LMFDB - Embedded newform 6336.2.d.j.3455.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6336,2,Mod(3455,6336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6336.3455"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$50.5932147207$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.46138325148368896.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 32x^{8} + 240x^{4} + 1 $ x^12 + 32x^8 + 240x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 3168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3455.12
Root			$-1.49547 + 1.49547i$ of defining polynomial
Character	$\chi$	$=$	6336.3455
Dual form			6336.2.d.j.3455.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.99093i q^{5} -2.08290i q^{7} +1.00000 q^{11} +1.57672 q^{13} +7.13784i q^{17} -3.45185i q^{19} -8.81890 q^{23} -3.94567 q^{25} -1.62388i q^{29} -1.93330i q^{31} +6.22982 q^{35} +8.09910 q^{37} +8.60593i q^{41} +0.713951i q^{43} +0.653098 q^{47} +2.66152 q^{49} +2.90040i q^{53} +2.99093i q^{55} +2.98763 q^{59} -11.2556 q^{61} +4.71585i q^{65} +4.53100i q^{67} -16.4510 q^{71} -1.72822 q^{73} -2.08290i q^{77} +3.57395i q^{79} -14.1935 q^{83} -21.3488 q^{85} +16.6429i q^{89} -3.28415i q^{91} +10.3243 q^{95} +10.5614 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{11} - 24 q^{23} - 4 q^{25} + 24 q^{35} + 16 q^{37} - 24 q^{47} - 4 q^{49} + 48 q^{59} + 8 q^{61} - 72 q^{71} - 16 q^{73} + 24 q^{83} - 40 q^{85} - 40 q^{95} - 24 q^{97}+O(q^{100}) $ 12 * q + 12 * q^11 - 24 * q^23 - 4 * q^25 + 24 * q^35 + 16 * q^37 - 24 * q^47 - 4 * q^49 + 48 * q^59 + 8 * q^61 - 72 * q^71 - 16 * q^73 + 24 * q^83 - 40 * q^85 - 40 * q^95 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times$.

$n$	$1729$	$3521$	$4159$	$4357$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$5$	2.99093i	1.33758i	0.743449	+	0.668792i	$0.233189\pi$
$5$	2.99093i	1.33758i	−0.743449	+	0.668792i	$0.766811\pi$
$6$	0	0
$7$	− 2.08290i	− 0.787263i	−0.919268	−	0.393631i	$-0.871219\pi$
$7$	− 2.08290i	− 0.787263i	0.919268	−	0.393631i	$-0.128781\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.57672	0.437303	0.218651	−	0.975803i	$-0.429834\pi$
$13$	1.57672	0.437303	0.218651	+	0.975803i	$0.429834\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.13784i	1.73118i	0.500752	+	0.865591i	$0.333057\pi$
$17$	7.13784i	1.73118i	−0.500752	+	0.865591i	$0.666943\pi$
$18$	0	0
$19$	− 3.45185i	− 0.791909i	−0.918270	−	0.395955i	$-0.870414\pi$
$19$	− 3.45185i	− 0.791909i	0.918270	−	0.395955i	$-0.129586\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.81890	−1.83887	−0.919434	−	0.393244i	$-0.871353\pi$
$23$	−8.81890	−1.83887	−0.919434	+	0.393244i	$0.871353\pi$
$24$	0	0
$25$	−3.94567	−0.789134
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1.62388i	− 0.301547i	−0.988568	−	0.150774i	$-0.951824\pi$
$29$	− 1.62388i	− 0.301547i	0.988568	−	0.150774i	$-0.0481765\pi$
$30$	0	0
$31$	− 1.93330i	− 0.347231i	−0.984814	−	0.173615i	$-0.944455\pi$
$31$	− 1.93330i	− 0.347231i	0.984814	−	0.173615i	$-0.0555450\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.22982	1.05303
$36$	0	0
$37$	8.09910	1.33148	0.665742	−	0.746182i	$-0.268115\pi$
$37$	8.09910	1.33148	0.665742	+	0.746182i	$0.268115\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.60593i	1.34402i	0.740542	+	0.672010i	$0.234569\pi$
$41$	8.60593i	1.34402i	−0.740542	+	0.672010i	$0.765431\pi$
$42$	0	0
$43$	0.713951i	0.108877i	0.998517	+	0.0544383i	$0.0173368\pi$
$43$	0.713951i	0.108877i	−0.998517	+	0.0544383i	$0.982663\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.653098	0.0952641	0.0476320	−	0.998865i	$-0.484833\pi$
$47$	0.653098	0.0952641	0.0476320	+	0.998865i	$0.484833\pi$
$48$	0	0
$49$	2.66152	0.380217
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.90040i	0.398401i	0.979959	+	0.199201i	$0.0638345\pi$
$53$	2.90040i	0.398401i	−0.979959	+	0.199201i	$0.936165\pi$
$54$	0	0
$55$	2.99093i	0.403297i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.98763	0.388956	0.194478	−	0.980907i	$-0.437699\pi$
$59$	2.98763	0.388956	0.194478	+	0.980907i	$0.437699\pi$
$60$	0	0
$61$	−11.2556	−1.44113	−0.720567	−	0.693385i	$-0.756119\pi$
$61$	−11.2556	−1.44113	−0.720567	+	0.693385i	$0.756119\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.71585i	0.584930i
$66$	0	0
$67$	4.53100i	0.553549i	0.960935	+	0.276775i	$0.0892656\pi$
$67$	4.53100i	0.553549i	−0.960935	+	0.276775i	$0.910734\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−16.4510	−1.95238	−0.976188	−	0.216925i	$-0.930397\pi$
$71$	−16.4510	−1.95238	−0.976188	+	0.216925i	$0.930397\pi$
$72$	0	0
$73$	−1.72822	−0.202273	−0.101137	−	0.994873i	$-0.532248\pi$
$73$	−1.72822	−0.202273	−0.101137	+	0.994873i	$0.532248\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.08290i	− 0.237369i
$78$	0	0
$79$	3.57395i	0.402101i	0.979581	+	0.201051i	$0.0644355\pi$
$79$	3.57395i	0.402101i	−0.979581	+	0.201051i	$0.935564\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.1935	−1.55794	−0.778972	−	0.627059i	$-0.784258\pi$
$83$	−14.1935	−1.55794	−0.778972	+	0.627059i	$0.784258\pi$
$84$	0	0
$85$	−21.3488	−2.31560
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.6429i	1.76415i	0.471112	+	0.882073i	$0.343853\pi$
$89$	16.6429i	1.76415i	−0.471112	+	0.882073i	$0.656147\pi$
$90$	0	0
$91$	− 3.28415i	− 0.344272i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.3243	1.05925
$96$	0	0
$97$	10.5614	1.07235	0.536175	−	0.844107i	$-0.319869\pi$
$97$	10.5614	1.07235	0.536175	+	0.844107i	$0.319869\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.74381i	− 0.173516i	−0.996229	−	0.0867579i	$-0.972349\pi$
$101$	− 1.74381i	− 0.173516i	0.996229	−	0.0867579i	$-0.0276507\pi$
$102$	0	0
$103$	18.3243i	1.80554i	0.430121	+	0.902771i	$0.358471\pi$
$103$	18.3243i	1.80554i	−0.430121	+	0.902771i	$0.641529\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.64717	0.835954	0.417977	−	0.908458i	$-0.362739\pi$
$107$	8.64717	0.835954	0.417977	+	0.908458i	$0.362739\pi$
$108$	0	0
$109$	−3.06777	−0.293839	−0.146919	−	0.989148i	$-0.546936\pi$
$109$	−3.06777	−0.293839	−0.146919	+	0.989148i	$0.546936\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 0.733452i	− 0.0689973i	−0.999405	−	0.0344987i	$-0.989017\pi$
$113$	− 0.733452i	− 0.0689973i	0.999405	−	0.0344987i	$-0.0109834\pi$
$114$	0	0
$115$	− 26.3767i	− 2.45964i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.8674	1.36289
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.15343i	0.282052i
$126$	0	0
$127$	− 13.1108i	− 1.16340i	−0.813404	−	0.581699i	$-0.802388\pi$
$127$	− 13.1108i	− 1.16340i	0.813404	−	0.581699i	$-0.197612\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.2395	0.981999	0.491000	−	0.871160i	$-0.336632\pi$
$131$	11.2395	0.981999	0.491000	+	0.871160i	$0.336632\pi$
$132$	0	0
$133$	−7.18987	−0.623441
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.7470i	− 1.51623i	−0.652120	−	0.758116i	$-0.726120\pi$
$137$	− 17.7470i	− 1.51623i	0.652120	−	0.758116i	$-0.273880\pi$
$138$	0	0
$139$	6.89911i	0.585175i	0.956239	+	0.292588i	$0.0945163\pi$
$139$	6.89911i	0.585175i	−0.956239	+	0.292588i	$0.905484\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.57672	0.131852
$144$	0	0
$145$	4.85692	0.403345
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 15.8174i	− 1.29581i	−0.761720	−	0.647907i	$-0.775645\pi$
$149$	− 15.8174i	− 1.29581i	0.761720	−	0.647907i	$-0.224355\pi$
$150$	0	0
$151$	7.05899i	0.574453i	0.957863	+	0.287226i	$0.0927332\pi$
$151$	7.05899i	0.574453i	−0.957863	+	0.287226i	$0.907267\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.78236	0.464451
$156$	0	0
$157$	−9.89885	−0.790014	−0.395007	−	0.918678i	$-0.629258\pi$
$157$	−9.89885	−0.790014	−0.395007	+	0.918678i	$0.629258\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.3689i	1.44767i
$162$	0	0
$163$	− 14.4078i	− 1.12851i	−0.825602	−	0.564253i	$-0.809164\pi$
$163$	− 14.4078i	− 1.12851i	0.825602	−	0.564253i	$-0.190836\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.0622	−1.62984	−0.814919	−	0.579575i	$-0.803219\pi$
$167$	−21.0622	−1.62984	−0.814919	+	0.579575i	$0.803219\pi$
$168$	0	0
$169$	−10.5140	−0.808766
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.0720i	1.52604i	0.646373	+	0.763022i	$0.276285\pi$
$173$	20.0720i	1.52604i	−0.646373	+	0.763022i	$0.723715\pi$
$174$	0	0
$175$	8.21844i	0.621256i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.1295	1.20558	0.602789	−	0.797901i	$-0.294056\pi$
$179$	16.1295	1.20558	0.602789	+	0.797901i	$0.294056\pi$
$180$	0	0
$181$	10.0964	0.750460	0.375230	−	0.926932i	$-0.377564\pi$
$181$	10.0964	0.750460	0.375230	+	0.926932i	$0.377564\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	24.2239i	1.78097i
$186$	0	0
$187$	7.13784i	0.521971i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.946925	0.0685171	0.0342585	−	0.999413i	$-0.489093\pi$
$191$	0.946925	0.0685171	0.0342585	+	0.999413i	$0.489093\pi$
$192$	0	0
$193$	−3.94817	−0.284195	−0.142098	−	0.989853i	$-0.545385\pi$
$193$	−3.94817	−0.284195	−0.142098	+	0.989853i	$0.545385\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.33147i	− 0.166110i	−0.996545	−	0.0830550i	$-0.973532\pi$
$197$	− 2.33147i	− 0.166110i	0.996545	−	0.0830550i	$-0.0264677\pi$
$198$	0	0
$199$	− 0.187545i	− 0.0132947i	−0.999978	−	0.00664734i	$-0.997884\pi$
$199$	− 0.187545i	− 0.0132947i	0.999978	−	0.00664734i	$-0.00211593\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.38239	−0.237397
$204$	0	0
$205$	−25.7398	−1.79774
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 3.45185i	− 0.238770i
$210$	0	0
$211$	13.1142i	0.902821i	0.892316	+	0.451411i	$0.149079\pi$
$211$	13.1142i	0.902821i	−0.892316	+	0.451411i	$0.850921\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.13538	−0.145632
$216$	0	0
$217$	−4.02687	−0.273362
$218$	0	0
$219$	0	0
$220$	0	0
$221$	11.2544i	0.757050i
$222$	0	0
$223$	11.3282i	0.758595i	0.925275	+	0.379298i	$0.123834\pi$
$223$	11.3282i	0.758595i	−0.925275	+	0.379298i	$0.876166\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.9096	0.790467	0.395233	−	0.918581i	$-0.370664\pi$
$227$	11.9096	0.790467	0.395233	+	0.918581i	$0.370664\pi$
$228$	0	0
$229$	−18.2109	−1.20341	−0.601706	−	0.798718i	$-0.705512\pi$
$229$	−18.2109	−1.20341	−0.601706	+	0.798718i	$0.705512\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2.13476i	− 0.139853i	−0.997552	−	0.0699265i	$-0.977724\pi$
$233$	− 2.13476i	− 0.139853i	0.997552	−	0.0699265i	$-0.0222765\pi$
$234$	0	0
$235$	1.95337i	0.127424i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.58635	−0.490720	−0.245360	−	0.969432i	$-0.578906\pi$
$239$	−7.58635	−0.490720	−0.245360	+	0.969432i	$0.578906\pi$
$240$	0	0
$241$	−16.4260	−1.05809	−0.529047	−	0.848592i	$-0.677451\pi$
$241$	−16.4260	−1.05809	−0.529047	+	0.848592i	$0.677451\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	7.96042i	0.508573i
$246$	0	0
$247$	− 5.44260i	− 0.346304i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.0399	−0.633714	−0.316857	−	0.948473i	$-0.602627\pi$
$251$	−10.0399	−0.633714	−0.316857	+	0.948473i	$0.602627\pi$
$252$	0	0
$253$	−8.81890	−0.554440
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.99868i	0.561322i	0.959807	+	0.280661i	$0.0905537\pi$
$257$	8.99868i	0.561322i	−0.959807	+	0.280661i	$0.909446\pi$
$258$	0	0
$259$	− 16.8696i	− 1.04823i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−21.5193	−1.32693	−0.663467	−	0.748205i	$-0.730916\pi$
$263$	−21.5193	−1.32693	−0.663467	+	0.748205i	$0.730916\pi$
$264$	0	0
$265$	−8.67491	−0.532895
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 8.37240i	− 0.510474i	−0.966878	−	0.255237i	$-0.917846\pi$
$269$	− 8.37240i	− 0.510474i	0.966878	−	0.255237i	$-0.0821536\pi$
$270$	0	0
$271$	21.2118i	1.28852i	0.764806	+	0.644261i	$0.222835\pi$
$271$	21.2118i	1.28852i	−0.764806	+	0.644261i	$0.777165\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.94567	−0.237933
$276$	0	0
$277$	−17.0184	−1.02254	−0.511270	−	0.859420i	$-0.670825\pi$
$277$	−17.0184	−1.02254	−0.511270	+	0.859420i	$0.670825\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	25.8020i	1.53922i	0.638516	+	0.769608i	$0.279548\pi$
$281$	25.8020i	1.53922i	−0.638516	+	0.769608i	$0.720452\pi$
$282$	0	0
$283$	2.33051i	0.138535i	0.997598	+	0.0692673i	$0.0220661\pi$
$283$	2.33051i	0.138535i	−0.997598	+	0.0692673i	$0.977934\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.9253	1.05810
$288$	0	0
$289$	−33.9488	−1.99699
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 26.2571i	− 1.53396i	−0.641672	−	0.766979i	$-0.721759\pi$
$293$	− 26.2571i	− 1.53396i	0.641672	−	0.766979i	$-0.278241\pi$
$294$	0	0
$295$	8.93580i	0.520262i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−13.9049	−0.804142
$300$	0	0
$301$	1.48709	0.0857145
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 33.6648i	− 1.92764i
$306$	0	0
$307$	25.6576i	1.46435i	0.681114	+	0.732177i	$0.261496\pi$
$307$	25.6576i	1.46435i	−0.681114	+	0.732177i	$0.738504\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.5969	−0.771012	−0.385506	−	0.922705i	$-0.625973\pi$
$311$	−13.5969	−0.771012	−0.385506	+	0.922705i	$0.625973\pi$
$312$	0	0
$313$	−13.0668	−0.738580	−0.369290	−	0.929314i	$-0.620399\pi$
$313$	−13.0668	−0.738580	−0.369290	+	0.929314i	$0.620399\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 11.9511i	− 0.671240i	−0.941997	−	0.335620i	$-0.891054\pi$
$317$	− 11.9511i	− 0.671240i	0.941997	−	0.335620i	$-0.108946\pi$
$318$	0	0
$319$	− 1.62388i	− 0.0909199i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.6388	1.37094
$324$	0	0
$325$	−6.22120	−0.345090
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.36034i	− 0.0749979i
$330$	0	0
$331$	29.9793i	1.64781i	0.566728	+	0.823905i	$0.308209\pi$
$331$	29.9793i	1.64781i	−0.566728	+	0.823905i	$0.691791\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−13.5519	−0.740419
$336$	0	0
$337$	−30.9771	−1.68743	−0.843715	−	0.536791i	$-0.819636\pi$
$337$	−30.9771	−1.68743	−0.843715	+	0.536791i	$0.819636\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.93330i	− 0.104694i
$342$	0	0
$343$	− 20.1240i	− 1.08659i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.1494	−0.544851	−0.272425	−	0.962177i	$-0.587826\pi$
$347$	−10.1494	−0.544851	−0.272425	+	0.962177i	$0.587826\pi$
$348$	0	0
$349$	8.71592	0.466553	0.233276	−	0.972410i	$-0.425055\pi$
$349$	8.71592	0.466553	0.233276	+	0.972410i	$0.425055\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 17.8052i	− 0.947673i	−0.880613	−	0.473837i	$-0.842869\pi$
$353$	− 17.8052i	− 0.947673i	0.880613	−	0.473837i	$-0.157131\pi$
$354$	0	0
$355$	− 49.2039i	− 2.61147i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.2943	1.12387	0.561935	−	0.827182i	$-0.310057\pi$
$359$	21.2943	1.12387	0.561935	+	0.827182i	$0.310057\pi$
$360$	0	0
$361$	7.08472	0.372880
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 5.16899i	− 0.270557i
$366$	0	0
$367$	20.5255i	1.07142i	0.844402	+	0.535710i	$0.179956\pi$
$367$	20.5255i	1.07142i	−0.844402	+	0.535710i	$0.820044\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.04126	0.313646
$372$	0	0
$373$	−4.05809	−0.210120	−0.105060	−	0.994466i	$-0.533503\pi$
$373$	−4.05809	−0.210120	−0.105060	+	0.994466i	$0.533503\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.56040i	− 0.131867i
$378$	0	0
$379$	15.3705i	0.789532i	0.918782	+	0.394766i	$0.129174\pi$
$379$	15.3705i	0.789532i	−0.918782	+	0.394766i	$0.870826\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.38151	0.0705919	0.0352959	−	0.999377i	$-0.488763\pi$
$383$	1.38151	0.0705919	0.0352959	+	0.999377i	$0.488763\pi$
$384$	0	0
$385$	6.22982	0.317501
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 2.33711i	− 0.118496i	−0.998243	−	0.0592482i	$-0.981130\pi$
$389$	− 2.33711i	− 0.118496i	0.998243	−	0.0592482i	$-0.0188703\pi$
$390$	0	0
$391$	− 62.9479i	− 3.18341i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.6894	−0.537844
$396$	0	0
$397$	4.67491	0.234627	0.117314	−	0.993095i	$-0.462572\pi$
$397$	4.67491	0.234627	0.117314	+	0.993095i	$0.462572\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.4421i	1.02083i	0.859928	+	0.510416i	$0.170509\pi$
$401$	20.4421i	1.02083i	−0.859928	+	0.510416i	$0.829491\pi$
$402$	0	0
$403$	− 3.04827i	− 0.151845i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.09910	0.401458
$408$	0	0
$409$	−10.7558	−0.531838	−0.265919	−	0.963995i	$-0.585675\pi$
$409$	−10.7558	−0.531838	−0.265919	+	0.963995i	$0.585675\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 6.22294i	− 0.306211i
$414$	0	0
$415$	− 42.4519i	− 2.08388i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.8737	1.21516	0.607579	−	0.794259i	$-0.292141\pi$
$419$	24.8737	1.21516	0.607579	+	0.794259i	$0.292141\pi$
$420$	0	0
$421$	21.2498	1.03565	0.517827	−	0.855485i	$-0.326741\pi$
$421$	21.2498	1.03565	0.517827	+	0.855485i	$0.326741\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 28.1636i	− 1.36613i
$426$	0	0
$427$	23.4443i	1.13455i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.2450	0.734327	0.367163	−	0.930156i	$-0.380329\pi$
$431$	15.2450	0.734327	0.367163	+	0.930156i	$0.380329\pi$
$432$	0	0
$433$	−5.23832	−0.251738	−0.125869	−	0.992047i	$-0.540172\pi$
$433$	−5.23832	−0.251738	−0.125869	+	0.992047i	$0.540172\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	30.4415i	1.45622i
$438$	0	0
$439$	− 28.8286i	− 1.37592i	−0.725751	−	0.687958i	$-0.758508\pi$
$439$	− 28.8286i	− 1.37592i	0.725751	−	0.687958i	$-0.241492\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.0107	0.713180	0.356590	−	0.934261i	$-0.383939\pi$
$443$	15.0107	0.713180	0.356590	+	0.934261i	$0.383939\pi$
$444$	0	0
$445$	−49.7778	−2.35970
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.1873i	0.952698i	0.879256	+	0.476349i	$0.158040\pi$
$449$	20.1873i	0.952698i	−0.879256	+	0.476349i	$0.841960\pi$
$450$	0	0
$451$	8.60593i	0.405238i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.82266	0.460493
$456$	0	0
$457$	−13.5340	−0.633095	−0.316548	−	0.948577i	$-0.602524\pi$
$457$	−13.5340	−0.633095	−0.316548	+	0.948577i	$0.602524\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.00376i	0.233048i	0.993188	+	0.116524i	$0.0371752\pi$
$461$	5.00376i	0.233048i	−0.993188	+	0.116524i	$0.962825\pi$
$462$	0	0
$463$	18.0976i	0.841066i	0.907277	+	0.420533i	$0.138157\pi$
$463$	18.0976i	0.841066i	−0.907277	+	0.420533i	$0.861843\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−41.4264	−1.91698	−0.958492	−	0.285118i	$-0.907967\pi$
$467$	−41.4264	−1.91698	−0.958492	+	0.285118i	$0.907967\pi$
$468$	0	0
$469$	9.43762	0.435789
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.713951i	0.0328275i
$474$	0	0
$475$	13.6199i	0.624922i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13.2146	0.603791	0.301895	−	0.953341i	$-0.402381\pi$
$479$	13.2146	0.603791	0.301895	+	0.953341i	$0.402381\pi$
$480$	0	0
$481$	12.7700	0.582262
$482$	0	0
$483$	0	0
$484$	0	0
$485$	31.5885i	1.43436i
$486$	0	0
$487$	23.3624i	1.05865i	0.848419	+	0.529325i	$0.177555\pi$
$487$	23.3624i	1.05865i	−0.848419	+	0.529325i	$0.822445\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.61387	0.253350	0.126675	−	0.991944i	$-0.459569\pi$
$491$	5.61387	0.253350	0.126675	+	0.991944i	$0.459569\pi$
$492$	0	0
$493$	11.5910	0.522033
$494$	0	0
$495$	0	0
$496$	0	0
$497$	34.2659i	1.53703i
$498$	0	0
$499$	− 34.0326i	− 1.52351i	−0.647867	−	0.761753i	$-0.724339\pi$
$499$	− 34.0326i	− 1.52351i	0.647867	−	0.761753i	$-0.275661\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−34.1962	−1.52473	−0.762366	−	0.647147i	$-0.775962\pi$
$503$	−34.1962	−1.52473	−0.762366	+	0.647147i	$0.775962\pi$
$504$	0	0
$505$	5.21562	0.232092
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.6870i	0.739639i	0.929104	+	0.369819i	$0.120580\pi$
$509$	16.6870i	0.739639i	−0.929104	+	0.369819i	$0.879420\pi$
$510$	0	0
$511$	3.59972i	0.159242i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−54.8066	−2.41507
$516$	0	0
$517$	0.653098	0.0287232
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.9579i	1.57534i	0.616096	+	0.787671i	$0.288713\pi$
$521$	35.9579i	1.57534i	−0.616096	+	0.787671i	$0.711287\pi$
$522$	0	0
$523$	− 23.7836i	− 1.03998i	−0.854171	−	0.519992i	$-0.825935\pi$
$523$	− 23.7836i	− 1.03998i	0.854171	−	0.519992i	$-0.174065\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.7996	0.601120
$528$	0	0
$529$	54.7730	2.38144
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13.5691i	0.587744i
$534$	0	0
$535$	25.8631i	1.11816i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.66152	0.114640
$540$	0	0
$541$	20.3186	0.873566	0.436783	−	0.899567i	$-0.356118\pi$
$541$	20.3186	0.873566	0.436783	+	0.899567i	$0.356118\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 9.17548i	− 0.393035i
$546$	0	0
$547$	− 27.0193i	− 1.15526i	−0.816298	−	0.577631i	$-0.803977\pi$
$547$	− 27.0193i	− 1.15526i	0.816298	−	0.577631i	$-0.196023\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.60540	−0.238798
$552$	0	0
$553$	7.44419	0.316559
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.0131i	0.720868i	0.932785	+	0.360434i	$0.117371\pi$
$557$	17.0131i	0.720868i	−0.932785	+	0.360434i	$0.882629\pi$
$558$	0	0
$559$	1.12570i	0.0476120i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−45.5313	−1.91891	−0.959457	−	0.281854i	$-0.909050\pi$
$563$	−45.5313	−1.91891	−0.959457	+	0.281854i	$0.909050\pi$
$564$	0	0
$565$	2.19370	0.0922898
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 32.1396i	− 1.34736i	−0.739022	−	0.673681i	$-0.764712\pi$
$569$	− 32.1396i	− 1.34736i	0.739022	−	0.673681i	$-0.235288\pi$
$570$	0	0
$571$	− 45.2462i	− 1.89349i	−0.321977	−	0.946747i	$-0.604347\pi$
$571$	− 45.2462i	− 1.89349i	0.321977	−	0.946747i	$-0.395653\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	34.7965	1.45111
$576$	0	0
$577$	−37.9294	−1.57902	−0.789511	−	0.613736i	$-0.789666\pi$
$577$	−37.9294	−1.57902	−0.789511	+	0.613736i	$0.789666\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	29.5637i	1.22651i
$582$	0	0
$583$	2.90040i	0.120122i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−42.1849	−1.74116	−0.870579	−	0.492028i	$-0.836256\pi$
$587$	−42.1849	−1.74116	−0.870579	+	0.492028i	$0.836256\pi$
$588$	0	0
$589$	−6.67346	−0.274975
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 8.00034i	− 0.328535i	−0.986416	−	0.164267i	$-0.947474\pi$
$593$	− 8.00034i	− 0.328535i	0.986416	−	0.164267i	$-0.0525260\pi$
$594$	0	0
$595$	44.4675i	1.82299i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−15.5125	−0.633823	−0.316911	−	0.948455i	$-0.602646\pi$
$599$	−15.5125	−0.633823	−0.316911	+	0.948455i	$0.602646\pi$
$600$	0	0
$601$	18.1148	0.738919	0.369459	−	0.929247i	$-0.379543\pi$
$601$	18.1148	0.738919	0.369459	+	0.929247i	$0.379543\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.99093i	0.121599i
$606$	0	0
$607$	14.4660i	0.587155i	0.955935	+	0.293578i	$0.0948459\pi$
$607$	14.4660i	0.587155i	−0.955935	+	0.293578i	$0.905154\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.02975	0.0416592
$612$	0	0
$613$	6.44130	0.260162	0.130081	−	0.991503i	$-0.458476\pi$
$613$	6.44130	0.260162	0.130081	+	0.991503i	$0.458476\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 19.3378i	− 0.778510i	−0.921130	−	0.389255i	$-0.872732\pi$
$617$	− 19.3378i	− 0.778510i	0.921130	−	0.389255i	$-0.127268\pi$
$618$	0	0
$619$	36.3327i	1.46033i	0.683268	+	0.730167i	$0.260558\pi$
$619$	36.3327i	1.46033i	−0.683268	+	0.730167i	$0.739442\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	34.6656	1.38885
$624$	0	0
$625$	−29.1600	−1.16640
$626$	0	0
$627$	0	0
$628$	0	0
$629$	57.8101i	2.30504i
$630$	0	0
$631$	− 27.0088i	− 1.07520i	−0.843199	−	0.537602i	$-0.819330\pi$
$631$	− 27.0088i	− 1.07520i	0.843199	−	0.537602i	$-0.180670\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	39.2136	1.55614
$636$	0	0
$637$	4.19647	0.166270
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 15.5982i	− 0.616090i	−0.951372	−	0.308045i	$-0.900325\pi$
$641$	− 15.5982i	− 0.616090i	0.951372	−	0.308045i	$-0.0996748\pi$
$642$	0	0
$643$	17.8524i	0.704032i	0.935994	+	0.352016i	$0.114504\pi$
$643$	17.8524i	0.704032i	−0.935994	+	0.352016i	$0.885496\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.2185	0.637615	0.318808	−	0.947819i	$-0.396718\pi$
$647$	16.2185	0.637615	0.318808	+	0.947819i	$0.396718\pi$
$648$	0	0
$649$	2.98763	0.117275
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.8194i	1.20606i	0.797720	+	0.603029i	$0.206040\pi$
$653$	30.8194i	1.20606i	−0.797720	+	0.603029i	$0.793960\pi$
$654$	0	0
$655$	33.6166i	1.31351i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.1757	−0.474297	−0.237148	−	0.971473i	$-0.576213\pi$
$659$	−12.1757	−0.474297	−0.237148	+	0.971473i	$0.576213\pi$
$660$	0	0
$661$	−29.0695	−1.13067	−0.565336	−	0.824860i	$-0.691254\pi$
$661$	−29.0695	−1.13067	−0.565336	+	0.824860i	$0.691254\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 21.5044i	− 0.833905i
$666$	0	0
$667$	14.3209i	0.554506i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.2556	−0.434518
$672$	0	0
$673$	24.2877	0.936221	0.468110	−	0.883670i	$-0.344935\pi$
$673$	24.2877	0.936221	0.468110	+	0.883670i	$0.344935\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.66133i	0.0638500i	0.999490	+	0.0319250i	$0.0101638\pi$
$677$	1.66133i	0.0638500i	−0.999490	+	0.0319250i	$0.989836\pi$
$678$	0	0
$679$	− 21.9984i	− 0.844221i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	42.1449	1.61263	0.806314	−	0.591488i	$-0.201459\pi$
$683$	42.1449	1.61263	0.806314	+	0.591488i	$0.201459\pi$
$684$	0	0
$685$	53.0802	2.02809
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.57312i	0.174222i
$690$	0	0
$691$	− 25.8144i	− 0.982028i	−0.871152	−	0.491014i	$-0.836626\pi$
$691$	− 25.8144i	− 0.982028i	0.871152	−	0.491014i	$-0.163374\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.6348	−0.782722
$696$	0	0
$697$	−61.4278	−2.32674
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 8.96034i	− 0.338428i	−0.985579	−	0.169214i	$-0.945877\pi$
$701$	− 8.96034i	− 0.338428i	0.985579	−	0.169214i	$-0.0541228\pi$
$702$	0	0
$703$	− 27.9569i	− 1.05441i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.63219	−0.136603
$708$	0	0
$709$	13.4121	0.503701	0.251851	−	0.967766i	$-0.418961\pi$
$709$	13.4121	0.503701	0.251851	+	0.967766i	$0.418961\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	17.0496i	0.638512i
$714$	0	0
$715$	4.71585i	0.176363i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11.6652	0.435037	0.217519	−	0.976056i	$-0.430204\pi$
$719$	11.6652	0.435037	0.217519	+	0.976056i	$0.430204\pi$
$720$	0	0
$721$	38.1676	1.42144
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.40730i	0.237961i
$726$	0	0
$727$	− 18.6737i	− 0.692571i	−0.938129	−	0.346285i	$-0.887443\pi$
$727$	− 18.6737i	− 0.692571i	0.938129	−	0.346285i	$-0.112557\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.09607	−0.188485
$732$	0	0
$733$	47.8113	1.76595	0.882975	−	0.469420i	$-0.155537\pi$
$733$	47.8113	1.76595	0.882975	+	0.469420i	$0.155537\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.53100i	0.166901i
$738$	0	0
$739$	22.8533i	0.840672i	0.907369	+	0.420336i	$0.138088\pi$
$739$	22.8533i	0.840672i	−0.907369	+	0.420336i	$0.861912\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.0857	0.443381	0.221690	−	0.975117i	$-0.428843\pi$
$743$	12.0857	0.443381	0.221690	+	0.975117i	$0.428843\pi$
$744$	0	0
$745$	47.3088	1.73326
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 18.0112i	− 0.658115i
$750$	0	0
$751$	18.6694i	0.681255i	0.940198	+	0.340628i	$0.110640\pi$
$751$	18.6694i	0.681255i	−0.940198	+	0.340628i	$0.889360\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−21.1130	−0.768379
$756$	0	0
$757$	34.5937	1.25733	0.628664	−	0.777677i	$-0.283602\pi$
$757$	34.5937	1.25733	0.628664	+	0.777677i	$0.283602\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 3.73270i	− 0.135310i	−0.997709	−	0.0676552i	$-0.978448\pi$
$761$	− 3.73270i	− 0.135310i	0.997709	−	0.0676552i	$-0.0215518\pi$
$762$	0	0
$763$	6.38986i	0.231328i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.71065	0.170092
$768$	0	0
$769$	−23.2455	−0.838254	−0.419127	−	0.907928i	$-0.637664\pi$
$769$	−23.2455	−0.838254	−0.419127	+	0.907928i	$0.637664\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5.99987i	0.215800i	0.994162	+	0.107900i	$0.0344127\pi$
$773$	5.99987i	0.215800i	−0.994162	+	0.107900i	$0.965587\pi$
$774$	0	0
$775$	7.62816i	0.274011i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29.7064	1.06434
$780$	0	0
$781$	−16.4510	−0.588664
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 29.6068i	− 1.05671i
$786$	0	0
$787$	23.1331i	0.824607i	0.911047	+	0.412303i	$0.135276\pi$
$787$	23.1331i	0.824607i	−0.911047	+	0.412303i	$0.864724\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.52771	−0.0543190
$792$	0	0
$793$	−17.7469	−0.630212
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.3172i	0.825939i	0.910745	+	0.412969i	$0.135508\pi$
$797$	23.3172i	0.825939i	−0.910745	+	0.412969i	$0.864492\pi$
$798$	0	0
$799$	4.66171i	0.164919i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.72822	−0.0609876
$804$	0	0
$805$	−54.9401	−1.93638
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 28.3590i	− 0.997049i	−0.866876	−	0.498525i	$-0.833875\pi$
$809$	− 28.3590i	− 0.997049i	0.866876	−	0.498525i	$-0.166125\pi$
$810$	0	0
$811$	29.6158i	1.03995i	0.854181	+	0.519976i	$0.174059\pi$
$811$	29.6158i	1.03995i	−0.854181	+	0.519976i	$0.825941\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	43.0927	1.50947
$816$	0	0
$817$	2.46445	0.0862203
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 51.8681i	− 1.81021i	−0.425188	−	0.905105i	$-0.639792\pi$
$821$	− 51.8681i	− 1.81021i	0.425188	−	0.905105i	$-0.360208\pi$
$822$	0	0
$823$	− 12.4583i	− 0.434269i	−0.976142	−	0.217134i	$-0.930329\pi$
$823$	− 12.4583i	− 0.434269i	0.976142	−	0.217134i	$-0.0696710\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	46.9241	1.63171	0.815856	−	0.578255i	$-0.196266\pi$
$827$	46.9241	1.63171	0.815856	+	0.578255i	$0.196266\pi$
$828$	0	0
$829$	47.7832	1.65958	0.829789	−	0.558077i	$-0.188461\pi$
$829$	47.7832	1.65958	0.829789	+	0.558077i	$0.188461\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18.9975i	0.658225i
$834$	0	0
$835$	− 62.9954i	− 2.18005i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	49.5836	1.71182	0.855908	−	0.517129i	$-0.172999\pi$
$839$	49.5836	1.71182	0.855908	+	0.517129i	$0.172999\pi$
$840$	0	0
$841$	26.3630	0.909069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 31.4465i	− 1.08179i
$846$	0	0
$847$	− 2.08290i	− 0.0715693i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−71.4252	−2.44842
$852$	0	0
$853$	55.9921	1.91713	0.958566	−	0.284871i	$-0.0919507\pi$
$853$	55.9921	1.91713	0.958566	+	0.284871i	$0.0919507\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 21.2358i	− 0.725400i	−0.931906	−	0.362700i	$-0.881855\pi$
$857$	− 21.2358i	− 0.725400i	0.931906	−	0.362700i	$-0.118145\pi$
$858$	0	0
$859$	22.3211i	0.761585i	0.924660	+	0.380793i	$0.124349\pi$
$859$	22.3211i	0.761585i	−0.924660	+	0.380793i	$0.875651\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.5705	−0.938512	−0.469256	−	0.883062i	$-0.655478\pi$
$863$	−27.5705	−0.938512	−0.469256	+	0.883062i	$0.655478\pi$
$864$	0	0
$865$	−60.0339	−2.04121
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.57395i	0.121238i
$870$	0	0
$871$	7.14410i	0.242069i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.56829	0.222049
$876$	0	0
$877$	−31.2115	−1.05394	−0.526969	−	0.849884i	$-0.676672\pi$
$877$	−31.2115	−1.05394	−0.526969	+	0.849884i	$0.676672\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	14.5165i	0.489073i	0.969640	+	0.244537i	$0.0786359\pi$
$881$	14.5165i	0.489073i	−0.969640	+	0.244537i	$0.921364\pi$
$882$	0	0
$883$	39.6025i	1.33273i	0.745626	+	0.666365i	$0.232151\pi$
$883$	39.6025i	1.33273i	−0.745626	+	0.666365i	$0.767849\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.5914	1.63154	0.815771	−	0.578376i	$-0.196313\pi$
$887$	48.5914	1.63154	0.815771	+	0.578376i	$0.196313\pi$
$888$	0	0
$889$	−27.3086	−0.915900
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2.25440i	− 0.0754405i
$894$	0	0
$895$	48.2423i	1.61256i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.13945	−0.104707
$900$	0	0
$901$	−20.7026	−0.689705
$902$	0	0
$903$	0	0
$904$	0	0
$905$	30.1977i	1.00380i
$906$	0	0
$907$	22.2447i	0.738623i	0.929306	+	0.369311i	$0.120406\pi$
$907$	22.2447i	0.738623i	−0.929306	+	0.369311i	$0.879594\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22.9203	−0.759383	−0.379692	−	0.925113i	$-0.623970\pi$
$911$	−22.9203	−0.759383	−0.379692	+	0.925113i	$0.623970\pi$
$912$	0	0
$913$	−14.1935	−0.469738
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 23.4108i	− 0.773091i
$918$	0	0
$919$	− 29.9724i	− 0.988699i	−0.869263	−	0.494349i	$-0.835406\pi$
$919$	− 29.9724i	− 0.988699i	0.869263	−	0.494349i	$-0.164594\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−25.9386	−0.853780
$924$	0	0
$925$	−31.9564	−1.05072
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 21.5654i	− 0.707537i	−0.935333	−	0.353768i	$-0.884900\pi$
$929$	− 21.5654i	− 0.707537i	0.935333	−	0.353768i	$-0.115100\pi$
$930$	0	0
$931$	− 9.18718i	− 0.301098i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−21.3488	−0.698180
$936$	0	0
$937$	11.5824	0.378381	0.189190	−	0.981940i	$-0.439414\pi$
$937$	11.5824	0.378381	0.189190	+	0.981940i	$0.439414\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	46.0713i	1.50188i	0.660369	+	0.750941i	$0.270400\pi$
$941$	46.0713i	1.50188i	−0.660369	+	0.750941i	$0.729600\pi$
$942$	0	0
$943$	− 75.8949i	− 2.47148i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.90636	0.321914	0.160957	−	0.986961i	$-0.448542\pi$
$947$	9.90636	0.321914	0.160957	+	0.986961i	$0.448542\pi$
$948$	0	0
$949$	−2.72492	−0.0884545
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 0.432238i	− 0.0140016i	−0.999975	−	0.00700078i	$-0.997772\pi$
$953$	− 0.432238i	− 0.0140016i	0.999975	−	0.00700078i	$-0.00222844\pi$
$954$	0	0
$955$	2.83219i	0.0916474i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36.9653	−1.19367
$960$	0	0
$961$	27.2624	0.879431
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 11.8087i	− 0.380135i
$966$	0	0
$967$	− 24.4147i	− 0.785123i	−0.919726	−	0.392561i	$-0.871589\pi$
$967$	− 24.4147i	− 0.785123i	0.919726	−	0.392561i	$-0.128411\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−0.827442	−0.0265539	−0.0132769	−	0.999912i	$-0.504226\pi$
$971$	−0.827442	−0.0265539	−0.0132769	+	0.999912i	$0.504226\pi$
$972$	0	0
$973$	14.3702	0.460687
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 40.3775i	− 1.29179i	−0.763426	−	0.645895i	$-0.776484\pi$
$977$	− 40.3775i	− 1.29179i	0.763426	−	0.645895i	$-0.223516\pi$
$978$	0	0
$979$	16.6429i	0.531910i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.6070	−0.880525	−0.440263	−	0.897869i	$-0.645115\pi$
$983$	−27.6070	−0.880525	−0.440263	+	0.897869i	$0.645115\pi$
$984$	0	0
$985$	6.97325	0.222186
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 6.29627i	− 0.200210i
$990$	0	0
$991$	23.8248i	0.756821i	0.925638	+	0.378411i	$0.123529\pi$
$991$	23.8248i	0.756821i	−0.925638	+	0.378411i	$0.876471\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.560933	0.0177828
$996$	0	0
$997$	0.380300	0.0120442	0.00602211	−	0.999982i	$-0.498083\pi$
$997$	0.380300	0.0120442	0.00602211	+	0.999982i	$0.498083\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.d.j.3455.12		12
3.2	odd	2		6336.2.d.i.3455.2		12
4.3	odd	2		6336.2.d.i.3455.12		12
8.3	odd	2		3168.2.d.d.287.1	yes	12
8.5	even	2		3168.2.d.c.287.1	✓	12
12.11	even	2	inner	6336.2.d.j.3455.2		12
24.5	odd	2		3168.2.d.d.287.11	yes	12
24.11	even	2		3168.2.d.c.287.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3168.2.d.c.287.1	✓	12	8.5	even	2
3168.2.d.c.287.11	yes	12	24.11	even	2
3168.2.d.d.287.1	yes	12	8.3	odd	2
3168.2.d.d.287.11	yes	12	24.5	odd	2
6336.2.d.i.3455.2		12	3.2	odd	2
6336.2.d.i.3455.12		12	4.3	odd	2
6336.2.d.j.3455.2		12	12.11	even	2	inner
6336.2.d.j.3455.12		12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.99093i q^{5} -2.08290i q^{7} +1.00000 q^{11} +1.57672 q^{13} +7.13784i q^{17} -3.45185i q^{19} -8.81890 q^{23} -3.94567 q^{25} -1.62388i q^{29} -1.93330i q^{31} +6.22982 q^{35} +8.09910 q^{37} +8.60593i q^{41} +0.713951i q^{43} +0.653098 q^{47} +2.66152 q^{49} +2.90040i q^{53} +2.99093i q^{55} +2.98763 q^{59} -11.2556 q^{61} +4.71585i q^{65} +4.53100i q^{67} -16.4510 q^{71} -1.72822 q^{73} -2.08290i q^{77} +3.57395i q^{79} -14.1935 q^{83} -21.3488 q^{85} +16.6429i q^{89} -3.28415i q^{91} +10.3243 q^{95} +10.5614 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{11} - 24 q^{23} - 4 q^{25} + 24 q^{35} + 16 q^{37} - 24 q^{47} - 4 q^{49} + 48 q^{59} + 8 q^{61} - 72 q^{71} - 16 q^{73} + 24 q^{83} - 40 q^{85} - 40 q^{95} - 24 q^{97}+O(q^{100}) \) 12 * q + 12 * q^11 - 24 * q^23 - 4 * q^25 + 24 * q^35 + 16 * q^37 - 24 * q^47 - 4 * q^49 + 48 * q^59 + 8 * q^61 - 72 * q^71 - 16 * q^73 + 24 * q^83 - 40 * q^85 - 40 * q^95 - 24 * q^97

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