$	6.88701	0.626092
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.6489i	0.952464i
$126$	0	0
$127$	−1.57175	−0.139471	−0.0697353	−	0.997566i	$-0.522215\pi$
$127$	−1.57175	−0.139471	−0.0697353	+	0.997566i	$0.522215\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 11.7947i	− 1.03051i	−0.857037	−	0.515256i	$-0.827697\pi$
$131$	− 11.7947i	− 1.03051i	0.857037	−	0.515256i	$-0.172303\pi$
$132$	0	0
$133$	3.28346i	0.284712i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.4006	−1.57207	−0.786033	−	0.618184i	$-0.787869\pi$
$137$	−18.4006	−1.57207	−0.786033	+	0.618184i	$0.787869\pi$
$138$	0	0
$139$	− 11.3851i	− 0.965674i	−0.875710	−	0.482837i	$-0.839606\pi$
$139$	− 11.3851i	− 0.965674i	0.875710	−	0.482837i	$-0.160394\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.22048	−0.687431
$144$	0	0
$145$	4.55483	0.378258
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 14.6966i	− 1.20400i	−0.798498	−	0.601998i	$-0.794372\pi$
$149$	− 14.6966i	− 1.20400i	0.798498	−	0.601998i	$-0.205628\pi$
$150$	0	0
$151$	0.220342	0.0179312	0.00896559	−	0.999960i	$-0.497146\pi$
$151$	0.220342	0.0179312	0.00896559	+	0.999960i	$0.497146\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.589430i	0.0473441i
$156$	0	0
$157$	10.4367i	0.832941i	0.909149	+	0.416471i	$0.136733\pi$
$157$	10.4367i	0.832941i	−0.909149	+	0.416471i	$0.863267\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.25680	0.177861
$162$	0	0
$163$	− 4.46955i	− 0.350082i	−0.984561	−	0.175041i	$-0.943994\pi$
$163$	− 4.46955i	− 0.350082i	0.984561	−	0.175041i	$-0.0560058\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.0617	−1.08812	−0.544062	−	0.839045i	$-0.683114\pi$
$167$	−14.0617	−1.08812	−0.544062	+	0.839045i	$0.683114\pi$
$168$	0	0
$169$	−3.42998	−0.263844
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.334570i	0.0254369i	0.999919	+	0.0127184i	$0.00404851\pi$
$173$	0.334570i	0.0254369i	−0.999919	+	0.0127184i	$0.995951\pi$
$174$	0	0
$175$	−3.38845	−0.256143
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.85419i	0.138589i	0.997596	+	0.0692944i	$0.0220748\pi$
$179$	1.85419i	0.138589i	−0.997596	+	0.0692944i	$0.977925\pi$
$180$	0	0
$181$	11.8864i	0.883512i	0.897135	+	0.441756i	$0.145644\pi$
$181$	11.8864i	0.883512i	−0.897135	+	0.441756i	$0.854356\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−15.0740	−1.10826
$186$	0	0
$187$	− 2.71064i	− 0.198222i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.4777	−1.91586	−0.957928	−	0.287008i	$-0.907339\pi$
$191$	−26.4777	−1.91586	−0.957928	+	0.287008i	$0.907339\pi$
$192$	0	0
$193$	−17.6831	−1.27285	−0.636427	−	0.771337i	$-0.719588\pi$
$193$	−17.6831	−1.27285	−0.636427	+	0.771337i	$0.719588\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.22153i	0.657007i	0.944503	+	0.328504i	$0.106544\pi$
$197$	9.22153i	0.657007i	−0.944503	+	0.328504i	$0.893456\pi$
$198$	0	0
$199$	8.46365	0.599973	0.299986	−	0.953944i	$-0.403018\pi$
$199$	8.46365	0.599973	0.299986	+	0.953944i	$0.403018\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.58798i	0.251827i
$204$	0	0
$205$	− 2.20581i	− 0.154060i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.65902	0.460614
$210$	0	0
$211$	− 4.66745i	− 0.321321i	−0.987010	−	0.160660i	$-0.948638\pi$
$211$	− 4.66745i	− 0.321321i	0.987010	−	0.160660i	$-0.0513624\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.16292	0.147510
$216$	0	0
$217$	−0.464312	−0.0315196
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 5.41767i	− 0.364432i
$222$	0	0
$223$	−7.85009	−0.525681	−0.262841	−	0.964839i	$-0.584659\pi$
$223$	−7.85009	−0.525681	−0.262841	+	0.964839i	$0.584659\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.2857i	1.28004i	0.768360	+	0.640018i	$0.221073\pi$
$227$	19.2857i	1.28004i	−0.768360	+	0.640018i	$0.778927\pi$
$228$	0	0
$229$	− 1.71588i	− 0.113389i	−0.998392	−	0.0566944i	$-0.981944\pi$
$229$	− 1.71588i	− 0.113389i	0.998392	−	0.0566944i	$-0.0180561\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.8734	1.56400	0.781999	−	0.623280i	$-0.214200\pi$
$233$	23.8734	1.56400	0.781999	+	0.623280i	$0.214200\pi$
$234$	0	0
$235$	10.7281i	0.699821i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.6697	−1.40170	−0.700848	−	0.713310i	$-0.747195\pi$
$239$	−21.6697	−1.40170	−0.700848	+	0.713310i	$0.747195\pi$
$240$	0	0
$241$	27.8432	1.79354	0.896770	−	0.442497i	$-0.145907\pi$
$241$	27.8432	1.79354	0.896770	+	0.442497i	$0.145907\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.26947i	0.0811033i
$246$	0	0
$247$	13.3091	0.846840
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.7106i	1.74908i	0.484957	+	0.874538i	$0.338835\pi$
$251$	27.7106i	1.74908i	−0.484957	+	0.874538i	$0.661165\pi$
$252$	0	0
$253$	− 4.57691i	− 0.287748i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.90462	−0.118807	−0.0594035	−	0.998234i	$-0.518920\pi$
$257$	−1.90462	−0.118807	−0.0594035	+	0.998234i	$0.518920\pi$
$258$	0	0
$259$	− 11.8743i	− 0.737831i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.2186	−1.18507	−0.592534	−	0.805545i	$-0.701872\pi$
$263$	−19.2186	−1.18507	−0.592534	+	0.805545i	$0.701872\pi$
$264$	0	0
$265$	−14.9218	−0.916642
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.2931i	1.11535i	0.830059	+	0.557676i	$0.188307\pi$
$269$	18.2931i	1.11535i	−0.830059	+	0.557676i	$0.811693\pi$
$270$	0	0
$271$	−19.1450	−1.16297	−0.581487	−	0.813555i	$-0.697529\pi$
$271$	−19.1450	−1.16297	−0.581487	+	0.813555i	$0.697529\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.87195i	0.414394i
$276$	0	0
$277$	− 17.6288i	− 1.05921i	−0.848244	−	0.529606i	$-0.822340\pi$
$277$	− 17.6288i	− 1.05921i	0.848244	−	0.529606i	$-0.177660\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.3772	−0.738362	−0.369181	−	0.929358i	$-0.620362\pi$
$281$	−12.3772	−0.738362	−0.369181	+	0.929358i	$0.620362\pi$
$282$	0	0
$283$	2.56225i	0.152310i	0.997096	+	0.0761550i	$0.0242644\pi$
$283$	2.56225i	0.152310i	−0.997096	+	0.0761550i	$0.975736\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.73758	0.102566
$288$	0	0
$289$	−15.2136	−0.894916
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.12608i	0.0657866i	0.999459	+	0.0328933i	$0.0104721\pi$
$293$	1.12608i	0.0657866i	−0.999459	+	0.0328933i	$0.989528\pi$
$294$	0	0
$295$	6.97559	0.406135
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 9.14770i	− 0.529025i
$300$	0	0
$301$	1.70380i	0.0982052i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7.64411	−0.437700
$306$	0	0
$307$	− 24.9543i	− 1.42422i	−0.702070	−	0.712108i	$-0.747741\pi$
$307$	− 24.9543i	− 1.42422i	0.702070	−	0.712108i	$-0.252259\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.4110	−1.15740	−0.578701	−	0.815540i	$-0.696440\pi$
$311$	−20.4110	−1.15740	−0.578701	+	0.815540i	$0.696440\pi$
$312$	0	0
$313$	18.5902	1.05078	0.525389	−	0.850862i	$-0.323920\pi$
$313$	18.5902	1.05078	0.525389	+	0.850862i	$0.323920\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.68959i	− 0.0948970i	−0.998874	−	0.0474485i	$-0.984891\pi$
$317$	− 1.68959i	− 0.0948970i	0.998874	−	0.0474485i	$-0.0151090\pi$
$318$	0	0
$319$	7.27661	0.407412
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.38859i	0.244188i
$324$	0	0
$325$	13.7347i	0.761865i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.45083	−0.465909
$330$	0	0
$331$	25.4187i	1.39714i	0.715542	+	0.698570i	$0.246180\pi$
$331$	25.4187i	1.39714i	−0.715542	+	0.698570i	$0.753820\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.484278	0.0264590
$336$	0	0
$337$	−10.5368	−0.573979	−0.286989	−	0.957934i	$-0.592654\pi$
$337$	−10.5368	−0.573979	−0.286989	+	0.957934i	$0.592654\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.941649i	0.0509932i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.6098i	− 0.676930i	−0.940979	−	0.338465i	$-0.890092\pi$
$347$	− 12.6098i	− 0.676930i	0.940979	−	0.338465i	$-0.109908\pi$
$348$	0	0
$349$	10.3681i	0.554990i	0.960727	+	0.277495i	$0.0895043\pi$
$349$	10.3681i	0.554990i	−0.960727	+	0.277495i	$0.910496\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.4319	−1.14071	−0.570353	−	0.821399i	$-0.693194\pi$
$353$	−21.4319	−1.14071	−0.570353	+	0.821399i	$0.693194\pi$
$354$	0	0
$355$	− 12.0298i	− 0.638473i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.5414	−0.925798	−0.462899	−	0.886411i	$-0.653191\pi$
$359$	−17.5414	−0.925798	−0.462899	+	0.886411i	$0.653191\pi$
$360$	0	0
$361$	8.21890	0.432574
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 12.7373i	− 0.666702i
$366$	0	0
$367$	−28.4730	−1.48628	−0.743140	−	0.669136i	$-0.766665\pi$
$367$	−28.4730	−1.48628	−0.743140	+	0.669136i	$0.766665\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 11.7544i	− 0.610259i
$372$	0	0
$373$	11.7775i	0.609818i	0.952382	+	0.304909i	$0.0986260\pi$
$373$	11.7775i	0.609818i	−0.952382	+	0.304909i	$0.901374\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.5435	0.749028
$378$	0	0
$379$	14.4058i	0.739975i	0.929037	+	0.369988i	$0.120638\pi$
$379$	14.4058i	0.739975i	−0.929037	+	0.369988i	$0.879362\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.8612	0.606078	0.303039	−	0.952978i	$-0.401999\pi$
$383$	11.8612	0.606078	0.303039	+	0.952978i	$0.401999\pi$
$384$	0	0
$385$	2.57454	0.131211
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 16.3633i	− 0.829655i	−0.909900	−	0.414827i	$-0.863842\pi$
$389$	− 16.3633i	− 0.829655i	0.909900	−	0.414827i	$-0.136158\pi$
$390$	0	0
$391$	3.01639	0.152545
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 6.22601i	− 0.313265i
$396$	0	0
$397$	− 10.7133i	− 0.537685i	−0.963184	−	0.268843i	$-0.913359\pi$
$397$	− 10.7133i	− 0.537685i	0.963184	−	0.268843i	$-0.0866411\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.7286	1.08507	0.542536	−	0.840032i	$-0.317464\pi$
$401$	21.7286	1.08507	0.542536	+	0.840032i	$0.317464\pi$
$402$	0	0
$403$	1.88204i	0.0937511i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.0816	−1.19368
$408$	0	0
$409$	−14.3607	−0.710092	−0.355046	−	0.934849i	$-0.615535\pi$
$409$	−14.3607	−0.710092	−0.355046	+	0.934849i	$0.615535\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.49489i	0.270386i
$414$	0	0
$415$	−8.83296	−0.433593
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 33.7826i	− 1.65039i	−0.564849	−	0.825194i	$-0.691065\pi$
$419$	− 33.7826i	− 1.65039i	0.564849	−	0.825194i	$-0.308935\pi$
$420$	0	0
$421$	− 10.6036i	− 0.516788i	−0.966040	−	0.258394i	$-0.916807\pi$
$421$	− 10.6036i	− 0.516788i	0.966040	−	0.258394i	$-0.0831933\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.52892	−0.219685
$426$	0	0
$427$	− 6.02150i	− 0.291401i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.2155	−0.925577	−0.462788	−	0.886469i	$-0.653151\pi$
$431$	−19.2155	−0.925577	−0.462788	+	0.886469i	$0.653151\pi$
$432$	0	0
$433$	39.8723	1.91614	0.958070	−	0.286534i	$-0.0925030\pi$
$433$	39.8723	1.91614	0.958070	+	0.286534i	$0.0925030\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.41011i	0.354474i
$438$	0	0
$439$	−2.96048	−0.141296	−0.0706481	−	0.997501i	$-0.522507\pi$
$439$	−2.96048	−0.141296	−0.0706481	+	0.997501i	$0.522507\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.7091i	0.603827i	0.953335	+	0.301914i	$0.0976254\pi$
$443$	12.7091i	0.603827i	−0.953335	+	0.301914i	$0.902375\pi$
$444$	0	0
$445$	− 22.1296i	− 1.04904i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−27.4353	−1.29475	−0.647375	−	0.762171i	$-0.724133\pi$
$449$	−27.4353	−1.29475	−0.647375	+	0.762171i	$0.724133\pi$
$450$	0	0
$451$	− 3.52391i	− 0.165934i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.14565	0.241232
$456$	0	0
$457$	18.9049	0.884335	0.442168	−	0.896932i	$-0.354210\pi$
$457$	18.9049	0.884335	0.442168	+	0.896932i	$0.354210\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 30.3182i	− 1.41206i	−0.708181	−	0.706030i	$-0.750484\pi$
$461$	− 30.3182i	− 1.41206i	0.708181	−	0.706030i	$-0.249516\pi$
$462$	0	0
$463$	29.2518	1.35944	0.679722	−	0.733470i	$-0.262100\pi$
$463$	29.2518	1.35944	0.679722	+	0.733470i	$0.262100\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.7692i	1.51638i	0.652036	+	0.758188i	$0.273915\pi$
$467$	32.7692i	1.51638i	−0.652036	+	0.758188i	$0.726085\pi$
$468$	0	0
$469$	0.381481i	0.0176152i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.45539	0.158879
$474$	0	0
$475$	− 11.1258i	− 0.510488i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25.8900	1.18294	0.591472	−	0.806326i	$-0.298547\pi$
$479$	25.8900	1.18294	0.591472	+	0.806326i	$0.298547\pi$
$480$	0	0
$481$	−48.1310	−2.19459
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.68342i	0.439701i
$486$	0	0
$487$	5.94127	0.269225	0.134612	−	0.990898i	$-0.457021\pi$
$487$	5.94127	0.269225	0.134612	+	0.990898i	$0.457021\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.77413i	0.170324i	0.996367	+	0.0851621i	$0.0271408\pi$
$491$	3.77413i	0.170324i	−0.996367	+	0.0851621i	$0.972859\pi$
$492$	0	0
$493$	4.79561i	0.215984i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.47622	0.425067
$498$	0	0
$499$	7.66921i	0.343321i	0.985156	+	0.171661i	$0.0549132\pi$
$499$	7.66921i	0.343321i	−0.985156	+	0.171661i	$0.945087\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.29637	0.325329	0.162664	−	0.986681i	$-0.447991\pi$
$503$	7.29637	0.325329	0.162664	+	0.986681i	$0.447991\pi$
$504$	0	0
$505$	6.69865	0.298086
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 7.32231i	− 0.324556i	−0.986745	−	0.162278i	$-0.948116\pi$
$509$	− 7.32231i	− 0.324556i	0.986745	−	0.162278i	$-0.0518841\pi$
$510$	0	0
$511$	10.0336	0.443860
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.43942i	0.283755i
$516$	0	0
$517$	17.1387i	0.753759i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−29.2300	−1.28059	−0.640295	−	0.768129i	$-0.721188\pi$
$521$	−29.2300	−1.28059	−0.640295	+	0.768129i	$0.721188\pi$
$522$	0	0
$523$	37.4676i	1.63835i	0.573547	+	0.819173i	$0.305567\pi$
$523$	37.4676i	1.63835i	−0.573547	+	0.819173i	$0.694433\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.620589	−0.0270333
$528$	0	0
$529$	−17.9068	−0.778559
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 7.04311i	− 0.305071i
$534$	0	0
$535$	6.63003	0.286641
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.02805i	0.0873543i
$540$	0	0
$541$	− 22.9143i	− 0.985164i	−0.870266	−	0.492582i	$-0.836053\pi$
$541$	− 22.9143i	− 0.985164i	0.870266	−	0.492582i	$-0.163947\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.7552	0.589208
$546$	0	0
$547$	− 14.0319i	− 0.599959i	−0.953946	−	0.299979i	$-0.903020\pi$
$547$	− 14.0319i	− 0.599959i	0.953946	−	0.299979i	$-0.0969798\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.7810	−0.501887
$552$	0	0
$553$	4.90442	0.208557
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.1958i	1.23706i	0.785760	+	0.618532i	$0.212272\pi$
$557$	29.1958i	1.23706i	−0.785760	+	0.618532i	$0.787728\pi$
$558$	0	0
$559$	6.90615	0.292099
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 29.5633i	− 1.24594i	−0.782245	−	0.622971i	$-0.785925\pi$
$563$	− 29.5633i	− 1.24594i	0.782245	−	0.622971i	$-0.214075\pi$
$564$	0	0
$565$	− 13.8096i	− 0.580976i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.2748	1.18534	0.592672	−	0.805444i	$-0.298073\pi$
$569$	28.2748	1.18534	0.592672	+	0.805444i	$0.298073\pi$
$570$	0	0
$571$	40.3483i	1.68852i	0.535932	+	0.844261i	$0.319960\pi$
$571$	40.3483i	1.68852i	−0.535932	+	0.844261i	$0.680040\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.64706	−0.318904
$576$	0	0
$577$	−0.0318593	−0.00132632	−0.000663161	−	1.00000i	$-0.500211\pi$
$577$	−0.0318593	−0.00132632	−0.000663161		1.00000i	$0.500211\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 6.95800i	− 0.288667i
$582$	0	0
$583$	−23.8385	−0.987291
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 22.9245i	− 0.946195i	−0.881010	−	0.473098i	$-0.843136\pi$
$587$	− 22.9245i	− 0.946195i	0.881010	−	0.473098i	$-0.156864\pi$
$588$	0	0
$589$	− 1.52455i	− 0.0628180i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.8895	0.816764	0.408382	−	0.912811i	$-0.366093\pi$
$593$	19.8895	0.816764	0.408382	+	0.912811i	$0.366093\pi$
$594$	0	0
$595$	1.69674i	0.0695596i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.01493	0.368340	0.184170	−	0.982894i	$-0.441040\pi$
$599$	9.01493	0.368340	0.184170	+	0.982894i	$0.441040\pi$
$600$	0	0
$601$	−30.3934	−1.23977	−0.619886	−	0.784692i	$-0.712821\pi$
$601$	−30.3934	−1.23977	−0.619886	+	0.784692i	$0.712821\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.74284i	0.355447i
$606$	0	0
$607$	35.5668	1.44361	0.721807	−	0.692095i	$-0.243312\pi$
$607$	35.5668	1.44361	0.721807	+	0.692095i	$0.243312\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	34.2545i	1.38579i
$612$	0	0
$613$	− 36.1874i	− 1.46159i	−0.682595	−	0.730797i	$-0.739149\pi$
$613$	− 36.1874i	− 1.46159i	0.682595	−	0.730797i	$-0.260851\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−27.7558	−1.11741	−0.558704	−	0.829367i	$-0.688701\pi$
$617$	−27.7558	−1.11741	−0.558704	+	0.829367i	$0.688701\pi$
$618$	0	0
$619$	30.0187i	1.20656i	0.797531	+	0.603278i	$0.206139\pi$
$619$	30.0187i	1.20656i	−0.797531	+	0.603278i	$0.793861\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	17.4322	0.698405
$624$	0	0
$625$	3.42386	0.136954
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 15.8709i	− 0.632813i
$630$	0	0
$631$	−46.3724	−1.84606	−0.923028	−	0.384732i	$-0.874294\pi$
$631$	−46.3724	−1.84606	−0.923028	+	0.384732i	$0.874294\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.99529i	− 0.0791807i
$636$	0	0
$637$	4.05339i	0.160601i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−16.0590	−0.634294	−0.317147	−	0.948376i	$-0.602725\pi$
$641$	−16.0590	−0.634294	−0.317147	+	0.948376i	$0.602725\pi$
$642$	0	0
$643$	− 11.8356i	− 0.466752i	−0.972387	−	0.233376i	$-0.925023\pi$
$643$	− 11.8356i	− 0.466752i	0.972387	−	0.233376i	$-0.0749774\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.05582	0.198765	0.0993824	−	0.995049i	$-0.468313\pi$
$647$	5.05582	0.198765	0.0993824	+	0.995049i	$0.468313\pi$
$648$	0	0
$649$	11.1439	0.437437
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 21.1804i	− 0.828851i	−0.910083	−	0.414426i	$-0.863982\pi$
$653$	− 21.1804i	− 0.828851i	0.910083	−	0.414426i	$-0.136018\pi$
$654$	0	0
$655$	14.9730	0.585045
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 18.6255i	− 0.725548i	−0.931877	−	0.362774i	$-0.881830\pi$
$659$	− 18.6255i	− 0.725548i	0.931877	−	0.362774i	$-0.118170\pi$
$660$	0	0
$661$	8.43660i	0.328146i	0.986448	+	0.164073i	$0.0524632\pi$
$661$	8.43660i	0.328146i	−0.986448	+	0.164073i	$0.947537\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.16824	−0.161638
$666$	0	0
$667$	8.09736i	0.313531i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.2119	−0.471436
$672$	0	0
$673$	−11.6372	−0.448581	−0.224291	−	0.974522i	$-0.572006\pi$
$673$	−11.6372	−0.448581	−0.224291	+	0.974522i	$0.572006\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.4251i	1.63053i	0.579089	+	0.815264i	$0.303408\pi$
$677$	42.4251i	1.63053i	−0.579089	+	0.815264i	$0.696592\pi$
$678$	0	0
$679$	−7.62794	−0.292733
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.8253i	1.02644i	0.858257	+	0.513221i	$0.171548\pi$
$683$	26.8253i	1.02644i	−0.858257	+	0.513221i	$0.828452\pi$
$684$	0	0
$685$	− 23.3589i	− 0.892499i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−47.6452	−1.81514
$690$	0	0
$691$	49.9065i	1.89853i	0.314476	+	0.949265i	$0.398171\pi$
$691$	49.9065i	1.89853i	−0.314476	+	0.949265i	$0.601829\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.4530	0.548235
$696$	0	0
$697$	2.32241	0.0879677
$698$	0	0
$699$	0	0
$700$	0	0
$701$	47.6375i	1.79924i	0.436671	+	0.899621i	$0.356157\pi$
$701$	47.6375i	1.79924i	−0.436671	+	0.899621i	$0.643843\pi$
$702$	0	0
$703$	38.9887	1.47049
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.27674i	0.198452i
$708$	0	0
$709$	− 24.0726i	− 0.904064i	−0.892002	−	0.452032i	$-0.850699\pi$
$709$	− 24.0726i	− 0.904064i	0.892002	−	0.452032i	$-0.149301\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.04786	−0.0392427
$714$	0	0
$715$	− 10.4356i	− 0.390271i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	49.0237	1.82827	0.914137	−	0.405406i	$-0.132870\pi$
$719$	49.0237	1.82827	0.914137	+	0.405406i	$0.132870\pi$
$720$	0	0
$721$	−5.07253	−0.188911
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 12.1577i	− 0.451526i
$726$	0	0
$727$	−18.0843	−0.670710	−0.335355	−	0.942092i	$-0.608856\pi$
$727$	−18.0843	−0.670710	−0.335355	+	0.942092i	$0.608856\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.27725i	0.0842273i
$732$	0	0
$733$	− 2.54526i	− 0.0940114i	−0.998895	−	0.0470057i	$-0.985032\pi$
$733$	− 2.54526i	− 0.0940114i	0.998895	−	0.0470057i	$-0.0149679\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.773664	0.0284983
$738$	0	0
$739$	36.7876i	1.35326i	0.736325	+	0.676628i	$0.236559\pi$
$739$	36.7876i	1.35326i	−0.736325	+	0.676628i	$0.763441\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−27.1051	−0.994391	−0.497196	−	0.867638i	$-0.665637\pi$
$743$	−27.1051	−0.994391	−0.497196	+	0.867638i	$0.665637\pi$
$744$	0	0
$745$	18.6569	0.683537
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.22269i	0.190833i
$750$	0	0
$751$	35.3786	1.29098	0.645491	−	0.763768i	$-0.276653\pi$
$751$	35.3786	1.29098	0.645491	+	0.763768i	$0.276653\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.279717i	0.0101799i
$756$	0	0
$757$	− 0.340328i	− 0.0123694i	−0.999981	−	0.00618471i	$-0.998031\pi$
$757$	− 0.340328i	− 0.0123694i	0.999981	−	0.00618471i	$-0.00196867\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.2601	1.09693	0.548464	−	0.836174i	$-0.315213\pi$
$761$	30.2601	1.09693	0.548464	+	0.836174i	$0.315213\pi$
$762$	0	0
$763$	10.8354i	0.392268i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22.2729	0.804230
$768$	0	0
$769$	18.0636	0.651389	0.325695	−	0.945475i	$-0.394402\pi$
$769$	18.0636	0.651389	0.325695	+	0.945475i	$0.394402\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 3.29847i	− 0.118638i	−0.998239	−	0.0593188i	$-0.981107\pi$
$773$	− 3.29847i	− 0.118638i	0.998239	−	0.0593188i	$-0.0188929\pi$
$774$	0	0
$775$	1.57330	0.0565146
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.70529i	0.204413i
$780$	0	0
$781$	− 19.2183i	− 0.687683i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13.2491	−0.472880
$786$	0	0
$787$	− 14.6607i	− 0.522597i	−0.965258	−	0.261298i	$-0.915849\pi$
$787$	− 14.6607i	− 0.522597i	0.965258	−	0.261298i	$-0.0841506\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.8783	0.386787
$792$	0	0
$793$	−24.4075	−0.866736
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.7001i	0.379016i	0.981879	+	0.189508i	$0.0606892\pi$
$797$	10.7001i	0.379016i	−0.981879	+	0.189508i	$0.939311\pi$
$798$	0	0
$799$	−11.2952	−0.399595
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 20.3486i	− 0.718087i
$804$	0	0
$805$	2.86494i	0.100976i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−19.7701	−0.695078	−0.347539	−	0.937666i	$-0.612983\pi$
$809$	−19.7701	−0.695078	−0.347539	+	0.937666i	$0.612983\pi$
$810$	0	0
$811$	29.8088i	1.04673i	0.852109	+	0.523364i	$0.175323\pi$
$811$	29.8088i	1.04673i	−0.852109	+	0.523364i	$0.824677\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.67395	0.198750
$816$	0	0
$817$	−5.59435	−0.195721
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.3014i	1.33673i	0.743834	+	0.668364i	$0.233005\pi$
$821$	38.3014i	1.33673i	−0.743834	+	0.668364i	$0.766995\pi$
$822$	0	0
$823$	−26.3640	−0.918990	−0.459495	−	0.888180i	$-0.651970\pi$
$823$	−26.3640	−0.918990	−0.459495	+	0.888180i	$0.651970\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.5542i	1.54930i	0.632390	+	0.774650i	$0.282074\pi$
$827$	44.5542i	1.54930i	−0.632390	+	0.774650i	$0.717926\pi$
$828$	0	0
$829$	48.8761i	1.69754i	0.528765	+	0.848768i	$0.322655\pi$
$829$	48.8761i	1.69754i	−0.528765	+	0.848768i	$0.677345\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.33658	−0.0463096
$834$	0	0
$835$	− 17.8508i	− 0.617753i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	39.1944	1.35314	0.676571	−	0.736377i	$-0.263465\pi$
$839$	39.1944	1.35314	0.676571	+	0.736377i	$0.263465\pi$
$840$	0	0
$841$	16.1264	0.556082
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 4.35425i	− 0.149791i
$846$	0	0
$847$	−6.88701	−0.236641
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 26.7979i	− 0.918619i
$852$	0	0
$853$	22.6315i	0.774887i	0.921893	+	0.387444i	$0.126642\pi$
$853$	22.6315i	0.774887i	−0.921893	+	0.387444i	$0.873358\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.6866	0.877436	0.438718	−	0.898625i	$-0.355433\pi$
$857$	25.6866	0.877436	0.438718	+	0.898625i	$0.355433\pi$
$858$	0	0
$859$	45.4628i	1.55117i	0.631244	+	0.775585i	$0.282545\pi$
$859$	45.4628i	1.55117i	−0.631244	+	0.775585i	$0.717455\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.81159	0.231869	0.115935	−	0.993257i	$-0.463014\pi$
$863$	6.81159	0.231869	0.115935	+	0.993257i	$0.463014\pi$
$864$	0	0
$865$	−0.424725	−0.0144411
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 9.94642i	− 0.337409i
$870$	0	0
$871$	1.54629	0.0523942
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 10.6489i	− 0.359997i
$876$	0	0
$877$	− 34.9922i	− 1.18160i	−0.806817	−	0.590802i	$-0.798812\pi$
$877$	− 34.9922i	− 1.18160i	0.806817	−	0.590802i	$-0.201188\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.1155	0.542946	0.271473	−	0.962446i	$-0.412489\pi$
$881$	16.1155	0.542946	0.271473	+	0.962446i	$0.412489\pi$
$882$	0	0
$883$	− 46.5075i	− 1.56510i	−0.622586	−	0.782551i	$-0.713918\pi$
$883$	− 46.5075i	− 1.56510i	0.622586	−	0.782551i	$-0.286082\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	43.0256	1.44466	0.722328	−	0.691550i	$-0.243072\pi$
$887$	43.0256	1.44466	0.722328	+	0.691550i	$0.243072\pi$
$888$	0	0
$889$	1.57175	0.0527149
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 27.7479i	− 0.928550i
$894$	0	0
$895$	−2.35384	−0.0786801
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.66595i	− 0.0555624i
$900$	0	0
$901$	− 15.7107i	− 0.523398i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15.0895	−0.501590
$906$	0	0
$907$	49.6912i	1.64997i	0.565156	+	0.824984i	$0.308816\pi$
$907$	49.6912i	1.64997i	−0.565156	+	0.824984i	$0.691184\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44.6723	−1.48006	−0.740029	−	0.672575i	$-0.765188\pi$
$911$	−44.6723	−1.48006	−0.740029	+	0.672575i	$0.765188\pi$
$912$	0	0
$913$	−14.1112	−0.467012
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.7947i	0.389497i
$918$	0	0
$919$	51.7662	1.70761	0.853804	−	0.520594i	$-0.174290\pi$
$919$	51.7662	1.70761	0.853804	+	0.520594i	$0.174290\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 38.4108i	− 1.26431i
$924$	0	0
$925$	40.2354i	1.32293i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16.4271	0.538956	0.269478	−	0.963006i	$-0.413149\pi$
$929$	16.4271	0.538956	0.269478	+	0.963006i	$0.413149\pi$
$930$	0	0
$931$	− 3.28346i	− 0.107611i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.44108	0.112535
$936$	0	0
$937$	−24.9154	−0.813951	−0.406976	−	0.913439i	$-0.633417\pi$
$937$	−24.9154	−0.813951	−0.406976	+	0.913439i	$0.633417\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 35.3088i	− 1.15103i	−0.817790	−	0.575517i	$-0.804801\pi$
$941$	− 35.3088i	− 1.15103i	0.817790	−	0.575517i	$-0.195199\pi$
$942$	0	0
$943$	3.92138	0.127698
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 41.3026i	− 1.34215i	−0.741388	−	0.671077i	$-0.765832\pi$
$947$	− 41.3026i	− 1.34215i	0.741388	−	0.671077i	$-0.234168\pi$
$948$	0	0
$949$	− 40.6701i	− 1.32021i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.2564	0.364630	0.182315	−	0.983240i	$-0.441641\pi$
$953$	11.2564	0.364630	0.182315	+	0.983240i	$0.441641\pi$
$954$	0	0
$955$	− 33.6125i	− 1.08768i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18.4006	0.594185
$960$	0	0
$961$	−30.7844	−0.993046
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 22.4481i	− 0.722629i
$966$	0	0
$967$	1.51158	0.0486093	0.0243046	−	0.999705i	$-0.492263\pi$
$967$	1.51158	0.0486093	0.0243046	+	0.999705i	$0.492263\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 9.03959i	− 0.290094i	−0.989425	−	0.145047i	$-0.953667\pi$
$971$	− 9.03959i	− 0.290094i	0.989425	−	0.145047i	$-0.0463334\pi$
$972$	0	0
$973$	11.3851i	0.364990i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.32341	0.0743325	0.0371662	−	0.999309i	$-0.488167\pi$
$977$	2.32341	0.0743325	0.0371662	+	0.999309i	$0.488167\pi$
$978$	0	0
$979$	− 35.3533i	− 1.12990i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−18.8206	−0.600284	−0.300142	−	0.953894i	$-0.597034\pi$
$983$	−18.8206	−0.600284	−0.300142	+	0.953894i	$0.597034\pi$
$984$	0	0
$985$	−11.7064	−0.372998
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.84513i	0.122268i
$990$	0	0
$991$	37.9699	1.20615	0.603077	−	0.797683i	$-0.293941\pi$
$991$	37.9699	1.20615	0.603077	+	0.797683i	$0.293941\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	10.7443i	0.340618i
$996$	0	0
$997$	− 9.55023i	− 0.302459i	−0.988499	−	0.151229i	$-0.951677\pi$
$997$	− 9.55023i	− 0.302459i	0.988499	−	0.151229i	$-0.0483232\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.g.3025.14		24
3.2	odd	2	inner	6048.2.c.g.3025.12		24
4.3	odd	2		1512.2.c.f.757.12	yes	24
8.3	odd	2		1512.2.c.f.757.11	✓	24
8.5	even	2	inner	6048.2.c.g.3025.11		24
12.11	even	2		1512.2.c.f.757.13	yes	24
24.5	odd	2	inner	6048.2.c.g.3025.13		24
24.11	even	2		1512.2.c.f.757.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.f.757.11	✓	24	8.3	odd	2
1512.2.c.f.757.12	yes	24	4.3	odd	2
1512.2.c.f.757.13	yes	24	12.11	even	2
1512.2.c.f.757.14	yes	24	24.11	even	2
6048.2.c.g.3025.11		24	8.5	even	2	inner
6048.2.c.g.3025.12		24	3.2	odd	2	inner
6048.2.c.g.3025.13		24	24.5	odd	2	inner
6048.2.c.g.3025.14		24	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)

\(f(q)\)	\(=\)	\(q+1.26947i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+1.26947i q^{5} -1.00000 q^{7} +2.02805i q^{11} +4.05339i q^{13} -1.33658 q^{17} -3.28346i q^{19} -2.25680 q^{23} +3.38845 q^{25} -3.58798i q^{29} +0.464312 q^{31} -1.26947i q^{35} +11.8743i q^{37} -1.73758 q^{41} -1.70380i q^{43} +8.45083 q^{47} +1.00000 q^{49} +11.7544i q^{53} -2.57454 q^{55} -5.49489i q^{59} +6.02150i q^{61} -5.14565 q^{65} -0.381481i q^{67} -9.47622 q^{71} -10.0336 q^{73} -2.02805i q^{77} -4.90442 q^{79} +6.95800i q^{83} -1.69674i q^{85} -17.4322 q^{89} -4.05339i q^{91} +4.16824 q^{95} +7.62794 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{7}+O(q^{10}) \) 24 * q - 24 * q^7	\( 24 q - 24 q^{7} - 24 q^{25} + 16 q^{31} + 24 q^{49} + 8 q^{55} - 32 q^{79}+O(q^{100}) \) 24 * q - 24 * q^7 - 24 * q^25 + 16 * q^31 + 24 * q^49 + 8 * q^55 - 32 * q^79

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