LMFDB - Embedded newform 7744.2.a.du.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7744,2,Mod(1,7744)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7744.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7744 = 2^{6} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7744.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-5,0,0,0,2,0,7,0,0,0,-4] 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:	yes
Analytic conductor:	$61.8361513253$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.19898000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 7x^{3} + 24x^{2} - 15x - 5 $ x^6 - x^5 - 10x^4 + 7x^3 + 24x^2 - 15x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 352)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.68692$ of defining polynomial
Character	$\chi$	$=$	7744.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.686920 q^{3} -0.750338 q^{5} -3.05529 q^{7} -2.52814 q^{9} +6.20933 q^{13} -0.515422 q^{15} -3.08177 q^{17} +4.23580 q^{19} -2.09874 q^{21} +0.264608 q^{23} -4.43699 q^{25} -3.79739 q^{27} +1.66044 q^{29} +4.19687 q^{31} +2.29250 q^{35} +2.36499 q^{37} +4.26531 q^{39} +5.67853 q^{41} -9.90575 q^{43} +1.89696 q^{45} +11.5274 q^{47} +2.33481 q^{49} -2.11693 q^{51} -4.40734 q^{53} +2.90966 q^{57} -0.836198 q^{59} -6.26228 q^{61} +7.72421 q^{63} -4.65909 q^{65} +7.76323 q^{67} +0.181764 q^{69} +10.3601 q^{71} -8.46823 q^{73} -3.04786 q^{75} -14.7211 q^{79} +4.97592 q^{81} -10.9957 q^{83} +2.31237 q^{85} +1.14059 q^{87} -15.9492 q^{89} -18.9713 q^{91} +2.88292 q^{93} -3.17828 q^{95} -0.881290 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{3} + 2 q^{7} + 7 q^{9} - 4 q^{13} + 8 q^{15} - 9 q^{17} - 5 q^{19} - 12 q^{21} + 6 q^{23} + 4 q^{25} - 26 q^{27} - 10 q^{29} + 12 q^{31} + 26 q^{35} + 12 q^{37} + 2 q^{39} - 17 q^{41} - 11 q^{43}+ \cdots - 21 q^{97}+O(q^{100}) $ 6 * q - 5 * q^3 + 2 * q^7 + 7 * q^9 - 4 * q^13 + 8 * q^15 - 9 * q^17 - 5 * q^19 - 12 * q^21 + 6 * q^23 + 4 * q^25 - 26 * q^27 - 10 * q^29 + 12 * q^31 + 26 * q^35 + 12 * q^37 + 2 * q^39 - 17 * q^41 - 11 * q^43 - 14 * q^45 + 22 * q^47 + 6 * q^51 - 18 * q^53 - 6 * q^57 - 7 * q^59 - 18 * q^61 + 26 * q^63 - 26 * q^65 - 31 * q^67 - 16 * q^69 + 22 * q^71 + q^73 - 11 * q^75 - 28 * q^79 + 34 * q^81 + 13 * q^83 - 2 * q^85 + 38 * q^87 - q^89 - 26 * q^91 - 34 * q^93 + 2 * q^95 - 21 * q^97

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
$3$	0.686920	0.396594	0.198297	−	0.980142i	$-0.436459\pi$
$3$	0.686920	0.396594	0.198297	+	0.980142i	$0.436459\pi$
$4$	0	0
$5$	−0.750338	−0.335561	−0.167781	−	0.985824i	$-0.553660\pi$
$5$	−0.750338	−0.335561	−0.167781	+	0.985824i	$0.553660\pi$
$6$	0	0
$7$	−3.05529	−1.15479	−0.577396	−	0.816464i	$-0.695931\pi$
$7$	−3.05529	−1.15479	−0.577396	+	0.816464i	$0.695931\pi$
$8$	0	0
$9$	−2.52814	−0.842713
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.20933	1.72216	0.861079	−	0.508472i	$-0.169789\pi$
$13$	6.20933	1.72216	0.861079	+	0.508472i	$0.169789\pi$
$14$	0	0
$15$	−0.515422	−0.133081
$16$	0	0
$17$	−3.08177	−0.747439	−0.373719	−	0.927542i	$-0.621918\pi$
$17$	−3.08177	−0.747439	−0.373719	+	0.927542i	$0.621918\pi$
$18$	0	0
$19$	4.23580	0.971760	0.485880	−	0.874026i	$-0.338499\pi$
$19$	4.23580	0.971760	0.485880	+	0.874026i	$0.338499\pi$
$20$	0	0
$21$	−2.09874	−0.457983
$22$	0	0
$23$	0.264608	0.0551745	0.0275873	−	0.999619i	$-0.491218\pi$
$23$	0.264608	0.0551745	0.0275873	+	0.999619i	$0.491218\pi$
$24$	0	0
$25$	−4.43699	−0.887399
$26$	0	0
$27$	−3.79739	−0.730808
$28$	0	0
$29$	1.66044	0.308337	0.154168	−	0.988045i	$-0.450730\pi$
$29$	1.66044	0.308337	0.154168	+	0.988045i	$0.450730\pi$
$30$	0	0
$31$	4.19687	0.753781	0.376890	−	0.926258i	$-0.376993\pi$
$31$	4.19687	0.753781	0.376890	+	0.926258i	$0.376993\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.29250	0.387503
$36$	0	0
$37$	2.36499	0.388802	0.194401	−	0.980922i	$-0.437724\pi$
$37$	2.36499	0.388802	0.194401	+	0.980922i	$0.437724\pi$
$38$	0	0
$39$	4.26531	0.682997
$40$	0	0
$41$	5.67853	0.886837	0.443419	−	0.896315i	$-0.353765\pi$
$41$	5.67853	0.886837	0.443419	+	0.896315i	$0.353765\pi$
$42$	0	0
$43$	−9.90575	−1.51061	−0.755306	−	0.655372i	$-0.772512\pi$
$43$	−9.90575	−1.51061	−0.755306	+	0.655372i	$0.772512\pi$
$44$	0	0
$45$	1.89696	0.282782
$46$	0	0
$47$	11.5274	1.68145	0.840724	−	0.541464i	$-0.182130\pi$
$47$	11.5274	1.68145	0.840724	+	0.541464i	$0.182130\pi$
$48$	0	0
$49$	2.33481	0.333544
$50$	0	0
$51$	−2.11693	−0.296429
$52$	0	0
$53$	−4.40734	−0.605395	−0.302697	−	0.953087i	$-0.597887\pi$
$53$	−4.40734	−0.605395	−0.302697	+	0.953087i	$0.597887\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.90966	0.385394
$58$	0	0
$59$	−0.836198	−0.108864	−0.0544319	−	0.998517i	$-0.517335\pi$
$59$	−0.836198	−0.108864	−0.0544319	+	0.998517i	$0.517335\pi$
$60$	0	0
$61$	−6.26228	−0.801803	−0.400901	−	0.916121i	$-0.631303\pi$
$61$	−6.26228	−0.801803	−0.400901	+	0.916121i	$0.631303\pi$
$62$	0	0
$63$	7.72421	0.973159
$64$	0	0
$65$	−4.65909	−0.577889
$66$	0	0
$67$	7.76323	0.948430	0.474215	−	0.880409i	$-0.342732\pi$
$67$	7.76323	0.948430	0.474215	+	0.880409i	$0.342732\pi$
$68$	0	0
$69$	0.181764	0.0218819
$70$	0	0
$71$	10.3601	1.22952	0.614758	−	0.788716i	$-0.289254\pi$
$71$	10.3601	1.22952	0.614758	+	0.788716i	$0.289254\pi$
$72$	0	0
$73$	−8.46823	−0.991131	−0.495565	−	0.868571i	$-0.665039\pi$
$73$	−8.46823	−0.991131	−0.495565	+	0.868571i	$0.665039\pi$
$74$	0	0
$75$	−3.04786	−0.351937
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.7211	−1.65625	−0.828127	−	0.560541i	$-0.810593\pi$
$79$	−14.7211	−1.65625	−0.828127	+	0.560541i	$0.810593\pi$
$80$	0	0
$81$	4.97592	0.552879
$82$	0	0
$83$	−10.9957	−1.20693	−0.603467	−	0.797388i	$-0.706215\pi$
$83$	−10.9957	−1.20693	−0.603467	+	0.797388i	$0.706215\pi$
$84$	0	0
$85$	2.31237	0.250811
$86$	0	0
$87$	1.14059	0.122284
$88$	0	0
$89$	−15.9492	−1.69061	−0.845306	−	0.534282i	$-0.820582\pi$
$89$	−15.9492	−1.69061	−0.845306	+	0.534282i	$0.820582\pi$
$90$	0	0
$91$	−18.9713	−1.98873
$92$	0	0
$93$	2.88292	0.298945
$94$	0	0
$95$	−3.17828	−0.326085
$96$	0	0
$97$	−0.881290	−0.0894815	−0.0447407	−	0.998999i	$-0.514246\pi$
$97$	−0.881290	−0.0894815	−0.0447407	+	0.998999i	$0.514246\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.4980	−1.14410	−0.572048	−	0.820220i	$-0.693851\pi$
$101$	−11.4980	−1.14410	−0.572048	+	0.820220i	$0.693851\pi$
$102$	0	0
$103$	10.9285	1.07681	0.538406	−	0.842685i	$-0.319027\pi$
$103$	10.9285	1.07681	0.538406	+	0.842685i	$0.319027\pi$
$104$	0	0
$105$	1.57477	0.153681
$106$	0	0
$107$	−2.08795	−0.201850	−0.100925	−	0.994894i	$-0.532180\pi$
$107$	−2.08795	−0.201850	−0.100925	+	0.994894i	$0.532180\pi$
$108$	0	0
$109$	3.88713	0.372320	0.186160	−	0.982519i	$-0.440396\pi$
$109$	3.88713	0.372320	0.186160	+	0.982519i	$0.440396\pi$
$110$	0	0
$111$	1.62456	0.154197
$112$	0	0
$113$	13.6144	1.28074	0.640368	−	0.768069i	$-0.278782\pi$
$113$	13.6144	1.28074	0.640368	+	0.768069i	$0.278782\pi$
$114$	0	0
$115$	−0.198545	−0.0185144
$116$	0	0
$117$	−15.6981	−1.45129
$118$	0	0
$119$	9.41570	0.863136
$120$	0	0
$121$	0	0
$122$	0	0
$123$	3.90070	0.351714
$124$	0	0
$125$	7.08093	0.633338
$126$	0	0
$127$	8.17419	0.725342	0.362671	−	0.931917i	$-0.381865\pi$
$127$	8.17419	0.725342	0.362671	+	0.931917i	$0.381865\pi$
$128$	0	0
$129$	−6.80446	−0.599099
$130$	0	0
$131$	5.90996	0.516356	0.258178	−	0.966097i	$-0.416878\pi$
$131$	5.90996	0.516356	0.258178	+	0.966097i	$0.416878\pi$
$132$	0	0
$133$	−12.9416	−1.12218
$134$	0	0
$135$	2.84933	0.245231
$136$	0	0
$137$	−8.29629	−0.708800	−0.354400	−	0.935094i	$-0.615315\pi$
$137$	−8.29629	−0.708800	−0.354400	+	0.935094i	$0.615315\pi$
$138$	0	0
$139$	18.1970	1.54345	0.771727	−	0.635954i	$-0.219393\pi$
$139$	18.1970	1.54345	0.771727	+	0.635954i	$0.219393\pi$
$140$	0	0
$141$	7.91843	0.666852
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.24589	−0.103466
$146$	0	0
$147$	1.60383	0.132282
$148$	0	0
$149$	−10.9906	−0.900384	−0.450192	−	0.892932i	$-0.648645\pi$
$149$	−10.9906	−0.900384	−0.450192	+	0.892932i	$0.648645\pi$
$150$	0	0
$151$	−17.5546	−1.42858	−0.714288	−	0.699852i	$-0.753249\pi$
$151$	−17.5546	−1.42858	−0.714288	+	0.699852i	$0.753249\pi$
$152$	0	0
$153$	7.79114	0.629877
$154$	0	0
$155$	−3.14907	−0.252940
$156$	0	0
$157$	13.2393	1.05661	0.528306	−	0.849054i	$-0.322827\pi$
$157$	13.2393	1.05661	0.528306	+	0.849054i	$0.322827\pi$
$158$	0	0
$159$	−3.02749	−0.240096
$160$	0	0
$161$	−0.808454	−0.0637151
$162$	0	0
$163$	−22.5899	−1.76938	−0.884689	−	0.466181i	$-0.845629\pi$
$163$	−22.5899	−1.76938	−0.884689	+	0.466181i	$0.845629\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.1022	−1.24603	−0.623014	−	0.782211i	$-0.714092\pi$
$167$	−16.1022	−1.24603	−0.623014	+	0.782211i	$0.714092\pi$
$168$	0	0
$169$	25.5557	1.96583
$170$	0	0
$171$	−10.7087	−0.818915
$172$	0	0
$173$	−8.63190	−0.656272	−0.328136	−	0.944631i	$-0.606420\pi$
$173$	−8.63190	−0.656272	−0.328136	+	0.944631i	$0.606420\pi$
$174$	0	0
$175$	13.5563	1.02476
$176$	0	0
$177$	−0.574402	−0.0431747
$178$	0	0
$179$	−13.4976	−1.00886	−0.504430	−	0.863452i	$-0.668297\pi$
$179$	−13.4976	−1.00886	−0.504430	+	0.863452i	$0.668297\pi$
$180$	0	0
$181$	8.19393	0.609050	0.304525	−	0.952504i	$-0.401502\pi$
$181$	8.19393	0.609050	0.304525	+	0.952504i	$0.401502\pi$
$182$	0	0
$183$	−4.30169	−0.317990
$184$	0	0
$185$	−1.77454	−0.130467
$186$	0	0
$187$	0	0
$188$	0	0
$189$	11.6021	0.843932
$190$	0	0
$191$	−19.6899	−1.42471	−0.712356	−	0.701819i	$-0.752372\pi$
$191$	−19.6899	−1.42471	−0.712356	+	0.701819i	$0.752372\pi$
$192$	0	0
$193$	−23.8601	−1.71749	−0.858745	−	0.512404i	$-0.828755\pi$
$193$	−23.8601	−1.71749	−0.858745	+	0.512404i	$0.828755\pi$
$194$	0	0
$195$	−3.20043	−0.229187
$196$	0	0
$197$	−18.9000	−1.34657	−0.673283	−	0.739385i	$-0.735116\pi$
$197$	−18.9000	−1.34657	−0.673283	+	0.739385i	$0.735116\pi$
$198$	0	0
$199$	−9.81826	−0.695998	−0.347999	−	0.937495i	$-0.613139\pi$
$199$	−9.81826	−0.695998	−0.347999	+	0.937495i	$0.613139\pi$
$200$	0	0
$201$	5.33272	0.376141
$202$	0	0
$203$	−5.07314	−0.356065
$204$	0	0
$205$	−4.26082	−0.297588
$206$	0	0
$207$	−0.668965	−0.0464963
$208$	0	0
$209$	0	0
$210$	0	0
$211$	13.0188	0.896253	0.448126	−	0.893970i	$-0.352091\pi$
$211$	13.0188	0.896253	0.448126	+	0.893970i	$0.352091\pi$
$212$	0	0
$213$	7.11655	0.487618
$214$	0	0
$215$	7.43266	0.506903
$216$	0	0
$217$	−12.8227	−0.870460
$218$	0	0
$219$	−5.81700	−0.393076
$220$	0	0
$221$	−19.1357	−1.28721
$222$	0	0
$223$	3.49698	0.234175	0.117088	−	0.993122i	$-0.462644\pi$
$223$	3.49698	0.234175	0.117088	+	0.993122i	$0.462644\pi$
$224$	0	0
$225$	11.2173	0.747823
$226$	0	0
$227$	−10.9286	−0.725354	−0.362677	−	0.931915i	$-0.618137\pi$
$227$	−10.9286	−0.725354	−0.362677	+	0.931915i	$0.618137\pi$
$228$	0	0
$229$	−11.6882	−0.772379	−0.386189	−	0.922419i	$-0.626209\pi$
$229$	−11.6882	−0.772379	−0.386189	+	0.922419i	$0.626209\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.71863	0.309128	0.154564	−	0.987983i	$-0.450603\pi$
$233$	4.71863	0.309128	0.154564	+	0.987983i	$0.450603\pi$
$234$	0	0
$235$	−8.64947	−0.564229
$236$	0	0
$237$	−10.1122	−0.656860
$238$	0	0
$239$	1.21129	0.0783520	0.0391760	−	0.999232i	$-0.487527\pi$
$239$	1.21129	0.0783520	0.0391760	+	0.999232i	$0.487527\pi$
$240$	0	0
$241$	−17.6151	−1.13469	−0.567344	−	0.823481i	$-0.692029\pi$
$241$	−17.6151	−1.13469	−0.567344	+	0.823481i	$0.692029\pi$
$242$	0	0
$243$	14.8102	0.950077
$244$	0	0
$245$	−1.75190	−0.111925
$246$	0	0
$247$	26.3015	1.67352
$248$	0	0
$249$	−7.55317	−0.478662
$250$	0	0
$251$	−0.796375	−0.0502667	−0.0251334	−	0.999684i	$-0.508001\pi$
$251$	−0.796375	−0.0502667	−0.0251334	+	0.999684i	$0.508001\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.58841	0.0994702
$256$	0	0
$257$	−15.4164	−0.961650	−0.480825	−	0.876817i	$-0.659663\pi$
$257$	−15.4164	−0.961650	−0.480825	+	0.876817i	$0.659663\pi$
$258$	0	0
$259$	−7.22574	−0.448986
$260$	0	0
$261$	−4.19784	−0.259840
$262$	0	0
$263$	23.1291	1.42620	0.713101	−	0.701062i	$-0.247290\pi$
$263$	23.1291	1.42620	0.713101	+	0.701062i	$0.247290\pi$
$264$	0	0
$265$	3.30700	0.203147
$266$	0	0
$267$	−10.9558	−0.670486
$268$	0	0
$269$	−24.1256	−1.47097	−0.735483	−	0.677544i	$-0.763045\pi$
$269$	−24.1256	−1.47097	−0.735483	+	0.677544i	$0.763045\pi$
$270$	0	0
$271$	5.06570	0.307719	0.153860	−	0.988093i	$-0.450830\pi$
$271$	5.06570	0.307719	0.153860	+	0.988093i	$0.450830\pi$
$272$	0	0
$273$	−13.0318	−0.788719
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.86711	0.352520	0.176260	−	0.984344i	$-0.443600\pi$
$277$	5.86711	0.352520	0.176260	+	0.984344i	$0.443600\pi$
$278$	0	0
$279$	−10.6103	−0.635221
$280$	0	0
$281$	−27.9832	−1.66934	−0.834669	−	0.550752i	$-0.814341\pi$
$281$	−27.9832	−1.66934	−0.834669	+	0.550752i	$0.814341\pi$
$282$	0	0
$283$	−7.60221	−0.451904	−0.225952	−	0.974138i	$-0.572549\pi$
$283$	−7.60221	−0.451904	−0.225952	+	0.974138i	$0.572549\pi$
$284$	0	0
$285$	−2.18323	−0.129323
$286$	0	0
$287$	−17.3496	−1.02411
$288$	0	0
$289$	−7.50270	−0.441336
$290$	0	0
$291$	−0.605376	−0.0354878
$292$	0	0
$293$	−3.94130	−0.230253	−0.115127	−	0.993351i	$-0.536727\pi$
$293$	−3.94130	−0.230253	−0.115127	+	0.993351i	$0.536727\pi$
$294$	0	0
$295$	0.627431	0.0365305
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.64304	0.0950192
$300$	0	0
$301$	30.2650	1.74444
$302$	0	0
$303$	−7.89822	−0.453741
$304$	0	0
$305$	4.69882	0.269054
$306$	0	0
$307$	−10.7552	−0.613829	−0.306915	−	0.951737i	$-0.599297\pi$
$307$	−10.7552	−0.613829	−0.306915	+	0.951737i	$0.599297\pi$
$308$	0	0
$309$	7.50698	0.427057
$310$	0	0
$311$	26.0833	1.47905	0.739525	−	0.673129i	$-0.235050\pi$
$311$	26.0833	1.47905	0.739525	+	0.673129i	$0.235050\pi$
$312$	0	0
$313$	−3.00837	−0.170043	−0.0850214	−	0.996379i	$-0.527096\pi$
$313$	−3.00837	−0.170043	−0.0850214	+	0.996379i	$0.527096\pi$
$314$	0	0
$315$	−5.79577	−0.326554
$316$	0	0
$317$	−32.3983	−1.81967	−0.909836	−	0.414969i	$-0.863792\pi$
$317$	−32.3983	−1.81967	−0.909836	+	0.414969i	$0.863792\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1.43425	−0.0800523
$322$	0	0
$323$	−13.0538	−0.726331
$324$	0	0
$325$	−27.5507	−1.52824
$326$	0	0
$327$	2.67015	0.147660
$328$	0	0
$329$	−35.2197	−1.94172
$330$	0	0
$331$	−19.2715	−1.05926	−0.529629	−	0.848229i	$-0.677669\pi$
$331$	−19.2715	−1.05926	−0.529629	+	0.848229i	$0.677669\pi$
$332$	0	0
$333$	−5.97903	−0.327649
$334$	0	0
$335$	−5.82505	−0.318256
$336$	0	0
$337$	−19.6589	−1.07089	−0.535445	−	0.844570i	$-0.679856\pi$
$337$	−19.6589	−1.07089	−0.535445	+	0.844570i	$0.679856\pi$
$338$	0	0
$339$	9.35201	0.507931
$340$	0	0
$341$	0	0
$342$	0	0
$343$	14.2535	0.769618
$344$	0	0
$345$	−0.136385	−0.00734271
$346$	0	0
$347$	−1.29421	−0.0694770	−0.0347385	−	0.999396i	$-0.511060\pi$
$347$	−1.29421	−0.0694770	−0.0347385	+	0.999396i	$0.511060\pi$
$348$	0	0
$349$	11.4717	0.614066	0.307033	−	0.951699i	$-0.400664\pi$
$349$	11.4717	0.614066	0.307033	+	0.951699i	$0.400664\pi$
$350$	0	0
$351$	−23.5792	−1.25857
$352$	0	0
$353$	−15.5104	−0.825534	−0.412767	−	0.910837i	$-0.635438\pi$
$353$	−15.5104	−0.825534	−0.412767	+	0.910837i	$0.635438\pi$
$354$	0	0
$355$	−7.77356	−0.412578
$356$	0	0
$357$	6.46784	0.342314
$358$	0	0
$359$	−13.9623	−0.736904	−0.368452	−	0.929647i	$-0.620112\pi$
$359$	−13.9623	−0.736904	−0.368452	+	0.929647i	$0.620112\pi$
$360$	0	0
$361$	−1.05797	−0.0556828
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.35403	0.332585
$366$	0	0
$367$	15.4530	0.806641	0.403320	−	0.915059i	$-0.367856\pi$
$367$	15.4530	0.806641	0.403320	+	0.915059i	$0.367856\pi$
$368$	0	0
$369$	−14.3561	−0.747350
$370$	0	0
$371$	13.4657	0.699105
$372$	0	0
$373$	−6.75664	−0.349846	−0.174923	−	0.984582i	$-0.555968\pi$
$373$	−6.75664	−0.349846	−0.174923	+	0.984582i	$0.555968\pi$
$374$	0	0
$375$	4.86404	0.251178
$376$	0	0
$377$	10.3102	0.531004
$378$	0	0
$379$	−23.1289	−1.18805	−0.594025	−	0.804446i	$-0.702462\pi$
$379$	−23.1289	−1.18805	−0.594025	+	0.804446i	$0.702462\pi$
$380$	0	0
$381$	5.61502	0.287666
$382$	0	0
$383$	23.9627	1.22444	0.612219	−	0.790689i	$-0.290277\pi$
$383$	23.9627	1.22444	0.612219	+	0.790689i	$0.290277\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	25.0431	1.27301
$388$	0	0
$389$	−13.3342	−0.676069	−0.338035	−	0.941134i	$-0.609762\pi$
$389$	−13.3342	−0.676069	−0.338035	+	0.941134i	$0.609762\pi$
$390$	0	0
$391$	−0.815460	−0.0412396
$392$	0	0
$393$	4.05967	0.204784
$394$	0	0
$395$	11.0458	0.555775
$396$	0	0
$397$	−23.8585	−1.19742	−0.598712	−	0.800964i	$-0.704321\pi$
$397$	−23.8585	−1.19742	−0.598712	+	0.800964i	$0.704321\pi$
$398$	0	0
$399$	−8.88986	−0.445050
$400$	0	0
$401$	13.5055	0.674432	0.337216	−	0.941427i	$-0.390515\pi$
$401$	13.5055	0.674432	0.337216	+	0.941427i	$0.390515\pi$
$402$	0	0
$403$	26.0598	1.29813
$404$	0	0
$405$	−3.73362	−0.185525
$406$	0	0
$407$	0	0
$408$	0	0
$409$	37.6546	1.86190	0.930950	−	0.365145i	$-0.118981\pi$
$409$	37.6546	1.86190	0.930950	+	0.365145i	$0.118981\pi$
$410$	0	0
$411$	−5.69889	−0.281105
$412$	0	0
$413$	2.55483	0.125715
$414$	0	0
$415$	8.25049	0.405000
$416$	0	0
$417$	12.4999	0.612124
$418$	0	0
$419$	−13.2409	−0.646862	−0.323431	−	0.946252i	$-0.604836\pi$
$419$	−13.2409	−0.646862	−0.323431	+	0.946252i	$0.604836\pi$
$420$	0	0
$421$	2.31827	0.112986	0.0564928	−	0.998403i	$-0.482008\pi$
$421$	2.31827	0.112986	0.0564928	+	0.998403i	$0.482008\pi$
$422$	0	0
$423$	−29.1430	−1.41698
$424$	0	0
$425$	13.6738	0.663276
$426$	0	0
$427$	19.1331	0.925915
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.90764	0.188225	0.0941123	−	0.995562i	$-0.469999\pi$
$431$	3.90764	0.188225	0.0941123	+	0.995562i	$0.469999\pi$
$432$	0	0
$433$	19.6112	0.942456	0.471228	−	0.882011i	$-0.343811\pi$
$433$	19.6112	0.942456	0.471228	+	0.882011i	$0.343811\pi$
$434$	0	0
$435$	−0.855830	−0.0410339
$436$	0	0
$437$	1.12083	0.0536164
$438$	0	0
$439$	−4.69615	−0.224135	−0.112068	−	0.993701i	$-0.535747\pi$
$439$	−4.69615	−0.224135	−0.112068	+	0.993701i	$0.535747\pi$
$440$	0	0
$441$	−5.90273	−0.281082
$442$	0	0
$443$	42.0488	1.99780	0.998899	−	0.0469116i	$-0.0149379\pi$
$443$	42.0488	1.99780	0.998899	+	0.0469116i	$0.0149379\pi$
$444$	0	0
$445$	11.9673	0.567304
$446$	0	0
$447$	−7.54966	−0.357087
$448$	0	0
$449$	−24.6953	−1.16544	−0.582722	−	0.812671i	$-0.698013\pi$
$449$	−24.6953	−1.16544	−0.582722	+	0.812671i	$0.698013\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−12.0586	−0.566564
$454$	0	0
$455$	14.2349	0.667342
$456$	0	0
$457$	17.8621	0.835555	0.417778	−	0.908549i	$-0.362809\pi$
$457$	17.8621	0.835555	0.417778	+	0.908549i	$0.362809\pi$
$458$	0	0
$459$	11.7027	0.546234
$460$	0	0
$461$	−29.9746	−1.39606	−0.698028	−	0.716071i	$-0.745939\pi$
$461$	−29.9746	−1.39606	−0.698028	+	0.716071i	$0.745939\pi$
$462$	0	0
$463$	29.0359	1.34941	0.674705	−	0.738087i	$-0.264271\pi$
$463$	29.0359	1.34941	0.674705	+	0.738087i	$0.264271\pi$
$464$	0	0
$465$	−2.16316	−0.100314
$466$	0	0
$467$	28.1602	1.30310	0.651549	−	0.758606i	$-0.274119\pi$
$467$	28.1602	1.30310	0.651549	+	0.758606i	$0.274119\pi$
$468$	0	0
$469$	−23.7189	−1.09524
$470$	0	0
$471$	9.09435	0.419045
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−18.7942	−0.862338
$476$	0	0
$477$	11.1424	0.510174
$478$	0	0
$479$	22.2165	1.01510	0.507548	−	0.861623i	$-0.330552\pi$
$479$	22.2165	1.01510	0.507548	+	0.861623i	$0.330552\pi$
$480$	0	0
$481$	14.6850	0.669579
$482$	0	0
$483$	−0.555343	−0.0252690
$484$	0	0
$485$	0.661265	0.0300265
$486$	0	0
$487$	−13.1020	−0.593710	−0.296855	−	0.954923i	$-0.595938\pi$
$487$	−13.1020	−0.593710	−0.296855	+	0.954923i	$0.595938\pi$
$488$	0	0
$489$	−15.5175	−0.701724
$490$	0	0
$491$	−13.5736	−0.612566	−0.306283	−	0.951940i	$-0.599085\pi$
$491$	−13.5736	−0.612566	−0.306283	+	0.951940i	$0.599085\pi$
$492$	0	0
$493$	−5.11710	−0.230463
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−31.6531	−1.41983
$498$	0	0
$499$	19.9633	0.893679	0.446840	−	0.894614i	$-0.352549\pi$
$499$	19.9633	0.893679	0.446840	+	0.894614i	$0.352549\pi$
$500$	0	0
$501$	−11.0610	−0.494167
$502$	0	0
$503$	8.09028	0.360728	0.180364	−	0.983600i	$-0.442272\pi$
$503$	8.09028	0.360728	0.180364	+	0.983600i	$0.442272\pi$
$504$	0	0
$505$	8.62740	0.383914
$506$	0	0
$507$	17.5548	0.779634
$508$	0	0
$509$	31.5349	1.39776	0.698879	−	0.715240i	$-0.253683\pi$
$509$	31.5349	1.39776	0.698879	+	0.715240i	$0.253683\pi$
$510$	0	0
$511$	25.8729	1.14455
$512$	0	0
$513$	−16.0850	−0.710170
$514$	0	0
$515$	−8.20004	−0.361337
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−5.92943	−0.260273
$520$	0	0
$521$	36.9841	1.62030	0.810151	−	0.586221i	$-0.199385\pi$
$521$	36.9841	1.62030	0.810151	+	0.586221i	$0.199385\pi$
$522$	0	0
$523$	7.41319	0.324156	0.162078	−	0.986778i	$-0.448180\pi$
$523$	7.41319	0.324156	0.162078	+	0.986778i	$0.448180\pi$
$524$	0	0
$525$	9.31211	0.406414
$526$	0	0
$527$	−12.9338	−0.563405
$528$	0	0
$529$	−22.9300	−0.996956
$530$	0	0
$531$	2.11403	0.0917410
$532$	0	0
$533$	35.2598	1.52727
$534$	0	0
$535$	1.56667	0.0677329
$536$	0	0
$537$	−9.27180	−0.400108
$538$	0	0
$539$	0	0
$540$	0	0
$541$	11.5798	0.497857	0.248928	−	0.968522i	$-0.419922\pi$
$541$	11.5798	0.497857	0.248928	+	0.968522i	$0.419922\pi$
$542$	0	0
$543$	5.62858	0.241545
$544$	0	0
$545$	−2.91666	−0.124936
$546$	0	0
$547$	−10.0361	−0.429113	−0.214557	−	0.976712i	$-0.568831\pi$
$547$	−10.0361	−0.429113	−0.214557	+	0.976712i	$0.568831\pi$
$548$	0	0
$549$	15.8319	0.675690
$550$	0	0
$551$	7.03331	0.299629
$552$	0	0
$553$	44.9773	1.91263
$554$	0	0
$555$	−1.21897	−0.0517424
$556$	0	0
$557$	−31.1186	−1.31854	−0.659269	−	0.751907i	$-0.729134\pi$
$557$	−31.1186	−1.31854	−0.659269	+	0.751907i	$0.729134\pi$
$558$	0	0
$559$	−61.5080	−2.60151
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22.2824	−0.939092	−0.469546	−	0.882908i	$-0.655582\pi$
$563$	−22.2824	−0.939092	−0.469546	+	0.882908i	$0.655582\pi$
$564$	0	0
$565$	−10.2154	−0.429765
$566$	0	0
$567$	−15.2029	−0.638461
$568$	0	0
$569$	−31.2700	−1.31091	−0.655453	−	0.755236i	$-0.727522\pi$
$569$	−31.2700	−1.31091	−0.655453	+	0.755236i	$0.727522\pi$
$570$	0	0
$571$	−5.93719	−0.248464	−0.124232	−	0.992253i	$-0.539647\pi$
$571$	−5.93719	−0.248464	−0.124232	+	0.992253i	$0.539647\pi$
$572$	0	0
$573$	−13.5254	−0.565032
$574$	0	0
$575$	−1.17406	−0.0489618
$576$	0	0
$577$	6.67861	0.278034	0.139017	−	0.990290i	$-0.455606\pi$
$577$	6.67861	0.278034	0.139017	+	0.990290i	$0.455606\pi$
$578$	0	0
$579$	−16.3900	−0.681145
$580$	0	0
$581$	33.5951	1.39376
$582$	0	0
$583$	0	0
$584$	0	0
$585$	11.7788	0.486995
$586$	0	0
$587$	41.0820	1.69563	0.847817	−	0.530289i	$-0.177916\pi$
$587$	41.0820	1.69563	0.847817	+	0.530289i	$0.177916\pi$
$588$	0	0
$589$	17.7771	0.732494
$590$	0	0
$591$	−12.9828	−0.534039
$592$	0	0
$593$	−0.283994	−0.0116622	−0.00583111	−	0.999983i	$-0.501856\pi$
$593$	−0.283994	−0.0116622	−0.00583111	+	0.999983i	$0.501856\pi$
$594$	0	0
$595$	−7.06496	−0.289635
$596$	0	0
$597$	−6.74436	−0.276028
$598$	0	0
$599$	−41.6379	−1.70128	−0.850638	−	0.525751i	$-0.823784\pi$
$599$	−41.6379	−1.70128	−0.850638	+	0.525751i	$0.823784\pi$
$600$	0	0
$601$	6.17280	0.251794	0.125897	−	0.992043i	$-0.459819\pi$
$601$	6.17280	0.251794	0.125897	+	0.992043i	$0.459819\pi$
$602$	0	0
$603$	−19.6265	−0.799254
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.6678	1.65066	0.825328	−	0.564653i	$-0.190990\pi$
$607$	40.6678	1.65066	0.825328	+	0.564653i	$0.190990\pi$
$608$	0	0
$609$	−3.48484	−0.141213
$610$	0	0
$611$	71.5776	2.89572
$612$	0	0
$613$	−17.2609	−0.697159	−0.348580	−	0.937279i	$-0.613336\pi$
$613$	−17.2609	−0.697159	−0.348580	+	0.937279i	$0.613336\pi$
$614$	0	0
$615$	−2.92684	−0.118022
$616$	0	0
$617$	13.2729	0.534346	0.267173	−	0.963649i	$-0.413911\pi$
$617$	13.2729	0.534346	0.267173	+	0.963649i	$0.413911\pi$
$618$	0	0
$619$	12.8183	0.515210	0.257605	−	0.966250i	$-0.417067\pi$
$619$	12.8183	0.515210	0.257605	+	0.966250i	$0.417067\pi$
$620$	0	0
$621$	−1.00482	−0.0403220
$622$	0	0
$623$	48.7295	1.95231
$624$	0	0
$625$	16.8719	0.674875
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.28836	−0.290606
$630$	0	0
$631$	6.44282	0.256485	0.128242	−	0.991743i	$-0.459066\pi$
$631$	6.44282	0.256485	0.128242	+	0.991743i	$0.459066\pi$
$632$	0	0
$633$	8.94290	0.355448
$634$	0	0
$635$	−6.13340	−0.243397
$636$	0	0
$637$	14.4976	0.574416
$638$	0	0
$639$	−26.1917	−1.03613
$640$	0	0
$641$	−20.0691	−0.792683	−0.396342	−	0.918103i	$-0.629720\pi$
$641$	−20.0691	−0.792683	−0.396342	+	0.918103i	$0.629720\pi$
$642$	0	0
$643$	−32.4578	−1.28001	−0.640005	−	0.768371i	$-0.721068\pi$
$643$	−32.4578	−1.28001	−0.640005	+	0.768371i	$0.721068\pi$
$644$	0	0
$645$	5.10564	0.201035
$646$	0	0
$647$	−15.8124	−0.621649	−0.310825	−	0.950467i	$-0.600605\pi$
$647$	−15.8124	−0.621649	−0.310825	+	0.950467i	$0.600605\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−8.80816	−0.345219
$652$	0	0
$653$	3.00274	0.117506	0.0587532	−	0.998273i	$-0.481287\pi$
$653$	3.00274	0.117506	0.0587532	+	0.998273i	$0.481287\pi$
$654$	0	0
$655$	−4.43447	−0.173269
$656$	0	0
$657$	21.4089	0.835239
$658$	0	0
$659$	−3.17515	−0.123686	−0.0618431	−	0.998086i	$-0.519698\pi$
$659$	−3.17515	−0.123686	−0.0618431	+	0.998086i	$0.519698\pi$
$660$	0	0
$661$	30.6169	1.19086	0.595430	−	0.803408i	$-0.296982\pi$
$661$	30.6169	1.19086	0.595430	+	0.803408i	$0.296982\pi$
$662$	0	0
$663$	−13.1447	−0.510498
$664$	0	0
$665$	9.71058	0.376560
$666$	0	0
$667$	0.439366	0.0170123
$668$	0	0
$669$	2.40215	0.0928725
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−17.4373	−0.672158	−0.336079	−	0.941834i	$-0.609101\pi$
$673$	−17.4373	−0.672158	−0.336079	+	0.941834i	$0.609101\pi$
$674$	0	0
$675$	16.8490	0.648518
$676$	0	0
$677$	−39.8843	−1.53288	−0.766439	−	0.642318i	$-0.777973\pi$
$677$	−39.8843	−1.53288	−0.766439	+	0.642318i	$0.777973\pi$
$678$	0	0
$679$	2.69260	0.103332
$680$	0	0
$681$	−7.50705	−0.287671
$682$	0	0
$683$	37.9702	1.45289	0.726445	−	0.687225i	$-0.241171\pi$
$683$	37.9702	1.45289	0.726445	+	0.687225i	$0.241171\pi$
$684$	0	0
$685$	6.22502	0.237846
$686$	0	0
$687$	−8.02887	−0.306321
$688$	0	0
$689$	−27.3666	−1.04259
$690$	0	0
$691$	30.3851	1.15590	0.577951	−	0.816071i	$-0.303852\pi$
$691$	30.3851	1.15590	0.577951	+	0.816071i	$0.303852\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.6539	−0.517923
$696$	0	0
$697$	−17.4999	−0.662856
$698$	0	0
$699$	3.24133	0.122598
$700$	0	0
$701$	−15.2756	−0.576950	−0.288475	−	0.957487i	$-0.593148\pi$
$701$	−15.2756	−0.576950	−0.288475	+	0.957487i	$0.593148\pi$
$702$	0	0
$703$	10.0176	0.377822
$704$	0	0
$705$	−5.94149	−0.223770
$706$	0	0
$707$	35.1298	1.32119
$708$	0	0
$709$	−34.9713	−1.31338	−0.656688	−	0.754162i	$-0.728043\pi$
$709$	−34.9713	−1.31338	−0.656688	+	0.754162i	$0.728043\pi$
$710$	0	0
$711$	37.2170	1.39575
$712$	0	0
$713$	1.11053	0.0415895
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.832062	0.0310739
$718$	0	0
$719$	10.5261	0.392557	0.196279	−	0.980548i	$-0.437114\pi$
$719$	10.5261	0.392557	0.196279	+	0.980548i	$0.437114\pi$
$720$	0	0
$721$	−33.3896	−1.24349
$722$	0	0
$723$	−12.1002	−0.450010
$724$	0	0
$725$	−7.36738	−0.273618
$726$	0	0
$727$	8.67301	0.321664	0.160832	−	0.986982i	$-0.448582\pi$
$727$	8.67301	0.321664	0.160832	+	0.986982i	$0.448582\pi$
$728$	0	0
$729$	−4.75429	−0.176085
$730$	0	0
$731$	30.5272	1.12909
$732$	0	0
$733$	42.6922	1.57687	0.788436	−	0.615116i	$-0.210891\pi$
$733$	42.6922	1.57687	0.788436	+	0.615116i	$0.210891\pi$
$734$	0	0
$735$	−1.20341	−0.0443886
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−41.2502	−1.51741	−0.758707	−	0.651432i	$-0.774168\pi$
$739$	−41.2502	−1.51741	−0.758707	+	0.651432i	$0.774168\pi$
$740$	0	0
$741$	18.0670	0.663709
$742$	0	0
$743$	−34.8323	−1.27787	−0.638937	−	0.769259i	$-0.720626\pi$
$743$	−34.8323	−1.27787	−0.638937	+	0.769259i	$0.720626\pi$
$744$	0	0
$745$	8.24666	0.302134
$746$	0	0
$747$	27.7987	1.01710
$748$	0	0
$749$	6.37930	0.233094
$750$	0	0
$751$	−35.8598	−1.30854	−0.654272	−	0.756259i	$-0.727025\pi$
$751$	−35.8598	−1.30854	−0.654272	+	0.756259i	$0.727025\pi$
$752$	0	0
$753$	−0.547046	−0.0199355
$754$	0	0
$755$	13.1719	0.479375
$756$	0	0
$757$	−11.0309	−0.400924	−0.200462	−	0.979702i	$-0.564244\pi$
$757$	−11.0309	−0.400924	−0.200462	+	0.979702i	$0.564244\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.74706	0.317081	0.158540	−	0.987352i	$-0.449321\pi$
$761$	8.74706	0.317081	0.158540	+	0.987352i	$0.449321\pi$
$762$	0	0
$763$	−11.8763	−0.429952
$764$	0	0
$765$	−5.84599	−0.211362
$766$	0	0
$767$	−5.19223	−0.187481
$768$	0	0
$769$	9.61551	0.346744	0.173372	−	0.984856i	$-0.444534\pi$
$769$	9.61551	0.346744	0.173372	+	0.984856i	$0.444534\pi$
$770$	0	0
$771$	−10.5899	−0.381384
$772$	0	0
$773$	6.57318	0.236421	0.118210	−	0.992989i	$-0.462284\pi$
$773$	6.57318	0.236421	0.118210	+	0.992989i	$0.462284\pi$
$774$	0	0
$775$	−18.6215	−0.668904
$776$	0	0
$777$	−4.96351	−0.178065
$778$	0	0
$779$	24.0531	0.861793
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.30536	−0.225335
$784$	0	0
$785$	−9.93395	−0.354558
$786$	0	0
$787$	−1.60753	−0.0573022	−0.0286511	−	0.999589i	$-0.509121\pi$
$787$	−1.60753	−0.0573022	−0.0286511	+	0.999589i	$0.509121\pi$
$788$	0	0
$789$	15.8879	0.565623
$790$	0	0
$791$	−41.5960	−1.47898
$792$	0	0
$793$	−38.8845	−1.38083
$794$	0	0
$795$	2.27164	0.0805669
$796$	0	0
$797$	−19.7184	−0.698463	−0.349231	−	0.937037i	$-0.613557\pi$
$797$	−19.7184	−0.698463	−0.349231	+	0.937037i	$0.613557\pi$
$798$	0	0
$799$	−35.5249	−1.25678
$800$	0	0
$801$	40.3218	1.42470
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0.606614	0.0213803
$806$	0	0
$807$	−16.5724	−0.583376
$808$	0	0
$809$	−4.57450	−0.160831	−0.0804154	−	0.996761i	$-0.525625\pi$
$809$	−4.57450	−0.160831	−0.0804154	+	0.996761i	$0.525625\pi$
$810$	0	0
$811$	3.52724	0.123858	0.0619291	−	0.998081i	$-0.480275\pi$
$811$	3.52724	0.123858	0.0619291	+	0.998081i	$0.480275\pi$
$812$	0	0
$813$	3.47973	0.122040
$814$	0	0
$815$	16.9501	0.593735
$816$	0	0
$817$	−41.9588	−1.46795
$818$	0	0
$819$	47.9621	1.67593
$820$	0	0
$821$	2.46739	0.0861126	0.0430563	−	0.999073i	$-0.486291\pi$
$821$	2.46739	0.0861126	0.0430563	+	0.999073i	$0.486291\pi$
$822$	0	0
$823$	−7.44148	−0.259394	−0.129697	−	0.991554i	$-0.541400\pi$
$823$	−7.44148	−0.259394	−0.129697	+	0.991554i	$0.541400\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.8067	0.375785	0.187893	−	0.982190i	$-0.439834\pi$
$827$	10.8067	0.375785	0.187893	+	0.982190i	$0.439834\pi$
$828$	0	0
$829$	22.3031	0.774619	0.387310	−	0.921950i	$-0.373404\pi$
$829$	22.3031	0.774619	0.387310	+	0.921950i	$0.373404\pi$
$830$	0	0
$831$	4.03024	0.139807
$832$	0	0
$833$	−7.19534	−0.249304
$834$	0	0
$835$	12.0821	0.418119
$836$	0	0
$837$	−15.9372	−0.550869
$838$	0	0
$839$	−31.9204	−1.10202	−0.551008	−	0.834500i	$-0.685757\pi$
$839$	−31.9204	−1.10202	−0.551008	+	0.834500i	$0.685757\pi$
$840$	0	0
$841$	−26.2429	−0.904928
$842$	0	0
$843$	−19.2222	−0.662049
$844$	0	0
$845$	−19.1754	−0.659655
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−5.22211	−0.179222
$850$	0	0
$851$	0.625795	0.0214520
$852$	0	0
$853$	16.7892	0.574851	0.287425	−	0.957803i	$-0.407201\pi$
$853$	16.7892	0.574851	0.287425	+	0.957803i	$0.407201\pi$
$854$	0	0
$855$	8.03515	0.274796
$856$	0	0
$857$	−11.2278	−0.383534	−0.191767	−	0.981440i	$-0.561422\pi$
$857$	−11.2278	−0.383534	−0.191767	+	0.981440i	$0.561422\pi$
$858$	0	0
$859$	−50.2349	−1.71399	−0.856996	−	0.515324i	$-0.827672\pi$
$859$	−50.2349	−1.71399	−0.856996	+	0.515324i	$0.827672\pi$
$860$	0	0
$861$	−11.9178	−0.406157
$862$	0	0
$863$	6.57659	0.223870	0.111935	−	0.993716i	$-0.464295\pi$
$863$	6.57659	0.223870	0.111935	+	0.993716i	$0.464295\pi$
$864$	0	0
$865$	6.47684	0.220219
$866$	0	0
$867$	−5.15376	−0.175031
$868$	0	0
$869$	0	0
$870$	0	0
$871$	48.2044	1.63335
$872$	0	0
$873$	2.22803	0.0754072
$874$	0	0
$875$	−21.6343	−0.731373
$876$	0	0
$877$	−44.2319	−1.49361	−0.746803	−	0.665045i	$-0.768412\pi$
$877$	−44.2319	−1.49361	−0.746803	+	0.665045i	$0.768412\pi$
$878$	0	0
$879$	−2.70736	−0.0913170
$880$	0	0
$881$	37.0273	1.24748	0.623741	−	0.781631i	$-0.285612\pi$
$881$	37.0273	1.24748	0.623741	+	0.781631i	$0.285612\pi$
$882$	0	0
$883$	27.8246	0.936371	0.468186	−	0.883630i	$-0.344908\pi$
$883$	27.8246	0.936371	0.468186	+	0.883630i	$0.344908\pi$
$884$	0	0
$885$	0.430995	0.0144877
$886$	0	0
$887$	11.8680	0.398488	0.199244	−	0.979950i	$-0.436151\pi$
$887$	11.8680	0.398488	0.199244	+	0.979950i	$0.436151\pi$
$888$	0	0
$889$	−24.9745	−0.837619
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48.8279	1.63396
$894$	0	0
$895$	10.1278	0.338535
$896$	0	0
$897$	1.12863	0.0376840
$898$	0	0
$899$	6.96868	0.232418
$900$	0	0
$901$	13.5824	0.452496
$902$	0	0
$903$	20.7896	0.691835
$904$	0	0
$905$	−6.14822	−0.204374
$906$	0	0
$907$	10.1825	0.338105	0.169052	−	0.985607i	$-0.445929\pi$
$907$	10.1825	0.338105	0.169052	+	0.985607i	$0.445929\pi$
$908$	0	0
$909$	29.0686	0.964145
$910$	0	0
$911$	16.1304	0.534425	0.267212	−	0.963638i	$-0.413897\pi$
$911$	16.1304	0.534425	0.267212	+	0.963638i	$0.413897\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.22772	0.106705
$916$	0	0
$917$	−18.0567	−0.596284
$918$	0	0
$919$	−18.8557	−0.621993	−0.310996	−	0.950411i	$-0.600663\pi$
$919$	−18.8557	−0.621993	−0.310996	+	0.950411i	$0.600663\pi$
$920$	0	0
$921$	−7.38794	−0.243441
$922$	0	0
$923$	64.3291	2.11742
$924$	0	0
$925$	−10.4935	−0.345023
$926$	0	0
$927$	−27.6287	−0.907445
$928$	0	0
$929$	12.7280	0.417593	0.208797	−	0.977959i	$-0.433045\pi$
$929$	12.7280	0.417593	0.208797	+	0.977959i	$0.433045\pi$
$930$	0	0
$931$	9.88980	0.324125
$932$	0	0
$933$	17.9172	0.586582
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.92146	0.291452	0.145726	−	0.989325i	$-0.453448\pi$
$937$	8.92146	0.291452	0.145726	+	0.989325i	$0.453448\pi$
$938$	0	0
$939$	−2.06651	−0.0674379
$940$	0	0
$941$	13.8416	0.451222	0.225611	−	0.974217i	$-0.427562\pi$
$941$	13.8416	0.451222	0.225611	+	0.974217i	$0.427562\pi$
$942$	0	0
$943$	1.50258	0.0489308
$944$	0	0
$945$	−8.70553	−0.283191
$946$	0	0
$947$	−34.5104	−1.12144	−0.560719	−	0.828006i	$-0.689475\pi$
$947$	−34.5104	−1.12144	−0.560719	+	0.828006i	$0.689475\pi$
$948$	0	0
$949$	−52.5820	−1.70688
$950$	0	0
$951$	−22.2551	−0.721670
$952$	0	0
$953$	26.3170	0.852492	0.426246	−	0.904607i	$-0.359836\pi$
$953$	26.3170	0.852492	0.426246	+	0.904607i	$0.359836\pi$
$954$	0	0
$955$	14.7741	0.478078
$956$	0	0
$957$	0	0
$958$	0	0
$959$	25.3476	0.818516
$960$	0	0
$961$	−13.3863	−0.431815
$962$	0	0
$963$	5.27863	0.170101
$964$	0	0
$965$	17.9032	0.576323
$966$	0	0
$967$	4.18054	0.134437	0.0672185	−	0.997738i	$-0.478588\pi$
$967$	4.18054	0.134437	0.0672185	+	0.997738i	$0.478588\pi$
$968$	0	0
$969$	−8.96690	−0.288058
$970$	0	0
$971$	17.5235	0.562356	0.281178	−	0.959656i	$-0.409275\pi$
$971$	17.5235	0.562356	0.281178	+	0.959656i	$0.409275\pi$
$972$	0	0
$973$	−55.5973	−1.78237
$974$	0	0
$975$	−18.9252	−0.606090
$976$	0	0
$977$	−21.9160	−0.701154	−0.350577	−	0.936534i	$-0.614015\pi$
$977$	−21.9160	−0.701154	−0.350577	+	0.936534i	$0.614015\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−9.82722	−0.313759
$982$	0	0
$983$	−33.4101	−1.06562	−0.532808	−	0.846236i	$-0.678863\pi$
$983$	−33.4101	−1.06562	−0.532808	+	0.846236i	$0.678863\pi$
$984$	0	0
$985$	14.1813	0.451855
$986$	0	0
$987$	−24.1931	−0.770075
$988$	0	0
$989$	−2.62114	−0.0833473
$990$	0	0
$991$	17.4811	0.555307	0.277653	−	0.960681i	$-0.410443\pi$
$991$	17.4811	0.555307	0.277653	+	0.960681i	$0.410443\pi$
$992$	0	0
$993$	−13.2380	−0.420095
$994$	0	0
$995$	7.36701	0.233550
$996$	0	0
$997$	−51.0447	−1.61660	−0.808301	−	0.588769i	$-0.799613\pi$
$997$	−51.0447	−1.61660	−0.808301	+	0.588769i	$0.799613\pi$
$998$	0	0
$999$	−8.98080	−0.284140

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.du.1.5		6
4.3	odd	2		7744.2.a.dv.1.2		6
8.3	odd	2		3872.2.a.bn.1.5		6
8.5	even	2		3872.2.a.bq.1.2		6
11.5	even	5		704.2.m.m.641.3		12
11.9	even	5		704.2.m.m.257.3		12
11.10	odd	2		7744.2.a.dt.1.5		6
44.27	odd	10		704.2.m.n.641.1		12
44.31	odd	10		704.2.m.n.257.1		12
44.43	even	2		7744.2.a.dw.1.2		6
88.5	even	10		352.2.m.e.289.1	yes	12
88.21	odd	2		3872.2.a.bp.1.2		6
88.27	odd	10		352.2.m.f.289.3	yes	12
88.43	even	2		3872.2.a.bo.1.5		6
88.53	even	10		352.2.m.e.257.1	✓	12
88.75	odd	10		352.2.m.f.257.3	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.m.e.257.1	✓	12	88.53	even	10
352.2.m.e.289.1	yes	12	88.5	even	10
352.2.m.f.257.3	yes	12	88.75	odd	10
352.2.m.f.289.3	yes	12	88.27	odd	10
704.2.m.m.257.3		12	11.9	even	5
704.2.m.m.641.3		12	11.5	even	5
704.2.m.n.257.1		12	44.31	odd	10
704.2.m.n.641.1		12	44.27	odd	10
3872.2.a.bn.1.5		6	8.3	odd	2
3872.2.a.bo.1.5		6	88.43	even	2
3872.2.a.bp.1.2		6	88.21	odd	2
3872.2.a.bq.1.2		6	8.5	even	2
7744.2.a.dt.1.5		6	11.10	odd	2
7744.2.a.du.1.5		6	1.1	even	1	trivial
7744.2.a.dv.1.2		6	4.3	odd	2
7744.2.a.dw.1.2		6	44.43	even	2

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:	\( N \)	\(=\)	\( 7744 = 2^{6} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7744.a (trivial)

\(f(q)\)	\(=\)	\(q+0.686920 q^{3} -0.750338 q^{5} -3.05529 q^{7} -2.52814 q^{9} +6.20933 q^{13} -0.515422 q^{15} -3.08177 q^{17} +4.23580 q^{19} -2.09874 q^{21} +0.264608 q^{23} -4.43699 q^{25} -3.79739 q^{27} +1.66044 q^{29} +4.19687 q^{31} +2.29250 q^{35} +2.36499 q^{37} +4.26531 q^{39} +5.67853 q^{41} -9.90575 q^{43} +1.89696 q^{45} +11.5274 q^{47} +2.33481 q^{49} -2.11693 q^{51} -4.40734 q^{53} +2.90966 q^{57} -0.836198 q^{59} -6.26228 q^{61} +7.72421 q^{63} -4.65909 q^{65} +7.76323 q^{67} +0.181764 q^{69} +10.3601 q^{71} -8.46823 q^{73} -3.04786 q^{75} -14.7211 q^{79} +4.97592 q^{81} -10.9957 q^{83} +2.31237 q^{85} +1.14059 q^{87} -15.9492 q^{89} -18.9713 q^{91} +2.88292 q^{93} -3.17828 q^{95} -0.881290 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} + 2 q^{7} + 7 q^{9} - 4 q^{13} + 8 q^{15} - 9 q^{17} - 5 q^{19} - 12 q^{21} + 6 q^{23} + 4 q^{25} - 26 q^{27} - 10 q^{29} + 12 q^{31} + 26 q^{35} + 12 q^{37} + 2 q^{39} - 17 q^{41} - 11 q^{43}+ \cdots - 21 q^{97}+O(q^{100}) \) 6 * q - 5 * q^3 + 2 * q^7 + 7 * q^9 - 4 * q^13 + 8 * q^15 - 9 * q^17 - 5 * q^19 - 12 * q^21 + 6 * q^23 + 4 * q^25 - 26 * q^27 - 10 * q^29 + 12 * q^31 + 26 * q^35 + 12 * q^37 + 2 * q^39 - 17 * q^41 - 11 * q^43 - 14 * q^45 + 22 * q^47 + 6 * q^51 - 18 * q^53 - 6 * q^57 - 7 * q^59 - 18 * q^61 + 26 * q^63 - 26 * q^65 - 31 * q^67 - 16 * q^69 + 22 * q^71 + q^73 - 11 * q^75 - 28 * q^79 + 34 * q^81 + 13 * q^83 - 2 * q^85 + 38 * q^87 - q^89 - 26 * q^91 - 34 * q^93 + 2 * q^95 - 21 * q^97

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