LMFDB - Embedded newform 6032.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5x^{10} - 9x^{9} + 65x^{8} + 19x^{7} - 298x^{6} + 17x^{5} + 541x^{4} - 60x^{3} - 287x^{2} + 16x + 16 $ x^11 - 5x^10 - 9x^9 + 65x^8 + 19x^7 - 298x^6 + 17x^5 + 541x^4 - 60x^3 - 287x^2 + 16x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.67951$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67951 q^{3} +1.05669 q^{5} +1.62834 q^{7} -0.179232 q^{9} +O(q^{10})$	$q-1.67951 q^{3} +1.05669 q^{5} +1.62834 q^{7} -0.179232 q^{9} +2.87781 q^{11} +1.00000 q^{13} -1.77473 q^{15} +4.55208 q^{17} +7.96080 q^{19} -2.73482 q^{21} +4.44103 q^{23} -3.88340 q^{25} +5.33957 q^{27} -1.00000 q^{29} +5.89630 q^{31} -4.83332 q^{33} +1.72065 q^{35} -5.01014 q^{37} -1.67951 q^{39} +10.8017 q^{41} +2.63770 q^{43} -0.189393 q^{45} -3.52295 q^{47} -4.34852 q^{49} -7.64528 q^{51} +0.493723 q^{53} +3.04096 q^{55} -13.3703 q^{57} -3.14185 q^{59} -7.35305 q^{61} -0.291849 q^{63} +1.05669 q^{65} -8.26867 q^{67} -7.45877 q^{69} -12.9531 q^{71} +2.43670 q^{73} +6.52222 q^{75} +4.68604 q^{77} +11.2895 q^{79} -8.43018 q^{81} +15.2096 q^{83} +4.81015 q^{85} +1.67951 q^{87} +6.35119 q^{89} +1.62834 q^{91} -9.90292 q^{93} +8.41212 q^{95} +12.3439 q^{97} -0.515794 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q + 6 * q^3 - 2 * q^5 + 3 * q^7 + 11 * q^9	$ 11 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 11 q^{9} + 13 q^{11} + 11 q^{13} + 8 q^{15} - 6 q^{17} + 12 q^{19} + q^{21} + 13 q^{23} + 11 q^{25} + 24 q^{27} - 11 q^{29} + 11 q^{31} + 17 q^{33} + 4 q^{35} + 11 q^{37} + 6 q^{39} - 9 q^{41} + 30 q^{43} - 16 q^{45} + q^{47} - 4 q^{49} + 13 q^{51} - 9 q^{53} + q^{55} + 2 q^{57} + 9 q^{59} - 5 q^{61} + 6 q^{63} - 2 q^{65} + 25 q^{67} + 26 q^{71} + 10 q^{73} + 41 q^{75} - 8 q^{77} + 14 q^{79} + 3 q^{81} + 6 q^{83} + 19 q^{85} - 6 q^{87} - 11 q^{89} + 3 q^{91} - 3 q^{93} + 31 q^{95} + 12 q^{97} + 35 q^{99}+O(q^{100}) $ 11 * q + 6 * q^3 - 2 * q^5 + 3 * q^7 + 11 * q^9 + 13 * q^11 + 11 * q^13 + 8 * q^15 - 6 * q^17 + 12 * q^19 + q^21 + 13 * q^23 + 11 * q^25 + 24 * q^27 - 11 * q^29 + 11 * q^31 + 17 * q^33 + 4 * q^35 + 11 * q^37 + 6 * q^39 - 9 * q^41 + 30 * q^43 - 16 * q^45 + q^47 - 4 * q^49 + 13 * q^51 - 9 * q^53 + q^55 + 2 * q^57 + 9 * q^59 - 5 * q^61 + 6 * q^63 - 2 * q^65 + 25 * q^67 + 26 * q^71 + 10 * q^73 + 41 * q^75 - 8 * q^77 + 14 * q^79 + 3 * q^81 + 6 * q^83 + 19 * q^85 - 6 * q^87 - 11 * q^89 + 3 * q^91 - 3 * q^93 + 31 * q^95 + 12 * q^97 + 35 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.67951	−0.969668	−0.484834	−	0.874606i	$-0.661120\pi$
$3$	−1.67951	−0.969668	−0.484834	+	0.874606i	$0.661120\pi$
$4$	0	0
$5$	1.05669	0.472568	0.236284	−	0.971684i	$-0.424070\pi$
$5$	1.05669	0.472568	0.236284	+	0.971684i	$0.424070\pi$
$6$	0	0
$7$	1.62834	0.615454	0.307727	−	0.951475i	$-0.400432\pi$
$7$	1.62834	0.615454	0.307727	+	0.951475i	$0.400432\pi$
$8$	0	0
$9$	−0.179232	−0.0597439
$10$	0	0
$11$	2.87781	0.867692	0.433846	−	0.900987i	$-0.357156\pi$
$11$	2.87781	0.867692	0.433846	+	0.900987i	$0.357156\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.77473	−0.458234
$16$	0	0
$17$	4.55208	1.10404	0.552020	−	0.833831i	$-0.313857\pi$
$17$	4.55208	1.10404	0.552020	+	0.833831i	$0.313857\pi$
$18$	0	0
$19$	7.96080	1.82633	0.913166	−	0.407588i	$-0.133630\pi$
$19$	7.96080	1.82633	0.913166	+	0.407588i	$0.133630\pi$
$20$	0	0
$21$	−2.73482	−0.596786
$22$	0	0
$23$	4.44103	0.926018	0.463009	−	0.886353i	$-0.346770\pi$
$23$	4.44103	0.926018	0.463009	+	0.886353i	$0.346770\pi$
$24$	0	0
$25$	−3.88340	−0.776680
$26$	0	0
$27$	5.33957	1.02760
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.89630	1.05901	0.529504	−	0.848308i	$-0.322378\pi$
$31$	5.89630	1.05901	0.529504	+	0.848308i	$0.322378\pi$
$32$	0	0
$33$	−4.83332	−0.841373
$34$	0	0
$35$	1.72065	0.290843
$36$	0	0
$37$	−5.01014	−0.823661	−0.411831	−	0.911260i	$-0.635111\pi$
$37$	−5.01014	−0.823661	−0.411831	+	0.911260i	$0.635111\pi$
$38$	0	0
$39$	−1.67951	−0.268938
$40$	0	0
$41$	10.8017	1.68694	0.843468	−	0.537179i	$-0.180510\pi$
$41$	10.8017	1.68694	0.843468	+	0.537179i	$0.180510\pi$
$42$	0	0
$43$	2.63770	0.402246	0.201123	−	0.979566i	$-0.435541\pi$
$43$	2.63770	0.402246	0.201123	+	0.979566i	$0.435541\pi$
$44$	0	0
$45$	−0.189393	−0.0282330
$46$	0	0
$47$	−3.52295	−0.513876	−0.256938	−	0.966428i	$-0.582714\pi$
$47$	−3.52295	−0.513876	−0.256938	+	0.966428i	$0.582714\pi$
$48$	0	0
$49$	−4.34852	−0.621217
$50$	0	0
$51$	−7.64528	−1.07055
$52$	0	0
$53$	0.493723	0.0678181	0.0339090	−	0.999425i	$-0.489204\pi$
$53$	0.493723	0.0678181	0.0339090	+	0.999425i	$0.489204\pi$
$54$	0	0
$55$	3.04096	0.410043
$56$	0	0
$57$	−13.3703	−1.77094
$58$	0	0
$59$	−3.14185	−0.409034	−0.204517	−	0.978863i	$-0.565562\pi$
$59$	−3.14185	−0.409034	−0.204517	+	0.978863i	$0.565562\pi$
$60$	0	0
$61$	−7.35305	−0.941462	−0.470731	−	0.882277i	$-0.656010\pi$
$61$	−7.35305	−0.941462	−0.470731	+	0.882277i	$0.656010\pi$
$62$	0	0
$63$	−0.291849	−0.0367696
$64$	0	0
$65$	1.05669	0.131067
$66$	0	0
$67$	−8.26867	−1.01018	−0.505089	−	0.863067i	$-0.668540\pi$
$67$	−8.26867	−1.01018	−0.505089	+	0.863067i	$0.668540\pi$
$68$	0	0
$69$	−7.45877	−0.897931
$70$	0	0
$71$	−12.9531	−1.53725	−0.768624	−	0.639701i	$-0.779058\pi$
$71$	−12.9531	−1.53725	−0.768624	+	0.639701i	$0.779058\pi$
$72$	0	0
$73$	2.43670	0.285194	0.142597	−	0.989781i	$-0.454455\pi$
$73$	2.43670	0.285194	0.142597	+	0.989781i	$0.454455\pi$
$74$	0	0
$75$	6.52222	0.753122
$76$	0	0
$77$	4.68604	0.534024
$78$	0	0
$79$	11.2895	1.27017	0.635087	−	0.772441i	$-0.280964\pi$
$79$	11.2895	1.27017	0.635087	+	0.772441i	$0.280964\pi$
$80$	0	0
$81$	−8.43018	−0.936687
$82$	0	0
$83$	15.2096	1.66947	0.834735	−	0.550653i	$-0.185621\pi$
$83$	15.2096	1.66947	0.834735	+	0.550653i	$0.185621\pi$
$84$	0	0
$85$	4.81015	0.521734
$86$	0	0
$87$	1.67951	0.180063
$88$	0	0
$89$	6.35119	0.673225	0.336612	−	0.941643i	$-0.390719\pi$
$89$	6.35119	0.673225	0.336612	+	0.941643i	$0.390719\pi$
$90$	0	0
$91$	1.62834	0.170696
$92$	0	0
$93$	−9.90292	−1.02689
$94$	0	0
$95$	8.41212	0.863065
$96$	0	0
$97$	12.3439	1.25334	0.626668	−	0.779286i	$-0.284418\pi$
$97$	12.3439	1.25334	0.626668	+	0.779286i	$0.284418\pi$
$98$	0	0
$99$	−0.515794	−0.0518392
$100$	0	0
$101$	−9.54860	−0.950121	−0.475061	−	0.879953i	$-0.657574\pi$
$101$	−9.54860	−0.950121	−0.475061	+	0.879953i	$0.657574\pi$
$102$	0	0
$103$	3.39132	0.334157	0.167079	−	0.985944i	$-0.446567\pi$
$103$	3.39132	0.334157	0.167079	+	0.985944i	$0.446567\pi$
$104$	0	0
$105$	−2.88986	−0.282022
$106$	0	0
$107$	17.6620	1.70745	0.853726	−	0.520723i	$-0.174338\pi$
$107$	17.6620	1.70745	0.853726	+	0.520723i	$0.174338\pi$
$108$	0	0
$109$	3.89873	0.373431	0.186715	−	0.982414i	$-0.440216\pi$
$109$	3.89873	0.373431	0.186715	+	0.982414i	$0.440216\pi$
$110$	0	0
$111$	8.41460	0.798678
$112$	0	0
$113$	−17.3734	−1.63435	−0.817177	−	0.576386i	$-0.804462\pi$
$113$	−17.3734	−1.63435	−0.817177	+	0.576386i	$0.804462\pi$
$114$	0	0
$115$	4.69281	0.437606
$116$	0	0
$117$	−0.179232	−0.0165700
$118$	0	0
$119$	7.41232	0.679486
$120$	0	0
$121$	−2.71823	−0.247111
$122$	0	0
$123$	−18.1416	−1.63577
$124$	0	0
$125$	−9.38703	−0.839601
$126$	0	0
$127$	−17.2648	−1.53201	−0.766003	−	0.642837i	$-0.777757\pi$
$127$	−17.2648	−1.53201	−0.766003	+	0.642837i	$0.777757\pi$
$128$	0	0
$129$	−4.43006	−0.390045
$130$	0	0
$131$	2.34755	0.205107	0.102553	−	0.994727i	$-0.467299\pi$
$131$	2.34755	0.205107	0.102553	+	0.994727i	$0.467299\pi$
$132$	0	0
$133$	12.9629	1.12402
$134$	0	0
$135$	5.64228	0.485610
$136$	0	0
$137$	2.00979	0.171708	0.0858540	−	0.996308i	$-0.472638\pi$
$137$	2.00979	0.171708	0.0858540	+	0.996308i	$0.472638\pi$
$138$	0	0
$139$	−7.00401	−0.594072	−0.297036	−	0.954866i	$-0.595998\pi$
$139$	−7.00401	−0.594072	−0.297036	+	0.954866i	$0.595998\pi$
$140$	0	0
$141$	5.91685	0.498289
$142$	0	0
$143$	2.87781	0.240654
$144$	0	0
$145$	−1.05669	−0.0877536
$146$	0	0
$147$	7.30340	0.602374
$148$	0	0
$149$	2.04116	0.167218	0.0836090	−	0.996499i	$-0.473355\pi$
$149$	2.04116	0.167218	0.0836090	+	0.996499i	$0.473355\pi$
$150$	0	0
$151$	−9.45335	−0.769302	−0.384651	−	0.923062i	$-0.625678\pi$
$151$	−9.45335	−0.769302	−0.384651	+	0.923062i	$0.625678\pi$
$152$	0	0
$153$	−0.815876	−0.0659597
$154$	0	0
$155$	6.23058	0.500452
$156$	0	0
$157$	3.68280	0.293919	0.146960	−	0.989142i	$-0.453051\pi$
$157$	3.68280	0.293919	0.146960	+	0.989142i	$0.453051\pi$
$158$	0	0
$159$	−0.829215	−0.0657610
$160$	0	0
$161$	7.23149	0.569921
$162$	0	0
$163$	−9.29214	−0.727817	−0.363908	−	0.931435i	$-0.618558\pi$
$163$	−9.29214	−0.727817	−0.363908	+	0.931435i	$0.618558\pi$
$164$	0	0
$165$	−5.10734	−0.397606
$166$	0	0
$167$	−22.6156	−1.75004	−0.875022	−	0.484083i	$-0.839153\pi$
$167$	−22.6156	−1.75004	−0.875022	+	0.484083i	$0.839153\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.42683	−0.109112
$172$	0	0
$173$	9.24953	0.703229	0.351614	−	0.936145i	$-0.385633\pi$
$173$	9.24953	0.703229	0.351614	+	0.936145i	$0.385633\pi$
$174$	0	0
$175$	−6.32348	−0.478010
$176$	0	0
$177$	5.27678	0.396627
$178$	0	0
$179$	−17.9435	−1.34116	−0.670580	−	0.741837i	$-0.733955\pi$
$179$	−17.9435	−1.34116	−0.670580	+	0.741837i	$0.733955\pi$
$180$	0	0
$181$	5.46888	0.406499	0.203249	−	0.979127i	$-0.434850\pi$
$181$	5.46888	0.406499	0.203249	+	0.979127i	$0.434850\pi$
$182$	0	0
$183$	12.3496	0.912906
$184$	0	0
$185$	−5.29418	−0.389236
$186$	0	0
$187$	13.1000	0.957967
$188$	0	0
$189$	8.69461	0.632440
$190$	0	0
$191$	19.9011	1.43999	0.719996	−	0.693978i	$-0.244144\pi$
$191$	19.9011	1.43999	0.719996	+	0.693978i	$0.244144\pi$
$192$	0	0
$193$	−23.6597	−1.70306	−0.851530	−	0.524306i	$-0.824325\pi$
$193$	−23.6597	−1.70306	−0.851530	+	0.524306i	$0.824325\pi$
$194$	0	0
$195$	−1.77473	−0.127091
$196$	0	0
$197$	1.22630	0.0873702	0.0436851	−	0.999045i	$-0.486090\pi$
$197$	1.22630	0.0873702	0.0436851	+	0.999045i	$0.486090\pi$
$198$	0	0
$199$	2.88291	0.204364	0.102182	−	0.994766i	$-0.467418\pi$
$199$	2.88291	0.204364	0.102182	+	0.994766i	$0.467418\pi$
$200$	0	0
$201$	13.8873	0.979538
$202$	0	0
$203$	−1.62834	−0.114287
$204$	0	0
$205$	11.4140	0.797192
$206$	0	0
$207$	−0.795973	−0.0553239
$208$	0	0
$209$	22.9096	1.58469
$210$	0	0
$211$	20.9617	1.44306	0.721531	−	0.692382i	$-0.243439\pi$
$211$	20.9617	1.44306	0.721531	+	0.692382i	$0.243439\pi$
$212$	0	0
$213$	21.7549	1.49062
$214$	0	0
$215$	2.78724	0.190089
$216$	0	0
$217$	9.60117	0.651770
$218$	0	0
$219$	−4.09247	−0.276543
$220$	0	0
$221$	4.55208	0.306206
$222$	0	0
$223$	15.3071	1.02504	0.512520	−	0.858675i	$-0.328712\pi$
$223$	15.3071	1.02504	0.512520	+	0.858675i	$0.328712\pi$
$224$	0	0
$225$	0.696028	0.0464019
$226$	0	0
$227$	26.0640	1.72993	0.864965	−	0.501832i	$-0.167341\pi$
$227$	26.0640	1.72993	0.864965	+	0.501832i	$0.167341\pi$
$228$	0	0
$229$	5.17798	0.342170	0.171085	−	0.985256i	$-0.445273\pi$
$229$	5.17798	0.342170	0.171085	+	0.985256i	$0.445273\pi$
$230$	0	0
$231$	−7.87027	−0.517826
$232$	0	0
$233$	−5.91944	−0.387795	−0.193898	−	0.981022i	$-0.562113\pi$
$233$	−5.91944	−0.387795	−0.193898	+	0.981022i	$0.562113\pi$
$234$	0	0
$235$	−3.72268	−0.242841
$236$	0	0
$237$	−18.9610	−1.23165
$238$	0	0
$239$	−3.22031	−0.208305	−0.104152	−	0.994561i	$-0.533213\pi$
$239$	−3.22031	−0.208305	−0.104152	+	0.994561i	$0.533213\pi$
$240$	0	0
$241$	−15.1579	−0.976403	−0.488202	−	0.872731i	$-0.662347\pi$
$241$	−15.1579	−0.976403	−0.488202	+	0.872731i	$0.662347\pi$
$242$	0	0
$243$	−1.86008	−0.119324
$244$	0	0
$245$	−4.59505	−0.293567
$246$	0	0
$247$	7.96080	0.506533
$248$	0	0
$249$	−25.5447	−1.61883
$250$	0	0
$251$	−16.8126	−1.06120	−0.530601	−	0.847621i	$-0.678034\pi$
$251$	−16.8126	−1.06120	−0.530601	+	0.847621i	$0.678034\pi$
$252$	0	0
$253$	12.7804	0.803498
$254$	0	0
$255$	−8.07872	−0.505909
$256$	0	0
$257$	−21.4515	−1.33811	−0.669055	−	0.743213i	$-0.733301\pi$
$257$	−21.4515	−1.33811	−0.669055	+	0.743213i	$0.733301\pi$
$258$	0	0
$259$	−8.15819	−0.506925
$260$	0	0
$261$	0.179232	0.0110942
$262$	0	0
$263$	−13.3707	−0.824474	−0.412237	−	0.911077i	$-0.635252\pi$
$263$	−13.3707	−0.824474	−0.412237	+	0.911077i	$0.635252\pi$
$264$	0	0
$265$	0.521714	0.0320486
$266$	0	0
$267$	−10.6669	−0.652805
$268$	0	0
$269$	18.1691	1.10779	0.553896	−	0.832586i	$-0.313141\pi$
$269$	18.1691	1.10779	0.553896	+	0.832586i	$0.313141\pi$
$270$	0	0
$271$	4.72677	0.287131	0.143565	−	0.989641i	$-0.454143\pi$
$271$	4.72677	0.287131	0.143565	+	0.989641i	$0.454143\pi$
$272$	0	0
$273$	−2.73482	−0.165519
$274$	0	0
$275$	−11.1757	−0.673919
$276$	0	0
$277$	28.8656	1.73436	0.867182	−	0.497991i	$-0.165929\pi$
$277$	28.8656	1.73436	0.867182	+	0.497991i	$0.165929\pi$
$278$	0	0
$279$	−1.05680	−0.0632692
$280$	0	0
$281$	8.79713	0.524793	0.262396	−	0.964960i	$-0.415487\pi$
$281$	8.79713	0.524793	0.262396	+	0.964960i	$0.415487\pi$
$282$	0	0
$283$	24.4181	1.45151	0.725754	−	0.687955i	$-0.241491\pi$
$283$	24.4181	1.45151	0.725754	+	0.687955i	$0.241491\pi$
$284$	0	0
$285$	−14.1283	−0.836887
$286$	0	0
$287$	17.5887	1.03823
$288$	0	0
$289$	3.72141	0.218907
$290$	0	0
$291$	−20.7318	−1.21532
$292$	0	0
$293$	17.2905	1.01012	0.505062	−	0.863083i	$-0.331470\pi$
$293$	17.2905	1.01012	0.505062	+	0.863083i	$0.331470\pi$
$294$	0	0
$295$	−3.31997	−0.193296
$296$	0	0
$297$	15.3662	0.891640
$298$	0	0
$299$	4.44103	0.256831
$300$	0	0
$301$	4.29507	0.247564
$302$	0	0
$303$	16.0370	0.921302
$304$	0	0
$305$	−7.76992	−0.444904
$306$	0	0
$307$	20.5631	1.17360	0.586799	−	0.809733i	$-0.300388\pi$
$307$	20.5631	1.17360	0.586799	+	0.809733i	$0.300388\pi$
$308$	0	0
$309$	−5.69578	−0.324022
$310$	0	0
$311$	20.5189	1.16352	0.581759	−	0.813361i	$-0.302365\pi$
$311$	20.5189	1.16352	0.581759	+	0.813361i	$0.302365\pi$
$312$	0	0
$313$	−13.4762	−0.761719	−0.380860	−	0.924633i	$-0.624372\pi$
$313$	−13.4762	−0.761719	−0.380860	+	0.924633i	$0.624372\pi$
$314$	0	0
$315$	−0.308395	−0.0173761
$316$	0	0
$317$	−30.9122	−1.73620	−0.868101	−	0.496388i	$-0.834659\pi$
$317$	−30.9122	−1.73620	−0.868101	+	0.496388i	$0.834659\pi$
$318$	0	0
$319$	−2.87781	−0.161126
$320$	0	0
$321$	−29.6636	−1.65566
$322$	0	0
$323$	36.2382	2.01635
$324$	0	0
$325$	−3.88340	−0.215412
$326$	0	0
$327$	−6.54798	−0.362104
$328$	0	0
$329$	−5.73656	−0.316267
$330$	0	0
$331$	−2.50567	−0.137724	−0.0688619	−	0.997626i	$-0.521937\pi$
$331$	−2.50567	−0.137724	−0.0688619	+	0.997626i	$0.521937\pi$
$332$	0	0
$333$	0.897975	0.0492087
$334$	0	0
$335$	−8.73744	−0.477378
$336$	0	0
$337$	−10.5862	−0.576669	−0.288335	−	0.957530i	$-0.593102\pi$
$337$	−10.5862	−0.576669	−0.288335	+	0.957530i	$0.593102\pi$
$338$	0	0
$339$	29.1789	1.58478
$340$	0	0
$341$	16.9684	0.918891
$342$	0	0
$343$	−18.4792	−0.997784
$344$	0	0
$345$	−7.88163	−0.424333
$346$	0	0
$347$	−0.537026	−0.0288291	−0.0144145	−	0.999896i	$-0.504588\pi$
$347$	−0.537026	−0.0288291	−0.0144145	+	0.999896i	$0.504588\pi$
$348$	0	0
$349$	17.6019	0.942207	0.471104	−	0.882078i	$-0.343856\pi$
$349$	17.6019	0.942207	0.471104	+	0.882078i	$0.343856\pi$
$350$	0	0
$351$	5.33957	0.285005
$352$	0	0
$353$	−4.53136	−0.241180	−0.120590	−	0.992702i	$-0.538479\pi$
$353$	−4.53136	−0.241180	−0.120590	+	0.992702i	$0.538479\pi$
$354$	0	0
$355$	−13.6874	−0.726453
$356$	0	0
$357$	−12.4491	−0.658876
$358$	0	0
$359$	−21.1864	−1.11818	−0.559088	−	0.829108i	$-0.688849\pi$
$359$	−21.1864	−1.11818	−0.559088	+	0.829108i	$0.688849\pi$
$360$	0	0
$361$	44.3743	2.33549
$362$	0	0
$363$	4.56530	0.239616
$364$	0	0
$365$	2.57484	0.134773
$366$	0	0
$367$	3.90486	0.203832	0.101916	−	0.994793i	$-0.467503\pi$
$367$	3.90486	0.203832	0.101916	+	0.994793i	$0.467503\pi$
$368$	0	0
$369$	−1.93600	−0.100784
$370$	0	0
$371$	0.803947	0.0417389
$372$	0	0
$373$	17.5480	0.908602	0.454301	−	0.890848i	$-0.349889\pi$
$373$	17.5480	0.908602	0.454301	+	0.890848i	$0.349889\pi$
$374$	0	0
$375$	15.7656	0.814135
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	1.54767	0.0794983	0.0397491	−	0.999210i	$-0.487344\pi$
$379$	1.54767	0.0794983	0.0397491	+	0.999210i	$0.487344\pi$
$380$	0	0
$381$	28.9965	1.48554
$382$	0	0
$383$	9.43517	0.482115	0.241057	−	0.970511i	$-0.422506\pi$
$383$	9.43517	0.482115	0.241057	+	0.970511i	$0.422506\pi$
$384$	0	0
$385$	4.95171	0.252362
$386$	0	0
$387$	−0.472760	−0.0240317
$388$	0	0
$389$	21.8485	1.10776	0.553881	−	0.832596i	$-0.313146\pi$
$389$	21.8485	1.10776	0.553881	+	0.832596i	$0.313146\pi$
$390$	0	0
$391$	20.2159	1.02236
$392$	0	0
$393$	−3.94275	−0.198886
$394$	0	0
$395$	11.9296	0.600243
$396$	0	0
$397$	17.1188	0.859166	0.429583	−	0.903027i	$-0.358661\pi$
$397$	17.1188	0.859166	0.429583	+	0.903027i	$0.358661\pi$
$398$	0	0
$399$	−21.7713	−1.08993
$400$	0	0
$401$	4.22517	0.210995	0.105498	−	0.994420i	$-0.466356\pi$
$401$	4.22517	0.210995	0.105498	+	0.994420i	$0.466356\pi$
$402$	0	0
$403$	5.89630	0.293716
$404$	0	0
$405$	−8.90812	−0.442648
$406$	0	0
$407$	−14.4182	−0.714684
$408$	0	0
$409$	17.6063	0.870573	0.435287	−	0.900292i	$-0.356647\pi$
$409$	17.6063	0.870573	0.435287	+	0.900292i	$0.356647\pi$
$410$	0	0
$411$	−3.37547	−0.166500
$412$	0	0
$413$	−5.11599	−0.251741
$414$	0	0
$415$	16.0719	0.788937
$416$	0	0
$417$	11.7633	0.576053
$418$	0	0
$419$	12.2556	0.598725	0.299363	−	0.954139i	$-0.403226\pi$
$419$	12.2556	0.598725	0.299363	+	0.954139i	$0.403226\pi$
$420$	0	0
$421$	−21.4145	−1.04368	−0.521839	−	0.853044i	$-0.674754\pi$
$421$	−21.4145	−1.04368	−0.521839	+	0.853044i	$0.674754\pi$
$422$	0	0
$423$	0.631425	0.0307009
$424$	0	0
$425$	−17.6775	−0.857486
$426$	0	0
$427$	−11.9732	−0.579426
$428$	0	0
$429$	−4.83332	−0.233355
$430$	0	0
$431$	−18.7964	−0.905392	−0.452696	−	0.891665i	$-0.649538\pi$
$431$	−18.7964	−0.905392	−0.452696	+	0.891665i	$0.649538\pi$
$432$	0	0
$433$	41.4697	1.99291	0.996453	−	0.0841486i	$-0.0268171\pi$
$433$	41.4697	1.99291	0.996453	+	0.0841486i	$0.0268171\pi$
$434$	0	0
$435$	1.77473	0.0850919
$436$	0	0
$437$	35.3541	1.69122
$438$	0	0
$439$	−16.4983	−0.787421	−0.393710	−	0.919235i	$-0.628809\pi$
$439$	−16.4983	−0.787421	−0.393710	+	0.919235i	$0.628809\pi$
$440$	0	0
$441$	0.779392	0.0371139
$442$	0	0
$443$	7.62821	0.362427	0.181214	−	0.983444i	$-0.441997\pi$
$443$	7.62821	0.362427	0.181214	+	0.983444i	$0.441997\pi$
$444$	0	0
$445$	6.71126	0.318144
$446$	0	0
$447$	−3.42815	−0.162146
$448$	0	0
$449$	7.79533	0.367885	0.183942	−	0.982937i	$-0.441114\pi$
$449$	7.79533	0.367885	0.183942	+	0.982937i	$0.441114\pi$
$450$	0	0
$451$	31.0851	1.46374
$452$	0	0
$453$	15.8770	0.745968
$454$	0	0
$455$	1.72065	0.0806654
$456$	0	0
$457$	−22.7697	−1.06512	−0.532562	−	0.846391i	$-0.678771\pi$
$457$	−22.7697	−1.06512	−0.532562	+	0.846391i	$0.678771\pi$
$458$	0	0
$459$	24.3061	1.13451
$460$	0	0
$461$	−6.73468	−0.313665	−0.156833	−	0.987625i	$-0.550128\pi$
$461$	−6.73468	−0.313665	−0.156833	+	0.987625i	$0.550128\pi$
$462$	0	0
$463$	26.9596	1.25292	0.626459	−	0.779454i	$-0.284504\pi$
$463$	26.9596	1.25292	0.626459	+	0.779454i	$0.284504\pi$
$464$	0	0
$465$	−10.4644	−0.485273
$466$	0	0
$467$	20.6418	0.955187	0.477593	−	0.878581i	$-0.341509\pi$
$467$	20.6418	0.955187	0.477593	+	0.878581i	$0.341509\pi$
$468$	0	0
$469$	−13.4642	−0.621718
$470$	0	0
$471$	−6.18531	−0.285004
$472$	0	0
$473$	7.59080	0.349026
$474$	0	0
$475$	−30.9149	−1.41848
$476$	0	0
$477$	−0.0884907	−0.00405171
$478$	0	0
$479$	−10.0402	−0.458750	−0.229375	−	0.973338i	$-0.573668\pi$
$479$	−10.0402	−0.458750	−0.229375	+	0.973338i	$0.573668\pi$
$480$	0	0
$481$	−5.01014	−0.228443
$482$	0	0
$483$	−12.1454	−0.552635
$484$	0	0
$485$	13.0437	0.592286
$486$	0	0
$487$	−10.0039	−0.453319	−0.226660	−	0.973974i	$-0.572781\pi$
$487$	−10.0039	−0.453319	−0.226660	+	0.973974i	$0.572781\pi$
$488$	0	0
$489$	15.6063	0.705741
$490$	0	0
$491$	−7.22008	−0.325837	−0.162919	−	0.986639i	$-0.552091\pi$
$491$	−7.22008	−0.325837	−0.162919	+	0.986639i	$0.552091\pi$
$492$	0	0
$493$	−4.55208	−0.205015
$494$	0	0
$495$	−0.545036	−0.0244975
$496$	0	0
$497$	−21.0920	−0.946104
$498$	0	0
$499$	−37.4314	−1.67566	−0.837829	−	0.545932i	$-0.816176\pi$
$499$	−37.4314	−1.67566	−0.837829	+	0.545932i	$0.816176\pi$
$500$	0	0
$501$	37.9831	1.69696
$502$	0	0
$503$	16.7577	0.747191	0.373595	−	0.927592i	$-0.378125\pi$
$503$	16.7577	0.747191	0.373595	+	0.927592i	$0.378125\pi$
$504$	0	0
$505$	−10.0899	−0.448996
$506$	0	0
$507$	−1.67951	−0.0745899
$508$	0	0
$509$	−8.51357	−0.377357	−0.188679	−	0.982039i	$-0.560420\pi$
$509$	−8.51357	−0.377357	−0.188679	+	0.982039i	$0.560420\pi$
$510$	0	0
$511$	3.96776	0.175524
$512$	0	0
$513$	42.5072	1.87674
$514$	0	0
$515$	3.58359	0.157912
$516$	0	0
$517$	−10.1384	−0.445886
$518$	0	0
$519$	−15.5347	−0.681898
$520$	0	0
$521$	−12.0222	−0.526702	−0.263351	−	0.964700i	$-0.584828\pi$
$521$	−12.0222	−0.526702	−0.263351	+	0.964700i	$0.584828\pi$
$522$	0	0
$523$	40.8799	1.78755	0.893776	−	0.448514i	$-0.148047\pi$
$523$	40.8799	1.78755	0.893776	+	0.448514i	$0.148047\pi$
$524$	0	0
$525$	10.6204	0.463511
$526$	0	0
$527$	26.8404	1.16919
$528$	0	0
$529$	−3.27726	−0.142490
$530$	0	0
$531$	0.563118	0.0244373
$532$	0	0
$533$	10.8017	0.467872
$534$	0	0
$535$	18.6633	0.806886
$536$	0	0
$537$	30.1364	1.30048
$538$	0	0
$539$	−12.5142	−0.539025
$540$	0	0
$541$	−37.9098	−1.62987	−0.814935	−	0.579552i	$-0.803228\pi$
$541$	−37.9098	−1.62987	−0.814935	+	0.579552i	$0.803228\pi$
$542$	0	0
$543$	−9.18506	−0.394169
$544$	0	0
$545$	4.11976	0.176471
$546$	0	0
$547$	−31.1530	−1.33200	−0.666002	−	0.745950i	$-0.731996\pi$
$547$	−31.1530	−1.33200	−0.666002	+	0.745950i	$0.731996\pi$
$548$	0	0
$549$	1.31790	0.0562466
$550$	0	0
$551$	−7.96080	−0.339141
$552$	0	0
$553$	18.3832	0.781733
$554$	0	0
$555$	8.89165	0.377429
$556$	0	0
$557$	−8.71322	−0.369191	−0.184595	−	0.982815i	$-0.559097\pi$
$557$	−8.71322	−0.369191	−0.184595	+	0.982815i	$0.559097\pi$
$558$	0	0
$559$	2.63770	0.111563
$560$	0	0
$561$	−22.0016	−0.928910
$562$	0	0
$563$	8.72398	0.367672	0.183836	−	0.982957i	$-0.441148\pi$
$563$	8.72398	0.367672	0.183836	+	0.982957i	$0.441148\pi$
$564$	0	0
$565$	−18.3584	−0.772343
$566$	0	0
$567$	−13.7272	−0.576487
$568$	0	0
$569$	−9.58363	−0.401767	−0.200883	−	0.979615i	$-0.564381\pi$
$569$	−9.58363	−0.401767	−0.200883	+	0.979615i	$0.564381\pi$
$570$	0	0
$571$	15.0740	0.630826	0.315413	−	0.948954i	$-0.397857\pi$
$571$	15.0740	0.630826	0.315413	+	0.948954i	$0.397857\pi$
$572$	0	0
$573$	−33.4242	−1.39631
$574$	0	0
$575$	−17.2463	−0.719220
$576$	0	0
$577$	0.730795	0.0304234	0.0152117	−	0.999884i	$-0.495158\pi$
$577$	0.730795	0.0304234	0.0152117	+	0.999884i	$0.495158\pi$
$578$	0	0
$579$	39.7367	1.65140
$580$	0	0
$581$	24.7663	1.02748
$582$	0	0
$583$	1.42084	0.0588452
$584$	0	0
$585$	−0.189393	−0.00783043
$586$	0	0
$587$	42.4518	1.75217	0.876087	−	0.482154i	$-0.160145\pi$
$587$	42.4518	1.75217	0.876087	+	0.482154i	$0.160145\pi$
$588$	0	0
$589$	46.9393	1.93410
$590$	0	0
$591$	−2.05959	−0.0847201
$592$	0	0
$593$	−5.15038	−0.211501	−0.105750	−	0.994393i	$-0.533724\pi$
$593$	−5.15038	−0.211501	−0.105750	+	0.994393i	$0.533724\pi$
$594$	0	0
$595$	7.83255	0.321103
$596$	0	0
$597$	−4.84189	−0.198165
$598$	0	0
$599$	48.4833	1.98098	0.990488	−	0.137602i	$-0.0439396\pi$
$599$	48.4833	1.98098	0.990488	+	0.137602i	$0.0439396\pi$
$600$	0	0
$601$	4.99225	0.203638	0.101819	−	0.994803i	$-0.467534\pi$
$601$	4.99225	0.203638	0.101819	+	0.994803i	$0.467534\pi$
$602$	0	0
$603$	1.48201	0.0603520
$604$	0	0
$605$	−2.87233	−0.116777
$606$	0	0
$607$	−12.8432	−0.521290	−0.260645	−	0.965435i	$-0.583935\pi$
$607$	−12.8432	−0.521290	−0.260645	+	0.965435i	$0.583935\pi$
$608$	0	0
$609$	2.73482	0.110820
$610$	0	0
$611$	−3.52295	−0.142524
$612$	0	0
$613$	39.9819	1.61485	0.807426	−	0.589968i	$-0.200860\pi$
$613$	39.9819	1.61485	0.807426	+	0.589968i	$0.200860\pi$
$614$	0	0
$615$	−19.1701	−0.773011
$616$	0	0
$617$	6.37143	0.256504	0.128252	−	0.991742i	$-0.459063\pi$
$617$	6.37143	0.256504	0.128252	+	0.991742i	$0.459063\pi$
$618$	0	0
$619$	−13.5704	−0.545440	−0.272720	−	0.962093i	$-0.587923\pi$
$619$	−13.5704	−0.545440	−0.272720	+	0.962093i	$0.587923\pi$
$620$	0	0
$621$	23.7132	0.951576
$622$	0	0
$623$	10.3419	0.414339
$624$	0	0
$625$	9.49779	0.379911
$626$	0	0
$627$	−38.4771	−1.53663
$628$	0	0
$629$	−22.8065	−0.909356
$630$	0	0
$631$	−16.7739	−0.667760	−0.333880	−	0.942616i	$-0.608358\pi$
$631$	−16.7739	−0.667760	−0.333880	+	0.942616i	$0.608358\pi$
$632$	0	0
$633$	−35.2055	−1.39929
$634$	0	0
$635$	−18.2436	−0.723977
$636$	0	0
$637$	−4.34852	−0.172295
$638$	0	0
$639$	2.32160	0.0918411
$640$	0	0
$641$	−43.2504	−1.70829	−0.854144	−	0.520036i	$-0.825919\pi$
$641$	−43.2504	−1.70829	−0.854144	+	0.520036i	$0.825919\pi$
$642$	0	0
$643$	37.9344	1.49598	0.747992	−	0.663707i	$-0.231018\pi$
$643$	37.9344	1.49598	0.747992	+	0.663707i	$0.231018\pi$
$644$	0	0
$645$	−4.68122	−0.184323
$646$	0	0
$647$	−48.7911	−1.91817	−0.959087	−	0.283111i	$-0.908633\pi$
$647$	−48.7911	−1.91817	−0.959087	+	0.283111i	$0.908633\pi$
$648$	0	0
$649$	−9.04163	−0.354915
$650$	0	0
$651$	−16.1253	−0.632000
$652$	0	0
$653$	9.63500	0.377047	0.188523	−	0.982069i	$-0.439630\pi$
$653$	9.63500	0.377047	0.188523	+	0.982069i	$0.439630\pi$
$654$	0	0
$655$	2.48065	0.0969268
$656$	0	0
$657$	−0.436733	−0.0170386
$658$	0	0
$659$	−29.4057	−1.14548	−0.572742	−	0.819736i	$-0.694120\pi$
$659$	−29.4057	−1.14548	−0.572742	+	0.819736i	$0.694120\pi$
$660$	0	0
$661$	0.0700922	0.00272627	0.00136313	−	0.999999i	$-0.499566\pi$
$661$	0.0700922	0.00272627	0.00136313	+	0.999999i	$0.499566\pi$
$662$	0	0
$663$	−7.64528	−0.296918
$664$	0	0
$665$	13.6978	0.531177
$666$	0	0
$667$	−4.44103	−0.171957
$668$	0	0
$669$	−25.7085	−0.993948
$670$	0	0
$671$	−21.1607	−0.816899
$672$	0	0
$673$	23.0643	0.889062	0.444531	−	0.895763i	$-0.353370\pi$
$673$	23.0643	0.889062	0.444531	+	0.895763i	$0.353370\pi$
$674$	0	0
$675$	−20.7357	−0.798116
$676$	0	0
$677$	44.1568	1.69708	0.848542	−	0.529128i	$-0.177481\pi$
$677$	44.1568	1.69708	0.848542	+	0.529128i	$0.177481\pi$
$678$	0	0
$679$	20.1001	0.771370
$680$	0	0
$681$	−43.7749	−1.67746
$682$	0	0
$683$	13.3193	0.509647	0.254824	−	0.966988i	$-0.417983\pi$
$683$	13.3193	0.509647	0.254824	+	0.966988i	$0.417983\pi$
$684$	0	0
$685$	2.12373	0.0811436
$686$	0	0
$687$	−8.69649	−0.331792
$688$	0	0
$689$	0.493723	0.0188093
$690$	0	0
$691$	−7.05362	−0.268333	−0.134166	−	0.990959i	$-0.542836\pi$
$691$	−7.05362	−0.268333	−0.134166	+	0.990959i	$0.542836\pi$
$692$	0	0
$693$	−0.839886	−0.0319046
$694$	0	0
$695$	−7.40109	−0.280739
$696$	0	0
$697$	49.1700	1.86245
$698$	0	0
$699$	9.94178	0.376033
$700$	0	0
$701$	−20.3112	−0.767143	−0.383571	−	0.923511i	$-0.625306\pi$
$701$	−20.3112	−0.767143	−0.383571	+	0.923511i	$0.625306\pi$
$702$	0	0
$703$	−39.8847	−1.50428
$704$	0	0
$705$	6.25230	0.235475
$706$	0	0
$707$	−15.5483	−0.584755
$708$	0	0
$709$	30.2402	1.13570	0.567848	−	0.823133i	$-0.307776\pi$
$709$	30.2402	1.13570	0.567848	+	0.823133i	$0.307776\pi$
$710$	0	0
$711$	−2.02344	−0.0758850
$712$	0	0
$713$	26.1856	0.980660
$714$	0	0
$715$	3.04096	0.113725
$716$	0	0
$717$	5.40856	0.201986
$718$	0	0
$719$	−24.0661	−0.897516	−0.448758	−	0.893653i	$-0.648133\pi$
$719$	−24.0661	−0.897516	−0.448758	+	0.893653i	$0.648133\pi$
$720$	0	0
$721$	5.52222	0.205658
$722$	0	0
$723$	25.4578	0.946787
$724$	0	0
$725$	3.88340	0.144226
$726$	0	0
$727$	−36.3860	−1.34948	−0.674740	−	0.738055i	$-0.735744\pi$
$727$	−36.3860	−1.34948	−0.674740	+	0.738055i	$0.735744\pi$
$728$	0	0
$729$	28.4146	1.05239
$730$	0	0
$731$	12.0070	0.444096
$732$	0	0
$733$	−27.0836	−1.00036	−0.500178	−	0.865922i	$-0.666732\pi$
$733$	−27.0836	−1.00036	−0.500178	+	0.865922i	$0.666732\pi$
$734$	0	0
$735$	7.71745	0.284663
$736$	0	0
$737$	−23.7956	−0.876523
$738$	0	0
$739$	−36.2282	−1.33268	−0.666339	−	0.745649i	$-0.732139\pi$
$739$	−36.2282	−1.33268	−0.666339	+	0.745649i	$0.732139\pi$
$740$	0	0
$741$	−13.3703	−0.491169
$742$	0	0
$743$	−4.65745	−0.170865	−0.0854327	−	0.996344i	$-0.527227\pi$
$743$	−4.65745	−0.170865	−0.0854327	+	0.996344i	$0.527227\pi$
$744$	0	0
$745$	2.15688	0.0790218
$746$	0	0
$747$	−2.72604	−0.0997405
$748$	0	0
$749$	28.7597	1.05086
$750$	0	0
$751$	38.7161	1.41277	0.706385	−	0.707828i	$-0.250325\pi$
$751$	38.7161	1.41277	0.706385	+	0.707828i	$0.250325\pi$
$752$	0	0
$753$	28.2370	1.02901
$754$	0	0
$755$	−9.98929	−0.363547
$756$	0	0
$757$	−30.6123	−1.11262	−0.556312	−	0.830973i	$-0.687784\pi$
$757$	−30.6123	−1.11262	−0.556312	+	0.830973i	$0.687784\pi$
$758$	0	0
$759$	−21.4649	−0.779127
$760$	0	0
$761$	52.8581	1.91610	0.958052	−	0.286594i	$-0.0925229\pi$
$761$	52.8581	1.91610	0.958052	+	0.286594i	$0.0925229\pi$
$762$	0	0
$763$	6.34845	0.229829
$764$	0	0
$765$	−0.862131	−0.0311704
$766$	0	0
$767$	−3.14185	−0.113446
$768$	0	0
$769$	8.50091	0.306551	0.153275	−	0.988184i	$-0.451018\pi$
$769$	8.50091	0.306551	0.153275	+	0.988184i	$0.451018\pi$
$770$	0	0
$771$	36.0282	1.29752
$772$	0	0
$773$	−50.3579	−1.81125	−0.905624	−	0.424081i	$-0.860597\pi$
$773$	−50.3579	−1.81125	−0.905624	+	0.424081i	$0.860597\pi$
$774$	0	0
$775$	−22.8977	−0.822509
$776$	0	0
$777$	13.7018	0.491549
$778$	0	0
$779$	85.9898	3.08091
$780$	0	0
$781$	−37.2765	−1.33386
$782$	0	0
$783$	−5.33957	−0.190820
$784$	0	0
$785$	3.89159	0.138897
$786$	0	0
$787$	5.45482	0.194443	0.0972217	−	0.995263i	$-0.469004\pi$
$787$	5.45482	0.194443	0.0972217	+	0.995263i	$0.469004\pi$
$788$	0	0
$789$	22.4563	0.799466
$790$	0	0
$791$	−28.2898	−1.00587
$792$	0	0
$793$	−7.35305	−0.261115
$794$	0	0
$795$	−0.876226	−0.0310765
$796$	0	0
$797$	−14.7960	−0.524102	−0.262051	−	0.965054i	$-0.584399\pi$
$797$	−14.7960	−0.524102	−0.262051	+	0.965054i	$0.584399\pi$
$798$	0	0
$799$	−16.0368	−0.567340
$800$	0	0
$801$	−1.13833	−0.0402211
$802$	0	0
$803$	7.01234	0.247460
$804$	0	0
$805$	7.64147	0.269326
$806$	0	0
$807$	−30.5153	−1.07419
$808$	0	0
$809$	4.40493	0.154869	0.0774345	−	0.996997i	$-0.475327\pi$
$809$	4.40493	0.154869	0.0774345	+	0.996997i	$0.475327\pi$
$810$	0	0
$811$	15.4668	0.543113	0.271557	−	0.962422i	$-0.412462\pi$
$811$	15.4668	0.543113	0.271557	+	0.962422i	$0.412462\pi$
$812$	0	0
$813$	−7.93867	−0.278421
$814$	0	0
$815$	−9.81895	−0.343943
$816$	0	0
$817$	20.9982	0.734635
$818$	0	0
$819$	−0.291849	−0.0101980
$820$	0	0
$821$	−52.2048	−1.82196	−0.910981	−	0.412449i	$-0.864674\pi$
$821$	−52.2048	−1.82196	−0.910981	+	0.412449i	$0.864674\pi$
$822$	0	0
$823$	−38.8665	−1.35480	−0.677401	−	0.735614i	$-0.736894\pi$
$823$	−38.8665	−1.35480	−0.677401	+	0.735614i	$0.736894\pi$
$824$	0	0
$825$	18.7697	0.653477
$826$	0	0
$827$	−26.4297	−0.919052	−0.459526	−	0.888164i	$-0.651981\pi$
$827$	−26.4297	−0.919052	−0.459526	+	0.888164i	$0.651981\pi$
$828$	0	0
$829$	39.4756	1.37104	0.685521	−	0.728053i	$-0.259574\pi$
$829$	39.4756	1.37104	0.685521	+	0.728053i	$0.259574\pi$
$830$	0	0
$831$	−48.4801	−1.68176
$832$	0	0
$833$	−19.7948	−0.685849
$834$	0	0
$835$	−23.8977	−0.827014
$836$	0	0
$837$	31.4837	1.08824
$838$	0	0
$839$	4.47244	0.154406	0.0772028	−	0.997015i	$-0.475401\pi$
$839$	4.47244	0.154406	0.0772028	+	0.997015i	$0.475401\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−14.7749	−0.508875
$844$	0	0
$845$	1.05669	0.0363514
$846$	0	0
$847$	−4.42619	−0.152086
$848$	0	0
$849$	−41.0106	−1.40748
$850$	0	0
$851$	−22.2502	−0.762726
$852$	0	0
$853$	−53.2224	−1.82230	−0.911150	−	0.412076i	$-0.864804\pi$
$853$	−53.2224	−1.82230	−0.911150	+	0.412076i	$0.864804\pi$
$854$	0	0
$855$	−1.50772	−0.0515628
$856$	0	0
$857$	−49.0250	−1.67466	−0.837330	−	0.546697i	$-0.815885\pi$
$857$	−49.0250	−1.67466	−0.837330	+	0.546697i	$0.815885\pi$
$858$	0	0
$859$	48.6773	1.66085	0.830424	−	0.557133i	$-0.188098\pi$
$859$	48.6773	1.66085	0.830424	+	0.557133i	$0.188098\pi$
$860$	0	0
$861$	−29.5406	−1.00674
$862$	0	0
$863$	23.2221	0.790489	0.395245	−	0.918576i	$-0.370660\pi$
$863$	23.2221	0.790489	0.395245	+	0.918576i	$0.370660\pi$
$864$	0	0
$865$	9.77392	0.332323
$866$	0	0
$867$	−6.25016	−0.212267
$868$	0	0
$869$	32.4891	1.10212
$870$	0	0
$871$	−8.26867	−0.280173
$872$	0	0
$873$	−2.21242	−0.0748791
$874$	0	0
$875$	−15.2852	−0.516736
$876$	0	0
$877$	15.4331	0.521137	0.260569	−	0.965455i	$-0.416090\pi$
$877$	15.4331	0.521137	0.260569	+	0.965455i	$0.416090\pi$
$878$	0	0
$879$	−29.0397	−0.979486
$880$	0	0
$881$	28.3257	0.954318	0.477159	−	0.878817i	$-0.341667\pi$
$881$	28.3257	0.954318	0.477159	+	0.878817i	$0.341667\pi$
$882$	0	0
$883$	7.38530	0.248535	0.124268	−	0.992249i	$-0.460342\pi$
$883$	7.38530	0.248535	0.124268	+	0.992249i	$0.460342\pi$
$884$	0	0
$885$	5.57594	0.187433
$886$	0	0
$887$	−24.3591	−0.817899	−0.408949	−	0.912557i	$-0.634105\pi$
$887$	−24.3591	−0.817899	−0.408949	+	0.912557i	$0.634105\pi$
$888$	0	0
$889$	−28.1130	−0.942879
$890$	0	0
$891$	−24.2604	−0.812755
$892$	0	0
$893$	−28.0455	−0.938508
$894$	0	0
$895$	−18.9608	−0.633789
$896$	0	0
$897$	−7.45877	−0.249041
$898$	0	0
$899$	−5.89630	−0.196653
$900$	0	0
$901$	2.24747	0.0748739
$902$	0	0
$903$	−7.21363	−0.240055
$904$	0	0
$905$	5.77893	0.192098
$906$	0	0
$907$	7.35771	0.244309	0.122154	−	0.992511i	$-0.461020\pi$
$907$	7.35771	0.244309	0.122154	+	0.992511i	$0.461020\pi$
$908$	0	0
$909$	1.71141	0.0567639
$910$	0	0
$911$	26.6537	0.883077	0.441538	−	0.897242i	$-0.354433\pi$
$911$	26.6537	0.883077	0.441538	+	0.897242i	$0.354433\pi$
$912$	0	0
$913$	43.7703	1.44858
$914$	0	0
$915$	13.0497	0.431410
$916$	0	0
$917$	3.82261	0.126234
$918$	0	0
$919$	−33.0321	−1.08963	−0.544814	−	0.838557i	$-0.683400\pi$
$919$	−33.0321	−1.08963	−0.544814	+	0.838557i	$0.683400\pi$
$920$	0	0
$921$	−34.5360	−1.13800
$922$	0	0
$923$	−12.9531	−0.426356
$924$	0	0
$925$	19.4564	0.639721
$926$	0	0
$927$	−0.607833	−0.0199638
$928$	0	0
$929$	20.3862	0.668849	0.334425	−	0.942423i	$-0.391458\pi$
$929$	20.3862	0.668849	0.334425	+	0.942423i	$0.391458\pi$
$930$	0	0
$931$	−34.6177	−1.13455
$932$	0	0
$933$	−34.4617	−1.12823
$934$	0	0
$935$	13.8427	0.452704
$936$	0	0
$937$	−4.64960	−0.151896	−0.0759479	−	0.997112i	$-0.524198\pi$
$937$	−4.64960	−0.151896	−0.0759479	+	0.997112i	$0.524198\pi$
$938$	0	0
$939$	22.6335	0.738615
$940$	0	0
$941$	−49.6203	−1.61758	−0.808788	−	0.588100i	$-0.799876\pi$
$941$	−49.6203	−1.61758	−0.808788	+	0.588100i	$0.799876\pi$
$942$	0	0
$943$	47.9705	1.56213
$944$	0	0
$945$	9.18754	0.298871
$946$	0	0
$947$	−51.5894	−1.67643	−0.838215	−	0.545340i	$-0.816401\pi$
$947$	−51.5894	−1.67643	−0.838215	+	0.545340i	$0.816401\pi$
$948$	0	0
$949$	2.43670	0.0790985
$950$	0	0
$951$	51.9175	1.68354
$952$	0	0
$953$	36.3612	1.17786	0.588928	−	0.808186i	$-0.299550\pi$
$953$	36.3612	1.17786	0.588928	+	0.808186i	$0.299550\pi$
$954$	0	0
$955$	21.0293	0.680494
$956$	0	0
$957$	4.83332	0.156239
$958$	0	0
$959$	3.27262	0.105678
$960$	0	0
$961$	3.76637	0.121496
$962$	0	0
$963$	−3.16559	−0.102010
$964$	0	0
$965$	−25.0010	−0.804811
$966$	0	0
$967$	−55.0098	−1.76900	−0.884498	−	0.466544i	$-0.845499\pi$
$967$	−55.0098	−1.76900	−0.884498	+	0.466544i	$0.845499\pi$
$968$	0	0
$969$	−60.8625	−1.95519
$970$	0	0
$971$	40.9802	1.31512	0.657559	−	0.753403i	$-0.271589\pi$
$971$	40.9802	1.31512	0.657559	+	0.753403i	$0.271589\pi$
$972$	0	0
$973$	−11.4049	−0.365624
$974$	0	0
$975$	6.52222	0.208878
$976$	0	0
$977$	11.6782	0.373617	0.186809	−	0.982396i	$-0.440186\pi$
$977$	11.6782	0.373617	0.186809	+	0.982396i	$0.440186\pi$
$978$	0	0
$979$	18.2775	0.584152
$980$	0	0
$981$	−0.698776	−0.0223102
$982$	0	0
$983$	−42.0896	−1.34245	−0.671225	−	0.741253i	$-0.734232\pi$
$983$	−42.0896	−1.34245	−0.671225	+	0.741253i	$0.734232\pi$
$984$	0	0
$985$	1.29582	0.0412883
$986$	0	0
$987$	9.63463	0.306674
$988$	0	0
$989$	11.7141	0.372487
$990$	0	0
$991$	41.0653	1.30448	0.652242	−	0.758011i	$-0.273829\pi$
$991$	41.0653	1.30448	0.652242	+	0.758011i	$0.273829\pi$
$992$	0	0
$993$	4.20830	0.133546
$994$	0	0
$995$	3.04635	0.0965759
$996$	0	0
$997$	18.2007	0.576421	0.288210	−	0.957567i	$-0.406940\pi$
$997$	18.2007	0.576421	0.288210	+	0.957567i	$0.406940\pi$
$998$	0	0
$999$	−26.7520	−0.846394

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.bc.1.2		11
4.3	odd	2		3016.2.a.i.1.10	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.i.1.10	✓	11	4.3	odd	2
6032.2.a.bc.1.2		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67951 q^{3} +1.05669 q^{5} +1.62834 q^{7} -0.179232 q^{9} +O(q^{10})\)	\(q-1.67951 q^{3} +1.05669 q^{5} +1.62834 q^{7} -0.179232 q^{9} +2.87781 q^{11} +1.00000 q^{13} -1.77473 q^{15} +4.55208 q^{17} +7.96080 q^{19} -2.73482 q^{21} +4.44103 q^{23} -3.88340 q^{25} +5.33957 q^{27} -1.00000 q^{29} +5.89630 q^{31} -4.83332 q^{33} +1.72065 q^{35} -5.01014 q^{37} -1.67951 q^{39} +10.8017 q^{41} +2.63770 q^{43} -0.189393 q^{45} -3.52295 q^{47} -4.34852 q^{49} -7.64528 q^{51} +0.493723 q^{53} +3.04096 q^{55} -13.3703 q^{57} -3.14185 q^{59} -7.35305 q^{61} -0.291849 q^{63} +1.05669 q^{65} -8.26867 q^{67} -7.45877 q^{69} -12.9531 q^{71} +2.43670 q^{73} +6.52222 q^{75} +4.68604 q^{77} +11.2895 q^{79} -8.43018 q^{81} +15.2096 q^{83} +4.81015 q^{85} +1.67951 q^{87} +6.35119 q^{89} +1.62834 q^{91} -9.90292 q^{93} +8.41212 q^{95} +12.3439 q^{97} -0.515794 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q + 6 * q^3 - 2 * q^5 + 3 * q^7 + 11 * q^9	\( 11 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 11 q^{9} + 13 q^{11} + 11 q^{13} + 8 q^{15} - 6 q^{17} + 12 q^{19} + q^{21} + 13 q^{23} + 11 q^{25} + 24 q^{27} - 11 q^{29} + 11 q^{31} + 17 q^{33} + 4 q^{35} + 11 q^{37} + 6 q^{39} - 9 q^{41} + 30 q^{43} - 16 q^{45} + q^{47} - 4 q^{49} + 13 q^{51} - 9 q^{53} + q^{55} + 2 q^{57} + 9 q^{59} - 5 q^{61} + 6 q^{63} - 2 q^{65} + 25 q^{67} + 26 q^{71} + 10 q^{73} + 41 q^{75} - 8 q^{77} + 14 q^{79} + 3 q^{81} + 6 q^{83} + 19 q^{85} - 6 q^{87} - 11 q^{89} + 3 q^{91} - 3 q^{93} + 31 q^{95} + 12 q^{97} + 35 q^{99}+O(q^{100}) \) 11 * q + 6 * q^3 - 2 * q^5 + 3 * q^7 + 11 * q^9 + 13 * q^11 + 11 * q^13 + 8 * q^15 - 6 * q^17 + 12 * q^19 + q^21 + 13 * q^23 + 11 * q^25 + 24 * q^27 - 11 * q^29 + 11 * q^31 + 17 * q^33 + 4 * q^35 + 11 * q^37 + 6 * q^39 - 9 * q^41 + 30 * q^43 - 16 * q^45 + q^47 - 4 * q^49 + 13 * q^51 - 9 * q^53 + q^55 + 2 * q^57 + 9 * q^59 - 5 * q^61 + 6 * q^63 - 2 * q^65 + 25 * q^67 + 26 * q^71 + 10 * q^73 + 41 * q^75 - 8 * q^77 + 14 * q^79 + 3 * q^81 + 6 * q^83 + 19 * q^85 - 6 * q^87 - 11 * q^89 + 3 * q^91 - 3 * q^93 + 31 * q^95 + 12 * q^97 + 35 * q^99

Introduction

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