LMFDB - Embedded newform 6864.2.a.bo.1.1

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:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.59774$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.59774 q^{5} -2.74823 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.59774 q^{5} -2.74823 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +2.59774 q^{15} +1.34596 q^{17} -2.00000 q^{19} +2.74823 q^{21} -3.94370 q^{23} +1.74823 q^{25} -1.00000 q^{27} -2.09419 q^{29} +1.34596 q^{31} -1.00000 q^{33} +7.13917 q^{35} +6.00000 q^{37} -1.00000 q^{39} +5.94370 q^{41} +5.28966 q^{43} -2.59774 q^{45} +6.69193 q^{47} +0.552758 q^{49} -1.34596 q^{51} +2.00000 q^{53} -2.59774 q^{55} +2.00000 q^{57} +8.63563 q^{59} +7.94370 q^{61} -2.74823 q^{63} -2.59774 q^{65} -14.9887 q^{67} +3.94370 q^{69} -7.88740 q^{71} +2.95079 q^{73} -1.74823 q^{75} -2.74823 q^{77} -7.34596 q^{79} +1.00000 q^{81} -1.49646 q^{83} -3.49646 q^{85} +2.09419 q^{87} +8.54143 q^{89} -2.74823 q^{91} -1.34596 q^{93} +5.19547 q^{95} +4.99291 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{13} + q^{15} - 6 q^{17} - 6 q^{19} + 5 q^{21} + 5 q^{23} + 2 q^{25} - 3 q^{27} + 7 q^{29} - 6 q^{31} - 3 q^{33} - 9 q^{35} + 18 q^{37} - 3 q^{39} + q^{41} - 11 q^{43} - q^{45} + 12 q^{49} + 6 q^{51} + 6 q^{53} - q^{55} + 6 q^{57} - 11 q^{59} + 7 q^{61} - 5 q^{63} - q^{65} - 11 q^{67} - 5 q^{69} + 10 q^{71} + 5 q^{73} - 2 q^{75} - 5 q^{77} - 12 q^{79} + 3 q^{81} + 2 q^{83} - 4 q^{85} - 7 q^{87} + 2 q^{89} - 5 q^{91} + 6 q^{93} + 2 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^13 + q^15 - 6 * q^17 - 6 * q^19 + 5 * q^21 + 5 * q^23 + 2 * q^25 - 3 * q^27 + 7 * q^29 - 6 * q^31 - 3 * q^33 - 9 * q^35 + 18 * q^37 - 3 * q^39 + q^41 - 11 * q^43 - q^45 + 12 * q^49 + 6 * q^51 + 6 * q^53 - q^55 + 6 * q^57 - 11 * q^59 + 7 * q^61 - 5 * q^63 - q^65 - 11 * q^67 - 5 * q^69 + 10 * q^71 + 5 * q^73 - 2 * q^75 - 5 * q^77 - 12 * q^79 + 3 * q^81 + 2 * q^83 - 4 * q^85 - 7 * q^87 + 2 * q^89 - 5 * q^91 + 6 * q^93 + 2 * q^95 + 2 * 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
$5$	−2.59774	−1.16174	−0.580871	−	0.813995i	$-0.697288\pi$
$5$	−2.59774	−1.16174	−0.580871	+	0.813995i	$0.697288\pi$
$6$	0	0
$7$	−2.74823	−1.03873	−0.519366	−	0.854552i	$-0.673832\pi$
$7$	−2.74823	−1.03873	−0.519366	+	0.854552i	$0.673832\pi$
$8$	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
$15$	2.59774	0.670732
$16$	0	0
$17$	1.34596	0.326444	0.163222	−	0.986589i	$-0.447811\pi$
$17$	1.34596	0.326444	0.163222	+	0.986589i	$0.447811\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	2.74823	0.599713
$22$	0	0
$23$	−3.94370	−0.822318	−0.411159	−	0.911564i	$-0.634876\pi$
$23$	−3.94370	−0.822318	−0.411159	+	0.911564i	$0.634876\pi$
$24$	0	0
$25$	1.74823	0.349646
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.09419	−0.388882	−0.194441	−	0.980914i	$-0.562289\pi$
$29$	−2.09419	−0.388882	−0.194441	+	0.980914i	$0.562289\pi$
$30$	0	0
$31$	1.34596	0.241742	0.120871	−	0.992668i	$-0.461431\pi$
$31$	1.34596	0.241742	0.120871	+	0.992668i	$0.461431\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	7.13917	1.20674
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	5.94370	0.928250	0.464125	−	0.885770i	$-0.346369\pi$
$41$	5.94370	0.928250	0.464125	+	0.885770i	$0.346369\pi$
$42$	0	0
$43$	5.28966	0.806666	0.403333	−	0.915053i	$-0.367852\pi$
$43$	5.28966	0.806666	0.403333	+	0.915053i	$0.367852\pi$
$44$	0	0
$45$	−2.59774	−0.387247
$46$	0	0
$47$	6.69193	0.976118	0.488059	−	0.872811i	$-0.337705\pi$
$47$	6.69193	0.976118	0.488059	+	0.872811i	$0.337705\pi$
$48$	0	0
$49$	0.552758	0.0789654
$50$	0	0
$51$	−1.34596	−0.188473
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−2.59774	−0.350279
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	8.63563	1.12426	0.562131	−	0.827048i	$-0.309982\pi$
$59$	8.63563	1.12426	0.562131	+	0.827048i	$0.309982\pi$
$60$	0	0
$61$	7.94370	1.01709	0.508543	−	0.861036i	$-0.330184\pi$
$61$	7.94370	1.01709	0.508543	+	0.861036i	$0.330184\pi$
$62$	0	0
$63$	−2.74823	−0.346244
$64$	0	0
$65$	−2.59774	−0.322209
$66$	0	0
$67$	−14.9887	−1.83116	−0.915579	−	0.402138i	$-0.868267\pi$
$67$	−14.9887	−1.83116	−0.915579	+	0.402138i	$0.868267\pi$
$68$	0	0
$69$	3.94370	0.474766
$70$	0	0
$71$	−7.88740	−0.936062	−0.468031	−	0.883712i	$-0.655036\pi$
$71$	−7.88740	−0.936062	−0.468031	+	0.883712i	$0.655036\pi$
$72$	0	0
$73$	2.95079	0.345363	0.172682	−	0.984978i	$-0.444757\pi$
$73$	2.95079	0.345363	0.172682	+	0.984978i	$0.444757\pi$
$74$	0	0
$75$	−1.74823	−0.201868
$76$	0	0
$77$	−2.74823	−0.313190
$78$	0	0
$79$	−7.34596	−0.826485	−0.413243	−	0.910621i	$-0.635604\pi$
$79$	−7.34596	−0.826485	−0.413243	+	0.910621i	$0.635604\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.49646	−0.164257	−0.0821287	−	0.996622i	$-0.526172\pi$
$83$	−1.49646	−0.164257	−0.0821287	+	0.996622i	$0.526172\pi$
$84$	0	0
$85$	−3.49646	−0.379244
$86$	0	0
$87$	2.09419	0.224521
$88$	0	0
$89$	8.54143	0.905390	0.452695	−	0.891665i	$-0.350463\pi$
$89$	8.54143	0.905390	0.452695	+	0.891665i	$0.350463\pi$
$90$	0	0
$91$	−2.74823	−0.288093
$92$	0	0
$93$	−1.34596	−0.139570
$94$	0	0
$95$	5.19547	0.533044
$96$	0	0
$97$	4.99291	0.506953	0.253477	−	0.967341i	$-0.418426\pi$
$97$	4.99291	0.506953	0.253477	+	0.967341i	$0.418426\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−10.8424	−1.07886	−0.539431	−	0.842030i	$-0.681360\pi$
$101$	−10.8424	−1.07886	−0.539431	+	0.842030i	$0.681360\pi$
$102$	0	0
$103$	−16.6356	−1.63916	−0.819578	−	0.572967i	$-0.805792\pi$
$103$	−16.6356	−1.63916	−0.819578	+	0.572967i	$0.805792\pi$
$104$	0	0
$105$	−7.13917	−0.696712
$106$	0	0
$107$	11.1392	1.07686	0.538432	−	0.842669i	$-0.319017\pi$
$107$	11.1392	1.07686	0.538432	+	0.842669i	$0.319017\pi$
$108$	0	0
$109$	13.3839	1.28194	0.640970	−	0.767566i	$-0.278532\pi$
$109$	13.3839	1.28194	0.640970	+	0.767566i	$0.278532\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	9.44015	0.888055	0.444028	−	0.896013i	$-0.353549\pi$
$113$	9.44015	0.888055	0.444028	+	0.896013i	$0.353549\pi$
$114$	0	0
$115$	10.2447	0.955322
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−3.69901	−0.339088
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.94370	−0.535925
$124$	0	0
$125$	8.44724	0.755544
$126$	0	0
$127$	17.0308	1.51124	0.755620	−	0.655011i	$-0.227336\pi$
$127$	17.0308	1.51124	0.755620	+	0.655011i	$0.227336\pi$
$128$	0	0
$129$	−5.28966	−0.465729
$130$	0	0
$131$	1.94370	0.169822	0.0849109	−	0.996389i	$-0.472939\pi$
$131$	1.94370	0.169822	0.0849109	+	0.996389i	$0.472939\pi$
$132$	0	0
$133$	5.49646	0.476603
$134$	0	0
$135$	2.59774	0.223577
$136$	0	0
$137$	−10.3389	−0.883310	−0.441655	−	0.897185i	$-0.645608\pi$
$137$	−10.3389	−0.883310	−0.441655	+	0.897185i	$0.645608\pi$
$138$	0	0
$139$	−5.84951	−0.496149	−0.248074	−	0.968741i	$-0.579798\pi$
$139$	−5.84951	−0.496149	−0.248074	+	0.968741i	$0.579798\pi$
$140$	0	0
$141$	−6.69193	−0.563562
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	5.44015	0.451780
$146$	0	0
$147$	−0.552758	−0.0455907
$148$	0	0
$149$	−0.300986	−0.0246577	−0.0123289	−	0.999924i	$-0.503924\pi$
$149$	−0.300986	−0.0246577	−0.0123289	+	0.999924i	$0.503924\pi$
$150$	0	0
$151$	−17.8874	−1.45566	−0.727828	−	0.685760i	$-0.759470\pi$
$151$	−17.8874	−1.45566	−0.727828	+	0.685760i	$0.759470\pi$
$152$	0	0
$153$	1.34596	0.108815
$154$	0	0
$155$	−3.49646	−0.280842
$156$	0	0
$157$	19.5864	1.56317	0.781583	−	0.623801i	$-0.214413\pi$
$157$	19.5864	1.56317	0.781583	+	0.623801i	$0.214413\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	10.8382	0.854168
$162$	0	0
$163$	7.28966	0.570970	0.285485	−	0.958383i	$-0.407845\pi$
$163$	7.28966	0.570970	0.285485	+	0.958383i	$0.407845\pi$
$164$	0	0
$165$	2.59774	0.202233
$166$	0	0
$167$	−7.55276	−0.584450	−0.292225	−	0.956350i	$-0.594396\pi$
$167$	−7.55276	−0.584450	−0.292225	+	0.956350i	$0.594396\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	6.09419	0.463333	0.231666	−	0.972795i	$-0.425582\pi$
$173$	6.09419	0.463333	0.231666	+	0.972795i	$0.425582\pi$
$174$	0	0
$175$	−4.80453	−0.363188
$176$	0	0
$177$	−8.63563	−0.649093
$178$	0	0
$179$	12.5035	0.934559	0.467279	−	0.884110i	$-0.345234\pi$
$179$	12.5035	0.934559	0.467279	+	0.884110i	$0.345234\pi$
$180$	0	0
$181$	−24.1884	−1.79791	−0.898954	−	0.438043i	$-0.855672\pi$
$181$	−24.1884	−1.79791	−0.898954	+	0.438043i	$0.855672\pi$
$182$	0	0
$183$	−7.94370	−0.587215
$184$	0	0
$185$	−15.5864	−1.14594
$186$	0	0
$187$	1.34596	0.0984266
$188$	0	0
$189$	2.74823	0.199904
$190$	0	0
$191$	−13.9437	−1.00893	−0.504465	−	0.863432i	$-0.668310\pi$
$191$	−13.9437	−1.00893	−0.504465	+	0.863432i	$0.668310\pi$
$192$	0	0
$193$	0.112603	0.00810534	0.00405267	−	0.999992i	$-0.498710\pi$
$193$	0.112603	0.00810534	0.00405267	+	0.999992i	$0.498710\pi$
$194$	0	0
$195$	2.59774	0.186028
$196$	0	0
$197$	−13.6848	−0.975004	−0.487502	−	0.873122i	$-0.662092\pi$
$197$	−13.6848	−0.975004	−0.487502	+	0.873122i	$0.662092\pi$
$198$	0	0
$199$	−20.7482	−1.47080	−0.735401	−	0.677632i	$-0.763006\pi$
$199$	−20.7482	−1.47080	−0.735401	+	0.677632i	$0.763006\pi$
$200$	0	0
$201$	14.9887	1.05722
$202$	0	0
$203$	5.75532	0.403944
$204$	0	0
$205$	−15.4402	−1.07839
$206$	0	0
$207$	−3.94370	−0.274106
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−14.2263	−0.979377	−0.489689	−	0.871897i	$-0.662889\pi$
$211$	−14.2263	−0.979377	−0.489689	+	0.871897i	$0.662889\pi$
$212$	0	0
$213$	7.88740	0.540436
$214$	0	0
$215$	−13.7411	−0.937138
$216$	0	0
$217$	−3.69901	−0.251105
$218$	0	0
$219$	−2.95079	−0.199396
$220$	0	0
$221$	1.34596	0.0905393
$222$	0	0
$223$	−12.7440	−0.853401	−0.426700	−	0.904393i	$-0.640324\pi$
$223$	−12.7440	−0.853401	−0.426700	+	0.904393i	$0.640324\pi$
$224$	0	0
$225$	1.74823	0.116549
$226$	0	0
$227$	−17.5864	−1.16725	−0.583626	−	0.812023i	$-0.698366\pi$
$227$	−17.5864	−1.16725	−0.583626	+	0.812023i	$0.698366\pi$
$228$	0	0
$229$	3.64271	0.240717	0.120359	−	0.992730i	$-0.461596\pi$
$229$	3.64271	0.240717	0.120359	+	0.992730i	$0.461596\pi$
$230$	0	0
$231$	2.74823	0.180820
$232$	0	0
$233$	3.14341	0.205931	0.102966	−	0.994685i	$-0.467167\pi$
$233$	3.14341	0.205931	0.102966	+	0.994685i	$0.467167\pi$
$234$	0	0
$235$	−17.3839	−1.13400
$236$	0	0
$237$	7.34596	0.477172
$238$	0	0
$239$	−14.6356	−0.946700	−0.473350	−	0.880874i	$-0.656955\pi$
$239$	−14.6356	−0.946700	−0.473350	+	0.880874i	$0.656955\pi$
$240$	0	0
$241$	−13.3980	−0.863043	−0.431521	−	0.902103i	$-0.642023\pi$
$241$	−13.3980	−0.863043	−0.431521	+	0.902103i	$0.642023\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.43592	−0.0917374
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	1.49646	0.0948341
$250$	0	0
$251$	−0.489369	−0.0308887	−0.0154443	−	0.999881i	$-0.504916\pi$
$251$	−0.489369	−0.0308887	−0.0154443	+	0.999881i	$0.504916\pi$
$252$	0	0
$253$	−3.94370	−0.247938
$254$	0	0
$255$	3.49646	0.218957
$256$	0	0
$257$	−4.35729	−0.271800	−0.135900	−	0.990723i	$-0.543393\pi$
$257$	−4.35729	−0.271800	−0.135900	+	0.990723i	$0.543393\pi$
$258$	0	0
$259$	−16.4894	−1.02460
$260$	0	0
$261$	−2.09419	−0.129627
$262$	0	0
$263$	9.68484	0.597193	0.298596	−	0.954379i	$-0.403482\pi$
$263$	9.68484	0.597193	0.298596	+	0.954379i	$0.403482\pi$
$264$	0	0
$265$	−5.19547	−0.319155
$266$	0	0
$267$	−8.54143	−0.522727
$268$	0	0
$269$	−14.5935	−0.889781	−0.444891	−	0.895585i	$-0.646757\pi$
$269$	−14.5935	−0.889781	−0.444891	+	0.895585i	$0.646757\pi$
$270$	0	0
$271$	−12.5035	−0.759536	−0.379768	−	0.925082i	$-0.623996\pi$
$271$	−12.5035	−0.759536	−0.379768	+	0.925082i	$0.623996\pi$
$272$	0	0
$273$	2.74823	0.166330
$274$	0	0
$275$	1.74823	0.105422
$276$	0	0
$277$	−13.4402	−0.807541	−0.403770	−	0.914860i	$-0.632301\pi$
$277$	−13.4402	−0.807541	−0.403770	+	0.914860i	$0.632301\pi$
$278$	0	0
$279$	1.34596	0.0805807
$280$	0	0
$281$	11.4402	0.682462	0.341231	−	0.939979i	$-0.389156\pi$
$281$	11.4402	0.682462	0.341231	+	0.939979i	$0.389156\pi$
$282$	0	0
$283$	21.0645	1.25215	0.626076	−	0.779762i	$-0.284660\pi$
$283$	21.0645	1.25215	0.626076	+	0.779762i	$0.284660\pi$
$284$	0	0
$285$	−5.19547	−0.307753
$286$	0	0
$287$	−16.3346	−0.964203
$288$	0	0
$289$	−15.1884	−0.893434
$290$	0	0
$291$	−4.99291	−0.292690
$292$	0	0
$293$	−1.79744	−0.105008	−0.0525038	−	0.998621i	$-0.516720\pi$
$293$	−1.79744	−0.105008	−0.0525038	+	0.998621i	$0.516720\pi$
$294$	0	0
$295$	−22.4331	−1.30610
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−3.94370	−0.228070
$300$	0	0
$301$	−14.5372	−0.837910
$302$	0	0
$303$	10.8424	0.622881
$304$	0	0
$305$	−20.6356	−1.18159
$306$	0	0
$307$	−8.39094	−0.478896	−0.239448	−	0.970909i	$-0.576967\pi$
$307$	−8.39094	−0.478896	−0.239448	+	0.970909i	$0.576967\pi$
$308$	0	0
$309$	16.6356	0.946368
$310$	0	0
$311$	−6.11260	−0.346614	−0.173307	−	0.984868i	$-0.555445\pi$
$311$	−6.11260	−0.346614	−0.173307	+	0.984868i	$0.555445\pi$
$312$	0	0
$313$	−6.24468	−0.352970	−0.176485	−	0.984303i	$-0.556473\pi$
$313$	−6.24468	−0.352970	−0.176485	+	0.984303i	$0.556473\pi$
$314$	0	0
$315$	7.13917	0.402247
$316$	0	0
$317$	−8.80029	−0.494274	−0.247137	−	0.968981i	$-0.579490\pi$
$317$	−8.80029	−0.494274	−0.247137	+	0.968981i	$0.579490\pi$
$318$	0	0
$319$	−2.09419	−0.117252
$320$	0	0
$321$	−11.1392	−0.621728
$322$	0	0
$323$	−2.69193	−0.149783
$324$	0	0
$325$	1.74823	0.0969742
$326$	0	0
$327$	−13.3839	−0.740129
$328$	0	0
$329$	−18.3909	−1.01393
$330$	0	0
$331$	11.7790	0.647434	0.323717	−	0.946154i	$-0.395067\pi$
$331$	11.7790	0.647434	0.323717	+	0.946154i	$0.395067\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	38.9366	2.12733
$336$	0	0
$337$	−15.0829	−0.821616	−0.410808	−	0.911722i	$-0.634753\pi$
$337$	−15.0829	−0.821616	−0.410808	+	0.911722i	$0.634753\pi$
$338$	0	0
$339$	−9.44015	−0.512719
$340$	0	0
$341$	1.34596	0.0728880
$342$	0	0
$343$	17.7185	0.956709
$344$	0	0
$345$	−10.2447	−0.551555
$346$	0	0
$347$	20.0758	1.07772	0.538862	−	0.842394i	$-0.318854\pi$
$347$	20.0758	1.07772	0.538862	+	0.842394i	$0.318854\pi$
$348$	0	0
$349$	−21.2854	−1.13938	−0.569691	−	0.821859i	$-0.692937\pi$
$349$	−21.2854	−1.13938	−0.569691	+	0.821859i	$0.692937\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	15.4217	0.820817	0.410408	−	0.911902i	$-0.365386\pi$
$353$	15.4217	0.820817	0.410408	+	0.911902i	$0.365386\pi$
$354$	0	0
$355$	20.4894	1.08746
$356$	0	0
$357$	3.69901	0.195773
$358$	0	0
$359$	−6.74823	−0.356158	−0.178079	−	0.984016i	$-0.556988\pi$
$359$	−6.74823	−0.356158	−0.178079	+	0.984016i	$0.556988\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−7.66536	−0.401223
$366$	0	0
$367$	−36.3768	−1.89885	−0.949426	−	0.313991i	$-0.898334\pi$
$367$	−36.3768	−1.89885	−0.949426	+	0.313991i	$0.898334\pi$
$368$	0	0
$369$	5.94370	0.309417
$370$	0	0
$371$	−5.49646	−0.285362
$372$	0	0
$373$	−15.5301	−0.804119	−0.402059	−	0.915614i	$-0.631705\pi$
$373$	−15.5301	−0.804119	−0.402059	+	0.915614i	$0.631705\pi$
$374$	0	0
$375$	−8.44724	−0.436214
$376$	0	0
$377$	−2.09419	−0.107856
$378$	0	0
$379$	25.4217	1.30583	0.652914	−	0.757432i	$-0.273546\pi$
$379$	25.4217	1.30583	0.652914	+	0.757432i	$0.273546\pi$
$380$	0	0
$381$	−17.0308	−0.872514
$382$	0	0
$383$	−5.19547	−0.265476	−0.132738	−	0.991151i	$-0.542377\pi$
$383$	−5.19547	−0.265476	−0.132738	+	0.991151i	$0.542377\pi$
$384$	0	0
$385$	7.13917	0.363846
$386$	0	0
$387$	5.28966	0.268889
$388$	0	0
$389$	8.46672	0.429280	0.214640	−	0.976693i	$-0.431142\pi$
$389$	8.46672	0.429280	0.214640	+	0.976693i	$0.431142\pi$
$390$	0	0
$391$	−5.30807	−0.268441
$392$	0	0
$393$	−1.94370	−0.0980466
$394$	0	0
$395$	19.0829	0.960163
$396$	0	0
$397$	−10.0421	−0.504000	−0.252000	−	0.967727i	$-0.581088\pi$
$397$	−10.0421	−0.504000	−0.252000	+	0.967727i	$0.581088\pi$
$398$	0	0
$399$	−5.49646	−0.275167
$400$	0	0
$401$	31.3091	1.56350	0.781752	−	0.623590i	$-0.214326\pi$
$401$	31.3091	1.56350	0.781752	+	0.623590i	$0.214326\pi$
$402$	0	0
$403$	1.34596	0.0670472
$404$	0	0
$405$	−2.59774	−0.129082
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	17.2291	0.851925	0.425963	−	0.904741i	$-0.359935\pi$
$409$	17.2291	0.851925	0.425963	+	0.904741i	$0.359935\pi$
$410$	0	0
$411$	10.3389	0.509979
$412$	0	0
$413$	−23.7327	−1.16781
$414$	0	0
$415$	3.88740	0.190825
$416$	0	0
$417$	5.84951	0.286452
$418$	0	0
$419$	−19.2713	−0.941462	−0.470731	−	0.882277i	$-0.656010\pi$
$419$	−19.2713	−0.941462	−0.470731	+	0.882277i	$0.656010\pi$
$420$	0	0
$421$	−2.44724	−0.119271	−0.0596356	−	0.998220i	$-0.518994\pi$
$421$	−2.44724	−0.119271	−0.0596356	+	0.998220i	$0.518994\pi$
$422$	0	0
$423$	6.69193	0.325373
$424$	0	0
$425$	2.35305	0.114140
$426$	0	0
$427$	−21.8311	−1.05648
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−1.60906	−0.0775057	−0.0387528	−	0.999249i	$-0.512338\pi$
$431$	−1.60906	−0.0775057	−0.0387528	+	0.999249i	$0.512338\pi$
$432$	0	0
$433$	−30.3346	−1.45779	−0.728895	−	0.684626i	$-0.759966\pi$
$433$	−30.3346	−1.45779	−0.728895	+	0.684626i	$0.759966\pi$
$434$	0	0
$435$	−5.44015	−0.260835
$436$	0	0
$437$	7.88740	0.377305
$438$	0	0
$439$	−15.5343	−0.741414	−0.370707	−	0.928750i	$-0.620885\pi$
$439$	−15.5343	−0.741414	−0.370707	+	0.928750i	$0.620885\pi$
$440$	0	0
$441$	0.552758	0.0263218
$442$	0	0
$443$	32.7819	1.55751	0.778757	−	0.627326i	$-0.215850\pi$
$443$	32.7819	1.55751	0.778757	+	0.627326i	$0.215850\pi$
$444$	0	0
$445$	−22.1884	−1.05183
$446$	0	0
$447$	0.300986	0.0142361
$448$	0	0
$449$	−7.15758	−0.337787	−0.168894	−	0.985634i	$-0.554019\pi$
$449$	−7.15758	−0.337787	−0.168894	+	0.985634i	$0.554019\pi$
$450$	0	0
$451$	5.94370	0.279878
$452$	0	0
$453$	17.8874	0.840423
$454$	0	0
$455$	7.13917	0.334689
$456$	0	0
$457$	−14.8240	−0.693438	−0.346719	−	0.937969i	$-0.612704\pi$
$457$	−14.8240	−0.693438	−0.346719	+	0.937969i	$0.612704\pi$
$458$	0	0
$459$	−1.34596	−0.0628242
$460$	0	0
$461$	−8.78188	−0.409013	−0.204506	−	0.978865i	$-0.565559\pi$
$461$	−8.78188	−0.409013	−0.204506	+	0.978865i	$0.565559\pi$
$462$	0	0
$463$	−10.5414	−0.489902	−0.244951	−	0.969535i	$-0.578772\pi$
$463$	−10.5414	−0.489902	−0.244951	+	0.969535i	$0.578772\pi$
$464$	0	0
$465$	3.49646	0.162144
$466$	0	0
$467$	26.2642	1.21536	0.607680	−	0.794182i	$-0.292100\pi$
$467$	26.2642	1.21536	0.607680	+	0.794182i	$0.292100\pi$
$468$	0	0
$469$	41.1923	1.90208
$470$	0	0
$471$	−19.5864	−0.902494
$472$	0	0
$473$	5.28966	0.243219
$474$	0	0
$475$	−3.49646	−0.160428
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	17.0266	0.777964	0.388982	−	0.921245i	$-0.372827\pi$
$479$	17.0266	0.777964	0.388982	+	0.921245i	$0.372827\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	−10.8382	−0.493154
$484$	0	0
$485$	−12.9703	−0.588949
$486$	0	0
$487$	−11.8495	−0.536952	−0.268476	−	0.963286i	$-0.586520\pi$
$487$	−11.8495	−0.536952	−0.268476	+	0.963286i	$0.586520\pi$
$488$	0	0
$489$	−7.28966	−0.329650
$490$	0	0
$491$	7.85374	0.354434	0.177217	−	0.984172i	$-0.443290\pi$
$491$	7.85374	0.354434	0.177217	+	0.984172i	$0.443290\pi$
$492$	0	0
$493$	−2.81870	−0.126948
$494$	0	0
$495$	−2.59774	−0.116760
$496$	0	0
$497$	21.6764	0.972318
$498$	0	0
$499$	14.6877	0.657511	0.328756	−	0.944415i	$-0.393371\pi$
$499$	14.6877	0.657511	0.328756	+	0.944415i	$0.393371\pi$
$500$	0	0
$501$	7.55276	0.337432
$502$	0	0
$503$	1.91005	0.0851647	0.0425824	−	0.999093i	$-0.486442\pi$
$503$	1.91005	0.0851647	0.0425824	+	0.999093i	$0.486442\pi$
$504$	0	0
$505$	28.1657	1.25336
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	1.36014	0.0602871	0.0301435	−	0.999546i	$-0.490404\pi$
$509$	1.36014	0.0602871	0.0301435	+	0.999546i	$0.490404\pi$
$510$	0	0
$511$	−8.10943	−0.358740
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	43.2149	1.90428
$516$	0	0
$517$	6.69193	0.294311
$518$	0	0
$519$	−6.09419	−0.267505
$520$	0	0
$521$	−27.0492	−1.18505	−0.592524	−	0.805553i	$-0.701868\pi$
$521$	−27.0492	−1.18505	−0.592524	+	0.805553i	$0.701868\pi$
$522$	0	0
$523$	10.9324	0.478039	0.239020	−	0.971015i	$-0.423174\pi$
$523$	10.9324	0.478039	0.239020	+	0.971015i	$0.423174\pi$
$524$	0	0
$525$	4.80453	0.209687
$526$	0	0
$527$	1.81162	0.0789153
$528$	0	0
$529$	−7.44724	−0.323793
$530$	0	0
$531$	8.63563	0.374754
$532$	0	0
$533$	5.94370	0.257450
$534$	0	0
$535$	−28.9366	−1.25104
$536$	0	0
$537$	−12.5035	−0.539568
$538$	0	0
$539$	0.552758	0.0238090
$540$	0	0
$541$	−7.38385	−0.317457	−0.158728	−	0.987322i	$-0.550739\pi$
$541$	−7.38385	−0.317457	−0.158728	+	0.987322i	$0.550739\pi$
$542$	0	0
$543$	24.1884	1.03802
$544$	0	0
$545$	−34.7677	−1.48928
$546$	0	0
$547$	−22.0800	−0.944073	−0.472037	−	0.881579i	$-0.656481\pi$
$547$	−22.0800	−0.944073	−0.472037	+	0.881579i	$0.656481\pi$
$548$	0	0
$549$	7.94370	0.339029
$550$	0	0
$551$	4.18838	0.178431
$552$	0	0
$553$	20.1884	0.858497
$554$	0	0
$555$	15.5864	0.661606
$556$	0	0
$557$	6.80453	0.288317	0.144159	−	0.989555i	$-0.453952\pi$
$557$	6.80453	0.288317	0.144159	+	0.989555i	$0.453952\pi$
$558$	0	0
$559$	5.28966	0.223729
$560$	0	0
$561$	−1.34596	−0.0568266
$562$	0	0
$563$	−39.5496	−1.66682	−0.833408	−	0.552658i	$-0.813614\pi$
$563$	−39.5496	−1.66682	−0.833408	+	0.552658i	$0.813614\pi$
$564$	0	0
$565$	−24.5230	−1.03169
$566$	0	0
$567$	−2.74823	−0.115415
$568$	0	0
$569$	−19.2192	−0.805710	−0.402855	−	0.915264i	$-0.631982\pi$
$569$	−19.2192	−0.805710	−0.402855	+	0.915264i	$0.631982\pi$
$570$	0	0
$571$	−38.9660	−1.63068	−0.815339	−	0.578984i	$-0.803449\pi$
$571$	−38.9660	−1.63068	−0.815339	+	0.578984i	$0.803449\pi$
$572$	0	0
$573$	13.9437	0.582506
$574$	0	0
$575$	−6.89448	−0.287520
$576$	0	0
$577$	−47.2713	−1.96793	−0.983964	−	0.178367i	$-0.942918\pi$
$577$	−47.2713	−1.96793	−0.983964	+	0.178367i	$0.942918\pi$
$578$	0	0
$579$	−0.112603	−0.00467962
$580$	0	0
$581$	4.11260	0.170620
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	−2.59774	−0.107403
$586$	0	0
$587$	−1.64271	−0.0678020	−0.0339010	−	0.999425i	$-0.510793\pi$
$587$	−1.64271	−0.0678020	−0.0339010	+	0.999425i	$0.510793\pi$
$588$	0	0
$589$	−2.69193	−0.110919
$590$	0	0
$591$	13.6848	0.562919
$592$	0	0
$593$	−18.5793	−0.762961	−0.381481	−	0.924377i	$-0.624586\pi$
$593$	−18.5793	−0.762961	−0.381481	+	0.924377i	$0.624586\pi$
$594$	0	0
$595$	9.60906	0.393933
$596$	0	0
$597$	20.7482	0.849168
$598$	0	0
$599$	−17.8311	−0.728559	−0.364279	−	0.931290i	$-0.618685\pi$
$599$	−17.8311	−0.728559	−0.364279	+	0.931290i	$0.618685\pi$
$600$	0	0
$601$	−7.90157	−0.322312	−0.161156	−	0.986929i	$-0.551522\pi$
$601$	−7.90157	−0.322312	−0.161156	+	0.986929i	$0.551522\pi$
$602$	0	0
$603$	−14.9887	−0.610386
$604$	0	0
$605$	−2.59774	−0.105613
$606$	0	0
$607$	37.2107	1.51034	0.755168	−	0.655531i	$-0.227555\pi$
$607$	37.2107	1.51034	0.755168	+	0.655531i	$0.227555\pi$
$608$	0	0
$609$	−5.75532	−0.233217
$610$	0	0
$611$	6.69193	0.270726
$612$	0	0
$613$	19.7748	0.798696	0.399348	−	0.916799i	$-0.369236\pi$
$613$	19.7748	0.798696	0.399348	+	0.916799i	$0.369236\pi$
$614$	0	0
$615$	15.4402	0.622607
$616$	0	0
$617$	27.1208	1.09184	0.545920	−	0.837837i	$-0.316180\pi$
$617$	27.1208	1.09184	0.545920	+	0.837837i	$0.316180\pi$
$618$	0	0
$619$	14.4710	0.581637	0.290818	−	0.956778i	$-0.406072\pi$
$619$	14.4710	0.581637	0.290818	+	0.956778i	$0.406072\pi$
$620$	0	0
$621$	3.94370	0.158255
$622$	0	0
$623$	−23.4738	−0.940458
$624$	0	0
$625$	−30.6848	−1.22739
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	8.07578	0.322002
$630$	0	0
$631$	3.62430	0.144281	0.0721406	−	0.997394i	$-0.477017\pi$
$631$	3.62430	0.144281	0.0721406	+	0.997394i	$0.477017\pi$
$632$	0	0
$633$	14.2263	0.565444
$634$	0	0
$635$	−44.2415	−1.75567
$636$	0	0
$637$	0.552758	0.0219011
$638$	0	0
$639$	−7.88740	−0.312021
$640$	0	0
$641$	17.9295	0.708174	0.354087	−	0.935213i	$-0.384792\pi$
$641$	17.9295	0.708174	0.354087	+	0.935213i	$0.384792\pi$
$642$	0	0
$643$	18.0521	0.711904	0.355952	−	0.934504i	$-0.384157\pi$
$643$	18.0521	0.711904	0.355952	+	0.934504i	$0.384157\pi$
$644$	0	0
$645$	13.7411	0.541057
$646$	0	0
$647$	29.9858	1.17886	0.589432	−	0.807818i	$-0.299352\pi$
$647$	29.9858	1.17886	0.589432	+	0.807818i	$0.299352\pi$
$648$	0	0
$649$	8.63563	0.338978
$650$	0	0
$651$	3.69901	0.144976
$652$	0	0
$653$	9.09704	0.355995	0.177997	−	0.984031i	$-0.443038\pi$
$653$	9.09704	0.355995	0.177997	+	0.984031i	$0.443038\pi$
$654$	0	0
$655$	−5.04921	−0.197289
$656$	0	0
$657$	2.95079	0.115121
$658$	0	0
$659$	−25.7606	−1.00349	−0.501746	−	0.865015i	$-0.667309\pi$
$659$	−25.7606	−1.00349	−0.501746	+	0.865015i	$0.667309\pi$
$660$	0	0
$661$	−3.68484	−0.143324	−0.0716618	−	0.997429i	$-0.522830\pi$
$661$	−3.68484	−0.143324	−0.0716618	+	0.997429i	$0.522830\pi$
$662$	0	0
$663$	−1.34596	−0.0522729
$664$	0	0
$665$	−14.2783	−0.553690
$666$	0	0
$667$	8.25886	0.319784
$668$	0	0
$669$	12.7440	0.492711
$670$	0	0
$671$	7.94370	0.306663
$672$	0	0
$673$	3.19547	0.123176	0.0615882	−	0.998102i	$-0.480383\pi$
$673$	3.19547	0.123176	0.0615882	+	0.998102i	$0.480383\pi$
$674$	0	0
$675$	−1.74823	−0.0672893
$676$	0	0
$677$	33.3091	1.28017	0.640087	−	0.768302i	$-0.278898\pi$
$677$	33.3091	1.28017	0.640087	+	0.768302i	$0.278898\pi$
$678$	0	0
$679$	−13.7217	−0.526589
$680$	0	0
$681$	17.5864	0.673913
$682$	0	0
$683$	12.9366	0.495006	0.247503	−	0.968887i	$-0.420390\pi$
$683$	12.9366	0.495006	0.247503	+	0.968887i	$0.420390\pi$
$684$	0	0
$685$	26.8577	1.02618
$686$	0	0
$687$	−3.64271	−0.138978
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	−14.0521	−0.534566	−0.267283	−	0.963618i	$-0.586126\pi$
$691$	−14.0521	−0.534566	−0.267283	+	0.963618i	$0.586126\pi$
$692$	0	0
$693$	−2.74823	−0.104397
$694$	0	0
$695$	15.1955	0.576397
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	−3.14341	−0.118895
$700$	0	0
$701$	−17.7932	−0.672040	−0.336020	−	0.941855i	$-0.609081\pi$
$701$	−17.7932	−0.672040	−0.336020	+	0.941855i	$0.609081\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	17.3839	0.654714
$706$	0	0
$707$	29.7974	1.12065
$708$	0	0
$709$	18.2305	0.684661	0.342331	−	0.939580i	$-0.388784\pi$
$709$	18.2305	0.684661	0.342331	+	0.939580i	$0.388784\pi$
$710$	0	0
$711$	−7.34596	−0.275495
$712$	0	0
$713$	−5.30807	−0.198789
$714$	0	0
$715$	−2.59774	−0.0971498
$716$	0	0
$717$	14.6356	0.546577
$718$	0	0
$719$	49.0039	1.82754	0.913769	−	0.406234i	$-0.133158\pi$
$719$	49.0039	1.82754	0.913769	+	0.406234i	$0.133158\pi$
$720$	0	0
$721$	45.7185	1.70265
$722$	0	0
$723$	13.3980	0.498278
$724$	0	0
$725$	−3.66112	−0.135971
$726$	0	0
$727$	−22.6919	−0.841597	−0.420798	−	0.907154i	$-0.638250\pi$
$727$	−22.6919	−0.841597	−0.420798	+	0.907154i	$0.638250\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.11969	0.263331
$732$	0	0
$733$	−20.8945	−0.771755	−0.385878	−	0.922550i	$-0.626101\pi$
$733$	−20.8945	−0.771755	−0.385878	+	0.922550i	$0.626101\pi$
$734$	0	0
$735$	1.43592	0.0529646
$736$	0	0
$737$	−14.9887	−0.552115
$738$	0	0
$739$	45.3612	1.66864	0.834319	−	0.551281i	$-0.185861\pi$
$739$	45.3612	1.66864	0.834319	+	0.551281i	$0.185861\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	−18.7256	−0.686975	−0.343487	−	0.939157i	$-0.611608\pi$
$743$	−18.7256	−0.686975	−0.343487	+	0.939157i	$0.611608\pi$
$744$	0	0
$745$	0.781882	0.0286459
$746$	0	0
$747$	−1.49646	−0.0547525
$748$	0	0
$749$	−30.6130	−1.11857
$750$	0	0
$751$	−28.4189	−1.03702	−0.518510	−	0.855072i	$-0.673513\pi$
$751$	−28.4189	−1.03702	−0.518510	+	0.855072i	$0.673513\pi$
$752$	0	0
$753$	0.489369	0.0178336
$754$	0	0
$755$	46.4667	1.69110
$756$	0	0
$757$	−38.9561	−1.41588	−0.707942	−	0.706271i	$-0.750376\pi$
$757$	−38.9561	−1.41588	−0.707942	+	0.706271i	$0.750376\pi$
$758$	0	0
$759$	3.94370	0.143147
$760$	0	0
$761$	−20.2305	−0.733355	−0.366678	−	0.930348i	$-0.619505\pi$
$761$	−20.2305	−0.733355	−0.366678	+	0.930348i	$0.619505\pi$
$762$	0	0
$763$	−36.7819	−1.33159
$764$	0	0
$765$	−3.49646	−0.126415
$766$	0	0
$767$	8.63563	0.311814
$768$	0	0
$769$	7.04921	0.254201	0.127101	−	0.991890i	$-0.459433\pi$
$769$	7.04921	0.254201	0.127101	+	0.991890i	$0.459433\pi$
$770$	0	0
$771$	4.35729	0.156924
$772$	0	0
$773$	23.6469	0.850522	0.425261	−	0.905071i	$-0.360182\pi$
$773$	23.6469	0.850522	0.425261	+	0.905071i	$0.360182\pi$
$774$	0	0
$775$	2.35305	0.0845241
$776$	0	0
$777$	16.4894	0.591553
$778$	0	0
$779$	−11.8874	−0.425910
$780$	0	0
$781$	−7.88740	−0.282233
$782$	0	0
$783$	2.09419	0.0748403
$784$	0	0
$785$	−50.8803	−1.81600
$786$	0	0
$787$	11.5722	0.412506	0.206253	−	0.978499i	$-0.433873\pi$
$787$	11.5722	0.412506	0.206253	+	0.978499i	$0.433873\pi$
$788$	0	0
$789$	−9.68484	−0.344789
$790$	0	0
$791$	−25.9437	−0.922452
$792$	0	0
$793$	7.94370	0.282089
$794$	0	0
$795$	5.19547	0.184264
$796$	0	0
$797$	−40.5793	−1.43739	−0.718697	−	0.695324i	$-0.755261\pi$
$797$	−40.5793	−1.43739	−0.718697	+	0.695324i	$0.755261\pi$
$798$	0	0
$799$	9.00709	0.318648
$800$	0	0
$801$	8.54143	0.301797
$802$	0	0
$803$	2.95079	0.104131
$804$	0	0
$805$	−28.1547	−0.992324
$806$	0	0
$807$	14.5935	0.513715
$808$	0	0
$809$	2.16467	0.0761057	0.0380528	−	0.999276i	$-0.487884\pi$
$809$	2.16467	0.0761057	0.0380528	+	0.999276i	$0.487884\pi$
$810$	0	0
$811$	−14.8945	−0.523016	−0.261508	−	0.965201i	$-0.584220\pi$
$811$	−14.8945	−0.523016	−0.261508	+	0.965201i	$0.584220\pi$
$812$	0	0
$813$	12.5035	0.438518
$814$	0	0
$815$	−18.9366	−0.663321
$816$	0	0
$817$	−10.5793	−0.370124
$818$	0	0
$819$	−2.74823	−0.0960309
$820$	0	0
$821$	49.2713	1.71958	0.859789	−	0.510649i	$-0.170595\pi$
$821$	49.2713	1.71958	0.859789	+	0.510649i	$0.170595\pi$
$822$	0	0
$823$	15.8537	0.552627	0.276313	−	0.961068i	$-0.410887\pi$
$823$	15.8537	0.552627	0.276313	+	0.961068i	$0.410887\pi$
$824$	0	0
$825$	−1.74823	−0.0608655
$826$	0	0
$827$	32.5793	1.13289	0.566447	−	0.824098i	$-0.308318\pi$
$827$	32.5793	1.13289	0.566447	+	0.824098i	$0.308318\pi$
$828$	0	0
$829$	5.77479	0.200567	0.100283	−	0.994959i	$-0.468025\pi$
$829$	5.77479	0.200567	0.100283	+	0.994959i	$0.468025\pi$
$830$	0	0
$831$	13.4402	0.466234
$832$	0	0
$833$	0.743992	0.0257778
$834$	0	0
$835$	19.6201	0.678980
$836$	0	0
$837$	−1.34596	−0.0465233
$838$	0	0
$839$	24.2925	0.838671	0.419335	−	0.907831i	$-0.362263\pi$
$839$	24.2925	0.838671	0.419335	+	0.907831i	$0.362263\pi$
$840$	0	0
$841$	−24.6144	−0.848771
$842$	0	0
$843$	−11.4402	−0.394020
$844$	0	0
$845$	−2.59774	−0.0893648
$846$	0	0
$847$	−2.74823	−0.0944302
$848$	0	0
$849$	−21.0645	−0.722930
$850$	0	0
$851$	−23.6622	−0.811129
$852$	0	0
$853$	45.0602	1.54283	0.771416	−	0.636331i	$-0.219549\pi$
$853$	45.0602	1.54283	0.771416	+	0.636331i	$0.219549\pi$
$854$	0	0
$855$	5.19547	0.177681
$856$	0	0
$857$	−18.0521	−0.616647	−0.308323	−	0.951282i	$-0.599768\pi$
$857$	−18.0521	−0.616647	−0.308323	+	0.951282i	$0.599768\pi$
$858$	0	0
$859$	−7.09704	−0.242148	−0.121074	−	0.992643i	$-0.538634\pi$
$859$	−7.09704	−0.242148	−0.121074	+	0.992643i	$0.538634\pi$
$860$	0	0
$861$	16.3346	0.556683
$862$	0	0
$863$	49.0829	1.67080	0.835400	−	0.549642i	$-0.185236\pi$
$863$	49.0829	1.67080	0.835400	+	0.549642i	$0.185236\pi$
$864$	0	0
$865$	−15.8311	−0.538273
$866$	0	0
$867$	15.1884	0.515825
$868$	0	0
$869$	−7.34596	−0.249195
$870$	0	0
$871$	−14.9887	−0.507872
$872$	0	0
$873$	4.99291	0.168984
$874$	0	0
$875$	−23.2149	−0.784809
$876$	0	0
$877$	23.2713	0.785814	0.392907	−	0.919578i	$-0.371469\pi$
$877$	23.2713	0.785814	0.392907	+	0.919578i	$0.371469\pi$
$878$	0	0
$879$	1.79744	0.0606262
$880$	0	0
$881$	−41.1023	−1.38477	−0.692387	−	0.721527i	$-0.743441\pi$
$881$	−41.1023	−1.38477	−0.692387	+	0.721527i	$0.743441\pi$
$882$	0	0
$883$	−7.58641	−0.255303	−0.127652	−	0.991819i	$-0.540744\pi$
$883$	−7.58641	−0.255303	−0.127652	+	0.991819i	$0.540744\pi$
$884$	0	0
$885$	22.4331	0.754079
$886$	0	0
$887$	−39.7748	−1.33551	−0.667753	−	0.744383i	$-0.732744\pi$
$887$	−39.7748	−1.33551	−0.667753	+	0.744383i	$0.732744\pi$
$888$	0	0
$889$	−46.8045	−1.56977
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−13.3839	−0.447874
$894$	0	0
$895$	−32.4809	−1.08572
$896$	0	0
$897$	3.94370	0.131676
$898$	0	0
$899$	−2.81870	−0.0940091
$900$	0	0
$901$	2.69193	0.0896811
$902$	0	0
$903$	14.5372	0.483768
$904$	0	0
$905$	62.8350	2.08871
$906$	0	0
$907$	8.22521	0.273113	0.136557	−	0.990632i	$-0.456396\pi$
$907$	8.22521	0.273113	0.136557	+	0.990632i	$0.456396\pi$
$908$	0	0
$909$	−10.8424	−0.359620
$910$	0	0
$911$	53.5354	1.77371	0.886854	−	0.462050i	$-0.152886\pi$
$911$	53.5354	1.77371	0.886854	+	0.462050i	$0.152886\pi$
$912$	0	0
$913$	−1.49646	−0.0495255
$914$	0	0
$915$	20.6356	0.682193
$916$	0	0
$917$	−5.34173	−0.176399
$918$	0	0
$919$	−31.2334	−1.03029	−0.515147	−	0.857102i	$-0.672263\pi$
$919$	−31.2334	−1.03029	−0.515147	+	0.857102i	$0.672263\pi$
$920$	0	0
$921$	8.39094	0.276491
$922$	0	0
$923$	−7.88740	−0.259617
$924$	0	0
$925$	10.4894	0.344888
$926$	0	0
$927$	−16.6356	−0.546386
$928$	0	0
$929$	−3.45857	−0.113472	−0.0567359	−	0.998389i	$-0.518069\pi$
$929$	−3.45857	−0.113472	−0.0567359	+	0.998389i	$0.518069\pi$
$930$	0	0
$931$	−1.10552	−0.0362318
$932$	0	0
$933$	6.11260	0.200118
$934$	0	0
$935$	−3.49646	−0.114346
$936$	0	0
$937$	−49.5864	−1.61992	−0.809959	−	0.586487i	$-0.800511\pi$
$937$	−49.5864	−1.61992	−0.809959	+	0.586487i	$0.800511\pi$
$938$	0	0
$939$	6.24468	0.203788
$940$	0	0
$941$	−22.3909	−0.729924	−0.364962	−	0.931022i	$-0.618918\pi$
$941$	−22.3909	−0.729924	−0.364962	+	0.931022i	$0.618918\pi$
$942$	0	0
$943$	−23.4402	−0.763316
$944$	0	0
$945$	−7.13917	−0.232237
$946$	0	0
$947$	25.4596	0.827327	0.413663	−	0.910430i	$-0.364249\pi$
$947$	25.4596	0.827327	0.413663	+	0.910430i	$0.364249\pi$
$948$	0	0
$949$	2.95079	0.0957866
$950$	0	0
$951$	8.80029	0.285369
$952$	0	0
$953$	−2.42883	−0.0786775	−0.0393388	−	0.999226i	$-0.512525\pi$
$953$	−2.42883	−0.0786775	−0.0393388	+	0.999226i	$0.512525\pi$
$954$	0	0
$955$	36.2220	1.17212
$956$	0	0
$957$	2.09419	0.0676956
$958$	0	0
$959$	28.4136	0.917523
$960$	0	0
$961$	−29.1884	−0.941561
$962$	0	0
$963$	11.1392	0.358955
$964$	0	0
$965$	−0.292513	−0.00941632
$966$	0	0
$967$	16.5372	0.531800	0.265900	−	0.964001i	$-0.414331\pi$
$967$	16.5372	0.531800	0.265900	+	0.964001i	$0.414331\pi$
$968$	0	0
$969$	2.69193	0.0864771
$970$	0	0
$971$	19.7890	0.635058	0.317529	−	0.948249i	$-0.397147\pi$
$971$	19.7890	0.635058	0.317529	+	0.948249i	$0.397147\pi$
$972$	0	0
$973$	16.0758	0.515366
$974$	0	0
$975$	−1.74823	−0.0559881
$976$	0	0
$977$	−10.9324	−0.349758	−0.174879	−	0.984590i	$-0.555953\pi$
$977$	−10.9324	−0.349758	−0.174879	+	0.984590i	$0.555953\pi$
$978$	0	0
$979$	8.54143	0.272985
$980$	0	0
$981$	13.3839	0.427314
$982$	0	0
$983$	−35.4370	−1.13026	−0.565132	−	0.825000i	$-0.691175\pi$
$983$	−35.4370	−1.13026	−0.565132	+	0.825000i	$0.691175\pi$
$984$	0	0
$985$	35.5496	1.13270
$986$	0	0
$987$	18.3909	0.585390
$988$	0	0
$989$	−20.8608	−0.663336
$990$	0	0
$991$	22.5372	0.715918	0.357959	−	0.933737i	$-0.383473\pi$
$991$	22.5372	0.715918	0.357959	+	0.933737i	$0.383473\pi$
$992$	0	0
$993$	−11.7790	−0.373796
$994$	0	0
$995$	53.8984	1.70869
$996$	0	0
$997$	9.54428	0.302271	0.151135	−	0.988513i	$-0.451707\pi$
$997$	9.54428	0.302271	0.151135	+	0.988513i	$0.451707\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bo.1.1		3
4.3	odd	2		3432.2.a.o.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.o.1.1	✓	3	4.3	odd	2
6864.2.a.bo.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.59774 q^{5} -2.74823 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.59774 q^{5} -2.74823 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +2.59774 q^{15} +1.34596 q^{17} -2.00000 q^{19} +2.74823 q^{21} -3.94370 q^{23} +1.74823 q^{25} -1.00000 q^{27} -2.09419 q^{29} +1.34596 q^{31} -1.00000 q^{33} +7.13917 q^{35} +6.00000 q^{37} -1.00000 q^{39} +5.94370 q^{41} +5.28966 q^{43} -2.59774 q^{45} +6.69193 q^{47} +0.552758 q^{49} -1.34596 q^{51} +2.00000 q^{53} -2.59774 q^{55} +2.00000 q^{57} +8.63563 q^{59} +7.94370 q^{61} -2.74823 q^{63} -2.59774 q^{65} -14.9887 q^{67} +3.94370 q^{69} -7.88740 q^{71} +2.95079 q^{73} -1.74823 q^{75} -2.74823 q^{77} -7.34596 q^{79} +1.00000 q^{81} -1.49646 q^{83} -3.49646 q^{85} +2.09419 q^{87} +8.54143 q^{89} -2.74823 q^{91} -1.34596 q^{93} +5.19547 q^{95} +4.99291 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{13} + q^{15} - 6 q^{17} - 6 q^{19} + 5 q^{21} + 5 q^{23} + 2 q^{25} - 3 q^{27} + 7 q^{29} - 6 q^{31} - 3 q^{33} - 9 q^{35} + 18 q^{37} - 3 q^{39} + q^{41} - 11 q^{43} - q^{45} + 12 q^{49} + 6 q^{51} + 6 q^{53} - q^{55} + 6 q^{57} - 11 q^{59} + 7 q^{61} - 5 q^{63} - q^{65} - 11 q^{67} - 5 q^{69} + 10 q^{71} + 5 q^{73} - 2 q^{75} - 5 q^{77} - 12 q^{79} + 3 q^{81} + 2 q^{83} - 4 q^{85} - 7 q^{87} + 2 q^{89} - 5 q^{91} + 6 q^{93} + 2 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^13 + q^15 - 6 * q^17 - 6 * q^19 + 5 * q^21 + 5 * q^23 + 2 * q^25 - 3 * q^27 + 7 * q^29 - 6 * q^31 - 3 * q^33 - 9 * q^35 + 18 * q^37 - 3 * q^39 + q^41 - 11 * q^43 - q^45 + 12 * q^49 + 6 * q^51 + 6 * q^53 - q^55 + 6 * q^57 - 11 * q^59 + 7 * q^61 - 5 * q^63 - q^65 - 11 * q^67 - 5 * q^69 + 10 * q^71 + 5 * q^73 - 2 * q^75 - 5 * q^77 - 12 * q^79 + 3 * q^81 + 2 * q^83 - 4 * q^85 - 7 * q^87 + 2 * q^89 - 5 * q^91 + 6 * q^93 + 2 * q^95 + 2 * q^97 + 3 * q^99

Introduction

L-functions

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