LMFDB - Embedded newform 6864.2.a.cc.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.23252.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2 $ x^4 - x^3 - 6*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.88474$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.43697 q^{5} +2.70832 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.43697 q^{5} +2.70832 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +3.43697 q^{15} -0.375819 q^{17} +2.70832 q^{19} +2.70832 q^{21} +0.938844 q^{23} +6.81279 q^{25} +1.00000 q^{27} +0.728654 q^{29} -6.85361 q^{31} -1.00000 q^{33} +9.30843 q^{35} +9.89179 q^{37} -1.00000 q^{39} -2.83063 q^{41} -1.62418 q^{43} +3.43697 q^{45} -1.10447 q^{47} +0.334998 q^{49} -0.375819 q^{51} +12.1683 q^{53} -3.43697 q^{55} +2.70832 q^{57} +2.62933 q^{59} -3.10447 q^{61} +2.70832 q^{63} -3.43697 q^{65} -4.73130 q^{67} +0.938844 q^{69} +0.108213 q^{71} +13.7046 q^{73} +6.81279 q^{75} -2.70832 q^{77} -8.03708 q^{79} +1.00000 q^{81} +4.99626 q^{83} -1.29168 q^{85} +0.728654 q^{87} +6.33250 q^{89} -2.70832 q^{91} -6.85361 q^{93} +9.30843 q^{95} +15.3951 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{17} - 2 q^{19} - 2 q^{21} + 8 q^{23} + 12 q^{25} + 4 q^{27} + 6 q^{29} + 8 q^{31} - 4 q^{33} - 14 q^{35} + 14 q^{37} - 4 q^{39} + 10 q^{41} - 12 q^{43} + 4 q^{45} - 2 q^{47} + 16 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{55} - 2 q^{57} + 8 q^{59} - 10 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} + 8 q^{69} + 26 q^{71} + 14 q^{73} + 12 q^{75} + 2 q^{77} + 16 q^{79} + 4 q^{81} - 8 q^{83} - 18 q^{85} + 6 q^{87} + 18 q^{89} + 2 q^{91} + 8 q^{93} - 14 q^{95} + 14 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^17 - 2 * q^19 - 2 * q^21 + 8 * q^23 + 12 * q^25 + 4 * q^27 + 6 * q^29 + 8 * q^31 - 4 * q^33 - 14 * q^35 + 14 * q^37 - 4 * q^39 + 10 * q^41 - 12 * q^43 + 4 * q^45 - 2 * q^47 + 16 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^55 - 2 * q^57 + 8 * q^59 - 10 * q^61 - 2 * q^63 - 4 * q^65 + 8 * q^67 + 8 * q^69 + 26 * q^71 + 14 * q^73 + 12 * q^75 + 2 * q^77 + 16 * q^79 + 4 * q^81 - 8 * q^83 - 18 * q^85 + 6 * q^87 + 18 * q^89 + 2 * q^91 + 8 * q^93 - 14 * q^95 + 14 * q^97 - 4 * 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
$5$	3.43697	1.53706	0.768531	−	0.639813i	$-0.220988\pi$
$5$	3.43697	1.53706	0.768531	+	0.639813i	$0.220988\pi$
$6$	0	0
$7$	2.70832	1.02365	0.511824	−	0.859090i	$-0.328970\pi$
$7$	2.70832	1.02365	0.511824	+	0.859090i	$0.328970\pi$
$8$	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
$15$	3.43697	0.887423
$16$	0	0
$17$	−0.375819	−0.0911495	−0.0455747	−	0.998961i	$-0.514512\pi$
$17$	−0.375819	−0.0911495	−0.0455747	+	0.998961i	$0.514512\pi$
$18$	0	0
$19$	2.70832	0.621331	0.310666	−	0.950519i	$-0.399448\pi$
$19$	2.70832	0.621331	0.310666	+	0.950519i	$0.399448\pi$
$20$	0	0
$21$	2.70832	0.591004
$22$	0	0
$23$	0.938844	0.195763	0.0978813	−	0.995198i	$-0.468793\pi$
$23$	0.938844	0.195763	0.0978813	+	0.995198i	$0.468793\pi$
$24$	0	0
$25$	6.81279	1.36256
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.728654	0.135308	0.0676539	−	0.997709i	$-0.478449\pi$
$29$	0.728654	0.135308	0.0676539	+	0.997709i	$0.478449\pi$
$30$	0	0
$31$	−6.85361	−1.23095	−0.615473	−	0.788158i	$-0.711035\pi$
$31$	−6.85361	−1.23095	−0.615473	+	0.788158i	$0.711035\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	9.30843	1.57341
$36$	0	0
$37$	9.89179	1.62620	0.813100	−	0.582124i	$-0.197778\pi$
$37$	9.89179	1.62620	0.813100	+	0.582124i	$0.197778\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.83063	−0.442070	−0.221035	−	0.975266i	$-0.570944\pi$
$41$	−2.83063	−0.442070	−0.221035	+	0.975266i	$0.570944\pi$
$42$	0	0
$43$	−1.62418	−0.247685	−0.123843	−	0.992302i	$-0.539522\pi$
$43$	−1.62418	−0.247685	−0.123843	+	0.992302i	$0.539522\pi$
$44$	0	0
$45$	3.43697	0.512354
$46$	0	0
$47$	−1.10447	−0.161104	−0.0805520	−	0.996750i	$-0.525668\pi$
$47$	−1.10447	−0.161104	−0.0805520	+	0.996750i	$0.525668\pi$
$48$	0	0
$49$	0.334998	0.0478568
$50$	0	0
$51$	−0.375819	−0.0526252
$52$	0	0
$53$	12.1683	1.67144	0.835721	−	0.549155i	$-0.185050\pi$
$53$	12.1683	1.67144	0.835721	+	0.549155i	$0.185050\pi$
$54$	0	0
$55$	−3.43697	−0.463442
$56$	0	0
$57$	2.70832	0.358726
$58$	0	0
$59$	2.62933	0.342309	0.171155	−	0.985244i	$-0.445250\pi$
$59$	2.62933	0.342309	0.171155	+	0.985244i	$0.445250\pi$
$60$	0	0
$61$	−3.10447	−0.397487	−0.198744	−	0.980052i	$-0.563686\pi$
$61$	−3.10447	−0.397487	−0.198744	+	0.980052i	$0.563686\pi$
$62$	0	0
$63$	2.70832	0.341216
$64$	0	0
$65$	−3.43697	−0.426304
$66$	0	0
$67$	−4.73130	−0.578021	−0.289010	−	0.957326i	$-0.593326\pi$
$67$	−4.73130	−0.578021	−0.289010	+	0.957326i	$0.593326\pi$
$68$	0	0
$69$	0.938844	0.113024
$70$	0	0
$71$	0.108213	0.0128426	0.00642128	−	0.999979i	$-0.497956\pi$
$71$	0.108213	0.0128426	0.00642128	+	0.999979i	$0.497956\pi$
$72$	0	0
$73$	13.7046	1.60400	0.802000	−	0.597324i	$-0.203769\pi$
$73$	13.7046	1.60400	0.802000	+	0.597324i	$0.203769\pi$
$74$	0	0
$75$	6.81279	0.786674
$76$	0	0
$77$	−2.70832	−0.308642
$78$	0	0
$79$	−8.03708	−0.904242	−0.452121	−	0.891957i	$-0.649333\pi$
$79$	−8.03708	−0.904242	−0.452121	+	0.891957i	$0.649333\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.99626	0.548411	0.274205	−	0.961671i	$-0.411585\pi$
$83$	4.99626	0.548411	0.274205	+	0.961671i	$0.411585\pi$
$84$	0	0
$85$	−1.29168	−0.140102
$86$	0	0
$87$	0.728654	0.0781199
$88$	0	0
$89$	6.33250	0.671244	0.335622	−	0.941997i	$-0.391054\pi$
$89$	6.33250	0.671244	0.335622	+	0.941997i	$0.391054\pi$
$90$	0	0
$91$	−2.70832	−0.283909
$92$	0	0
$93$	−6.85361	−0.710687
$94$	0	0
$95$	9.30843	0.955024
$96$	0	0
$97$	15.3951	1.56313	0.781566	−	0.623823i	$-0.214421\pi$
$97$	15.3951	1.56313	0.781566	+	0.623823i	$0.214421\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	0.750232	0.0746509	0.0373254	−	0.999303i	$-0.488116\pi$
$101$	0.750232	0.0746509	0.0373254	+	0.999303i	$0.488116\pi$
$102$	0	0
$103$	−1.17202	−0.115482	−0.0577412	−	0.998332i	$-0.518390\pi$
$103$	−1.17202	−0.115482	−0.0577412	+	0.998332i	$0.518390\pi$
$104$	0	0
$105$	9.30843	0.908409
$106$	0	0
$107$	10.4129	1.00665	0.503327	−	0.864096i	$-0.332109\pi$
$107$	10.4129	1.00665	0.503327	+	0.864096i	$0.332109\pi$
$108$	0	0
$109$	−8.99891	−0.861939	−0.430970	−	0.902366i	$-0.641828\pi$
$109$	−8.99891	−0.861939	−0.430970	+	0.902366i	$0.641828\pi$
$110$	0	0
$111$	9.89179	0.938887
$112$	0	0
$113$	8.78731	0.826641	0.413320	−	0.910586i	$-0.364369\pi$
$113$	8.78731	0.826641	0.413320	+	0.910586i	$0.364369\pi$
$114$	0	0
$115$	3.22678	0.300899
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−1.01784	−0.0933050
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.83063	−0.255229
$124$	0	0
$125$	6.23052	0.557275
$126$	0	0
$127$	15.6842	1.39175	0.695876	−	0.718162i	$-0.255016\pi$
$127$	15.6842	1.39175	0.695876	+	0.718162i	$0.255016\pi$
$128$	0	0
$129$	−1.62418	−0.143001
$130$	0	0
$131$	−13.6613	−1.19359	−0.596795	−	0.802394i	$-0.703559\pi$
$131$	−13.6613	−1.19359	−0.596795	+	0.802394i	$0.703559\pi$
$132$	0	0
$133$	7.33500	0.636025
$134$	0	0
$135$	3.43697	0.295808
$136$	0	0
$137$	−10.9543	−0.935893	−0.467947	−	0.883757i	$-0.655006\pi$
$137$	−10.9543	−0.935893	−0.467947	+	0.883757i	$0.655006\pi$
$138$	0	0
$139$	−5.51597	−0.467858	−0.233929	−	0.972254i	$-0.575158\pi$
$139$	−5.51597	−0.467858	−0.233929	+	0.972254i	$0.575158\pi$
$140$	0	0
$141$	−1.10447	−0.0930134
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	2.50437	0.207976
$146$	0	0
$147$	0.334998	0.0276301
$148$	0	0
$149$	−4.95294	−0.405761	−0.202880	−	0.979204i	$-0.565030\pi$
$149$	−4.95294	−0.405761	−0.202880	+	0.979204i	$0.565030\pi$
$150$	0	0
$151$	−0.124960	−0.0101691	−0.00508456	−	0.999987i	$-0.501618\pi$
$151$	−0.124960	−0.0101691	−0.00508456	+	0.999987i	$0.501618\pi$
$152$	0	0
$153$	−0.375819	−0.0303832
$154$	0	0
$155$	−23.5557	−1.89204
$156$	0	0
$157$	−13.3517	−1.06559	−0.532793	−	0.846246i	$-0.678857\pi$
$157$	−13.3517	−1.06559	−0.532793	+	0.846246i	$0.678857\pi$
$158$	0	0
$159$	12.1683	0.965007
$160$	0	0
$161$	2.54269	0.200392
$162$	0	0
$163$	14.0804	1.10286	0.551431	−	0.834221i	$-0.314082\pi$
$163$	14.0804	1.10286	0.551431	+	0.834221i	$0.314082\pi$
$164$	0	0
$165$	−3.43697	−0.267568
$166$	0	0
$167$	−19.1645	−1.48300	−0.741498	−	0.670955i	$-0.765884\pi$
$167$	−19.1645	−1.48300	−0.741498	+	0.670955i	$0.765884\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.70832	0.207110
$172$	0	0
$173$	10.4115	0.791571	0.395786	−	0.918343i	$-0.370472\pi$
$173$	10.4115	0.791571	0.395786	+	0.918343i	$0.370472\pi$
$174$	0	0
$175$	18.4512	1.39478
$176$	0	0
$177$	2.62933	0.197632
$178$	0	0
$179$	−0.273842	−0.0204679	−0.0102340	−	0.999948i	$-0.503258\pi$
$179$	−0.273842	−0.0204679	−0.0102340	+	0.999948i	$0.503258\pi$
$180$	0	0
$181$	17.3517	1.28975	0.644873	−	0.764290i	$-0.276910\pi$
$181$	17.3517	1.28975	0.644873	+	0.764290i	$0.276910\pi$
$182$	0	0
$183$	−3.10447	−0.229489
$184$	0	0
$185$	33.9978	2.49957
$186$	0	0
$187$	0.375819	0.0274826
$188$	0	0
$189$	2.70832	0.197001
$190$	0	0
$191$	3.49298	0.252743	0.126372	−	0.991983i	$-0.459667\pi$
$191$	3.49298	0.252743	0.126372	+	0.991983i	$0.459667\pi$
$192$	0	0
$193$	3.33765	0.240249	0.120125	−	0.992759i	$-0.461671\pi$
$193$	3.33765	0.240249	0.120125	+	0.992759i	$0.461671\pi$
$194$	0	0
$195$	−3.43697	−0.246127
$196$	0	0
$197$	22.4103	1.59666	0.798332	−	0.602217i	$-0.205716\pi$
$197$	22.4103	1.59666	0.798332	+	0.602217i	$0.205716\pi$
$198$	0	0
$199$	−6.12231	−0.433999	−0.217000	−	0.976172i	$-0.569627\pi$
$199$	−6.12231	−0.433999	−0.217000	+	0.976172i	$0.569627\pi$
$200$	0	0
$201$	−4.73130	−0.333720
$202$	0	0
$203$	1.97343	0.138508
$204$	0	0
$205$	−9.72881	−0.679489
$206$	0	0
$207$	0.938844	0.0652542
$208$	0	0
$209$	−2.70832	−0.187338
$210$	0	0
$211$	−17.6983	−1.21840	−0.609202	−	0.793015i	$-0.708510\pi$
$211$	−17.6983	−1.21840	−0.609202	+	0.793015i	$0.708510\pi$
$212$	0	0
$213$	0.108213	0.00741465
$214$	0	0
$215$	−5.58227	−0.380708
$216$	0	0
$217$	−18.5618	−1.26006
$218$	0	0
$219$	13.7046	0.926070
$220$	0	0
$221$	0.375819	0.0252803
$222$	0	0
$223$	−25.5111	−1.70835	−0.854176	−	0.519984i	$-0.825938\pi$
$223$	−25.5111	−1.70835	−0.854176	+	0.519984i	$0.825938\pi$
$224$	0	0
$225$	6.81279	0.454186
$226$	0	0
$227$	−14.3339	−0.951375	−0.475687	−	0.879614i	$-0.657801\pi$
$227$	−14.3339	−0.951375	−0.475687	+	0.879614i	$0.657801\pi$
$228$	0	0
$229$	−17.2321	−1.13873	−0.569364	−	0.822086i	$-0.692810\pi$
$229$	−17.2321	−1.13873	−0.569364	+	0.822086i	$0.692810\pi$
$230$	0	0
$231$	−2.70832	−0.178194
$232$	0	0
$233$	7.64576	0.500890	0.250445	−	0.968131i	$-0.419423\pi$
$233$	7.64576	0.500890	0.250445	+	0.968131i	$0.419423\pi$
$234$	0	0
$235$	−3.79605	−0.247627
$236$	0	0
$237$	−8.03708	−0.522065
$238$	0	0
$239$	−10.4919	−0.678664	−0.339332	−	0.940667i	$-0.610201\pi$
$239$	−10.4919	−0.678664	−0.339332	+	0.940667i	$0.610201\pi$
$240$	0	0
$241$	−10.3772	−0.668456	−0.334228	−	0.942492i	$-0.608476\pi$
$241$	−10.3772	−0.668456	−0.334228	+	0.942492i	$0.608476\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.15138	0.0735589
$246$	0	0
$247$	−2.70832	−0.172326
$248$	0	0
$249$	4.99626	0.316625
$250$	0	0
$251$	8.53521	0.538738	0.269369	−	0.963037i	$-0.413185\pi$
$251$	8.53521	0.538738	0.269369	+	0.963037i	$0.413185\pi$
$252$	0	0
$253$	−0.938844	−0.0590246
$254$	0	0
$255$	−1.29168	−0.0808881
$256$	0	0
$257$	1.29433	0.0807380	0.0403690	−	0.999185i	$-0.487147\pi$
$257$	1.29433	0.0807380	0.0403690	+	0.999185i	$0.487147\pi$
$258$	0	0
$259$	26.7901	1.66466
$260$	0	0
$261$	0.728654	0.0451026
$262$	0	0
$263$	−12.4204	−0.765873	−0.382937	−	0.923775i	$-0.625087\pi$
$263$	−12.4204	−0.765873	−0.382937	+	0.923775i	$0.625087\pi$
$264$	0	0
$265$	41.8221	2.56911
$266$	0	0
$267$	6.33250	0.387543
$268$	0	0
$269$	−23.5774	−1.43754	−0.718771	−	0.695247i	$-0.755295\pi$
$269$	−23.5774	−1.43754	−0.718771	+	0.695247i	$0.755295\pi$
$270$	0	0
$271$	−0.871300	−0.0529277	−0.0264638	−	0.999650i	$-0.508425\pi$
$271$	−0.871300	−0.0529277	−0.0264638	+	0.999650i	$0.508425\pi$
$272$	0	0
$273$	−2.70832	−0.163915
$274$	0	0
$275$	−6.81279	−0.410827
$276$	0	0
$277$	18.2499	1.09653	0.548266	−	0.836304i	$-0.315288\pi$
$277$	18.2499	1.09653	0.548266	+	0.836304i	$0.315288\pi$
$278$	0	0
$279$	−6.85361	−0.410315
$280$	0	0
$281$	−31.4578	−1.87661	−0.938307	−	0.345804i	$-0.887606\pi$
$281$	−31.4578	−1.87661	−0.938307	+	0.345804i	$0.887606\pi$
$282$	0	0
$283$	8.89693	0.528868	0.264434	−	0.964404i	$-0.414815\pi$
$283$	8.89693	0.528868	0.264434	+	0.964404i	$0.414815\pi$
$284$	0	0
$285$	9.30843	0.551384
$286$	0	0
$287$	−7.66626	−0.452525
$288$	0	0
$289$	−16.8588	−0.991692
$290$	0	0
$291$	15.3951	0.902475
$292$	0	0
$293$	10.1656	0.593882	0.296941	−	0.954896i	$-0.404033\pi$
$293$	10.1656	0.593882	0.296941	+	0.954896i	$0.404033\pi$
$294$	0	0
$295$	9.03693	0.526150
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−0.938844	−0.0542948
$300$	0	0
$301$	−4.39880	−0.253543
$302$	0	0
$303$	0.750232	0.0430997
$304$	0	0
$305$	−10.6700	−0.610962
$306$	0	0
$307$	−12.2116	−0.696953	−0.348476	−	0.937318i	$-0.613301\pi$
$307$	−12.2116	−0.696953	−0.348476	+	0.937318i	$0.613301\pi$
$308$	0	0
$309$	−1.17202	−0.0666738
$310$	0	0
$311$	−13.9811	−0.792794	−0.396397	−	0.918079i	$-0.629740\pi$
$311$	−13.9811	−0.792794	−0.396397	+	0.918079i	$0.629740\pi$
$312$	0	0
$313$	10.1465	0.573516	0.286758	−	0.958003i	$-0.407422\pi$
$313$	10.1465	0.573516	0.286758	+	0.958003i	$0.407422\pi$
$314$	0	0
$315$	9.30843	0.524470
$316$	0	0
$317$	−8.74540	−0.491191	−0.245595	−	0.969372i	$-0.578983\pi$
$317$	−8.74540	−0.491191	−0.245595	+	0.969372i	$0.578983\pi$
$318$	0	0
$319$	−0.728654	−0.0407968
$320$	0	0
$321$	10.4129	0.581191
$322$	0	0
$323$	−1.01784	−0.0566340
$324$	0	0
$325$	−6.81279	−0.377906
$326$	0	0
$327$	−8.99891	−0.497641
$328$	0	0
$329$	−2.99127	−0.164914
$330$	0	0
$331$	−10.8320	−0.595383	−0.297691	−	0.954662i	$-0.596217\pi$
$331$	−10.8320	−0.595383	−0.297691	+	0.954662i	$0.596217\pi$
$332$	0	0
$333$	9.89179	0.542067
$334$	0	0
$335$	−16.2614	−0.888454
$336$	0	0
$337$	−15.3707	−0.837294	−0.418647	−	0.908149i	$-0.637496\pi$
$337$	−15.3707	−0.837294	−0.418647	+	0.908149i	$0.637496\pi$
$338$	0	0
$339$	8.78731	0.477261
$340$	0	0
$341$	6.85361	0.371144
$342$	0	0
$343$	−18.0510	−0.974660
$344$	0	0
$345$	3.22678	0.173724
$346$	0	0
$347$	−28.7176	−1.54164	−0.770820	−	0.637052i	$-0.780153\pi$
$347$	−28.7176	−1.54164	−0.770820	+	0.637052i	$0.780153\pi$
$348$	0	0
$349$	−22.4486	−1.20164	−0.600822	−	0.799383i	$-0.705160\pi$
$349$	−22.4486	−1.20164	−0.600822	+	0.799383i	$0.705160\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	22.3109	1.18749	0.593745	−	0.804653i	$-0.297649\pi$
$353$	22.3109	1.18749	0.593745	+	0.804653i	$0.297649\pi$
$354$	0	0
$355$	0.371926	0.0197398
$356$	0	0
$357$	−1.01784	−0.0538697
$358$	0	0
$359$	5.75803	0.303897	0.151949	−	0.988388i	$-0.451445\pi$
$359$	5.75803	0.303897	0.151949	+	0.988388i	$0.451445\pi$
$360$	0	0
$361$	−11.6650	−0.613947
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	47.1023	2.46545
$366$	0	0
$367$	−2.07634	−0.108384	−0.0541921	−	0.998531i	$-0.517258\pi$
$367$	−2.07634	−0.108384	−0.0541921	+	0.998531i	$0.517258\pi$
$368$	0	0
$369$	−2.83063	−0.147357
$370$	0	0
$371$	32.9556	1.71097
$372$	0	0
$373$	−28.4129	−1.47116	−0.735582	−	0.677436i	$-0.763091\pi$
$373$	−28.4129	−1.47116	−0.735582	+	0.677436i	$0.763091\pi$
$374$	0	0
$375$	6.23052	0.321743
$376$	0	0
$377$	−0.728654	−0.0375276
$378$	0	0
$379$	5.24213	0.269270	0.134635	−	0.990895i	$-0.457014\pi$
$379$	5.24213	0.269270	0.134635	+	0.990895i	$0.457014\pi$
$380$	0	0
$381$	15.6842	0.803528
$382$	0	0
$383$	5.83702	0.298258	0.149129	−	0.988818i	$-0.452353\pi$
$383$	5.83702	0.298258	0.149129	+	0.988818i	$0.452353\pi$
$384$	0	0
$385$	−9.30843	−0.474401
$386$	0	0
$387$	−1.62418	−0.0825618
$388$	0	0
$389$	32.9941	1.67287	0.836433	−	0.548069i	$-0.184637\pi$
$389$	32.9941	1.67287	0.836433	+	0.548069i	$0.184637\pi$
$390$	0	0
$391$	−0.352835	−0.0178437
$392$	0	0
$393$	−13.6613	−0.689120
$394$	0	0
$395$	−27.6232	−1.38988
$396$	0	0
$397$	13.6472	0.684932	0.342466	−	0.939530i	$-0.388738\pi$
$397$	13.6472	0.684932	0.342466	+	0.939530i	$0.388738\pi$
$398$	0	0
$399$	7.33500	0.367209
$400$	0	0
$401$	14.4689	0.722543	0.361271	−	0.932461i	$-0.382343\pi$
$401$	14.4689	0.722543	0.361271	+	0.932461i	$0.382343\pi$
$402$	0	0
$403$	6.85361	0.341403
$404$	0	0
$405$	3.43697	0.170785
$406$	0	0
$407$	−9.89179	−0.490318
$408$	0	0
$409$	−12.2879	−0.607600	−0.303800	−	0.952736i	$-0.598255\pi$
$409$	−12.2879	−0.607600	−0.303800	+	0.952736i	$0.598255\pi$
$410$	0	0
$411$	−10.9543	−0.540338
$412$	0	0
$413$	7.12106	0.350404
$414$	0	0
$415$	17.1720	0.842941
$416$	0	0
$417$	−5.51597	−0.270118
$418$	0	0
$419$	6.17973	0.301899	0.150950	−	0.988541i	$-0.451767\pi$
$419$	6.17973	0.301899	0.150950	+	0.988541i	$0.451767\pi$
$420$	0	0
$421$	−13.0726	−0.637120	−0.318560	−	0.947903i	$-0.603199\pi$
$421$	−13.0726	−0.637120	−0.318560	+	0.947903i	$0.603199\pi$
$422$	0	0
$423$	−1.10447	−0.0537013
$424$	0	0
$425$	−2.56038	−0.124196
$426$	0	0
$427$	−8.40791	−0.406887
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	15.4092	0.742233	0.371117	−	0.928586i	$-0.378975\pi$
$431$	15.4092	0.742233	0.371117	+	0.928586i	$0.378975\pi$
$432$	0	0
$433$	−5.07790	−0.244028	−0.122014	−	0.992528i	$-0.538935\pi$
$433$	−5.07790	−0.244028	−0.122014	+	0.992528i	$0.538935\pi$
$434$	0	0
$435$	2.50437	0.120075
$436$	0	0
$437$	2.54269	0.121633
$438$	0	0
$439$	21.6270	1.03220	0.516100	−	0.856528i	$-0.327383\pi$
$439$	21.6270	1.03220	0.516100	+	0.856528i	$0.327383\pi$
$440$	0	0
$441$	0.334998	0.0159523
$442$	0	0
$443$	−37.2437	−1.76950	−0.884751	−	0.466065i	$-0.845671\pi$
$443$	−37.2437	−1.76950	−0.884751	+	0.466065i	$0.845671\pi$
$444$	0	0
$445$	21.7646	1.03174
$446$	0	0
$447$	−4.95294	−0.234266
$448$	0	0
$449$	−16.4332	−0.775532	−0.387766	−	0.921758i	$-0.626753\pi$
$449$	−16.4332	−0.775532	−0.387766	+	0.921758i	$0.626753\pi$
$450$	0	0
$451$	2.83063	0.133289
$452$	0	0
$453$	−0.124960	−0.00587114
$454$	0	0
$455$	−9.30843	−0.436386
$456$	0	0
$457$	23.9059	1.11827	0.559135	−	0.829077i	$-0.311133\pi$
$457$	23.9059	1.11827	0.559135	+	0.829077i	$0.311133\pi$
$458$	0	0
$459$	−0.375819	−0.0175417
$460$	0	0
$461$	11.1619	0.519861	0.259930	−	0.965627i	$-0.416300\pi$
$461$	11.1619	0.519861	0.259930	+	0.965627i	$0.416300\pi$
$462$	0	0
$463$	11.2524	0.522944	0.261472	−	0.965211i	$-0.415792\pi$
$463$	11.2524	0.522944	0.261472	+	0.965211i	$0.415792\pi$
$464$	0	0
$465$	−23.5557	−1.09237
$466$	0	0
$467$	23.9736	1.10937	0.554683	−	0.832062i	$-0.312840\pi$
$467$	23.9736	1.10937	0.554683	+	0.832062i	$0.312840\pi$
$468$	0	0
$469$	−12.8139	−0.591690
$470$	0	0
$471$	−13.3517	−0.615216
$472$	0	0
$473$	1.62418	0.0746799
$474$	0	0
$475$	18.4512	0.846600
$476$	0	0
$477$	12.1683	0.557147
$478$	0	0
$479$	−5.16719	−0.236095	−0.118047	−	0.993008i	$-0.537663\pi$
$479$	−5.16719	−0.236095	−0.118047	+	0.993008i	$0.537663\pi$
$480$	0	0
$481$	−9.89179	−0.451027
$482$	0	0
$483$	2.54269	0.115696
$484$	0	0
$485$	52.9124	2.40263
$486$	0	0
$487$	2.38065	0.107878	0.0539388	−	0.998544i	$-0.482822\pi$
$487$	2.38065	0.107878	0.0539388	+	0.998544i	$0.482822\pi$
$488$	0	0
$489$	14.0804	0.636738
$490$	0	0
$491$	−3.48918	−0.157464	−0.0787322	−	0.996896i	$-0.525087\pi$
$491$	−3.48918	−0.157464	−0.0787322	+	0.996896i	$0.525087\pi$
$492$	0	0
$493$	−0.273842	−0.0123332
$494$	0	0
$495$	−3.43697	−0.154481
$496$	0	0
$497$	0.293076	0.0131463
$498$	0	0
$499$	−34.1048	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$499$	−34.1048	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$500$	0	0
$501$	−19.1645	−0.856208
$502$	0	0
$503$	32.6218	1.45454	0.727268	−	0.686353i	$-0.240790\pi$
$503$	32.6218	1.45454	0.727268	+	0.686353i	$0.240790\pi$
$504$	0	0
$505$	2.57853	0.114743
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	38.8299	1.72110	0.860552	−	0.509363i	$-0.170119\pi$
$509$	38.8299	1.72110	0.860552	+	0.509363i	$0.170119\pi$
$510$	0	0
$511$	37.1164	1.64193
$512$	0	0
$513$	2.70832	0.119575
$514$	0	0
$515$	−4.02820	−0.177504
$516$	0	0
$517$	1.10447	0.0485747
$518$	0	0
$519$	10.4115	0.457014
$520$	0	0
$521$	14.3366	0.628096	0.314048	−	0.949407i	$-0.398315\pi$
$521$	14.3366	0.628096	0.314048	+	0.949407i	$0.398315\pi$
$522$	0	0
$523$	14.4500	0.631854	0.315927	−	0.948784i	$-0.397685\pi$
$523$	14.4500	0.631854	0.315927	+	0.948784i	$0.397685\pi$
$524$	0	0
$525$	18.4512	0.805277
$526$	0	0
$527$	2.57572	0.112200
$528$	0	0
$529$	−22.1186	−0.961677
$530$	0	0
$531$	2.62933	0.114103
$532$	0	0
$533$	2.83063	0.122608
$534$	0	0
$535$	35.7889	1.54729
$536$	0	0
$537$	−0.273842	−0.0118172
$538$	0	0
$539$	−0.334998	−0.0144294
$540$	0	0
$541$	−42.0770	−1.80903	−0.904515	−	0.426441i	$-0.859767\pi$
$541$	−42.0770	−1.80903	−0.904515	+	0.426441i	$0.859767\pi$
$542$	0	0
$543$	17.3517	0.744635
$544$	0	0
$545$	−30.9290	−1.32485
$546$	0	0
$547$	−23.3098	−0.996656	−0.498328	−	0.866988i	$-0.666053\pi$
$547$	−23.3098	−0.996656	−0.498328	+	0.866988i	$0.666053\pi$
$548$	0	0
$549$	−3.10447	−0.132496
$550$	0	0
$551$	1.97343	0.0840709
$552$	0	0
$553$	−21.7670	−0.925627
$554$	0	0
$555$	33.9978	1.44313
$556$	0	0
$557$	−39.0374	−1.65407	−0.827034	−	0.562152i	$-0.809973\pi$
$557$	−39.0374	−1.65407	−0.827034	+	0.562152i	$0.809973\pi$
$558$	0	0
$559$	1.62418	0.0686955
$560$	0	0
$561$	0.375819	0.0158671
$562$	0	0
$563$	−5.08008	−0.214100	−0.107050	−	0.994254i	$-0.534140\pi$
$563$	−5.08008	−0.214100	−0.107050	+	0.994254i	$0.534140\pi$
$564$	0	0
$565$	30.2018	1.27060
$566$	0	0
$567$	2.70832	0.113739
$568$	0	0
$569$	−16.6664	−0.698692	−0.349346	−	0.936994i	$-0.613596\pi$
$569$	−16.6664	−0.698692	−0.349346	+	0.936994i	$0.613596\pi$
$570$	0	0
$571$	29.5863	1.23815	0.619075	−	0.785332i	$-0.287508\pi$
$571$	29.5863	1.23815	0.619075	+	0.785332i	$0.287508\pi$
$572$	0	0
$573$	3.49298	0.145922
$574$	0	0
$575$	6.39615	0.266738
$576$	0	0
$577$	4.41009	0.183594	0.0917972	−	0.995778i	$-0.470739\pi$
$577$	4.41009	0.183594	0.0917972	+	0.995778i	$0.470739\pi$
$578$	0	0
$579$	3.33765	0.138708
$580$	0	0
$581$	13.5315	0.561380
$582$	0	0
$583$	−12.1683	−0.503959
$584$	0	0
$585$	−3.43697	−0.142101
$586$	0	0
$587$	3.91618	0.161638	0.0808189	−	0.996729i	$-0.474246\pi$
$587$	3.91618	0.161638	0.0808189	+	0.996729i	$0.474246\pi$
$588$	0	0
$589$	−18.5618	−0.764825
$590$	0	0
$591$	22.4103	0.921835
$592$	0	0
$593$	6.96323	0.285946	0.142973	−	0.989727i	$-0.454334\pi$
$593$	6.96323	0.285946	0.142973	+	0.989727i	$0.454334\pi$
$594$	0	0
$595$	−3.49828	−0.143416
$596$	0	0
$597$	−6.12231	−0.250570
$598$	0	0
$599$	33.8221	1.38193	0.690966	−	0.722887i	$-0.257185\pi$
$599$	33.8221	1.38193	0.690966	+	0.722887i	$0.257185\pi$
$600$	0	0
$601$	30.2934	1.23569	0.617847	−	0.786299i	$-0.288005\pi$
$601$	30.2934	1.23569	0.617847	+	0.786299i	$0.288005\pi$
$602$	0	0
$603$	−4.73130	−0.192674
$604$	0	0
$605$	3.43697	0.139733
$606$	0	0
$607$	8.60634	0.349321	0.174660	−	0.984629i	$-0.444117\pi$
$607$	8.60634	0.349321	0.174660	+	0.984629i	$0.444117\pi$
$608$	0	0
$609$	1.97343	0.0799674
$610$	0	0
$611$	1.10447	0.0446822
$612$	0	0
$613$	−9.58975	−0.387326	−0.193663	−	0.981068i	$-0.562037\pi$
$613$	−9.58975	−0.387326	−0.193663	+	0.981068i	$0.562037\pi$
$614$	0	0
$615$	−9.72881	−0.392303
$616$	0	0
$617$	−28.1073	−1.13156	−0.565778	−	0.824558i	$-0.691424\pi$
$617$	−28.1073	−1.13156	−0.565778	+	0.824558i	$0.691424\pi$
$618$	0	0
$619$	−4.33780	−0.174351	−0.0871754	−	0.996193i	$-0.527784\pi$
$619$	−4.33780	−0.174351	−0.0871754	+	0.996193i	$0.527784\pi$
$620$	0	0
$621$	0.938844	0.0376745
$622$	0	0
$623$	17.1504	0.687118
$624$	0	0
$625$	−12.6498	−0.505993
$626$	0	0
$627$	−2.70832	−0.108160
$628$	0	0
$629$	−3.71752	−0.148227
$630$	0	0
$631$	23.6321	0.940780	0.470390	−	0.882459i	$-0.344113\pi$
$631$	23.6321	0.940780	0.470390	+	0.882459i	$0.344113\pi$
$632$	0	0
$633$	−17.6983	−0.703446
$634$	0	0
$635$	53.9064	2.13921
$636$	0	0
$637$	−0.334998	−0.0132731
$638$	0	0
$639$	0.108213	0.00428085
$640$	0	0
$641$	−2.92740	−0.115625	−0.0578126	−	0.998327i	$-0.518413\pi$
$641$	−2.92740	−0.115625	−0.0578126	+	0.998327i	$0.518413\pi$
$642$	0	0
$643$	−19.6334	−0.774268	−0.387134	−	0.922024i	$-0.626535\pi$
$643$	−19.6334	−0.774268	−0.387134	+	0.922024i	$0.626535\pi$
$644$	0	0
$645$	−5.58227	−0.219802
$646$	0	0
$647$	29.4855	1.15919	0.579597	−	0.814903i	$-0.303210\pi$
$647$	29.4855	1.15919	0.579597	+	0.814903i	$0.303210\pi$
$648$	0	0
$649$	−2.62933	−0.103210
$650$	0	0
$651$	−18.5618	−0.727494
$652$	0	0
$653$	13.7554	0.538290	0.269145	−	0.963100i	$-0.413259\pi$
$653$	13.7554	0.538290	0.269145	+	0.963100i	$0.413259\pi$
$654$	0	0
$655$	−46.9534	−1.83462
$656$	0	0
$657$	13.7046	0.534667
$658$	0	0
$659$	−3.35658	−0.130754	−0.0653768	−	0.997861i	$-0.520825\pi$
$659$	−3.35658	−0.130754	−0.0653768	+	0.997861i	$0.520825\pi$
$660$	0	0
$661$	−1.49298	−0.0580704	−0.0290352	−	0.999578i	$-0.509243\pi$
$661$	−1.49298	−0.0580704	−0.0290352	+	0.999578i	$0.509243\pi$
$662$	0	0
$663$	0.375819	0.0145956
$664$	0	0
$665$	25.2102	0.977610
$666$	0	0
$667$	0.684093	0.0264882
$668$	0	0
$669$	−25.5111	−0.986318
$670$	0	0
$671$	3.10447	0.119847
$672$	0	0
$673$	−0.0725376	−0.00279612	−0.00139806	−	0.999999i	$-0.500445\pi$
$673$	−0.0725376	−0.00279612	−0.00139806	+	0.999999i	$0.500445\pi$
$674$	0	0
$675$	6.81279	0.262225
$676$	0	0
$677$	−9.81404	−0.377184	−0.188592	−	0.982056i	$-0.560392\pi$
$677$	−9.81404	−0.377184	−0.188592	+	0.982056i	$0.560392\pi$
$678$	0	0
$679$	41.6948	1.60010
$680$	0	0
$681$	−14.3339	−0.549277
$682$	0	0
$683$	25.6115	0.979996	0.489998	−	0.871723i	$-0.336997\pi$
$683$	25.6115	0.979996	0.489998	+	0.871723i	$0.336997\pi$
$684$	0	0
$685$	−37.6498	−1.43853
$686$	0	0
$687$	−17.2321	−0.657445
$688$	0	0
$689$	−12.1683	−0.463574
$690$	0	0
$691$	36.6920	1.39583	0.697914	−	0.716182i	$-0.254112\pi$
$691$	36.6920	1.39583	0.697914	+	0.716182i	$0.254112\pi$
$692$	0	0
$693$	−2.70832	−0.102881
$694$	0	0
$695$	−18.9582	−0.719127
$696$	0	0
$697$	1.06380	0.0402945
$698$	0	0
$699$	7.64576	0.289189
$700$	0	0
$701$	−6.94141	−0.262173	−0.131087	−	0.991371i	$-0.541847\pi$
$701$	−6.94141	−0.262173	−0.131087	+	0.991371i	$0.541847\pi$
$702$	0	0
$703$	26.7901	1.01041
$704$	0	0
$705$	−3.79605	−0.142967
$706$	0	0
$707$	2.03187	0.0764163
$708$	0	0
$709$	23.1098	0.867906	0.433953	−	0.900936i	$-0.357118\pi$
$709$	23.1098	0.867906	0.433953	+	0.900936i	$0.357118\pi$
$710$	0	0
$711$	−8.03708	−0.301414
$712$	0	0
$713$	−6.43448	−0.240973
$714$	0	0
$715$	3.43697	0.128536
$716$	0	0
$717$	−10.4919	−0.391827
$718$	0	0
$719$	29.2868	1.09222	0.546108	−	0.837715i	$-0.316109\pi$
$719$	29.2868	1.09222	0.546108	+	0.837715i	$0.316109\pi$
$720$	0	0
$721$	−3.17420	−0.118213
$722$	0	0
$723$	−10.3772	−0.385933
$724$	0	0
$725$	4.96417	0.184365
$726$	0	0
$727$	38.5277	1.42891	0.714457	−	0.699679i	$-0.246674\pi$
$727$	38.5277	1.42891	0.714457	+	0.699679i	$0.246674\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.610398	0.0225764
$732$	0	0
$733$	−41.9545	−1.54962	−0.774812	−	0.632191i	$-0.782156\pi$
$733$	−41.9545	−1.54962	−0.774812	+	0.632191i	$0.782156\pi$
$734$	0	0
$735$	1.15138	0.0424692
$736$	0	0
$737$	4.73130	0.174280
$738$	0	0
$739$	−34.5407	−1.27060	−0.635300	−	0.772266i	$-0.719123\pi$
$739$	−34.5407	−1.27060	−0.635300	+	0.772266i	$0.719123\pi$
$740$	0	0
$741$	−2.70832	−0.0994926
$742$	0	0
$743$	−0.881429	−0.0323365	−0.0161682	−	0.999869i	$-0.505147\pi$
$743$	−0.881429	−0.0323365	−0.0161682	+	0.999869i	$0.505147\pi$
$744$	0	0
$745$	−17.0231	−0.623679
$746$	0	0
$747$	4.99626	0.182804
$748$	0	0
$749$	28.2015	1.03046
$750$	0	0
$751$	36.2040	1.32110	0.660551	−	0.750782i	$-0.270323\pi$
$751$	36.2040	1.32110	0.660551	+	0.750782i	$0.270323\pi$
$752$	0	0
$753$	8.53521	0.311040
$754$	0	0
$755$	−0.429485	−0.0156306
$756$	0	0
$757$	−37.1353	−1.34971	−0.674853	−	0.737952i	$-0.735793\pi$
$757$	−37.1353	−1.34971	−0.674853	+	0.737952i	$0.735793\pi$
$758$	0	0
$759$	−0.938844	−0.0340779
$760$	0	0
$761$	−20.7825	−0.753364	−0.376682	−	0.926343i	$-0.622935\pi$
$761$	−20.7825	−0.753364	−0.376682	+	0.926343i	$0.622935\pi$
$762$	0	0
$763$	−24.3719	−0.882323
$764$	0	0
$765$	−1.29168	−0.0467008
$766$	0	0
$767$	−2.62933	−0.0949395
$768$	0	0
$769$	−51.4501	−1.85534	−0.927670	−	0.373402i	$-0.878191\pi$
$769$	−51.4501	−1.85534	−0.927670	+	0.373402i	$0.878191\pi$
$770$	0	0
$771$	1.29433	0.0466141
$772$	0	0
$773$	−42.8730	−1.54204	−0.771018	−	0.636814i	$-0.780252\pi$
$773$	−42.8730	−1.54204	−0.771018	+	0.636814i	$0.780252\pi$
$774$	0	0
$775$	−46.6923	−1.67724
$776$	0	0
$777$	26.7901	0.961090
$778$	0	0
$779$	−7.66626	−0.274672
$780$	0	0
$781$	−0.108213	−0.00387217
$782$	0	0
$783$	0.728654	0.0260400
$784$	0	0
$785$	−45.8896	−1.63787
$786$	0	0
$787$	28.5429	1.01744	0.508721	−	0.860931i	$-0.330118\pi$
$787$	28.5429	1.01744	0.508721	+	0.860931i	$0.330118\pi$
$788$	0	0
$789$	−12.4204	−0.442177
$790$	0	0
$791$	23.7989	0.846190
$792$	0	0
$793$	3.10447	0.110243
$794$	0	0
$795$	41.8221	1.48328
$796$	0	0
$797$	37.8702	1.34143	0.670716	−	0.741714i	$-0.265987\pi$
$797$	37.8702	1.34143	0.670716	+	0.741714i	$0.265987\pi$
$798$	0	0
$799$	0.415082	0.0146845
$800$	0	0
$801$	6.33250	0.223748
$802$	0	0
$803$	−13.7046	−0.483624
$804$	0	0
$805$	8.73916	0.308015
$806$	0	0
$807$	−23.5774	−0.829965
$808$	0	0
$809$	−3.13494	−0.110219	−0.0551093	−	0.998480i	$-0.517551\pi$
$809$	−3.13494	−0.110219	−0.0551093	+	0.998480i	$0.517551\pi$
$810$	0	0
$811$	−48.9455	−1.71871	−0.859354	−	0.511381i	$-0.829134\pi$
$811$	−48.9455	−1.71871	−0.859354	+	0.511381i	$0.829134\pi$
$812$	0	0
$813$	−0.871300	−0.0305578
$814$	0	0
$815$	48.3940	1.69517
$816$	0	0
$817$	−4.39880	−0.153895
$818$	0	0
$819$	−2.70832	−0.0946364
$820$	0	0
$821$	47.7077	1.66501	0.832505	−	0.554017i	$-0.186906\pi$
$821$	47.7077	1.66501	0.832505	+	0.554017i	$0.186906\pi$
$822$	0	0
$823$	38.8361	1.35374	0.676871	−	0.736102i	$-0.263336\pi$
$823$	38.8361	1.35374	0.676871	+	0.736102i	$0.263336\pi$
$824$	0	0
$825$	−6.81279	−0.237191
$826$	0	0
$827$	−1.78092	−0.0619288	−0.0309644	−	0.999520i	$-0.509858\pi$
$827$	−1.78092	−0.0619288	−0.0309644	+	0.999520i	$0.509858\pi$
$828$	0	0
$829$	−42.3216	−1.46989	−0.734945	−	0.678127i	$-0.762792\pi$
$829$	−42.3216	−1.46989	−0.734945	+	0.678127i	$0.762792\pi$
$830$	0	0
$831$	18.2499	0.633083
$832$	0	0
$833$	−0.125898	−0.00436212
$834$	0	0
$835$	−65.8680	−2.27946
$836$	0	0
$837$	−6.85361	−0.236896
$838$	0	0
$839$	8.83546	0.305034	0.152517	−	0.988301i	$-0.451262\pi$
$839$	8.83546	0.305034	0.152517	+	0.988301i	$0.451262\pi$
$840$	0	0
$841$	−28.4691	−0.981692
$842$	0	0
$843$	−31.4578	−1.08346
$844$	0	0
$845$	3.43697	0.118236
$846$	0	0
$847$	2.70832	0.0930590
$848$	0	0
$849$	8.89693	0.305342
$850$	0	0
$851$	9.28685	0.318349
$852$	0	0
$853$	12.6753	0.433994	0.216997	−	0.976172i	$-0.430374\pi$
$853$	12.6753	0.433994	0.216997	+	0.976172i	$0.430374\pi$
$854$	0	0
$855$	9.30843	0.318341
$856$	0	0
$857$	0.756850	0.0258535	0.0129268	−	0.999916i	$-0.495885\pi$
$857$	0.756850	0.0258535	0.0129268	+	0.999916i	$0.495885\pi$
$858$	0	0
$859$	−15.8727	−0.541569	−0.270785	−	0.962640i	$-0.587283\pi$
$859$	−15.8727	−0.541569	−0.270785	+	0.962640i	$0.587283\pi$
$860$	0	0
$861$	−7.66626	−0.261265
$862$	0	0
$863$	−2.47134	−0.0841254	−0.0420627	−	0.999115i	$-0.513393\pi$
$863$	−2.47134	−0.0841254	−0.0420627	+	0.999115i	$0.513393\pi$
$864$	0	0
$865$	35.7840	1.21669
$866$	0	0
$867$	−16.8588	−0.572554
$868$	0	0
$869$	8.03708	0.272639
$870$	0	0
$871$	4.73130	0.160314
$872$	0	0
$873$	15.3951	0.521044
$874$	0	0
$875$	16.8743	0.570454
$876$	0	0
$877$	7.66407	0.258797	0.129399	−	0.991593i	$-0.458695\pi$
$877$	7.66407	0.258797	0.129399	+	0.991593i	$0.458695\pi$
$878$	0	0
$879$	10.1656	0.342878
$880$	0	0
$881$	−14.7544	−0.497090	−0.248545	−	0.968620i	$-0.579952\pi$
$881$	−14.7544	−0.497090	−0.248545	+	0.968620i	$0.579952\pi$
$882$	0	0
$883$	6.45887	0.217358	0.108679	−	0.994077i	$-0.465338\pi$
$883$	6.45887	0.217358	0.108679	+	0.994077i	$0.465338\pi$
$884$	0	0
$885$	9.03693	0.303773
$886$	0	0
$887$	31.1889	1.04722	0.523611	−	0.851958i	$-0.324585\pi$
$887$	31.1889	1.04722	0.523611	+	0.851958i	$0.324585\pi$
$888$	0	0
$889$	42.4780	1.42466
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−2.99127	−0.100099
$894$	0	0
$895$	−0.941188	−0.0314605
$896$	0	0
$897$	−0.938844	−0.0313471
$898$	0	0
$899$	−4.99392	−0.166556
$900$	0	0
$901$	−4.57307	−0.152351
$902$	0	0
$903$	−4.39880	−0.146383
$904$	0	0
$905$	59.6375	1.98242
$906$	0	0
$907$	−32.4204	−1.07650	−0.538251	−	0.842785i	$-0.680915\pi$
$907$	−32.4204	−1.07650	−0.538251	+	0.842785i	$0.680915\pi$
$908$	0	0
$909$	0.750232	0.0248836
$910$	0	0
$911$	−49.2083	−1.63034	−0.815172	−	0.579219i	$-0.803358\pi$
$911$	−49.2083	−1.63034	−0.815172	+	0.579219i	$0.803358\pi$
$912$	0	0
$913$	−4.99626	−0.165352
$914$	0	0
$915$	−10.6700	−0.352739
$916$	0	0
$917$	−36.9991	−1.22182
$918$	0	0
$919$	−15.2692	−0.503683	−0.251842	−	0.967768i	$-0.581036\pi$
$919$	−15.2692	−0.503683	−0.251842	+	0.967768i	$0.581036\pi$
$920$	0	0
$921$	−12.2116	−0.402386
$922$	0	0
$923$	−0.108213	−0.00356188
$924$	0	0
$925$	67.3907	2.21579
$926$	0	0
$927$	−1.17202	−0.0384941
$928$	0	0
$929$	−21.3644	−0.700944	−0.350472	−	0.936573i	$-0.613979\pi$
$929$	−21.3644	−0.700944	−0.350472	+	0.936573i	$0.613979\pi$
$930$	0	0
$931$	0.907281	0.0297349
$932$	0	0
$933$	−13.9811	−0.457720
$934$	0	0
$935$	1.29168	0.0422425
$936$	0	0
$937$	3.74128	0.122222	0.0611111	−	0.998131i	$-0.480536\pi$
$937$	3.74128	0.122222	0.0611111	+	0.998131i	$0.480536\pi$
$938$	0	0
$939$	10.1465	0.331120
$940$	0	0
$941$	−5.50063	−0.179315	−0.0896576	−	0.995973i	$-0.528577\pi$
$941$	−5.50063	−0.179315	−0.0896576	+	0.995973i	$0.528577\pi$
$942$	0	0
$943$	−2.65752	−0.0865408
$944$	0	0
$945$	9.30843	0.302803
$946$	0	0
$947$	46.7939	1.52060	0.760299	−	0.649573i	$-0.225052\pi$
$947$	46.7939	1.52060	0.760299	+	0.649573i	$0.225052\pi$
$948$	0	0
$949$	−13.7046	−0.444869
$950$	0	0
$951$	−8.74540	−0.283589
$952$	0	0
$953$	−21.5138	−0.696900	−0.348450	−	0.937327i	$-0.613292\pi$
$953$	−21.5138	−0.696900	−0.348450	+	0.937327i	$0.613292\pi$
$954$	0	0
$955$	12.0053	0.388482
$956$	0	0
$957$	−0.728654	−0.0235541
$958$	0	0
$959$	−29.6679	−0.958026
$960$	0	0
$961$	15.9720	0.515227
$962$	0	0
$963$	10.4129	0.335551
$964$	0	0
$965$	11.4714	0.369278
$966$	0	0
$967$	−55.0271	−1.76955	−0.884776	−	0.466016i	$-0.845689\pi$
$967$	−55.0271	−1.76955	−0.884776	+	0.466016i	$0.845689\pi$
$968$	0	0
$969$	−1.01784	−0.0326977
$970$	0	0
$971$	−20.4346	−0.655779	−0.327889	−	0.944716i	$-0.606337\pi$
$971$	−20.4346	−0.655779	−0.327889	+	0.944716i	$0.606337\pi$
$972$	0	0
$973$	−14.9390	−0.478923
$974$	0	0
$975$	−6.81279	−0.218184
$976$	0	0
$977$	30.1567	0.964800	0.482400	−	0.875951i	$-0.339765\pi$
$977$	30.1567	0.964800	0.482400	+	0.875951i	$0.339765\pi$
$978$	0	0
$979$	−6.33250	−0.202388
$980$	0	0
$981$	−8.99891	−0.287313
$982$	0	0
$983$	26.4871	0.844806	0.422403	−	0.906408i	$-0.361187\pi$
$983$	26.4871	0.844806	0.422403	+	0.906408i	$0.361187\pi$
$984$	0	0
$985$	77.0235	2.45417
$986$	0	0
$987$	−2.99127	−0.0952131
$988$	0	0
$989$	−1.52485	−0.0484875
$990$	0	0
$991$	29.7072	0.943681	0.471841	−	0.881684i	$-0.343590\pi$
$991$	29.7072	0.943681	0.471841	+	0.881684i	$0.343590\pi$
$992$	0	0
$993$	−10.8320	−0.343744
$994$	0	0
$995$	−21.0422	−0.667083
$996$	0	0
$997$	13.9444	0.441623	0.220811	−	0.975317i	$-0.429129\pi$
$997$	13.9444	0.441623	0.220811	+	0.975317i	$0.429129\pi$
$998$	0	0
$999$	9.89179	0.312962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.cc.1.3		4
4.3	odd	2		3432.2.a.t.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.t.1.3	✓	4	4.3	odd	2
6864.2.a.cc.1.3		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.43697 q^{5} +2.70832 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.43697 q^{5} +2.70832 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +3.43697 q^{15} -0.375819 q^{17} +2.70832 q^{19} +2.70832 q^{21} +0.938844 q^{23} +6.81279 q^{25} +1.00000 q^{27} +0.728654 q^{29} -6.85361 q^{31} -1.00000 q^{33} +9.30843 q^{35} +9.89179 q^{37} -1.00000 q^{39} -2.83063 q^{41} -1.62418 q^{43} +3.43697 q^{45} -1.10447 q^{47} +0.334998 q^{49} -0.375819 q^{51} +12.1683 q^{53} -3.43697 q^{55} +2.70832 q^{57} +2.62933 q^{59} -3.10447 q^{61} +2.70832 q^{63} -3.43697 q^{65} -4.73130 q^{67} +0.938844 q^{69} +0.108213 q^{71} +13.7046 q^{73} +6.81279 q^{75} -2.70832 q^{77} -8.03708 q^{79} +1.00000 q^{81} +4.99626 q^{83} -1.29168 q^{85} +0.728654 q^{87} +6.33250 q^{89} -2.70832 q^{91} -6.85361 q^{93} +9.30843 q^{95} +15.3951 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{17} - 2 q^{19} - 2 q^{21} + 8 q^{23} + 12 q^{25} + 4 q^{27} + 6 q^{29} + 8 q^{31} - 4 q^{33} - 14 q^{35} + 14 q^{37} - 4 q^{39} + 10 q^{41} - 12 q^{43} + 4 q^{45} - 2 q^{47} + 16 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{55} - 2 q^{57} + 8 q^{59} - 10 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} + 8 q^{69} + 26 q^{71} + 14 q^{73} + 12 q^{75} + 2 q^{77} + 16 q^{79} + 4 q^{81} - 8 q^{83} - 18 q^{85} + 6 q^{87} + 18 q^{89} + 2 q^{91} + 8 q^{93} - 14 q^{95} + 14 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^17 - 2 * q^19 - 2 * q^21 + 8 * q^23 + 12 * q^25 + 4 * q^27 + 6 * q^29 + 8 * q^31 - 4 * q^33 - 14 * q^35 + 14 * q^37 - 4 * q^39 + 10 * q^41 - 12 * q^43 + 4 * q^45 - 2 * q^47 + 16 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^55 - 2 * q^57 + 8 * q^59 - 10 * q^61 - 2 * q^63 - 4 * q^65 + 8 * q^67 + 8 * q^69 + 26 * q^71 + 14 * q^73 + 12 * q^75 + 2 * q^77 + 16 * q^79 + 4 * q^81 - 8 * q^83 - 18 * q^85 + 6 * q^87 + 18 * q^89 + 2 * q^91 + 8 * q^93 - 14 * q^95 + 14 * q^97 - 4 * q^99

Introduction

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