LMFDB - Embedded newform 3024.2.h.b.2591.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(2591,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("3024.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.h (of order $2$, degree $1$, 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:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.19752615936.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} + 195x^{4} - 976x^{2} + 3721 $ x^8 - 16x^6 + 195x^4 - 976*x^2 + 3721
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.6
Root			$-2.16817 - 1.25179i$ of defining polynomial
Character	$\chi$	$=$	3024.2591
Dual form			3024.2.h.b.2591.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.50359i q^{5} +1.00000i q^{7} +O(q^{10})$	$q+2.50359i q^{5} +1.00000i q^{7} -2.50359 q^{11} +0.732051 q^{13} -6.83993i q^{17} -8.46410i q^{19} -2.50359 q^{23} -1.26795 q^{25} +6.83993i q^{29} +0.464102i q^{31} -2.50359 q^{35} -1.00000 q^{37} -11.1763i q^{41} -6.73205i q^{43} -11.8471 q^{47} -1.00000 q^{49} -5.00717i q^{53} -6.26795i q^{55} +6.83993 q^{59} +10.3923 q^{61} +1.83275i q^{65} -1.26795i q^{67} +0.670834 q^{71} +1.80385 q^{73} -2.50359i q^{77} -1.26795i q^{79} +6.83993 q^{83} +17.1244 q^{85} +12.5179i q^{89} +0.732051i q^{91} +21.1906 q^{95} +18.3923 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{13} - 24 q^{25} - 8 q^{37} - 8 q^{49} + 56 q^{73} + 40 q^{85} + 64 q^{97}+O(q^{100}) $ 8 * q - 8 * q^13 - 24 * q^25 - 8 * q^37 - 8 * q^49 + 56 * q^73 + 40 * q^85 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$5$	2.50359i	1.11964i	0.828615	+	0.559819i	$0.189129\pi$
$5$	2.50359i	1.11964i	−0.828615	+	0.559819i	$0.810871\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.50359	−0.754860	−0.377430	−	0.926038i	$-0.623192\pi$
$11$	−2.50359	−0.754860	−0.377430	+	0.926038i	$0.623192\pi$
$12$	0	0
$13$	0.732051	0.203034	0.101517	−	0.994834i	$-0.467630\pi$
$13$	0.732051	0.203034	0.101517	+	0.994834i	$0.467630\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.83993i	− 1.65893i	−0.558562	−	0.829463i	$-0.688647\pi$
$17$	− 6.83993i	− 1.65893i	0.558562	−	0.829463i	$-0.311353\pi$
$18$	0	0
$19$	− 8.46410i	− 1.94180i	−0.239489	−	0.970899i	$-0.576980\pi$
$19$	− 8.46410i	− 1.94180i	0.239489	−	0.970899i	$-0.423020\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.50359	−0.522034	−0.261017	−	0.965334i	$-0.584058\pi$
$23$	−2.50359	−0.522034	−0.261017	+	0.965334i	$0.584058\pi$
$24$	0	0
$25$	−1.26795	−0.253590
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.83993i	1.27014i	0.772453	+	0.635071i	$0.219029\pi$
$29$	6.83993i	1.27014i	−0.772453	+	0.635071i	$0.780971\pi$
$30$	0	0
$31$	0.464102i	0.0833551i	0.999131	+	0.0416776i	$0.0132702\pi$
$31$	0.464102i	0.0833551i	−0.999131	+	0.0416776i	$0.986730\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.50359	−0.423183
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 11.1763i	− 1.74544i	−0.488221	−	0.872720i	$-0.662354\pi$
$41$	− 11.1763i	− 1.74544i	0.488221	−	0.872720i	$-0.337646\pi$
$42$	0	0
$43$	− 6.73205i	− 1.02663i	−0.858201	−	0.513314i	$-0.828418\pi$
$43$	− 6.73205i	− 1.02663i	0.858201	−	0.513314i	$-0.171582\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.8471	−1.72808	−0.864039	−	0.503425i	$-0.832073\pi$
$47$	−11.8471	−1.72808	−0.864039	+	0.503425i	$0.832073\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 5.00717i	− 0.687788i	−0.939008	−	0.343894i	$-0.888254\pi$
$53$	− 5.00717i	− 0.687788i	0.939008	−	0.343894i	$-0.111746\pi$
$54$	0	0
$55$	− 6.26795i	− 0.845170i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.83993	0.890483	0.445241	−	0.895411i	$-0.353118\pi$
$59$	6.83993	0.890483	0.445241	+	0.895411i	$0.353118\pi$
$60$	0	0
$61$	10.3923	1.33060	0.665299	−	0.746577i	$-0.268304\pi$
$61$	10.3923	1.33060	0.665299	+	0.746577i	$0.268304\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.83275i	0.227325i
$66$	0	0
$67$	− 1.26795i	− 0.154905i	−0.996996	−	0.0774523i	$-0.975321\pi$
$67$	− 1.26795i	− 0.154905i	0.996996	−	0.0774523i	$-0.0246786\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.670834	0.0796134	0.0398067	−	0.999207i	$-0.487326\pi$
$71$	0.670834	0.0796134	0.0398067	+	0.999207i	$0.487326\pi$
$72$	0	0
$73$	1.80385	0.211124	0.105562	−	0.994413i	$-0.466336\pi$
$73$	1.80385	0.211124	0.105562	+	0.994413i	$0.466336\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.50359i	− 0.285310i
$78$	0	0
$79$	− 1.26795i	− 0.142655i	−0.997453	−	0.0713277i	$-0.977276\pi$
$79$	− 1.26795i	− 0.142655i	0.997453	−	0.0713277i	$-0.0227236\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.83993	0.750780	0.375390	−	0.926867i	$-0.377509\pi$
$83$	6.83993	0.750780	0.375390	+	0.926867i	$0.377509\pi$
$84$	0	0
$85$	17.1244	1.85740
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.5179i	1.32690i	0.748221	+	0.663449i	$0.230908\pi$
$89$	12.5179i	1.32690i	−0.748221	+	0.663449i	$0.769092\pi$
$90$	0	0
$91$	0.732051i	0.0767398i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	21.1906	2.17411
$96$	0	0
$97$	18.3923	1.86746	0.933728	−	0.357984i	$-0.116536\pi$
$97$	18.3923	1.86746	0.933728	+	0.357984i	$0.116536\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.5054i	1.04533i	0.852538	+	0.522665i	$0.175062\pi$
$101$	10.5054i	1.04533i	−0.852538	+	0.522665i	$0.824938\pi$
$102$	0	0
$103$	− 14.4641i	− 1.42519i	−0.701575	−	0.712595i	$-0.747520\pi$
$103$	− 14.4641i	− 1.42519i	0.701575	−	0.712595i	$-0.252480\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.83993	0.661241	0.330620	−	0.943764i	$-0.392742\pi$
$107$	6.83993	0.661241	0.330620	+	0.943764i	$0.392742\pi$
$108$	0	0
$109$	−8.26795	−0.791926	−0.395963	−	0.918266i	$-0.629589\pi$
$109$	−8.26795	−0.791926	−0.395963	+	0.918266i	$0.629589\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.0143i	0.942071i	0.882114	+	0.471035i	$0.156120\pi$
$113$	10.0143i	0.942071i	−0.882114	+	0.471035i	$0.843880\pi$
$114$	0	0
$115$	− 6.26795i	− 0.584489i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.83993	0.627015
$120$	0	0
$121$	−4.73205	−0.430186
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.34351i	0.835709i
$126$	0	0
$127$	− 5.80385i	− 0.515008i	−0.966277	−	0.257504i	$-0.917100\pi$
$127$	− 5.80385i	− 0.515008i	0.966277	−	0.257504i	$-0.0829001\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.5054	−0.917864	−0.458932	−	0.888471i	$-0.651768\pi$
$131$	−10.5054	−0.917864	−0.458932	+	0.888471i	$0.651768\pi$
$132$	0	0
$133$	8.46410	0.733931
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 16.8543i	− 1.43996i	−0.693996	−	0.719979i	$-0.744151\pi$
$137$	− 16.8543i	− 1.43996i	0.693996	−	0.719979i	$-0.255849\pi$
$138$	0	0
$139$	− 5.46410i	− 0.463459i	−0.972780	−	0.231730i	$-0.925562\pi$
$139$	− 5.46410i	− 0.463459i	0.972780	−	0.231730i	$-0.0744384\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.83275	−0.153263
$144$	0	0
$145$	−17.1244	−1.42210
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 10.0143i	− 0.820407i	−0.911994	−	0.410204i	$-0.865458\pi$
$149$	− 10.0143i	− 0.820407i	0.911994	−	0.410204i	$-0.134542\pi$
$150$	0	0
$151$	− 19.3205i	− 1.57228i	−0.618048	−	0.786140i	$-0.712076\pi$
$151$	− 19.3205i	− 1.57228i	0.618048	−	0.786140i	$-0.287924\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.16192	−0.0933276
$156$	0	0
$157$	18.5885	1.48352	0.741760	−	0.670665i	$-0.233991\pi$
$157$	18.5885	1.48352	0.741760	+	0.670665i	$0.233991\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2.50359i	− 0.197310i
$162$	0	0
$163$	− 20.7321i	− 1.62386i	−0.583755	−	0.811930i	$-0.698417\pi$
$163$	− 20.7321i	− 1.62386i	0.583755	−	0.811930i	$-0.301583\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.3525	−1.72969	−0.864846	−	0.502038i	$-0.832584\pi$
$167$	−22.3525	−1.72969	−0.864846	+	0.502038i	$0.832584\pi$
$168$	0	0
$169$	−12.4641	−0.958777
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.6852i	0.812379i	0.913789	+	0.406190i	$0.133143\pi$
$173$	10.6852i	0.812379i	−0.913789	+	0.406190i	$0.866857\pi$
$174$	0	0
$175$	− 1.26795i	− 0.0958479i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.83275	0.136986	0.0684932	−	0.997652i	$-0.478181\pi$
$179$	1.83275	0.136986	0.0684932	+	0.997652i	$0.478181\pi$
$180$	0	0
$181$	−4.39230	−0.326477	−0.163239	−	0.986587i	$-0.552194\pi$
$181$	−4.39230	−0.326477	−0.163239	+	0.986587i	$0.552194\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.50359i	− 0.184067i
$186$	0	0
$187$	17.1244i	1.25226i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.16192	0.0840735	0.0420368	−	0.999116i	$-0.486615\pi$
$191$	1.16192	0.0840735	0.0420368	+	0.999116i	$0.486615\pi$
$192$	0	0
$193$	−1.60770	−0.115724	−0.0578622	−	0.998325i	$-0.518428\pi$
$193$	−1.60770	−0.115724	−0.0578622	+	0.998325i	$0.518428\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.67268i	0.617903i	0.951078	+	0.308951i	$0.0999781\pi$
$197$	8.67268i	0.617903i	−0.951078	+	0.308951i	$0.900022\pi$
$198$	0	0
$199$	− 11.7321i	− 0.831663i	−0.909442	−	0.415832i	$-0.863491\pi$
$199$	− 11.7321i	− 0.831663i	0.909442	−	0.415832i	$-0.136509\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.83993	−0.480069
$204$	0	0
$205$	27.9808	1.95426
$206$	0	0
$207$	0	0
$208$	0	0
$209$	21.1906i	1.46579i
$210$	0	0
$211$	8.00000i	0.550743i	0.961338	+	0.275371i	$0.0888008\pi$
$211$	8.00000i	0.550743i	−0.961338	+	0.275371i	$0.911199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.8543	1.14945
$216$	0	0
$217$	−0.464102	−0.0315053
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 5.00717i	− 0.336819i
$222$	0	0
$223$	17.5885i	1.17781i	0.808202	+	0.588905i	$0.200441\pi$
$223$	17.5885i	1.17781i	−0.808202	+	0.588905i	$0.799559\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.6799	0.907964	0.453982	−	0.891011i	$-0.350003\pi$
$227$	13.6799	0.907964	0.453982	+	0.891011i	$0.350003\pi$
$228$	0	0
$229$	−3.07180	−0.202990	−0.101495	−	0.994836i	$-0.532363\pi$
$229$	−3.07180	−0.202990	−0.101495	+	0.994836i	$0.532363\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 27.3597i	− 1.79239i	−0.443656	−	0.896197i	$-0.646319\pi$
$233$	− 27.3597i	− 1.79239i	0.443656	−	0.896197i	$-0.353681\pi$
$234$	0	0
$235$	− 29.6603i	− 1.93482i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.1781	1.24053	0.620265	−	0.784393i	$-0.287025\pi$
$239$	19.1781	1.24053	0.620265	+	0.784393i	$0.287025\pi$
$240$	0	0
$241$	12.1962	0.785623	0.392812	−	0.919619i	$-0.371502\pi$
$241$	12.1962	0.785623	0.392812	+	0.919619i	$0.371502\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.50359i	− 0.159948i
$246$	0	0
$247$	− 6.19615i	− 0.394252i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.34884	−0.400735	−0.200368	−	0.979721i	$-0.564214\pi$
$251$	−6.34884	−0.400735	−0.200368	+	0.979721i	$0.564214\pi$
$252$	0	0
$253$	6.26795	0.394063
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.34351i	0.582832i	0.956596	+	0.291416i	$0.0941265\pi$
$257$	9.34351i	0.582832i	−0.956596	+	0.291416i	$0.905874\pi$
$258$	0	0
$259$	− 1.00000i	− 0.0621370i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−21.1906	−1.30667	−0.653335	−	0.757069i	$-0.726631\pi$
$263$	−21.1906	−1.30667	−0.653335	+	0.757069i	$0.726631\pi$
$264$	0	0
$265$	12.5359	0.770074
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 23.0234i	− 1.40376i	−0.712295	−	0.701880i	$-0.752344\pi$
$269$	− 23.0234i	− 1.40376i	0.712295	−	0.701880i	$-0.247656\pi$
$270$	0	0
$271$	− 4.92820i	− 0.299367i	−0.988734	−	0.149684i	$-0.952175\pi$
$271$	− 4.92820i	− 0.299367i	0.988734	−	0.149684i	$-0.0478255\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.17442	0.191425
$276$	0	0
$277$	−2.60770	−0.156681	−0.0783406	−	0.996927i	$-0.524962\pi$
$277$	−2.60770	−0.156681	−0.0783406	+	0.996927i	$0.524962\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 24.1853i	− 1.44277i	−0.692532	−	0.721387i	$-0.743505\pi$
$281$	− 24.1853i	− 1.44277i	0.692532	−	0.721387i	$-0.256495\pi$
$282$	0	0
$283$	10.3923i	0.617758i	0.951101	+	0.308879i	$0.0999539\pi$
$283$	10.3923i	0.617758i	−0.951101	+	0.308879i	$0.900046\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.1763	0.659714
$288$	0	0
$289$	−29.7846	−1.75204
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.5126i	0.906256i	0.891445	+	0.453128i	$0.149692\pi$
$293$	15.5126i	0.906256i	−0.891445	+	0.453128i	$0.850308\pi$
$294$	0	0
$295$	17.1244i	0.997019i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.83275	−0.105991
$300$	0	0
$301$	6.73205	0.388029
$302$	0	0
$303$	0	0
$304$	0	0
$305$	26.0180i	1.48979i
$306$	0	0
$307$	1.39230i	0.0794630i	0.999210	+	0.0397315i	$0.0126503\pi$
$307$	1.39230i	0.0794630i	−0.999210	+	0.0397315i	$0.987350\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.1959	−1.03180	−0.515899	−	0.856650i	$-0.672542\pi$
$311$	−18.1959	−1.03180	−0.515899	+	0.856650i	$0.672542\pi$
$312$	0	0
$313$	−19.2679	−1.08909	−0.544544	−	0.838732i	$-0.683297\pi$
$313$	−19.2679	−1.08909	−0.544544	+	0.838732i	$0.683297\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 20.5198i	− 1.15251i	−0.817271	−	0.576253i	$-0.804514\pi$
$317$	− 20.5198i	− 1.15251i	0.817271	−	0.576253i	$-0.195486\pi$
$318$	0	0
$319$	− 17.1244i	− 0.958780i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−57.8938	−3.22130
$324$	0	0
$325$	−0.928203	−0.0514875
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 11.8471i	− 0.653152i
$330$	0	0
$331$	− 12.7846i	− 0.702706i	−0.936243	−	0.351353i	$-0.885722\pi$
$331$	− 12.7846i	− 0.702706i	0.936243	−	0.351353i	$-0.114278\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.17442	0.173437
$336$	0	0
$337$	8.60770	0.468891	0.234446	−	0.972129i	$-0.424673\pi$
$337$	8.60770	0.468891	0.234446	+	0.972129i	$0.424673\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.16192i	− 0.0629214i
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.01250	−0.108037	−0.0540184	−	0.998540i	$-0.517203\pi$
$347$	−2.01250	−0.108037	−0.0540184	+	0.998540i	$0.517203\pi$
$348$	0	0
$349$	21.3205	1.14126	0.570630	−	0.821207i	$-0.306699\pi$
$349$	21.3205	1.14126	0.570630	+	0.821207i	$0.306699\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 11.1763i	− 0.594853i	−0.954745	−	0.297426i	$-0.903872\pi$
$353$	− 11.1763i	− 0.594853i	0.954745	−	0.297426i	$-0.0961283\pi$
$354$	0	0
$355$	1.67949i	0.0891382i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.5126	0.818724	0.409362	−	0.912372i	$-0.365751\pi$
$359$	15.5126	0.818724	0.409362	+	0.912372i	$0.365751\pi$
$360$	0	0
$361$	−52.6410	−2.77058
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.51609i	0.236383i
$366$	0	0
$367$	13.3397i	0.696329i	0.937433	+	0.348165i	$0.113195\pi$
$367$	13.3397i	0.696329i	−0.937433	+	0.348165i	$0.886805\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.00717	0.259960
$372$	0	0
$373$	4.66025	0.241299	0.120649	−	0.992695i	$-0.461502\pi$
$373$	4.66025	0.241299	0.120649	+	0.992695i	$0.461502\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.00717i	0.257883i
$378$	0	0
$379$	− 1.26795i	− 0.0651302i	−0.999470	−	0.0325651i	$-0.989632\pi$
$379$	− 1.26795i	− 0.0651302i	0.999470	−	0.0325651i	$-0.0103676\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−27.3597	−1.39802	−0.699008	−	0.715114i	$-0.746375\pi$
$383$	−27.3597	−1.39802	−0.699008	+	0.715114i	$0.746375\pi$
$384$	0	0
$385$	6.26795	0.319444
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 12.3382i	− 0.625571i	−0.949824	−	0.312785i	$-0.898738\pi$
$389$	− 12.3382i	− 0.625571i	0.949824	−	0.312785i	$-0.101262\pi$
$390$	0	0
$391$	17.1244i	0.866016i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.17442	0.159722
$396$	0	0
$397$	−8.92820	−0.448094	−0.224047	−	0.974578i	$-0.571927\pi$
$397$	−8.92820	−0.448094	−0.224047	+	0.974578i	$0.571927\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.1781i	0.957709i	0.877894	+	0.478855i	$0.158948\pi$
$401$	19.1781i	0.957709i	−0.877894	+	0.478855i	$0.841052\pi$
$402$	0	0
$403$	0.339746i	0.0169240i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.50359	0.124098
$408$	0	0
$409$	13.2679	0.656058	0.328029	−	0.944668i	$-0.393616\pi$
$409$	13.2679	0.656058	0.328029	+	0.944668i	$0.393616\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.83993i	0.336571i
$414$	0	0
$415$	17.1244i	0.840602i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.0143	0.489233	0.244616	−	0.969620i	$-0.421338\pi$
$419$	10.0143	0.489233	0.244616	+	0.969620i	$0.421338\pi$
$420$	0	0
$421$	30.8564	1.50385	0.751925	−	0.659249i	$-0.229126\pi$
$421$	30.8564	1.50385	0.751925	+	0.659249i	$0.229126\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.67268i	0.420687i
$426$	0	0
$427$	10.3923i	0.502919i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.16909	−0.297155	−0.148577	−	0.988901i	$-0.547469\pi$
$431$	−6.16909	−0.297155	−0.148577	+	0.988901i	$0.547469\pi$
$432$	0	0
$433$	−2.58846	−0.124393	−0.0621967	−	0.998064i	$-0.519811\pi$
$433$	−2.58846	−0.124393	−0.0621967	+	0.998064i	$0.519811\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21.1906i	1.01368i
$438$	0	0
$439$	9.46410i	0.451697i	0.974162	+	0.225848i	$0.0725154\pi$
$439$	9.46410i	0.451697i	−0.974162	+	0.225848i	$0.927485\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.16192	−0.0552044	−0.0276022	−	0.999619i	$-0.508787\pi$
$443$	−1.16192	−0.0552044	−0.0276022	+	0.999619i	$0.508787\pi$
$444$	0	0
$445$	−31.3397	−1.48565
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 15.0215i	− 0.708910i	−0.935073	−	0.354455i	$-0.884666\pi$
$449$	− 15.0215i	− 0.708910i	0.935073	−	0.354455i	$-0.115334\pi$
$450$	0	0
$451$	27.9808i	1.31756i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.83275	−0.0859208
$456$	0	0
$457$	−13.3923	−0.626466	−0.313233	−	0.949676i	$-0.601412\pi$
$457$	−13.3923	−0.626466	−0.313233	+	0.949676i	$0.601412\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 31.6961i	− 1.47623i	−0.674674	−	0.738116i	$-0.735716\pi$
$461$	− 31.6961i	− 1.47623i	0.674674	−	0.738116i	$-0.264284\pi$
$462$	0	0
$463$	− 4.53590i	− 0.210801i	−0.994430	−	0.105401i	$-0.966388\pi$
$463$	− 4.53590i	− 0.210801i	0.994430	−	0.105401i	$-0.0336125\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.1888	−0.610304	−0.305152	−	0.952304i	$-0.598707\pi$
$467$	−13.1888	−0.610304	−0.305152	+	0.952304i	$0.598707\pi$
$468$	0	0
$469$	1.26795	0.0585485
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.8543i	0.774960i
$474$	0	0
$475$	10.7321i	0.492420i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−27.3597	−1.25010	−0.625049	−	0.780586i	$-0.714921\pi$
$479$	−27.3597	−1.25010	−0.625049	+	0.780586i	$0.714921\pi$
$480$	0	0
$481$	−0.732051	−0.0333786
$482$	0	0
$483$	0	0
$484$	0	0
$485$	46.0467i	2.09087i
$486$	0	0
$487$	17.8038i	0.806769i	0.915030	+	0.403385i	$0.132166\pi$
$487$	17.8038i	0.806769i	−0.915030	+	0.403385i	$0.867834\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	35.3616	1.59584	0.797922	−	0.602760i	$-0.205932\pi$
$491$	35.3616	1.59584	0.797922	+	0.602760i	$0.205932\pi$
$492$	0	0
$493$	46.7846	2.10707
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.670834i	0.0300910i
$498$	0	0
$499$	32.7321i	1.46529i	0.680612	+	0.732644i	$0.261714\pi$
$499$	32.7321i	1.46529i	−0.680612	+	0.732644i	$0.738286\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.1959	0.811317	0.405659	−	0.914025i	$-0.367042\pi$
$503$	18.1959	0.811317	0.405659	+	0.914025i	$0.367042\pi$
$504$	0	0
$505$	−26.3013	−1.17039
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 13.1888i	− 0.584582i	−0.956329	−	0.292291i	$-0.905582\pi$
$509$	− 13.1888i	− 0.584582i	0.956329	−	0.292291i	$-0.0944176\pi$
$510$	0	0
$511$	1.80385i	0.0797975i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	36.2121	1.59570
$516$	0	0
$517$	29.6603	1.30446
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.0162i	0.789304i	0.918831	+	0.394652i	$0.129135\pi$
$521$	18.0162i	0.789304i	−0.918831	+	0.394652i	$0.870865\pi$
$522$	0	0
$523$	− 39.0526i	− 1.70765i	−0.520561	−	0.853825i	$-0.674277\pi$
$523$	− 39.0526i	− 1.70765i	0.520561	−	0.853825i	$-0.325723\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.17442	0.138280
$528$	0	0
$529$	−16.7321	−0.727480
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 8.18160i	− 0.354384i
$534$	0	0
$535$	17.1244i	0.740350i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.50359	0.107837
$540$	0	0
$541$	−17.0000	−0.730887	−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000	−0.730887	−0.365444	+	0.930834i	$0.619083\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 20.6995i	− 0.886671i
$546$	0	0
$547$	28.7846i	1.23074i	0.788238	+	0.615371i	$0.210994\pi$
$547$	28.7846i	1.23074i	−0.788238	+	0.615371i	$0.789006\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	57.8938	2.46636
$552$	0	0
$553$	1.26795	0.0539187
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 36.0324i	− 1.52674i	−0.645961	−	0.763371i	$-0.723543\pi$
$557$	− 36.0324i	− 1.52674i	0.645961	−	0.763371i	$-0.276457\pi$
$558$	0	0
$559$	− 4.92820i	− 0.208441i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.8651	1.59583	0.797913	−	0.602773i	$-0.205938\pi$
$563$	37.8651	1.59583	0.797913	+	0.602773i	$0.205938\pi$
$564$	0	0
$565$	−25.0718	−1.05478
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.3669i	1.35689i	0.734651	+	0.678445i	$0.237346\pi$
$569$	32.3669i	1.35689i	−0.734651	+	0.678445i	$0.762654\pi$
$570$	0	0
$571$	40.0000i	1.67395i	0.547243	+	0.836974i	$0.315677\pi$
$571$	40.0000i	1.67395i	−0.547243	+	0.836974i	$0.684323\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.17442	0.132383
$576$	0	0
$577$	−35.1769	−1.46443	−0.732217	−	0.681071i	$-0.761514\pi$
$577$	−35.1769	−1.46443	−0.732217	+	0.681071i	$0.761514\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.83993i	0.283768i
$582$	0	0
$583$	12.5359i	0.519184i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.0252	1.28055	0.640274	−	0.768147i	$-0.278821\pi$
$587$	31.0252	1.28055	0.640274	+	0.768147i	$0.278821\pi$
$588$	0	0
$589$	3.92820	0.161859
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.00185i	0.328596i	0.986411	+	0.164298i	$0.0525359\pi$
$593$	8.00185i	0.328596i	−0.986411	+	0.164298i	$0.947464\pi$
$594$	0	0
$595$	17.1244i	0.702030i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.8561	−1.01559	−0.507797	−	0.861477i	$-0.669540\pi$
$599$	−24.8561	−1.01559	−0.507797	+	0.861477i	$0.669540\pi$
$600$	0	0
$601$	−25.3731	−1.03499	−0.517494	−	0.855687i	$-0.673135\pi$
$601$	−25.3731	−1.03499	−0.517494	+	0.855687i	$0.673135\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 11.8471i	− 0.481653i
$606$	0	0
$607$	− 46.1051i	− 1.87135i	−0.352864	−	0.935675i	$-0.614792\pi$
$607$	− 46.1051i	− 1.87135i	0.352864	−	0.935675i	$-0.385208\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.67268	−0.350859
$612$	0	0
$613$	−41.5885	−1.67974	−0.839871	−	0.542786i	$-0.817369\pi$
$613$	−41.5885	−1.67974	−0.839871	+	0.542786i	$0.817369\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.1996i	1.37683i	0.725319	+	0.688413i	$0.241692\pi$
$617$	34.1996i	1.37683i	−0.725319	+	0.688413i	$0.758308\pi$
$618$	0	0
$619$	21.1962i	0.851945i	0.904736	+	0.425973i	$0.140068\pi$
$619$	21.1962i	0.851945i	−0.904736	+	0.425973i	$0.859932\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.5179	−0.501521
$624$	0	0
$625$	−29.7321	−1.18928
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.83993i	0.272726i
$630$	0	0
$631$	− 4.05256i	− 0.161330i	−0.996741	−	0.0806649i	$-0.974296\pi$
$631$	− 4.05256i	− 0.161330i	0.996741	−	0.0806649i	$-0.0257044\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.5304	0.576623
$636$	0	0
$637$	−0.732051	−0.0290049
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.66551i	0.144779i	0.997376	+	0.0723894i	$0.0230624\pi$
$641$	3.66551i	0.144779i	−0.997376	+	0.0723894i	$0.976938\pi$
$642$	0	0
$643$	26.4641i	1.04364i	0.853055	+	0.521821i	$0.174747\pi$
$643$	26.4641i	1.04364i	−0.853055	+	0.521821i	$0.825253\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.5054	−0.413011	−0.206506	−	0.978445i	$-0.566209\pi$
$647$	−10.5054	−0.413011	−0.206506	+	0.978445i	$0.566209\pi$
$648$	0	0
$649$	−17.1244	−0.672190
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 32.8580i	− 1.28583i	−0.765937	−	0.642916i	$-0.777724\pi$
$653$	− 32.8580i	− 1.28583i	0.765937	−	0.642916i	$-0.222276\pi$
$654$	0	0
$655$	− 26.3013i	− 1.02768i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−43.5432	−1.69620	−0.848100	−	0.529836i	$-0.822254\pi$
$659$	−43.5432	−1.69620	−0.848100	+	0.529836i	$0.822254\pi$
$660$	0	0
$661$	−16.9808	−0.660475	−0.330238	−	0.943898i	$-0.607129\pi$
$661$	−16.9808	−0.660475	−0.330238	+	0.943898i	$0.607129\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	21.1906i	0.821737i
$666$	0	0
$667$	− 17.1244i	− 0.663058i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−26.0180	−1.00442
$672$	0	0
$673$	44.6410	1.72078	0.860392	−	0.509632i	$-0.170219\pi$
$673$	44.6410	1.72078	0.860392	+	0.509632i	$0.170219\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.16192i	0.0446562i	0.999751	+	0.0223281i	$0.00710784\pi$
$677$	1.16192i	0.0446562i	−0.999751	+	0.0223281i	$0.992892\pi$
$678$	0	0
$679$	18.3923i	0.705832i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.50359	0.0957971	0.0478986	−	0.998852i	$-0.484748\pi$
$683$	2.50359	0.0957971	0.0478986	+	0.998852i	$0.484748\pi$
$684$	0	0
$685$	42.1962	1.61223
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 3.66551i	− 0.139645i
$690$	0	0
$691$	30.2487i	1.15072i	0.817902	+	0.575358i	$0.195137\pi$
$691$	30.2487i	1.15072i	−0.817902	+	0.575358i	$0.804863\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.6799	0.518906
$696$	0	0
$697$	−76.4449	−2.89556
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 21.8615i	− 0.825696i	−0.910800	−	0.412848i	$-0.864534\pi$
$701$	− 21.8615i	− 0.825696i	0.910800	−	0.412848i	$-0.135466\pi$
$702$	0	0
$703$	8.46410i	0.319230i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.5054	−0.395098
$708$	0	0
$709$	−32.9090	−1.23592	−0.617961	−	0.786209i	$-0.712041\pi$
$709$	−32.9090	−1.23592	−0.617961	+	0.786209i	$0.712041\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1.16192i	− 0.0435142i
$714$	0	0
$715$	− 4.58846i	− 0.171599i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.16377	0.341751	0.170875	−	0.985293i	$-0.445340\pi$
$719$	9.16377	0.341751	0.170875	+	0.985293i	$0.445340\pi$
$720$	0	0
$721$	14.4641	0.538671
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 8.67268i	− 0.322095i
$726$	0	0
$727$	43.7128i	1.62122i	0.585587	+	0.810609i	$0.300864\pi$
$727$	43.7128i	1.62122i	−0.585587	+	0.810609i	$0.699136\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−46.0467	−1.70310
$732$	0	0
$733$	38.9808	1.43979	0.719894	−	0.694084i	$-0.244191\pi$
$733$	38.9808	1.43979	0.719894	+	0.694084i	$0.244191\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.17442i	0.116931i
$738$	0	0
$739$	− 38.2487i	− 1.40700i	−0.710694	−	0.703501i	$-0.751619\pi$
$739$	− 38.2487i	− 1.40700i	0.710694	−	0.703501i	$-0.248381\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.3400	0.746203	0.373102	−	0.927790i	$-0.378294\pi$
$743$	20.3400	0.746203	0.373102	+	0.927790i	$0.378294\pi$
$744$	0	0
$745$	25.0718	0.918560
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.83993i	0.249926i
$750$	0	0
$751$	34.8372i	1.27123i	0.772008	+	0.635613i	$0.219253\pi$
$751$	34.8372i	1.27123i	−0.772008	+	0.635613i	$0.780747\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	48.3706	1.76039
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 45.5557i	− 1.65139i	−0.564116	−	0.825696i	$-0.690783\pi$
$761$	− 45.5557i	− 1.65139i	0.564116	−	0.825696i	$-0.309217\pi$
$762$	0	0
$763$	− 8.26795i	− 0.299320i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.00717	0.180799
$768$	0	0
$769$	28.1962	1.01678	0.508390	−	0.861127i	$-0.330241\pi$
$769$	28.1962	1.01678	0.508390	+	0.861127i	$0.330241\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 14.8418i	− 0.533822i	−0.963721	−	0.266911i	$-0.913997\pi$
$773$	− 14.8418i	− 0.533822i	0.963721	−	0.266911i	$-0.0860029\pi$
$774$	0	0
$775$	− 0.588457i	− 0.0211380i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−94.5971	−3.38929
$780$	0	0
$781$	−1.67949	−0.0600969
$782$	0	0
$783$	0	0
$784$	0	0
$785$	46.5378i	1.66101i
$786$	0	0
$787$	47.3205i	1.68679i	0.537291	+	0.843397i	$0.319448\pi$
$787$	47.3205i	1.68679i	−0.537291	+	0.843397i	$0.680552\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−10.0143	−0.356069
$792$	0	0
$793$	7.60770	0.270157
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 11.1763i	− 0.395884i	−0.980214	−	0.197942i	$-0.936574\pi$
$797$	− 11.1763i	− 0.395884i	0.980214	−	0.197942i	$-0.0634257\pi$
$798$	0	0
$799$	81.0333i	2.86675i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.51609	−0.159369
$804$	0	0
$805$	6.26795	0.220916
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.15659i	− 0.146138i	−0.997327	−	0.0730690i	$-0.976721\pi$
$809$	− 4.15659i	− 0.146138i	0.997327	−	0.0730690i	$-0.0232793\pi$
$810$	0	0
$811$	29.1962i	1.02522i	0.858623	+	0.512608i	$0.171320\pi$
$811$	29.1962i	1.02522i	−0.858623	+	0.512608i	$0.828680\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	51.9045	1.81814
$816$	0	0
$817$	−56.9808	−1.99350
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.8615i	0.762970i	0.924375	+	0.381485i	$0.124587\pi$
$821$	21.8615i	0.762970i	−0.924375	+	0.381485i	$0.875413\pi$
$822$	0	0
$823$	− 30.0000i	− 1.04573i	−0.852414	−	0.522867i	$-0.824862\pi$
$823$	− 30.0000i	− 1.04573i	0.852414	−	0.522867i	$-0.175138\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.7067	0.893910	0.446955	−	0.894556i	$-0.352508\pi$
$827$	25.7067	0.893910	0.446955	+	0.894556i	$0.352508\pi$
$828$	0	0
$829$	37.6603	1.30799	0.653997	−	0.756497i	$-0.273091\pi$
$829$	37.6603	1.30799	0.653997	+	0.756497i	$0.273091\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.83993i	0.236989i
$834$	0	0
$835$	− 55.9615i	− 1.93663i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−54.2283	−1.87217	−0.936085	−	0.351774i	$-0.885579\pi$
$839$	−54.2283	−1.87217	−0.936085	+	0.351774i	$0.885579\pi$
$840$	0	0
$841$	−17.7846	−0.613262
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 31.2050i	− 1.07348i
$846$	0	0
$847$	− 4.73205i	− 0.162595i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.50359	0.0858219
$852$	0	0
$853$	21.3205	0.730000	0.365000	−	0.931007i	$-0.381069\pi$
$853$	21.3205	0.730000	0.365000	+	0.931007i	$0.381069\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.5179i	0.427605i	0.976877	+	0.213802i	$0.0685848\pi$
$857$	12.5179i	0.427605i	−0.976877	+	0.213802i	$0.931415\pi$
$858$	0	0
$859$	24.0718i	0.821319i	0.911789	+	0.410660i	$0.134702\pi$
$859$	24.0718i	0.821319i	−0.911789	+	0.410660i	$0.865298\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19.1781	0.652831	0.326415	−	0.945226i	$-0.394159\pi$
$863$	19.1781	0.652831	0.326415	+	0.945226i	$0.394159\pi$
$864$	0	0
$865$	−26.7513	−0.909571
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.17442i	0.107685i
$870$	0	0
$871$	− 0.928203i	− 0.0314510i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.34351	−0.315868
$876$	0	0
$877$	10.9282	0.369019	0.184510	−	0.982831i	$-0.440930\pi$
$877$	10.9282	0.369019	0.184510	+	0.982831i	$0.440930\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 33.5288i	− 1.12961i	−0.825223	−	0.564807i	$-0.808951\pi$
$881$	− 33.5288i	− 1.12961i	0.825223	−	0.564807i	$-0.191049\pi$
$882$	0	0
$883$	− 18.0000i	− 0.605748i	−0.953031	−	0.302874i	$-0.902054\pi$
$883$	− 18.0000i	− 0.605748i	0.953031	−	0.302874i	$-0.0979462\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.5376	−0.656009	−0.328004	−	0.944676i	$-0.606376\pi$
$887$	−19.5376	−0.656009	−0.328004	+	0.944676i	$0.606376\pi$
$888$	0	0
$889$	5.80385	0.194655
$890$	0	0
$891$	0	0
$892$	0	0
$893$	100.275i	3.35558i
$894$	0	0
$895$	4.58846i	0.153375i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.17442	−0.105873
$900$	0	0
$901$	−34.2487	−1.14099
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 10.9965i	− 0.365537i
$906$	0	0
$907$	− 27.5692i	− 0.915421i	−0.889101	−	0.457710i	$-0.848670\pi$
$907$	− 27.5692i	− 0.915421i	0.889101	−	0.457710i	$-0.151330\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−13.1888	−0.436963	−0.218482	−	0.975841i	$-0.570110\pi$
$911$	−13.1888	−0.436963	−0.218482	+	0.975841i	$0.570110\pi$
$912$	0	0
$913$	−17.1244	−0.566733
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 10.5054i	− 0.346920i
$918$	0	0
$919$	56.2487i	1.85547i	0.373235	+	0.927737i	$0.378249\pi$
$919$	56.2487i	1.85547i	−0.373235	+	0.927737i	$0.621751\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.491085	0.0161643
$924$	0	0
$925$	1.26795	0.0416899
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 49.2212i	− 1.61489i	−0.589940	−	0.807447i	$-0.700849\pi$
$929$	− 49.2212i	− 1.61489i	0.589940	−	0.807447i	$-0.299151\pi$
$930$	0	0
$931$	8.46410i	0.277400i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−42.8723	−1.40207
$936$	0	0
$937$	−35.1769	−1.14918	−0.574590	−	0.818442i	$-0.694838\pi$
$937$	−35.1769	−1.14918	−0.574590	+	0.818442i	$0.694838\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	54.8992i	1.78966i	0.446405	+	0.894831i	$0.352704\pi$
$941$	54.8992i	1.78966i	−0.446405	+	0.894831i	$0.647296\pi$
$942$	0	0
$943$	27.9808i	0.911179i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10.6852	−0.347222	−0.173611	−	0.984814i	$-0.555544\pi$
$947$	−10.6852	−0.347222	−0.173611	+	0.984814i	$0.555544\pi$
$948$	0	0
$949$	1.32051	0.0428655
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 35.5413i	− 1.15130i	−0.817698	−	0.575648i	$-0.804750\pi$
$953$	− 35.5413i	− 1.15130i	0.817698	−	0.575648i	$-0.195250\pi$
$954$	0	0
$955$	2.90897i	0.0941319i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	16.8543	0.544253
$960$	0	0
$961$	30.7846	0.993052
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 4.02501i	− 0.129569i
$966$	0	0
$967$	− 18.3923i	− 0.591457i	−0.955272	−	0.295728i	$-0.904438\pi$
$967$	− 18.3923i	− 0.591457i	0.955272	−	0.295728i	$-0.0955623\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	15.5126	0.497823	0.248912	−	0.968526i	$-0.419927\pi$
$971$	15.5126	0.497823	0.248912	+	0.968526i	$0.419927\pi$
$972$	0	0
$973$	5.46410	0.175171
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.68334i	0.0858475i	0.999078	+	0.0429238i	$0.0136673\pi$
$977$	2.68334i	0.0858475i	−0.999078	+	0.0429238i	$0.986333\pi$
$978$	0	0
$979$	− 31.3397i	− 1.00162i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	30.5341	0.973888	0.486944	−	0.873433i	$-0.338112\pi$
$983$	30.5341	0.973888	0.486944	+	0.873433i	$0.338112\pi$
$984$	0	0
$985$	−21.7128	−0.691828
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.8543i	0.535935i
$990$	0	0
$991$	18.7321i	0.595043i	0.954715	+	0.297522i	$0.0961600\pi$
$991$	18.7321i	0.595043i	−0.954715	+	0.297522i	$0.903840\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	29.3722	0.931162
$996$	0	0
$997$	15.0718	0.477329	0.238664	−	0.971102i	$-0.423290\pi$
$997$	15.0718	0.477329	0.238664	+	0.971102i	$0.423290\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.h.b.2591.6	yes	8
3.2	odd	2	inner	3024.2.h.b.2591.4	yes	8
4.3	odd	2	inner	3024.2.h.b.2591.5	yes	8
12.11	even	2	inner	3024.2.h.b.2591.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3024.2.h.b.2591.3	✓	8	12.11	even	2	inner
3024.2.h.b.2591.4	yes	8	3.2	odd	2	inner
3024.2.h.b.2591.5	yes	8	4.3	odd	2	inner
3024.2.h.b.2591.6	yes	8	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:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.50359i q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+2.50359i q^{5} +1.00000i q^{7} -2.50359 q^{11} +0.732051 q^{13} -6.83993i q^{17} -8.46410i q^{19} -2.50359 q^{23} -1.26795 q^{25} +6.83993i q^{29} +0.464102i q^{31} -2.50359 q^{35} -1.00000 q^{37} -11.1763i q^{41} -6.73205i q^{43} -11.8471 q^{47} -1.00000 q^{49} -5.00717i q^{53} -6.26795i q^{55} +6.83993 q^{59} +10.3923 q^{61} +1.83275i q^{65} -1.26795i q^{67} +0.670834 q^{71} +1.80385 q^{73} -2.50359i q^{77} -1.26795i q^{79} +6.83993 q^{83} +17.1244 q^{85} +12.5179i q^{89} +0.732051i q^{91} +21.1906 q^{95} +18.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{13} - 24 q^{25} - 8 q^{37} - 8 q^{49} + 56 q^{73} + 40 q^{85} + 64 q^{97}+O(q^{100}) \) 8 * q - 8 * q^13 - 24 * q^25 - 8 * q^37 - 8 * q^49 + 56 * q^73 + 40 * q^85 + 64 * q^97

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