LMFDB - Embedded newform 6080.2.a.ch.1.2

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.78165$ 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)

$f(q)$	$=$	$q+0.296842 q^{3} +1.00000 q^{5} +3.56331 q^{7} -2.91188 q^{9} +O(q^{10})$	$q+0.296842 q^{3} +1.00000 q^{5} +3.56331 q^{7} -2.91188 q^{9} +5.56331 q^{11} -5.26647 q^{13} +0.296842 q^{15} +1.40632 q^{17} +1.00000 q^{19} +1.05774 q^{21} +6.96962 q^{23} +1.00000 q^{25} -1.75489 q^{27} -1.40632 q^{29} -1.75489 q^{31} +1.65142 q^{33} +3.56331 q^{35} -3.61504 q^{37} -1.56331 q^{39} +4.34858 q^{41} -3.56331 q^{43} -2.91188 q^{45} +8.26046 q^{47} +5.69716 q^{49} +0.417453 q^{51} +7.61504 q^{53} +5.56331 q^{55} +0.296842 q^{57} +9.47519 q^{59} -9.21473 q^{61} -10.3759 q^{63} -5.26647 q^{65} -4.76090 q^{67} +2.06888 q^{69} +14.0689 q^{71} +6.59368 q^{73} +0.296842 q^{75} +19.8238 q^{77} -5.47519 q^{79} +8.21473 q^{81} -4.15699 q^{83} +1.40632 q^{85} -0.417453 q^{87} -9.23009 q^{89} -18.7660 q^{91} -0.520926 q^{93} +1.00000 q^{95} +11.5116 q^{97} -16.1997 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 8 * q^9	$ 4 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 8 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 4 q^{25} - 4 q^{27} - 4 q^{29} - 4 q^{31} + 8 q^{33} - 4 q^{35} + 6 q^{37} + 12 q^{39} + 16 q^{41} + 4 q^{43} + 8 q^{45} + 12 q^{47} + 20 q^{49} - 36 q^{51} + 10 q^{53} + 4 q^{55} + 2 q^{57} - 20 q^{61} - 20 q^{63} - 2 q^{65} - 18 q^{67} - 28 q^{69} + 20 q^{71} + 28 q^{73} + 2 q^{75} + 40 q^{77} + 16 q^{79} + 16 q^{81} + 4 q^{85} + 36 q^{87} + 4 q^{89} - 36 q^{91} + 40 q^{93} + 4 q^{95} + 30 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 8 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^15 + 4 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 4 * q^25 - 4 * q^27 - 4 * q^29 - 4 * q^31 + 8 * q^33 - 4 * q^35 + 6 * q^37 + 12 * q^39 + 16 * q^41 + 4 * q^43 + 8 * q^45 + 12 * q^47 + 20 * q^49 - 36 * q^51 + 10 * q^53 + 4 * q^55 + 2 * q^57 - 20 * q^61 - 20 * q^63 - 2 * q^65 - 18 * q^67 - 28 * q^69 + 20 * q^71 + 28 * q^73 + 2 * q^75 + 40 * q^77 + 16 * q^79 + 16 * q^81 + 4 * q^85 + 36 * q^87 + 4 * q^89 - 36 * q^91 + 40 * q^93 + 4 * q^95 + 30 * 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
$3$	0.296842	0.171382	0.0856908	−	0.996322i	$-0.472690\pi$
$3$	0.296842	0.171382	0.0856908	+	0.996322i	$0.472690\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.56331	1.34680	0.673402	−	0.739277i	$-0.264832\pi$
$7$	3.56331	1.34680	0.673402	+	0.739277i	$0.264832\pi$
$8$	0	0
$9$	−2.91188	−0.970628
$10$	0	0
$11$	5.56331	1.67740	0.838700	−	0.544594i	$-0.183316\pi$
$11$	5.56331	1.67740	0.838700	+	0.544594i	$0.183316\pi$
$12$	0	0
$13$	−5.26647	−1.46065	−0.730327	−	0.683097i	$-0.760632\pi$
$13$	−5.26647	−1.46065	−0.730327	+	0.683097i	$0.760632\pi$
$14$	0	0
$15$	0.296842	0.0766442
$16$	0	0
$17$	1.40632	0.341082	0.170541	−	0.985351i	$-0.445448\pi$
$17$	1.40632	0.341082	0.170541	+	0.985351i	$0.445448\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.05774	0.230817
$22$	0	0
$23$	6.96962	1.45327	0.726633	−	0.687025i	$-0.241084\pi$
$23$	6.96962	1.45327	0.726633	+	0.687025i	$0.241084\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.75489	−0.337730
$28$	0	0
$29$	−1.40632	−0.261146	−0.130573	−	0.991439i	$-0.541682\pi$
$29$	−1.40632	−0.261146	−0.130573	+	0.991439i	$0.541682\pi$
$30$	0	0
$31$	−1.75489	−0.315188	−0.157594	−	0.987504i	$-0.550374\pi$
$31$	−1.75489	−0.315188	−0.157594	+	0.987504i	$0.550374\pi$
$32$	0	0
$33$	1.65142	0.287476
$34$	0	0
$35$	3.56331	0.602309
$36$	0	0
$37$	−3.61504	−0.594309	−0.297155	−	0.954829i	$-0.596038\pi$
$37$	−3.61504	−0.594309	−0.297155	+	0.954829i	$0.596038\pi$
$38$	0	0
$39$	−1.56331	−0.250329
$40$	0	0
$41$	4.34858	0.679134	0.339567	−	0.940582i	$-0.389720\pi$
$41$	4.34858	0.679134	0.339567	+	0.940582i	$0.389720\pi$
$42$	0	0
$43$	−3.56331	−0.543399	−0.271700	−	0.962382i	$-0.587586\pi$
$43$	−3.56331	−0.543399	−0.271700	+	0.962382i	$0.587586\pi$
$44$	0	0
$45$	−2.91188	−0.434078
$46$	0	0
$47$	8.26046	1.20491	0.602456	−	0.798152i	$-0.294189\pi$
$47$	8.26046	1.20491	0.602456	+	0.798152i	$0.294189\pi$
$48$	0	0
$49$	5.69716	0.813879
$50$	0	0
$51$	0.417453	0.0584552
$52$	0	0
$53$	7.61504	1.04601	0.523003	−	0.852331i	$-0.324812\pi$
$53$	7.61504	1.04601	0.523003	+	0.852331i	$0.324812\pi$
$54$	0	0
$55$	5.56331	0.750156
$56$	0	0
$57$	0.296842	0.0393177
$58$	0	0
$59$	9.47519	1.23356	0.616782	−	0.787134i	$-0.288436\pi$
$59$	9.47519	1.23356	0.616782	+	0.787134i	$0.288436\pi$
$60$	0	0
$61$	−9.21473	−1.17983	−0.589913	−	0.807467i	$-0.700838\pi$
$61$	−9.21473	−1.17983	−0.589913	+	0.807467i	$0.700838\pi$
$62$	0	0
$63$	−10.3759	−1.30725
$64$	0	0
$65$	−5.26647	−0.653225
$66$	0	0
$67$	−4.76090	−0.581636	−0.290818	−	0.956778i	$-0.593927\pi$
$67$	−4.76090	−0.581636	−0.290818	+	0.956778i	$0.593927\pi$
$68$	0	0
$69$	2.06888	0.249063
$70$	0	0
$71$	14.0689	1.66967	0.834834	−	0.550502i	$-0.185563\pi$
$71$	14.0689	1.66967	0.834834	+	0.550502i	$0.185563\pi$
$72$	0	0
$73$	6.59368	0.771732	0.385866	−	0.922555i	$-0.373903\pi$
$73$	6.59368	0.771732	0.385866	+	0.922555i	$0.373903\pi$
$74$	0	0
$75$	0.296842	0.0342763
$76$	0	0
$77$	19.8238	2.25913
$78$	0	0
$79$	−5.47519	−0.616007	−0.308004	−	0.951385i	$-0.599661\pi$
$79$	−5.47519	−0.616007	−0.308004	+	0.951385i	$0.599661\pi$
$80$	0	0
$81$	8.21473	0.912748
$82$	0	0
$83$	−4.15699	−0.456289	−0.228144	−	0.973627i	$-0.573266\pi$
$83$	−4.15699	−0.456289	−0.228144	+	0.973627i	$0.573266\pi$
$84$	0	0
$85$	1.40632	0.152536
$86$	0	0
$87$	−0.417453	−0.0447557
$88$	0	0
$89$	−9.23009	−0.978387	−0.489194	−	0.872175i	$-0.662709\pi$
$89$	−9.23009	−0.978387	−0.489194	+	0.872175i	$0.662709\pi$
$90$	0	0
$91$	−18.7660	−1.96721
$92$	0	0
$93$	−0.520926	−0.0540175
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	11.5116	1.16882	0.584411	−	0.811457i	$-0.301325\pi$
$97$	11.5116	1.16882	0.584411	+	0.811457i	$0.301325\pi$
$98$	0	0
$99$	−16.1997	−1.62813
$100$	0	0
$101$	11.8511	1.17923	0.589616	−	0.807684i	$-0.299279\pi$
$101$	11.8511	1.17923	0.589616	+	0.807684i	$0.299279\pi$
$102$	0	0
$103$	−1.35458	−0.133471	−0.0667354	−	0.997771i	$-0.521258\pi$
$103$	−1.35458	−0.133471	−0.0667354	+	0.997771i	$0.521258\pi$
$104$	0	0
$105$	1.05774	0.103225
$106$	0	0
$107$	−7.06287	−0.682794	−0.341397	−	0.939919i	$-0.610900\pi$
$107$	−7.06287	−0.682794	−0.341397	+	0.939919i	$0.610900\pi$
$108$	0	0
$109$	−10.1844	−0.975484	−0.487742	−	0.872988i	$-0.662179\pi$
$109$	−10.1844	−0.975484	−0.487742	+	0.872988i	$0.662179\pi$
$110$	0	0
$111$	−1.07310	−0.101854
$112$	0	0
$113$	1.86015	0.174988	0.0874940	−	0.996165i	$-0.472114\pi$
$113$	1.86015	0.174988	0.0874940	+	0.996165i	$0.472114\pi$
$114$	0	0
$115$	6.96962	0.649921
$116$	0	0
$117$	15.3353	1.41775
$118$	0	0
$119$	5.01114	0.459370
$120$	0	0
$121$	19.9504	1.81367
$122$	0	0
$123$	1.29084	0.116391
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.89053	0.433964	0.216982	−	0.976176i	$-0.430379\pi$
$127$	4.89053	0.433964	0.216982	+	0.976176i	$0.430379\pi$
$128$	0	0
$129$	−1.05774	−0.0931287
$130$	0	0
$131$	2.81263	0.245741	0.122870	−	0.992423i	$-0.460790\pi$
$131$	2.81263	0.245741	0.122870	+	0.992423i	$0.460790\pi$
$132$	0	0
$133$	3.56331	0.308978
$134$	0	0
$135$	−1.75489	−0.151037
$136$	0	0
$137$	9.23009	0.788579	0.394290	−	0.918986i	$-0.370991\pi$
$137$	9.23009	0.788579	0.394290	+	0.918986i	$0.370991\pi$
$138$	0	0
$139$	−3.67878	−0.312030	−0.156015	−	0.987755i	$-0.549865\pi$
$139$	−3.67878	−0.312030	−0.156015	+	0.987755i	$0.549865\pi$
$140$	0	0
$141$	2.45205	0.206500
$142$	0	0
$143$	−29.2990	−2.45010
$144$	0	0
$145$	−1.40632	−0.116788
$146$	0	0
$147$	1.69115	0.139484
$148$	0	0
$149$	−7.09925	−0.581593	−0.290797	−	0.956785i	$-0.593920\pi$
$149$	−7.09925	−0.581593	−0.290797	+	0.956785i	$0.593920\pi$
$150$	0	0
$151$	18.3567	1.49385	0.746924	−	0.664910i	$-0.231530\pi$
$151$	18.3567	1.49385	0.746924	+	0.664910i	$0.231530\pi$
$152$	0	0
$153$	−4.09503	−0.331064
$154$	0	0
$155$	−1.75489	−0.140957
$156$	0	0
$157$	−17.2301	−1.37511	−0.687555	−	0.726132i	$-0.741316\pi$
$157$	−17.2301	−1.37511	−0.687555	+	0.726132i	$0.741316\pi$
$158$	0	0
$159$	2.26046	0.179266
$160$	0	0
$161$	24.8349	1.95726
$162$	0	0
$163$	10.8662	0.851103	0.425551	−	0.904934i	$-0.360080\pi$
$163$	10.8662	0.851103	0.425551	+	0.904934i	$0.360080\pi$
$164$	0	0
$165$	1.65142	0.128563
$166$	0	0
$167$	−2.82977	−0.218974	−0.109487	−	0.993988i	$-0.534921\pi$
$167$	−2.82977	−0.218974	−0.109487	+	0.993988i	$0.534921\pi$
$168$	0	0
$169$	14.7357	1.13351
$170$	0	0
$171$	−2.91188	−0.222677
$172$	0	0
$173$	9.26647	0.704516	0.352258	−	0.935903i	$-0.385414\pi$
$173$	9.26647	0.704516	0.352258	+	0.935903i	$0.385414\pi$
$174$	0	0
$175$	3.56331	0.269361
$176$	0	0
$177$	2.81263	0.211410
$178$	0	0
$179$	−3.59067	−0.268379	−0.134190	−	0.990956i	$-0.542843\pi$
$179$	−3.59067	−0.268379	−0.134190	+	0.990956i	$0.542843\pi$
$180$	0	0
$181$	19.7630	1.46897	0.734487	−	0.678623i	$-0.237423\pi$
$181$	19.7630	1.46897	0.734487	+	0.678623i	$0.237423\pi$
$182$	0	0
$183$	−2.73532	−0.202200
$184$	0	0
$185$	−3.61504	−0.265783
$186$	0	0
$187$	7.82377	0.572131
$188$	0	0
$189$	−6.25323	−0.454855
$190$	0	0
$191$	−8.31398	−0.601579	−0.300789	−	0.953691i	$-0.597250\pi$
$191$	−8.31398	−0.601579	−0.300789	+	0.953691i	$0.597250\pi$
$192$	0	0
$193$	−22.2514	−1.60169	−0.800847	−	0.598869i	$-0.795617\pi$
$193$	−22.2514	−1.60169	−0.800847	+	0.598869i	$0.795617\pi$
$194$	0	0
$195$	−1.56331	−0.111951
$196$	0	0
$197$	8.81263	0.627874	0.313937	−	0.949444i	$-0.398352\pi$
$197$	8.81263	0.627874	0.313937	+	0.949444i	$0.398352\pi$
$198$	0	0
$199$	−21.0659	−1.49332	−0.746660	−	0.665206i	$-0.768344\pi$
$199$	−21.0659	−1.49332	−0.746660	+	0.665206i	$0.768344\pi$
$200$	0	0
$201$	−1.41323	−0.0996818
$202$	0	0
$203$	−5.01114	−0.351713
$204$	0	0
$205$	4.34858	0.303718
$206$	0	0
$207$	−20.2947	−1.41058
$208$	0	0
$209$	5.56331	0.384822
$210$	0	0
$211$	5.34556	0.368004	0.184002	−	0.982926i	$-0.441095\pi$
$211$	5.34556	0.368004	0.184002	+	0.982926i	$0.441095\pi$
$212$	0	0
$213$	4.17623	0.286151
$214$	0	0
$215$	−3.56331	−0.243016
$216$	0	0
$217$	−6.25323	−0.424497
$218$	0	0
$219$	1.95728	0.132261
$220$	0	0
$221$	−7.40632	−0.498203
$222$	0	0
$223$	3.10947	0.208226	0.104113	−	0.994565i	$-0.466800\pi$
$223$	3.10947	0.208226	0.104113	+	0.994565i	$0.466800\pi$
$224$	0	0
$225$	−2.91188	−0.194126
$226$	0	0
$227$	14.4692	0.960354	0.480177	−	0.877172i	$-0.340572\pi$
$227$	14.4692	0.960354	0.480177	+	0.877172i	$0.340572\pi$
$228$	0	0
$229$	5.21473	0.344599	0.172299	−	0.985045i	$-0.444880\pi$
$229$	5.21473	0.344599	0.172299	+	0.985045i	$0.444880\pi$
$230$	0	0
$231$	5.88452	0.387173
$232$	0	0
$233$	−3.18737	−0.208811	−0.104406	−	0.994535i	$-0.533294\pi$
$233$	−3.18737	−0.208811	−0.104406	+	0.994535i	$0.533294\pi$
$234$	0	0
$235$	8.26046	0.538853
$236$	0	0
$237$	−1.62527	−0.105572
$238$	0	0
$239$	−16.5209	−1.06865	−0.534325	−	0.845279i	$-0.679434\pi$
$239$	−16.5209	−1.06865	−0.534325	+	0.845279i	$0.679434\pi$
$240$	0	0
$241$	−12.2271	−0.787615	−0.393807	−	0.919193i	$-0.628842\pi$
$241$	−12.2271	−0.787615	−0.393807	+	0.919193i	$0.628842\pi$
$242$	0	0
$243$	7.70316	0.494158
$244$	0	0
$245$	5.69716	0.363978
$246$	0	0
$247$	−5.26647	−0.335097
$248$	0	0
$249$	−1.23397	−0.0781996
$250$	0	0
$251$	4.52093	0.285358	0.142679	−	0.989769i	$-0.454428\pi$
$251$	4.52093	0.285358	0.142679	+	0.989769i	$0.454428\pi$
$252$	0	0
$253$	38.7742	2.43771
$254$	0	0
$255$	0.417453	0.0261420
$256$	0	0
$257$	15.9290	0.993625	0.496813	−	0.867858i	$-0.334504\pi$
$257$	15.9290	0.993625	0.496813	+	0.867858i	$0.334504\pi$
$258$	0	0
$259$	−12.8815	−0.800418
$260$	0	0
$261$	4.09503	0.253476
$262$	0	0
$263$	−0.854147	−0.0526689	−0.0263345	−	0.999653i	$-0.508383\pi$
$263$	−0.854147	−0.0526689	−0.0263345	+	0.999653i	$0.508383\pi$
$264$	0	0
$265$	7.61504	0.467788
$266$	0	0
$267$	−2.73988	−0.167678
$268$	0	0
$269$	−10.3913	−0.633569	−0.316784	−	0.948498i	$-0.602603\pi$
$269$	−10.3913	−0.633569	−0.316784	+	0.948498i	$0.602603\pi$
$270$	0	0
$271$	8.19971	0.498097	0.249048	−	0.968491i	$-0.419882\pi$
$271$	8.19971	0.498097	0.249048	+	0.968491i	$0.419882\pi$
$272$	0	0
$273$	−5.57054	−0.337145
$274$	0	0
$275$	5.56331	0.335480
$276$	0	0
$277$	−14.6484	−0.880137	−0.440069	−	0.897964i	$-0.645046\pi$
$277$	−14.6484	−0.880137	−0.440069	+	0.897964i	$0.645046\pi$
$278$	0	0
$279$	5.11005	0.305931
$280$	0	0
$281$	−31.6129	−1.88587	−0.942935	−	0.332977i	$-0.891947\pi$
$281$	−31.6129	−1.88587	−0.942935	+	0.332977i	$0.891947\pi$
$282$	0	0
$283$	24.7326	1.47020	0.735101	−	0.677957i	$-0.237135\pi$
$283$	24.7326	1.47020	0.735101	+	0.677957i	$0.237135\pi$
$284$	0	0
$285$	0.296842	0.0175834
$286$	0	0
$287$	15.4953	0.914660
$288$	0	0
$289$	−15.0223	−0.883663
$290$	0	0
$291$	3.41712	0.200315
$292$	0	0
$293$	−30.2857	−1.76931	−0.884655	−	0.466245i	$-0.845606\pi$
$293$	−30.2857	−1.76931	−0.884655	+	0.466245i	$0.845606\pi$
$294$	0	0
$295$	9.47519	0.551667
$296$	0	0
$297$	−9.76302	−0.566508
$298$	0	0
$299$	−36.7053	−2.12272
$300$	0	0
$301$	−12.6972	−0.731852
$302$	0	0
$303$	3.51791	0.202099
$304$	0	0
$305$	−9.21473	−0.527634
$306$	0	0
$307$	−23.0629	−1.31627	−0.658134	−	0.752901i	$-0.728654\pi$
$307$	−23.0629	−1.31627	−0.658134	+	0.752901i	$0.728654\pi$
$308$	0	0
$309$	−0.402096	−0.0228744
$310$	0	0
$311$	10.3152	0.584921	0.292460	−	0.956278i	$-0.405526\pi$
$311$	10.3152	0.584921	0.292460	+	0.956278i	$0.405526\pi$
$312$	0	0
$313$	−4.95038	−0.279812	−0.139906	−	0.990165i	$-0.544680\pi$
$313$	−4.95038	−0.279812	−0.139906	+	0.990165i	$0.544680\pi$
$314$	0	0
$315$	−10.3759	−0.584618
$316$	0	0
$317$	14.2433	0.799985	0.399992	−	0.916518i	$-0.369013\pi$
$317$	14.2433	0.799985	0.399992	+	0.916518i	$0.369013\pi$
$318$	0	0
$319$	−7.82377	−0.438047
$320$	0	0
$321$	−2.09656	−0.117018
$322$	0	0
$323$	1.40632	0.0782495
$324$	0	0
$325$	−5.26647	−0.292131
$326$	0	0
$327$	−3.02314	−0.167180
$328$	0	0
$329$	29.4346	1.62278
$330$	0	0
$331$	2.04272	0.112278	0.0561390	−	0.998423i	$-0.482121\pi$
$331$	2.04272	0.112278	0.0561390	+	0.998423i	$0.482121\pi$
$332$	0	0
$333$	10.5266	0.576854
$334$	0	0
$335$	−4.76090	−0.260116
$336$	0	0
$337$	34.6951	1.88996	0.944980	−	0.327128i	$-0.106081\pi$
$337$	34.6951	1.88996	0.944980	+	0.327128i	$0.106081\pi$
$338$	0	0
$339$	0.552170	0.0299898
$340$	0	0
$341$	−9.76302	−0.528697
$342$	0	0
$343$	−4.64243	−0.250668
$344$	0	0
$345$	2.06888	0.111385
$346$	0	0
$347$	7.35280	0.394719	0.197359	−	0.980331i	$-0.436763\pi$
$347$	7.35280	0.394719	0.197359	+	0.980331i	$0.436763\pi$
$348$	0	0
$349$	31.7630	1.70024	0.850118	−	0.526593i	$-0.176531\pi$
$349$	31.7630	1.70024	0.850118	+	0.526593i	$0.176531\pi$
$350$	0	0
$351$	9.24209	0.493306
$352$	0	0
$353$	−7.52179	−0.400345	−0.200172	−	0.979761i	$-0.564150\pi$
$353$	−7.52179	−0.400345	−0.200172	+	0.979761i	$0.564150\pi$
$354$	0	0
$355$	14.0689	0.746698
$356$	0	0
$357$	1.48751	0.0787276
$358$	0	0
$359$	30.3982	1.60436	0.802178	−	0.597085i	$-0.203674\pi$
$359$	30.3982	1.60436	0.802178	+	0.597085i	$0.203674\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.92211	0.310830
$364$	0	0
$365$	6.59368	0.345129
$366$	0	0
$367$	−3.91577	−0.204401	−0.102201	−	0.994764i	$-0.532588\pi$
$367$	−3.91577	−0.204401	−0.102201	+	0.994764i	$0.532588\pi$
$368$	0	0
$369$	−12.6626	−0.659186
$370$	0	0
$371$	27.1347	1.40877
$372$	0	0
$373$	−26.7759	−1.38641	−0.693203	−	0.720743i	$-0.743801\pi$
$373$	−26.7759	−1.38641	−0.693203	+	0.720743i	$0.743801\pi$
$374$	0	0
$375$	0.296842	0.0153288
$376$	0	0
$377$	7.40632	0.381445
$378$	0	0
$379$	14.9504	0.767950	0.383975	−	0.923344i	$-0.374555\pi$
$379$	14.9504	0.767950	0.383975	+	0.923344i	$0.374555\pi$
$380$	0	0
$381$	1.45171	0.0743735
$382$	0	0
$383$	27.9910	1.43027	0.715136	−	0.698985i	$-0.246365\pi$
$383$	27.9910	1.43027	0.715136	+	0.698985i	$0.246365\pi$
$384$	0	0
$385$	19.8238	1.01031
$386$	0	0
$387$	10.3759	0.527439
$388$	0	0
$389$	35.2036	1.78489	0.892447	−	0.451152i	$-0.148987\pi$
$389$	35.2036	1.78489	0.892447	+	0.451152i	$0.148987\pi$
$390$	0	0
$391$	9.80150	0.495683
$392$	0	0
$393$	0.834907	0.0421155
$394$	0	0
$395$	−5.47519	−0.275487
$396$	0	0
$397$	35.9735	1.80546	0.902730	−	0.430208i	$-0.141560\pi$
$397$	35.9735	1.80546	0.902730	+	0.430208i	$0.141560\pi$
$398$	0	0
$399$	1.05774	0.0529532
$400$	0	0
$401$	23.2421	1.16065	0.580327	−	0.814383i	$-0.302925\pi$
$401$	23.2421	1.16065	0.580327	+	0.814383i	$0.302925\pi$
$402$	0	0
$403$	9.24209	0.460381
$404$	0	0
$405$	8.21473	0.408193
$406$	0	0
$407$	−20.1116	−0.996895
$408$	0	0
$409$	−31.8926	−1.57699	−0.788495	−	0.615041i	$-0.789139\pi$
$409$	−31.8926	−1.57699	−0.788495	+	0.615041i	$0.789139\pi$
$410$	0	0
$411$	2.73988	0.135148
$412$	0	0
$413$	33.7630	1.66137
$414$	0	0
$415$	−4.15699	−0.204059
$416$	0	0
$417$	−1.09202	−0.0534763
$418$	0	0
$419$	31.8238	1.55469	0.777346	−	0.629073i	$-0.216565\pi$
$419$	31.8238	1.55469	0.777346	+	0.629073i	$0.216565\pi$
$420$	0	0
$421$	−0.348578	−0.0169887	−0.00849433	−	0.999964i	$-0.502704\pi$
$421$	−0.348578	−0.0169887	−0.00849433	+	0.999964i	$0.502704\pi$
$422$	0	0
$423$	−24.0535	−1.16952
$424$	0	0
$425$	1.40632	0.0682164
$426$	0	0
$427$	−32.8349	−1.58899
$428$	0	0
$429$	−8.69716	−0.419903
$430$	0	0
$431$	−29.2764	−1.41019	−0.705097	−	0.709111i	$-0.749096\pi$
$431$	−29.2764	−1.41019	−0.705097	+	0.709111i	$0.749096\pi$
$432$	0	0
$433$	−0.883290	−0.0424482	−0.0212241	−	0.999775i	$-0.506756\pi$
$433$	−0.883290	−0.0424482	−0.0212241	+	0.999775i	$0.506756\pi$
$434$	0	0
$435$	−0.417453	−0.0200154
$436$	0	0
$437$	6.96962	0.333402
$438$	0	0
$439$	13.8584	0.661424	0.330712	−	0.943732i	$-0.392711\pi$
$439$	13.8584	0.661424	0.330712	+	0.943732i	$0.392711\pi$
$440$	0	0
$441$	−16.5895	−0.789974
$442$	0	0
$443$	−13.5753	−0.644982	−0.322491	−	0.946572i	$-0.604520\pi$
$443$	−13.5753	−0.644982	−0.322491	+	0.946572i	$0.604520\pi$
$444$	0	0
$445$	−9.23009	−0.437548
$446$	0	0
$447$	−2.10735	−0.0996745
$448$	0	0
$449$	−15.6334	−0.737785	−0.368893	−	0.929472i	$-0.620263\pi$
$449$	−15.6334	−0.737785	−0.368893	+	0.929472i	$0.620263\pi$
$450$	0	0
$451$	24.1925	1.13918
$452$	0	0
$453$	5.44904	0.256018
$454$	0	0
$455$	−18.7660	−0.879765
$456$	0	0
$457$	5.30284	0.248057	0.124028	−	0.992279i	$-0.460419\pi$
$457$	5.30284	0.248057	0.124028	+	0.992279i	$0.460419\pi$
$458$	0	0
$459$	−2.46794	−0.115193
$460$	0	0
$461$	−0.374734	−0.0174531	−0.00872656	−	0.999962i	$-0.502778\pi$
$461$	−0.374734	−0.0174531	−0.00872656	+	0.999962i	$0.502778\pi$
$462$	0	0
$463$	−6.65564	−0.309314	−0.154657	−	0.987968i	$-0.549427\pi$
$463$	−6.65564	−0.309314	−0.154657	+	0.987968i	$0.549427\pi$
$464$	0	0
$465$	−0.520926	−0.0241574
$466$	0	0
$467$	−0.854147	−0.0395252	−0.0197626	−	0.999805i	$-0.506291\pi$
$467$	−0.854147	−0.0395252	−0.0197626	+	0.999805i	$0.506291\pi$
$468$	0	0
$469$	−16.9645	−0.783349
$470$	0	0
$471$	−5.11461	−0.235669
$472$	0	0
$473$	−19.8238	−0.911498
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−22.1741	−1.01528
$478$	0	0
$479$	17.0731	0.780090	0.390045	−	0.920796i	$-0.372460\pi$
$479$	17.0731	0.780090	0.390045	+	0.920796i	$0.372460\pi$
$480$	0	0
$481$	19.0385	0.868081
$482$	0	0
$483$	7.37204	0.335439
$484$	0	0
$485$	11.5116	0.522713
$486$	0	0
$487$	−12.8259	−0.581197	−0.290598	−	0.956845i	$-0.593854\pi$
$487$	−12.8259	−0.581197	−0.290598	+	0.956845i	$0.593854\pi$
$488$	0	0
$489$	3.22553	0.145863
$490$	0	0
$491$	−10.4054	−0.469591	−0.234796	−	0.972045i	$-0.575442\pi$
$491$	−10.4054	−0.469591	−0.234796	+	0.972045i	$0.575442\pi$
$492$	0	0
$493$	−1.97773	−0.0890723
$494$	0	0
$495$	−16.1997	−0.728123
$496$	0	0
$497$	50.1317	2.24872
$498$	0	0
$499$	36.1612	1.61880	0.809399	−	0.587258i	$-0.199793\pi$
$499$	36.1612	1.61880	0.809399	+	0.587258i	$0.199793\pi$
$500$	0	0
$501$	−0.839995	−0.0375282
$502$	0	0
$503$	−33.2536	−1.48270	−0.741352	−	0.671117i	$-0.765815\pi$
$503$	−33.2536	−1.48270	−0.741352	+	0.671117i	$0.765815\pi$
$504$	0	0
$505$	11.8511	0.527368
$506$	0	0
$507$	4.37416	0.194263
$508$	0	0
$509$	−7.29084	−0.323161	−0.161580	−	0.986860i	$-0.551659\pi$
$509$	−7.29084	−0.323161	−0.161580	+	0.986860i	$0.551659\pi$
$510$	0	0
$511$	23.4953	1.03937
$512$	0	0
$513$	−1.75489	−0.0774805
$514$	0	0
$515$	−1.35458	−0.0596899
$516$	0	0
$517$	45.9555	2.02112
$518$	0	0
$519$	2.75067	0.120741
$520$	0	0
$521$	−9.87849	−0.432785	−0.216392	−	0.976306i	$-0.569429\pi$
$521$	−9.87849	−0.432785	−0.216392	+	0.976306i	$0.569429\pi$
$522$	0	0
$523$	−11.1925	−0.489414	−0.244707	−	0.969597i	$-0.578692\pi$
$523$	−11.1925	−0.489414	−0.244707	+	0.969597i	$0.578692\pi$
$524$	0	0
$525$	1.05774	0.0461635
$526$	0	0
$527$	−2.46794	−0.107505
$528$	0	0
$529$	25.5756	1.11198
$530$	0	0
$531$	−27.5907	−1.19733
$532$	0	0
$533$	−22.9016	−0.991980
$534$	0	0
$535$	−7.06287	−0.305355
$536$	0	0
$537$	−1.06586	−0.0459953
$538$	0	0
$539$	31.6950	1.36520
$540$	0	0
$541$	−2.22587	−0.0956974	−0.0478487	−	0.998855i	$-0.515237\pi$
$541$	−2.22587	−0.0956974	−0.0478487	+	0.998855i	$0.515237\pi$
$542$	0	0
$543$	5.86649	0.251755
$544$	0	0
$545$	−10.1844	−0.436250
$546$	0	0
$547$	−34.9675	−1.49510	−0.747552	−	0.664204i	$-0.768771\pi$
$547$	−34.9675	−1.49510	−0.747552	+	0.664204i	$0.768771\pi$
$548$	0	0
$549$	26.8322	1.14517
$550$	0	0
$551$	−1.40632	−0.0599111
$552$	0	0
$553$	−19.5098	−0.829641
$554$	0	0
$555$	−1.07310	−0.0455504
$556$	0	0
$557$	8.88539	0.376486	0.188243	−	0.982122i	$-0.439721\pi$
$557$	8.88539	0.376486	0.188243	+	0.982122i	$0.439721\pi$
$558$	0	0
$559$	18.7660	0.793719
$560$	0	0
$561$	2.32242	0.0980527
$562$	0	0
$563$	−29.6767	−1.25072	−0.625362	−	0.780335i	$-0.715048\pi$
$563$	−29.6767	−1.25072	−0.625362	+	0.780335i	$0.715048\pi$
$564$	0	0
$565$	1.86015	0.0782570
$566$	0	0
$567$	29.2716	1.22929
$568$	0	0
$569$	−22.1152	−0.927116	−0.463558	−	0.886067i	$-0.653427\pi$
$569$	−22.1152	−0.927116	−0.463558	+	0.886067i	$0.653427\pi$
$570$	0	0
$571$	−33.2656	−1.39212	−0.696060	−	0.717983i	$-0.745065\pi$
$571$	−33.2656	−1.39212	−0.696060	+	0.717983i	$0.745065\pi$
$572$	0	0
$573$	−2.46794	−0.103100
$574$	0	0
$575$	6.96962	0.290653
$576$	0	0
$577$	0.934140	0.0388887	0.0194444	−	0.999811i	$-0.493810\pi$
$577$	0.934140	0.0388887	0.0194444	+	0.999811i	$0.493810\pi$
$578$	0	0
$579$	−6.60516	−0.274501
$580$	0	0
$581$	−14.8126	−0.614532
$582$	0	0
$583$	42.3648	1.75457
$584$	0	0
$585$	15.3353	0.634038
$586$	0	0
$587$	1.57531	0.0650200	0.0325100	−	0.999471i	$-0.489650\pi$
$587$	1.57531	0.0650200	0.0325100	+	0.999471i	$0.489650\pi$
$588$	0	0
$589$	−1.75489	−0.0723092
$590$	0	0
$591$	2.61596	0.107606
$592$	0	0
$593$	26.0162	1.06836	0.534180	−	0.845371i	$-0.320621\pi$
$593$	26.0162	1.06836	0.534180	+	0.845371i	$0.320621\pi$
$594$	0	0
$595$	5.01114	0.205437
$596$	0	0
$597$	−6.25323	−0.255928
$598$	0	0
$599$	−19.5359	−0.798217	−0.399109	−	0.916904i	$-0.630680\pi$
$599$	−19.5359	−0.798217	−0.399109	+	0.916904i	$0.630680\pi$
$600$	0	0
$601$	43.5299	1.77562	0.887811	−	0.460208i	$-0.152225\pi$
$601$	43.5299	1.77562	0.887811	+	0.460208i	$0.152225\pi$
$602$	0	0
$603$	13.8632	0.564552
$604$	0	0
$605$	19.9504	0.811098
$606$	0	0
$607$	12.5847	0.510796	0.255398	−	0.966836i	$-0.417794\pi$
$607$	12.5847	0.510796	0.255398	+	0.966836i	$0.417794\pi$
$608$	0	0
$609$	−1.48751	−0.0602771
$610$	0	0
$611$	−43.5034	−1.75996
$612$	0	0
$613$	18.2412	0.736756	0.368378	−	0.929676i	$-0.379913\pi$
$613$	18.2412	0.736756	0.368378	+	0.929676i	$0.379913\pi$
$614$	0	0
$615$	1.29084	0.0520517
$616$	0	0
$617$	26.7314	1.07617	0.538084	−	0.842892i	$-0.319148\pi$
$617$	26.7314	1.07617	0.538084	+	0.842892i	$0.319148\pi$
$618$	0	0
$619$	2.92690	0.117642	0.0588211	−	0.998269i	$-0.481266\pi$
$619$	2.92690	0.117642	0.0588211	+	0.998269i	$0.481266\pi$
$620$	0	0
$621$	−12.2310	−0.490811
$622$	0	0
$623$	−32.8896	−1.31770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.65142	0.0659514
$628$	0	0
$629$	−5.08389	−0.202708
$630$	0	0
$631$	13.2493	0.527447	0.263724	−	0.964598i	$-0.415049\pi$
$631$	13.2493	0.527447	0.263724	+	0.964598i	$0.415049\pi$
$632$	0	0
$633$	1.58679	0.0630691
$634$	0	0
$635$	4.89053	0.194075
$636$	0	0
$637$	−30.0039	−1.18880
$638$	0	0
$639$	−40.9669	−1.62063
$640$	0	0
$641$	6.93026	0.273729	0.136864	−	0.990590i	$-0.456298\pi$
$641$	6.93026	0.273729	0.136864	+	0.990590i	$0.456298\pi$
$642$	0	0
$643$	14.4794	0.571012	0.285506	−	0.958377i	$-0.407838\pi$
$643$	14.4794	0.571012	0.285506	+	0.958377i	$0.407838\pi$
$644$	0	0
$645$	−1.05774	−0.0416484
$646$	0	0
$647$	−6.35549	−0.249860	−0.124930	−	0.992166i	$-0.539871\pi$
$647$	−6.35549	−0.249860	−0.124930	+	0.992166i	$0.539871\pi$
$648$	0	0
$649$	52.7134	2.06918
$650$	0	0
$651$	−1.85622	−0.0727510
$652$	0	0
$653$	34.4030	1.34629	0.673146	−	0.739509i	$-0.264942\pi$
$653$	34.4030	1.34629	0.673146	+	0.739509i	$0.264942\pi$
$654$	0	0
$655$	2.81263	0.109899
$656$	0	0
$657$	−19.2000	−0.749065
$658$	0	0
$659$	−19.6214	−0.764341	−0.382170	−	0.924092i	$-0.624823\pi$
$659$	−19.6214	−0.764341	−0.382170	+	0.924092i	$0.624823\pi$
$660$	0	0
$661$	−39.8054	−1.54825	−0.774126	−	0.633032i	$-0.781810\pi$
$661$	−39.8054	−1.54825	−0.774126	+	0.633032i	$0.781810\pi$
$662$	0	0
$663$	−2.19850	−0.0853828
$664$	0	0
$665$	3.56331	0.138179
$666$	0	0
$667$	−9.80150	−0.379515
$668$	0	0
$669$	0.923022	0.0356861
$670$	0	0
$671$	−51.2644	−1.97904
$672$	0	0
$673$	−8.90374	−0.343214	−0.171607	−	0.985166i	$-0.554896\pi$
$673$	−8.90374	−0.343214	−0.171607	+	0.985166i	$0.554896\pi$
$674$	0	0
$675$	−1.75489	−0.0675459
$676$	0	0
$677$	−22.2695	−0.855886	−0.427943	−	0.903806i	$-0.640762\pi$
$677$	−22.2695	−0.855886	−0.427943	+	0.903806i	$0.640762\pi$
$678$	0	0
$679$	41.0193	1.57417
$680$	0	0
$681$	4.29506	0.164587
$682$	0	0
$683$	15.4054	0.589472	0.294736	−	0.955579i	$-0.404768\pi$
$683$	15.4054	0.589472	0.294736	+	0.955579i	$0.404768\pi$
$684$	0	0
$685$	9.23009	0.352663
$686$	0	0
$687$	1.54795	0.0590580
$688$	0	0
$689$	−40.1044	−1.52785
$690$	0	0
$691$	−17.8773	−0.680084	−0.340042	−	0.940410i	$-0.610441\pi$
$691$	−17.8773	−0.680084	−0.340042	+	0.940410i	$0.610441\pi$
$692$	0	0
$693$	−57.7245	−2.19277
$694$	0	0
$695$	−3.67878	−0.139544
$696$	0	0
$697$	6.11548	0.231640
$698$	0	0
$699$	−0.946144	−0.0357864
$700$	0	0
$701$	−35.3609	−1.33556	−0.667782	−	0.744357i	$-0.732756\pi$
$701$	−35.3609	−1.33556	−0.667782	+	0.744357i	$0.732756\pi$
$702$	0	0
$703$	−3.61504	−0.136344
$704$	0	0
$705$	2.45205	0.0923496
$706$	0	0
$707$	42.2292	1.58819
$708$	0	0
$709$	−6.90410	−0.259289	−0.129644	−	0.991561i	$-0.541384\pi$
$709$	−6.90410	−0.259289	−0.129644	+	0.991561i	$0.541384\pi$
$710$	0	0
$711$	15.9431	0.597914
$712$	0	0
$713$	−12.2310	−0.458053
$714$	0	0
$715$	−29.2990	−1.09572
$716$	0	0
$717$	−4.90410	−0.183147
$718$	0	0
$719$	38.9431	1.45233	0.726167	−	0.687518i	$-0.241300\pi$
$719$	38.9431	1.45233	0.726167	+	0.687518i	$0.241300\pi$
$720$	0	0
$721$	−4.82678	−0.179759
$722$	0	0
$723$	−3.62951	−0.134983
$724$	0	0
$725$	−1.40632	−0.0522293
$726$	0	0
$727$	6.18347	0.229332	0.114666	−	0.993404i	$-0.463420\pi$
$727$	6.18347	0.229332	0.114666	+	0.993404i	$0.463420\pi$
$728$	0	0
$729$	−22.3576	−0.828058
$730$	0	0
$731$	−5.01114	−0.185344
$732$	0	0
$733$	0.688715	0.0254383	0.0127191	−	0.999919i	$-0.495951\pi$
$733$	0.688715	0.0254383	0.0127191	+	0.999919i	$0.495951\pi$
$734$	0	0
$735$	1.69115	0.0623792
$736$	0	0
$737$	−26.4863	−0.975636
$738$	0	0
$739$	−26.2532	−0.965741	−0.482870	−	0.875692i	$-0.660406\pi$
$739$	−26.2532	−0.965741	−0.482870	+	0.875692i	$0.660406\pi$
$740$	0	0
$741$	−1.56331	−0.0574295
$742$	0	0
$743$	32.5688	1.19483	0.597416	−	0.801931i	$-0.296194\pi$
$743$	32.5688	1.19483	0.597416	+	0.801931i	$0.296194\pi$
$744$	0	0
$745$	−7.09925	−0.260096
$746$	0	0
$747$	12.1047	0.442887
$748$	0	0
$749$	−25.1672	−0.919589
$750$	0	0
$751$	−45.4833	−1.65971	−0.829855	−	0.557979i	$-0.811577\pi$
$751$	−45.4833	−1.65971	−0.829855	+	0.557979i	$0.811577\pi$
$752$	0	0
$753$	1.34200	0.0489052
$754$	0	0
$755$	18.3567	0.668069
$756$	0	0
$757$	−2.74947	−0.0999311	−0.0499656	−	0.998751i	$-0.515911\pi$
$757$	−2.74947	−0.0999311	−0.0499656	+	0.998751i	$0.515911\pi$
$758$	0	0
$759$	11.5098	0.417779
$760$	0	0
$761$	−33.2978	−1.20704	−0.603521	−	0.797347i	$-0.706236\pi$
$761$	−33.2978	−1.20704	−0.603521	+	0.797347i	$0.706236\pi$
$762$	0	0
$763$	−36.2900	−1.31379
$764$	0	0
$765$	−4.09503	−0.148056
$766$	0	0
$767$	−49.9008	−1.80181
$768$	0	0
$769$	−19.1540	−0.690710	−0.345355	−	0.938472i	$-0.612241\pi$
$769$	−19.1540	−0.690710	−0.345355	+	0.938472i	$0.612241\pi$
$770$	0	0
$771$	4.72840	0.170289
$772$	0	0
$773$	−0.569309	−0.0204766	−0.0102383	−	0.999948i	$-0.503259\pi$
$773$	−0.569309	−0.0204766	−0.0102383	+	0.999948i	$0.503259\pi$
$774$	0	0
$775$	−1.75489	−0.0630377
$776$	0	0
$777$	−3.82377	−0.137177
$778$	0	0
$779$	4.34858	0.155804
$780$	0	0
$781$	78.2695	2.80070
$782$	0	0
$783$	2.46794	0.0881969
$784$	0	0
$785$	−17.2301	−0.614968
$786$	0	0
$787$	15.9991	0.570307	0.285153	−	0.958482i	$-0.407956\pi$
$787$	15.9991	0.570307	0.285153	+	0.958482i	$0.407956\pi$
$788$	0	0
$789$	−0.253546	−0.00902649
$790$	0	0
$791$	6.62828	0.235675
$792$	0	0
$793$	48.5290	1.72332
$794$	0	0
$795$	2.26046	0.0801704
$796$	0	0
$797$	35.7528	1.26643	0.633214	−	0.773976i	$-0.281735\pi$
$797$	35.7528	1.26643	0.633214	+	0.773976i	$0.281735\pi$
$798$	0	0
$799$	11.6168	0.410974
$800$	0	0
$801$	26.8769	0.949650
$802$	0	0
$803$	36.6827	1.29450
$804$	0	0
$805$	24.8349	0.875315
$806$	0	0
$807$	−3.08457	−0.108582
$808$	0	0
$809$	23.2036	0.815796	0.407898	−	0.913028i	$-0.366262\pi$
$809$	23.2036	0.815796	0.407898	+	0.913028i	$0.366262\pi$
$810$	0	0
$811$	−21.7549	−0.763918	−0.381959	−	0.924179i	$-0.624750\pi$
$811$	−21.7549	−0.763918	−0.381959	+	0.924179i	$0.624750\pi$
$812$	0	0
$813$	2.43402	0.0853647
$814$	0	0
$815$	10.8662	0.380625
$816$	0	0
$817$	−3.56331	−0.124664
$818$	0	0
$819$	54.6445	1.90943
$820$	0	0
$821$	−52.2532	−1.82365	−0.911825	−	0.410579i	$-0.865327\pi$
$821$	−52.2532	−1.82365	−0.911825	+	0.410579i	$0.865327\pi$
$822$	0	0
$823$	9.44783	0.329331	0.164665	−	0.986349i	$-0.447346\pi$
$823$	9.44783	0.329331	0.164665	+	0.986349i	$0.447346\pi$
$824$	0	0
$825$	1.65142	0.0574951
$826$	0	0
$827$	−50.2216	−1.74638	−0.873188	−	0.487383i	$-0.837952\pi$
$827$	−50.2216	−1.74638	−0.873188	+	0.487383i	$0.837952\pi$
$828$	0	0
$829$	−44.2066	−1.53536	−0.767680	−	0.640834i	$-0.778589\pi$
$829$	−44.2066	−1.53536	−0.767680	+	0.640834i	$0.778589\pi$
$830$	0	0
$831$	−4.34826	−0.150839
$832$	0	0
$833$	8.01200	0.277599
$834$	0	0
$835$	−2.82977	−0.0979283
$836$	0	0
$837$	3.07965	0.106448
$838$	0	0
$839$	30.4033	1.04964	0.524819	−	0.851214i	$-0.324133\pi$
$839$	30.4033	1.04964	0.524819	+	0.851214i	$0.324133\pi$
$840$	0	0
$841$	−27.0223	−0.931803
$842$	0	0
$843$	−9.38404	−0.323204
$844$	0	0
$845$	14.7357	0.506922
$846$	0	0
$847$	71.0893	2.44266
$848$	0	0
$849$	7.34168	0.251966
$850$	0	0
$851$	−25.1955	−0.863690
$852$	0	0
$853$	−3.93925	−0.134877	−0.0674386	−	0.997723i	$-0.521483\pi$
$853$	−3.93925	−0.134877	−0.0674386	+	0.997723i	$0.521483\pi$
$854$	0	0
$855$	−2.91188	−0.0995844
$856$	0	0
$857$	27.4388	0.937292	0.468646	−	0.883386i	$-0.344742\pi$
$857$	27.4388	0.937292	0.468646	+	0.883386i	$0.344742\pi$
$858$	0	0
$859$	32.4517	1.10724	0.553619	−	0.832770i	$-0.313246\pi$
$859$	32.4517	1.10724	0.553619	+	0.832770i	$0.313246\pi$
$860$	0	0
$861$	4.59966	0.156756
$862$	0	0
$863$	2.10861	0.0717778	0.0358889	−	0.999356i	$-0.488574\pi$
$863$	2.10861	0.0717778	0.0358889	+	0.999356i	$0.488574\pi$
$864$	0	0
$865$	9.26647	0.315069
$866$	0	0
$867$	−4.45924	−0.151444
$868$	0	0
$869$	−30.4602	−1.03329
$870$	0	0
$871$	25.0731	0.849569
$872$	0	0
$873$	−33.5204	−1.13449
$874$	0	0
$875$	3.56331	0.120462
$876$	0	0
$877$	37.6613	1.27173	0.635866	−	0.771799i	$-0.280643\pi$
$877$	37.6613	1.27173	0.635866	+	0.771799i	$0.280643\pi$
$878$	0	0
$879$	−8.99007	−0.303227
$880$	0	0
$881$	−39.8818	−1.34365	−0.671827	−	0.740708i	$-0.734490\pi$
$881$	−39.8818	−1.34365	−0.671827	+	0.740708i	$0.734490\pi$
$882$	0	0
$883$	36.0458	1.21304	0.606518	−	0.795070i	$-0.292566\pi$
$883$	36.0458	1.21304	0.606518	+	0.795070i	$0.292566\pi$
$884$	0	0
$885$	2.81263	0.0945456
$886$	0	0
$887$	26.2057	0.879902	0.439951	−	0.898022i	$-0.354996\pi$
$887$	26.2057	0.879902	0.439951	+	0.898022i	$0.354996\pi$
$888$	0	0
$889$	17.4264	0.584464
$890$	0	0
$891$	45.7011	1.53104
$892$	0	0
$893$	8.26046	0.276426
$894$	0	0
$895$	−3.59067	−0.120023
$896$	0	0
$897$	−10.8957	−0.363796
$898$	0	0
$899$	2.46794	0.0823103
$900$	0	0
$901$	10.7092	0.356774
$902$	0	0
$903$	−3.76905	−0.125426
$904$	0	0
$905$	19.7630	0.656945
$906$	0	0
$907$	−22.8036	−0.757182	−0.378591	−	0.925564i	$-0.623591\pi$
$907$	−22.8036	−0.757182	−0.378591	+	0.925564i	$0.623591\pi$
$908$	0	0
$909$	−34.5091	−1.14460
$910$	0	0
$911$	−4.44146	−0.147152	−0.0735761	−	0.997290i	$-0.523441\pi$
$911$	−4.44146	−0.147152	−0.0735761	+	0.997290i	$0.523441\pi$
$912$	0	0
$913$	−23.1266	−0.765379
$914$	0	0
$915$	−2.73532	−0.0904268
$916$	0	0
$917$	10.0223	0.330965
$918$	0	0
$919$	−37.0111	−1.22088	−0.610442	−	0.792061i	$-0.709008\pi$
$919$	−37.0111	−1.22088	−0.610442	+	0.792061i	$0.709008\pi$
$920$	0	0
$921$	−6.84602	−0.225584
$922$	0	0
$923$	−74.0932	−2.43881
$924$	0	0
$925$	−3.61504	−0.118862
$926$	0	0
$927$	3.94438	0.129550
$928$	0	0
$929$	−3.04185	−0.0997999	−0.0499000	−	0.998754i	$-0.515890\pi$
$929$	−3.04185	−0.0997999	−0.0499000	+	0.998754i	$0.515890\pi$
$930$	0	0
$931$	5.69716	0.186717
$932$	0	0
$933$	3.06198	0.100245
$934$	0	0
$935$	7.82377	0.255865
$936$	0	0
$937$	38.6099	1.26133	0.630666	−	0.776055i	$-0.282782\pi$
$937$	38.6099	1.26133	0.630666	+	0.776055i	$0.282782\pi$
$938$	0	0
$939$	−1.46948	−0.0479547
$940$	0	0
$941$	−2.69716	−0.0879248	−0.0439624	−	0.999033i	$-0.513998\pi$
$941$	−2.69716	−0.0879248	−0.0439624	+	0.999033i	$0.513998\pi$
$942$	0	0
$943$	30.3080	0.986963
$944$	0	0
$945$	−6.25323	−0.203418
$946$	0	0
$947$	−20.1877	−0.656012	−0.328006	−	0.944676i	$-0.606377\pi$
$947$	−20.1877	−0.656012	−0.328006	+	0.944676i	$0.606377\pi$
$948$	0	0
$949$	−34.7254	−1.12723
$950$	0	0
$951$	4.22801	0.137103
$952$	0	0
$953$	5.18559	0.167978	0.0839888	−	0.996467i	$-0.473234\pi$
$953$	5.18559	0.167978	0.0839888	+	0.996467i	$0.473234\pi$
$954$	0	0
$955$	−8.31398	−0.269034
$956$	0	0
$957$	−2.32242	−0.0750732
$958$	0	0
$959$	32.8896	1.06206
$960$	0	0
$961$	−27.9203	−0.900656
$962$	0	0
$963$	20.5663	0.662739
$964$	0	0
$965$	−22.2514	−0.716299
$966$	0	0
$967$	−58.0054	−1.86533	−0.932665	−	0.360744i	$-0.882523\pi$
$967$	−58.0054	−1.86533	−0.932665	+	0.360744i	$0.882523\pi$
$968$	0	0
$969$	0.417453	0.0134105
$970$	0	0
$971$	−24.3221	−0.780533	−0.390267	−	0.920702i	$-0.627617\pi$
$971$	−24.3221	−0.780533	−0.390267	+	0.920702i	$0.627617\pi$
$972$	0	0
$973$	−13.1086	−0.420244
$974$	0	0
$975$	−1.56331	−0.0500659
$976$	0	0
$977$	−36.1134	−1.15537	−0.577685	−	0.816260i	$-0.696044\pi$
$977$	−36.1134	−1.15537	−0.577685	+	0.816260i	$0.696044\pi$
$978$	0	0
$979$	−51.3498	−1.64115
$980$	0	0
$981$	29.6557	0.946832
$982$	0	0
$983$	−21.1603	−0.674909	−0.337455	−	0.941342i	$-0.609566\pi$
$983$	−21.1603	−0.674909	−0.337455	+	0.941342i	$0.609566\pi$
$984$	0	0
$985$	8.81263	0.280794
$986$	0	0
$987$	8.73741	0.278115
$988$	0	0
$989$	−24.8349	−0.789704
$990$	0	0
$991$	−20.2652	−0.643746	−0.321873	−	0.946783i	$-0.604312\pi$
$991$	−20.2652	−0.643746	−0.321873	+	0.946783i	$0.604312\pi$
$992$	0	0
$993$	0.606364	0.0192424
$994$	0	0
$995$	−21.0659	−0.667833
$996$	0	0
$997$	−24.2875	−0.769193	−0.384597	−	0.923085i	$-0.625659\pi$
$997$	−24.2875	−0.769193	−0.384597	+	0.923085i	$0.625659\pi$
$998$	0	0
$999$	6.34402	0.200716

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.ch.1.2		4
4.3	odd	2		6080.2.a.cc.1.3		4
8.3	odd	2		95.2.a.b.1.4	✓	4
8.5	even	2		1520.2.a.t.1.3		4
24.11	even	2		855.2.a.m.1.1		4
40.3	even	4		475.2.b.e.324.3		8
40.19	odd	2		475.2.a.i.1.1		4
40.27	even	4		475.2.b.e.324.6		8
40.29	even	2		7600.2.a.cf.1.2		4
56.27	even	2		4655.2.a.y.1.4		4
120.59	even	2		4275.2.a.bo.1.4		4
152.75	even	2		1805.2.a.p.1.1		4
760.379	even	2		9025.2.a.bf.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.4	✓	4	8.3	odd	2
475.2.a.i.1.1		4	40.19	odd	2
475.2.b.e.324.3		8	40.3	even	4
475.2.b.e.324.6		8	40.27	even	4
855.2.a.m.1.1		4	24.11	even	2
1520.2.a.t.1.3		4	8.5	even	2
1805.2.a.p.1.1		4	152.75	even	2
4275.2.a.bo.1.4		4	120.59	even	2
4655.2.a.y.1.4		4	56.27	even	2
6080.2.a.cc.1.3		4	4.3	odd	2
6080.2.a.ch.1.2		4	1.1	even	1	trivial
7600.2.a.cf.1.2		4	40.29	even	2
9025.2.a.bf.1.4		4	760.379	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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q+0.296842 q^{3} +1.00000 q^{5} +3.56331 q^{7} -2.91188 q^{9} +O(q^{10})\)	\(q+0.296842 q^{3} +1.00000 q^{5} +3.56331 q^{7} -2.91188 q^{9} +5.56331 q^{11} -5.26647 q^{13} +0.296842 q^{15} +1.40632 q^{17} +1.00000 q^{19} +1.05774 q^{21} +6.96962 q^{23} +1.00000 q^{25} -1.75489 q^{27} -1.40632 q^{29} -1.75489 q^{31} +1.65142 q^{33} +3.56331 q^{35} -3.61504 q^{37} -1.56331 q^{39} +4.34858 q^{41} -3.56331 q^{43} -2.91188 q^{45} +8.26046 q^{47} +5.69716 q^{49} +0.417453 q^{51} +7.61504 q^{53} +5.56331 q^{55} +0.296842 q^{57} +9.47519 q^{59} -9.21473 q^{61} -10.3759 q^{63} -5.26647 q^{65} -4.76090 q^{67} +2.06888 q^{69} +14.0689 q^{71} +6.59368 q^{73} +0.296842 q^{75} +19.8238 q^{77} -5.47519 q^{79} +8.21473 q^{81} -4.15699 q^{83} +1.40632 q^{85} -0.417453 q^{87} -9.23009 q^{89} -18.7660 q^{91} -0.520926 q^{93} +1.00000 q^{95} +11.5116 q^{97} -16.1997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 8 * q^9	\( 4 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 8 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 4 q^{25} - 4 q^{27} - 4 q^{29} - 4 q^{31} + 8 q^{33} - 4 q^{35} + 6 q^{37} + 12 q^{39} + 16 q^{41} + 4 q^{43} + 8 q^{45} + 12 q^{47} + 20 q^{49} - 36 q^{51} + 10 q^{53} + 4 q^{55} + 2 q^{57} - 20 q^{61} - 20 q^{63} - 2 q^{65} - 18 q^{67} - 28 q^{69} + 20 q^{71} + 28 q^{73} + 2 q^{75} + 40 q^{77} + 16 q^{79} + 16 q^{81} + 4 q^{85} + 36 q^{87} + 4 q^{89} - 36 q^{91} + 40 q^{93} + 4 q^{95} + 30 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 8 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^15 + 4 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 4 * q^25 - 4 * q^27 - 4 * q^29 - 4 * q^31 + 8 * q^33 - 4 * q^35 + 6 * q^37 + 12 * q^39 + 16 * q^41 + 4 * q^43 + 8 * q^45 + 12 * q^47 + 20 * q^49 - 36 * q^51 + 10 * q^53 + 4 * q^55 + 2 * q^57 - 20 * q^61 - 20 * q^63 - 2 * q^65 - 18 * q^67 - 28 * q^69 + 20 * q^71 + 28 * q^73 + 2 * q^75 + 40 * q^77 + 16 * q^79 + 16 * q^81 + 4 * q^85 + 36 * q^87 + 4 * q^89 - 36 * q^91 + 40 * q^93 + 4 * q^95 + 30 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more