LMFDB - Embedded newform 6032.2.a.ba.1.6

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 21x^{8} + 40x^{7} + 138x^{6} - 243x^{5} - 318x^{4} + 448x^{3} + 312x^{2} - 240x - 128 $ x^10 - 2x^9 - 21x^8 + 40x^7 + 138x^6 - 243x^5 - 318x^4 + 448x^3 + 312x^2 - 240*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.478552$ 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)

$f(q)$	$=$	$q+0.478552 q^{3} -1.65690 q^{5} +2.65348 q^{7} -2.77099 q^{9} +O(q^{10})$	$q+0.478552 q^{3} -1.65690 q^{5} +2.65348 q^{7} -2.77099 q^{9} +1.18185 q^{11} +1.00000 q^{13} -0.792914 q^{15} -6.91522 q^{17} -7.76220 q^{19} +1.26983 q^{21} +2.47137 q^{23} -2.25467 q^{25} -2.76172 q^{27} -1.00000 q^{29} +5.89476 q^{31} +0.565577 q^{33} -4.39657 q^{35} -4.93539 q^{37} +0.478552 q^{39} +8.79335 q^{41} +9.88758 q^{43} +4.59126 q^{45} +3.73017 q^{47} +0.0409796 q^{49} -3.30929 q^{51} +12.4868 q^{53} -1.95821 q^{55} -3.71462 q^{57} +3.88836 q^{59} +15.0581 q^{61} -7.35277 q^{63} -1.65690 q^{65} +13.3709 q^{67} +1.18268 q^{69} -10.4408 q^{71} -1.54266 q^{73} -1.07898 q^{75} +3.13603 q^{77} +0.273172 q^{79} +6.99134 q^{81} -12.3104 q^{83} +11.4578 q^{85} -0.478552 q^{87} +9.81419 q^{89} +2.65348 q^{91} +2.82095 q^{93} +12.8612 q^{95} +7.44492 q^{97} -3.27490 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9	$ 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9} - 4 q^{11} + 10 q^{13} - 8 q^{15} + 10 q^{17} + q^{19} + q^{21} - 23 q^{23} + 25 q^{25} - 2 q^{27} - 10 q^{29} + 13 q^{31} + 15 q^{33} + 12 q^{35} - 7 q^{37} - 2 q^{39} + 16 q^{41} + 12 q^{43} + 55 q^{45} - 11 q^{47} + 25 q^{51} + 11 q^{53} - 22 q^{55} - 6 q^{57} + 11 q^{59} + 34 q^{61} - 37 q^{63} + 5 q^{65} + 23 q^{67} + 2 q^{69} + 4 q^{71} + 39 q^{73} - 11 q^{75} + 32 q^{77} - 5 q^{79} + 38 q^{81} - 6 q^{83} + 45 q^{85} + 2 q^{87} - 24 q^{89} + 2 q^{91} + 13 q^{93} - 33 q^{95} + 19 q^{97}+O(q^{100}) $ 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9 - 4 * q^11 + 10 * q^13 - 8 * q^15 + 10 * q^17 + q^19 + q^21 - 23 * q^23 + 25 * q^25 - 2 * q^27 - 10 * q^29 + 13 * q^31 + 15 * q^33 + 12 * q^35 - 7 * q^37 - 2 * q^39 + 16 * q^41 + 12 * q^43 + 55 * q^45 - 11 * q^47 + 25 * q^51 + 11 * q^53 - 22 * q^55 - 6 * q^57 + 11 * q^59 + 34 * q^61 - 37 * q^63 + 5 * q^65 + 23 * q^67 + 2 * q^69 + 4 * q^71 + 39 * q^73 - 11 * q^75 + 32 * q^77 - 5 * q^79 + 38 * q^81 - 6 * q^83 + 45 * q^85 + 2 * q^87 - 24 * q^89 + 2 * q^91 + 13 * q^93 - 33 * q^95 + 19 * q^97

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
$3$	0.478552	0.276292	0.138146	−	0.990412i	$-0.455886\pi$
$3$	0.478552	0.276292	0.138146	+	0.990412i	$0.455886\pi$
$4$	0	0
$5$	−1.65690	−0.740989	−0.370495	−	0.928835i	$-0.620812\pi$
$5$	−1.65690	−0.740989	−0.370495	+	0.928835i	$0.620812\pi$
$6$	0	0
$7$	2.65348	1.00292	0.501461	−	0.865180i	$-0.332796\pi$
$7$	2.65348	1.00292	0.501461	+	0.865180i	$0.332796\pi$
$8$	0	0
$9$	−2.77099	−0.923663
$10$	0	0
$11$	1.18185	0.356342	0.178171	−	0.984000i	$-0.442982\pi$
$11$	1.18185	0.356342	0.178171	+	0.984000i	$0.442982\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.792914	−0.204729
$16$	0	0
$17$	−6.91522	−1.67719	−0.838594	−	0.544757i	$-0.816622\pi$
$17$	−6.91522	−1.67719	−0.838594	+	0.544757i	$0.816622\pi$
$18$	0	0
$19$	−7.76220	−1.78077	−0.890386	−	0.455207i	$-0.849565\pi$
$19$	−7.76220	−1.78077	−0.890386	+	0.455207i	$0.849565\pi$
$20$	0	0
$21$	1.26983	0.277100
$22$	0	0
$23$	2.47137	0.515316	0.257658	−	0.966236i	$-0.417049\pi$
$23$	2.47137	0.515316	0.257658	+	0.966236i	$0.417049\pi$
$24$	0	0
$25$	−2.25467	−0.450935
$26$	0	0
$27$	−2.76172	−0.531493
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.89476	1.05873	0.529365	−	0.848394i	$-0.322430\pi$
$31$	5.89476	1.05873	0.529365	+	0.848394i	$0.322430\pi$
$32$	0	0
$33$	0.565577	0.0984544
$34$	0	0
$35$	−4.39657	−0.743155
$36$	0	0
$37$	−4.93539	−0.811373	−0.405686	−	0.914012i	$-0.632968\pi$
$37$	−4.93539	−0.811373	−0.405686	+	0.914012i	$0.632968\pi$
$38$	0	0
$39$	0.478552	0.0766296
$40$	0	0
$41$	8.79335	1.37329	0.686645	−	0.726993i	$-0.259083\pi$
$41$	8.79335	1.37329	0.686645	+	0.726993i	$0.259083\pi$
$42$	0	0
$43$	9.88758	1.50784	0.753921	−	0.656965i	$-0.228160\pi$
$43$	9.88758	1.50784	0.753921	+	0.656965i	$0.228160\pi$
$44$	0	0
$45$	4.59126	0.684424
$46$	0	0
$47$	3.73017	0.544101	0.272050	−	0.962283i	$-0.412298\pi$
$47$	3.73017	0.544101	0.272050	+	0.962283i	$0.412298\pi$
$48$	0	0
$49$	0.0409796	0.00585423
$50$	0	0
$51$	−3.30929	−0.463393
$52$	0	0
$53$	12.4868	1.71519	0.857597	−	0.514323i	$-0.171957\pi$
$53$	12.4868	1.71519	0.857597	+	0.514323i	$0.171957\pi$
$54$	0	0
$55$	−1.95821	−0.264045
$56$	0	0
$57$	−3.71462	−0.492013
$58$	0	0
$59$	3.88836	0.506222	0.253111	−	0.967437i	$-0.418546\pi$
$59$	3.88836	0.506222	0.253111	+	0.967437i	$0.418546\pi$
$60$	0	0
$61$	15.0581	1.92800	0.963999	−	0.265907i	$-0.0856714\pi$
$61$	15.0581	1.92800	0.963999	+	0.265907i	$0.0856714\pi$
$62$	0	0
$63$	−7.35277	−0.926362
$64$	0	0
$65$	−1.65690	−0.205513
$66$	0	0
$67$	13.3709	1.63352	0.816759	−	0.576979i	$-0.195768\pi$
$67$	13.3709	1.63352	0.816759	+	0.576979i	$0.195768\pi$
$68$	0	0
$69$	1.18268	0.142378
$70$	0	0
$71$	−10.4408	−1.23910	−0.619548	−	0.784959i	$-0.712684\pi$
$71$	−10.4408	−1.23910	−0.619548	+	0.784959i	$0.712684\pi$
$72$	0	0
$73$	−1.54266	−0.180554	−0.0902771	−	0.995917i	$-0.528775\pi$
$73$	−1.54266	−0.180554	−0.0902771	+	0.995917i	$0.528775\pi$
$74$	0	0
$75$	−1.07898	−0.124590
$76$	0	0
$77$	3.13603	0.357383
$78$	0	0
$79$	0.273172	0.0307342	0.0153671	−	0.999882i	$-0.495108\pi$
$79$	0.273172	0.0307342	0.0153671	+	0.999882i	$0.495108\pi$
$80$	0	0
$81$	6.99134	0.776816
$82$	0	0
$83$	−12.3104	−1.35124	−0.675621	−	0.737249i	$-0.736125\pi$
$83$	−12.3104	−1.35124	−0.675621	+	0.737249i	$0.736125\pi$
$84$	0	0
$85$	11.4578	1.24278
$86$	0	0
$87$	−0.478552	−0.0513061
$88$	0	0
$89$	9.81419	1.04030	0.520151	−	0.854074i	$-0.325876\pi$
$89$	9.81419	1.04030	0.520151	+	0.854074i	$0.325876\pi$
$90$	0	0
$91$	2.65348	0.278161
$92$	0	0
$93$	2.82095	0.292519
$94$	0	0
$95$	12.8612	1.31953
$96$	0	0
$97$	7.44492	0.755917	0.377958	−	0.925823i	$-0.376626\pi$
$97$	7.44492	0.755917	0.377958	+	0.925823i	$0.376626\pi$
$98$	0	0
$99$	−3.27490	−0.329140
$100$	0	0
$101$	18.2671	1.81764	0.908820	−	0.417187i	$-0.136984\pi$
$101$	18.2671	1.81764	0.908820	+	0.417187i	$0.136984\pi$
$102$	0	0
$103$	1.00882	0.0994015	0.0497008	−	0.998764i	$-0.484173\pi$
$103$	1.00882	0.0994015	0.0497008	+	0.998764i	$0.484173\pi$
$104$	0	0
$105$	−2.10398	−0.205328
$106$	0	0
$107$	−4.80810	−0.464816	−0.232408	−	0.972618i	$-0.574660\pi$
$107$	−4.80810	−0.464816	−0.232408	+	0.972618i	$0.574660\pi$
$108$	0	0
$109$	−1.50979	−0.144612	−0.0723060	−	0.997382i	$-0.523036\pi$
$109$	−1.50979	−0.144612	−0.0723060	+	0.997382i	$0.523036\pi$
$110$	0	0
$111$	−2.36184	−0.224176
$112$	0	0
$113$	−6.97663	−0.656306	−0.328153	−	0.944625i	$-0.606426\pi$
$113$	−6.97663	−0.656306	−0.328153	+	0.944625i	$0.606426\pi$
$114$	0	0
$115$	−4.09482	−0.381844
$116$	0	0
$117$	−2.77099	−0.256178
$118$	0	0
$119$	−18.3494	−1.68209
$120$	0	0
$121$	−9.60323	−0.873021
$122$	0	0
$123$	4.20807	0.379429
$124$	0	0
$125$	12.0203	1.07513
$126$	0	0
$127$	6.46685	0.573840	0.286920	−	0.957955i	$-0.407369\pi$
$127$	6.46685	0.573840	0.286920	+	0.957955i	$0.407369\pi$
$128$	0	0
$129$	4.73172	0.416604
$130$	0	0
$131$	−9.73614	−0.850651	−0.425325	−	0.905041i	$-0.639840\pi$
$131$	−9.73614	−0.850651	−0.425325	+	0.905041i	$0.639840\pi$
$132$	0	0
$133$	−20.5969	−1.78598
$134$	0	0
$135$	4.57589	0.393830
$136$	0	0
$137$	−0.512647	−0.0437984	−0.0218992	−	0.999760i	$-0.506971\pi$
$137$	−0.512647	−0.0437984	−0.0218992	+	0.999760i	$0.506971\pi$
$138$	0	0
$139$	−4.77183	−0.404742	−0.202371	−	0.979309i	$-0.564865\pi$
$139$	−4.77183	−0.404742	−0.202371	+	0.979309i	$0.564865\pi$
$140$	0	0
$141$	1.78508	0.150331
$142$	0	0
$143$	1.18185	0.0988314
$144$	0	0
$145$	1.65690	0.137598
$146$	0	0
$147$	0.0196109	0.00161748
$148$	0	0
$149$	−18.3705	−1.50497	−0.752486	−	0.658608i	$-0.771146\pi$
$149$	−18.3705	−1.50497	−0.752486	+	0.658608i	$0.771146\pi$
$150$	0	0
$151$	−11.5181	−0.937328	−0.468664	−	0.883377i	$-0.655264\pi$
$151$	−11.5181	−0.937328	−0.468664	+	0.883377i	$0.655264\pi$
$152$	0	0
$153$	19.1620	1.54916
$154$	0	0
$155$	−9.76704	−0.784508
$156$	0	0
$157$	6.25865	0.499494	0.249747	−	0.968311i	$-0.419653\pi$
$157$	6.25865	0.499494	0.249747	+	0.968311i	$0.419653\pi$
$158$	0	0
$159$	5.97558	0.473894
$160$	0	0
$161$	6.55774	0.516822
$162$	0	0
$163$	0.778114	0.0609466	0.0304733	−	0.999536i	$-0.490299\pi$
$163$	0.778114	0.0609466	0.0304733	+	0.999536i	$0.490299\pi$
$164$	0	0
$165$	−0.937106	−0.0729536
$166$	0	0
$167$	0.334678	0.0258981	0.0129491	−	0.999916i	$-0.495878\pi$
$167$	0.334678	0.0258981	0.0129491	+	0.999916i	$0.495878\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	21.5090	1.64483
$172$	0	0
$173$	−0.667995	−0.0507867	−0.0253934	−	0.999678i	$-0.508084\pi$
$173$	−0.667995	−0.0507867	−0.0253934	+	0.999678i	$0.508084\pi$
$174$	0	0
$175$	−5.98274	−0.452253
$176$	0	0
$177$	1.86078	0.139865
$178$	0	0
$179$	0.316324	0.0236432	0.0118216	−	0.999930i	$-0.496237\pi$
$179$	0.316324	0.0236432	0.0118216	+	0.999930i	$0.496237\pi$
$180$	0	0
$181$	7.93326	0.589674	0.294837	−	0.955547i	$-0.404735\pi$
$181$	7.93326	0.589674	0.294837	+	0.955547i	$0.404735\pi$
$182$	0	0
$183$	7.20610	0.532690
$184$	0	0
$185$	8.17746	0.601219
$186$	0	0
$187$	−8.17277	−0.597652
$188$	0	0
$189$	−7.32817	−0.533046
$190$	0	0
$191$	2.91354	0.210816	0.105408	−	0.994429i	$-0.466385\pi$
$191$	2.91354	0.210816	0.105408	+	0.994429i	$0.466385\pi$
$192$	0	0
$193$	12.9759	0.934025	0.467012	−	0.884251i	$-0.345330\pi$
$193$	12.9759	0.934025	0.467012	+	0.884251i	$0.345330\pi$
$194$	0	0
$195$	−0.792914	−0.0567817
$196$	0	0
$197$	14.5148	1.03414	0.517069	−	0.855944i	$-0.327023\pi$
$197$	14.5148	1.03414	0.517069	+	0.855944i	$0.327023\pi$
$198$	0	0
$199$	17.4125	1.23434	0.617171	−	0.786829i	$-0.288279\pi$
$199$	17.4125	1.23434	0.617171	+	0.786829i	$0.288279\pi$
$200$	0	0
$201$	6.39868	0.451328
$202$	0	0
$203$	−2.65348	−0.186238
$204$	0	0
$205$	−14.5697	−1.01759
$206$	0	0
$207$	−6.84813	−0.475978
$208$	0	0
$209$	−9.17377	−0.634563
$210$	0	0
$211$	19.2956	1.32836	0.664181	−	0.747572i	$-0.268780\pi$
$211$	19.2956	1.32836	0.664181	+	0.747572i	$0.268780\pi$
$212$	0	0
$213$	−4.99646	−0.342352
$214$	0	0
$215$	−16.3828	−1.11729
$216$	0	0
$217$	15.6417	1.06182
$218$	0	0
$219$	−0.738240	−0.0498856
$220$	0	0
$221$	−6.91522	−0.465168
$222$	0	0
$223$	0.383421	0.0256757	0.0128379	−	0.999918i	$-0.495913\pi$
$223$	0.383421	0.0256757	0.0128379	+	0.999918i	$0.495913\pi$
$224$	0	0
$225$	6.24767	0.416512
$226$	0	0
$227$	23.0637	1.53079	0.765397	−	0.643558i	$-0.222543\pi$
$227$	23.0637	1.53079	0.765397	+	0.643558i	$0.222543\pi$
$228$	0	0
$229$	−15.1519	−1.00127	−0.500633	−	0.865660i	$-0.666899\pi$
$229$	−15.1519	−1.00127	−0.500633	+	0.865660i	$0.666899\pi$
$230$	0	0
$231$	1.50075	0.0987421
$232$	0	0
$233$	8.08372	0.529582	0.264791	−	0.964306i	$-0.414697\pi$
$233$	8.08372	0.529582	0.264791	+	0.964306i	$0.414697\pi$
$234$	0	0
$235$	−6.18052	−0.403173
$236$	0	0
$237$	0.130727	0.00849162
$238$	0	0
$239$	−14.0096	−0.906206	−0.453103	−	0.891458i	$-0.649683\pi$
$239$	−14.0096	−0.906206	−0.453103	+	0.891458i	$0.649683\pi$
$240$	0	0
$241$	9.30184	0.599184	0.299592	−	0.954067i	$-0.403149\pi$
$241$	9.30184	0.599184	0.299592	+	0.954067i	$0.403149\pi$
$242$	0	0
$243$	11.6309	0.746120
$244$	0	0
$245$	−0.0678992	−0.00433792
$246$	0	0
$247$	−7.76220	−0.493897
$248$	0	0
$249$	−5.89117	−0.373337
$250$	0	0
$251$	5.82303	0.367547	0.183773	−	0.982969i	$-0.441169\pi$
$251$	5.82303	0.367547	0.183773	+	0.982969i	$0.441169\pi$
$252$	0	0
$253$	2.92079	0.183629
$254$	0	0
$255$	5.48317	0.343370
$256$	0	0
$257$	2.07193	0.129244	0.0646219	−	0.997910i	$-0.479416\pi$
$257$	2.07193	0.129244	0.0646219	+	0.997910i	$0.479416\pi$
$258$	0	0
$259$	−13.0960	−0.813744
$260$	0	0
$261$	2.77099	0.171520
$262$	0	0
$263$	−27.0167	−1.66592	−0.832959	−	0.553335i	$-0.813355\pi$
$263$	−27.0167	−1.66592	−0.832959	+	0.553335i	$0.813355\pi$
$264$	0	0
$265$	−20.6894	−1.27094
$266$	0	0
$267$	4.69660	0.287427
$268$	0	0
$269$	−2.85167	−0.173869	−0.0869346	−	0.996214i	$-0.527707\pi$
$269$	−2.85167	−0.173869	−0.0869346	+	0.996214i	$0.527707\pi$
$270$	0	0
$271$	15.7377	0.955998	0.477999	−	0.878360i	$-0.341362\pi$
$271$	15.7377	0.955998	0.477999	+	0.878360i	$0.341362\pi$
$272$	0	0
$273$	1.26983	0.0768536
$274$	0	0
$275$	−2.66469	−0.160687
$276$	0	0
$277$	21.6018	1.29792	0.648962	−	0.760821i	$-0.275203\pi$
$277$	21.6018	1.29792	0.648962	+	0.760821i	$0.275203\pi$
$278$	0	0
$279$	−16.3343	−0.977910
$280$	0	0
$281$	20.1203	1.20028	0.600139	−	0.799896i	$-0.295112\pi$
$281$	20.1203	1.20028	0.600139	+	0.799896i	$0.295112\pi$
$282$	0	0
$283$	19.1470	1.13817	0.569085	−	0.822279i	$-0.307297\pi$
$283$	19.1470	1.13817	0.569085	+	0.822279i	$0.307297\pi$
$284$	0	0
$285$	6.15476	0.364576
$286$	0	0
$287$	23.3330	1.37730
$288$	0	0
$289$	30.8203	1.81296
$290$	0	0
$291$	3.56278	0.208854
$292$	0	0
$293$	−14.6890	−0.858139	−0.429070	−	0.903271i	$-0.641158\pi$
$293$	−14.6890	−0.858139	−0.429070	+	0.903271i	$0.641158\pi$
$294$	0	0
$295$	−6.44264	−0.375105
$296$	0	0
$297$	−3.26394	−0.189393
$298$	0	0
$299$	2.47137	0.142923
$300$	0	0
$301$	26.2365	1.51225
$302$	0	0
$303$	8.74174	0.502200
$304$	0	0
$305$	−24.9499	−1.42863
$306$	0	0
$307$	9.10821	0.519833	0.259917	−	0.965631i	$-0.416305\pi$
$307$	9.10821	0.519833	0.259917	+	0.965631i	$0.416305\pi$
$308$	0	0
$309$	0.482770	0.0274638
$310$	0	0
$311$	1.37680	0.0780710	0.0390355	−	0.999238i	$-0.487571\pi$
$311$	1.37680	0.0780710	0.0390355	+	0.999238i	$0.487571\pi$
$312$	0	0
$313$	−7.88032	−0.445422	−0.222711	−	0.974885i	$-0.571491\pi$
$313$	−7.88032	−0.445422	−0.222711	+	0.974885i	$0.571491\pi$
$314$	0	0
$315$	12.1828	0.686425
$316$	0	0
$317$	−23.8668	−1.34049	−0.670247	−	0.742138i	$-0.733812\pi$
$317$	−23.8668	−1.34049	−0.670247	+	0.742138i	$0.733812\pi$
$318$	0	0
$319$	−1.18185	−0.0661710
$320$	0	0
$321$	−2.30092	−0.128425
$322$	0	0
$323$	53.6774	2.98669
$324$	0	0
$325$	−2.25467	−0.125067
$326$	0	0
$327$	−0.722514	−0.0399551
$328$	0	0
$329$	9.89794	0.545691
$330$	0	0
$331$	14.9801	0.823383	0.411691	−	0.911323i	$-0.364938\pi$
$331$	14.9801	0.823383	0.411691	+	0.911323i	$0.364938\pi$
$332$	0	0
$333$	13.6759	0.749435
$334$	0	0
$335$	−22.1543	−1.21042
$336$	0	0
$337$	16.2865	0.887181	0.443590	−	0.896230i	$-0.353705\pi$
$337$	16.2865	0.887181	0.443590	+	0.896230i	$0.353705\pi$
$338$	0	0
$339$	−3.33868	−0.181332
$340$	0	0
$341$	6.96673	0.377270
$342$	0	0
$343$	−18.4657	−0.997052
$344$	0	0
$345$	−1.95958	−0.105500
$346$	0	0
$347$	−7.34245	−0.394163	−0.197082	−	0.980387i	$-0.563146\pi$
$347$	−7.34245	−0.394163	−0.197082	+	0.980387i	$0.563146\pi$
$348$	0	0
$349$	29.6383	1.58650	0.793250	−	0.608895i	$-0.208387\pi$
$349$	29.6383	1.58650	0.793250	+	0.608895i	$0.208387\pi$
$350$	0	0
$351$	−2.76172	−0.147410
$352$	0	0
$353$	−24.9000	−1.32529	−0.662647	−	0.748932i	$-0.730567\pi$
$353$	−24.9000	−1.32529	−0.662647	+	0.748932i	$0.730567\pi$
$354$	0	0
$355$	17.2994	0.918156
$356$	0	0
$357$	−8.78115	−0.464748
$358$	0	0
$359$	18.1365	0.957206	0.478603	−	0.878031i	$-0.341143\pi$
$359$	18.1365	0.957206	0.478603	+	0.878031i	$0.341143\pi$
$360$	0	0
$361$	41.2518	2.17115
$362$	0	0
$363$	−4.59564	−0.241209
$364$	0	0
$365$	2.55603	0.133789
$366$	0	0
$367$	−11.2702	−0.588300	−0.294150	−	0.955759i	$-0.595037\pi$
$367$	−11.2702	−0.588300	−0.294150	+	0.955759i	$0.595037\pi$
$368$	0	0
$369$	−24.3663	−1.26846
$370$	0	0
$371$	33.1335	1.72021
$372$	0	0
$373$	4.61317	0.238861	0.119430	−	0.992843i	$-0.461893\pi$
$373$	4.61317	0.238861	0.119430	+	0.992843i	$0.461893\pi$
$374$	0	0
$375$	5.75233	0.297049
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	18.0063	0.924924	0.462462	−	0.886639i	$-0.346966\pi$
$379$	18.0063	0.924924	0.462462	+	0.886639i	$0.346966\pi$
$380$	0	0
$381$	3.09472	0.158547
$382$	0	0
$383$	−21.9473	−1.12145	−0.560727	−	0.828001i	$-0.689478\pi$
$383$	−21.9473	−1.12145	−0.560727	+	0.828001i	$0.689478\pi$
$384$	0	0
$385$	−5.19609	−0.264817
$386$	0	0
$387$	−27.3984	−1.39274
$388$	0	0
$389$	6.26022	0.317406	0.158703	−	0.987326i	$-0.449269\pi$
$389$	6.26022	0.317406	0.158703	+	0.987326i	$0.449269\pi$
$390$	0	0
$391$	−17.0901	−0.864282
$392$	0	0
$393$	−4.65925	−0.235028
$394$	0	0
$395$	−0.452619	−0.0227737
$396$	0	0
$397$	18.0505	0.905931	0.452966	−	0.891528i	$-0.350366\pi$
$397$	18.0505	0.905931	0.452966	+	0.891528i	$0.350366\pi$
$398$	0	0
$399$	−9.85667	−0.493451
$400$	0	0
$401$	2.49757	0.124723	0.0623614	−	0.998054i	$-0.480137\pi$
$401$	2.49757	0.124723	0.0623614	+	0.998054i	$0.480137\pi$
$402$	0	0
$403$	5.89476	0.293639
$404$	0	0
$405$	−11.5840	−0.575612
$406$	0	0
$407$	−5.83290	−0.289126
$408$	0	0
$409$	−35.5972	−1.76017	−0.880085	−	0.474816i	$-0.842514\pi$
$409$	−35.5972	−1.76017	−0.880085	+	0.474816i	$0.842514\pi$
$410$	0	0
$411$	−0.245328	−0.0121011
$412$	0	0
$413$	10.3177	0.507701
$414$	0	0
$415$	20.3971	1.00126
$416$	0	0
$417$	−2.28357	−0.111827
$418$	0	0
$419$	−24.0735	−1.17607	−0.588034	−	0.808836i	$-0.700098\pi$
$419$	−24.0735	−1.17607	−0.588034	+	0.808836i	$0.700098\pi$
$420$	0	0
$421$	34.2449	1.66900	0.834498	−	0.551011i	$-0.185758\pi$
$421$	34.2449	1.66900	0.834498	+	0.551011i	$0.185758\pi$
$422$	0	0
$423$	−10.3363	−0.502566
$424$	0	0
$425$	15.5916	0.756302
$426$	0	0
$427$	39.9565	1.93363
$428$	0	0
$429$	0.565577	0.0273063
$430$	0	0
$431$	7.64902	0.368440	0.184220	−	0.982885i	$-0.441024\pi$
$431$	7.64902	0.368440	0.184220	+	0.982885i	$0.441024\pi$
$432$	0	0
$433$	−23.6012	−1.13420	−0.567101	−	0.823648i	$-0.691935\pi$
$433$	−23.6012	−1.13420	−0.567101	+	0.823648i	$0.691935\pi$
$434$	0	0
$435$	0.792914	0.0380173
$436$	0	0
$437$	−19.1833	−0.917660
$438$	0	0
$439$	−29.0892	−1.38835	−0.694176	−	0.719805i	$-0.744231\pi$
$439$	−29.0892	−1.38835	−0.694176	+	0.719805i	$0.744231\pi$
$440$	0	0
$441$	−0.113554	−0.00540733
$442$	0	0
$443$	−4.84411	−0.230150	−0.115075	−	0.993357i	$-0.536711\pi$
$443$	−4.84411	−0.230150	−0.115075	+	0.993357i	$0.536711\pi$
$444$	0	0
$445$	−16.2612	−0.770852
$446$	0	0
$447$	−8.79125	−0.415812
$448$	0	0
$449$	−1.83144	−0.0864310	−0.0432155	−	0.999066i	$-0.513760\pi$
$449$	−1.83144	−0.0864310	−0.0432155	+	0.999066i	$0.513760\pi$
$450$	0	0
$451$	10.3924	0.489361
$452$	0	0
$453$	−5.51199	−0.258976
$454$	0	0
$455$	−4.39657	−0.206114
$456$	0	0
$457$	16.3091	0.762907	0.381454	−	0.924388i	$-0.375424\pi$
$457$	16.3091	0.762907	0.381454	+	0.924388i	$0.375424\pi$
$458$	0	0
$459$	19.0979	0.891413
$460$	0	0
$461$	5.25371	0.244690	0.122345	−	0.992488i	$-0.460959\pi$
$461$	5.25371	0.244690	0.122345	+	0.992488i	$0.460959\pi$
$462$	0	0
$463$	−7.98444	−0.371068	−0.185534	−	0.982638i	$-0.559402\pi$
$463$	−7.98444	−0.371068	−0.185534	+	0.982638i	$0.559402\pi$
$464$	0	0
$465$	−4.67404	−0.216753
$466$	0	0
$467$	34.2685	1.58576	0.792879	−	0.609379i	$-0.208581\pi$
$467$	34.2685	1.58576	0.792879	+	0.609379i	$0.208581\pi$
$468$	0	0
$469$	35.4795	1.63829
$470$	0	0
$471$	2.99509	0.138006
$472$	0	0
$473$	11.6857	0.537307
$474$	0	0
$475$	17.5012	0.803012
$476$	0	0
$477$	−34.6008	−1.58426
$478$	0	0
$479$	6.67710	0.305085	0.152542	−	0.988297i	$-0.451254\pi$
$479$	6.67710	0.305085	0.152542	+	0.988297i	$0.451254\pi$
$480$	0	0
$481$	−4.93539	−0.225034
$482$	0	0
$483$	3.13822	0.142794
$484$	0	0
$485$	−12.3355	−0.560126
$486$	0	0
$487$	28.3157	1.28310	0.641552	−	0.767079i	$-0.278291\pi$
$487$	28.3157	1.28310	0.641552	+	0.767079i	$0.278291\pi$
$488$	0	0
$489$	0.372368	0.0168391
$490$	0	0
$491$	−26.5455	−1.19798	−0.598990	−	0.800757i	$-0.704431\pi$
$491$	−26.5455	−1.19798	−0.598990	+	0.800757i	$0.704431\pi$
$492$	0	0
$493$	6.91522	0.311446
$494$	0	0
$495$	5.42619	0.243889
$496$	0	0
$497$	−27.7045	−1.24272
$498$	0	0
$499$	18.4702	0.826838	0.413419	−	0.910541i	$-0.364334\pi$
$499$	18.4702	0.826838	0.413419	+	0.910541i	$0.364334\pi$
$500$	0	0
$501$	0.160160	0.00715544
$502$	0	0
$503$	−17.8035	−0.793821	−0.396910	−	0.917857i	$-0.629918\pi$
$503$	−17.8035	−0.793821	−0.396910	+	0.917857i	$0.629918\pi$
$504$	0	0
$505$	−30.2667	−1.34685
$506$	0	0
$507$	0.478552	0.0212532
$508$	0	0
$509$	−36.7033	−1.62684	−0.813422	−	0.581674i	$-0.802398\pi$
$509$	−36.7033	−1.62684	−0.813422	+	0.581674i	$0.802398\pi$
$510$	0	0
$511$	−4.09341	−0.181082
$512$	0	0
$513$	21.4370	0.946467
$514$	0	0
$515$	−1.67151	−0.0736555
$516$	0	0
$517$	4.40851	0.193886
$518$	0	0
$519$	−0.319670	−0.0140320
$520$	0	0
$521$	8.67533	0.380073	0.190036	−	0.981777i	$-0.439139\pi$
$521$	8.67533	0.380073	0.190036	+	0.981777i	$0.439139\pi$
$522$	0	0
$523$	−27.2952	−1.19353	−0.596767	−	0.802415i	$-0.703548\pi$
$523$	−27.2952	−1.19353	−0.596767	+	0.802415i	$0.703548\pi$
$524$	0	0
$525$	−2.86305	−0.124954
$526$	0	0
$527$	−40.7636	−1.77569
$528$	0	0
$529$	−16.8923	−0.734449
$530$	0	0
$531$	−10.7746	−0.467578
$532$	0	0
$533$	8.79335	0.380882
$534$	0	0
$535$	7.96655	0.344424
$536$	0	0
$537$	0.151377	0.00653241
$538$	0	0
$539$	0.0484318	0.00208611
$540$	0	0
$541$	−26.4100	−1.13546	−0.567728	−	0.823216i	$-0.692178\pi$
$541$	−26.4100	−1.13546	−0.567728	+	0.823216i	$0.692178\pi$
$542$	0	0
$543$	3.79647	0.162922
$544$	0	0
$545$	2.50158	0.107156
$546$	0	0
$547$	−2.24012	−0.0957808	−0.0478904	−	0.998853i	$-0.515250\pi$
$547$	−2.24012	−0.0957808	−0.0478904	+	0.998853i	$0.515250\pi$
$548$	0	0
$549$	−41.7259	−1.78082
$550$	0	0
$551$	7.76220	0.330681
$552$	0	0
$553$	0.724857	0.0308241
$554$	0	0
$555$	3.91334	0.166112
$556$	0	0
$557$	−24.1214	−1.02205	−0.511027	−	0.859564i	$-0.670735\pi$
$557$	−24.1214	−1.02205	−0.511027	+	0.859564i	$0.670735\pi$
$558$	0	0
$559$	9.88758	0.418200
$560$	0	0
$561$	−3.91109	−0.165126
$562$	0	0
$563$	−19.7971	−0.834350	−0.417175	−	0.908826i	$-0.636980\pi$
$563$	−19.7971	−0.834350	−0.417175	+	0.908826i	$0.636980\pi$
$564$	0	0
$565$	11.5596	0.486316
$566$	0	0
$567$	18.5514	0.779086
$568$	0	0
$569$	27.6980	1.16116	0.580580	−	0.814203i	$-0.302826\pi$
$569$	27.6980	1.16116	0.580580	+	0.814203i	$0.302826\pi$
$570$	0	0
$571$	26.9965	1.12977	0.564884	−	0.825170i	$-0.308921\pi$
$571$	26.9965	1.12977	0.564884	+	0.825170i	$0.308921\pi$
$572$	0	0
$573$	1.39428	0.0582469
$574$	0	0
$575$	−5.57213	−0.232374
$576$	0	0
$577$	−29.5320	−1.22943	−0.614716	−	0.788749i	$-0.710729\pi$
$577$	−29.5320	−1.22943	−0.614716	+	0.788749i	$0.710729\pi$
$578$	0	0
$579$	6.20963	0.258064
$580$	0	0
$581$	−32.6655	−1.35519
$582$	0	0
$583$	14.7575	0.611195
$584$	0	0
$585$	4.59126	0.189825
$586$	0	0
$587$	0.536914	0.0221608	0.0110804	−	0.999939i	$-0.496473\pi$
$587$	0.536914	0.0221608	0.0110804	+	0.999939i	$0.496473\pi$
$588$	0	0
$589$	−45.7563	−1.88536
$590$	0	0
$591$	6.94609	0.285724
$592$	0	0
$593$	21.0490	0.864378	0.432189	−	0.901783i	$-0.357741\pi$
$593$	21.0490	0.864378	0.432189	+	0.901783i	$0.357741\pi$
$594$	0	0
$595$	30.4032	1.24641
$596$	0	0
$597$	8.33280	0.341039
$598$	0	0
$599$	−8.90608	−0.363893	−0.181946	−	0.983308i	$-0.558240\pi$
$599$	−8.90608	−0.363893	−0.181946	+	0.983308i	$0.558240\pi$
$600$	0	0
$601$	−28.7847	−1.17415	−0.587075	−	0.809532i	$-0.699721\pi$
$601$	−28.7847	−1.17415	−0.587075	+	0.809532i	$0.699721\pi$
$602$	0	0
$603$	−37.0507	−1.50882
$604$	0	0
$605$	15.9116	0.646899
$606$	0	0
$607$	−10.5518	−0.428283	−0.214142	−	0.976803i	$-0.568695\pi$
$607$	−10.5518	−0.428283	−0.214142	+	0.976803i	$0.568695\pi$
$608$	0	0
$609$	−1.26983	−0.0514561
$610$	0	0
$611$	3.73017	0.150906
$612$	0	0
$613$	21.8421	0.882195	0.441097	−	0.897459i	$-0.354589\pi$
$613$	21.8421	0.882195	0.441097	+	0.897459i	$0.354589\pi$
$614$	0	0
$615$	−6.97236	−0.281153
$616$	0	0
$617$	−27.4544	−1.10527	−0.552636	−	0.833423i	$-0.686378\pi$
$617$	−27.4544	−1.10527	−0.552636	+	0.833423i	$0.686378\pi$
$618$	0	0
$619$	−22.7762	−0.915453	−0.457726	−	0.889093i	$-0.651336\pi$
$619$	−22.7762	−0.915453	−0.457726	+	0.889093i	$0.651336\pi$
$620$	0	0
$621$	−6.82522	−0.273887
$622$	0	0
$623$	26.0418	1.04334
$624$	0	0
$625$	−8.64308	−0.345723
$626$	0	0
$627$	−4.39013	−0.175325
$628$	0	0
$629$	34.1293	1.36082
$630$	0	0
$631$	37.8881	1.50830	0.754151	−	0.656701i	$-0.228048\pi$
$631$	37.8881	1.50830	0.754151	+	0.656701i	$0.228048\pi$
$632$	0	0
$633$	9.23393	0.367016
$634$	0	0
$635$	−10.7149	−0.425209
$636$	0	0
$637$	0.0409796	0.00162367
$638$	0	0
$639$	28.9313	1.14451
$640$	0	0
$641$	0.814089	0.0321546	0.0160773	−	0.999871i	$-0.494882\pi$
$641$	0.814089	0.0321546	0.0160773	+	0.999871i	$0.494882\pi$
$642$	0	0
$643$	−21.8305	−0.860912	−0.430456	−	0.902611i	$-0.641647\pi$
$643$	−21.8305	−0.860912	−0.430456	+	0.902611i	$0.641647\pi$
$644$	0	0
$645$	−7.83999	−0.308699
$646$	0	0
$647$	25.0213	0.983689	0.491845	−	0.870683i	$-0.336323\pi$
$647$	25.0213	0.983689	0.491845	+	0.870683i	$0.336323\pi$
$648$	0	0
$649$	4.59547	0.180388
$650$	0	0
$651$	7.48534	0.293374
$652$	0	0
$653$	6.44775	0.252320	0.126160	−	0.992010i	$-0.459735\pi$
$653$	6.44775	0.252320	0.126160	+	0.992010i	$0.459735\pi$
$654$	0	0
$655$	16.1318	0.630323
$656$	0	0
$657$	4.27468	0.166771
$658$	0	0
$659$	−28.4071	−1.10658	−0.553291	−	0.832988i	$-0.686628\pi$
$659$	−28.4071	−1.10658	−0.553291	+	0.832988i	$0.686628\pi$
$660$	0	0
$661$	15.7145	0.611223	0.305611	−	0.952156i	$-0.401139\pi$
$661$	15.7145	0.611223	0.305611	+	0.952156i	$0.401139\pi$
$662$	0	0
$663$	−3.30929	−0.128522
$664$	0	0
$665$	34.1270	1.32339
$666$	0	0
$667$	−2.47137	−0.0956918
$668$	0	0
$669$	0.183487	0.00709400
$670$	0	0
$671$	17.7965	0.687026
$672$	0	0
$673$	12.0308	0.463754	0.231877	−	0.972745i	$-0.425513\pi$
$673$	12.0308	0.463754	0.231877	+	0.972745i	$0.425513\pi$
$674$	0	0
$675$	6.22677	0.239668
$676$	0	0
$677$	13.9167	0.534863	0.267432	−	0.963577i	$-0.413825\pi$
$677$	13.9167	0.534863	0.267432	+	0.963577i	$0.413825\pi$
$678$	0	0
$679$	19.7550	0.758126
$680$	0	0
$681$	11.0372	0.422946
$682$	0	0
$683$	3.14624	0.120388	0.0601938	−	0.998187i	$-0.480828\pi$
$683$	3.14624	0.120388	0.0601938	+	0.998187i	$0.480828\pi$
$684$	0	0
$685$	0.849406	0.0324541
$686$	0	0
$687$	−7.25097	−0.276642
$688$	0	0
$689$	12.4868	0.475709
$690$	0	0
$691$	1.23986	0.0471666	0.0235833	−	0.999722i	$-0.492493\pi$
$691$	1.23986	0.0471666	0.0235833	+	0.999722i	$0.492493\pi$
$692$	0	0
$693$	−8.68989	−0.330102
$694$	0	0
$695$	7.90647	0.299909
$696$	0	0
$697$	−60.8080	−2.30327
$698$	0	0
$699$	3.86848	0.146319
$700$	0	0
$701$	−0.300174	−0.0113374	−0.00566870	−	0.999984i	$-0.501804\pi$
$701$	−0.300174	−0.0113374	−0.00566870	+	0.999984i	$0.501804\pi$
$702$	0	0
$703$	38.3095	1.44487
$704$	0	0
$705$	−2.95770	−0.111393
$706$	0	0
$707$	48.4714	1.82295
$708$	0	0
$709$	16.8542	0.632974	0.316487	−	0.948597i	$-0.397497\pi$
$709$	16.8542	0.632974	0.316487	+	0.948597i	$0.397497\pi$
$710$	0	0
$711$	−0.756956	−0.0283881
$712$	0	0
$713$	14.5681	0.545581
$714$	0	0
$715$	−1.95821	−0.0732330
$716$	0	0
$717$	−6.70432	−0.250378
$718$	0	0
$719$	−14.5781	−0.543671	−0.271836	−	0.962344i	$-0.587631\pi$
$719$	−14.5781	−0.543671	−0.271836	+	0.962344i	$0.587631\pi$
$720$	0	0
$721$	2.67688	0.0996921
$722$	0	0
$723$	4.45141	0.165550
$724$	0	0
$725$	2.25467	0.0837365
$726$	0	0
$727$	−13.1900	−0.489192	−0.244596	−	0.969625i	$-0.578655\pi$
$727$	−13.1900	−0.489192	−0.244596	+	0.969625i	$0.578655\pi$
$728$	0	0
$729$	−15.4081	−0.570669
$730$	0	0
$731$	−68.3748	−2.52893
$732$	0	0
$733$	−5.38709	−0.198977	−0.0994884	−	0.995039i	$-0.531721\pi$
$733$	−5.38709	−0.198977	−0.0994884	+	0.995039i	$0.531721\pi$
$734$	0	0
$735$	−0.0324933	−0.00119853
$736$	0	0
$737$	15.8025	0.582091
$738$	0	0
$739$	48.5889	1.78737	0.893685	−	0.448695i	$-0.148111\pi$
$739$	48.5889	1.78737	0.893685	+	0.448695i	$0.148111\pi$
$740$	0	0
$741$	−3.71462	−0.136460
$742$	0	0
$743$	−44.3417	−1.62674	−0.813370	−	0.581747i	$-0.802369\pi$
$743$	−44.3417	−1.62674	−0.813370	+	0.581747i	$0.802369\pi$
$744$	0	0
$745$	30.4382	1.11517
$746$	0	0
$747$	34.1120	1.24809
$748$	0	0
$749$	−12.7582	−0.466175
$750$	0	0
$751$	−32.6271	−1.19058	−0.595290	−	0.803511i	$-0.702963\pi$
$751$	−32.6271	−1.19058	−0.595290	+	0.803511i	$0.702963\pi$
$752$	0	0
$753$	2.78662	0.101550
$754$	0	0
$755$	19.0843	0.694550
$756$	0	0
$757$	−38.1109	−1.38517	−0.692583	−	0.721338i	$-0.743527\pi$
$757$	−38.1109	−1.38517	−0.692583	+	0.721338i	$0.743527\pi$
$758$	0	0
$759$	1.39775	0.0507351
$760$	0	0
$761$	18.3195	0.664081	0.332041	−	0.943265i	$-0.392263\pi$
$761$	18.3195	0.664081	0.332041	+	0.943265i	$0.392263\pi$
$762$	0	0
$763$	−4.00621	−0.145035
$764$	0	0
$765$	−31.7496	−1.14791
$766$	0	0
$767$	3.88836	0.140401
$768$	0	0
$769$	4.11074	0.148237	0.0741185	−	0.997249i	$-0.476386\pi$
$769$	4.11074	0.148237	0.0741185	+	0.997249i	$0.476386\pi$
$770$	0	0
$771$	0.991528	0.0357090
$772$	0	0
$773$	6.54157	0.235284	0.117642	−	0.993056i	$-0.462466\pi$
$773$	6.54157	0.235284	0.117642	+	0.993056i	$0.462466\pi$
$774$	0	0
$775$	−13.2908	−0.477418
$776$	0	0
$777$	−6.26710	−0.224831
$778$	0	0
$779$	−68.2558	−2.44552
$780$	0	0
$781$	−12.3395	−0.441541
$782$	0	0
$783$	2.76172	0.0986957
$784$	0	0
$785$	−10.3700	−0.370120
$786$	0	0
$787$	41.9982	1.49707	0.748537	−	0.663093i	$-0.230756\pi$
$787$	41.9982	1.49707	0.748537	+	0.663093i	$0.230756\pi$
$788$	0	0
$789$	−12.9289	−0.460280
$790$	0	0
$791$	−18.5124	−0.658224
$792$	0	0
$793$	15.0581	0.534730
$794$	0	0
$795$	−9.90095	−0.351150
$796$	0	0
$797$	32.2115	1.14099	0.570496	−	0.821301i	$-0.306751\pi$
$797$	32.2115	1.14099	0.570496	+	0.821301i	$0.306751\pi$
$798$	0	0
$799$	−25.7949	−0.912559
$800$	0	0
$801$	−27.1950	−0.960888
$802$	0	0
$803$	−1.82319	−0.0643390
$804$	0	0
$805$	−10.8655	−0.382960
$806$	0	0
$807$	−1.36467	−0.0480387
$808$	0	0
$809$	38.9696	1.37010	0.685049	−	0.728497i	$-0.259781\pi$
$809$	38.9696	1.37010	0.685049	+	0.728497i	$0.259781\pi$
$810$	0	0
$811$	34.1541	1.19931	0.599656	−	0.800258i	$-0.295304\pi$
$811$	34.1541	1.19931	0.599656	+	0.800258i	$0.295304\pi$
$812$	0	0
$813$	7.53131	0.264135
$814$	0	0
$815$	−1.28926	−0.0451608
$816$	0	0
$817$	−76.7494	−2.68512
$818$	0	0
$819$	−7.35277	−0.256927
$820$	0	0
$821$	49.0307	1.71118	0.855591	−	0.517652i	$-0.173194\pi$
$821$	49.0307	1.71118	0.855591	+	0.517652i	$0.173194\pi$
$822$	0	0
$823$	18.3070	0.638144	0.319072	−	0.947731i	$-0.396629\pi$
$823$	18.3070	0.638144	0.319072	+	0.947731i	$0.396629\pi$
$824$	0	0
$825$	−1.27519	−0.0443965
$826$	0	0
$827$	−14.7846	−0.514110	−0.257055	−	0.966397i	$-0.582752\pi$
$827$	−14.7846	−0.514110	−0.257055	+	0.966397i	$0.582752\pi$
$828$	0	0
$829$	41.9457	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$829$	41.9457	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$830$	0	0
$831$	10.3376	0.358606
$832$	0	0
$833$	−0.283383	−0.00981864
$834$	0	0
$835$	−0.554528	−0.0191902
$836$	0	0
$837$	−16.2797	−0.562707
$838$	0	0
$839$	−21.5444	−0.743794	−0.371897	−	0.928274i	$-0.621293\pi$
$839$	−21.5444	−0.743794	−0.371897	+	0.928274i	$0.621293\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	9.62861	0.331627
$844$	0	0
$845$	−1.65690	−0.0569992
$846$	0	0
$847$	−25.4820	−0.875572
$848$	0	0
$849$	9.16283	0.314467
$850$	0	0
$851$	−12.1972	−0.418113
$852$	0	0
$853$	−26.0432	−0.891701	−0.445850	−	0.895107i	$-0.647099\pi$
$853$	−26.0432	−0.891701	−0.445850	+	0.895107i	$0.647099\pi$
$854$	0	0
$855$	−35.6383	−1.21880
$856$	0	0
$857$	−33.6686	−1.15010	−0.575048	−	0.818120i	$-0.695017\pi$
$857$	−33.6686	−1.15010	−0.575048	+	0.818120i	$0.695017\pi$
$858$	0	0
$859$	53.9991	1.84243	0.921213	−	0.389058i	$-0.127199\pi$
$859$	53.9991	1.84243	0.921213	+	0.389058i	$0.127199\pi$
$860$	0	0
$861$	11.1661	0.380538
$862$	0	0
$863$	1.69560	0.0577190	0.0288595	−	0.999583i	$-0.490812\pi$
$863$	1.69560	0.0577190	0.0288595	+	0.999583i	$0.490812\pi$
$864$	0	0
$865$	1.10680	0.0376324
$866$	0	0
$867$	14.7491	0.500906
$868$	0	0
$869$	0.322849	0.0109519
$870$	0	0
$871$	13.3709	0.453057
$872$	0	0
$873$	−20.6298	−0.698212
$874$	0	0
$875$	31.8956	1.07827
$876$	0	0
$877$	22.6911	0.766223	0.383112	−	0.923702i	$-0.374852\pi$
$877$	22.6911	0.766223	0.383112	+	0.923702i	$0.374852\pi$
$878$	0	0
$879$	−7.02943	−0.237097
$880$	0	0
$881$	−31.9409	−1.07612	−0.538059	−	0.842907i	$-0.680842\pi$
$881$	−31.9409	−1.07612	−0.538059	+	0.842907i	$0.680842\pi$
$882$	0	0
$883$	42.8167	1.44090	0.720448	−	0.693509i	$-0.243936\pi$
$883$	42.8167	1.44090	0.720448	+	0.693509i	$0.243936\pi$
$884$	0	0
$885$	−3.08314	−0.103638
$886$	0	0
$887$	−40.1094	−1.34674	−0.673370	−	0.739305i	$-0.735154\pi$
$887$	−40.1094	−1.34674	−0.673370	+	0.739305i	$0.735154\pi$
$888$	0	0
$889$	17.1597	0.575517
$890$	0	0
$891$	8.26273	0.276812
$892$	0	0
$893$	−28.9543	−0.968919
$894$	0	0
$895$	−0.524118	−0.0175193
$896$	0	0
$897$	1.18268	0.0394885
$898$	0	0
$899$	−5.89476	−0.196601
$900$	0	0
$901$	−86.3490	−2.87670
$902$	0	0
$903$	12.5555	0.417822
$904$	0	0
$905$	−13.1446	−0.436942
$906$	0	0
$907$	−58.8045	−1.95257	−0.976285	−	0.216488i	$-0.930540\pi$
$907$	−58.8045	−1.95257	−0.976285	+	0.216488i	$0.930540\pi$
$908$	0	0
$909$	−50.6178	−1.67889
$910$	0	0
$911$	−18.4718	−0.611998	−0.305999	−	0.952032i	$-0.598990\pi$
$911$	−18.4718	−0.611998	−0.305999	+	0.952032i	$0.598990\pi$
$912$	0	0
$913$	−14.5491	−0.481504
$914$	0	0
$915$	−11.9398	−0.394718
$916$	0	0
$917$	−25.8347	−0.853137
$918$	0	0
$919$	−2.09415	−0.0690796	−0.0345398	−	0.999403i	$-0.510997\pi$
$919$	−2.09415	−0.0690796	−0.0345398	+	0.999403i	$0.510997\pi$
$920$	0	0
$921$	4.35875	0.143626
$922$	0	0
$923$	−10.4408	−0.343663
$924$	0	0
$925$	11.1277	0.365876
$926$	0	0
$927$	−2.79542	−0.0918135
$928$	0	0
$929$	−15.0428	−0.493540	−0.246770	−	0.969074i	$-0.579369\pi$
$929$	−15.0428	−0.493540	−0.246770	+	0.969074i	$0.579369\pi$
$930$	0	0
$931$	−0.318092	−0.0104250
$932$	0	0
$933$	0.658868	0.0215704
$934$	0	0
$935$	13.5415	0.442854
$936$	0	0
$937$	−5.83276	−0.190548	−0.0952740	−	0.995451i	$-0.530373\pi$
$937$	−5.83276	−0.190548	−0.0952740	+	0.995451i	$0.530373\pi$
$938$	0	0
$939$	−3.77114	−0.123066
$940$	0	0
$941$	−9.53419	−0.310806	−0.155403	−	0.987851i	$-0.549668\pi$
$941$	−9.53419	−0.310806	−0.155403	+	0.987851i	$0.549668\pi$
$942$	0	0
$943$	21.7316	0.707679
$944$	0	0
$945$	12.1421	0.394981
$946$	0	0
$947$	20.4409	0.664241	0.332121	−	0.943237i	$-0.392236\pi$
$947$	20.4409	0.664241	0.332121	+	0.943237i	$0.392236\pi$
$948$	0	0
$949$	−1.54266	−0.0500767
$950$	0	0
$951$	−11.4215	−0.370368
$952$	0	0
$953$	35.6972	1.15635	0.578173	−	0.815914i	$-0.303766\pi$
$953$	35.6972	1.15635	0.578173	+	0.815914i	$0.303766\pi$
$954$	0	0
$955$	−4.82745	−0.156213
$956$	0	0
$957$	−0.565577	−0.0182825
$958$	0	0
$959$	−1.36030	−0.0439264
$960$	0	0
$961$	3.74820	0.120910
$962$	0	0
$963$	13.3232	0.429333
$964$	0	0
$965$	−21.4998	−0.692103
$966$	0	0
$967$	−21.8134	−0.701471	−0.350735	−	0.936475i	$-0.614068\pi$
$967$	−21.8134	−0.701471	−0.350735	+	0.936475i	$0.614068\pi$
$968$	0	0
$969$	25.6874	0.825198
$970$	0	0
$971$	20.2527	0.649941	0.324970	−	0.945724i	$-0.394646\pi$
$971$	20.2527	0.649941	0.324970	+	0.945724i	$0.394646\pi$
$972$	0	0
$973$	−12.6620	−0.405925
$974$	0	0
$975$	−1.07898	−0.0345550
$976$	0	0
$977$	−25.2708	−0.808484	−0.404242	−	0.914652i	$-0.632465\pi$
$977$	−25.2708	−0.808484	−0.404242	+	0.914652i	$0.632465\pi$
$978$	0	0
$979$	11.5989	0.370703
$980$	0	0
$981$	4.18362	0.133573
$982$	0	0
$983$	23.6372	0.753909	0.376955	−	0.926232i	$-0.376971\pi$
$983$	23.6372	0.753909	0.376955	+	0.926232i	$0.376971\pi$
$984$	0	0
$985$	−24.0496	−0.766285
$986$	0	0
$987$	4.73668	0.150770
$988$	0	0
$989$	24.4359	0.777015
$990$	0	0
$991$	20.6022	0.654449	0.327225	−	0.944947i	$-0.393887\pi$
$991$	20.6022	0.654449	0.327225	+	0.944947i	$0.393887\pi$
$992$	0	0
$993$	7.16877	0.227494
$994$	0	0
$995$	−28.8509	−0.914634
$996$	0	0
$997$	25.5560	0.809368	0.404684	−	0.914457i	$-0.367382\pi$
$997$	25.5560	0.809368	0.404684	+	0.914457i	$0.367382\pi$
$998$	0	0
$999$	13.6301	0.431239

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.ba.1.6		10
4.3	odd	2		3016.2.a.g.1.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.g.1.5	✓	10	4.3	odd	2
6032.2.a.ba.1.6		10	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)

\(f(q)\)	\(=\)	\(q+0.478552 q^{3} -1.65690 q^{5} +2.65348 q^{7} -2.77099 q^{9} +O(q^{10})\)	\(q+0.478552 q^{3} -1.65690 q^{5} +2.65348 q^{7} -2.77099 q^{9} +1.18185 q^{11} +1.00000 q^{13} -0.792914 q^{15} -6.91522 q^{17} -7.76220 q^{19} +1.26983 q^{21} +2.47137 q^{23} -2.25467 q^{25} -2.76172 q^{27} -1.00000 q^{29} +5.89476 q^{31} +0.565577 q^{33} -4.39657 q^{35} -4.93539 q^{37} +0.478552 q^{39} +8.79335 q^{41} +9.88758 q^{43} +4.59126 q^{45} +3.73017 q^{47} +0.0409796 q^{49} -3.30929 q^{51} +12.4868 q^{53} -1.95821 q^{55} -3.71462 q^{57} +3.88836 q^{59} +15.0581 q^{61} -7.35277 q^{63} -1.65690 q^{65} +13.3709 q^{67} +1.18268 q^{69} -10.4408 q^{71} -1.54266 q^{73} -1.07898 q^{75} +3.13603 q^{77} +0.273172 q^{79} +6.99134 q^{81} -12.3104 q^{83} +11.4578 q^{85} -0.478552 q^{87} +9.81419 q^{89} +2.65348 q^{91} +2.82095 q^{93} +12.8612 q^{95} +7.44492 q^{97} -3.27490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9	\( 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9} - 4 q^{11} + 10 q^{13} - 8 q^{15} + 10 q^{17} + q^{19} + q^{21} - 23 q^{23} + 25 q^{25} - 2 q^{27} - 10 q^{29} + 13 q^{31} + 15 q^{33} + 12 q^{35} - 7 q^{37} - 2 q^{39} + 16 q^{41} + 12 q^{43} + 55 q^{45} - 11 q^{47} + 25 q^{51} + 11 q^{53} - 22 q^{55} - 6 q^{57} + 11 q^{59} + 34 q^{61} - 37 q^{63} + 5 q^{65} + 23 q^{67} + 2 q^{69} + 4 q^{71} + 39 q^{73} - 11 q^{75} + 32 q^{77} - 5 q^{79} + 38 q^{81} - 6 q^{83} + 45 q^{85} + 2 q^{87} - 24 q^{89} + 2 q^{91} + 13 q^{93} - 33 q^{95} + 19 q^{97}+O(q^{100}) \) 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9 - 4 * q^11 + 10 * q^13 - 8 * q^15 + 10 * q^17 + q^19 + q^21 - 23 * q^23 + 25 * q^25 - 2 * q^27 - 10 * q^29 + 13 * q^31 + 15 * q^33 + 12 * q^35 - 7 * q^37 - 2 * q^39 + 16 * q^41 + 12 * q^43 + 55 * q^45 - 11 * q^47 + 25 * q^51 + 11 * q^53 - 22 * q^55 - 6 * q^57 + 11 * q^59 + 34 * q^61 - 37 * q^63 + 5 * q^65 + 23 * q^67 + 2 * q^69 + 4 * q^71 + 39 * q^73 - 11 * q^75 + 32 * q^77 - 5 * q^79 + 38 * q^81 - 6 * q^83 + 45 * q^85 + 2 * q^87 - 24 * q^89 + 2 * q^91 + 13 * q^93 - 33 * q^95 + 19 * q^97

Introduction

L-functions

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Database

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