LMFDB - Embedded newform 6864.2.a.bb.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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.41421 q^{15} -6.24264 q^{17} +2.82843 q^{21} +4.82843 q^{23} +6.65685 q^{25} -1.00000 q^{27} -0.585786 q^{29} +5.41421 q^{31} -1.00000 q^{33} +9.65685 q^{35} +0.828427 q^{37} +1.00000 q^{39} +3.65685 q^{41} +0.242641 q^{43} -3.41421 q^{45} +8.00000 q^{47} +1.00000 q^{49} +6.24264 q^{51} +9.31371 q^{53} -3.41421 q^{55} -2.82843 q^{59} -3.17157 q^{61} -2.82843 q^{63} +3.41421 q^{65} -4.24264 q^{67} -4.82843 q^{69} +10.8284 q^{71} -5.65685 q^{73} -6.65685 q^{75} -2.82843 q^{77} +7.07107 q^{79} +1.00000 q^{81} +17.6569 q^{83} +21.3137 q^{85} +0.585786 q^{87} -11.8995 q^{89} +2.82843 q^{91} -5.41421 q^{93} -12.8284 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + 2 q^{11} - 2 q^{13} + 4 q^{15} - 4 q^{17} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 8 q^{31} - 2 q^{33} + 8 q^{35} - 4 q^{37} + 2 q^{39} - 4 q^{41} - 8 q^{43} - 4 q^{45} + 16 q^{47} + 2 q^{49} + 4 q^{51} - 4 q^{53} - 4 q^{55} - 12 q^{61} + 4 q^{65} - 4 q^{69} + 16 q^{71} - 2 q^{75} + 2 q^{81} + 24 q^{83} + 20 q^{85} + 4 q^{87} - 4 q^{89} - 8 q^{93} - 20 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 + 2 * q^11 - 2 * q^13 + 4 * q^15 - 4 * q^17 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 + 8 * q^31 - 2 * q^33 + 8 * q^35 - 4 * q^37 + 2 * q^39 - 4 * q^41 - 8 * q^43 - 4 * q^45 + 16 * q^47 + 2 * q^49 + 4 * q^51 - 4 * q^53 - 4 * q^55 - 12 * q^61 + 4 * q^65 - 4 * q^69 + 16 * q^71 - 2 * q^75 + 2 * q^81 + 24 * q^83 + 20 * q^85 + 4 * q^87 - 4 * q^89 - 8 * q^93 - 20 * q^97 + 2 * 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
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	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
$15$	3.41421	0.881546
$16$	0	0
$17$	−6.24264	−1.51406	−0.757031	−	0.653379i	$-0.773351\pi$
$17$	−6.24264	−1.51406	−0.757031	+	0.653379i	$0.773351\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	4.82843	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$23$	4.82843	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.585786	−0.108778	−0.0543889	−	0.998520i	$-0.517321\pi$
$29$	−0.585786	−0.108778	−0.0543889	+	0.998520i	$0.517321\pi$
$30$	0	0
$31$	5.41421	0.972421	0.486211	−	0.873842i	$-0.338379\pi$
$31$	5.41421	0.972421	0.486211	+	0.873842i	$0.338379\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	9.65685	1.63231
$36$	0	0
$37$	0.828427	0.136193	0.0680963	−	0.997679i	$-0.478307\pi$
$37$	0.828427	0.136193	0.0680963	+	0.997679i	$0.478307\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	3.65685	0.571105	0.285552	−	0.958363i	$-0.407823\pi$
$41$	3.65685	0.571105	0.285552	+	0.958363i	$0.407823\pi$
$42$	0	0
$43$	0.242641	0.0370024	0.0185012	−	0.999829i	$-0.494111\pi$
$43$	0.242641	0.0370024	0.0185012	+	0.999829i	$0.494111\pi$
$44$	0	0
$45$	−3.41421	−0.508961
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.24264	0.874145
$52$	0	0
$53$	9.31371	1.27934	0.639668	−	0.768651i	$-0.279072\pi$
$53$	9.31371	1.27934	0.639668	+	0.768651i	$0.279072\pi$
$54$	0	0
$55$	−3.41421	−0.460372
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.82843	−0.368230	−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843	−0.368230	−0.184115	+	0.982905i	$0.558942\pi$
$60$	0	0
$61$	−3.17157	−0.406078	−0.203039	−	0.979171i	$-0.565082\pi$
$61$	−3.17157	−0.406078	−0.203039	+	0.979171i	$0.565082\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	3.41421	0.423481
$66$	0	0
$67$	−4.24264	−0.518321	−0.259161	−	0.965834i	$-0.583446\pi$
$67$	−4.24264	−0.518321	−0.259161	+	0.965834i	$0.583446\pi$
$68$	0	0
$69$	−4.82843	−0.581274
$70$	0	0
$71$	10.8284	1.28510	0.642549	−	0.766245i	$-0.277877\pi$
$71$	10.8284	1.28510	0.642549	+	0.766245i	$0.277877\pi$
$72$	0	0
$73$	−5.65685	−0.662085	−0.331042	−	0.943616i	$-0.607400\pi$
$73$	−5.65685	−0.662085	−0.331042	+	0.943616i	$0.607400\pi$
$74$	0	0
$75$	−6.65685	−0.768667
$76$	0	0
$77$	−2.82843	−0.322329
$78$	0	0
$79$	7.07107	0.795557	0.397779	−	0.917481i	$-0.369781\pi$
$79$	7.07107	0.795557	0.397779	+	0.917481i	$0.369781\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	17.6569	1.93809	0.969046	−	0.246881i	$-0.0794057\pi$
$83$	17.6569	1.93809	0.969046	+	0.246881i	$0.0794057\pi$
$84$	0	0
$85$	21.3137	2.31180
$86$	0	0
$87$	0.585786	0.0628029
$88$	0	0
$89$	−11.8995	−1.26134	−0.630672	−	0.776050i	$-0.717221\pi$
$89$	−11.8995	−1.26134	−0.630672	+	0.776050i	$0.717221\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	−5.41421	−0.561428
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.8284	−1.30253	−0.651265	−	0.758851i	$-0.725761\pi$
$97$	−12.8284	−1.30253	−0.651265	+	0.758851i	$0.725761\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−18.7279	−1.86350	−0.931749	−	0.363103i	$-0.881717\pi$
$101$	−18.7279	−1.86350	−0.931749	+	0.363103i	$0.881717\pi$
$102$	0	0
$103$	2.82843	0.278693	0.139347	−	0.990244i	$-0.455500\pi$
$103$	2.82843	0.278693	0.139347	+	0.990244i	$0.455500\pi$
$104$	0	0
$105$	−9.65685	−0.942412
$106$	0	0
$107$	−1.17157	−0.113260	−0.0566301	−	0.998395i	$-0.518036\pi$
$107$	−1.17157	−0.113260	−0.0566301	+	0.998395i	$0.518036\pi$
$108$	0	0
$109$	−15.3137	−1.46679	−0.733394	−	0.679804i	$-0.762065\pi$
$109$	−15.3137	−1.46679	−0.733394	+	0.679804i	$0.762065\pi$
$110$	0	0
$111$	−0.828427	−0.0786308
$112$	0	0
$113$	18.9706	1.78460	0.892300	−	0.451442i	$-0.149090\pi$
$113$	18.9706	1.78460	0.892300	+	0.451442i	$0.149090\pi$
$114$	0	0
$115$	−16.4853	−1.53726
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	17.6569	1.61860
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.65685	−0.329727
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	7.07107	0.627456	0.313728	−	0.949513i	$-0.398422\pi$
$127$	7.07107	0.627456	0.313728	+	0.949513i	$0.398422\pi$
$128$	0	0
$129$	−0.242641	−0.0213633
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.41421	0.293849
$136$	0	0
$137$	13.0711	1.11674	0.558368	−	0.829593i	$-0.311428\pi$
$137$	13.0711	1.11674	0.558368	+	0.829593i	$0.311428\pi$
$138$	0	0
$139$	−8.24264	−0.699132	−0.349566	−	0.936912i	$-0.613671\pi$
$139$	−8.24264	−0.699132	−0.349566	+	0.936912i	$0.613671\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	2.48528	0.203602	0.101801	−	0.994805i	$-0.467539\pi$
$149$	2.48528	0.203602	0.101801	+	0.994805i	$0.467539\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	−6.24264	−0.504688
$154$	0	0
$155$	−18.4853	−1.48477
$156$	0	0
$157$	6.34315	0.506238	0.253119	−	0.967435i	$-0.418544\pi$
$157$	6.34315	0.506238	0.253119	+	0.967435i	$0.418544\pi$
$158$	0	0
$159$	−9.31371	−0.738625
$160$	0	0
$161$	−13.6569	−1.07631
$162$	0	0
$163$	12.7279	0.996928	0.498464	−	0.866910i	$-0.333898\pi$
$163$	12.7279	0.996928	0.498464	+	0.866910i	$0.333898\pi$
$164$	0	0
$165$	3.41421	0.265796
$166$	0	0
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.8995	1.51293	0.756465	−	0.654034i	$-0.226925\pi$
$173$	19.8995	1.51293	0.756465	+	0.654034i	$0.226925\pi$
$174$	0	0
$175$	−18.8284	−1.42330
$176$	0	0
$177$	2.82843	0.212598
$178$	0	0
$179$	−4.82843	−0.360894	−0.180447	−	0.983585i	$-0.557754\pi$
$179$	−4.82843	−0.360894	−0.180447	+	0.983585i	$0.557754\pi$
$180$	0	0
$181$	24.9706	1.85605	0.928024	−	0.372521i	$-0.121507\pi$
$181$	24.9706	1.85605	0.928024	+	0.372521i	$0.121507\pi$
$182$	0	0
$183$	3.17157	0.234449
$184$	0	0
$185$	−2.82843	−0.207950
$186$	0	0
$187$	−6.24264	−0.456507
$188$	0	0
$189$	2.82843	0.205738
$190$	0	0
$191$	2.34315	0.169544	0.0847720	−	0.996400i	$-0.472984\pi$
$191$	2.34315	0.169544	0.0847720	+	0.996400i	$0.472984\pi$
$192$	0	0
$193$	−5.65685	−0.407189	−0.203595	−	0.979055i	$-0.565262\pi$
$193$	−5.65685	−0.407189	−0.203595	+	0.979055i	$0.565262\pi$
$194$	0	0
$195$	−3.41421	−0.244497
$196$	0	0
$197$	−12.1421	−0.865091	−0.432546	−	0.901612i	$-0.642385\pi$
$197$	−12.1421	−0.865091	−0.432546	+	0.901612i	$0.642385\pi$
$198$	0	0
$199$	4.48528	0.317953	0.158977	−	0.987282i	$-0.449181\pi$
$199$	4.48528	0.317953	0.158977	+	0.987282i	$0.449181\pi$
$200$	0	0
$201$	4.24264	0.299253
$202$	0	0
$203$	1.65685	0.116288
$204$	0	0
$205$	−12.4853	−0.872010
$206$	0	0
$207$	4.82843	0.335599
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−10.5858	−0.728756	−0.364378	−	0.931251i	$-0.618718\pi$
$211$	−10.5858	−0.728756	−0.364378	+	0.931251i	$0.618718\pi$
$212$	0	0
$213$	−10.8284	−0.741952
$214$	0	0
$215$	−0.828427	−0.0564983
$216$	0	0
$217$	−15.3137	−1.03956
$218$	0	0
$219$	5.65685	0.382255
$220$	0	0
$221$	6.24264	0.419925
$222$	0	0
$223$	21.8995	1.46650	0.733249	−	0.679960i	$-0.238003\pi$
$223$	21.8995	1.46650	0.733249	+	0.679960i	$0.238003\pi$
$224$	0	0
$225$	6.65685	0.443790
$226$	0	0
$227$	−12.1421	−0.805902	−0.402951	−	0.915222i	$-0.632015\pi$
$227$	−12.1421	−0.805902	−0.402951	+	0.915222i	$0.632015\pi$
$228$	0	0
$229$	1.31371	0.0868123	0.0434062	−	0.999058i	$-0.486179\pi$
$229$	1.31371	0.0868123	0.0434062	+	0.999058i	$0.486179\pi$
$230$	0	0
$231$	2.82843	0.186097
$232$	0	0
$233$	8.58579	0.562474	0.281237	−	0.959638i	$-0.409255\pi$
$233$	8.58579	0.562474	0.281237	+	0.959638i	$0.409255\pi$
$234$	0	0
$235$	−27.3137	−1.78175
$236$	0	0
$237$	−7.07107	−0.459315
$238$	0	0
$239$	−20.1421	−1.30289	−0.651443	−	0.758697i	$-0.725836\pi$
$239$	−20.1421	−1.30289	−0.651443	+	0.758697i	$0.725836\pi$
$240$	0	0
$241$	−0.343146	−0.0221040	−0.0110520	−	0.999939i	$-0.503518\pi$
$241$	−0.343146	−0.0221040	−0.0110520	+	0.999939i	$0.503518\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.41421	−0.218126
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−17.6569	−1.11896
$250$	0	0
$251$	−25.6569	−1.61945	−0.809723	−	0.586812i	$-0.800383\pi$
$251$	−25.6569	−1.61945	−0.809723	+	0.586812i	$0.800383\pi$
$252$	0	0
$253$	4.82843	0.303561
$254$	0	0
$255$	−21.3137	−1.33472
$256$	0	0
$257$	−0.828427	−0.0516759	−0.0258379	−	0.999666i	$-0.508225\pi$
$257$	−0.828427	−0.0516759	−0.0258379	+	0.999666i	$0.508225\pi$
$258$	0	0
$259$	−2.34315	−0.145596
$260$	0	0
$261$	−0.585786	−0.0362593
$262$	0	0
$263$	−8.48528	−0.523225	−0.261612	−	0.965173i	$-0.584254\pi$
$263$	−8.48528	−0.523225	−0.261612	+	0.965173i	$0.584254\pi$
$264$	0	0
$265$	−31.7990	−1.95340
$266$	0	0
$267$	11.8995	0.728237
$268$	0	0
$269$	−8.82843	−0.538279	−0.269139	−	0.963101i	$-0.586739\pi$
$269$	−8.82843	−0.538279	−0.269139	+	0.963101i	$0.586739\pi$
$270$	0	0
$271$	−1.65685	−0.100647	−0.0503234	−	0.998733i	$-0.516025\pi$
$271$	−1.65685	−0.100647	−0.0503234	+	0.998733i	$0.516025\pi$
$272$	0	0
$273$	−2.82843	−0.171184
$274$	0	0
$275$	6.65685	0.401423
$276$	0	0
$277$	14.9706	0.899494	0.449747	−	0.893156i	$-0.351514\pi$
$277$	14.9706	0.899494	0.449747	+	0.893156i	$0.351514\pi$
$278$	0	0
$279$	5.41421	0.324140
$280$	0	0
$281$	−23.6569	−1.41125	−0.705625	−	0.708586i	$-0.749334\pi$
$281$	−23.6569	−1.41125	−0.705625	+	0.708586i	$0.749334\pi$
$282$	0	0
$283$	−27.0711	−1.60921	−0.804604	−	0.593812i	$-0.797622\pi$
$283$	−27.0711	−1.60921	−0.804604	+	0.593812i	$0.797622\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.3431	−0.610537
$288$	0	0
$289$	21.9706	1.29239
$290$	0	0
$291$	12.8284	0.752016
$292$	0	0
$293$	−1.51472	−0.0884908	−0.0442454	−	0.999021i	$-0.514088\pi$
$293$	−1.51472	−0.0884908	−0.0442454	+	0.999021i	$0.514088\pi$
$294$	0	0
$295$	9.65685	0.562244
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−4.82843	−0.279235
$300$	0	0
$301$	−0.686292	−0.0395572
$302$	0	0
$303$	18.7279	1.07589
$304$	0	0
$305$	10.8284	0.620034
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	−2.82843	−0.160904
$310$	0	0
$311$	24.8284	1.40789	0.703945	−	0.710254i	$-0.251420\pi$
$311$	24.8284	1.40789	0.703945	+	0.710254i	$0.251420\pi$
$312$	0	0
$313$	−22.6274	−1.27898	−0.639489	−	0.768801i	$-0.720854\pi$
$313$	−22.6274	−1.27898	−0.639489	+	0.768801i	$0.720854\pi$
$314$	0	0
$315$	9.65685	0.544102
$316$	0	0
$317$	−6.92893	−0.389168	−0.194584	−	0.980886i	$-0.562336\pi$
$317$	−6.92893	−0.389168	−0.194584	+	0.980886i	$0.562336\pi$
$318$	0	0
$319$	−0.585786	−0.0327977
$320$	0	0
$321$	1.17157	0.0653908
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−6.65685	−0.369256
$326$	0	0
$327$	15.3137	0.846850
$328$	0	0
$329$	−22.6274	−1.24749
$330$	0	0
$331$	−31.5563	−1.73449	−0.867247	−	0.497878i	$-0.834113\pi$
$331$	−31.5563	−1.73449	−0.867247	+	0.497878i	$0.834113\pi$
$332$	0	0
$333$	0.828427	0.0453975
$334$	0	0
$335$	14.4853	0.791415
$336$	0	0
$337$	9.51472	0.518300	0.259150	−	0.965837i	$-0.416558\pi$
$337$	9.51472	0.518300	0.259150	+	0.965837i	$0.416558\pi$
$338$	0	0
$339$	−18.9706	−1.03034
$340$	0	0
$341$	5.41421	0.293196
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	16.4853	0.887538
$346$	0	0
$347$	25.6569	1.37733	0.688666	−	0.725079i	$-0.258197\pi$
$347$	25.6569	1.37733	0.688666	+	0.725079i	$0.258197\pi$
$348$	0	0
$349$	−35.9411	−1.92388	−0.961942	−	0.273253i	$-0.911900\pi$
$349$	−35.9411	−1.92388	−0.961942	+	0.273253i	$0.911900\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	14.7279	0.783888	0.391944	−	0.919989i	$-0.371803\pi$
$353$	14.7279	0.783888	0.391944	+	0.919989i	$0.371803\pi$
$354$	0	0
$355$	−36.9706	−1.96219
$356$	0	0
$357$	−17.6569	−0.934500
$358$	0	0
$359$	16.1421	0.851949	0.425975	−	0.904735i	$-0.359931\pi$
$359$	16.1421	0.851949	0.425975	+	0.904735i	$0.359931\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	19.3137	1.01093
$366$	0	0
$367$	−8.68629	−0.453421	−0.226710	−	0.973962i	$-0.572797\pi$
$367$	−8.68629	−0.453421	−0.226710	+	0.973962i	$0.572797\pi$
$368$	0	0
$369$	3.65685	0.190368
$370$	0	0
$371$	−26.3431	−1.36767
$372$	0	0
$373$	19.4558	1.00739	0.503693	−	0.863883i	$-0.331974\pi$
$373$	19.4558	1.00739	0.503693	+	0.863883i	$0.331974\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	0	0
$377$	0.585786	0.0301695
$378$	0	0
$379$	−29.6985	−1.52551	−0.762754	−	0.646688i	$-0.776153\pi$
$379$	−29.6985	−1.52551	−0.762754	+	0.646688i	$0.776153\pi$
$380$	0	0
$381$	−7.07107	−0.362262
$382$	0	0
$383$	19.7990	1.01168	0.505841	−	0.862627i	$-0.331182\pi$
$383$	19.7990	1.01168	0.505841	+	0.862627i	$0.331182\pi$
$384$	0	0
$385$	9.65685	0.492159
$386$	0	0
$387$	0.242641	0.0123341
$388$	0	0
$389$	−17.7990	−0.902445	−0.451222	−	0.892412i	$-0.649012\pi$
$389$	−17.7990	−0.902445	−0.451222	+	0.892412i	$0.649012\pi$
$390$	0	0
$391$	−30.1421	−1.52435
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	−24.1421	−1.21472
$396$	0	0
$397$	−37.7990	−1.89708	−0.948538	−	0.316662i	$-0.897438\pi$
$397$	−37.7990	−1.89708	−0.948538	+	0.316662i	$0.897438\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.2132	−0.959462	−0.479731	−	0.877416i	$-0.659266\pi$
$401$	−19.2132	−0.959462	−0.479731	+	0.877416i	$0.659266\pi$
$402$	0	0
$403$	−5.41421	−0.269701
$404$	0	0
$405$	−3.41421	−0.169654
$406$	0	0
$407$	0.828427	0.0410636
$408$	0	0
$409$	2.34315	0.115861	0.0579306	−	0.998321i	$-0.481550\pi$
$409$	2.34315	0.115861	0.0579306	+	0.998321i	$0.481550\pi$
$410$	0	0
$411$	−13.0711	−0.644748
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	−60.2843	−2.95924
$416$	0	0
$417$	8.24264	0.403644
$418$	0	0
$419$	−28.8284	−1.40836	−0.704180	−	0.710021i	$-0.748685\pi$
$419$	−28.8284	−1.40836	−0.704180	+	0.710021i	$0.748685\pi$
$420$	0	0
$421$	−37.3137	−1.81856	−0.909279	−	0.416186i	$-0.863366\pi$
$421$	−37.3137	−1.81856	−0.909279	+	0.416186i	$0.863366\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−41.5563	−2.01578
$426$	0	0
$427$	8.97056	0.434116
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	13.6569	0.657828	0.328914	−	0.944360i	$-0.393317\pi$
$431$	13.6569	0.657828	0.328914	+	0.944360i	$0.393317\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−9.41421	−0.449316	−0.224658	−	0.974438i	$-0.572126\pi$
$439$	−9.41421	−0.449316	−0.224658	+	0.974438i	$0.572126\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	23.3137	1.10767	0.553834	−	0.832627i	$-0.313164\pi$
$443$	23.3137	1.10767	0.553834	+	0.832627i	$0.313164\pi$
$444$	0	0
$445$	40.6274	1.92592
$446$	0	0
$447$	−2.48528	−0.117550
$448$	0	0
$449$	8.58579	0.405188	0.202594	−	0.979263i	$-0.435063\pi$
$449$	8.58579	0.405188	0.202594	+	0.979263i	$0.435063\pi$
$450$	0	0
$451$	3.65685	0.172195
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	−9.65685	−0.452720
$456$	0	0
$457$	25.3137	1.18413	0.592063	−	0.805892i	$-0.298314\pi$
$457$	25.3137	1.18413	0.592063	+	0.805892i	$0.298314\pi$
$458$	0	0
$459$	6.24264	0.291382
$460$	0	0
$461$	10.9706	0.510950	0.255475	−	0.966816i	$-0.417768\pi$
$461$	10.9706	0.510950	0.255475	+	0.966816i	$0.417768\pi$
$462$	0	0
$463$	−12.9289	−0.600858	−0.300429	−	0.953804i	$-0.597130\pi$
$463$	−12.9289	−0.600858	−0.300429	+	0.953804i	$0.597130\pi$
$464$	0	0
$465$	18.4853	0.857234
$466$	0	0
$467$	23.4558	1.08541	0.542704	−	0.839924i	$-0.317401\pi$
$467$	23.4558	1.08541	0.542704	+	0.839924i	$0.317401\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	−6.34315	−0.292277
$472$	0	0
$473$	0.242641	0.0111566
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.31371	0.426445
$478$	0	0
$479$	−24.1421	−1.10308	−0.551541	−	0.834148i	$-0.685960\pi$
$479$	−24.1421	−1.10308	−0.551541	+	0.834148i	$0.685960\pi$
$480$	0	0
$481$	−0.828427	−0.0377730
$482$	0	0
$483$	13.6569	0.621408
$484$	0	0
$485$	43.7990	1.98881
$486$	0	0
$487$	11.5563	0.523668	0.261834	−	0.965113i	$-0.415673\pi$
$487$	11.5563	0.523668	0.261834	+	0.965113i	$0.415673\pi$
$488$	0	0
$489$	−12.7279	−0.575577
$490$	0	0
$491$	−20.9706	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$491$	−20.9706	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$492$	0	0
$493$	3.65685	0.164696
$494$	0	0
$495$	−3.41421	−0.153457
$496$	0	0
$497$	−30.6274	−1.37383
$498$	0	0
$499$	−9.89949	−0.443162	−0.221581	−	0.975142i	$-0.571122\pi$
$499$	−9.89949	−0.443162	−0.221581	+	0.975142i	$0.571122\pi$
$500$	0	0
$501$	11.3137	0.505459
$502$	0	0
$503$	24.4853	1.09174	0.545872	−	0.837868i	$-0.316198\pi$
$503$	24.4853	1.09174	0.545872	+	0.837868i	$0.316198\pi$
$504$	0	0
$505$	63.9411	2.84534
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	9.55635	0.423578	0.211789	−	0.977315i	$-0.432071\pi$
$509$	9.55635	0.423578	0.211789	+	0.977315i	$0.432071\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.65685	−0.425532
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	−19.8995	−0.873491
$520$	0	0
$521$	14.9706	0.655872	0.327936	−	0.944700i	$-0.393647\pi$
$521$	14.9706	0.655872	0.327936	+	0.944700i	$0.393647\pi$
$522$	0	0
$523$	2.10051	0.0918487	0.0459243	−	0.998945i	$-0.485377\pi$
$523$	2.10051	0.0918487	0.0459243	+	0.998945i	$0.485377\pi$
$524$	0	0
$525$	18.8284	0.821740
$526$	0	0
$527$	−33.7990	−1.47231
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	−2.82843	−0.122743
$532$	0	0
$533$	−3.65685	−0.158396
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	4.82843	0.208362
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	−24.9706	−1.07159
$544$	0	0
$545$	52.2843	2.23961
$546$	0	0
$547$	−26.1005	−1.11598	−0.557989	−	0.829849i	$-0.688427\pi$
$547$	−26.1005	−1.11598	−0.557989	+	0.829849i	$0.688427\pi$
$548$	0	0
$549$	−3.17157	−0.135359
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−20.0000	−0.850487
$554$	0	0
$555$	2.82843	0.120060
$556$	0	0
$557$	26.4853	1.12222	0.561109	−	0.827742i	$-0.310375\pi$
$557$	26.4853	1.12222	0.561109	+	0.827742i	$0.310375\pi$
$558$	0	0
$559$	−0.242641	−0.0102626
$560$	0	0
$561$	6.24264	0.263564
$562$	0	0
$563$	1.65685	0.0698281	0.0349140	−	0.999390i	$-0.488884\pi$
$563$	1.65685	0.0698281	0.0349140	+	0.999390i	$0.488884\pi$
$564$	0	0
$565$	−64.7696	−2.72488
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	13.5563	0.568312	0.284156	−	0.958778i	$-0.408287\pi$
$569$	13.5563	0.568312	0.284156	+	0.958778i	$0.408287\pi$
$570$	0	0
$571$	28.0416	1.17351	0.586753	−	0.809766i	$-0.300406\pi$
$571$	28.0416	1.17351	0.586753	+	0.809766i	$0.300406\pi$
$572$	0	0
$573$	−2.34315	−0.0978863
$574$	0	0
$575$	32.1421	1.34042
$576$	0	0
$577$	26.9706	1.12280	0.561400	−	0.827545i	$-0.310263\pi$
$577$	26.9706	1.12280	0.561400	+	0.827545i	$0.310263\pi$
$578$	0	0
$579$	5.65685	0.235091
$580$	0	0
$581$	−49.9411	−2.07191
$582$	0	0
$583$	9.31371	0.385734
$584$	0	0
$585$	3.41421	0.141160
$586$	0	0
$587$	31.7990	1.31248	0.656242	−	0.754550i	$-0.272145\pi$
$587$	31.7990	1.31248	0.656242	+	0.754550i	$0.272145\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	12.1421	0.499461
$592$	0	0
$593$	−19.1716	−0.787282	−0.393641	−	0.919264i	$-0.628785\pi$
$593$	−19.1716	−0.787282	−0.393641	+	0.919264i	$0.628785\pi$
$594$	0	0
$595$	−60.2843	−2.47141
$596$	0	0
$597$	−4.48528	−0.183570
$598$	0	0
$599$	13.6569	0.558004	0.279002	−	0.960291i	$-0.409996\pi$
$599$	13.6569	0.558004	0.279002	+	0.960291i	$0.409996\pi$
$600$	0	0
$601$	43.2548	1.76440	0.882201	−	0.470874i	$-0.156061\pi$
$601$	43.2548	1.76440	0.882201	+	0.470874i	$0.156061\pi$
$602$	0	0
$603$	−4.24264	−0.172774
$604$	0	0
$605$	−3.41421	−0.138808
$606$	0	0
$607$	−13.2132	−0.536307	−0.268154	−	0.963376i	$-0.586414\pi$
$607$	−13.2132	−0.536307	−0.268154	+	0.963376i	$0.586414\pi$
$608$	0	0
$609$	−1.65685	−0.0671391
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−37.9411	−1.53243	−0.766214	−	0.642586i	$-0.777862\pi$
$613$	−37.9411	−1.53243	−0.766214	+	0.642586i	$0.777862\pi$
$614$	0	0
$615$	12.4853	0.503455
$616$	0	0
$617$	26.9289	1.08412	0.542059	−	0.840340i	$-0.317645\pi$
$617$	26.9289	1.08412	0.542059	+	0.840340i	$0.317645\pi$
$618$	0	0
$619$	−9.41421	−0.378389	−0.189195	−	0.981940i	$-0.560588\pi$
$619$	−9.41421	−0.378389	−0.189195	+	0.981940i	$0.560588\pi$
$620$	0	0
$621$	−4.82843	−0.193758
$622$	0	0
$623$	33.6569	1.34843
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.17157	−0.206204
$630$	0	0
$631$	43.0711	1.71463	0.857316	−	0.514790i	$-0.172130\pi$
$631$	43.0711	1.71463	0.857316	+	0.514790i	$0.172130\pi$
$632$	0	0
$633$	10.5858	0.420747
$634$	0	0
$635$	−24.1421	−0.958051
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	10.8284	0.428366
$640$	0	0
$641$	37.5980	1.48503	0.742515	−	0.669829i	$-0.233633\pi$
$641$	37.5980	1.48503	0.742515	+	0.669829i	$0.233633\pi$
$642$	0	0
$643$	7.55635	0.297993	0.148997	−	0.988838i	$-0.452396\pi$
$643$	7.55635	0.297993	0.148997	+	0.988838i	$0.452396\pi$
$644$	0	0
$645$	0.828427	0.0326193
$646$	0	0
$647$	−29.6569	−1.16593	−0.582966	−	0.812497i	$-0.698108\pi$
$647$	−29.6569	−1.16593	−0.582966	+	0.812497i	$0.698108\pi$
$648$	0	0
$649$	−2.82843	−0.111025
$650$	0	0
$651$	15.3137	0.600192
$652$	0	0
$653$	−28.8284	−1.12814	−0.564072	−	0.825726i	$-0.690766\pi$
$653$	−28.8284	−1.12814	−0.564072	+	0.825726i	$0.690766\pi$
$654$	0	0
$655$	27.3137	1.06723
$656$	0	0
$657$	−5.65685	−0.220695
$658$	0	0
$659$	31.1127	1.21198	0.605989	−	0.795473i	$-0.292777\pi$
$659$	31.1127	1.21198	0.605989	+	0.795473i	$0.292777\pi$
$660$	0	0
$661$	8.82843	0.343386	0.171693	−	0.985151i	$-0.445076\pi$
$661$	8.82843	0.343386	0.171693	+	0.985151i	$0.445076\pi$
$662$	0	0
$663$	−6.24264	−0.242444
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.82843	−0.109517
$668$	0	0
$669$	−21.8995	−0.846683
$670$	0	0
$671$	−3.17157	−0.122437
$672$	0	0
$673$	−11.6569	−0.449339	−0.224669	−	0.974435i	$-0.572130\pi$
$673$	−11.6569	−0.449339	−0.224669	+	0.974435i	$0.572130\pi$
$674$	0	0
$675$	−6.65685	−0.256222
$676$	0	0
$677$	−13.7574	−0.528738	−0.264369	−	0.964422i	$-0.585164\pi$
$677$	−13.7574	−0.528738	−0.264369	+	0.964422i	$0.585164\pi$
$678$	0	0
$679$	36.2843	1.39246
$680$	0	0
$681$	12.1421	0.465288
$682$	0	0
$683$	−6.82843	−0.261283	−0.130641	−	0.991430i	$-0.541704\pi$
$683$	−6.82843	−0.261283	−0.130641	+	0.991430i	$0.541704\pi$
$684$	0	0
$685$	−44.6274	−1.70513
$686$	0	0
$687$	−1.31371	−0.0501211
$688$	0	0
$689$	−9.31371	−0.354824
$690$	0	0
$691$	8.44365	0.321212	0.160606	−	0.987019i	$-0.448655\pi$
$691$	8.44365	0.321212	0.160606	+	0.987019i	$0.448655\pi$
$692$	0	0
$693$	−2.82843	−0.107443
$694$	0	0
$695$	28.1421	1.06749
$696$	0	0
$697$	−22.8284	−0.864688
$698$	0	0
$699$	−8.58579	−0.324744
$700$	0	0
$701$	30.2426	1.14225	0.571124	−	0.820864i	$-0.306507\pi$
$701$	30.2426	1.14225	0.571124	+	0.820864i	$0.306507\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	27.3137	1.02869
$706$	0	0
$707$	52.9706	1.99216
$708$	0	0
$709$	−48.6274	−1.82624	−0.913120	−	0.407690i	$-0.866334\pi$
$709$	−48.6274	−1.82624	−0.913120	+	0.407690i	$0.866334\pi$
$710$	0	0
$711$	7.07107	0.265186
$712$	0	0
$713$	26.1421	0.979031
$714$	0	0
$715$	3.41421	0.127684
$716$	0	0
$717$	20.1421	0.752222
$718$	0	0
$719$	35.3137	1.31698	0.658490	−	0.752590i	$-0.271196\pi$
$719$	35.3137	1.31698	0.658490	+	0.752590i	$0.271196\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0.343146	0.0127617
$724$	0	0
$725$	−3.89949	−0.144824
$726$	0	0
$727$	17.1716	0.636858	0.318429	−	0.947947i	$-0.396845\pi$
$727$	17.1716	0.636858	0.318429	+	0.947947i	$0.396845\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.51472	−0.0560239
$732$	0	0
$733$	14.6274	0.540276	0.270138	−	0.962822i	$-0.412931\pi$
$733$	14.6274	0.540276	0.270138	+	0.962822i	$0.412931\pi$
$734$	0	0
$735$	3.41421	0.125935
$736$	0	0
$737$	−4.24264	−0.156280
$738$	0	0
$739$	−9.17157	−0.337382	−0.168691	−	0.985669i	$-0.553954\pi$
$739$	−9.17157	−0.337382	−0.168691	+	0.985669i	$0.553954\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−25.9411	−0.951688	−0.475844	−	0.879530i	$-0.657857\pi$
$743$	−25.9411	−0.951688	−0.475844	+	0.879530i	$0.657857\pi$
$744$	0	0
$745$	−8.48528	−0.310877
$746$	0	0
$747$	17.6569	0.646031
$748$	0	0
$749$	3.31371	0.121080
$750$	0	0
$751$	19.3137	0.704767	0.352384	−	0.935856i	$-0.385371\pi$
$751$	19.3137	0.704767	0.352384	+	0.935856i	$0.385371\pi$
$752$	0	0
$753$	25.6569	0.934988
$754$	0	0
$755$	40.9706	1.49107
$756$	0	0
$757$	−24.2843	−0.882627	−0.441313	−	0.897353i	$-0.645487\pi$
$757$	−24.2843	−0.882627	−0.441313	+	0.897353i	$0.645487\pi$
$758$	0	0
$759$	−4.82843	−0.175261
$760$	0	0
$761$	−19.1716	−0.694969	−0.347484	−	0.937686i	$-0.612964\pi$
$761$	−19.1716	−0.694969	−0.347484	+	0.937686i	$0.612964\pi$
$762$	0	0
$763$	43.3137	1.56806
$764$	0	0
$765$	21.3137	0.770599
$766$	0	0
$767$	2.82843	0.102129
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	0.828427	0.0298351
$772$	0	0
$773$	−4.58579	−0.164939	−0.0824696	−	0.996594i	$-0.526281\pi$
$773$	−4.58579	−0.164939	−0.0824696	+	0.996594i	$0.526281\pi$
$774$	0	0
$775$	36.0416	1.29465
$776$	0	0
$777$	2.34315	0.0840599
$778$	0	0
$779$	0	0
$780$	0	0
$781$	10.8284	0.387472
$782$	0	0
$783$	0.585786	0.0209343
$784$	0	0
$785$	−21.6569	−0.772966
$786$	0	0
$787$	15.7990	0.563173	0.281587	−	0.959536i	$-0.409139\pi$
$787$	15.7990	0.563173	0.281587	+	0.959536i	$0.409139\pi$
$788$	0	0
$789$	8.48528	0.302084
$790$	0	0
$791$	−53.6569	−1.90782
$792$	0	0
$793$	3.17157	0.112626
$794$	0	0
$795$	31.7990	1.12779
$796$	0	0
$797$	5.51472	0.195341	0.0976707	−	0.995219i	$-0.468861\pi$
$797$	5.51472	0.195341	0.0976707	+	0.995219i	$0.468861\pi$
$798$	0	0
$799$	−49.9411	−1.76679
$800$	0	0
$801$	−11.8995	−0.420448
$802$	0	0
$803$	−5.65685	−0.199626
$804$	0	0
$805$	46.6274	1.64340
$806$	0	0
$807$	8.82843	0.310775
$808$	0	0
$809$	−20.3848	−0.716691	−0.358345	−	0.933589i	$-0.616659\pi$
$809$	−20.3848	−0.716691	−0.358345	+	0.933589i	$0.616659\pi$
$810$	0	0
$811$	−16.2843	−0.571818	−0.285909	−	0.958257i	$-0.592296\pi$
$811$	−16.2843	−0.571818	−0.285909	+	0.958257i	$0.592296\pi$
$812$	0	0
$813$	1.65685	0.0581084
$814$	0	0
$815$	−43.4558	−1.52219
$816$	0	0
$817$	0	0
$818$	0	0
$819$	2.82843	0.0988332
$820$	0	0
$821$	−40.6274	−1.41791	−0.708953	−	0.705255i	$-0.750832\pi$
$821$	−40.6274	−1.41791	−0.708953	+	0.705255i	$0.750832\pi$
$822$	0	0
$823$	−1.45584	−0.0507475	−0.0253738	−	0.999678i	$-0.508078\pi$
$823$	−1.45584	−0.0507475	−0.0253738	+	0.999678i	$0.508078\pi$
$824$	0	0
$825$	−6.65685	−0.231762
$826$	0	0
$827$	−42.7696	−1.48724	−0.743622	−	0.668601i	$-0.766894\pi$
$827$	−42.7696	−1.48724	−0.743622	+	0.668601i	$0.766894\pi$
$828$	0	0
$829$	−41.3137	−1.43488	−0.717442	−	0.696618i	$-0.754687\pi$
$829$	−41.3137	−1.43488	−0.717442	+	0.696618i	$0.754687\pi$
$830$	0	0
$831$	−14.9706	−0.519323
$832$	0	0
$833$	−6.24264	−0.216295
$834$	0	0
$835$	38.6274	1.33676
$836$	0	0
$837$	−5.41421	−0.187143
$838$	0	0
$839$	41.6569	1.43815	0.719077	−	0.694930i	$-0.244565\pi$
$839$	41.6569	1.43815	0.719077	+	0.694930i	$0.244565\pi$
$840$	0	0
$841$	−28.6569	−0.988167
$842$	0	0
$843$	23.6569	0.814785
$844$	0	0
$845$	−3.41421	−0.117453
$846$	0	0
$847$	−2.82843	−0.0971859
$848$	0	0
$849$	27.0711	0.929077
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	−17.3137	−0.592810	−0.296405	−	0.955062i	$-0.595788\pi$
$853$	−17.3137	−0.592810	−0.296405	+	0.955062i	$0.595788\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.0711	0.856411	0.428206	−	0.903681i	$-0.359146\pi$
$857$	25.0711	0.856411	0.428206	+	0.903681i	$0.359146\pi$
$858$	0	0
$859$	22.8284	0.778896	0.389448	−	0.921048i	$-0.372666\pi$
$859$	22.8284	0.778896	0.389448	+	0.921048i	$0.372666\pi$
$860$	0	0
$861$	10.3431	0.352493
$862$	0	0
$863$	−20.7696	−0.707004	−0.353502	−	0.935434i	$-0.615009\pi$
$863$	−20.7696	−0.707004	−0.353502	+	0.935434i	$0.615009\pi$
$864$	0	0
$865$	−67.9411	−2.31007
$866$	0	0
$867$	−21.9706	−0.746159
$868$	0	0
$869$	7.07107	0.239870
$870$	0	0
$871$	4.24264	0.143756
$872$	0	0
$873$	−12.8284	−0.434176
$874$	0	0
$875$	16.0000	0.540899
$876$	0	0
$877$	27.6569	0.933906	0.466953	−	0.884282i	$-0.345352\pi$
$877$	27.6569	0.933906	0.466953	+	0.884282i	$0.345352\pi$
$878$	0	0
$879$	1.51472	0.0510902
$880$	0	0
$881$	13.0294	0.438973	0.219486	−	0.975616i	$-0.429562\pi$
$881$	13.0294	0.438973	0.219486	+	0.975616i	$0.429562\pi$
$882$	0	0
$883$	34.8284	1.17207	0.586035	−	0.810286i	$-0.300688\pi$
$883$	34.8284	1.17207	0.586035	+	0.810286i	$0.300688\pi$
$884$	0	0
$885$	−9.65685	−0.324612
$886$	0	0
$887$	−8.48528	−0.284908	−0.142454	−	0.989801i	$-0.545499\pi$
$887$	−8.48528	−0.284908	−0.142454	+	0.989801i	$0.545499\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	16.4853	0.551042
$896$	0	0
$897$	4.82843	0.161216
$898$	0	0
$899$	−3.17157	−0.105778
$900$	0	0
$901$	−58.1421	−1.93700
$902$	0	0
$903$	0.686292	0.0228384
$904$	0	0
$905$	−85.2548	−2.83397
$906$	0	0
$907$	−11.3137	−0.375666	−0.187833	−	0.982201i	$-0.560146\pi$
$907$	−11.3137	−0.375666	−0.187833	+	0.982201i	$0.560146\pi$
$908$	0	0
$909$	−18.7279	−0.621166
$910$	0	0
$911$	−44.2843	−1.46720	−0.733602	−	0.679580i	$-0.762162\pi$
$911$	−44.2843	−1.46720	−0.733602	+	0.679580i	$0.762162\pi$
$912$	0	0
$913$	17.6569	0.584357
$914$	0	0
$915$	−10.8284	−0.357977
$916$	0	0
$917$	22.6274	0.747223
$918$	0	0
$919$	41.4142	1.36613	0.683064	−	0.730358i	$-0.260647\pi$
$919$	41.4142	1.36613	0.683064	+	0.730358i	$0.260647\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	0	0
$923$	−10.8284	−0.356422
$924$	0	0
$925$	5.51472	0.181323
$926$	0	0
$927$	2.82843	0.0928977
$928$	0	0
$929$	−18.9289	−0.621038	−0.310519	−	0.950567i	$-0.600503\pi$
$929$	−18.9289	−0.621038	−0.310519	+	0.950567i	$0.600503\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−24.8284	−0.812846
$934$	0	0
$935$	21.3137	0.697033
$936$	0	0
$937$	22.2843	0.727995	0.363998	−	0.931400i	$-0.381412\pi$
$937$	22.2843	0.727995	0.363998	+	0.931400i	$0.381412\pi$
$938$	0	0
$939$	22.6274	0.738418
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	17.6569	0.574986
$944$	0	0
$945$	−9.65685	−0.314137
$946$	0	0
$947$	10.6274	0.345345	0.172672	−	0.984979i	$-0.444760\pi$
$947$	10.6274	0.345345	0.172672	+	0.984979i	$0.444760\pi$
$948$	0	0
$949$	5.65685	0.183629
$950$	0	0
$951$	6.92893	0.224686
$952$	0	0
$953$	35.4142	1.14718	0.573589	−	0.819143i	$-0.305550\pi$
$953$	35.4142	1.14718	0.573589	+	0.819143i	$0.305550\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	0.585786	0.0189358
$958$	0	0
$959$	−36.9706	−1.19384
$960$	0	0
$961$	−1.68629	−0.0543965
$962$	0	0
$963$	−1.17157	−0.0377534
$964$	0	0
$965$	19.3137	0.621730
$966$	0	0
$967$	−46.4264	−1.49297	−0.746486	−	0.665401i	$-0.768261\pi$
$967$	−46.4264	−1.49297	−0.746486	+	0.665401i	$0.768261\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.1716	1.00034	0.500172	−	0.865926i	$-0.333270\pi$
$971$	31.1716	1.00034	0.500172	+	0.865926i	$0.333270\pi$
$972$	0	0
$973$	23.3137	0.747403
$974$	0	0
$975$	6.65685	0.213190
$976$	0	0
$977$	13.5563	0.433706	0.216853	−	0.976204i	$-0.430421\pi$
$977$	13.5563	0.433706	0.216853	+	0.976204i	$0.430421\pi$
$978$	0	0
$979$	−11.8995	−0.380310
$980$	0	0
$981$	−15.3137	−0.488929
$982$	0	0
$983$	−27.5980	−0.880239	−0.440119	−	0.897939i	$-0.645064\pi$
$983$	−27.5980	−0.880239	−0.440119	+	0.897939i	$0.645064\pi$
$984$	0	0
$985$	41.4558	1.32089
$986$	0	0
$987$	22.6274	0.720239
$988$	0	0
$989$	1.17157	0.0372539
$990$	0	0
$991$	25.4558	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$991$	25.4558	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$992$	0	0
$993$	31.5563	1.00141
$994$	0	0
$995$	−15.3137	−0.485477
$996$	0	0
$997$	−3.85786	−0.122180	−0.0610899	−	0.998132i	$-0.519458\pi$
$997$	−3.85786	−0.122180	−0.0610899	+	0.998132i	$0.519458\pi$
$998$	0	0
$999$	−0.828427	−0.0262103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bb.1.1		2
4.3	odd	2		1716.2.a.e.1.1	✓	2
12.11	even	2		5148.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.e.1.1	✓	2	4.3	odd	2
5148.2.a.j.1.2		2	12.11	even	2
6864.2.a.bb.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.41421 q^{15} -6.24264 q^{17} +2.82843 q^{21} +4.82843 q^{23} +6.65685 q^{25} -1.00000 q^{27} -0.585786 q^{29} +5.41421 q^{31} -1.00000 q^{33} +9.65685 q^{35} +0.828427 q^{37} +1.00000 q^{39} +3.65685 q^{41} +0.242641 q^{43} -3.41421 q^{45} +8.00000 q^{47} +1.00000 q^{49} +6.24264 q^{51} +9.31371 q^{53} -3.41421 q^{55} -2.82843 q^{59} -3.17157 q^{61} -2.82843 q^{63} +3.41421 q^{65} -4.24264 q^{67} -4.82843 q^{69} +10.8284 q^{71} -5.65685 q^{73} -6.65685 q^{75} -2.82843 q^{77} +7.07107 q^{79} +1.00000 q^{81} +17.6569 q^{83} +21.3137 q^{85} +0.585786 q^{87} -11.8995 q^{89} +2.82843 q^{91} -5.41421 q^{93} -12.8284 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + 2 q^{11} - 2 q^{13} + 4 q^{15} - 4 q^{17} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 8 q^{31} - 2 q^{33} + 8 q^{35} - 4 q^{37} + 2 q^{39} - 4 q^{41} - 8 q^{43} - 4 q^{45} + 16 q^{47} + 2 q^{49} + 4 q^{51} - 4 q^{53} - 4 q^{55} - 12 q^{61} + 4 q^{65} - 4 q^{69} + 16 q^{71} - 2 q^{75} + 2 q^{81} + 24 q^{83} + 20 q^{85} + 4 q^{87} - 4 q^{89} - 8 q^{93} - 20 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 + 2 * q^11 - 2 * q^13 + 4 * q^15 - 4 * q^17 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 + 8 * q^31 - 2 * q^33 + 8 * q^35 - 4 * q^37 + 2 * q^39 - 4 * q^41 - 8 * q^43 - 4 * q^45 + 16 * q^47 + 2 * q^49 + 4 * q^51 - 4 * q^53 - 4 * q^55 - 12 * q^61 + 4 * q^65 - 4 * q^69 + 16 * q^71 - 2 * q^75 + 2 * q^81 + 24 * q^83 + 20 * q^85 + 4 * q^87 - 4 * q^89 - 8 * q^93 - 20 * q^97 + 2 * q^99

Introduction

L-functions

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Database

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