LMFDB - Embedded newform 6864.2.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.a (trivial)

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:	yes
Analytic conductor:	$54.8093159474$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.21432 q^{5} -1.52543 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.21432 q^{5} -1.52543 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -2.21432 q^{15} -1.73975 q^{17} -4.28100 q^{19} +1.52543 q^{21} -0.474572 q^{23} -0.0967881 q^{25} -1.00000 q^{27} +5.11753 q^{29} +4.21432 q^{31} +1.00000 q^{33} -3.37778 q^{35} +0.428639 q^{37} -1.00000 q^{39} +6.28100 q^{41} +8.16839 q^{43} +2.21432 q^{45} -5.05086 q^{47} -4.67307 q^{49} +1.73975 q^{51} -14.0415 q^{53} -2.21432 q^{55} +4.28100 q^{57} +1.67307 q^{59} -5.80642 q^{61} -1.52543 q^{63} +2.21432 q^{65} +6.02074 q^{67} +0.474572 q^{69} -5.18421 q^{71} -8.08742 q^{73} +0.0967881 q^{75} +1.52543 q^{77} -0.260253 q^{79} +1.00000 q^{81} +1.37778 q^{83} -3.85236 q^{85} -5.11753 q^{87} -10.1541 q^{89} -1.52543 q^{91} -4.21432 q^{93} -9.47949 q^{95} +0.428639 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 8 q^{17} - 6 q^{19} - 2 q^{21} - 8 q^{23} - 7 q^{25} - 3 q^{27} + 2 q^{29} + 6 q^{31} + 3 q^{33} - 10 q^{35} - 12 q^{37} - 3 q^{39} + 12 q^{41} - 2 q^{43} - 2 q^{47} - q^{49} - 8 q^{51} - 2 q^{53} + 6 q^{57} - 8 q^{59} - 4 q^{61} + 2 q^{63} - 2 q^{67} + 8 q^{69} - 2 q^{71} - 4 q^{73} + 7 q^{75} - 2 q^{77} - 14 q^{79} + 3 q^{81} + 4 q^{83} - 18 q^{85} - 2 q^{87} - 10 q^{89} + 2 q^{91} - 6 q^{93} - 2 q^{95} - 12 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 8 * q^17 - 6 * q^19 - 2 * q^21 - 8 * q^23 - 7 * q^25 - 3 * q^27 + 2 * q^29 + 6 * q^31 + 3 * q^33 - 10 * q^35 - 12 * q^37 - 3 * q^39 + 12 * q^41 - 2 * q^43 - 2 * q^47 - q^49 - 8 * q^51 - 2 * q^53 + 6 * q^57 - 8 * q^59 - 4 * q^61 + 2 * q^63 - 2 * q^67 + 8 * q^69 - 2 * q^71 - 4 * q^73 + 7 * q^75 - 2 * q^77 - 14 * q^79 + 3 * q^81 + 4 * q^83 - 18 * q^85 - 2 * q^87 - 10 * q^89 + 2 * q^91 - 6 * q^93 - 2 * q^95 - 12 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.21432	0.990274	0.495137	−	0.868815i	$-0.335118\pi$
$5$	2.21432	0.990274	0.495137	+	0.868815i	$0.335118\pi$
$6$	0	0
$7$	−1.52543	−0.576557	−0.288279	−	0.957547i	$-0.593083\pi$
$7$	−1.52543	−0.576557	−0.288279	+	0.957547i	$0.593083\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.21432	−0.571735
$16$	0	0
$17$	−1.73975	−0.421951	−0.210975	−	0.977491i	$-0.567664\pi$
$17$	−1.73975	−0.421951	−0.210975	+	0.977491i	$0.567664\pi$
$18$	0	0
$19$	−4.28100	−0.982128	−0.491064	−	0.871124i	$-0.663392\pi$
$19$	−4.28100	−0.982128	−0.491064	+	0.871124i	$0.663392\pi$
$20$	0	0
$21$	1.52543	0.332876
$22$	0	0
$23$	−0.474572	−0.0989552	−0.0494776	−	0.998775i	$-0.515756\pi$
$23$	−0.474572	−0.0989552	−0.0494776	+	0.998775i	$0.515756\pi$
$24$	0	0
$25$	−0.0967881	−0.0193576
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.11753	0.950302	0.475151	−	0.879904i	$-0.342393\pi$
$29$	5.11753	0.950302	0.475151	+	0.879904i	$0.342393\pi$
$30$	0	0
$31$	4.21432	0.756914	0.378457	−	0.925619i	$-0.376455\pi$
$31$	4.21432	0.756914	0.378457	+	0.925619i	$0.376455\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−3.37778	−0.570950
$36$	0	0
$37$	0.428639	0.0704679	0.0352339	−	0.999379i	$-0.488782\pi$
$37$	0.428639	0.0704679	0.0352339	+	0.999379i	$0.488782\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.28100	0.980927	0.490463	−	0.871462i	$-0.336828\pi$
$41$	6.28100	0.980927	0.490463	+	0.871462i	$0.336828\pi$
$42$	0	0
$43$	8.16839	1.24567	0.622834	−	0.782354i	$-0.285981\pi$
$43$	8.16839	1.24567	0.622834	+	0.782354i	$0.285981\pi$
$44$	0	0
$45$	2.21432	0.330091
$46$	0	0
$47$	−5.05086	−0.736743	−0.368371	−	0.929679i	$-0.620085\pi$
$47$	−5.05086	−0.736743	−0.368371	+	0.929679i	$0.620085\pi$
$48$	0	0
$49$	−4.67307	−0.667582
$50$	0	0
$51$	1.73975	0.243613
$52$	0	0
$53$	−14.0415	−1.92875	−0.964373	−	0.264545i	$-0.914778\pi$
$53$	−14.0415	−1.92875	−0.964373	+	0.264545i	$0.914778\pi$
$54$	0	0
$55$	−2.21432	−0.298579
$56$	0	0
$57$	4.28100	0.567032
$58$	0	0
$59$	1.67307	0.217815	0.108908	−	0.994052i	$-0.465265\pi$
$59$	1.67307	0.217815	0.108908	+	0.994052i	$0.465265\pi$
$60$	0	0
$61$	−5.80642	−0.743436	−0.371718	−	0.928346i	$-0.621231\pi$
$61$	−5.80642	−0.743436	−0.371718	+	0.928346i	$0.621231\pi$
$62$	0	0
$63$	−1.52543	−0.192186
$64$	0	0
$65$	2.21432	0.274653
$66$	0	0
$67$	6.02074	0.735551	0.367775	−	0.929915i	$-0.380120\pi$
$67$	6.02074	0.735551	0.367775	+	0.929915i	$0.380120\pi$
$68$	0	0
$69$	0.474572	0.0571318
$70$	0	0
$71$	−5.18421	−0.615252	−0.307626	−	0.951507i	$-0.599535\pi$
$71$	−5.18421	−0.615252	−0.307626	+	0.951507i	$0.599535\pi$
$72$	0	0
$73$	−8.08742	−0.946561	−0.473280	−	0.880912i	$-0.656930\pi$
$73$	−8.08742	−0.946561	−0.473280	+	0.880912i	$0.656930\pi$
$74$	0	0
$75$	0.0967881	0.0111761
$76$	0	0
$77$	1.52543	0.173839
$78$	0	0
$79$	−0.260253	−0.0292807	−0.0146404	−	0.999893i	$-0.504660\pi$
$79$	−0.260253	−0.0292807	−0.0146404	+	0.999893i	$0.504660\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.37778	0.151231	0.0756157	−	0.997137i	$-0.475908\pi$
$83$	1.37778	0.151231	0.0756157	+	0.997137i	$0.475908\pi$
$84$	0	0
$85$	−3.85236	−0.417847
$86$	0	0
$87$	−5.11753	−0.548657
$88$	0	0
$89$	−10.1541	−1.07633	−0.538166	−	0.842839i	$-0.680883\pi$
$89$	−10.1541	−1.07633	−0.538166	+	0.842839i	$0.680883\pi$
$90$	0	0
$91$	−1.52543	−0.159908
$92$	0	0
$93$	−4.21432	−0.437005
$94$	0	0
$95$	−9.47949	−0.972576
$96$	0	0
$97$	0.428639	0.0435217	0.0217609	−	0.999763i	$-0.493073\pi$
$97$	0.428639	0.0435217	0.0217609	+	0.999763i	$0.493073\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	3.93332	0.391380	0.195690	−	0.980666i	$-0.437305\pi$
$101$	3.93332	0.391380	0.195690	+	0.980666i	$0.437305\pi$
$102$	0	0
$103$	13.8064	1.36039	0.680194	−	0.733032i	$-0.261896\pi$
$103$	13.8064	1.36039	0.680194	+	0.733032i	$0.261896\pi$
$104$	0	0
$105$	3.37778	0.329638
$106$	0	0
$107$	−9.67307	−0.935131	−0.467566	−	0.883958i	$-0.654869\pi$
$107$	−9.67307	−0.935131	−0.467566	+	0.883958i	$0.654869\pi$
$108$	0	0
$109$	−17.6271	−1.68837	−0.844187	−	0.536049i	$-0.819916\pi$
$109$	−17.6271	−1.68837	−0.844187	+	0.536049i	$0.819916\pi$
$110$	0	0
$111$	−0.428639	−0.0406847
$112$	0	0
$113$	13.4795	1.26804	0.634022	−	0.773315i	$-0.281403\pi$
$113$	13.4795	1.26804	0.634022	+	0.773315i	$0.281403\pi$
$114$	0	0
$115$	−1.05086	−0.0979927
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	2.65386	0.243279
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.28100	−0.566338
$124$	0	0
$125$	−11.2859	−1.00944
$126$	0	0
$127$	−20.2415	−1.79614	−0.898072	−	0.439848i	$-0.855032\pi$
$127$	−20.2415	−1.79614	−0.898072	+	0.439848i	$0.855032\pi$
$128$	0	0
$129$	−8.16839	−0.719186
$130$	0	0
$131$	−8.29529	−0.724763	−0.362381	−	0.932030i	$-0.618036\pi$
$131$	−8.29529	−0.724763	−0.362381	+	0.932030i	$0.618036\pi$
$132$	0	0
$133$	6.53035	0.566253
$134$	0	0
$135$	−2.21432	−0.190578
$136$	0	0
$137$	−14.7447	−1.25972	−0.629861	−	0.776708i	$-0.716888\pi$
$137$	−14.7447	−1.25972	−0.629861	+	0.776708i	$0.716888\pi$
$138$	0	0
$139$	9.05731	0.768231	0.384115	−	0.923285i	$-0.374506\pi$
$139$	9.05731	0.768231	0.384115	+	0.923285i	$0.374506\pi$
$140$	0	0
$141$	5.05086	0.425359
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	11.3319	0.941059
$146$	0	0
$147$	4.67307	0.385428
$148$	0	0
$149$	−2.01429	−0.165017	−0.0825085	−	0.996590i	$-0.526293\pi$
$149$	−2.01429	−0.165017	−0.0825085	+	0.996590i	$0.526293\pi$
$150$	0	0
$151$	−0.668149	−0.0543732	−0.0271866	−	0.999630i	$-0.508655\pi$
$151$	−0.668149	−0.0543732	−0.0271866	+	0.999630i	$0.508655\pi$
$152$	0	0
$153$	−1.73975	−0.140650
$154$	0	0
$155$	9.33185	0.749552
$156$	0	0
$157$	4.70964	0.375870	0.187935	−	0.982181i	$-0.439821\pi$
$157$	4.70964	0.375870	0.187935	+	0.982181i	$0.439821\pi$
$158$	0	0
$159$	14.0415	1.11356
$160$	0	0
$161$	0.723926	0.0570534
$162$	0	0
$163$	2.73483	0.214208	0.107104	−	0.994248i	$-0.465842\pi$
$163$	2.73483	0.214208	0.107104	+	0.994248i	$0.465842\pi$
$164$	0	0
$165$	2.21432	0.172385
$166$	0	0
$167$	0.561993	0.0434883	0.0217441	−	0.999764i	$-0.493078\pi$
$167$	0.561993	0.0434883	0.0217441	+	0.999764i	$0.493078\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.28100	−0.327376
$172$	0	0
$173$	8.88247	0.675322	0.337661	−	0.941268i	$-0.390364\pi$
$173$	8.88247	0.675322	0.337661	+	0.941268i	$0.390364\pi$
$174$	0	0
$175$	0.147643	0.0111608
$176$	0	0
$177$	−1.67307	−0.125756
$178$	0	0
$179$	4.51606	0.337546	0.168773	−	0.985655i	$-0.446019\pi$
$179$	4.51606	0.337546	0.168773	+	0.985655i	$0.446019\pi$
$180$	0	0
$181$	−24.7096	−1.83665	−0.918326	−	0.395824i	$-0.870459\pi$
$181$	−24.7096	−1.83665	−0.918326	+	0.395824i	$0.870459\pi$
$182$	0	0
$183$	5.80642	0.429223
$184$	0	0
$185$	0.949145	0.0697825
$186$	0	0
$187$	1.73975	0.127223
$188$	0	0
$189$	1.52543	0.110959
$190$	0	0
$191$	−12.9906	−0.939969	−0.469985	−	0.882675i	$-0.655741\pi$
$191$	−12.9906	−0.939969	−0.469985	+	0.882675i	$0.655741\pi$
$192$	0	0
$193$	15.0464	1.08306	0.541532	−	0.840680i	$-0.317844\pi$
$193$	15.0464	1.08306	0.541532	+	0.840680i	$0.317844\pi$
$194$	0	0
$195$	−2.21432	−0.158571
$196$	0	0
$197$	15.8622	1.13014	0.565068	−	0.825045i	$-0.308850\pi$
$197$	15.8622	1.13014	0.565068	+	0.825045i	$0.308850\pi$
$198$	0	0
$199$	−13.4193	−0.951267	−0.475633	−	0.879644i	$-0.657781\pi$
$199$	−13.4193	−0.951267	−0.475633	+	0.879644i	$0.657781\pi$
$200$	0	0
$201$	−6.02074	−0.424671
$202$	0	0
$203$	−7.80642	−0.547904
$204$	0	0
$205$	13.9081	0.971386
$206$	0	0
$207$	−0.474572	−0.0329851
$208$	0	0
$209$	4.28100	0.296123
$210$	0	0
$211$	−24.1684	−1.66382	−0.831910	−	0.554910i	$-0.812753\pi$
$211$	−24.1684	−1.66382	−0.831910	+	0.554910i	$0.812753\pi$
$212$	0	0
$213$	5.18421	0.355216
$214$	0	0
$215$	18.0874	1.23355
$216$	0	0
$217$	−6.42864	−0.436404
$218$	0	0
$219$	8.08742	0.546497
$220$	0	0
$221$	−1.73975	−0.117028
$222$	0	0
$223$	9.50024	0.636183	0.318092	−	0.948060i	$-0.396958\pi$
$223$	9.50024	0.636183	0.318092	+	0.948060i	$0.396958\pi$
$224$	0	0
$225$	−0.0967881	−0.00645254
$226$	0	0
$227$	−8.41435	−0.558480	−0.279240	−	0.960221i	$-0.590083\pi$
$227$	−8.41435	−0.558480	−0.279240	+	0.960221i	$0.590083\pi$
$228$	0	0
$229$	29.0321	1.91850	0.959248	−	0.282565i	$-0.0911852\pi$
$229$	29.0321	1.91850	0.959248	+	0.282565i	$0.0911852\pi$
$230$	0	0
$231$	−1.52543	−0.100366
$232$	0	0
$233$	12.3017	0.805914	0.402957	−	0.915219i	$-0.367982\pi$
$233$	12.3017	0.805914	0.402957	+	0.915219i	$0.367982\pi$
$234$	0	0
$235$	−11.1842	−0.729577
$236$	0	0
$237$	0.260253	0.0169052
$238$	0	0
$239$	−21.5669	−1.39505	−0.697524	−	0.716562i	$-0.745715\pi$
$239$	−21.5669	−1.39505	−0.697524	+	0.716562i	$0.745715\pi$
$240$	0	0
$241$	−25.0005	−1.61042	−0.805211	−	0.592988i	$-0.797948\pi$
$241$	−25.0005	−1.61042	−0.805211	+	0.592988i	$0.797948\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−10.3477	−0.661089
$246$	0	0
$247$	−4.28100	−0.272393
$248$	0	0
$249$	−1.37778	−0.0873135
$250$	0	0
$251$	−11.4795	−0.724579	−0.362290	−	0.932066i	$-0.618005\pi$
$251$	−11.4795	−0.724579	−0.362290	+	0.932066i	$0.618005\pi$
$252$	0	0
$253$	0.474572	0.0298361
$254$	0	0
$255$	3.85236	0.241244
$256$	0	0
$257$	12.1017	0.754884	0.377442	−	0.926033i	$-0.376804\pi$
$257$	12.1017	0.754884	0.377442	+	0.926033i	$0.376804\pi$
$258$	0	0
$259$	−0.653858	−0.0406288
$260$	0	0
$261$	5.11753	0.316767
$262$	0	0
$263$	19.6543	1.21194	0.605969	−	0.795488i	$-0.292785\pi$
$263$	19.6543	1.21194	0.605969	+	0.795488i	$0.292785\pi$
$264$	0	0
$265$	−31.0923	−1.90999
$266$	0	0
$267$	10.1541	0.621421
$268$	0	0
$269$	−4.62222	−0.281821	−0.140911	−	0.990022i	$-0.545003\pi$
$269$	−4.62222	−0.281821	−0.140911	+	0.990022i	$0.545003\pi$
$270$	0	0
$271$	7.37334	0.447898	0.223949	−	0.974601i	$-0.428105\pi$
$271$	7.37334	0.447898	0.223949	+	0.974601i	$0.428105\pi$
$272$	0	0
$273$	1.52543	0.0923231
$274$	0	0
$275$	0.0967881	0.00583654
$276$	0	0
$277$	24.0098	1.44261	0.721306	−	0.692617i	$-0.243542\pi$
$277$	24.0098	1.44261	0.721306	+	0.692617i	$0.243542\pi$
$278$	0	0
$279$	4.21432	0.252305
$280$	0	0
$281$	10.6178	0.633403	0.316702	−	0.948525i	$-0.397425\pi$
$281$	10.6178	0.633403	0.316702	+	0.948525i	$0.397425\pi$
$282$	0	0
$283$	−8.88247	−0.528008	−0.264004	−	0.964522i	$-0.585043\pi$
$283$	−8.88247	−0.528008	−0.264004	+	0.964522i	$0.585043\pi$
$284$	0	0
$285$	9.47949	0.561517
$286$	0	0
$287$	−9.58120	−0.565561
$288$	0	0
$289$	−13.9733	−0.821958
$290$	0	0
$291$	−0.428639	−0.0251273
$292$	0	0
$293$	−21.1985	−1.23843	−0.619215	−	0.785222i	$-0.712549\pi$
$293$	−21.1985	−1.23843	−0.619215	+	0.785222i	$0.712549\pi$
$294$	0	0
$295$	3.70471	0.215697
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−0.474572	−0.0274452
$300$	0	0
$301$	−12.4603	−0.718199
$302$	0	0
$303$	−3.93332	−0.225964
$304$	0	0
$305$	−12.8573	−0.736206
$306$	0	0
$307$	−10.0874	−0.575719	−0.287860	−	0.957673i	$-0.592944\pi$
$307$	−10.0874	−0.575719	−0.287860	+	0.957673i	$0.592944\pi$
$308$	0	0
$309$	−13.8064	−0.785420
$310$	0	0
$311$	−17.0366	−0.966055	−0.483027	−	0.875605i	$-0.660463\pi$
$311$	−17.0366	−0.966055	−0.483027	+	0.875605i	$0.660463\pi$
$312$	0	0
$313$	−19.5986	−1.10778	−0.553888	−	0.832591i	$-0.686856\pi$
$313$	−19.5986	−1.10778	−0.553888	+	0.832591i	$0.686856\pi$
$314$	0	0
$315$	−3.37778	−0.190317
$316$	0	0
$317$	−31.8687	−1.78992	−0.894961	−	0.446144i	$-0.852797\pi$
$317$	−31.8687	−1.78992	−0.894961	+	0.446144i	$0.852797\pi$
$318$	0	0
$319$	−5.11753	−0.286527
$320$	0	0
$321$	9.67307	0.539898
$322$	0	0
$323$	7.44785	0.414410
$324$	0	0
$325$	−0.0967881	−0.00536884
$326$	0	0
$327$	17.6271	0.974783
$328$	0	0
$329$	7.70471	0.424775
$330$	0	0
$331$	−1.09033	−0.0599302	−0.0299651	−	0.999551i	$-0.509540\pi$
$331$	−1.09033	−0.0599302	−0.0299651	+	0.999551i	$0.509540\pi$
$332$	0	0
$333$	0.428639	0.0234893
$334$	0	0
$335$	13.3319	0.728397
$336$	0	0
$337$	−11.0825	−0.603702	−0.301851	−	0.953355i	$-0.597605\pi$
$337$	−11.0825	−0.603702	−0.301851	+	0.953355i	$0.597605\pi$
$338$	0	0
$339$	−13.4795	−0.732106
$340$	0	0
$341$	−4.21432	−0.228218
$342$	0	0
$343$	17.8064	0.961457
$344$	0	0
$345$	1.05086	0.0565761
$346$	0	0
$347$	0.326929	0.0175505	0.00877524	−	0.999961i	$-0.497207\pi$
$347$	0.326929	0.0175505	0.00877524	+	0.999961i	$0.497207\pi$
$348$	0	0
$349$	−3.93978	−0.210891	−0.105446	−	0.994425i	$-0.533627\pi$
$349$	−3.93978	−0.210891	−0.105446	+	0.994425i	$0.533627\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−36.7862	−1.95793	−0.978965	−	0.204029i	$-0.934596\pi$
$353$	−36.7862	−1.95793	−0.978965	+	0.204029i	$0.934596\pi$
$354$	0	0
$355$	−11.4795	−0.609268
$356$	0	0
$357$	−2.65386	−0.140457
$358$	0	0
$359$	5.48394	0.289431	0.144716	−	0.989473i	$-0.453773\pi$
$359$	5.48394	0.289431	0.144716	+	0.989473i	$0.453773\pi$
$360$	0	0
$361$	−0.673071	−0.0354248
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−17.9081	−0.937355
$366$	0	0
$367$	−5.28592	−0.275923	−0.137961	−	0.990438i	$-0.544055\pi$
$367$	−5.28592	−0.275923	−0.137961	+	0.990438i	$0.544055\pi$
$368$	0	0
$369$	6.28100	0.326976
$370$	0	0
$371$	21.4193	1.11203
$372$	0	0
$373$	−13.5714	−0.702698	−0.351349	−	0.936244i	$-0.614277\pi$
$373$	−13.5714	−0.702698	−0.351349	+	0.936244i	$0.614277\pi$
$374$	0	0
$375$	11.2859	0.582802
$376$	0	0
$377$	5.11753	0.263566
$378$	0	0
$379$	−0.836535	−0.0429699	−0.0214850	−	0.999769i	$-0.506839\pi$
$379$	−0.836535	−0.0429699	−0.0214850	+	0.999769i	$0.506839\pi$
$380$	0	0
$381$	20.2415	1.03700
$382$	0	0
$383$	12.1334	0.619985	0.309993	−	0.950739i	$-0.399673\pi$
$383$	12.1334	0.619985	0.309993	+	0.950739i	$0.399673\pi$
$384$	0	0
$385$	3.37778	0.172148
$386$	0	0
$387$	8.16839	0.415222
$388$	0	0
$389$	−20.9175	−1.06056	−0.530280	−	0.847823i	$-0.677913\pi$
$389$	−20.9175	−1.06056	−0.530280	+	0.847823i	$0.677913\pi$
$390$	0	0
$391$	0.825636	0.0417542
$392$	0	0
$393$	8.29529	0.418442
$394$	0	0
$395$	−0.576283	−0.0289959
$396$	0	0
$397$	3.45091	0.173196	0.0865982	−	0.996243i	$-0.472400\pi$
$397$	3.45091	0.173196	0.0865982	+	0.996243i	$0.472400\pi$
$398$	0	0
$399$	−6.53035	−0.326926
$400$	0	0
$401$	38.1639	1.90582	0.952908	−	0.303259i	$-0.0980748\pi$
$401$	38.1639	1.90582	0.952908	+	0.303259i	$0.0980748\pi$
$402$	0	0
$403$	4.21432	0.209930
$404$	0	0
$405$	2.21432	0.110030
$406$	0	0
$407$	−0.428639	−0.0212469
$408$	0	0
$409$	38.0973	1.88379	0.941894	−	0.335910i	$-0.109044\pi$
$409$	38.0973	1.88379	0.941894	+	0.335910i	$0.109044\pi$
$410$	0	0
$411$	14.7447	0.727301
$412$	0	0
$413$	−2.55215	−0.125583
$414$	0	0
$415$	3.05086	0.149761
$416$	0	0
$417$	−9.05731	−0.443538
$418$	0	0
$419$	−9.73191	−0.475435	−0.237717	−	0.971334i	$-0.576399\pi$
$419$	−9.73191	−0.475435	−0.237717	+	0.971334i	$0.576399\pi$
$420$	0	0
$421$	−25.0509	−1.22090	−0.610452	−	0.792053i	$-0.709012\pi$
$421$	−25.0509	−1.22090	−0.610452	+	0.792053i	$0.709012\pi$
$422$	0	0
$423$	−5.05086	−0.245581
$424$	0	0
$425$	0.168387	0.00816796
$426$	0	0
$427$	8.85728	0.428634
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	23.2543	1.12012	0.560060	−	0.828452i	$-0.310778\pi$
$431$	23.2543	1.12012	0.560060	+	0.828452i	$0.310778\pi$
$432$	0	0
$433$	−20.1017	−0.966027	−0.483013	−	0.875613i	$-0.660458\pi$
$433$	−20.1017	−0.966027	−0.483013	+	0.875613i	$0.660458\pi$
$434$	0	0
$435$	−11.3319	−0.543321
$436$	0	0
$437$	2.03164	0.0971867
$438$	0	0
$439$	−32.1180	−1.53291	−0.766454	−	0.642299i	$-0.777981\pi$
$439$	−32.1180	−1.53291	−0.766454	+	0.642299i	$0.777981\pi$
$440$	0	0
$441$	−4.67307	−0.222527
$442$	0	0
$443$	−30.8385	−1.46518	−0.732592	−	0.680668i	$-0.761689\pi$
$443$	−30.8385	−1.46518	−0.732592	+	0.680668i	$0.761689\pi$
$444$	0	0
$445$	−22.4844	−1.06586
$446$	0	0
$447$	2.01429	0.0952727
$448$	0	0
$449$	−19.3254	−0.912022	−0.456011	−	0.889974i	$-0.650722\pi$
$449$	−19.3254	−0.912022	−0.456011	+	0.889974i	$0.650722\pi$
$450$	0	0
$451$	−6.28100	−0.295761
$452$	0	0
$453$	0.668149	0.0313924
$454$	0	0
$455$	−3.37778	−0.158353
$456$	0	0
$457$	−8.58073	−0.401390	−0.200695	−	0.979654i	$-0.564320\pi$
$457$	−8.58073	−0.401390	−0.200695	+	0.979654i	$0.564320\pi$
$458$	0	0
$459$	1.73975	0.0812045
$460$	0	0
$461$	−4.94470	−0.230298	−0.115149	−	0.993348i	$-0.536734\pi$
$461$	−4.94470	−0.230298	−0.115149	+	0.993348i	$0.536734\pi$
$462$	0	0
$463$	−19.9289	−0.926173	−0.463087	−	0.886313i	$-0.653258\pi$
$463$	−19.9289	−0.926173	−0.463087	+	0.886313i	$0.653258\pi$
$464$	0	0
$465$	−9.33185	−0.432754
$466$	0	0
$467$	−18.5763	−0.859608	−0.429804	−	0.902922i	$-0.641417\pi$
$467$	−18.5763	−0.859608	−0.429804	+	0.902922i	$0.641417\pi$
$468$	0	0
$469$	−9.18421	−0.424087
$470$	0	0
$471$	−4.70964	−0.217009
$472$	0	0
$473$	−8.16839	−0.375583
$474$	0	0
$475$	0.414349	0.0190117
$476$	0	0
$477$	−14.0415	−0.642916
$478$	0	0
$479$	1.09679	0.0501135	0.0250568	−	0.999686i	$-0.492023\pi$
$479$	1.09679	0.0501135	0.0250568	+	0.999686i	$0.492023\pi$
$480$	0	0
$481$	0.428639	0.0195443
$482$	0	0
$483$	−0.723926	−0.0329398
$484$	0	0
$485$	0.949145	0.0430984
$486$	0	0
$487$	28.6113	1.29650	0.648251	−	0.761427i	$-0.275501\pi$
$487$	28.6113	1.29650	0.648251	+	0.761427i	$0.275501\pi$
$488$	0	0
$489$	−2.73483	−0.123673
$490$	0	0
$491$	19.4608	0.878252	0.439126	−	0.898426i	$-0.355288\pi$
$491$	19.4608	0.878252	0.439126	+	0.898426i	$0.355288\pi$
$492$	0	0
$493$	−8.90321	−0.400980
$494$	0	0
$495$	−2.21432	−0.0995263
$496$	0	0
$497$	7.90813	0.354728
$498$	0	0
$499$	0.601472	0.0269256	0.0134628	−	0.999909i	$-0.495715\pi$
$499$	0.601472	0.0269256	0.0134628	+	0.999909i	$0.495715\pi$
$500$	0	0
$501$	−0.561993	−0.0251080
$502$	0	0
$503$	−8.69535	−0.387706	−0.193853	−	0.981031i	$-0.562099\pi$
$503$	−8.69535	−0.387706	−0.193853	+	0.981031i	$0.562099\pi$
$504$	0	0
$505$	8.70964	0.387574
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−3.94761	−0.174975	−0.0874874	−	0.996166i	$-0.527884\pi$
$509$	−3.94761	−0.174975	−0.0874874	+	0.996166i	$0.527884\pi$
$510$	0	0
$511$	12.3368	0.545747
$512$	0	0
$513$	4.28100	0.189011
$514$	0	0
$515$	30.5718	1.34716
$516$	0	0
$517$	5.05086	0.222136
$518$	0	0
$519$	−8.88247	−0.389897
$520$	0	0
$521$	22.4701	0.984434	0.492217	−	0.870472i	$-0.336187\pi$
$521$	22.4701	0.984434	0.492217	+	0.870472i	$0.336187\pi$
$522$	0	0
$523$	21.4257	0.936882	0.468441	−	0.883495i	$-0.344816\pi$
$523$	21.4257	0.936882	0.468441	+	0.883495i	$0.344816\pi$
$524$	0	0
$525$	−0.147643	−0.00644368
$526$	0	0
$527$	−7.33185	−0.319380
$528$	0	0
$529$	−22.7748	−0.990208
$530$	0	0
$531$	1.67307	0.0726051
$532$	0	0
$533$	6.28100	0.272060
$534$	0	0
$535$	−21.4193	−0.926036
$536$	0	0
$537$	−4.51606	−0.194882
$538$	0	0
$539$	4.67307	0.201283
$540$	0	0
$541$	−26.6735	−1.14679	−0.573393	−	0.819281i	$-0.694373\pi$
$541$	−26.6735	−1.14679	−0.573393	+	0.819281i	$0.694373\pi$
$542$	0	0
$543$	24.7096	1.06039
$544$	0	0
$545$	−39.0321	−1.67195
$546$	0	0
$547$	16.1497	0.690509	0.345255	−	0.938509i	$-0.387793\pi$
$547$	16.1497	0.690509	0.345255	+	0.938509i	$0.387793\pi$
$548$	0	0
$549$	−5.80642	−0.247812
$550$	0	0
$551$	−21.9081	−0.933318
$552$	0	0
$553$	0.396997	0.0168820
$554$	0	0
$555$	−0.949145	−0.0402890
$556$	0	0
$557$	−25.2400	−1.06945	−0.534726	−	0.845025i	$-0.679585\pi$
$557$	−25.2400	−1.06945	−0.534726	+	0.845025i	$0.679585\pi$
$558$	0	0
$559$	8.16839	0.345486
$560$	0	0
$561$	−1.73975	−0.0734522
$562$	0	0
$563$	8.37826	0.353102	0.176551	−	0.984292i	$-0.443506\pi$
$563$	8.37826	0.353102	0.176551	+	0.984292i	$0.443506\pi$
$564$	0	0
$565$	29.8479	1.25571
$566$	0	0
$567$	−1.52543	−0.0640619
$568$	0	0
$569$	37.9432	1.59066	0.795330	−	0.606176i	$-0.207297\pi$
$569$	37.9432	1.59066	0.795330	+	0.606176i	$0.207297\pi$
$570$	0	0
$571$	−30.1782	−1.26292	−0.631460	−	0.775409i	$-0.717544\pi$
$571$	−30.1782	−1.26292	−0.631460	+	0.775409i	$0.717544\pi$
$572$	0	0
$573$	12.9906	0.542691
$574$	0	0
$575$	0.0459330	0.00191554
$576$	0	0
$577$	−24.1936	−1.00719	−0.503596	−	0.863939i	$-0.667990\pi$
$577$	−24.1936	−1.00719	−0.503596	+	0.863939i	$0.667990\pi$
$578$	0	0
$579$	−15.0464	−0.625307
$580$	0	0
$581$	−2.10171	−0.0871936
$582$	0	0
$583$	14.0415	0.581539
$584$	0	0
$585$	2.21432	0.0915509
$586$	0	0
$587$	30.7971	1.27113	0.635565	−	0.772047i	$-0.280767\pi$
$587$	30.7971	1.27113	0.635565	+	0.772047i	$0.280767\pi$
$588$	0	0
$589$	−18.0415	−0.743387
$590$	0	0
$591$	−15.8622	−0.652484
$592$	0	0
$593$	14.7382	0.605226	0.302613	−	0.953114i	$-0.402141\pi$
$593$	14.7382	0.605226	0.302613	+	0.953114i	$0.402141\pi$
$594$	0	0
$595$	5.87649	0.240913
$596$	0	0
$597$	13.4193	0.549214
$598$	0	0
$599$	−11.5999	−0.473961	−0.236980	−	0.971514i	$-0.576158\pi$
$599$	−11.5999	−0.473961	−0.236980	+	0.971514i	$0.576158\pi$
$600$	0	0
$601$	27.3274	1.11471	0.557354	−	0.830275i	$-0.311817\pi$
$601$	27.3274	1.11471	0.557354	+	0.830275i	$0.311817\pi$
$602$	0	0
$603$	6.02074	0.245184
$604$	0	0
$605$	2.21432	0.0900249
$606$	0	0
$607$	−0.0952567	−0.00386635	−0.00193318	−	0.999998i	$-0.500615\pi$
$607$	−0.0952567	−0.00386635	−0.00193318	+	0.999998i	$0.500615\pi$
$608$	0	0
$609$	7.80642	0.316332
$610$	0	0
$611$	−5.05086	−0.204336
$612$	0	0
$613$	33.3230	1.34590	0.672951	−	0.739687i	$-0.265027\pi$
$613$	33.3230	1.34590	0.672951	+	0.739687i	$0.265027\pi$
$614$	0	0
$615$	−13.9081	−0.560830
$616$	0	0
$617$	41.7353	1.68020	0.840100	−	0.542432i	$-0.182496\pi$
$617$	41.7353	1.68020	0.840100	+	0.542432i	$0.182496\pi$
$618$	0	0
$619$	−29.8687	−1.20052	−0.600261	−	0.799804i	$-0.704937\pi$
$619$	−29.8687	−1.20052	−0.600261	+	0.799804i	$0.704937\pi$
$620$	0	0
$621$	0.474572	0.0190439
$622$	0	0
$623$	15.4893	0.620567
$624$	0	0
$625$	−24.5067	−0.980268
$626$	0	0
$627$	−4.28100	−0.170967
$628$	0	0
$629$	−0.745724	−0.0297340
$630$	0	0
$631$	−12.3160	−0.490293	−0.245147	−	0.969486i	$-0.578836\pi$
$631$	−12.3160	−0.490293	−0.245147	+	0.969486i	$0.578836\pi$
$632$	0	0
$633$	24.1684	0.960607
$634$	0	0
$635$	−44.8212	−1.77867
$636$	0	0
$637$	−4.67307	−0.185154
$638$	0	0
$639$	−5.18421	−0.205084
$640$	0	0
$641$	44.0928	1.74156	0.870781	−	0.491671i	$-0.163614\pi$
$641$	44.0928	1.74156	0.870781	+	0.491671i	$0.163614\pi$
$642$	0	0
$643$	−32.1225	−1.26679	−0.633393	−	0.773830i	$-0.718338\pi$
$643$	−32.1225	−1.26679	−0.633393	+	0.773830i	$0.718338\pi$
$644$	0	0
$645$	−18.0874	−0.712191
$646$	0	0
$647$	−7.47949	−0.294049	−0.147025	−	0.989133i	$-0.546970\pi$
$647$	−7.47949	−0.294049	−0.147025	+	0.989133i	$0.546970\pi$
$648$	0	0
$649$	−1.67307	−0.0656738
$650$	0	0
$651$	6.42864	0.251958
$652$	0	0
$653$	41.3087	1.61653	0.808267	−	0.588817i	$-0.200406\pi$
$653$	41.3087	1.61653	0.808267	+	0.588817i	$0.200406\pi$
$654$	0	0
$655$	−18.3684	−0.717713
$656$	0	0
$657$	−8.08742	−0.315520
$658$	0	0
$659$	12.6637	0.493308	0.246654	−	0.969104i	$-0.420669\pi$
$659$	12.6637	0.493308	0.246654	+	0.969104i	$0.420669\pi$
$660$	0	0
$661$	13.5999	0.528976	0.264488	−	0.964389i	$-0.414797\pi$
$661$	13.5999	0.528976	0.264488	+	0.964389i	$0.414797\pi$
$662$	0	0
$663$	1.73975	0.0675662
$664$	0	0
$665$	14.4603	0.560746
$666$	0	0
$667$	−2.42864	−0.0940373
$668$	0	0
$669$	−9.50024	−0.367300
$670$	0	0
$671$	5.80642	0.224155
$672$	0	0
$673$	−12.9621	−0.499650	−0.249825	−	0.968291i	$-0.580373\pi$
$673$	−12.9621	−0.499650	−0.249825	+	0.968291i	$0.580373\pi$
$674$	0	0
$675$	0.0967881	0.00372537
$676$	0	0
$677$	−19.0988	−0.734026	−0.367013	−	0.930216i	$-0.619620\pi$
$677$	−19.0988	−0.734026	−0.367013	+	0.930216i	$0.619620\pi$
$678$	0	0
$679$	−0.653858	−0.0250928
$680$	0	0
$681$	8.41435	0.322439
$682$	0	0
$683$	−47.0450	−1.80013	−0.900064	−	0.435758i	$-0.856480\pi$
$683$	−47.0450	−1.80013	−0.900064	+	0.435758i	$0.856480\pi$
$684$	0	0
$685$	−32.6494	−1.24747
$686$	0	0
$687$	−29.0321	−1.10764
$688$	0	0
$689$	−14.0415	−0.534938
$690$	0	0
$691$	−44.5827	−1.69601	−0.848004	−	0.529990i	$-0.822196\pi$
$691$	−44.5827	−1.69601	−0.848004	+	0.529990i	$0.822196\pi$
$692$	0	0
$693$	1.52543	0.0579462
$694$	0	0
$695$	20.0558	0.760759
$696$	0	0
$697$	−10.9273	−0.413903
$698$	0	0
$699$	−12.3017	−0.465295
$700$	0	0
$701$	43.3624	1.63778	0.818888	−	0.573953i	$-0.194591\pi$
$701$	43.3624	1.63778	0.818888	+	0.573953i	$0.194591\pi$
$702$	0	0
$703$	−1.83500	−0.0692085
$704$	0	0
$705$	11.1842	0.421222
$706$	0	0
$707$	−6.00000	−0.225653
$708$	0	0
$709$	−3.34614	−0.125667	−0.0628335	−	0.998024i	$-0.520014\pi$
$709$	−3.34614	−0.125667	−0.0628335	+	0.998024i	$0.520014\pi$
$710$	0	0
$711$	−0.260253	−0.00976024
$712$	0	0
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	−2.21432	−0.0828109
$716$	0	0
$717$	21.5669	0.805431
$718$	0	0
$719$	45.3274	1.69043	0.845213	−	0.534429i	$-0.179473\pi$
$719$	45.3274	1.69043	0.845213	+	0.534429i	$0.179473\pi$
$720$	0	0
$721$	−21.0607	−0.784341
$722$	0	0
$723$	25.0005	0.929778
$724$	0	0
$725$	−0.495316	−0.0183956
$726$	0	0
$727$	24.2163	0.898134	0.449067	−	0.893498i	$-0.351756\pi$
$727$	24.2163	0.898134	0.449067	+	0.893498i	$0.351756\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.2109	−0.525610
$732$	0	0
$733$	−31.9037	−1.17839	−0.589195	−	0.807991i	$-0.700555\pi$
$733$	−31.9037	−1.17839	−0.589195	+	0.807991i	$0.700555\pi$
$734$	0	0
$735$	10.3477	0.381680
$736$	0	0
$737$	−6.02074	−0.221777
$738$	0	0
$739$	−44.3926	−1.63301	−0.816503	−	0.577341i	$-0.804090\pi$
$739$	−44.3926	−1.63301	−0.816503	+	0.577341i	$0.804090\pi$
$740$	0	0
$741$	4.28100	0.157266
$742$	0	0
$743$	52.2449	1.91668	0.958340	−	0.285630i	$-0.0922029\pi$
$743$	52.2449	1.91668	0.958340	+	0.285630i	$0.0922029\pi$
$744$	0	0
$745$	−4.46028	−0.163412
$746$	0	0
$747$	1.37778	0.0504105
$748$	0	0
$749$	14.7556	0.539157
$750$	0	0
$751$	9.81933	0.358312	0.179156	−	0.983821i	$-0.442663\pi$
$751$	9.81933	0.358312	0.179156	+	0.983821i	$0.442663\pi$
$752$	0	0
$753$	11.4795	0.418336
$754$	0	0
$755$	−1.47949	−0.0538443
$756$	0	0
$757$	−22.5990	−0.821376	−0.410688	−	0.911776i	$-0.634711\pi$
$757$	−22.5990	−0.821376	−0.410688	+	0.911776i	$0.634711\pi$
$758$	0	0
$759$	−0.474572	−0.0172259
$760$	0	0
$761$	0.0745132	0.00270110	0.00135055	−	0.999999i	$-0.499570\pi$
$761$	0.0745132	0.00270110	0.00135055	+	0.999999i	$0.499570\pi$
$762$	0	0
$763$	26.8889	0.973444
$764$	0	0
$765$	−3.85236	−0.139282
$766$	0	0
$767$	1.67307	0.0604111
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	−12.1017	−0.435832
$772$	0	0
$773$	−2.66170	−0.0957345	−0.0478673	−	0.998854i	$-0.515242\pi$
$773$	−2.66170	−0.0957345	−0.0478673	+	0.998854i	$0.515242\pi$
$774$	0	0
$775$	−0.407896	−0.0146521
$776$	0	0
$777$	0.653858	0.0234570
$778$	0	0
$779$	−26.8889	−0.963396
$780$	0	0
$781$	5.18421	0.185506
$782$	0	0
$783$	−5.11753	−0.182886
$784$	0	0
$785$	10.4286	0.372214
$786$	0	0
$787$	−28.5763	−1.01863	−0.509317	−	0.860579i	$-0.670102\pi$
$787$	−28.5763	−1.01863	−0.509317	+	0.860579i	$0.670102\pi$
$788$	0	0
$789$	−19.6543	−0.699713
$790$	0	0
$791$	−20.5620	−0.731100
$792$	0	0
$793$	−5.80642	−0.206192
$794$	0	0
$795$	31.0923	1.10273
$796$	0	0
$797$	−27.6860	−0.980688	−0.490344	−	0.871529i	$-0.663129\pi$
$797$	−27.6860	−0.980688	−0.490344	+	0.871529i	$0.663129\pi$
$798$	0	0
$799$	8.78721	0.310869
$800$	0	0
$801$	−10.1541	−0.358777
$802$	0	0
$803$	8.08742	0.285399
$804$	0	0
$805$	1.60300	0.0564984
$806$	0	0
$807$	4.62222	0.162710
$808$	0	0
$809$	−43.7911	−1.53961	−0.769806	−	0.638278i	$-0.779647\pi$
$809$	−43.7911	−1.53961	−0.769806	+	0.638278i	$0.779647\pi$
$810$	0	0
$811$	−42.4371	−1.49017	−0.745084	−	0.666971i	$-0.767591\pi$
$811$	−42.4371	−1.49017	−0.745084	+	0.666971i	$0.767591\pi$
$812$	0	0
$813$	−7.37334	−0.258594
$814$	0	0
$815$	6.05578	0.212125
$816$	0	0
$817$	−34.9688	−1.22340
$818$	0	0
$819$	−1.52543	−0.0533028
$820$	0	0
$821$	−44.5705	−1.55552	−0.777760	−	0.628562i	$-0.783644\pi$
$821$	−44.5705	−1.55552	−0.777760	+	0.628562i	$0.783644\pi$
$822$	0	0
$823$	19.8666	0.692508	0.346254	−	0.938141i	$-0.387454\pi$
$823$	19.8666	0.692508	0.346254	+	0.938141i	$0.387454\pi$
$824$	0	0
$825$	−0.0967881	−0.00336973
$826$	0	0
$827$	23.3822	0.813080	0.406540	−	0.913633i	$-0.366735\pi$
$827$	23.3822	0.813080	0.406540	+	0.913633i	$0.366735\pi$
$828$	0	0
$829$	9.52987	0.330986	0.165493	−	0.986211i	$-0.447078\pi$
$829$	9.52987	0.330986	0.165493	+	0.986211i	$0.447078\pi$
$830$	0	0
$831$	−24.0098	−0.832892
$832$	0	0
$833$	8.12996	0.281686
$834$	0	0
$835$	1.24443	0.0430653
$836$	0	0
$837$	−4.21432	−0.145668
$838$	0	0
$839$	6.58073	0.227192	0.113596	−	0.993527i	$-0.463763\pi$
$839$	6.58073	0.227192	0.113596	+	0.993527i	$0.463763\pi$
$840$	0	0
$841$	−2.81087	−0.0969265
$842$	0	0
$843$	−10.6178	−0.365695
$844$	0	0
$845$	2.21432	0.0761749
$846$	0	0
$847$	−1.52543	−0.0524143
$848$	0	0
$849$	8.88247	0.304846
$850$	0	0
$851$	−0.203420	−0.00697316
$852$	0	0
$853$	0.726989	0.0248916	0.0124458	−	0.999923i	$-0.496038\pi$
$853$	0.726989	0.0248916	0.0124458	+	0.999923i	$0.496038\pi$
$854$	0	0
$855$	−9.47949	−0.324192
$856$	0	0
$857$	28.7304	0.981411	0.490706	−	0.871325i	$-0.336739\pi$
$857$	28.7304	0.981411	0.490706	+	0.871325i	$0.336739\pi$
$858$	0	0
$859$	−56.9273	−1.94234	−0.971168	−	0.238396i	$-0.923378\pi$
$859$	−56.9273	−1.94234	−0.971168	+	0.238396i	$0.923378\pi$
$860$	0	0
$861$	9.58120	0.326527
$862$	0	0
$863$	−7.86665	−0.267784	−0.133892	−	0.990996i	$-0.542747\pi$
$863$	−7.86665	−0.267784	−0.133892	+	0.990996i	$0.542747\pi$
$864$	0	0
$865$	19.6686	0.668753
$866$	0	0
$867$	13.9733	0.474557
$868$	0	0
$869$	0.260253	0.00882847
$870$	0	0
$871$	6.02074	0.204005
$872$	0	0
$873$	0.428639	0.0145072
$874$	0	0
$875$	17.2159	0.582002
$876$	0	0
$877$	−5.18421	−0.175058	−0.0875291	−	0.996162i	$-0.527897\pi$
$877$	−5.18421	−0.175058	−0.0875291	+	0.996162i	$0.527897\pi$
$878$	0	0
$879$	21.1985	0.715008
$880$	0	0
$881$	23.5111	0.792110	0.396055	−	0.918227i	$-0.370379\pi$
$881$	23.5111	0.792110	0.396055	+	0.918227i	$0.370379\pi$
$882$	0	0
$883$	6.07313	0.204377	0.102189	−	0.994765i	$-0.467415\pi$
$883$	6.07313	0.204377	0.102189	+	0.994765i	$0.467415\pi$
$884$	0	0
$885$	−3.70471	−0.124533
$886$	0	0
$887$	−46.4612	−1.56002	−0.780008	−	0.625770i	$-0.784785\pi$
$887$	−46.4612	−1.56002	−0.780008	+	0.625770i	$0.784785\pi$
$888$	0	0
$889$	30.8770	1.03558
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	21.6227	0.723576
$894$	0	0
$895$	10.0000	0.334263
$896$	0	0
$897$	0.474572	0.0158455
$898$	0	0
$899$	21.5669	0.719297
$900$	0	0
$901$	24.4286	0.813836
$902$	0	0
$903$	12.4603	0.414652
$904$	0	0
$905$	−54.7150	−1.81879
$906$	0	0
$907$	−44.3912	−1.47398	−0.736992	−	0.675901i	$-0.763755\pi$
$907$	−44.3912	−1.47398	−0.736992	+	0.675901i	$0.763755\pi$
$908$	0	0
$909$	3.93332	0.130460
$910$	0	0
$911$	−10.4385	−0.345842	−0.172921	−	0.984936i	$-0.555321\pi$
$911$	−10.4385	−0.345842	−0.172921	+	0.984936i	$0.555321\pi$
$912$	0	0
$913$	−1.37778	−0.0455980
$914$	0	0
$915$	12.8573	0.425049
$916$	0	0
$917$	12.6539	0.417867
$918$	0	0
$919$	20.4953	0.676078	0.338039	−	0.941132i	$-0.390236\pi$
$919$	20.4953	0.676078	0.338039	+	0.941132i	$0.390236\pi$
$920$	0	0
$921$	10.0874	0.332392
$922$	0	0
$923$	−5.18421	−0.170640
$924$	0	0
$925$	−0.0414872	−0.00136409
$926$	0	0
$927$	13.8064	0.453462
$928$	0	0
$929$	−12.8178	−0.420538	−0.210269	−	0.977644i	$-0.567434\pi$
$929$	−12.8178	−0.420538	−0.210269	+	0.977644i	$0.567434\pi$
$930$	0	0
$931$	20.0054	0.655650
$932$	0	0
$933$	17.0366	0.557752
$934$	0	0
$935$	3.85236	0.125986
$936$	0	0
$937$	−33.3145	−1.08834	−0.544169	−	0.838976i	$-0.683155\pi$
$937$	−33.3145	−1.08834	−0.544169	+	0.838976i	$0.683155\pi$
$938$	0	0
$939$	19.5986	0.639575
$940$	0	0
$941$	55.1008	1.79623	0.898117	−	0.439756i	$-0.144935\pi$
$941$	55.1008	1.79623	0.898117	+	0.439756i	$0.144935\pi$
$942$	0	0
$943$	−2.98079	−0.0970678
$944$	0	0
$945$	3.37778	0.109879
$946$	0	0
$947$	13.6958	0.445054	0.222527	−	0.974926i	$-0.428569\pi$
$947$	13.6958	0.445054	0.222527	+	0.974926i	$0.428569\pi$
$948$	0	0
$949$	−8.08742	−0.262529
$950$	0	0
$951$	31.8687	1.03341
$952$	0	0
$953$	−12.0350	−0.389853	−0.194926	−	0.980818i	$-0.562447\pi$
$953$	−12.0350	−0.389853	−0.194926	+	0.980818i	$0.562447\pi$
$954$	0	0
$955$	−28.7654	−0.930827
$956$	0	0
$957$	5.11753	0.165426
$958$	0	0
$959$	22.4919	0.726302
$960$	0	0
$961$	−13.2395	−0.427081
$962$	0	0
$963$	−9.67307	−0.311710
$964$	0	0
$965$	33.3176	1.07253
$966$	0	0
$967$	59.5768	1.91586	0.957930	−	0.287003i	$-0.0926589\pi$
$967$	59.5768	1.91586	0.957930	+	0.287003i	$0.0926589\pi$
$968$	0	0
$969$	−7.44785	−0.239259
$970$	0	0
$971$	53.2944	1.71030	0.855149	−	0.518382i	$-0.173466\pi$
$971$	53.2944	1.71030	0.855149	+	0.518382i	$0.173466\pi$
$972$	0	0
$973$	−13.8163	−0.442929
$974$	0	0
$975$	0.0967881	0.00309970
$976$	0	0
$977$	−0.481026	−0.0153894	−0.00769469	−	0.999970i	$-0.502449\pi$
$977$	−0.481026	−0.0153894	−0.00769469	+	0.999970i	$0.502449\pi$
$978$	0	0
$979$	10.1541	0.324526
$980$	0	0
$981$	−17.6271	−0.562791
$982$	0	0
$983$	−32.4612	−1.03535	−0.517676	−	0.855577i	$-0.673203\pi$
$983$	−32.4612	−1.03535	−0.517676	+	0.855577i	$0.673203\pi$
$984$	0	0
$985$	35.1240	1.11914
$986$	0	0
$987$	−7.70471	−0.245244
$988$	0	0
$989$	−3.87649	−0.123265
$990$	0	0
$991$	−61.3056	−1.94744	−0.973718	−	0.227755i	$-0.926861\pi$
$991$	−61.3056	−1.94744	−0.973718	+	0.227755i	$0.926861\pi$
$992$	0	0
$993$	1.09033	0.0346007
$994$	0	0
$995$	−29.7146	−0.942015
$996$	0	0
$997$	−3.20294	−0.101438	−0.0507191	−	0.998713i	$-0.516151\pi$
$997$	−3.20294	−0.101438	−0.0507191	+	0.998713i	$0.516151\pi$
$998$	0	0
$999$	−0.428639	−0.0135616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bp.1.3		3
4.3	odd	2		429.2.a.f.1.2	✓	3
12.11	even	2		1287.2.a.i.1.2		3
44.43	even	2		4719.2.a.t.1.2		3
52.51	odd	2		5577.2.a.k.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.f.1.2	✓	3	4.3	odd	2
1287.2.a.i.1.2		3	12.11	even	2
4719.2.a.t.1.2		3	44.43	even	2
5577.2.a.k.1.2		3	52.51	odd	2
6864.2.a.bp.1.3		3	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.21432 q^{5} -1.52543 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.21432 q^{5} -1.52543 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -2.21432 q^{15} -1.73975 q^{17} -4.28100 q^{19} +1.52543 q^{21} -0.474572 q^{23} -0.0967881 q^{25} -1.00000 q^{27} +5.11753 q^{29} +4.21432 q^{31} +1.00000 q^{33} -3.37778 q^{35} +0.428639 q^{37} -1.00000 q^{39} +6.28100 q^{41} +8.16839 q^{43} +2.21432 q^{45} -5.05086 q^{47} -4.67307 q^{49} +1.73975 q^{51} -14.0415 q^{53} -2.21432 q^{55} +4.28100 q^{57} +1.67307 q^{59} -5.80642 q^{61} -1.52543 q^{63} +2.21432 q^{65} +6.02074 q^{67} +0.474572 q^{69} -5.18421 q^{71} -8.08742 q^{73} +0.0967881 q^{75} +1.52543 q^{77} -0.260253 q^{79} +1.00000 q^{81} +1.37778 q^{83} -3.85236 q^{85} -5.11753 q^{87} -10.1541 q^{89} -1.52543 q^{91} -4.21432 q^{93} -9.47949 q^{95} +0.428639 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 8 q^{17} - 6 q^{19} - 2 q^{21} - 8 q^{23} - 7 q^{25} - 3 q^{27} + 2 q^{29} + 6 q^{31} + 3 q^{33} - 10 q^{35} - 12 q^{37} - 3 q^{39} + 12 q^{41} - 2 q^{43} - 2 q^{47} - q^{49} - 8 q^{51} - 2 q^{53} + 6 q^{57} - 8 q^{59} - 4 q^{61} + 2 q^{63} - 2 q^{67} + 8 q^{69} - 2 q^{71} - 4 q^{73} + 7 q^{75} - 2 q^{77} - 14 q^{79} + 3 q^{81} + 4 q^{83} - 18 q^{85} - 2 q^{87} - 10 q^{89} + 2 q^{91} - 6 q^{93} - 2 q^{95} - 12 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 8 * q^17 - 6 * q^19 - 2 * q^21 - 8 * q^23 - 7 * q^25 - 3 * q^27 + 2 * q^29 + 6 * q^31 + 3 * q^33 - 10 * q^35 - 12 * q^37 - 3 * q^39 + 12 * q^41 - 2 * q^43 - 2 * q^47 - q^49 - 8 * q^51 - 2 * q^53 + 6 * q^57 - 8 * q^59 - 4 * q^61 + 2 * q^63 - 2 * q^67 + 8 * q^69 - 2 * q^71 - 4 * q^73 + 7 * q^75 - 2 * q^77 - 14 * q^79 + 3 * q^81 + 4 * q^83 - 18 * q^85 - 2 * q^87 - 10 * q^89 + 2 * q^91 - 6 * q^93 - 2 * q^95 - 12 * q^97 - 3 * q^99

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