LMFDB - Embedded newform 6080.2.a.bw.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
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			1.3
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.12489 q^{3} +1.00000 q^{5} -1.51514 q^{7} +6.76491 q^{9} -4.24977 q^{11} -4.15516 q^{13} +3.12489 q^{15} -3.51514 q^{17} +1.00000 q^{19} -4.73463 q^{21} -8.73463 q^{23} +1.00000 q^{25} +11.7649 q^{27} -1.45459 q^{29} -4.96972 q^{31} -13.2800 q^{33} -1.51514 q^{35} -7.60975 q^{37} -12.9844 q^{39} -9.21949 q^{41} +8.31032 q^{43} +6.76491 q^{45} +5.28005 q^{47} -4.70436 q^{49} -10.9844 q^{51} -0.155162 q^{53} -4.24977 q^{55} +3.12489 q^{57} +2.48486 q^{59} +4.49954 q^{61} -10.2498 q^{63} -4.15516 q^{65} -7.43521 q^{67} -27.2947 q^{69} +8.49954 q^{71} -15.0450 q^{73} +3.12489 q^{75} +6.43899 q^{77} -0.310323 q^{79} +16.4693 q^{81} +8.96972 q^{83} -3.51514 q^{85} -4.54541 q^{87} +0.719953 q^{89} +6.29564 q^{91} -15.5298 q^{93} +1.00000 q^{95} +17.3893 q^{97} -28.7493 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 3 q^{5} - 5 q^{7} + 4 q^{9} + 4 q^{11} - 5 q^{13} + q^{15} - 11 q^{17} + 3 q^{19} + 3 q^{21} - 9 q^{23} + 3 q^{25} + 19 q^{27} - 3 q^{29} - 14 q^{31} - 24 q^{33} - 5 q^{35} - 14 q^{37} - 5 q^{39}+ \cdots - 36 q^{99}+O(q^{100}) $ 3 * q + q^3 + 3 * q^5 - 5 * q^7 + 4 * q^9 + 4 * q^11 - 5 * q^13 + q^15 - 11 * q^17 + 3 * q^19 + 3 * q^21 - 9 * q^23 + 3 * q^25 + 19 * q^27 - 3 * q^29 - 14 * q^31 - 24 * q^33 - 5 * q^35 - 14 * q^37 - 5 * q^39 - 10 * q^41 + 10 * q^43 + 4 * q^45 + 4 * q^49 + q^51 + 7 * q^53 + 4 * q^55 + q^57 + 7 * q^59 - 20 * q^61 - 14 * q^63 - 5 * q^65 + q^67 - 33 * q^69 - 8 * q^71 - 13 * q^73 + q^75 - 16 * q^77 + 14 * q^79 + 15 * q^81 + 26 * q^83 - 11 * q^85 - 15 * q^87 + 18 * q^89 + 37 * q^91 - 14 * q^93 + 3 * q^95 - 6 * q^97 - 36 * q^99

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
$3$	3.12489	1.80415	0.902077	−	0.431576i	$-0.142042\pi$
$3$	3.12489	1.80415	0.902077	+	0.431576i	$0.142042\pi$
$4$	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
$9$	6.76491	2.25497
$10$	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	$-0.695472\pi$
$13$	−4.15516	−1.15243	−0.576217	+	0.817297i	$0.695472\pi$
$14$	0	0
$15$	3.12489	0.806842
$16$	0	0
$17$	−3.51514	−0.852546	−0.426273	−	0.904595i	$-0.640174\pi$
$17$	−3.51514	−0.852546	−0.426273	+	0.904595i	$0.640174\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.73463	−1.03318
$22$	0	0
$23$	−8.73463	−1.82130	−0.910648	−	0.413182i	$-0.864417\pi$
$23$	−8.73463	−1.82130	−0.910648	+	0.413182i	$0.864417\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	11.7649	2.26416
$28$	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
$33$	−13.2800	−2.31176
$34$	0	0
$35$	−1.51514	−0.256105
$36$	0	0
$37$	−7.60975	−1.25103	−0.625517	−	0.780210i	$-0.715112\pi$
$37$	−7.60975	−1.25103	−0.625517	+	0.780210i	$0.715112\pi$
$38$	0	0
$39$	−12.9844	−2.07917
$40$	0	0
$41$	−9.21949	−1.43984	−0.719922	−	0.694055i	$-0.755822\pi$
$41$	−9.21949	−1.43984	−0.719922	+	0.694055i	$0.755822\pi$
$42$	0	0
$43$	8.31032	1.26731	0.633656	−	0.773615i	$-0.281553\pi$
$43$	8.31032	1.26731	0.633656	+	0.773615i	$0.281553\pi$
$44$	0	0
$45$	6.76491	1.00845
$46$	0	0
$47$	5.28005	0.770174	0.385087	−	0.922880i	$-0.374171\pi$
$47$	5.28005	0.770174	0.385087	+	0.922880i	$0.374171\pi$
$48$	0	0
$49$	−4.70436	−0.672051
$50$	0	0
$51$	−10.9844	−1.53812
$52$	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
$57$	3.12489	0.413901
$58$	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	$-0.406992\pi$
$61$	4.49954	0.576107	0.288054	+	0.957614i	$0.406992\pi$
$62$	0	0
$63$	−10.2498	−1.29135
$64$	0	0
$65$	−4.15516	−0.515384
$66$	0	0
$67$	−7.43521	−0.908355	−0.454178	−	0.890911i	$-0.650067\pi$
$67$	−7.43521	−0.908355	−0.454178	+	0.890911i	$0.650067\pi$
$68$	0	0
$69$	−27.2947	−3.28590
$70$	0	0
$71$	8.49954	1.00871	0.504355	−	0.863496i	$-0.331730\pi$
$71$	8.49954	1.00871	0.504355	+	0.863496i	$0.331730\pi$
$72$	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
$75$	3.12489	0.360831
$76$	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
$81$	16.4693	1.82992
$82$	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
$87$	−4.54541	−0.487320
$88$	0	0
$89$	0.719953	0.0763149	0.0381574	−	0.999272i	$-0.487851\pi$
$89$	0.719953	0.0763149	0.0381574	+	0.999272i	$0.487851\pi$
$90$	0	0
$91$	6.29564	0.659963
$92$	0	0
$93$	−15.5298	−1.61037
$94$	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
$99$	−28.7493	−2.88941
$100$	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
$105$	−4.73463	−0.462053
$106$	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	$-0.122421\pi$
$109$	19.3553	1.85390	0.926950	+	0.375186i	$0.122421\pi$
$110$	0	0
$111$	−23.7796	−2.25706
$112$	0	0
$113$	−17.8595	−1.68008	−0.840041	−	0.542523i	$-0.817469\pi$
$113$	−17.8595	−1.68008	−0.840041	+	0.542523i	$0.817469\pi$
$114$	0	0
$115$	−8.73463	−0.814509
$116$	0	0
$117$	−28.1093	−2.59870
$118$	0	0
$119$	5.32592	0.488226
$120$	0	0
$121$	7.06055	0.641868
$122$	0	0
$123$	−28.8099	−2.59770
$124$	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
$129$	25.9688	2.28643
$130$	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
$135$	11.7649	1.01256
$136$	0	0
$137$	−7.51514	−0.642062	−0.321031	−	0.947069i	$-0.604029\pi$
$137$	−7.51514	−0.642062	−0.321031	+	0.947069i	$0.604029\pi$
$138$	0	0
$139$	−16.7493	−1.42066	−0.710329	−	0.703870i	$-0.751454\pi$
$139$	−16.7493	−1.42066	−0.710329	+	0.703870i	$0.751454\pi$
$140$	0	0
$141$	16.4995	1.38951
$142$	0	0
$143$	17.6585	1.47668
$144$	0	0
$145$	−1.45459	−0.120797
$146$	0	0
$147$	−14.7006	−1.21248
$148$	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
$153$	−23.7796	−1.92247
$154$	0	0
$155$	−4.96972	−0.399178
$156$	0	0
$157$	−4.71995	−0.376693	−0.188347	−	0.982103i	$-0.560313\pi$
$157$	−4.71995	−0.376693	−0.188347	+	0.982103i	$0.560313\pi$
$158$	0	0
$159$	−0.484862	−0.0384521
$160$	0	0
$161$	13.2342	1.04300
$162$	0	0
$163$	−10.7493	−0.841951	−0.420976	−	0.907072i	$-0.638312\pi$
$163$	−10.7493	−0.841951	−0.420976	+	0.907072i	$0.638312\pi$
$164$	0	0
$165$	−13.2800	−1.03385
$166$	0	0
$167$	−10.7006	−0.828035	−0.414018	−	0.910269i	$-0.635875\pi$
$167$	−10.7006	−0.828035	−0.414018	+	0.910269i	$0.635875\pi$
$168$	0	0
$169$	4.26537	0.328105
$170$	0	0
$171$	6.76491	0.517326
$172$	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
$177$	7.76491	0.583646
$178$	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	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	14.0606	1.03939
$184$	0	0
$185$	−7.60975	−0.559480
$186$	0	0
$187$	14.9385	1.09241
$188$	0	0
$189$	−17.8255	−1.29661
$190$	0	0
$191$	13.2342	0.957591	0.478796	−	0.877926i	$-0.341073\pi$
$191$	13.2342	0.957591	0.478796	+	0.877926i	$0.341073\pi$
$192$	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
$195$	−12.9844	−0.929832
$196$	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
$201$	−23.2342	−1.63881
$202$	0	0
$203$	2.20390	0.154683
$204$	0	0
$205$	−9.21949	−0.643917
$206$	0	0
$207$	−59.0890	−4.10697
$208$	0	0
$209$	−4.24977	−0.293963
$210$	0	0
$211$	6.98440	0.480826	0.240413	−	0.970671i	$-0.422717\pi$
$211$	6.98440	0.480826	0.240413	+	0.970671i	$0.422717\pi$
$212$	0	0
$213$	26.5601	1.81987
$214$	0	0
$215$	8.31032	0.566759
$216$	0	0
$217$	7.52982	0.511157
$218$	0	0
$219$	−47.0138	−3.17690
$220$	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
$225$	6.76491	0.450994
$226$	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	$-0.547187\pi$
$229$	−4.47018	−0.295398	−0.147699	+	0.989032i	$0.547187\pi$
$230$	0	0
$231$	20.1211	1.32387
$232$	0	0
$233$	−3.93945	−0.258082	−0.129041	−	0.991639i	$-0.541190\pi$
$233$	−3.93945	−0.258082	−0.129041	+	0.991639i	$0.541190\pi$
$234$	0	0
$235$	5.28005	0.344432
$236$	0	0
$237$	−0.969724	−0.0629903
$238$	0	0
$239$	−17.8255	−1.15303	−0.576517	−	0.817085i	$-0.695588\pi$
$239$	−17.8255	−1.15303	−0.576517	+	0.817085i	$0.695588\pi$
$240$	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
$243$	16.1698	1.03730
$244$	0	0
$245$	−4.70436	−0.300550
$246$	0	0
$247$	−4.15516	−0.264387
$248$	0	0
$249$	28.0294	1.77629
$250$	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
$255$	−10.9844	−0.687870
$256$	0	0
$257$	−0.889794	−0.0555038	−0.0277519	−	0.999615i	$-0.508835\pi$
$257$	−0.889794	−0.0555038	−0.0277519	+	0.999615i	$0.508835\pi$
$258$	0	0
$259$	11.5298	0.716428
$260$	0	0
$261$	−9.84014	−0.609089
$262$	0	0
$263$	−12.6206	−0.778222	−0.389111	−	0.921191i	$-0.627218\pi$
$263$	−12.6206	−0.778222	−0.389111	+	0.921191i	$0.627218\pi$
$264$	0	0
$265$	−0.155162	−0.00953150
$266$	0	0
$267$	2.24977	0.137684
$268$	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
$273$	19.6732	1.19067
$274$	0	0
$275$	−4.24977	−0.256271
$276$	0	0
$277$	−0.749313	−0.0450218	−0.0225109	−	0.999747i	$-0.507166\pi$
$277$	−0.749313	−0.0450218	−0.0225109	+	0.999747i	$0.507166\pi$
$278$	0	0
$279$	−33.6197	−2.01276
$280$	0	0
$281$	3.77959	0.225471	0.112736	−	0.993625i	$-0.464039\pi$
$281$	3.77959	0.225471	0.112736	+	0.993625i	$0.464039\pi$
$282$	0	0
$283$	9.46927	0.562889	0.281445	−	0.959577i	$-0.409186\pi$
$283$	9.46927	0.562889	0.281445	+	0.959577i	$0.409186\pi$
$284$	0	0
$285$	3.12489	0.185102
$286$	0	0
$287$	13.9688	0.824553
$288$	0	0
$289$	−4.64380	−0.273165
$290$	0	0
$291$	54.3397	3.18545
$292$	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
$297$	−49.9982	−2.90119
$298$	0	0
$299$	36.2938	2.09893
$300$	0	0
$301$	−12.5913	−0.725750
$302$	0	0
$303$	39.0596	2.24392
$304$	0	0
$305$	4.49954	0.257643
$306$	0	0
$307$	−13.3288	−0.760714	−0.380357	−	0.924840i	$-0.624199\pi$
$307$	−13.3288	−0.760714	−0.380357	+	0.924840i	$0.624199\pi$
$308$	0	0
$309$	−34.0294	−1.93586
$310$	0	0
$311$	−13.4546	−0.762940	−0.381470	−	0.924381i	$-0.624582\pi$
$311$	−13.4546	−0.762940	−0.381470	+	0.924381i	$0.624582\pi$
$312$	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
$315$	−10.2498	−0.577509
$316$	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
$321$	23.2342	1.29681
$322$	0	0
$323$	−3.51514	−0.195588
$324$	0	0
$325$	−4.15516	−0.230487
$326$	0	0
$327$	60.4830	3.34472
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	11.9541	0.657058	0.328529	−	0.944494i	$-0.393447\pi$
$331$	11.9541	0.657058	0.328529	+	0.944494i	$0.393447\pi$
$332$	0	0
$333$	−51.4792	−2.82105
$334$	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
$339$	−55.8089	−3.03113
$340$	0	0
$341$	21.1202	1.14372
$342$	0	0
$343$	17.7337	0.957531
$344$	0	0
$345$	−27.2947	−1.46950
$346$	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	$-0.538946\pi$
$349$	−4.56009	−0.244096	−0.122048	+	0.992524i	$0.538946\pi$
$350$	0	0
$351$	−48.8851	−2.60929
$352$	0	0
$353$	−24.4849	−1.30320	−0.651599	−	0.758564i	$-0.725901\pi$
$353$	−24.4849	−1.30320	−0.651599	+	0.758564i	$0.725901\pi$
$354$	0	0
$355$	8.49954	0.451109
$356$	0	0
$357$	16.6429	0.880835
$358$	0	0
$359$	12.0147	0.634111	0.317055	−	0.948407i	$-0.397306\pi$
$359$	12.0147	0.634111	0.317055	+	0.948407i	$0.397306\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	22.0634	1.15803
$364$	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
$369$	−62.3690	−3.24680
$370$	0	0
$371$	0.235091	0.0122053
$372$	0	0
$373$	−9.62443	−0.498334	−0.249167	−	0.968461i	$-0.580157\pi$
$373$	−9.62443	−0.498334	−0.249167	+	0.968461i	$0.580157\pi$
$374$	0	0
$375$	3.12489	0.161368
$376$	0	0
$377$	6.04404	0.311284
$378$	0	0
$379$	−19.7943	−1.01676	−0.508382	−	0.861132i	$-0.669756\pi$
$379$	−19.7943	−1.01676	−0.508382	+	0.861132i	$0.669756\pi$
$380$	0	0
$381$	8.43899	0.432343
$382$	0	0
$383$	33.3288	1.70302	0.851511	−	0.524337i	$-0.175687\pi$
$383$	33.3288	1.70302	0.851511	+	0.524337i	$0.175687\pi$
$384$	0	0
$385$	6.43899	0.328161
$386$	0	0
$387$	56.2186	2.85775
$388$	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
$393$	12.4995	0.630518
$394$	0	0
$395$	−0.310323	−0.0156140
$396$	0	0
$397$	8.74931	0.439115	0.219558	−	0.975600i	$-0.429539\pi$
$397$	8.74931	0.439115	0.219558	+	0.975600i	$0.429539\pi$
$398$	0	0
$399$	−4.73463	−0.237028
$400$	0	0
$401$	−18.6206	−0.929871	−0.464935	−	0.885345i	$-0.653922\pi$
$401$	−18.6206	−0.929871	−0.464935	+	0.885345i	$0.653922\pi$
$402$	0	0
$403$	20.6500	1.02865
$404$	0	0
$405$	16.4693	0.818364
$406$	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
$411$	−23.4839	−1.15838
$412$	0	0
$413$	−3.76491	−0.185259
$414$	0	0
$415$	8.96972	0.440306
$416$	0	0
$417$	−52.3397	−2.56309
$418$	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	$-0.661099\pi$
$421$	−19.8936	−0.969554	−0.484777	+	0.874638i	$0.661099\pi$
$422$	0	0
$423$	35.7190	1.73672
$424$	0	0
$425$	−3.51514	−0.170509
$426$	0	0
$427$	−6.81743	−0.329918
$428$	0	0
$429$	55.1807	2.66415
$430$	0	0
$431$	−33.7796	−1.62711	−0.813553	−	0.581491i	$-0.802470\pi$
$431$	−33.7796	−1.62711	−0.813553	+	0.581491i	$0.802470\pi$
$432$	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
$435$	−4.54541	−0.217936
$436$	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
$441$	−31.8245	−1.51545
$442$	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
$447$	39.0596	1.84746
$448$	0	0
$449$	22.9991	1.08539	0.542697	−	0.839929i	$-0.317403\pi$
$449$	22.9991	1.08539	0.542697	+	0.839929i	$0.317403\pi$
$450$	0	0
$451$	39.1807	1.84495
$452$	0	0
$453$	44.5289	2.09215
$454$	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
$459$	−41.3553	−1.93030
$460$	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
$465$	−15.5298	−0.720178
$466$	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
$471$	−14.7493	−0.679612
$472$	0	0
$473$	−35.3170	−1.62388
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−1.04965	−0.0480603
$478$	0	0
$479$	16.8704	0.770829	0.385415	−	0.922744i	$-0.374058\pi$
$479$	16.8704	0.770829	0.385415	+	0.922744i	$0.374058\pi$
$480$	0	0
$481$	31.6197	1.44174
$482$	0	0
$483$	41.3553	1.88173
$484$	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
$489$	−33.5904	−1.51901
$490$	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
$495$	−28.7493	−1.29219
$496$	0	0
$497$	−12.8780	−0.577656
$498$	0	0
$499$	36.7787	1.64644	0.823220	−	0.567723i	$-0.192175\pi$
$499$	36.7787	1.64644	0.823220	+	0.567723i	$0.192175\pi$
$500$	0	0
$501$	−33.4381	−1.49390
$502$	0	0
$503$	15.8860	0.708322	0.354161	−	0.935184i	$-0.384766\pi$
$503$	15.8860	0.708322	0.354161	+	0.935184i	$0.384766\pi$
$504$	0	0
$505$	12.4995	0.556222
$506$	0	0
$507$	13.3288	0.591952
$508$	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
$513$	11.7649	0.519433
$514$	0	0
$515$	−10.8898	−0.479862
$516$	0	0
$517$	−22.4390	−0.986866
$518$	0	0
$519$	13.7190	0.602199
$520$	0	0
$521$	−23.4693	−1.02821	−0.514104	−	0.857728i	$-0.671875\pi$
$521$	−23.4693	−1.02821	−0.514104	+	0.857728i	$0.671875\pi$
$522$	0	0
$523$	26.3737	1.15324	0.576622	−	0.817011i	$-0.304371\pi$
$523$	26.3737	1.15324	0.576622	+	0.817011i	$0.304371\pi$
$524$	0	0
$525$	−4.73463	−0.206636
$526$	0	0
$527$	17.4693	0.760973
$528$	0	0
$529$	53.2938	2.31712
$530$	0	0
$531$	16.8099	0.729486
$532$	0	0
$533$	38.3085	1.65932
$534$	0	0
$535$	7.43521	0.321452
$536$	0	0
$537$	10.0606	0.434145
$538$	0	0
$539$	19.9924	0.861135
$540$	0	0
$541$	−22.5601	−0.969934	−0.484967	−	0.874532i	$-0.661168\pi$
$541$	−22.5601	−0.969934	−0.484967	+	0.874532i	$0.661168\pi$
$542$	0	0
$543$	−68.7475	−2.95024
$544$	0	0
$545$	19.3553	0.829089
$546$	0	0
$547$	−20.6694	−0.883759	−0.441879	−	0.897074i	$-0.645688\pi$
$547$	−20.6694	−0.883759	−0.441879	+	0.897074i	$0.645688\pi$
$548$	0	0
$549$	30.4390	1.29910
$550$	0	0
$551$	−1.45459	−0.0619674
$552$	0	0
$553$	0.470182	0.0199942
$554$	0	0
$555$	−23.7796	−1.00939
$556$	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
$561$	46.6812	1.97088
$562$	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
$567$	−24.9532	−1.04794
$568$	0	0
$569$	2.84862	0.119420	0.0597102	−	0.998216i	$-0.480982\pi$
$569$	2.84862	0.119420	0.0597102	+	0.998216i	$0.480982\pi$
$570$	0	0
$571$	−28.4002	−1.18851	−0.594256	−	0.804276i	$-0.702554\pi$
$571$	−28.4002	−1.18851	−0.594256	+	0.804276i	$0.702554\pi$
$572$	0	0
$573$	41.3553	1.72764
$574$	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
$579$	54.3397	2.25828
$580$	0	0
$581$	−13.5904	−0.563824
$582$	0	0
$583$	0.659401	0.0273096
$584$	0	0
$585$	−28.1093	−1.16218
$586$	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
$591$	6.24977	0.257081
$592$	0	0
$593$	16.0606	0.659528	0.329764	−	0.944063i	$-0.393031\pi$
$593$	16.0606	0.659528	0.329764	+	0.944063i	$0.393031\pi$
$594$	0	0
$595$	5.32592	0.218341
$596$	0	0
$597$	−44.7640	−1.83207
$598$	0	0
$599$	−23.2489	−0.949922	−0.474961	−	0.880007i	$-0.657538\pi$
$599$	−23.2489	−0.949922	−0.474961	+	0.880007i	$0.657538\pi$
$600$	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
$603$	−50.2985	−2.04831
$604$	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
$609$	6.88693	0.279072
$610$	0	0
$611$	−21.9394	−0.887575
$612$	0	0
$613$	19.4305	0.784791	0.392395	−	0.919797i	$-0.371646\pi$
$613$	19.4305	0.784791	0.392395	+	0.919797i	$0.371646\pi$
$614$	0	0
$615$	−28.8099	−1.16173
$616$	0	0
$617$	20.7787	0.836518	0.418259	−	0.908328i	$-0.362640\pi$
$617$	20.7787	0.836518	0.418259	+	0.908328i	$0.362640\pi$
$618$	0	0
$619$	−18.9310	−0.760900	−0.380450	−	0.924802i	$-0.624231\pi$
$619$	−18.9310	−0.760900	−0.380450	+	0.924802i	$0.624231\pi$
$620$	0	0
$621$	−102.762	−4.12370
$622$	0	0
$623$	−1.09083	−0.0437031
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−13.2800	−0.530354
$628$	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
$633$	21.8255	0.867484
$634$	0	0
$635$	2.70058	0.107169
$636$	0	0
$637$	19.5474	0.774495
$638$	0	0
$639$	57.4986	2.27461
$640$	0	0
$641$	26.8099	1.05893	0.529463	−	0.848333i	$-0.322393\pi$
$641$	26.8099	1.05893	0.529463	+	0.848333i	$0.322393\pi$
$642$	0	0
$643$	0.969724	0.0382422	0.0191211	−	0.999817i	$-0.493913\pi$
$643$	0.969724	0.0382422	0.0191211	+	0.999817i	$0.493913\pi$
$644$	0	0
$645$	25.9688	1.02252
$646$	0	0
$647$	1.20482	0.0473662	0.0236831	−	0.999720i	$-0.492461\pi$
$647$	1.20482	0.0473662	0.0236831	+	0.999720i	$0.492461\pi$
$648$	0	0
$649$	−10.5601	−0.414520
$650$	0	0
$651$	23.5298	0.922206
$652$	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
$657$	−101.778	−3.97073
$658$	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	$-0.620773\pi$
$661$	−19.0450	−0.740763	−0.370381	+	0.928880i	$0.620773\pi$
$662$	0	0
$663$	45.6420	1.77259
$664$	0	0
$665$	−1.51514	−0.0587545
$666$	0	0
$667$	12.7053	0.491950
$668$	0	0
$669$	76.4977	2.95757
$670$	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
$675$	11.7649	0.452832
$676$	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
$681$	77.8236	2.98221
$682$	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
$687$	−13.9688	−0.532943
$688$	0	0
$689$	0.644721	0.0245619
$690$	0	0
$691$	21.5979	0.821624	0.410812	−	0.911720i	$-0.365245\pi$
$691$	21.5979	0.821624	0.410812	+	0.911720i	$0.365245\pi$
$692$	0	0
$693$	43.5592	1.65468
$694$	0	0
$695$	−16.7493	−0.635338
$696$	0	0
$697$	32.4078	1.22753
$698$	0	0
$699$	−12.3103	−0.465619
$700$	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
$705$	16.4995	0.621409
$706$	0	0
$707$	−18.9385	−0.712257
$708$	0	0
$709$	−47.4986	−1.78385	−0.891924	−	0.452185i	$-0.850645\pi$
$709$	−47.4986	−1.78385	−0.891924	+	0.452185i	$0.850645\pi$
$710$	0	0
$711$	−2.09931	−0.0787302
$712$	0	0
$713$	43.4087	1.62567
$714$	0	0
$715$	17.6585	0.660390
$716$	0	0
$717$	−55.7025	−2.08025
$718$	0	0
$719$	−25.4546	−0.949296	−0.474648	−	0.880176i	$-0.657425\pi$
$719$	−25.4546	−0.949296	−0.474648	+	0.880176i	$0.657425\pi$
$720$	0	0
$721$	16.4995	0.614475
$722$	0	0
$723$	−85.3388	−3.17378
$724$	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
$729$	1.12110	0.0415224
$730$	0	0
$731$	−29.2119	−1.08044
$732$	0	0
$733$	−24.2110	−0.894254	−0.447127	−	0.894470i	$-0.647553\pi$
$733$	−24.2110	−0.894254	−0.447127	+	0.894470i	$0.647553\pi$
$734$	0	0
$735$	−14.7006	−0.542239
$736$	0	0
$737$	31.5979	1.16392
$738$	0	0
$739$	37.4693	1.37833	0.689165	−	0.724605i	$-0.257978\pi$
$739$	37.4693	1.37833	0.689165	+	0.724605i	$0.257978\pi$
$740$	0	0
$741$	−12.9844	−0.476994
$742$	0	0
$743$	−15.8595	−0.581829	−0.290915	−	0.956749i	$-0.593960\pi$
$743$	−15.8595	−0.581829	−0.290915	+	0.956749i	$0.593960\pi$
$744$	0	0
$745$	12.4995	0.457948
$746$	0	0
$747$	60.6794	2.22014
$748$	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
$753$	57.6197	2.09978
$754$	0	0
$755$	14.2498	0.518602
$756$	0	0
$757$	−37.4381	−1.36071	−0.680355	−	0.732883i	$-0.738174\pi$
$757$	−37.4381	−1.36071	−0.680355	+	0.732883i	$0.738174\pi$
$758$	0	0
$759$	115.996	4.21040
$760$	0	0
$761$	21.5833	0.782392	0.391196	−	0.920307i	$-0.372061\pi$
$761$	21.5833	0.782392	0.391196	+	0.920307i	$0.372061\pi$
$762$	0	0
$763$	−29.3259	−1.06167
$764$	0	0
$765$	−23.7796	−0.859753
$766$	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
$771$	−2.78051	−0.100137
$772$	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
$777$	36.0294	1.29255
$778$	0	0
$779$	−9.21949	−0.330323
$780$	0	0
$781$	−36.1211	−1.29251
$782$	0	0
$783$	−17.1131	−0.611571
$784$	0	0
$785$	−4.71995	−0.168462
$786$	0	0
$787$	−27.6925	−0.987132	−0.493566	−	0.869708i	$-0.664307\pi$
$787$	−27.6925	−0.987132	−0.493566	+	0.869708i	$0.664307\pi$
$788$	0	0
$789$	−39.4381	−1.40403
$790$	0	0
$791$	27.0596	0.962130
$792$	0	0
$793$	−18.6963	−0.663926
$794$	0	0
$795$	−0.484862	−0.0171963
$796$	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
$801$	4.87042	0.172088
$802$	0	0
$803$	63.9376	2.25631
$804$	0	0
$805$	13.2342	0.466443
$806$	0	0
$807$	51.3700	1.80831
$808$	0	0
$809$	−23.2342	−0.816870	−0.408435	−	0.912787i	$-0.633925\pi$
$809$	−23.2342	−0.816870	−0.408435	+	0.912787i	$0.633925\pi$
$810$	0	0
$811$	−32.1433	−1.12871	−0.564353	−	0.825534i	$-0.690874\pi$
$811$	−32.1433	−1.12871	−0.564353	+	0.825534i	$0.690874\pi$
$812$	0	0
$813$	80.0128	2.80617
$814$	0	0
$815$	−10.7493	−0.376532
$816$	0	0
$817$	8.31032	0.290741
$818$	0	0
$819$	42.5895	1.48820
$820$	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
$825$	−13.2800	−0.462352
$826$	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	$-0.605651\pi$
$829$	−18.7640	−0.651700	−0.325850	+	0.945421i	$0.605651\pi$
$830$	0	0
$831$	−2.34152	−0.0812263
$832$	0	0
$833$	16.5365	0.572954
$834$	0	0
$835$	−10.7006	−0.370309
$836$	0	0
$837$	−58.4683	−2.02096
$838$	0	0
$839$	−7.84014	−0.270672	−0.135336	−	0.990800i	$-0.543211\pi$
$839$	−7.84014	−0.270672	−0.135336	+	0.990800i	$0.543211\pi$
$840$	0	0
$841$	−26.8842	−0.927041
$842$	0	0
$843$	11.8108	0.406785
$844$	0	0
$845$	4.26537	0.146733
$846$	0	0
$847$	−10.6977	−0.367578
$848$	0	0
$849$	29.5904	1.01554
$850$	0	0
$851$	66.4683	2.27851
$852$	0	0
$853$	−36.8174	−1.26060	−0.630302	−	0.776350i	$-0.717069\pi$
$853$	−36.8174	−1.26060	−0.630302	+	0.776350i	$0.717069\pi$
$854$	0	0
$855$	6.76491	0.231355
$856$	0	0
$857$	−5.23887	−0.178956	−0.0894782	−	0.995989i	$-0.528520\pi$
$857$	−5.23887	−0.178956	−0.0894782	+	0.995989i	$0.528520\pi$
$858$	0	0
$859$	29.6197	1.01061	0.505306	−	0.862940i	$-0.331380\pi$
$859$	29.6197	1.01061	0.505306	+	0.862940i	$0.331380\pi$
$860$	0	0
$861$	43.6509	1.48762
$862$	0	0
$863$	−35.8889	−1.22167	−0.610836	−	0.791757i	$-0.709166\pi$
$863$	−35.8889	−1.22167	−0.610836	+	0.791757i	$0.709166\pi$
$864$	0	0
$865$	4.39025	0.149273
$866$	0	0
$867$	−14.5114	−0.492832
$868$	0	0
$869$	1.31880	0.0447373
$870$	0	0
$871$	30.8945	1.04682
$872$	0	0
$873$	117.637	3.98142
$874$	0	0
$875$	−1.51514	−0.0512210
$876$	0	0
$877$	29.8136	1.00674	0.503368	−	0.864072i	$-0.332094\pi$
$877$	29.8136	1.00674	0.503368	+	0.864072i	$0.332094\pi$
$878$	0	0
$879$	−77.1349	−2.60169
$880$	0	0
$881$	−38.9679	−1.31286	−0.656431	−	0.754386i	$-0.727935\pi$
$881$	−38.9679	−1.31286	−0.656431	+	0.754386i	$0.727935\pi$
$882$	0	0
$883$	28.5677	0.961378	0.480689	−	0.876891i	$-0.340387\pi$
$883$	28.5677	0.961378	0.480689	+	0.876891i	$0.340387\pi$
$884$	0	0
$885$	7.76491	0.261015
$886$	0	0
$887$	27.5104	0.923710	0.461855	−	0.886955i	$-0.347184\pi$
$887$	27.5104	0.923710	0.461855	+	0.886955i	$0.347184\pi$
$888$	0	0
$889$	−4.09174	−0.137233
$890$	0	0
$891$	−69.9906	−2.34477
$892$	0	0
$893$	5.28005	0.176690
$894$	0	0
$895$	3.21949	0.107616
$896$	0	0
$897$	113.414	3.78678
$898$	0	0
$899$	7.22889	0.241097
$900$	0	0
$901$	0.545414	0.0181704
$902$	0	0
$903$	−39.3463	−1.30936
$904$	0	0
$905$	−22.0000	−0.731305
$906$	0	0
$907$	55.4939	1.84265	0.921323	−	0.388799i	$-0.127110\pi$
$907$	55.4939	1.84265	0.921323	+	0.388799i	$0.127110\pi$
$908$	0	0
$909$	84.5583	2.80462
$910$	0	0
$911$	6.84106	0.226654	0.113327	−	0.993558i	$-0.463849\pi$
$911$	6.84106	0.226654	0.113327	+	0.993558i	$0.463849\pi$
$912$	0	0
$913$	−38.1193	−1.26156
$914$	0	0
$915$	14.0606	0.464828
$916$	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
$921$	−41.6509	−1.37244
$922$	0	0
$923$	−35.3170	−1.16247
$924$	0	0
$925$	−7.60975	−0.250207
$926$	0	0
$927$	−73.6685	−2.41959
$928$	0	0
$929$	44.7034	1.46667	0.733336	−	0.679866i	$-0.237962\pi$
$929$	44.7034	1.46667	0.733336	+	0.679866i	$0.237962\pi$
$930$	0	0
$931$	−4.70436	−0.154179
$932$	0	0
$933$	−42.0440	−1.37646
$934$	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
$939$	−68.0128	−2.21952
$940$	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
$945$	−17.8255	−0.579862
$946$	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
$951$	−73.6344	−2.38776
$952$	0	0
$953$	−21.5398	−0.697743	−0.348871	−	0.937171i	$-0.613435\pi$
$953$	−21.5398	−0.697743	−0.348871	+	0.937171i	$0.613435\pi$
$954$	0	0
$955$	13.2342	0.428248
$956$	0	0
$957$	19.3170	0.624429
$958$	0	0
$959$	11.3865	0.367688
$960$	0	0
$961$	−6.30184	−0.203285
$962$	0	0
$963$	50.2985	1.62085
$964$	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
$969$	−10.9844	−0.352870
$970$	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
$975$	−12.9844	−0.415834
$976$	0	0
$977$	−58.6694	−1.87700	−0.938500	−	0.345279i	$-0.887784\pi$
$977$	−58.6694	−1.87700	−0.938500	+	0.345279i	$0.887784\pi$
$978$	0	0
$979$	−3.05964	−0.0977864
$980$	0	0
$981$	130.937	4.18049
$982$	0	0
$983$	11.7384	0.374397	0.187199	−	0.982322i	$-0.440059\pi$
$983$	11.7384	0.374397	0.187199	+	0.982322i	$0.440059\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	−24.9991	−0.795730
$988$	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
$993$	37.3553	1.18543
$994$	0	0
$995$	−14.3250	−0.454133
$996$	0	0
$997$	−45.8401	−1.45177	−0.725886	−	0.687815i	$-0.758570\pi$
$997$	−45.8401	−1.45177	−0.725886	+	0.687815i	$0.758570\pi$
$998$	0	0
$999$	−89.5280	−2.83254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.bw.1.3		3
4.3	odd	2		6080.2.a.bs.1.1		3
8.3	odd	2		1520.2.a.r.1.3		3
8.5	even	2		760.2.a.h.1.1	✓	3
24.5	odd	2		6840.2.a.bj.1.2		3
40.13	odd	4		3800.2.d.k.3649.1		6
40.19	odd	2		7600.2.a.bo.1.1		3
40.29	even	2		3800.2.a.y.1.3		3
40.37	odd	4		3800.2.d.k.3649.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.h.1.1	✓	3	8.5	even	2
1520.2.a.r.1.3		3	8.3	odd	2
3800.2.a.y.1.3		3	40.29	even	2
3800.2.d.k.3649.1		6	40.13	odd	4
3800.2.d.k.3649.6		6	40.37	odd	4
6080.2.a.bs.1.1		3	4.3	odd	2
6080.2.a.bw.1.3		3	1.1	even	1	trivial
6840.2.a.bj.1.2		3	24.5	odd	2
7600.2.a.bo.1.1		3	40.19	odd	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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q+3.12489 q^{3} +1.00000 q^{5} -1.51514 q^{7} +6.76491 q^{9} -4.24977 q^{11} -4.15516 q^{13} +3.12489 q^{15} -3.51514 q^{17} +1.00000 q^{19} -4.73463 q^{21} -8.73463 q^{23} +1.00000 q^{25} +11.7649 q^{27} -1.45459 q^{29} -4.96972 q^{31} -13.2800 q^{33} -1.51514 q^{35} -7.60975 q^{37} -12.9844 q^{39} -9.21949 q^{41} +8.31032 q^{43} +6.76491 q^{45} +5.28005 q^{47} -4.70436 q^{49} -10.9844 q^{51} -0.155162 q^{53} -4.24977 q^{55} +3.12489 q^{57} +2.48486 q^{59} +4.49954 q^{61} -10.2498 q^{63} -4.15516 q^{65} -7.43521 q^{67} -27.2947 q^{69} +8.49954 q^{71} -15.0450 q^{73} +3.12489 q^{75} +6.43899 q^{77} -0.310323 q^{79} +16.4693 q^{81} +8.96972 q^{83} -3.51514 q^{85} -4.54541 q^{87} +0.719953 q^{89} +6.29564 q^{91} -15.5298 q^{93} +1.00000 q^{95} +17.3893 q^{97} -28.7493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 3 q^{5} - 5 q^{7} + 4 q^{9} + 4 q^{11} - 5 q^{13} + q^{15} - 11 q^{17} + 3 q^{19} + 3 q^{21} - 9 q^{23} + 3 q^{25} + 19 q^{27} - 3 q^{29} - 14 q^{31} - 24 q^{33} - 5 q^{35} - 14 q^{37} - 5 q^{39}+ \cdots - 36 q^{99}+O(q^{100}) \) 3 * q + q^3 + 3 * q^5 - 5 * q^7 + 4 * q^9 + 4 * q^11 - 5 * q^13 + q^15 - 11 * q^17 + 3 * q^19 + 3 * q^21 - 9 * q^23 + 3 * q^25 + 19 * q^27 - 3 * q^29 - 14 * q^31 - 24 * q^33 - 5 * q^35 - 14 * q^37 - 5 * q^39 - 10 * q^41 + 10 * q^43 + 4 * q^45 + 4 * q^49 + q^51 + 7 * q^53 + 4 * q^55 + q^57 + 7 * q^59 - 20 * q^61 - 14 * q^63 - 5 * q^65 + q^67 - 33 * q^69 - 8 * q^71 - 13 * q^73 + q^75 - 16 * q^77 + 14 * q^79 + 15 * q^81 + 26 * q^83 - 11 * q^85 - 15 * q^87 + 18 * q^89 + 37 * q^91 - 14 * q^93 + 3 * q^95 - 6 * q^97 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more