LMFDB - Embedded newform 6048.2.j.c.5615.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(5615,6048)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6048.5615");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.j (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:	$32$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.8
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.c.5615.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.09311 q^{5} -1.00000i q^{7} +O(q^{10})$	$q-2.09311 q^{5} -1.00000i q^{7} -0.961211i q^{11} -4.60210i q^{13} -3.28066i q^{17} -3.66977 q^{19} +2.40322 q^{23} -0.618872 q^{25} -9.31771 q^{29} +1.84927i q^{31} +2.09311i q^{35} -2.29364i q^{37} +0.314406i q^{41} +8.64261 q^{43} +2.34305 q^{47} -1.00000 q^{49} -7.73354 q^{53} +2.01192i q^{55} +7.99530i q^{59} -10.6321i q^{61} +9.63272i q^{65} -4.12203 q^{67} -6.79440 q^{71} +16.0267 q^{73} -0.961211 q^{77} +1.26231i q^{79} -2.99495i q^{83} +6.86680i q^{85} +12.8990i q^{89} -4.60210 q^{91} +7.68126 q^{95} +4.98346 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 64 q^{19} - 16 q^{25} + 48 q^{43} - 32 q^{49} - 16 q^{67} - 16 q^{73} - 16 q^{91} + 64 q^{97}+O(q^{100}) $ 32 * q + 64 * q^19 - 16 * q^25 + 48 * q^43 - 32 * q^49 - 16 * q^67 - 16 * q^73 - 16 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.09311	−0.936069	−0.468035	−	0.883710i	$-0.655038\pi$
$5$	−2.09311	−0.936069	−0.468035	+	0.883710i	$0.655038\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 0.961211i	− 0.289816i	−0.989445	−	0.144908i	$-0.953711\pi$
$11$	− 0.961211i	− 0.289816i	0.989445	−	0.144908i	$-0.0462886\pi$
$12$	0	0
$13$	− 4.60210i	− 1.27639i	−0.769874	−	0.638196i	$-0.779681\pi$
$13$	− 4.60210i	− 1.27639i	0.769874	−	0.638196i	$-0.220319\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.28066i	− 0.795677i	−0.917455	−	0.397839i	$-0.869760\pi$
$17$	− 3.28066i	− 0.795677i	0.917455	−	0.397839i	$-0.130240\pi$
$18$	0	0
$19$	−3.66977	−0.841904	−0.420952	−	0.907083i	$-0.638304\pi$
$19$	−3.66977	−0.841904	−0.420952	+	0.907083i	$0.638304\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.40322	0.501105	0.250553	−	0.968103i	$-0.419388\pi$
$23$	2.40322	0.501105	0.250553	+	0.968103i	$0.419388\pi$
$24$	0	0
$25$	−0.618872	−0.123774
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.31771	−1.73026	−0.865128	−	0.501552i	$-0.832763\pi$
$29$	−9.31771	−1.73026	−0.865128	+	0.501552i	$0.832763\pi$
$30$	0	0
$31$	1.84927i	0.332138i	0.986114	+	0.166069i	$0.0531075\pi$
$31$	1.84927i	0.332138i	−0.986114	+	0.166069i	$0.946892\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.09311i	0.353801i
$36$	0	0
$37$	− 2.29364i	− 0.377072i	−0.982066	−	0.188536i	$-0.939626\pi$
$37$	− 2.29364i	− 0.377072i	0.982066	−	0.188536i	$-0.0603742\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.314406i	0.0491020i	0.999699	+	0.0245510i	$0.00781561\pi$
$41$	0.314406i	0.0491020i	−0.999699	+	0.0245510i	$0.992184\pi$
$42$	0	0
$43$	8.64261	1.31799	0.658993	−	0.752149i	$-0.270983\pi$
$43$	8.64261	1.31799	0.658993	+	0.752149i	$0.270983\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.34305	0.341769	0.170884	−	0.985291i	$-0.445338\pi$
$47$	2.34305	0.341769	0.170884	+	0.985291i	$0.445338\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.73354	−1.06228	−0.531142	−	0.847283i	$-0.678237\pi$
$53$	−7.73354	−1.06228	−0.531142	+	0.847283i	$0.678237\pi$
$54$	0	0
$55$	2.01192i	0.271288i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.99530i	1.04090i	0.853892	+	0.520450i	$0.174236\pi$
$59$	7.99530i	1.04090i	−0.853892	+	0.520450i	$0.825764\pi$
$60$	0	0
$61$	− 10.6321i	− 1.36130i	−0.732610	−	0.680649i	$-0.761698\pi$
$61$	− 10.6321i	− 1.36130i	0.732610	−	0.680649i	$-0.238302\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.63272i	1.19479i
$66$	0	0
$67$	−4.12203	−0.503586	−0.251793	−	0.967781i	$-0.581020\pi$
$67$	−4.12203	−0.503586	−0.251793	+	0.967781i	$0.581020\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.79440	−0.806347	−0.403174	−	0.915124i	$-0.632093\pi$
$71$	−6.79440	−0.806347	−0.403174	+	0.915124i	$0.632093\pi$
$72$	0	0
$73$	16.0267	1.87579	0.937894	−	0.346922i	$-0.112773\pi$
$73$	16.0267	1.87579	0.937894	+	0.346922i	$0.112773\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.961211	−0.109540
$78$	0	0
$79$	1.26231i	0.142021i	0.997476	+	0.0710106i	$0.0226224\pi$
$79$	1.26231i	0.142021i	−0.997476	+	0.0710106i	$0.977378\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2.99495i	− 0.328738i	−0.986399	−	0.164369i	$-0.947441\pi$
$83$	− 2.99495i	− 0.328738i	0.986399	−	0.164369i	$-0.0525589\pi$
$84$	0	0
$85$	6.86680i	0.744809i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.8990i	1.36729i	0.729815	+	0.683645i	$0.239606\pi$
$89$	12.8990i	1.36729i	−0.729815	+	0.683645i	$0.760394\pi$
$90$	0	0
$91$	−4.60210	−0.482431
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.68126	0.788080
$96$	0	0
$97$	4.98346	0.505994	0.252997	−	0.967467i	$-0.418584\pi$
$97$	4.98346	0.505994	0.252997	+	0.967467i	$0.418584\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.18764	0.317182	0.158591	−	0.987344i	$-0.449305\pi$
$101$	3.18764	0.317182	0.158591	+	0.987344i	$0.449305\pi$
$102$	0	0
$103$	− 0.535425i	− 0.0527570i	−0.999652	−	0.0263785i	$-0.991602\pi$
$103$	− 0.535425i	− 0.0527570i	0.999652	−	0.0263785i	$-0.00839750\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.70673i	0.938385i	0.883096	+	0.469193i	$0.155455\pi$
$107$	9.70673i	0.938385i	−0.883096	+	0.469193i	$0.844545\pi$
$108$	0	0
$109$	− 8.03576i	− 0.769686i	−0.922982	−	0.384843i	$-0.874256\pi$
$109$	− 8.03576i	− 0.769686i	0.922982	−	0.384843i	$-0.125744\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.7809i	1.20233i	0.799126	+	0.601163i	$0.205296\pi$
$113$	12.7809i	1.20233i	−0.799126	+	0.601163i	$0.794704\pi$
$114$	0	0
$115$	−5.03021	−0.469069
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.28066	−0.300738
$120$	0	0
$121$	10.0761	0.916007
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.7609	1.05193
$126$	0	0
$127$	− 16.6260i	− 1.47532i	−0.675175	−	0.737658i	$-0.735932\pi$
$127$	− 16.6260i	− 1.47532i	0.675175	−	0.737658i	$-0.264068\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.5603i	1.27214i	0.771632	+	0.636069i	$0.219440\pi$
$131$	14.5603i	1.27214i	−0.771632	+	0.636069i	$0.780560\pi$
$132$	0	0
$133$	3.66977i	0.318210i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.2672i	− 1.47524i	−0.675216	−	0.737620i	$-0.735950\pi$
$137$	− 17.2672i	− 1.47524i	0.675216	−	0.737620i	$-0.264050\pi$
$138$	0	0
$139$	−14.5580	−1.23480	−0.617398	−	0.786651i	$-0.711813\pi$
$139$	−14.5580	−1.23480	−0.617398	+	0.786651i	$0.711813\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.42358	−0.369919
$144$	0	0
$145$	19.5030	1.61964
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.1739	−1.65271	−0.826356	−	0.563148i	$-0.809590\pi$
$149$	−20.1739	−1.65271	−0.826356	+	0.563148i	$0.809590\pi$
$150$	0	0
$151$	11.9560i	0.972966i	0.873690	+	0.486483i	$0.161720\pi$
$151$	11.9560i	0.972966i	−0.873690	+	0.486483i	$0.838280\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3.87073i	− 0.310905i
$156$	0	0
$157$	22.1011i	1.76386i	0.471381	+	0.881930i	$0.343756\pi$
$157$	22.1011i	1.76386i	−0.471381	+	0.881930i	$0.656244\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2.40322i	− 0.189400i
$162$	0	0
$163$	−11.2054	−0.877674	−0.438837	−	0.898567i	$-0.644609\pi$
$163$	−11.2054	−0.877674	−0.438837	+	0.898567i	$0.644609\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.76628	−0.136679	−0.0683395	−	0.997662i	$-0.521770\pi$
$167$	−1.76628	−0.136679	−0.0683395	+	0.997662i	$0.521770\pi$
$168$	0	0
$169$	−8.17930	−0.629177
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.37240	0.332428	0.166214	−	0.986090i	$-0.446846\pi$
$173$	4.37240	0.332428	0.166214	+	0.986090i	$0.446846\pi$
$174$	0	0
$175$	0.618872i	0.0467823i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.0068i	1.27115i	0.772041	+	0.635573i	$0.219236\pi$
$179$	17.0068i	1.27115i	−0.772041	+	0.635573i	$0.780764\pi$
$180$	0	0
$181$	− 8.50624i	− 0.632264i	−0.948715	−	0.316132i	$-0.897616\pi$
$181$	− 8.50624i	− 0.632264i	0.948715	−	0.316132i	$-0.102384\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.80085i	0.352965i
$186$	0	0
$187$	−3.15341	−0.230600
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.725194	0.0524732	0.0262366	−	0.999656i	$-0.491648\pi$
$191$	0.725194	0.0524732	0.0262366	+	0.999656i	$0.491648\pi$
$192$	0	0
$193$	−15.7648	−1.13478	−0.567388	−	0.823450i	$-0.692046\pi$
$193$	−15.7648	−1.13478	−0.567388	+	0.823450i	$0.692046\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.76032	0.410406	0.205203	−	0.978719i	$-0.434215\pi$
$197$	5.76032	0.410406	0.205203	+	0.978719i	$0.434215\pi$
$198$	0	0
$199$	11.7773i	0.834872i	0.908706	+	0.417436i	$0.137071\pi$
$199$	11.7773i	0.834872i	−0.908706	+	0.417436i	$0.862929\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.31771i	0.653975i
$204$	0	0
$205$	− 0.658088i	− 0.0459629i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.52742i	0.243997i
$210$	0	0
$211$	−19.9971	−1.37666	−0.688328	−	0.725399i	$-0.741655\pi$
$211$	−19.9971	−1.37666	−0.688328	+	0.725399i	$0.741655\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−18.0900	−1.23373
$216$	0	0
$217$	1.84927	0.125537
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−15.0979	−1.01560
$222$	0	0
$223$	− 6.84047i	− 0.458071i	−0.973418	−	0.229036i	$-0.926443\pi$
$223$	− 6.84047i	− 0.458071i	0.973418	−	0.229036i	$-0.0735573\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.9061i	0.922981i	0.887145	+	0.461490i	$0.152685\pi$
$227$	13.9061i	0.922981i	−0.887145	+	0.461490i	$0.847315\pi$
$228$	0	0
$229$	3.87934i	0.256354i	0.991751	+	0.128177i	$0.0409126\pi$
$229$	3.87934i	0.256354i	−0.991751	+	0.128177i	$0.959087\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.8307i	1.43017i	0.699035	+	0.715087i	$0.253613\pi$
$233$	21.8307i	1.43017i	−0.699035	+	0.715087i	$0.746387\pi$
$234$	0	0
$235$	−4.90427	−0.319919
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−27.8956	−1.80441	−0.902207	−	0.431303i	$-0.858054\pi$
$239$	−27.8956	−1.80441	−0.902207	+	0.431303i	$0.858054\pi$
$240$	0	0
$241$	−6.17606	−0.397835	−0.198918	−	0.980016i	$-0.563743\pi$
$241$	−6.17606	−0.397835	−0.198918	+	0.980016i	$0.563743\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.09311	0.133724
$246$	0	0
$247$	16.8887i	1.07460i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 18.6366i	− 1.17633i	−0.808740	−	0.588167i	$-0.799850\pi$
$251$	− 18.6366i	− 1.17633i	0.808740	−	0.588167i	$-0.200150\pi$
$252$	0	0
$253$	− 2.31000i	− 0.145228i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.25845i	0.515148i	0.966259	+	0.257574i	$0.0829231\pi$
$257$	8.25845i	0.515148i	−0.966259	+	0.257574i	$0.917077\pi$
$258$	0	0
$259$	−2.29364	−0.142520
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.34158	−0.576027	−0.288013	−	0.957626i	$-0.592995\pi$
$263$	−9.34158	−0.576027	−0.288013	+	0.957626i	$0.592995\pi$
$264$	0	0
$265$	16.1872	0.994371
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.9972	−0.731484	−0.365742	−	0.930716i	$-0.619185\pi$
$269$	−11.9972	−0.731484	−0.365742	+	0.930716i	$0.619185\pi$
$270$	0	0
$271$	23.7030i	1.43986i	0.694049	+	0.719928i	$0.255825\pi$
$271$	23.7030i	1.43986i	−0.694049	+	0.719928i	$0.744175\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.594866i	0.0358718i
$276$	0	0
$277$	− 13.7688i	− 0.827284i	−0.910440	−	0.413642i	$-0.864256\pi$
$277$	− 13.7688i	− 0.827284i	0.910440	−	0.413642i	$-0.135744\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.69078i	0.518448i	0.965817	+	0.259224i	$0.0834669\pi$
$281$	8.69078i	0.518448i	−0.965817	+	0.259224i	$0.916533\pi$
$282$	0	0
$283$	32.7518	1.94689	0.973447	−	0.228913i	$-0.0735170\pi$
$283$	32.7518	1.94689	0.973447	+	0.228913i	$0.0735170\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.314406	0.0185588
$288$	0	0
$289$	6.23727	0.366898
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−12.3629	−0.722249	−0.361124	−	0.932518i	$-0.617607\pi$
$293$	−12.3629	−0.722249	−0.361124	+	0.932518i	$0.617607\pi$
$294$	0	0
$295$	− 16.7351i	− 0.974354i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 11.0598i	− 0.639607i
$300$	0	0
$301$	− 8.64261i	− 0.498152i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	22.2542i	1.27427i
$306$	0	0
$307$	−15.2600	−0.870935	−0.435468	−	0.900204i	$-0.643417\pi$
$307$	−15.2600	−0.870935	−0.435468	+	0.900204i	$0.643417\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.12076	−0.290372	−0.145186	−	0.989404i	$-0.546378\pi$
$311$	−5.12076	−0.290372	−0.145186	+	0.989404i	$0.546378\pi$
$312$	0	0
$313$	−11.7058	−0.661649	−0.330825	−	0.943692i	$-0.607327\pi$
$313$	−11.7058	−0.661649	−0.330825	+	0.943692i	$0.607327\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.5302	1.04076	0.520379	−	0.853935i	$-0.325791\pi$
$317$	18.5302	1.04076	0.520379	+	0.853935i	$0.325791\pi$
$318$	0	0
$319$	8.95628i	0.501456i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.0393i	0.669883i
$324$	0	0
$325$	2.84811i	0.157985i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 2.34305i	− 0.129176i
$330$	0	0
$331$	20.4688	1.12507	0.562534	−	0.826774i	$-0.309826\pi$
$331$	20.4688	1.12507	0.562534	+	0.826774i	$0.309826\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.62789	0.471392
$336$	0	0
$337$	−19.9419	−1.08631	−0.543153	−	0.839634i	$-0.682769\pi$
$337$	−19.9419	−1.08631	−0.543153	+	0.839634i	$0.682769\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.77754	0.0962590
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.1743i	0.546187i	0.961987	+	0.273094i	$0.0880469\pi$
$347$	10.1743i	0.546187i	−0.961987	+	0.273094i	$0.911953\pi$
$348$	0	0
$349$	31.9386i	1.70963i	0.518930	+	0.854817i	$0.326331\pi$
$349$	31.9386i	1.70963i	−0.518930	+	0.854817i	$0.673669\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 24.3932i	− 1.29832i	−0.760653	−	0.649159i	$-0.775121\pi$
$353$	− 24.3932i	− 1.29832i	0.760653	−	0.649159i	$-0.224879\pi$
$354$	0	0
$355$	14.2215	0.754797
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.59571	−0.453664	−0.226832	−	0.973934i	$-0.572837\pi$
$359$	−8.59571	−0.453664	−0.226832	+	0.973934i	$0.572837\pi$
$360$	0	0
$361$	−5.53276	−0.291198
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−33.5458	−1.75587
$366$	0	0
$367$	− 29.1285i	− 1.52050i	−0.649632	−	0.760249i	$-0.725077\pi$
$367$	− 29.1285i	− 1.52050i	0.649632	−	0.760249i	$-0.274923\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.73354i	0.401505i
$372$	0	0
$373$	4.56533i	0.236384i	0.992991	+	0.118192i	$0.0377098\pi$
$373$	4.56533i	0.236384i	−0.992991	+	0.118192i	$0.962290\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	42.8810i	2.20848i
$378$	0	0
$379$	−15.5673	−0.799641	−0.399820	−	0.916593i	$-0.630928\pi$
$379$	−15.5673	−0.799641	−0.399820	+	0.916593i	$0.630928\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.08947	0.157865	0.0789323	−	0.996880i	$-0.474849\pi$
$383$	3.08947	0.157865	0.0789323	+	0.996880i	$0.474849\pi$
$384$	0	0
$385$	2.01192	0.102537
$386$	0	0
$387$	0	0
$388$	0	0
$389$	30.6022	1.55160	0.775798	−	0.630982i	$-0.217348\pi$
$389$	30.6022	1.55160	0.775798	+	0.630982i	$0.217348\pi$
$390$	0	0
$391$	− 7.88413i	− 0.398718i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 2.64216i	− 0.132942i
$396$	0	0
$397$	− 0.403742i	− 0.0202632i	−0.999949	−	0.0101316i	$-0.996775\pi$
$397$	− 0.403742i	− 0.0202632i	0.999949	−	0.0101316i	$-0.00322505\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.3779i	1.51700i	0.651672	+	0.758501i	$0.274068\pi$
$401$	30.3779i	1.51700i	−0.651672	+	0.758501i	$0.725932\pi$
$402$	0	0
$403$	8.51051	0.423939
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.20467	−0.109281
$408$	0	0
$409$	−17.8549	−0.882869	−0.441434	−	0.897293i	$-0.645530\pi$
$409$	−17.8549	−0.882869	−0.441434	+	0.897293i	$0.645530\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.99530	0.393423
$414$	0	0
$415$	6.26877i	0.307722i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	31.1609i	1.52231i	0.648569	+	0.761156i	$0.275368\pi$
$419$	31.1609i	1.52231i	−0.648569	+	0.761156i	$0.724632\pi$
$420$	0	0
$421$	7.19292i	0.350562i	0.984518	+	0.175281i	$0.0560833\pi$
$421$	7.19292i	0.350562i	−0.984518	+	0.175281i	$0.943917\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.03031i	0.0984844i
$426$	0	0
$427$	−10.6321	−0.514522
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.7598	−1.24081	−0.620404	−	0.784283i	$-0.713031\pi$
$431$	−25.7598	−1.24081	−0.620404	+	0.784283i	$0.713031\pi$
$432$	0	0
$433$	−3.66449	−0.176104	−0.0880521	−	0.996116i	$-0.528064\pi$
$433$	−3.66449	−0.176104	−0.0880521	+	0.996116i	$0.528064\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.81926	−0.421882
$438$	0	0
$439$	39.4350i	1.88213i	0.338227	+	0.941064i	$0.390173\pi$
$439$	39.4350i	1.88213i	−0.338227	+	0.941064i	$0.609827\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 36.1437i	− 1.71724i	−0.512612	−	0.858620i	$-0.671322\pi$
$443$	− 36.1437i	− 1.71724i	0.512612	−	0.858620i	$-0.328678\pi$
$444$	0	0
$445$	− 26.9990i	− 1.27988i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 24.9465i	− 1.17730i	−0.808389	−	0.588649i	$-0.799660\pi$
$449$	− 24.9465i	− 1.17730i	0.808389	−	0.588649i	$-0.200340\pi$
$450$	0	0
$451$	0.302210	0.0142305
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.63272	0.451589
$456$	0	0
$457$	15.1856	0.710354	0.355177	−	0.934799i	$-0.384421\pi$
$457$	15.1856	0.710354	0.355177	+	0.934799i	$0.384421\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.29479	0.432902	0.216451	−	0.976294i	$-0.430552\pi$
$461$	9.29479	0.432902	0.216451	+	0.976294i	$0.430552\pi$
$462$	0	0
$463$	− 2.13438i	− 0.0991930i	−0.998769	−	0.0495965i	$-0.984206\pi$
$463$	− 2.13438i	− 0.0991930i	0.998769	−	0.0495965i	$-0.0157935\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	41.6077i	1.92538i	0.270612	+	0.962688i	$0.412774\pi$
$467$	41.6077i	1.92538i	−0.270612	+	0.962688i	$0.587226\pi$
$468$	0	0
$469$	4.12203i	0.190338i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 8.30736i	− 0.381973i
$474$	0	0
$475$	2.27112	0.104206
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−40.9442	−1.87079	−0.935394	−	0.353608i	$-0.884955\pi$
$479$	−40.9442	−1.87079	−0.935394	+	0.353608i	$0.884955\pi$
$480$	0	0
$481$	−10.5556	−0.481292
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.4310	−0.473645
$486$	0	0
$487$	14.2039i	0.643639i	0.946801	+	0.321820i	$0.104294\pi$
$487$	14.2039i	0.643639i	−0.946801	+	0.321820i	$0.895706\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 39.3013i	− 1.77364i	−0.462113	−	0.886821i	$-0.652909\pi$
$491$	− 39.3013i	− 1.77364i	0.462113	−	0.886821i	$-0.347091\pi$
$492$	0	0
$493$	30.5682i	1.37672i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.79440i	0.304771i
$498$	0	0
$499$	32.3257	1.44710	0.723548	−	0.690274i	$-0.242510\pi$
$499$	32.3257	1.44710	0.723548	+	0.690274i	$0.242510\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19.7861	0.882217	0.441108	−	0.897454i	$-0.354585\pi$
$503$	19.7861	0.882217	0.441108	+	0.897454i	$0.354585\pi$
$504$	0	0
$505$	−6.67210	−0.296904
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.7174	1.49450	0.747248	−	0.664545i	$-0.231375\pi$
$509$	33.7174	1.49450	0.747248	+	0.664545i	$0.231375\pi$
$510$	0	0
$511$	− 16.0267i	− 0.708981i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.12070i	0.0493842i
$516$	0	0
$517$	− 2.25216i	− 0.0990500i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 28.4958i	− 1.24842i	−0.781255	−	0.624212i	$-0.785420\pi$
$521$	− 28.4958i	− 1.24842i	0.781255	−	0.624212i	$-0.214580\pi$
$522$	0	0
$523$	4.42637	0.193552	0.0967758	−	0.995306i	$-0.469147\pi$
$523$	4.42637	0.193552	0.0967758	+	0.995306i	$0.469147\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.06682	0.264275
$528$	0	0
$529$	−17.2246	−0.748894
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.44693	0.0626734
$534$	0	0
$535$	− 20.3173i	− 0.878394i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.961211i	0.0414023i
$540$	0	0
$541$	− 33.7548i	− 1.45123i	−0.688101	−	0.725615i	$-0.741555\pi$
$541$	− 33.7548i	− 1.45123i	0.688101	−	0.725615i	$-0.258445\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16.8198i	0.720480i
$546$	0	0
$547$	14.6641	0.626992	0.313496	−	0.949589i	$-0.398500\pi$
$547$	14.6641	0.626992	0.313496	+	0.949589i	$0.398500\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	34.1939	1.45671
$552$	0	0
$553$	1.26231	0.0536790
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15.4373	−0.654099	−0.327050	−	0.945007i	$-0.606054\pi$
$557$	−15.4373	−0.654099	−0.327050	+	0.945007i	$0.606054\pi$
$558$	0	0
$559$	− 39.7741i	− 1.68227i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26.9472i	1.13569i	0.823135	+	0.567845i	$0.192223\pi$
$563$	26.9472i	1.13569i	−0.823135	+	0.567845i	$0.807777\pi$
$564$	0	0
$565$	− 26.7519i	− 1.12546i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 3.63013i	− 0.152183i	−0.997101	−	0.0760915i	$-0.975756\pi$
$569$	− 3.63013i	− 0.152183i	0.997101	−	0.0760915i	$-0.0242441\pi$
$570$	0	0
$571$	16.2293	0.679174	0.339587	−	0.940575i	$-0.389713\pi$
$571$	16.2293	0.679174	0.339587	+	0.940575i	$0.389713\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.48728	−0.0620240
$576$	0	0
$577$	42.1511	1.75477	0.877386	−	0.479786i	$-0.159286\pi$
$577$	42.1511	1.75477	0.877386	+	0.479786i	$0.159286\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.99495	−0.124251
$582$	0	0
$583$	7.43356i	0.307867i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.7097i	0.937328i	0.883377	+	0.468664i	$0.155264\pi$
$587$	22.7097i	0.937328i	−0.883377	+	0.468664i	$0.844736\pi$
$588$	0	0
$589$	− 6.78640i	− 0.279629i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 22.8921i	− 0.940068i	−0.882648	−	0.470034i	$-0.844242\pi$
$593$	− 22.8921i	− 0.940068i	0.882648	−	0.470034i	$-0.155758\pi$
$594$	0	0
$595$	6.86680	0.281511
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.0216	−1.26751	−0.633755	−	0.773534i	$-0.718487\pi$
$599$	−31.0216	−1.26751	−0.633755	+	0.773534i	$0.718487\pi$
$600$	0	0
$601$	−36.3657	−1.48339	−0.741693	−	0.670739i	$-0.765977\pi$
$601$	−36.3657	−1.48339	−0.741693	+	0.670739i	$0.765977\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−21.0904	−0.857446
$606$	0	0
$607$	− 27.1019i	− 1.10003i	−0.835154	−	0.550016i	$-0.814622\pi$
$607$	− 27.1019i	− 1.10003i	0.835154	−	0.550016i	$-0.185378\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 10.7829i	− 0.436231i
$612$	0	0
$613$	− 21.7164i	− 0.877118i	−0.898702	−	0.438559i	$-0.855489\pi$
$613$	− 21.7164i	− 0.877118i	0.898702	−	0.438559i	$-0.144511\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.0335i	0.605224i	0.953114	+	0.302612i	$0.0978587\pi$
$617$	15.0335i	0.605224i	−0.953114	+	0.302612i	$0.902141\pi$
$618$	0	0
$619$	12.2249	0.491359	0.245680	−	0.969351i	$-0.420989\pi$
$619$	12.2249	0.491359	0.245680	+	0.969351i	$0.420989\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.8990	0.516787
$624$	0	0
$625$	−21.5226	−0.860906
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.52465	−0.300028
$630$	0	0
$631$	− 0.290118i	− 0.0115494i	−0.999983	−	0.00577472i	$-0.998162\pi$
$631$	− 0.290118i	− 0.0115494i	0.999983	−	0.00577472i	$-0.00183816\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	34.8000i	1.38100i
$636$	0	0
$637$	4.60210i	0.182342i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 32.2612i	− 1.27424i	−0.770764	−	0.637120i	$-0.780125\pi$
$641$	− 32.2612i	− 1.27424i	0.770764	−	0.637120i	$-0.219875\pi$
$642$	0	0
$643$	25.2641	0.996317	0.498159	−	0.867086i	$-0.334010\pi$
$643$	25.2641	0.996317	0.498159	+	0.867086i	$0.334010\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.6735	−0.616188	−0.308094	−	0.951356i	$-0.599691\pi$
$647$	−15.6735	−0.616188	−0.308094	+	0.951356i	$0.599691\pi$
$648$	0	0
$649$	7.68517	0.301669
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.4580	0.683183	0.341591	−	0.939849i	$-0.389034\pi$
$653$	17.4580	0.683183	0.341591	+	0.939849i	$0.389034\pi$
$654$	0	0
$655$	− 30.4763i	− 1.19081i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 34.8692i	− 1.35831i	−0.733995	−	0.679155i	$-0.762347\pi$
$659$	− 34.8692i	− 1.35831i	0.733995	−	0.679155i	$-0.237653\pi$
$660$	0	0
$661$	13.7607i	0.535230i	0.963526	+	0.267615i	$0.0862355\pi$
$661$	13.7607i	0.535230i	−0.963526	+	0.267615i	$0.913765\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 7.68126i	− 0.297866i
$666$	0	0
$667$	−22.3925	−0.867040
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.2197	−0.394526
$672$	0	0
$673$	−8.50938	−0.328013	−0.164006	−	0.986459i	$-0.552442\pi$
$673$	−8.50938	−0.328013	−0.164006	+	0.986459i	$0.552442\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.14515	−0.120878	−0.0604390	−	0.998172i	$-0.519250\pi$
$677$	−3.14515	−0.120878	−0.0604390	+	0.998172i	$0.519250\pi$
$678$	0	0
$679$	− 4.98346i	− 0.191248i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.12590i	0.349193i	0.984640	+	0.174596i	$0.0558621\pi$
$683$	9.12590i	0.349193i	−0.984640	+	0.174596i	$0.944138\pi$
$684$	0	0
$685$	36.1423i	1.38093i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	35.5905i	1.35589i
$690$	0	0
$691$	15.6298	0.594586	0.297293	−	0.954786i	$-0.403916\pi$
$691$	15.6298	0.594586	0.297293	+	0.954786i	$0.403916\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	30.4716	1.15585
$696$	0	0
$697$	1.03146	0.0390693
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−50.4853	−1.90680	−0.953402	−	0.301701i	$-0.902445\pi$
$701$	−50.4853	−1.90680	−0.953402	+	0.301701i	$0.902445\pi$
$702$	0	0
$703$	8.41714i	0.317458i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.18764i	− 0.119884i
$708$	0	0
$709$	28.7706i	1.08050i	0.841504	+	0.540251i	$0.181671\pi$
$709$	28.7706i	1.08050i	−0.841504	+	0.540251i	$0.818329\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.44419i	0.166436i
$714$	0	0
$715$	9.25907	0.346270
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.582170	−0.0217113	−0.0108556	−	0.999941i	$-0.503456\pi$
$719$	−0.582170	−0.0217113	−0.0108556	+	0.999941i	$0.503456\pi$
$720$	0	0
$721$	−0.535425	−0.0199403
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.76647	0.214161
$726$	0	0
$727$	− 6.92907i	− 0.256985i	−0.991710	−	0.128492i	$-0.958986\pi$
$727$	− 6.92907i	− 0.256985i	0.991710	−	0.128492i	$-0.0410138\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 28.3535i	− 1.04869i
$732$	0	0
$733$	− 1.86770i	− 0.0689850i	−0.999405	−	0.0344925i	$-0.989019\pi$
$733$	− 1.86770i	− 0.0689850i	0.999405	−	0.0344925i	$-0.0109815\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.96214i	0.145947i
$738$	0	0
$739$	38.7036	1.42373	0.711867	−	0.702314i	$-0.247850\pi$
$739$	38.7036	1.42373	0.711867	+	0.702314i	$0.247850\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23.3088	−0.855118	−0.427559	−	0.903987i	$-0.640626\pi$
$743$	−23.3088	−0.855118	−0.427559	+	0.903987i	$0.640626\pi$
$744$	0	0
$745$	42.2263	1.54705
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.70673	0.354676
$750$	0	0
$751$	− 26.6345i	− 0.971908i	−0.873984	−	0.485954i	$-0.838472\pi$
$751$	− 26.6345i	− 0.971908i	0.873984	−	0.485954i	$-0.161528\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 25.0253i	− 0.910764i
$756$	0	0
$757$	26.4285i	0.960562i	0.877115	+	0.480281i	$0.159465\pi$
$757$	26.4285i	0.960562i	−0.877115	+	0.480281i	$0.840535\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 13.2617i	− 0.480736i	−0.970682	−	0.240368i	$-0.922732\pi$
$761$	− 13.2617i	− 0.480736i	0.970682	−	0.240368i	$-0.0772681\pi$
$762$	0	0
$763$	−8.03576	−0.290914
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.7951	1.32860
$768$	0	0
$769$	−4.34986	−0.156860	−0.0784300	−	0.996920i	$-0.524991\pi$
$769$	−4.34986	−0.156860	−0.0784300	+	0.996920i	$0.524991\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−34.8861	−1.25477	−0.627383	−	0.778711i	$-0.715874\pi$
$773$	−34.8861	−1.25477	−0.627383	+	0.778711i	$0.715874\pi$
$774$	0	0
$775$	− 1.14446i	− 0.0411102i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.15380i	− 0.0413391i
$780$	0	0
$781$	6.53085i	0.233692i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 46.2601i	− 1.65109i
$786$	0	0
$787$	−12.9317	−0.460964	−0.230482	−	0.973077i	$-0.574030\pi$
$787$	−12.9317	−0.460964	−0.230482	+	0.973077i	$0.574030\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.7809	0.454437
$792$	0	0
$793$	−48.9299	−1.73755
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.1981	−0.396656	−0.198328	−	0.980136i	$-0.563551\pi$
$797$	−11.1981	−0.396656	−0.198328	+	0.980136i	$0.563551\pi$
$798$	0	0
$799$	− 7.68675i	− 0.271938i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 15.4051i	− 0.543633i
$804$	0	0
$805$	5.03021i	0.177291i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.6434i	0.831258i	0.909534	+	0.415629i	$0.136439\pi$
$809$	23.6434i	0.831258i	−0.909534	+	0.415629i	$0.863561\pi$
$810$	0	0
$811$	−34.8213	−1.22274	−0.611371	−	0.791344i	$-0.709382\pi$
$811$	−34.8213	−1.22274	−0.611371	+	0.791344i	$0.709382\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	23.4542	0.821564
$816$	0	0
$817$	−31.7164	−1.10962
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.361884	0.0126298	0.00631491	−	0.999980i	$-0.497990\pi$
$821$	0.361884	0.0126298	0.00631491	+	0.999980i	$0.497990\pi$
$822$	0	0
$823$	49.2033i	1.71512i	0.514386	+	0.857559i	$0.328020\pi$
$823$	49.2033i	1.71512i	−0.514386	+	0.857559i	$0.671980\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.2920i	0.566529i	0.959042	+	0.283264i	$0.0914174\pi$
$827$	16.2920i	0.566529i	−0.959042	+	0.283264i	$0.908583\pi$
$828$	0	0
$829$	6.40526i	0.222464i	0.993794	+	0.111232i	$0.0354796\pi$
$829$	6.40526i	0.222464i	−0.993794	+	0.111232i	$0.964520\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.28066i	0.113668i
$834$	0	0
$835$	3.69703	0.127941
$836$	0	0
$837$	0	0
$838$	0	0
$839$	23.3877	0.807432	0.403716	−	0.914884i	$-0.367718\pi$
$839$	23.3877	0.807432	0.403716	+	0.914884i	$0.367718\pi$
$840$	0	0
$841$	57.8197	1.99378
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17.1202	0.588953
$846$	0	0
$847$	− 10.0761i	− 0.346218i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 5.51211i	− 0.188953i
$852$	0	0
$853$	− 22.3667i	− 0.765821i	−0.923786	−	0.382910i	$-0.874922\pi$
$853$	− 22.3667i	− 0.765821i	0.923786	−	0.382910i	$-0.125078\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 34.7703i	− 1.18773i	−0.804564	−	0.593866i	$-0.797601\pi$
$857$	− 34.7703i	− 1.18773i	0.804564	−	0.593866i	$-0.202399\pi$
$858$	0	0
$859$	−16.1287	−0.550305	−0.275152	−	0.961401i	$-0.588728\pi$
$859$	−16.1287	−0.550305	−0.275152	+	0.961401i	$0.588728\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34.3951	1.17082	0.585412	−	0.810736i	$-0.300933\pi$
$863$	34.3951	1.17082	0.585412	+	0.810736i	$0.300933\pi$
$864$	0	0
$865$	−9.15194	−0.311175
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.21335	0.0411600
$870$	0	0
$871$	18.9700i	0.642774i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 11.7609i	− 0.397592i
$876$	0	0
$877$	− 14.0555i	− 0.474620i	−0.971434	−	0.237310i	$-0.923734\pi$
$877$	− 14.0555i	− 0.474620i	0.971434	−	0.237310i	$-0.0762658\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41.1209i	1.38540i	0.721226	+	0.692700i	$0.243579\pi$
$881$	41.1209i	1.38540i	−0.721226	+	0.692700i	$0.756421\pi$
$882$	0	0
$883$	18.8276	0.633598	0.316799	−	0.948493i	$-0.397392\pi$
$883$	18.8276	0.633598	0.316799	+	0.948493i	$0.397392\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.8728	0.365071	0.182536	−	0.983199i	$-0.441570\pi$
$887$	10.8728	0.365071	0.182536	+	0.983199i	$0.441570\pi$
$888$	0	0
$889$	−16.6260	−0.557617
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.59846	−0.287736
$894$	0	0
$895$	− 35.5971i	− 1.18988i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 17.2310i	− 0.574684i
$900$	0	0
$901$	25.3711i	0.845235i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.8045i	0.591843i
$906$	0	0
$907$	−15.4589	−0.513304	−0.256652	−	0.966504i	$-0.582619\pi$
$907$	−15.4589	−0.513304	−0.256652	+	0.966504i	$0.582619\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23.9492	0.793473	0.396737	−	0.917933i	$-0.370143\pi$
$911$	23.9492	0.793473	0.396737	+	0.917933i	$0.370143\pi$
$912$	0	0
$913$	−2.87878	−0.0952736
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.5603	0.480823
$918$	0	0
$919$	24.4853i	0.807694i	0.914826	+	0.403847i	$0.132327\pi$
$919$	24.4853i	0.807694i	−0.914826	+	0.403847i	$0.867673\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	31.2685i	1.02922i
$924$	0	0
$925$	1.41947i	0.0466719i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 35.7122i	− 1.17168i	−0.810428	−	0.585839i	$-0.800765\pi$
$929$	− 35.7122i	− 1.17168i	0.810428	−	0.585839i	$-0.199235\pi$
$930$	0	0
$931$	3.66977	0.120272
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.60044	0.215857
$936$	0	0
$937$	0.263959	0.00862315	0.00431157	−	0.999991i	$-0.498628\pi$
$937$	0.263959	0.00862315	0.00431157	+	0.999991i	$0.498628\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.9124	1.33371	0.666853	−	0.745189i	$-0.267641\pi$
$941$	40.9124	1.33371	0.666853	+	0.745189i	$0.267641\pi$
$942$	0	0
$943$	0.755586i	0.0246053i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 43.3236i	− 1.40783i	−0.710286	−	0.703913i	$-0.751434\pi$
$947$	− 43.3236i	− 1.40783i	0.710286	−	0.703913i	$-0.248566\pi$
$948$	0	0
$949$	− 73.7566i	− 2.39424i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 52.0756i	− 1.68690i	−0.537211	−	0.843448i	$-0.680522\pi$
$953$	− 52.0756i	− 1.68690i	0.537211	−	0.843448i	$-0.319478\pi$
$954$	0	0
$955$	−1.51791	−0.0491185
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17.2672	−0.557588
$960$	0	0
$961$	27.5802	0.889684
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.9976	1.06223
$966$	0	0
$967$	17.3116i	0.556705i	0.960479	+	0.278352i	$0.0897883\pi$
$967$	17.3116i	0.556705i	−0.960479	+	0.278352i	$0.910212\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.86074i	0.284355i	0.989841	+	0.142177i	$0.0454103\pi$
$971$	8.86074i	0.284355i	−0.989841	+	0.142177i	$0.954590\pi$
$972$	0	0
$973$	14.5580i	0.466709i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.48509i	0.271462i	0.990746	+	0.135731i	$0.0433383\pi$
$977$	8.48509i	0.271462i	−0.990746	+	0.135731i	$0.956662\pi$
$978$	0	0
$979$	12.3986	0.396262
$980$	0	0
$981$	0	0
$982$	0	0
$983$	43.6181	1.39120	0.695600	−	0.718429i	$-0.255138\pi$
$983$	43.6181	1.39120	0.695600	+	0.718429i	$0.255138\pi$
$984$	0	0
$985$	−12.0570	−0.384168
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20.7700	0.660449
$990$	0	0
$991$	51.0937i	1.62304i	0.584322	+	0.811522i	$0.301361\pi$
$991$	51.0937i	1.62304i	−0.584322	+	0.811522i	$0.698639\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 24.6513i	− 0.781498i
$996$	0	0
$997$	1.59455i	0.0504998i	0.999681	+	0.0252499i	$0.00803815\pi$
$997$	1.59455i	0.0504998i	−0.999681	+	0.0252499i	$0.991962\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.c.5615.8		32
3.2	odd	2	inner	6048.2.j.c.5615.25		32
4.3	odd	2		1512.2.j.c.323.25	yes	32
8.3	odd	2	inner	6048.2.j.c.5615.26		32
8.5	even	2		1512.2.j.c.323.7	✓	32
12.11	even	2		1512.2.j.c.323.8	yes	32
24.5	odd	2		1512.2.j.c.323.26	yes	32
24.11	even	2	inner	6048.2.j.c.5615.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.c.323.7	✓	32	8.5	even	2
1512.2.j.c.323.8	yes	32	12.11	even	2
1512.2.j.c.323.25	yes	32	4.3	odd	2
1512.2.j.c.323.26	yes	32	24.5	odd	2
6048.2.j.c.5615.7		32	24.11	even	2	inner
6048.2.j.c.5615.8		32	1.1	even	1	trivial
6048.2.j.c.5615.25		32	3.2	odd	2	inner
6048.2.j.c.5615.26		32	8.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.09311 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-2.09311 q^{5} -1.00000i q^{7} -0.961211i q^{11} -4.60210i q^{13} -3.28066i q^{17} -3.66977 q^{19} +2.40322 q^{23} -0.618872 q^{25} -9.31771 q^{29} +1.84927i q^{31} +2.09311i q^{35} -2.29364i q^{37} +0.314406i q^{41} +8.64261 q^{43} +2.34305 q^{47} -1.00000 q^{49} -7.73354 q^{53} +2.01192i q^{55} +7.99530i q^{59} -10.6321i q^{61} +9.63272i q^{65} -4.12203 q^{67} -6.79440 q^{71} +16.0267 q^{73} -0.961211 q^{77} +1.26231i q^{79} -2.99495i q^{83} +6.86680i q^{85} +12.8990i q^{89} -4.60210 q^{91} +7.68126 q^{95} +4.98346 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 64 q^{19} - 16 q^{25} + 48 q^{43} - 32 q^{49} - 16 q^{67} - 16 q^{73} - 16 q^{91} + 64 q^{97}+O(q^{100}) \) 32 * q + 64 * q^19 - 16 * q^25 + 48 * q^43 - 32 * q^49 - 16 * q^67 - 16 * q^73 - 16 * q^91 + 64 * q^97

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