LMFDB - Embedded newform 6840.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6840.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:	$54.6176749826$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	6840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -1.51514 q^{7} -4.24977 q^{11} +4.15516 q^{13} +3.51514 q^{17} -1.00000 q^{19} +8.73463 q^{23} +1.00000 q^{25} -1.45459 q^{29} -4.96972 q^{31} -1.51514 q^{35} +7.60975 q^{37} +9.21949 q^{41} -8.31032 q^{43} -5.28005 q^{47} -4.70436 q^{49} -0.155162 q^{53} -4.24977 q^{55} +2.48486 q^{59} -4.49954 q^{61} +4.15516 q^{65} +7.43521 q^{67} -8.49954 q^{71} -15.0450 q^{73} +6.43899 q^{77} -0.310323 q^{79} +8.96972 q^{83} +3.51514 q^{85} -0.719953 q^{89} -6.29564 q^{91} -1.00000 q^{95} +17.3893 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 5 q^{7} + 4 q^{11} + 5 q^{13} + 11 q^{17} - 3 q^{19} + 9 q^{23} + 3 q^{25} - 3 q^{29} - 14 q^{31} - 5 q^{35} + 14 q^{37} + 10 q^{41} - 10 q^{43} + 4 q^{49} + 7 q^{53} + 4 q^{55} + 7 q^{59}+ \cdots - 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 5 * q^7 + 4 * q^11 + 5 * q^13 + 11 * q^17 - 3 * q^19 + 9 * q^23 + 3 * q^25 - 3 * q^29 - 14 * q^31 - 5 * q^35 + 14 * q^37 + 10 * q^41 - 10 * q^43 + 4 * q^49 + 7 * q^53 + 4 * q^55 + 7 * q^59 + 20 * q^61 + 5 * q^65 - q^67 + 8 * q^71 - 13 * q^73 - 16 * q^77 + 14 * q^79 + 26 * q^83 + 11 * q^85 - 18 * q^89 - 37 * q^91 - 3 * q^95 - 6 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.51514	−0.572668	−0.286334	−	0.958130i	$-0.592437\pi$
$7$	−1.51514	−0.572668	−0.286334	+	0.958130i	$0.592437\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.24977	−1.28135	−0.640677	−	0.767810i	$-0.721346\pi$
$11$	−4.24977	−1.28135	−0.640677	+	0.767810i	$0.721346\pi$
$12$	0	0
$13$	4.15516	1.15243	0.576217	−	0.817297i	$-0.304528\pi$
$13$	4.15516	1.15243	0.576217	+	0.817297i	$0.304528\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.51514	0.852546	0.426273	−	0.904595i	$-0.359826\pi$
$17$	3.51514	0.852546	0.426273	+	0.904595i	$0.359826\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.73463	1.82130	0.910648	−	0.413182i	$-0.135583\pi$
$23$	8.73463	1.82130	0.910648	+	0.413182i	$0.135583\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.45459	−0.270110	−0.135055	−	0.990838i	$-0.543121\pi$
$29$	−1.45459	−0.270110	−0.135055	+	0.990838i	$0.543121\pi$
$30$	0	0
$31$	−4.96972	−0.892589	−0.446294	−	0.894886i	$-0.647257\pi$
$31$	−4.96972	−0.892589	−0.446294	+	0.894886i	$0.647257\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.51514	−0.256105
$36$	0	0
$37$	7.60975	1.25103	0.625517	−	0.780210i	$-0.284888\pi$
$37$	7.60975	1.25103	0.625517	+	0.780210i	$0.284888\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.21949	1.43984	0.719922	−	0.694055i	$-0.244178\pi$
$41$	9.21949	1.43984	0.719922	+	0.694055i	$0.244178\pi$
$42$	0	0
$43$	−8.31032	−1.26731	−0.633656	−	0.773615i	$-0.718447\pi$
$43$	−8.31032	−1.26731	−0.633656	+	0.773615i	$0.718447\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.28005	−0.770174	−0.385087	−	0.922880i	$-0.625829\pi$
$47$	−5.28005	−0.770174	−0.385087	+	0.922880i	$0.625829\pi$
$48$	0	0
$49$	−4.70436	−0.672051
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.155162	−0.0213131	−0.0106565	−	0.999943i	$-0.503392\pi$
$53$	−0.155162	−0.0213131	−0.0106565	+	0.999943i	$0.503392\pi$
$54$	0	0
$55$	−4.24977	−0.573039
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.48486	0.323501	0.161751	−	0.986832i	$-0.448286\pi$
$59$	2.48486	0.323501	0.161751	+	0.986832i	$0.448286\pi$
$60$	0	0
$61$	−4.49954	−0.576107	−0.288054	−	0.957614i	$-0.593008\pi$
$61$	−4.49954	−0.576107	−0.288054	+	0.957614i	$0.593008\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.15516	0.515384
$66$	0	0
$67$	7.43521	0.908355	0.454178	−	0.890911i	$-0.349933\pi$
$67$	7.43521	0.908355	0.454178	+	0.890911i	$0.349933\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.49954	−1.00871	−0.504355	−	0.863496i	$-0.668270\pi$
$71$	−8.49954	−1.00871	−0.504355	+	0.863496i	$0.668270\pi$
$72$	0	0
$73$	−15.0450	−1.76088	−0.880439	−	0.474159i	$-0.842752\pi$
$73$	−15.0450	−1.76088	−0.880439	+	0.474159i	$0.842752\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.43899	0.733791
$78$	0	0
$79$	−0.310323	−0.0349141	−0.0174570	−	0.999848i	$-0.505557\pi$
$79$	−0.310323	−0.0349141	−0.0174570	+	0.999848i	$0.505557\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.96972	0.984555	0.492278	−	0.870438i	$-0.336165\pi$
$83$	8.96972	0.984555	0.492278	+	0.870438i	$0.336165\pi$
$84$	0	0
$85$	3.51514	0.381270
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.719953	−0.0763149	−0.0381574	−	0.999272i	$-0.512149\pi$
$89$	−0.719953	−0.0763149	−0.0381574	+	0.999272i	$0.512149\pi$
$90$	0	0
$91$	−6.29564	−0.659963
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	17.3893	1.76562	0.882810	−	0.469731i	$-0.155649\pi$
$97$	17.3893	1.76562	0.882810	+	0.469731i	$0.155649\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.4995	1.24375	0.621875	−	0.783116i	$-0.286371\pi$
$101$	12.4995	1.24375	0.621875	+	0.783116i	$0.286371\pi$
$102$	0	0
$103$	−10.8898	−1.07300	−0.536502	−	0.843899i	$-0.680254\pi$
$103$	−10.8898	−1.07300	−0.536502	+	0.843899i	$0.680254\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.43521	0.718789	0.359394	−	0.933186i	$-0.382983\pi$
$107$	7.43521	0.718789	0.359394	+	0.933186i	$0.382983\pi$
$108$	0	0
$109$	−19.3553	−1.85390	−0.926950	−	0.375186i	$-0.877579\pi$
$109$	−19.3553	−1.85390	−0.926950	+	0.375186i	$0.877579\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.8595	1.68008	0.840041	−	0.542523i	$-0.182531\pi$
$113$	17.8595	1.68008	0.840041	+	0.542523i	$0.182531\pi$
$114$	0	0
$115$	8.73463	0.814509
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.32592	−0.488226
$120$	0	0
$121$	7.06055	0.641868
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.70058	0.239637	0.119819	−	0.992796i	$-0.461769\pi$
$127$	2.70058	0.239637	0.119819	+	0.992796i	$0.461769\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	1.51514	0.131379
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.51514	0.642062	0.321031	−	0.947069i	$-0.395971\pi$
$137$	7.51514	0.642062	0.321031	+	0.947069i	$0.395971\pi$
$138$	0	0
$139$	16.7493	1.42066	0.710329	−	0.703870i	$-0.248546\pi$
$139$	16.7493	1.42066	0.710329	+	0.703870i	$0.248546\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−17.6585	−1.47668
$144$	0	0
$145$	−1.45459	−0.120797
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.4995	1.02400	0.512001	−	0.858985i	$-0.328904\pi$
$149$	12.4995	1.02400	0.512001	+	0.858985i	$0.328904\pi$
$150$	0	0
$151$	14.2498	1.15963	0.579815	−	0.814748i	$-0.303125\pi$
$151$	14.2498	1.15963	0.579815	+	0.814748i	$0.303125\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.96972	−0.399178
$156$	0	0
$157$	4.71995	0.376693	0.188347	−	0.982103i	$-0.439687\pi$
$157$	4.71995	0.376693	0.188347	+	0.982103i	$0.439687\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13.2342	−1.04300
$162$	0	0
$163$	10.7493	0.841951	0.420976	−	0.907072i	$-0.361688\pi$
$163$	10.7493	0.841951	0.420976	+	0.907072i	$0.361688\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.7006	0.828035	0.414018	−	0.910269i	$-0.364125\pi$
$167$	10.7006	0.828035	0.414018	+	0.910269i	$0.364125\pi$
$168$	0	0
$169$	4.26537	0.328105
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.39025	0.333785	0.166892	−	0.985975i	$-0.446627\pi$
$173$	4.39025	0.333785	0.166892	+	0.985975i	$0.446627\pi$
$174$	0	0
$175$	−1.51514	−0.114534
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.21949	0.240636	0.120318	−	0.992735i	$-0.461609\pi$
$179$	3.21949	0.240636	0.120318	+	0.992735i	$0.461609\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.60975	0.559480
$186$	0	0
$187$	−14.9385	−1.09241
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.2342	−0.957591	−0.478796	−	0.877926i	$-0.658927\pi$
$191$	−13.2342	−0.957591	−0.478796	+	0.877926i	$0.658927\pi$
$192$	0	0
$193$	17.3893	1.25171	0.625856	−	0.779939i	$-0.284750\pi$
$193$	17.3893	1.25171	0.625856	+	0.779939i	$0.284750\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−14.3250	−1.01547	−0.507736	−	0.861513i	$-0.669518\pi$
$199$	−14.3250	−1.01547	−0.507736	+	0.861513i	$0.669518\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.20390	0.154683
$204$	0	0
$205$	9.21949	0.643917
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.24977	0.293963
$210$	0	0
$211$	−6.98440	−0.480826	−0.240413	−	0.970671i	$-0.577283\pi$
$211$	−6.98440	−0.480826	−0.240413	+	0.970671i	$0.577283\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.31032	−0.566759
$216$	0	0
$217$	7.52982	0.511157
$218$	0	0
$219$	0	0
$220$	0	0
$221$	14.6060	0.982504
$222$	0	0
$223$	24.4802	1.63931	0.819657	−	0.572855i	$-0.194164\pi$
$223$	24.4802	1.63931	0.819657	+	0.572855i	$0.194164\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.9045	1.65297	0.826484	−	0.562960i	$-0.190338\pi$
$227$	24.9045	1.65297	0.826484	+	0.562960i	$0.190338\pi$
$228$	0	0
$229$	4.47018	0.295398	0.147699	−	0.989032i	$-0.452813\pi$
$229$	4.47018	0.295398	0.147699	+	0.989032i	$0.452813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.93945	0.258082	0.129041	−	0.991639i	$-0.458810\pi$
$233$	3.93945	0.258082	0.129041	+	0.991639i	$0.458810\pi$
$234$	0	0
$235$	−5.28005	−0.344432
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.8255	1.15303	0.576517	−	0.817085i	$-0.304412\pi$
$239$	17.8255	1.15303	0.576517	+	0.817085i	$0.304412\pi$
$240$	0	0
$241$	−27.3094	−1.75915	−0.879577	−	0.475757i	$-0.842174\pi$
$241$	−27.3094	−1.75915	−0.879577	+	0.475757i	$0.842174\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.70436	−0.300550
$246$	0	0
$247$	−4.15516	−0.264387
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.4390	1.16386	0.581929	−	0.813239i	$-0.302298\pi$
$251$	18.4390	1.16386	0.581929	+	0.813239i	$0.302298\pi$
$252$	0	0
$253$	−37.1202	−2.33373
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.889794	0.0555038	0.0277519	−	0.999615i	$-0.491165\pi$
$257$	0.889794	0.0555038	0.0277519	+	0.999615i	$0.491165\pi$
$258$	0	0
$259$	−11.5298	−0.716428
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.6206	0.778222	0.389111	−	0.921191i	$-0.372782\pi$
$263$	12.6206	0.778222	0.389111	+	0.921191i	$0.372782\pi$
$264$	0	0
$265$	−0.155162	−0.00953150
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.4390	1.00230	0.501151	−	0.865360i	$-0.332910\pi$
$269$	16.4390	1.00230	0.501151	+	0.865360i	$0.332910\pi$
$270$	0	0
$271$	25.6050	1.55540	0.777698	−	0.628638i	$-0.216387\pi$
$271$	25.6050	1.55540	0.777698	+	0.628638i	$0.216387\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.24977	−0.256271
$276$	0	0
$277$	0.749313	0.0450218	0.0225109	−	0.999747i	$-0.492834\pi$
$277$	0.749313	0.0450218	0.0225109	+	0.999747i	$0.492834\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.77959	−0.225471	−0.112736	−	0.993625i	$-0.535961\pi$
$281$	−3.77959	−0.225471	−0.112736	+	0.993625i	$0.535961\pi$
$282$	0	0
$283$	−9.46927	−0.562889	−0.281445	−	0.959577i	$-0.590814\pi$
$283$	−9.46927	−0.562889	−0.281445	+	0.959577i	$0.590814\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−13.9688	−0.824553
$288$	0	0
$289$	−4.64380	−0.273165
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.6841	−1.44206	−0.721029	−	0.692905i	$-0.756331\pi$
$293$	−24.6841	−1.44206	−0.721029	+	0.692905i	$0.756331\pi$
$294$	0	0
$295$	2.48486	0.144674
$296$	0	0
$297$	0	0
$298$	0	0
$299$	36.2938	2.09893
$300$	0	0
$301$	12.5913	0.725750
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.49954	−0.257643
$306$	0	0
$307$	13.3288	0.760714	0.380357	−	0.924840i	$-0.375801\pi$
$307$	13.3288	0.760714	0.380357	+	0.924840i	$0.375801\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.4546	0.762940	0.381470	−	0.924381i	$-0.375418\pi$
$311$	13.4546	0.762940	0.381470	+	0.924381i	$0.375418\pi$
$312$	0	0
$313$	−21.7649	−1.23023	−0.615113	−	0.788439i	$-0.710889\pi$
$313$	−21.7649	−1.23023	−0.615113	+	0.788439i	$0.710889\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.5639	−1.32348	−0.661740	−	0.749734i	$-0.730182\pi$
$317$	−23.5639	−1.32348	−0.661740	+	0.749734i	$0.730182\pi$
$318$	0	0
$319$	6.18166	0.346106
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.51514	−0.195588
$324$	0	0
$325$	4.15516	0.230487
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−11.9541	−0.657058	−0.328529	−	0.944494i	$-0.606553\pi$
$331$	−11.9541	−0.657058	−0.328529	+	0.944494i	$0.606553\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.43521	0.406229
$336$	0	0
$337$	−22.8292	−1.24359	−0.621794	−	0.783181i	$-0.713596\pi$
$337$	−22.8292	−1.24359	−0.621794	+	0.783181i	$0.713596\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.1202	1.14372
$342$	0	0
$343$	17.7337	0.957531
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.4693	−1.58199	−0.790997	−	0.611821i	$-0.790437\pi$
$347$	−29.4693	−1.58199	−0.790997	+	0.611821i	$0.790437\pi$
$348$	0	0
$349$	4.56009	0.244096	0.122048	−	0.992524i	$-0.461054\pi$
$349$	4.56009	0.244096	0.122048	+	0.992524i	$0.461054\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.4849	1.30320	0.651599	−	0.758564i	$-0.274099\pi$
$353$	24.4849	1.30320	0.651599	+	0.758564i	$0.274099\pi$
$354$	0	0
$355$	−8.49954	−0.451109
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.0147	−0.634111	−0.317055	−	0.948407i	$-0.602694\pi$
$359$	−12.0147	−0.634111	−0.317055	+	0.948407i	$0.602694\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15.0450	−0.787489
$366$	0	0
$367$	−25.1589	−1.31329	−0.656643	−	0.754201i	$-0.728024\pi$
$367$	−25.1589	−1.31329	−0.656643	+	0.754201i	$0.728024\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.235091	0.0122053
$372$	0	0
$373$	9.62443	0.498334	0.249167	−	0.968461i	$-0.419843\pi$
$373$	9.62443	0.498334	0.249167	+	0.968461i	$0.419843\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.04404	−0.311284
$378$	0	0
$379$	19.7943	1.01676	0.508382	−	0.861132i	$-0.330244\pi$
$379$	19.7943	1.01676	0.508382	+	0.861132i	$0.330244\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−33.3288	−1.70302	−0.851511	−	0.524337i	$-0.824313\pi$
$383$	−33.3288	−1.70302	−0.851511	+	0.524337i	$0.824313\pi$
$384$	0	0
$385$	6.43899	0.328161
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11.5904	−0.587655	−0.293828	−	0.955858i	$-0.594929\pi$
$389$	−11.5904	−0.587655	−0.293828	+	0.955858i	$0.594929\pi$
$390$	0	0
$391$	30.7034	1.55274
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.310323	−0.0156140
$396$	0	0
$397$	−8.74931	−0.439115	−0.219558	−	0.975600i	$-0.570461\pi$
$397$	−8.74931	−0.439115	−0.219558	+	0.975600i	$0.570461\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.6206	0.929871	0.464935	−	0.885345i	$-0.346078\pi$
$401$	18.6206	0.929871	0.464935	+	0.885345i	$0.346078\pi$
$402$	0	0
$403$	−20.6500	−1.02865
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−32.3397	−1.60302
$408$	0	0
$409$	7.93945	0.392580	0.196290	−	0.980546i	$-0.437111\pi$
$409$	7.93945	0.392580	0.196290	+	0.980546i	$0.437111\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.76491	−0.185259
$414$	0	0
$415$	8.96972	0.440306
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13.0908	0.639529	0.319764	−	0.947497i	$-0.396396\pi$
$419$	13.0908	0.639529	0.319764	+	0.947497i	$0.396396\pi$
$420$	0	0
$421$	19.8936	0.969554	0.484777	−	0.874638i	$-0.338901\pi$
$421$	19.8936	0.969554	0.484777	+	0.874638i	$0.338901\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.51514	0.170509
$426$	0	0
$427$	6.81743	0.329918
$428$	0	0
$429$	0	0
$430$	0	0
$431$	33.7796	1.62711	0.813553	−	0.581491i	$-0.197530\pi$
$431$	33.7796	1.62711	0.813553	+	0.581491i	$0.197530\pi$
$432$	0	0
$433$	9.07901	0.436310	0.218155	−	0.975914i	$-0.429996\pi$
$433$	9.07901	0.436310	0.218155	+	0.975914i	$0.429996\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.73463	−0.417834
$438$	0	0
$439$	−12.3103	−0.587540	−0.293770	−	0.955876i	$-0.594910\pi$
$439$	−12.3103	−0.587540	−0.293770	+	0.955876i	$0.594910\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.56009	−0.121634	−0.0608169	−	0.998149i	$-0.519371\pi$
$443$	−2.56009	−0.121634	−0.0608169	+	0.998149i	$0.519371\pi$
$444$	0	0
$445$	−0.719953	−0.0341290
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−22.9991	−1.08539	−0.542697	−	0.839929i	$-0.682597\pi$
$449$	−22.9991	−1.08539	−0.542697	+	0.839929i	$0.682597\pi$
$450$	0	0
$451$	−39.1807	−1.84495
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.29564	−0.295144
$456$	0	0
$457$	20.6741	0.967093	0.483546	−	0.875319i	$-0.339348\pi$
$457$	20.6741	0.967093	0.483546	+	0.875319i	$0.339348\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.87890	0.273807	0.136904	−	0.990584i	$-0.456285\pi$
$461$	5.87890	0.273807	0.136904	+	0.990584i	$0.456285\pi$
$462$	0	0
$463$	35.5592	1.65258	0.826288	−	0.563249i	$-0.190449\pi$
$463$	35.5592	1.65258	0.826288	+	0.563249i	$0.190449\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−41.7408	−1.93154	−0.965768	−	0.259408i	$-0.916472\pi$
$467$	−41.7408	−1.93154	−0.965768	+	0.259408i	$0.916472\pi$
$468$	0	0
$469$	−11.2654	−0.520186
$470$	0	0
$471$	0	0
$472$	0	0
$473$	35.3170	1.62388
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.8704	−0.770829	−0.385415	−	0.922744i	$-0.625942\pi$
$479$	−16.8704	−0.770829	−0.385415	+	0.922744i	$0.625942\pi$
$480$	0	0
$481$	31.6197	1.44174
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.3893	0.789609
$486$	0	0
$487$	3.64094	0.164987	0.0824934	−	0.996592i	$-0.473712\pi$
$487$	3.64094	0.164987	0.0824934	+	0.996592i	$0.473712\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.96881	0.0888510	0.0444255	−	0.999013i	$-0.485854\pi$
$491$	1.96881	0.0888510	0.0444255	+	0.999013i	$0.485854\pi$
$492$	0	0
$493$	−5.11307	−0.230281
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.8780	0.577656
$498$	0	0
$499$	−36.7787	−1.64644	−0.823220	−	0.567723i	$-0.807825\pi$
$499$	−36.7787	−1.64644	−0.823220	+	0.567723i	$0.807825\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15.8860	−0.708322	−0.354161	−	0.935184i	$-0.615234\pi$
$503$	−15.8860	−0.708322	−0.354161	+	0.935184i	$0.615234\pi$
$504$	0	0
$505$	12.4995	0.556222
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.2791	−0.898857	−0.449428	−	0.893316i	$-0.648372\pi$
$509$	−20.2791	−0.898857	−0.449428	+	0.893316i	$0.648372\pi$
$510$	0	0
$511$	22.7952	1.00840
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.8898	−0.479862
$516$	0	0
$517$	22.4390	0.986866
$518$	0	0
$519$	0	0
$520$	0	0
$521$	23.4693	1.02821	0.514104	−	0.857728i	$-0.328125\pi$
$521$	23.4693	1.02821	0.514104	+	0.857728i	$0.328125\pi$
$522$	0	0
$523$	−26.3737	−1.15324	−0.576622	−	0.817011i	$-0.695629\pi$
$523$	−26.3737	−1.15324	−0.576622	+	0.817011i	$0.695629\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.4693	−0.760973
$528$	0	0
$529$	53.2938	2.31712
$530$	0	0
$531$	0	0
$532$	0	0
$533$	38.3085	1.65932
$534$	0	0
$535$	7.43521	0.321452
$536$	0	0
$537$	0	0
$538$	0	0
$539$	19.9924	0.861135
$540$	0	0
$541$	22.5601	0.969934	0.484967	−	0.874532i	$-0.338832\pi$
$541$	22.5601	0.969934	0.484967	+	0.874532i	$0.338832\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−19.3553	−0.829089
$546$	0	0
$547$	20.6694	0.883759	0.441879	−	0.897074i	$-0.354312\pi$
$547$	20.6694	0.883759	0.441879	+	0.897074i	$0.354312\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.45459	0.0619674
$552$	0	0
$553$	0.470182	0.0199942
$554$	0	0
$555$	0	0
$556$	0	0
$557$	37.7484	1.59945	0.799725	−	0.600366i	$-0.204978\pi$
$557$	37.7484	1.59945	0.799725	+	0.600366i	$0.204978\pi$
$558$	0	0
$559$	−34.5307	−1.46049
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.6694	1.54543	0.772715	−	0.634753i	$-0.218898\pi$
$563$	36.6694	1.54543	0.772715	+	0.634753i	$0.218898\pi$
$564$	0	0
$565$	17.8595	0.751356
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.84862	−0.119420	−0.0597102	−	0.998216i	$-0.519018\pi$
$569$	−2.84862	−0.119420	−0.0597102	+	0.998216i	$0.519018\pi$
$570$	0	0
$571$	28.4002	1.18851	0.594256	−	0.804276i	$-0.297446\pi$
$571$	28.4002	1.18851	0.594256	+	0.804276i	$0.297446\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.73463	0.364259
$576$	0	0
$577$	−29.2947	−1.21956	−0.609778	−	0.792572i	$-0.708741\pi$
$577$	−29.2947	−1.21956	−0.609778	+	0.792572i	$0.708741\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−13.5904	−0.563824
$582$	0	0
$583$	0.659401	0.0273096
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.4683	−0.927368	−0.463684	−	0.886001i	$-0.653473\pi$
$587$	−22.4683	−0.927368	−0.463684	+	0.886001i	$0.653473\pi$
$588$	0	0
$589$	4.96972	0.204774
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.0606	−0.659528	−0.329764	−	0.944063i	$-0.606969\pi$
$593$	−16.0606	−0.659528	−0.329764	+	0.944063i	$0.606969\pi$
$594$	0	0
$595$	−5.32592	−0.218341
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23.2489	0.949922	0.474961	−	0.880007i	$-0.342462\pi$
$599$	23.2489	0.949922	0.474961	+	0.880007i	$0.342462\pi$
$600$	0	0
$601$	−0.841057	−0.0343074	−0.0171537	−	0.999853i	$-0.505460\pi$
$601$	−0.841057	−0.0343074	−0.0171537	+	0.999853i	$0.505460\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.06055	0.287052
$606$	0	0
$607$	−32.4802	−1.31833	−0.659165	−	0.751999i	$-0.729090\pi$
$607$	−32.4802	−1.31833	−0.659165	+	0.751999i	$0.729090\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.9394	−0.887575
$612$	0	0
$613$	−19.4305	−0.784791	−0.392395	−	0.919797i	$-0.628354\pi$
$613$	−19.4305	−0.784791	−0.392395	+	0.919797i	$0.628354\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.7787	−0.836518	−0.418259	−	0.908328i	$-0.637360\pi$
$617$	−20.7787	−0.836518	−0.418259	+	0.908328i	$0.637360\pi$
$618$	0	0
$619$	18.9310	0.760900	0.380450	−	0.924802i	$-0.375769\pi$
$619$	18.9310	0.760900	0.380450	+	0.924802i	$0.375769\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.09083	0.0437031
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	26.7493	1.06656
$630$	0	0
$631$	−0.370875	−0.0147643	−0.00738215	−	0.999973i	$-0.502350\pi$
$631$	−0.370875	−0.0147643	−0.00738215	+	0.999973i	$0.502350\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.70058	0.107169
$636$	0	0
$637$	−19.5474	−0.774495
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−26.8099	−1.05893	−0.529463	−	0.848333i	$-0.677607\pi$
$641$	−26.8099	−1.05893	−0.529463	+	0.848333i	$0.677607\pi$
$642$	0	0
$643$	−0.969724	−0.0382422	−0.0191211	−	0.999817i	$-0.506087\pi$
$643$	−0.969724	−0.0382422	−0.0191211	+	0.999817i	$0.506087\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.20482	−0.0473662	−0.0236831	−	0.999720i	$-0.507539\pi$
$647$	−1.20482	−0.0473662	−0.0236831	+	0.999720i	$0.507539\pi$
$648$	0	0
$649$	−10.5601	−0.414520
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.9310	0.897358	0.448679	−	0.893693i	$-0.351895\pi$
$653$	22.9310	0.897358	0.448679	+	0.893693i	$0.351895\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.1055	−0.588427	−0.294214	−	0.955740i	$-0.595058\pi$
$659$	−15.1055	−0.588427	−0.294214	+	0.955740i	$0.595058\pi$
$660$	0	0
$661$	19.0450	0.740763	0.370381	−	0.928880i	$-0.379227\pi$
$661$	19.0450	0.740763	0.370381	+	0.928880i	$0.379227\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.51514	0.0587545
$666$	0	0
$667$	−12.7053	−0.491950
$668$	0	0
$669$	0	0
$670$	0	0
$671$	19.1220	0.738197
$672$	0	0
$673$	15.4205	0.594418	0.297209	−	0.954812i	$-0.403944\pi$
$673$	15.4205	0.594418	0.297209	+	0.954812i	$0.403944\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.62534	−0.331499	−0.165749	−	0.986168i	$-0.553004\pi$
$677$	−8.62534	−0.331499	−0.165749	+	0.986168i	$0.553004\pi$
$678$	0	0
$679$	−26.3472	−1.01111
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−38.8898	−1.48808	−0.744038	−	0.668137i	$-0.767092\pi$
$683$	−38.8898	−1.48808	−0.744038	+	0.668137i	$0.767092\pi$
$684$	0	0
$685$	7.51514	0.287139
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−0.644721	−0.0245619
$690$	0	0
$691$	−21.5979	−0.821624	−0.410812	−	0.911720i	$-0.634755\pi$
$691$	−21.5979	−0.821624	−0.410812	+	0.911720i	$0.634755\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16.7493	0.635338
$696$	0	0
$697$	32.4078	1.22753
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17.8183	−0.672990	−0.336495	−	0.941685i	$-0.609241\pi$
$701$	−17.8183	−0.672990	−0.336495	+	0.941685i	$0.609241\pi$
$702$	0	0
$703$	−7.60975	−0.287007
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18.9385	−0.712257
$708$	0	0
$709$	47.4986	1.78385	0.891924	−	0.452185i	$-0.149355\pi$
$709$	47.4986	1.78385	0.891924	+	0.452185i	$0.149355\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−43.4087	−1.62567
$714$	0	0
$715$	−17.6585	−0.660390
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.4546	0.949296	0.474648	−	0.880176i	$-0.342575\pi$
$719$	25.4546	0.949296	0.474648	+	0.880176i	$0.342575\pi$
$720$	0	0
$721$	16.4995	0.614475
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.45459	−0.0540220
$726$	0	0
$727$	21.5445	0.799041	0.399521	−	0.916724i	$-0.369177\pi$
$727$	21.5445	0.799041	0.399521	+	0.916724i	$0.369177\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−29.2119	−1.08044
$732$	0	0
$733$	24.2110	0.894254	0.447127	−	0.894470i	$-0.352447\pi$
$733$	24.2110	0.894254	0.447127	+	0.894470i	$0.352447\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.5979	−1.16392
$738$	0	0
$739$	−37.4693	−1.37833	−0.689165	−	0.724605i	$-0.742022\pi$
$739$	−37.4693	−1.37833	−0.689165	+	0.724605i	$0.742022\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.8595	0.581829	0.290915	−	0.956749i	$-0.406040\pi$
$743$	15.8595	0.581829	0.290915	+	0.956749i	$0.406040\pi$
$744$	0	0
$745$	12.4995	0.457948
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−11.2654	−0.411628
$750$	0	0
$751$	5.12958	0.187181	0.0935906	−	0.995611i	$-0.470166\pi$
$751$	5.12958	0.187181	0.0935906	+	0.995611i	$0.470166\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.2498	0.518602
$756$	0	0
$757$	37.4381	1.36071	0.680355	−	0.732883i	$-0.261826\pi$
$757$	37.4381	1.36071	0.680355	+	0.732883i	$0.261826\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.5833	−0.782392	−0.391196	−	0.920307i	$-0.627939\pi$
$761$	−21.5833	−0.782392	−0.391196	+	0.920307i	$0.627939\pi$
$762$	0	0
$763$	29.3259	1.06167
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.3250	0.372814
$768$	0	0
$769$	−40.7640	−1.46999	−0.734994	−	0.678074i	$-0.762815\pi$
$769$	−40.7640	−1.46999	−0.734994	+	0.678074i	$0.762815\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−48.1845	−1.73308	−0.866538	−	0.499110i	$-0.833660\pi$
$773$	−48.1845	−1.73308	−0.866538	+	0.499110i	$0.833660\pi$
$774$	0	0
$775$	−4.96972	−0.178518
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.21949	−0.330323
$780$	0	0
$781$	36.1211	1.29251
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.71995	0.168462
$786$	0	0
$787$	27.6925	0.987132	0.493566	−	0.869708i	$-0.335693\pi$
$787$	27.6925	0.987132	0.493566	+	0.869708i	$0.335693\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−27.0596	−0.962130
$792$	0	0
$793$	−18.6963	−0.663926
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.3368	−0.366149	−0.183074	−	0.983099i	$-0.558605\pi$
$797$	−10.3368	−0.366149	−0.183074	+	0.983099i	$0.558605\pi$
$798$	0	0
$799$	−18.5601	−0.656609
$800$	0	0
$801$	0	0
$802$	0	0
$803$	63.9376	2.25631
$804$	0	0
$805$	−13.2342	−0.466443
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.2342	0.816870	0.408435	−	0.912787i	$-0.366075\pi$
$809$	23.2342	0.816870	0.408435	+	0.912787i	$0.366075\pi$
$810$	0	0
$811$	32.1433	1.12871	0.564353	−	0.825534i	$-0.309126\pi$
$811$	32.1433	1.12871	0.564353	+	0.825534i	$0.309126\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	10.7493	0.376532
$816$	0	0
$817$	8.31032	0.290741
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27.0908	0.945476	0.472738	−	0.881203i	$-0.343266\pi$
$821$	27.0908	0.945476	0.472738	+	0.881203i	$0.343266\pi$
$822$	0	0
$823$	−14.4849	−0.504911	−0.252455	−	0.967609i	$-0.581238\pi$
$823$	−14.4849	−0.504911	−0.252455	+	0.967609i	$0.581238\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.0247	1.87862	0.939311	−	0.343067i	$-0.111466\pi$
$827$	54.0247	1.87862	0.939311	+	0.343067i	$0.111466\pi$
$828$	0	0
$829$	18.7640	0.651700	0.325850	−	0.945421i	$-0.394349\pi$
$829$	18.7640	0.651700	0.325850	+	0.945421i	$0.394349\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−16.5365	−0.572954
$834$	0	0
$835$	10.7006	0.370309
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7.84014	0.270672	0.135336	−	0.990800i	$-0.456789\pi$
$839$	7.84014	0.270672	0.135336	+	0.990800i	$0.456789\pi$
$840$	0	0
$841$	−26.8842	−0.927041
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.26537	0.146733
$846$	0	0
$847$	−10.6977	−0.367578
$848$	0	0
$849$	0	0
$850$	0	0
$851$	66.4683	2.27851
$852$	0	0
$853$	36.8174	1.26060	0.630302	−	0.776350i	$-0.282931\pi$
$853$	36.8174	1.26060	0.630302	+	0.776350i	$0.282931\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.23887	0.178956	0.0894782	−	0.995989i	$-0.471480\pi$
$857$	5.23887	0.178956	0.0894782	+	0.995989i	$0.471480\pi$
$858$	0	0
$859$	−29.6197	−1.01061	−0.505306	−	0.862940i	$-0.668620\pi$
$859$	−29.6197	−1.01061	−0.505306	+	0.862940i	$0.668620\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	35.8889	1.22167	0.610836	−	0.791757i	$-0.290834\pi$
$863$	35.8889	1.22167	0.610836	+	0.791757i	$0.290834\pi$
$864$	0	0
$865$	4.39025	0.149273
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.31880	0.0447373
$870$	0	0
$871$	30.8945	1.04682
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.51514	−0.0512210
$876$	0	0
$877$	−29.8136	−1.00674	−0.503368	−	0.864072i	$-0.667906\pi$
$877$	−29.8136	−1.00674	−0.503368	+	0.864072i	$0.667906\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38.9679	1.31286	0.656431	−	0.754386i	$-0.272065\pi$
$881$	38.9679	1.31286	0.656431	+	0.754386i	$0.272065\pi$
$882$	0	0
$883$	−28.5677	−0.961378	−0.480689	−	0.876891i	$-0.659613\pi$
$883$	−28.5677	−0.961378	−0.480689	+	0.876891i	$0.659613\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−27.5104	−0.923710	−0.461855	−	0.886955i	$-0.652816\pi$
$887$	−27.5104	−0.923710	−0.461855	+	0.886955i	$0.652816\pi$
$888$	0	0
$889$	−4.09174	−0.137233
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.28005	0.176690
$894$	0	0
$895$	3.21949	0.107616
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.22889	0.241097
$900$	0	0
$901$	−0.545414	−0.0181704
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	−55.4939	−1.84265	−0.921323	−	0.388799i	$-0.872890\pi$
$907$	−55.4939	−1.84265	−0.921323	+	0.388799i	$0.872890\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.84106	−0.226654	−0.113327	−	0.993558i	$-0.536151\pi$
$911$	−6.84106	−0.226654	−0.113327	+	0.993558i	$0.536151\pi$
$912$	0	0
$913$	−38.1193	−1.26156
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.06055	−0.200137
$918$	0	0
$919$	−6.17454	−0.203679	−0.101840	−	0.994801i	$-0.532473\pi$
$919$	−6.17454	−0.203679	−0.101840	+	0.994801i	$0.532473\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−35.3170	−1.16247
$924$	0	0
$925$	7.60975	0.250207
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−44.7034	−1.46667	−0.733336	−	0.679866i	$-0.762038\pi$
$929$	−44.7034	−1.46667	−0.733336	+	0.679866i	$0.762038\pi$
$930$	0	0
$931$	4.70436	0.154179
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14.9385	−0.488542
$936$	0	0
$937$	−12.1064	−0.395500	−0.197750	−	0.980253i	$-0.563363\pi$
$937$	−12.1064	−0.395500	−0.197750	+	0.980253i	$0.563363\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26.2645	0.856197	0.428098	−	0.903732i	$-0.359184\pi$
$941$	26.2645	0.856197	0.428098	+	0.903732i	$0.359184\pi$
$942$	0	0
$943$	80.5289	2.62238
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.37935	−0.239797	−0.119898	−	0.992786i	$-0.538257\pi$
$947$	−7.37935	−0.239797	−0.119898	+	0.992786i	$0.538257\pi$
$948$	0	0
$949$	−62.5142	−2.02930
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.5398	0.697743	0.348871	−	0.937171i	$-0.386565\pi$
$953$	21.5398	0.697743	0.348871	+	0.937171i	$0.386565\pi$
$954$	0	0
$955$	−13.2342	−0.428248
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11.3865	−0.367688
$960$	0	0
$961$	−6.30184	−0.203285
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17.3893	0.559783
$966$	0	0
$967$	−39.8789	−1.28242	−0.641209	−	0.767366i	$-0.721567\pi$
$967$	−39.8789	−1.28242	−0.641209	+	0.767366i	$0.721567\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	29.7408	0.954429	0.477214	−	0.878787i	$-0.341647\pi$
$971$	29.7408	0.954429	0.477214	+	0.878787i	$0.341647\pi$
$972$	0	0
$973$	−25.3775	−0.813566
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.6694	1.87700	0.938500	−	0.345279i	$-0.112216\pi$
$977$	58.6694	1.87700	0.938500	+	0.345279i	$0.112216\pi$
$978$	0	0
$979$	3.05964	0.0977864
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−11.7384	−0.374397	−0.187199	−	0.982322i	$-0.559941\pi$
$983$	−11.7384	−0.374397	−0.187199	+	0.982322i	$0.559941\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−72.5876	−2.30815
$990$	0	0
$991$	−26.7493	−0.849720	−0.424860	−	0.905259i	$-0.639677\pi$
$991$	−26.7493	−0.849720	−0.424860	+	0.905259i	$0.639677\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14.3250	−0.454133
$996$	0	0
$997$	45.8401	1.45177	0.725886	−	0.687815i	$-0.241430\pi$
$997$	45.8401	1.45177	0.725886	+	0.687815i	$0.241430\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.bj.1.2		3
3.2	odd	2		760.2.a.h.1.1	✓	3
12.11	even	2		1520.2.a.r.1.3		3
15.2	even	4		3800.2.d.k.3649.6		6
15.8	even	4		3800.2.d.k.3649.1		6
15.14	odd	2		3800.2.a.y.1.3		3
24.5	odd	2		6080.2.a.bw.1.3		3
24.11	even	2		6080.2.a.bs.1.1		3
60.59	even	2		7600.2.a.bo.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.h.1.1	✓	3	3.2	odd	2
1520.2.a.r.1.3		3	12.11	even	2
3800.2.a.y.1.3		3	15.14	odd	2
3800.2.d.k.3649.1		6	15.8	even	4
3800.2.d.k.3649.6		6	15.2	even	4
6080.2.a.bs.1.1		3	24.11	even	2
6080.2.a.bw.1.3		3	24.5	odd	2
6840.2.a.bj.1.2		3	1.1	even	1	trivial
7600.2.a.bo.1.1		3	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.51514 q^{7} -4.24977 q^{11} +4.15516 q^{13} +3.51514 q^{17} -1.00000 q^{19} +8.73463 q^{23} +1.00000 q^{25} -1.45459 q^{29} -4.96972 q^{31} -1.51514 q^{35} +7.60975 q^{37} +9.21949 q^{41} -8.31032 q^{43} -5.28005 q^{47} -4.70436 q^{49} -0.155162 q^{53} -4.24977 q^{55} +2.48486 q^{59} -4.49954 q^{61} +4.15516 q^{65} +7.43521 q^{67} -8.49954 q^{71} -15.0450 q^{73} +6.43899 q^{77} -0.310323 q^{79} +8.96972 q^{83} +3.51514 q^{85} -0.719953 q^{89} -6.29564 q^{91} -1.00000 q^{95} +17.3893 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 5 q^{7} + 4 q^{11} + 5 q^{13} + 11 q^{17} - 3 q^{19} + 9 q^{23} + 3 q^{25} - 3 q^{29} - 14 q^{31} - 5 q^{35} + 14 q^{37} + 10 q^{41} - 10 q^{43} + 4 q^{49} + 7 q^{53} + 4 q^{55} + 7 q^{59}+ \cdots - 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 5 * q^7 + 4 * q^11 + 5 * q^13 + 11 * q^17 - 3 * q^19 + 9 * q^23 + 3 * q^25 - 3 * q^29 - 14 * q^31 - 5 * q^35 + 14 * q^37 + 10 * q^41 - 10 * q^43 + 4 * q^49 + 7 * q^53 + 4 * q^55 + 7 * q^59 + 20 * q^61 + 5 * q^65 - q^67 + 8 * q^71 - 13 * q^73 - 16 * q^77 + 14 * q^79 + 26 * q^83 + 11 * q^85 - 18 * q^89 - 37 * q^91 - 3 * q^95 - 6 * q^97

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