LMFDB - Embedded newform 8040.2.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8040,2,Mod(1,8040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8040.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:	$64.1997232251$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 3x^{4} + 43x^{3} - 6x^{2} - 29x + 6 $ x^7 - x^6 - 15x^5 + 3x^4 + 43x^3 - 6x^2 - 29*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.95767$ of defining polynomial
Character	$\chi$	$=$	8040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -0.212435 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -0.212435 q^{7} +1.00000 q^{9} -2.52948 q^{11} -3.33132 q^{13} -1.00000 q^{15} -7.70550 q^{17} +7.01222 q^{19} -0.212435 q^{21} -1.74388 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.69465 q^{29} -2.26698 q^{31} -2.52948 q^{33} +0.212435 q^{35} +9.86329 q^{37} -3.33132 q^{39} +4.26698 q^{41} -1.58745 q^{43} -1.00000 q^{45} -5.44041 q^{47} -6.95487 q^{49} -7.70550 q^{51} +6.91104 q^{53} +2.52948 q^{55} +7.01222 q^{57} -7.97332 q^{59} +8.27524 q^{61} -0.212435 q^{63} +3.33132 q^{65} +1.00000 q^{67} -1.74388 q^{69} +4.11298 q^{71} +10.0883 q^{73} +1.00000 q^{75} +0.537349 q^{77} -6.71856 q^{79} +1.00000 q^{81} +6.07709 q^{83} +7.70550 q^{85} +1.69465 q^{87} -15.7411 q^{89} +0.707688 q^{91} -2.26698 q^{93} -7.01222 q^{95} +7.28158 q^{97} -2.52948 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9	$ 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9} - q^{13} - 7 q^{15} - 2 q^{17} + 9 q^{19} + 10 q^{21} + 2 q^{23} + 7 q^{25} + 7 q^{27} - q^{29} + 9 q^{31} - 10 q^{35} + 23 q^{37} - q^{39} + 5 q^{41} - 3 q^{43} - 7 q^{45} + 11 q^{47} + 13 q^{49} - 2 q^{51} + 13 q^{53} + 9 q^{57} + q^{59} + 4 q^{61} + 10 q^{63} + q^{65} + 7 q^{67} + 2 q^{69} + q^{71} + 14 q^{73} + 7 q^{75} + 18 q^{77} + 25 q^{79} + 7 q^{81} - 29 q^{83} + 2 q^{85} - q^{87} + 7 q^{89} + 27 q^{91} + 9 q^{93} - 9 q^{95} + 38 q^{97}+O(q^{100}) $ 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9 - q^13 - 7 * q^15 - 2 * q^17 + 9 * q^19 + 10 * q^21 + 2 * q^23 + 7 * q^25 + 7 * q^27 - q^29 + 9 * q^31 - 10 * q^35 + 23 * q^37 - q^39 + 5 * q^41 - 3 * q^43 - 7 * q^45 + 11 * q^47 + 13 * q^49 - 2 * q^51 + 13 * q^53 + 9 * q^57 + q^59 + 4 * q^61 + 10 * q^63 + q^65 + 7 * q^67 + 2 * q^69 + q^71 + 14 * q^73 + 7 * q^75 + 18 * q^77 + 25 * q^79 + 7 * q^81 - 29 * q^83 + 2 * q^85 - q^87 + 7 * q^89 + 27 * q^91 + 9 * q^93 - 9 * q^95 + 38 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.212435	−0.0802927	−0.0401464	−	0.999194i	$-0.512782\pi$
$7$	−0.212435	−0.0802927	−0.0401464	+	0.999194i	$0.512782\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.52948	−0.762667	−0.381333	−	0.924438i	$-0.624535\pi$
$11$	−2.52948	−0.762667	−0.381333	+	0.924438i	$0.624535\pi$
$12$	0	0
$13$	−3.33132	−0.923943	−0.461971	−	0.886895i	$-0.652858\pi$
$13$	−3.33132	−0.923943	−0.461971	+	0.886895i	$0.652858\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−7.70550	−1.86886	−0.934429	−	0.356148i	$-0.884090\pi$
$17$	−7.70550	−1.86886	−0.934429	+	0.356148i	$0.884090\pi$
$18$	0	0
$19$	7.01222	1.60871	0.804356	−	0.594147i	$-0.202510\pi$
$19$	7.01222	1.60871	0.804356	+	0.594147i	$0.202510\pi$
$20$	0	0
$21$	−0.212435	−0.0463570
$22$	0	0
$23$	−1.74388	−0.363623	−0.181812	−	0.983333i	$-0.558196\pi$
$23$	−1.74388	−0.363623	−0.181812	+	0.983333i	$0.558196\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.69465	0.314688	0.157344	−	0.987544i	$-0.449707\pi$
$29$	1.69465	0.314688	0.157344	+	0.987544i	$0.449707\pi$
$30$	0	0
$31$	−2.26698	−0.407161	−0.203581	−	0.979058i	$-0.565258\pi$
$31$	−2.26698	−0.407161	−0.203581	+	0.979058i	$0.565258\pi$
$32$	0	0
$33$	−2.52948	−0.440326
$34$	0	0
$35$	0.212435	0.0359080
$36$	0	0
$37$	9.86329	1.62151	0.810757	−	0.585383i	$-0.199056\pi$
$37$	9.86329	1.62151	0.810757	+	0.585383i	$0.199056\pi$
$38$	0	0
$39$	−3.33132	−0.533439
$40$	0	0
$41$	4.26698	0.666390	0.333195	−	0.942858i	$-0.391873\pi$
$41$	4.26698	0.666390	0.333195	+	0.942858i	$0.391873\pi$
$42$	0	0
$43$	−1.58745	−0.242083	−0.121042	−	0.992647i	$-0.538623\pi$
$43$	−1.58745	−0.242083	−0.121042	+	0.992647i	$0.538623\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−5.44041	−0.793565	−0.396783	−	0.917913i	$-0.629873\pi$
$47$	−5.44041	−0.793565	−0.396783	+	0.917913i	$0.629873\pi$
$48$	0	0
$49$	−6.95487	−0.993553
$50$	0	0
$51$	−7.70550	−1.07899
$52$	0	0
$53$	6.91104	0.949304	0.474652	−	0.880174i	$-0.342574\pi$
$53$	6.91104	0.949304	0.474652	+	0.880174i	$0.342574\pi$
$54$	0	0
$55$	2.52948	0.341075
$56$	0	0
$57$	7.01222	0.928791
$58$	0	0
$59$	−7.97332	−1.03804	−0.519019	−	0.854763i	$-0.673703\pi$
$59$	−7.97332	−1.03804	−0.519019	+	0.854763i	$0.673703\pi$
$60$	0	0
$61$	8.27524	1.05954	0.529768	−	0.848143i	$-0.322279\pi$
$61$	8.27524	1.05954	0.529768	+	0.848143i	$0.322279\pi$
$62$	0	0
$63$	−0.212435	−0.0267642
$64$	0	0
$65$	3.33132	0.413200
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−1.74388	−0.209938
$70$	0	0
$71$	4.11298	0.488121	0.244060	−	0.969760i	$-0.421520\pi$
$71$	4.11298	0.488121	0.244060	+	0.969760i	$0.421520\pi$
$72$	0	0
$73$	10.0883	1.18074	0.590370	−	0.807132i	$-0.298982\pi$
$73$	10.0883	1.18074	0.590370	+	0.807132i	$0.298982\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0.537349	0.0612366
$78$	0	0
$79$	−6.71856	−0.755897	−0.377948	−	0.925827i	$-0.623370\pi$
$79$	−6.71856	−0.755897	−0.377948	+	0.925827i	$0.623370\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.07709	0.667047	0.333524	−	0.942742i	$-0.391762\pi$
$83$	6.07709	0.667047	0.333524	+	0.942742i	$0.391762\pi$
$84$	0	0
$85$	7.70550	0.835779
$86$	0	0
$87$	1.69465	0.181685
$88$	0	0
$89$	−15.7411	−1.66855	−0.834277	−	0.551346i	$-0.814114\pi$
$89$	−15.7411	−1.66855	−0.834277	+	0.551346i	$0.814114\pi$
$90$	0	0
$91$	0.707688	0.0741859
$92$	0	0
$93$	−2.26698	−0.235075
$94$	0	0
$95$	−7.01222	−0.719438
$96$	0	0
$97$	7.28158	0.739332	0.369666	−	0.929165i	$-0.379472\pi$
$97$	7.28158	0.739332	0.369666	+	0.929165i	$0.379472\pi$
$98$	0	0
$99$	−2.52948	−0.254222
$100$	0	0
$101$	−5.69269	−0.566444	−0.283222	−	0.959054i	$-0.591403\pi$
$101$	−5.69269	−0.566444	−0.283222	+	0.959054i	$0.591403\pi$
$102$	0	0
$103$	15.5449	1.53169	0.765843	−	0.643028i	$-0.222322\pi$
$103$	15.5449	1.53169	0.765843	+	0.643028i	$0.222322\pi$
$104$	0	0
$105$	0.212435	0.0207315
$106$	0	0
$107$	5.04456	0.487676	0.243838	−	0.969816i	$-0.421594\pi$
$107$	5.04456	0.487676	0.243838	+	0.969816i	$0.421594\pi$
$108$	0	0
$109$	12.4786	1.19523	0.597615	−	0.801783i	$-0.296115\pi$
$109$	12.4786	1.19523	0.597615	+	0.801783i	$0.296115\pi$
$110$	0	0
$111$	9.86329	0.936182
$112$	0	0
$113$	14.5800	1.37157	0.685784	−	0.727805i	$-0.259460\pi$
$113$	14.5800	1.37157	0.685784	+	0.727805i	$0.259460\pi$
$114$	0	0
$115$	1.74388	0.162617
$116$	0	0
$117$	−3.33132	−0.307981
$118$	0	0
$119$	1.63691	0.150056
$120$	0	0
$121$	−4.60173	−0.418339
$122$	0	0
$123$	4.26698	0.384740
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.9783	−1.15163	−0.575817	−	0.817578i	$-0.695316\pi$
$127$	−12.9783	−1.15163	−0.575817	+	0.817578i	$0.695316\pi$
$128$	0	0
$129$	−1.58745	−0.139767
$130$	0	0
$131$	22.1129	1.93201	0.966007	−	0.258517i	$-0.0832339\pi$
$131$	22.1129	1.93201	0.966007	+	0.258517i	$0.0832339\pi$
$132$	0	0
$133$	−1.48964	−0.129168
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	11.9140	1.01788	0.508940	−	0.860802i	$-0.330037\pi$
$137$	11.9140	1.01788	0.508940	+	0.860802i	$0.330037\pi$
$138$	0	0
$139$	9.97144	0.845766	0.422883	−	0.906184i	$-0.361018\pi$
$139$	9.97144	0.845766	0.422883	+	0.906184i	$0.361018\pi$
$140$	0	0
$141$	−5.44041	−0.458165
$142$	0	0
$143$	8.42651	0.704660
$144$	0	0
$145$	−1.69465	−0.140733
$146$	0	0
$147$	−6.95487	−0.573628
$148$	0	0
$149$	−19.1380	−1.56785	−0.783925	−	0.620856i	$-0.786785\pi$
$149$	−19.1380	−1.56785	−0.783925	+	0.620856i	$0.786785\pi$
$150$	0	0
$151$	6.08188	0.494936	0.247468	−	0.968896i	$-0.420401\pi$
$151$	6.08188	0.494936	0.247468	+	0.968896i	$0.420401\pi$
$152$	0	0
$153$	−7.70550	−0.622953
$154$	0	0
$155$	2.26698	0.182088
$156$	0	0
$157$	24.5267	1.95744	0.978722	−	0.205191i	$-0.0657815\pi$
$157$	24.5267	1.95744	0.978722	+	0.205191i	$0.0657815\pi$
$158$	0	0
$159$	6.91104	0.548081
$160$	0	0
$161$	0.370459	0.0291963
$162$	0	0
$163$	17.6879	1.38543	0.692713	−	0.721213i	$-0.256415\pi$
$163$	17.6879	1.38543	0.692713	+	0.721213i	$0.256415\pi$
$164$	0	0
$165$	2.52948	0.196920
$166$	0	0
$167$	6.23694	0.482629	0.241314	−	0.970447i	$-0.422422\pi$
$167$	6.23694	0.482629	0.241314	+	0.970447i	$0.422422\pi$
$168$	0	0
$169$	−1.90229	−0.146330
$170$	0	0
$171$	7.01222	0.536238
$172$	0	0
$173$	10.0057	0.760719	0.380360	−	0.924839i	$-0.375800\pi$
$173$	10.0057	0.760719	0.380360	+	0.924839i	$0.375800\pi$
$174$	0	0
$175$	−0.212435	−0.0160585
$176$	0	0
$177$	−7.97332	−0.599311
$178$	0	0
$179$	−3.75325	−0.280531	−0.140265	−	0.990114i	$-0.544796\pi$
$179$	−3.75325	−0.280531	−0.140265	+	0.990114i	$0.544796\pi$
$180$	0	0
$181$	−1.67917	−0.124812	−0.0624058	−	0.998051i	$-0.519877\pi$
$181$	−1.67917	−0.124812	−0.0624058	+	0.998051i	$0.519877\pi$
$182$	0	0
$183$	8.27524	0.611723
$184$	0	0
$185$	−9.86329	−0.725163
$186$	0	0
$187$	19.4909	1.42532
$188$	0	0
$189$	−0.212435	−0.0154523
$190$	0	0
$191$	−12.2025	−0.882944	−0.441472	−	0.897275i	$-0.645543\pi$
$191$	−12.2025	−0.882944	−0.441472	+	0.897275i	$0.645543\pi$
$192$	0	0
$193$	−3.73803	−0.269069	−0.134535	−	0.990909i	$-0.542954\pi$
$193$	−3.73803	−0.269069	−0.134535	+	0.990909i	$0.542954\pi$
$194$	0	0
$195$	3.33132	0.238561
$196$	0	0
$197$	14.8662	1.05918	0.529588	−	0.848255i	$-0.322347\pi$
$197$	14.8662	1.05918	0.529588	+	0.848255i	$0.322347\pi$
$198$	0	0
$199$	19.6990	1.39642	0.698212	−	0.715891i	$-0.253979\pi$
$199$	19.6990	1.39642	0.698212	+	0.715891i	$0.253979\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−0.360002	−0.0252672
$204$	0	0
$205$	−4.26698	−0.298019
$206$	0	0
$207$	−1.74388	−0.121208
$208$	0	0
$209$	−17.7373	−1.22691
$210$	0	0
$211$	−12.1149	−0.834026	−0.417013	−	0.908901i	$-0.636923\pi$
$211$	−12.1149	−0.834026	−0.417013	+	0.908901i	$0.636923\pi$
$212$	0	0
$213$	4.11298	0.281817
$214$	0	0
$215$	1.58745	0.108263
$216$	0	0
$217$	0.481584	0.0326921
$218$	0	0
$219$	10.0883	0.681701
$220$	0	0
$221$	25.6695	1.72672
$222$	0	0
$223$	16.7125	1.11915	0.559576	−	0.828779i	$-0.310964\pi$
$223$	16.7125	1.11915	0.559576	+	0.828779i	$0.310964\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−12.8830	−0.855072	−0.427536	−	0.903998i	$-0.640618\pi$
$227$	−12.8830	−0.855072	−0.427536	+	0.903998i	$0.640618\pi$
$228$	0	0
$229$	−19.3688	−1.27992	−0.639962	−	0.768406i	$-0.721050\pi$
$229$	−19.3688	−1.27992	−0.639962	+	0.768406i	$0.721050\pi$
$230$	0	0
$231$	0.537349	0.0353550
$232$	0	0
$233$	17.3340	1.13559	0.567793	−	0.823171i	$-0.307797\pi$
$233$	17.3340	1.13559	0.567793	+	0.823171i	$0.307797\pi$
$234$	0	0
$235$	5.44041	0.354893
$236$	0	0
$237$	−6.71856	−0.436417
$238$	0	0
$239$	14.5029	0.938115	0.469058	−	0.883168i	$-0.344594\pi$
$239$	14.5029	0.938115	0.469058	+	0.883168i	$0.344594\pi$
$240$	0	0
$241$	−29.9192	−1.92727	−0.963634	−	0.267227i	$-0.913893\pi$
$241$	−29.9192	−1.92727	−0.963634	+	0.267227i	$0.913893\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.95487	0.444330
$246$	0	0
$247$	−23.3600	−1.48636
$248$	0	0
$249$	6.07709	0.385120
$250$	0	0
$251$	−19.3907	−1.22393	−0.611966	−	0.790884i	$-0.709621\pi$
$251$	−19.3907	−1.22393	−0.611966	+	0.790884i	$0.709621\pi$
$252$	0	0
$253$	4.41110	0.277323
$254$	0	0
$255$	7.70550	0.482537
$256$	0	0
$257$	−20.2810	−1.26509	−0.632547	−	0.774522i	$-0.717991\pi$
$257$	−20.2810	−1.26509	−0.632547	+	0.774522i	$0.717991\pi$
$258$	0	0
$259$	−2.09530	−0.130196
$260$	0	0
$261$	1.69465	0.104896
$262$	0	0
$263$	22.1346	1.36488	0.682440	−	0.730942i	$-0.260919\pi$
$263$	22.1346	1.36488	0.682440	+	0.730942i	$0.260919\pi$
$264$	0	0
$265$	−6.91104	−0.424542
$266$	0	0
$267$	−15.7411	−0.963340
$268$	0	0
$269$	18.6744	1.13860	0.569299	−	0.822131i	$-0.307215\pi$
$269$	18.6744	1.13860	0.569299	+	0.822131i	$0.307215\pi$
$270$	0	0
$271$	1.49646	0.0909035	0.0454518	−	0.998967i	$-0.485527\pi$
$271$	1.49646	0.0909035	0.0454518	+	0.998967i	$0.485527\pi$
$272$	0	0
$273$	0.707688	0.0428312
$274$	0	0
$275$	−2.52948	−0.152533
$276$	0	0
$277$	−23.2529	−1.39713	−0.698565	−	0.715547i	$-0.746177\pi$
$277$	−23.2529	−1.39713	−0.698565	+	0.715547i	$0.746177\pi$
$278$	0	0
$279$	−2.26698	−0.135720
$280$	0	0
$281$	−7.66461	−0.457232	−0.228616	−	0.973517i	$-0.573420\pi$
$281$	−7.66461	−0.457232	−0.228616	+	0.973517i	$0.573420\pi$
$282$	0	0
$283$	−23.6825	−1.40778	−0.703888	−	0.710311i	$-0.748554\pi$
$283$	−23.6825	−1.40778	−0.703888	+	0.710311i	$0.748554\pi$
$284$	0	0
$285$	−7.01222	−0.415368
$286$	0	0
$287$	−0.906454	−0.0535063
$288$	0	0
$289$	42.3748	2.49263
$290$	0	0
$291$	7.28158	0.426854
$292$	0	0
$293$	−9.94969	−0.581267	−0.290634	−	0.956834i	$-0.593866\pi$
$293$	−9.94969	−0.581267	−0.290634	+	0.956834i	$0.593866\pi$
$294$	0	0
$295$	7.97332	0.464225
$296$	0	0
$297$	−2.52948	−0.146775
$298$	0	0
$299$	5.80941	0.335967
$300$	0	0
$301$	0.337229	0.0194375
$302$	0	0
$303$	−5.69269	−0.327037
$304$	0	0
$305$	−8.27524	−0.473839
$306$	0	0
$307$	0.901961	0.0514776	0.0257388	−	0.999669i	$-0.491806\pi$
$307$	0.901961	0.0514776	0.0257388	+	0.999669i	$0.491806\pi$
$308$	0	0
$309$	15.5449	0.884319
$310$	0	0
$311$	30.1842	1.71159	0.855795	−	0.517316i	$-0.173069\pi$
$311$	30.1842	1.71159	0.855795	+	0.517316i	$0.173069\pi$
$312$	0	0
$313$	−30.1003	−1.70137	−0.850685	−	0.525676i	$-0.823812\pi$
$313$	−30.1003	−1.70137	−0.850685	+	0.525676i	$0.823812\pi$
$314$	0	0
$315$	0.212435	0.0119693
$316$	0	0
$317$	18.3980	1.03334	0.516669	−	0.856185i	$-0.327172\pi$
$317$	18.3980	1.03334	0.516669	+	0.856185i	$0.327172\pi$
$318$	0	0
$319$	−4.28658	−0.240002
$320$	0	0
$321$	5.04456	0.281560
$322$	0	0
$323$	−54.0327	−3.00646
$324$	0	0
$325$	−3.33132	−0.184789
$326$	0	0
$327$	12.4786	0.690067
$328$	0	0
$329$	1.15573	0.0637175
$330$	0	0
$331$	−21.1271	−1.16125	−0.580624	−	0.814171i	$-0.697192\pi$
$331$	−21.1271	−1.16125	−0.580624	+	0.814171i	$0.697192\pi$
$332$	0	0
$333$	9.86329	0.540505
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	29.0141	1.58050	0.790250	−	0.612784i	$-0.209950\pi$
$337$	29.0141	1.58050	0.790250	+	0.612784i	$0.209950\pi$
$338$	0	0
$339$	14.5800	0.791875
$340$	0	0
$341$	5.73428	0.310528
$342$	0	0
$343$	2.96450	0.160068
$344$	0	0
$345$	1.74388	0.0938871
$346$	0	0
$347$	27.8380	1.49442	0.747212	−	0.664586i	$-0.231392\pi$
$347$	27.8380	1.49442	0.747212	+	0.664586i	$0.231392\pi$
$348$	0	0
$349$	−2.90755	−0.155638	−0.0778189	−	0.996968i	$-0.524796\pi$
$349$	−2.90755	−0.155638	−0.0778189	+	0.996968i	$0.524796\pi$
$350$	0	0
$351$	−3.33132	−0.177813
$352$	0	0
$353$	21.0904	1.12253	0.561266	−	0.827636i	$-0.310315\pi$
$353$	21.0904	1.12253	0.561266	+	0.827636i	$0.310315\pi$
$354$	0	0
$355$	−4.11298	−0.218294
$356$	0	0
$357$	1.63691	0.0866347
$358$	0	0
$359$	12.0546	0.636216	0.318108	−	0.948055i	$-0.396953\pi$
$359$	12.0546	0.636216	0.318108	+	0.948055i	$0.396953\pi$
$360$	0	0
$361$	30.1712	1.58796
$362$	0	0
$363$	−4.60173	−0.241528
$364$	0	0
$365$	−10.0883	−0.528043
$366$	0	0
$367$	31.2712	1.63234	0.816171	−	0.577811i	$-0.196093\pi$
$367$	31.2712	1.63234	0.816171	+	0.577811i	$0.196093\pi$
$368$	0	0
$369$	4.26698	0.222130
$370$	0	0
$371$	−1.46814	−0.0762222
$372$	0	0
$373$	−29.9857	−1.55260	−0.776301	−	0.630363i	$-0.782906\pi$
$373$	−29.9857	−1.55260	−0.776301	+	0.630363i	$0.782906\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−5.64542	−0.290754
$378$	0	0
$379$	25.0562	1.28705	0.643525	−	0.765426i	$-0.277471\pi$
$379$	25.0562	1.28705	0.643525	+	0.765426i	$0.277471\pi$
$380$	0	0
$381$	−12.9783	−0.664897
$382$	0	0
$383$	−7.30875	−0.373460	−0.186730	−	0.982411i	$-0.559789\pi$
$383$	−7.30875	−0.373460	−0.186730	+	0.982411i	$0.559789\pi$
$384$	0	0
$385$	−0.537349	−0.0273858
$386$	0	0
$387$	−1.58745	−0.0806945
$388$	0	0
$389$	−8.12284	−0.411844	−0.205922	−	0.978568i	$-0.566019\pi$
$389$	−8.12284	−0.411844	−0.205922	+	0.978568i	$0.566019\pi$
$390$	0	0
$391$	13.4374	0.679561
$392$	0	0
$393$	22.1129	1.11545
$394$	0	0
$395$	6.71856	0.338047
$396$	0	0
$397$	21.1178	1.05987	0.529937	−	0.848037i	$-0.322215\pi$
$397$	21.1178	1.05987	0.529937	+	0.848037i	$0.322215\pi$
$398$	0	0
$399$	−1.48964	−0.0745751
$400$	0	0
$401$	−31.8005	−1.58804	−0.794020	−	0.607891i	$-0.792016\pi$
$401$	−31.8005	−1.58804	−0.794020	+	0.607891i	$0.792016\pi$
$402$	0	0
$403$	7.55204	0.376194
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−24.9490	−1.23668
$408$	0	0
$409$	14.3081	0.707488	0.353744	−	0.935342i	$-0.384908\pi$
$409$	14.3081	0.707488	0.353744	+	0.935342i	$0.384908\pi$
$410$	0	0
$411$	11.9140	0.587674
$412$	0	0
$413$	1.69381	0.0833469
$414$	0	0
$415$	−6.07709	−0.298313
$416$	0	0
$417$	9.97144	0.488303
$418$	0	0
$419$	−1.46362	−0.0715026	−0.0357513	−	0.999361i	$-0.511382\pi$
$419$	−1.46362	−0.0715026	−0.0357513	+	0.999361i	$0.511382\pi$
$420$	0	0
$421$	−16.3850	−0.798557	−0.399278	−	0.916830i	$-0.630739\pi$
$421$	−16.3850	−0.798557	−0.399278	+	0.916830i	$0.630739\pi$
$422$	0	0
$423$	−5.44041	−0.264522
$424$	0	0
$425$	−7.70550	−0.373772
$426$	0	0
$427$	−1.75795	−0.0850730
$428$	0	0
$429$	8.42651	0.406836
$430$	0	0
$431$	−13.5448	−0.652431	−0.326216	−	0.945295i	$-0.605774\pi$
$431$	−13.5448	−0.652431	−0.326216	+	0.945295i	$0.605774\pi$
$432$	0	0
$433$	38.2212	1.83679	0.918397	−	0.395659i	$-0.129484\pi$
$433$	38.2212	1.83679	0.918397	+	0.395659i	$0.129484\pi$
$434$	0	0
$435$	−1.69465	−0.0812522
$436$	0	0
$437$	−12.2284	−0.584965
$438$	0	0
$439$	−6.90056	−0.329346	−0.164673	−	0.986348i	$-0.552657\pi$
$439$	−6.90056	−0.329346	−0.164673	+	0.986348i	$0.552657\pi$
$440$	0	0
$441$	−6.95487	−0.331184
$442$	0	0
$443$	−29.2928	−1.39174	−0.695871	−	0.718167i	$-0.744981\pi$
$443$	−29.2928	−1.39174	−0.695871	+	0.718167i	$0.744981\pi$
$444$	0	0
$445$	15.7411	0.746200
$446$	0	0
$447$	−19.1380	−0.905198
$448$	0	0
$449$	−10.4498	−0.493156	−0.246578	−	0.969123i	$-0.579306\pi$
$449$	−10.4498	−0.493156	−0.246578	+	0.969123i	$0.579306\pi$
$450$	0	0
$451$	−10.7932	−0.508234
$452$	0	0
$453$	6.08188	0.285752
$454$	0	0
$455$	−0.707688	−0.0331769
$456$	0	0
$457$	−19.0231	−0.889864	−0.444932	−	0.895564i	$-0.646772\pi$
$457$	−19.0231	−0.889864	−0.444932	+	0.895564i	$0.646772\pi$
$458$	0	0
$459$	−7.70550	−0.359662
$460$	0	0
$461$	7.17635	0.334236	0.167118	−	0.985937i	$-0.446554\pi$
$461$	7.17635	0.334236	0.167118	+	0.985937i	$0.446554\pi$
$462$	0	0
$463$	1.71960	0.0799167	0.0399583	−	0.999201i	$-0.487277\pi$
$463$	1.71960	0.0799167	0.0399583	+	0.999201i	$0.487277\pi$
$464$	0	0
$465$	2.26698	0.105129
$466$	0	0
$467$	−19.9954	−0.925277	−0.462639	−	0.886547i	$-0.653097\pi$
$467$	−19.9954	−0.925277	−0.462639	+	0.886547i	$0.653097\pi$
$468$	0	0
$469$	−0.212435	−0.00980932
$470$	0	0
$471$	24.5267	1.13013
$472$	0	0
$473$	4.01541	0.184629
$474$	0	0
$475$	7.01222	0.321743
$476$	0	0
$477$	6.91104	0.316435
$478$	0	0
$479$	34.5385	1.57811	0.789053	−	0.614325i	$-0.210572\pi$
$479$	34.5385	1.57811	0.789053	+	0.614325i	$0.210572\pi$
$480$	0	0
$481$	−32.8578	−1.49819
$482$	0	0
$483$	0.370459	0.0168565
$484$	0	0
$485$	−7.28158	−0.330640
$486$	0	0
$487$	0.00256256	0.000116121 0	5.80603e−5	−	1.00000i	$-0.499982\pi$
$487$	0.00256256	0.000116121 0	5.80603e−5		1.00000i	$0.499982\pi$
$488$	0	0
$489$	17.6879	0.799876
$490$	0	0
$491$	−0.220395	−0.00994629	−0.00497315	−	0.999988i	$-0.501583\pi$
$491$	−0.220395	−0.00994629	−0.00497315	+	0.999988i	$0.501583\pi$
$492$	0	0
$493$	−13.0581	−0.588108
$494$	0	0
$495$	2.52948	0.113692
$496$	0	0
$497$	−0.873739	−0.0391926
$498$	0	0
$499$	−0.0784875	−0.00351358	−0.00175679	−	0.999998i	$-0.500559\pi$
$499$	−0.0784875	−0.00351358	−0.00175679	+	0.999998i	$0.500559\pi$
$500$	0	0
$501$	6.23694	0.278646
$502$	0	0
$503$	18.5231	0.825903	0.412951	−	0.910753i	$-0.364498\pi$
$503$	18.5231	0.825903	0.412951	+	0.910753i	$0.364498\pi$
$504$	0	0
$505$	5.69269	0.253322
$506$	0	0
$507$	−1.90229	−0.0844836
$508$	0	0
$509$	30.7175	1.36153	0.680765	−	0.732502i	$-0.261648\pi$
$509$	30.7175	1.36153	0.680765	+	0.732502i	$0.261648\pi$
$510$	0	0
$511$	−2.14309	−0.0948049
$512$	0	0
$513$	7.01222	0.309597
$514$	0	0
$515$	−15.5449	−0.684991
$516$	0	0
$517$	13.7614	0.605226
$518$	0	0
$519$	10.0057	0.439201
$520$	0	0
$521$	14.9739	0.656017	0.328009	−	0.944675i	$-0.393622\pi$
$521$	14.9739	0.656017	0.328009	+	0.944675i	$0.393622\pi$
$522$	0	0
$523$	41.2942	1.80567	0.902834	−	0.429990i	$-0.141483\pi$
$523$	41.2942	1.80567	0.902834	+	0.429990i	$0.141483\pi$
$524$	0	0
$525$	−0.212435	−0.00927140
$526$	0	0
$527$	17.4682	0.760927
$528$	0	0
$529$	−19.9589	−0.867778
$530$	0	0
$531$	−7.97332	−0.346013
$532$	0	0
$533$	−14.2147	−0.615706
$534$	0	0
$535$	−5.04456	−0.218095
$536$	0	0
$537$	−3.75325	−0.161964
$538$	0	0
$539$	17.5922	0.757750
$540$	0	0
$541$	−5.13542	−0.220789	−0.110394	−	0.993888i	$-0.535211\pi$
$541$	−5.13542	−0.220789	−0.110394	+	0.993888i	$0.535211\pi$
$542$	0	0
$543$	−1.67917	−0.0720600
$544$	0	0
$545$	−12.4786	−0.534523
$546$	0	0
$547$	12.1938	0.521371	0.260686	−	0.965424i	$-0.416051\pi$
$547$	12.1938	0.521371	0.260686	+	0.965424i	$0.416051\pi$
$548$	0	0
$549$	8.27524	0.353179
$550$	0	0
$551$	11.8832	0.506243
$552$	0	0
$553$	1.42725	0.0606930
$554$	0	0
$555$	−9.86329	−0.418673
$556$	0	0
$557$	24.0906	1.02075	0.510375	−	0.859952i	$-0.329506\pi$
$557$	24.0906	1.02075	0.510375	+	0.859952i	$0.329506\pi$
$558$	0	0
$559$	5.28830	0.223671
$560$	0	0
$561$	19.4909	0.822907
$562$	0	0
$563$	−44.4180	−1.87199	−0.935997	−	0.352007i	$-0.885499\pi$
$563$	−44.4180	−1.87199	−0.935997	+	0.352007i	$0.885499\pi$
$564$	0	0
$565$	−14.5800	−0.613384
$566$	0	0
$567$	−0.212435	−0.00892141
$568$	0	0
$569$	34.6946	1.45447	0.727236	−	0.686387i	$-0.240805\pi$
$569$	34.6946	1.45447	0.727236	+	0.686387i	$0.240805\pi$
$570$	0	0
$571$	7.84002	0.328095	0.164047	−	0.986452i	$-0.447545\pi$
$571$	7.84002	0.328095	0.164047	+	0.986452i	$0.447545\pi$
$572$	0	0
$573$	−12.2025	−0.509768
$574$	0	0
$575$	−1.74388	−0.0727247
$576$	0	0
$577$	26.7169	1.11224	0.556119	−	0.831103i	$-0.312290\pi$
$577$	26.7169	1.11224	0.556119	+	0.831103i	$0.312290\pi$
$578$	0	0
$579$	−3.73803	−0.155347
$580$	0	0
$581$	−1.29098	−0.0535590
$582$	0	0
$583$	−17.4813	−0.724003
$584$	0	0
$585$	3.33132	0.137733
$586$	0	0
$587$	−40.0388	−1.65258	−0.826289	−	0.563246i	$-0.809552\pi$
$587$	−40.0388	−1.65258	−0.826289	+	0.563246i	$0.809552\pi$
$588$	0	0
$589$	−15.8965	−0.655006
$590$	0	0
$591$	14.8662	0.611515
$592$	0	0
$593$	−13.1721	−0.540914	−0.270457	−	0.962732i	$-0.587175\pi$
$593$	−13.1721	−0.540914	−0.270457	+	0.962732i	$0.587175\pi$
$594$	0	0
$595$	−1.63691	−0.0671070
$596$	0	0
$597$	19.6990	0.806226
$598$	0	0
$599$	13.3011	0.543467	0.271734	−	0.962373i	$-0.412403\pi$
$599$	13.3011	0.543467	0.271734	+	0.962373i	$0.412403\pi$
$600$	0	0
$601$	45.1751	1.84273	0.921366	−	0.388696i	$-0.127074\pi$
$601$	45.1751	1.84273	0.921366	+	0.388696i	$0.127074\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	4.60173	0.187087
$606$	0	0
$607$	29.1545	1.18334	0.591672	−	0.806179i	$-0.298468\pi$
$607$	29.1545	1.18334	0.591672	+	0.806179i	$0.298468\pi$
$608$	0	0
$609$	−0.360002	−0.0145880
$610$	0	0
$611$	18.1238	0.733209
$612$	0	0
$613$	−22.2457	−0.898495	−0.449247	−	0.893407i	$-0.648308\pi$
$613$	−22.2457	−0.898495	−0.449247	+	0.893407i	$0.648308\pi$
$614$	0	0
$615$	−4.26698	−0.172061
$616$	0	0
$617$	−3.64339	−0.146677	−0.0733387	−	0.997307i	$-0.523365\pi$
$617$	−3.64339	−0.146677	−0.0733387	+	0.997307i	$0.523365\pi$
$618$	0	0
$619$	42.5751	1.71124	0.855619	−	0.517606i	$-0.173177\pi$
$619$	42.5751	1.71124	0.855619	+	0.517606i	$0.173177\pi$
$620$	0	0
$621$	−1.74388	−0.0699793
$622$	0	0
$623$	3.34395	0.133973
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−17.7373	−0.708358
$628$	0	0
$629$	−76.0016	−3.03038
$630$	0	0
$631$	−0.367702	−0.0146380	−0.00731898	−	0.999973i	$-0.502330\pi$
$631$	−0.367702	−0.0146380	−0.00731898	+	0.999973i	$0.502330\pi$
$632$	0	0
$633$	−12.1149	−0.481525
$634$	0	0
$635$	12.9783	0.515027
$636$	0	0
$637$	23.1689	0.917986
$638$	0	0
$639$	4.11298	0.162707
$640$	0	0
$641$	−25.6375	−1.01262	−0.506311	−	0.862351i	$-0.668991\pi$
$641$	−25.6375	−1.01262	−0.506311	+	0.862351i	$0.668991\pi$
$642$	0	0
$643$	32.7395	1.29112	0.645560	−	0.763710i	$-0.276624\pi$
$643$	32.7395	1.29112	0.645560	+	0.763710i	$0.276624\pi$
$644$	0	0
$645$	1.58745	0.0625057
$646$	0	0
$647$	29.6166	1.16435	0.582174	−	0.813064i	$-0.302202\pi$
$647$	29.6166	1.16435	0.582174	+	0.813064i	$0.302202\pi$
$648$	0	0
$649$	20.1684	0.791677
$650$	0	0
$651$	0.481584	0.0188748
$652$	0	0
$653$	−31.9237	−1.24927	−0.624636	−	0.780916i	$-0.714752\pi$
$653$	−31.9237	−1.24927	−0.624636	+	0.780916i	$0.714752\pi$
$654$	0	0
$655$	−22.1129	−0.864023
$656$	0	0
$657$	10.0883	0.393580
$658$	0	0
$659$	36.5538	1.42394	0.711968	−	0.702212i	$-0.247804\pi$
$659$	36.5538	1.42394	0.711968	+	0.702212i	$0.247804\pi$
$660$	0	0
$661$	8.58271	0.333829	0.166914	−	0.985971i	$-0.446620\pi$
$661$	8.58271	0.333829	0.166914	+	0.985971i	$0.446620\pi$
$662$	0	0
$663$	25.6695	0.996921
$664$	0	0
$665$	1.48964	0.0577656
$666$	0	0
$667$	−2.95526	−0.114428
$668$	0	0
$669$	16.7125	0.646143
$670$	0	0
$671$	−20.9321	−0.808073
$672$	0	0
$673$	−24.7150	−0.952693	−0.476347	−	0.879258i	$-0.658039\pi$
$673$	−24.7150	−0.952693	−0.476347	+	0.879258i	$0.658039\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	2.85042	0.109550	0.0547752	−	0.998499i	$-0.482556\pi$
$677$	2.85042	0.109550	0.0547752	+	0.998499i	$0.482556\pi$
$678$	0	0
$679$	−1.54686	−0.0593630
$680$	0	0
$681$	−12.8830	−0.493676
$682$	0	0
$683$	−13.7574	−0.526411	−0.263206	−	0.964740i	$-0.584780\pi$
$683$	−13.7574	−0.526411	−0.263206	+	0.964740i	$0.584780\pi$
$684$	0	0
$685$	−11.9140	−0.455210
$686$	0	0
$687$	−19.3688	−0.738965
$688$	0	0
$689$	−23.0229	−0.877102
$690$	0	0
$691$	−23.1176	−0.879436	−0.439718	−	0.898136i	$-0.644922\pi$
$691$	−23.1176	−0.879436	−0.439718	+	0.898136i	$0.644922\pi$
$692$	0	0
$693$	0.537349	0.0204122
$694$	0	0
$695$	−9.97144	−0.378238
$696$	0	0
$697$	−32.8792	−1.24539
$698$	0	0
$699$	17.3340	0.655631
$700$	0	0
$701$	−15.6124	−0.589671	−0.294836	−	0.955548i	$-0.595265\pi$
$701$	−15.6124	−0.589671	−0.294836	+	0.955548i	$0.595265\pi$
$702$	0	0
$703$	69.1635	2.60855
$704$	0	0
$705$	5.44041	0.204898
$706$	0	0
$707$	1.20933	0.0454813
$708$	0	0
$709$	−21.6799	−0.814205	−0.407102	−	0.913383i	$-0.633461\pi$
$709$	−21.6799	−0.814205	−0.407102	+	0.913383i	$0.633461\pi$
$710$	0	0
$711$	−6.71856	−0.251966
$712$	0	0
$713$	3.95333	0.148053
$714$	0	0
$715$	−8.42651	−0.315134
$716$	0	0
$717$	14.5029	0.541621
$718$	0	0
$719$	−33.9102	−1.26464	−0.632318	−	0.774709i	$-0.717897\pi$
$719$	−33.9102	−1.26464	−0.632318	+	0.774709i	$0.717897\pi$
$720$	0	0
$721$	−3.30228	−0.122983
$722$	0	0
$723$	−29.9192	−1.11271
$724$	0	0
$725$	1.69465	0.0629377
$726$	0	0
$727$	47.0138	1.74364	0.871822	−	0.489823i	$-0.162938\pi$
$727$	47.0138	1.74364	0.871822	+	0.489823i	$0.162938\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.2321	0.452420
$732$	0	0
$733$	40.6811	1.50259	0.751295	−	0.659966i	$-0.229429\pi$
$733$	40.6811	1.50259	0.751295	+	0.659966i	$0.229429\pi$
$734$	0	0
$735$	6.95487	0.256534
$736$	0	0
$737$	−2.52948	−0.0931746
$738$	0	0
$739$	18.7355	0.689198	0.344599	−	0.938750i	$-0.388015\pi$
$739$	18.7355	0.689198	0.344599	+	0.938750i	$0.388015\pi$
$740$	0	0
$741$	−23.3600	−0.858149
$742$	0	0
$743$	−39.8737	−1.46282	−0.731412	−	0.681936i	$-0.761138\pi$
$743$	−39.8737	−1.46282	−0.731412	+	0.681936i	$0.761138\pi$
$744$	0	0
$745$	19.1380	0.701164
$746$	0	0
$747$	6.07709	0.222349
$748$	0	0
$749$	−1.07164	−0.0391568
$750$	0	0
$751$	45.3132	1.65350	0.826751	−	0.562567i	$-0.190186\pi$
$751$	45.3132	1.65350	0.826751	+	0.562567i	$0.190186\pi$
$752$	0	0
$753$	−19.3907	−0.706638
$754$	0	0
$755$	−6.08188	−0.221342
$756$	0	0
$757$	−46.4498	−1.68825	−0.844123	−	0.536150i	$-0.819878\pi$
$757$	−46.4498	−1.68825	−0.844123	+	0.536150i	$0.819878\pi$
$758$	0	0
$759$	4.41110	0.160113
$760$	0	0
$761$	−19.7862	−0.717248	−0.358624	−	0.933482i	$-0.616754\pi$
$761$	−19.7862	−0.717248	−0.358624	+	0.933482i	$0.616754\pi$
$762$	0	0
$763$	−2.65088	−0.0959683
$764$	0	0
$765$	7.70550	0.278593
$766$	0	0
$767$	26.5617	0.959088
$768$	0	0
$769$	−26.6491	−0.960990	−0.480495	−	0.876997i	$-0.659543\pi$
$769$	−26.6491	−0.960990	−0.480495	+	0.876997i	$0.659543\pi$
$770$	0	0
$771$	−20.2810	−0.730403
$772$	0	0
$773$	−9.66514	−0.347631	−0.173815	−	0.984778i	$-0.555610\pi$
$773$	−9.66514	−0.347631	−0.173815	+	0.984778i	$0.555610\pi$
$774$	0	0
$775$	−2.26698	−0.0814323
$776$	0	0
$777$	−2.09530	−0.0751686
$778$	0	0
$779$	29.9210	1.07203
$780$	0	0
$781$	−10.4037	−0.372274
$782$	0	0
$783$	1.69465	0.0605618
$784$	0	0
$785$	−24.5267	−0.875396
$786$	0	0
$787$	−18.4042	−0.656038	−0.328019	−	0.944671i	$-0.606381\pi$
$787$	−18.4042	−0.656038	−0.328019	+	0.944671i	$0.606381\pi$
$788$	0	0
$789$	22.1346	0.788013
$790$	0	0
$791$	−3.09729	−0.110127
$792$	0	0
$793$	−27.5675	−0.978951
$794$	0	0
$795$	−6.91104	−0.245109
$796$	0	0
$797$	−27.2176	−0.964097	−0.482048	−	0.876145i	$-0.660107\pi$
$797$	−27.2176	−0.964097	−0.482048	+	0.876145i	$0.660107\pi$
$798$	0	0
$799$	41.9211	1.48306
$800$	0	0
$801$	−15.7411	−0.556184
$802$	0	0
$803$	−25.5180	−0.900512
$804$	0	0
$805$	−0.370459	−0.0130570
$806$	0	0
$807$	18.6744	0.657370
$808$	0	0
$809$	6.61720	0.232649	0.116324	−	0.993211i	$-0.462889\pi$
$809$	6.61720	0.232649	0.116324	+	0.993211i	$0.462889\pi$
$810$	0	0
$811$	−8.00432	−0.281070	−0.140535	−	0.990076i	$-0.544882\pi$
$811$	−8.00432	−0.281070	−0.140535	+	0.990076i	$0.544882\pi$
$812$	0	0
$813$	1.49646	0.0524832
$814$	0	0
$815$	−17.6879	−0.619581
$816$	0	0
$817$	−11.1315	−0.389443
$818$	0	0
$819$	0.707688	0.0247286
$820$	0	0
$821$	6.23807	0.217710	0.108855	−	0.994058i	$-0.465282\pi$
$821$	6.23807	0.217710	0.108855	+	0.994058i	$0.465282\pi$
$822$	0	0
$823$	52.7005	1.83702	0.918512	−	0.395394i	$-0.129392\pi$
$823$	52.7005	1.83702	0.918512	+	0.395394i	$0.129392\pi$
$824$	0	0
$825$	−2.52948	−0.0880652
$826$	0	0
$827$	−49.9701	−1.73763	−0.868816	−	0.495135i	$-0.835118\pi$
$827$	−49.9701	−1.73763	−0.868816	+	0.495135i	$0.835118\pi$
$828$	0	0
$829$	−55.9098	−1.94183	−0.970914	−	0.239427i	$-0.923041\pi$
$829$	−55.9098	−1.94183	−0.970914	+	0.239427i	$0.923041\pi$
$830$	0	0
$831$	−23.2529	−0.806633
$832$	0	0
$833$	53.5908	1.85681
$834$	0	0
$835$	−6.23694	−0.215838
$836$	0	0
$837$	−2.26698	−0.0783582
$838$	0	0
$839$	−22.8083	−0.787428	−0.393714	−	0.919233i	$-0.628810\pi$
$839$	−22.8083	−0.787428	−0.393714	+	0.919233i	$0.628810\pi$
$840$	0	0
$841$	−26.1282	−0.900971
$842$	0	0
$843$	−7.66461	−0.263983
$844$	0	0
$845$	1.90229	0.0654407
$846$	0	0
$847$	0.977567	0.0335896
$848$	0	0
$849$	−23.6825	−0.812780
$850$	0	0
$851$	−17.2003	−0.589620
$852$	0	0
$853$	8.14727	0.278957	0.139479	−	0.990225i	$-0.455457\pi$
$853$	8.14727	0.278957	0.139479	+	0.990225i	$0.455457\pi$
$854$	0	0
$855$	−7.01222	−0.239813
$856$	0	0
$857$	−43.0389	−1.47018	−0.735091	−	0.677968i	$-0.762861\pi$
$857$	−43.0389	−1.47018	−0.735091	+	0.677968i	$0.762861\pi$
$858$	0	0
$859$	44.2750	1.51064	0.755322	−	0.655354i	$-0.227480\pi$
$859$	44.2750	1.51064	0.755322	+	0.655354i	$0.227480\pi$
$860$	0	0
$861$	−0.906454	−0.0308919
$862$	0	0
$863$	3.63083	0.123595	0.0617974	−	0.998089i	$-0.480317\pi$
$863$	3.63083	0.123595	0.0617974	+	0.998089i	$0.480317\pi$
$864$	0	0
$865$	−10.0057	−0.340204
$866$	0	0
$867$	42.3748	1.43912
$868$	0	0
$869$	16.9945	0.576498
$870$	0	0
$871$	−3.33132	−0.112878
$872$	0	0
$873$	7.28158	0.246444
$874$	0	0
$875$	0.212435	0.00718160
$876$	0	0
$877$	−50.7268	−1.71292	−0.856461	−	0.516212i	$-0.827341\pi$
$877$	−50.7268	−1.71292	−0.856461	+	0.516212i	$0.827341\pi$
$878$	0	0
$879$	−9.94969	−0.335595
$880$	0	0
$881$	−30.5139	−1.02804	−0.514019	−	0.857779i	$-0.671844\pi$
$881$	−30.5139	−1.02804	−0.514019	+	0.857779i	$0.671844\pi$
$882$	0	0
$883$	−1.88298	−0.0633674	−0.0316837	−	0.999498i	$-0.510087\pi$
$883$	−1.88298	−0.0633674	−0.0316837	+	0.999498i	$0.510087\pi$
$884$	0	0
$885$	7.97332	0.268020
$886$	0	0
$887$	41.0893	1.37964	0.689822	−	0.723979i	$-0.257689\pi$
$887$	41.0893	1.37964	0.689822	+	0.723979i	$0.257689\pi$
$888$	0	0
$889$	2.75703	0.0924679
$890$	0	0
$891$	−2.52948	−0.0847408
$892$	0	0
$893$	−38.1493	−1.27662
$894$	0	0
$895$	3.75325	0.125457
$896$	0	0
$897$	5.80941	0.193971
$898$	0	0
$899$	−3.84173	−0.128129
$900$	0	0
$901$	−53.2530	−1.77412
$902$	0	0
$903$	0.337229	0.0112223
$904$	0	0
$905$	1.67917	0.0558174
$906$	0	0
$907$	39.8574	1.32344	0.661721	−	0.749750i	$-0.269826\pi$
$907$	39.8574	1.32344	0.661721	+	0.749750i	$0.269826\pi$
$908$	0	0
$909$	−5.69269	−0.188815
$910$	0	0
$911$	−34.2419	−1.13449	−0.567243	−	0.823551i	$-0.691990\pi$
$911$	−34.2419	−1.13449	−0.567243	+	0.823551i	$0.691990\pi$
$912$	0	0
$913$	−15.3719	−0.508735
$914$	0	0
$915$	−8.27524	−0.273571
$916$	0	0
$917$	−4.69754	−0.155127
$918$	0	0
$919$	−20.0814	−0.662425	−0.331212	−	0.943556i	$-0.607458\pi$
$919$	−20.0814	−0.662425	−0.331212	+	0.943556i	$0.607458\pi$
$920$	0	0
$921$	0.901961	0.0297206
$922$	0	0
$923$	−13.7017	−0.450996
$924$	0	0
$925$	9.86329	0.324303
$926$	0	0
$927$	15.5449	0.510562
$928$	0	0
$929$	5.81521	0.190791	0.0953955	−	0.995439i	$-0.469588\pi$
$929$	5.81521	0.190791	0.0953955	+	0.995439i	$0.469588\pi$
$930$	0	0
$931$	−48.7691	−1.59834
$932$	0	0
$933$	30.1842	0.988187
$934$	0	0
$935$	−19.4909	−0.637421
$936$	0	0
$937$	20.1697	0.658914	0.329457	−	0.944171i	$-0.393134\pi$
$937$	20.1697	0.658914	0.329457	+	0.944171i	$0.393134\pi$
$938$	0	0
$939$	−30.1003	−0.982286
$940$	0	0
$941$	55.8940	1.82209	0.911046	−	0.412304i	$-0.135276\pi$
$941$	55.8940	1.82209	0.911046	+	0.412304i	$0.135276\pi$
$942$	0	0
$943$	−7.44108	−0.242315
$944$	0	0
$945$	0.212435	0.00691050
$946$	0	0
$947$	48.9753	1.59148	0.795742	−	0.605636i	$-0.207081\pi$
$947$	48.9753	1.59148	0.795742	+	0.605636i	$0.207081\pi$
$948$	0	0
$949$	−33.6072	−1.09094
$950$	0	0
$951$	18.3980	0.596598
$952$	0	0
$953$	−7.58428	−0.245679	−0.122839	−	0.992427i	$-0.539200\pi$
$953$	−7.58428	−0.245679	−0.122839	+	0.992427i	$0.539200\pi$
$954$	0	0
$955$	12.2025	0.394864
$956$	0	0
$957$	−4.28658	−0.138565
$958$	0	0
$959$	−2.53094	−0.0817284
$960$	0	0
$961$	−25.8608	−0.834220
$962$	0	0
$963$	5.04456	0.162559
$964$	0	0
$965$	3.73803	0.120331
$966$	0	0
$967$	−42.5001	−1.36671	−0.683355	−	0.730086i	$-0.739480\pi$
$967$	−42.5001	−1.36671	−0.683355	+	0.730086i	$0.739480\pi$
$968$	0	0
$969$	−54.0327	−1.73578
$970$	0	0
$971$	−40.8770	−1.31180	−0.655902	−	0.754846i	$-0.727712\pi$
$971$	−40.8770	−1.31180	−0.655902	+	0.754846i	$0.727712\pi$
$972$	0	0
$973$	−2.11828	−0.0679089
$974$	0	0
$975$	−3.33132	−0.106688
$976$	0	0
$977$	27.5634	0.881830	0.440915	−	0.897549i	$-0.354654\pi$
$977$	27.5634	0.881830	0.440915	+	0.897549i	$0.354654\pi$
$978$	0	0
$979$	39.8168	1.27255
$980$	0	0
$981$	12.4786	0.398410
$982$	0	0
$983$	21.7873	0.694906	0.347453	−	0.937697i	$-0.387047\pi$
$983$	21.7873	0.694906	0.347453	+	0.937697i	$0.387047\pi$
$984$	0	0
$985$	−14.8662	−0.473678
$986$	0	0
$987$	1.15573	0.0367873
$988$	0	0
$989$	2.76831	0.0880271
$990$	0	0
$991$	−50.4050	−1.60117	−0.800583	−	0.599222i	$-0.795477\pi$
$991$	−50.4050	−1.60117	−0.800583	+	0.599222i	$0.795477\pi$
$992$	0	0
$993$	−21.1271	−0.670447
$994$	0	0
$995$	−19.6990	−0.624500
$996$	0	0
$997$	5.37551	0.170244	0.0851221	−	0.996371i	$-0.472872\pi$
$997$	5.37551	0.170244	0.0851221	+	0.996371i	$0.472872\pi$
$998$	0	0
$999$	9.86329	0.312061

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.t.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.t.1.3	✓	7	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.212435 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.212435 q^{7} +1.00000 q^{9} -2.52948 q^{11} -3.33132 q^{13} -1.00000 q^{15} -7.70550 q^{17} +7.01222 q^{19} -0.212435 q^{21} -1.74388 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.69465 q^{29} -2.26698 q^{31} -2.52948 q^{33} +0.212435 q^{35} +9.86329 q^{37} -3.33132 q^{39} +4.26698 q^{41} -1.58745 q^{43} -1.00000 q^{45} -5.44041 q^{47} -6.95487 q^{49} -7.70550 q^{51} +6.91104 q^{53} +2.52948 q^{55} +7.01222 q^{57} -7.97332 q^{59} +8.27524 q^{61} -0.212435 q^{63} +3.33132 q^{65} +1.00000 q^{67} -1.74388 q^{69} +4.11298 q^{71} +10.0883 q^{73} +1.00000 q^{75} +0.537349 q^{77} -6.71856 q^{79} +1.00000 q^{81} +6.07709 q^{83} +7.70550 q^{85} +1.69465 q^{87} -15.7411 q^{89} +0.707688 q^{91} -2.26698 q^{93} -7.01222 q^{95} +7.28158 q^{97} -2.52948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9	\( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9} - q^{13} - 7 q^{15} - 2 q^{17} + 9 q^{19} + 10 q^{21} + 2 q^{23} + 7 q^{25} + 7 q^{27} - q^{29} + 9 q^{31} - 10 q^{35} + 23 q^{37} - q^{39} + 5 q^{41} - 3 q^{43} - 7 q^{45} + 11 q^{47} + 13 q^{49} - 2 q^{51} + 13 q^{53} + 9 q^{57} + q^{59} + 4 q^{61} + 10 q^{63} + q^{65} + 7 q^{67} + 2 q^{69} + q^{71} + 14 q^{73} + 7 q^{75} + 18 q^{77} + 25 q^{79} + 7 q^{81} - 29 q^{83} + 2 q^{85} - q^{87} + 7 q^{89} + 27 q^{91} + 9 q^{93} - 9 q^{95} + 38 q^{97}+O(q^{100}) \) 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9 - q^13 - 7 * q^15 - 2 * q^17 + 9 * q^19 + 10 * q^21 + 2 * q^23 + 7 * q^25 + 7 * q^27 - q^29 + 9 * q^31 - 10 * q^35 + 23 * q^37 - q^39 + 5 * q^41 - 3 * q^43 - 7 * q^45 + 11 * q^47 + 13 * q^49 - 2 * q^51 + 13 * q^53 + 9 * q^57 + q^59 + 4 * q^61 + 10 * q^63 + q^65 + 7 * q^67 + 2 * q^69 + q^71 + 14 * q^73 + 7 * q^75 + 18 * q^77 + 25 * q^79 + 7 * q^81 - 29 * q^83 + 2 * q^85 - q^87 + 7 * q^89 + 27 * q^91 + 9 * q^93 - 9 * q^95 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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