LMFDB - Embedded newform 3120.2.a.bj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, 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("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.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:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	3120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +2.71982 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +2.71982 q^{7} +1.00000 q^{9} +2.71982 q^{11} +1.00000 q^{13} -1.00000 q^{15} -2.83709 q^{17} +3.55691 q^{19} +2.71982 q^{21} +4.83709 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -7.55691 q^{31} +2.71982 q^{33} -2.71982 q^{35} -4.27674 q^{37} +1.00000 q^{39} +2.83709 q^{41} -11.1138 q^{43} -1.00000 q^{45} +11.5569 q^{47} +0.397442 q^{49} -2.83709 q^{51} +1.16291 q^{53} -2.71982 q^{55} +3.55691 q^{57} +2.11727 q^{59} +6.60256 q^{61} +2.71982 q^{63} -1.00000 q^{65} -1.88273 q^{67} +4.83709 q^{69} +6.71982 q^{71} +9.11383 q^{73} +1.00000 q^{75} +7.39744 q^{77} -10.2767 q^{79} +1.00000 q^{81} -2.11727 q^{83} +2.83709 q^{85} +6.00000 q^{87} +1.16291 q^{89} +2.71982 q^{91} -7.55691 q^{93} -3.55691 q^{95} -10.8371 q^{97} +2.71982 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - 3 q^{15} - q^{17} - 6 q^{19} - q^{21} + 7 q^{23} + 3 q^{25} + 3 q^{27} + 18 q^{29} - 6 q^{31} - q^{33} + q^{35} + 13 q^{37} + 3 q^{39} + q^{41} - 3 q^{45} + 18 q^{47} + 12 q^{49} - q^{51} + 11 q^{53} + q^{55} - 6 q^{57} + 8 q^{59} + 9 q^{61} - q^{63} - 3 q^{65} - 4 q^{67} + 7 q^{69} + 11 q^{71} - 6 q^{73} + 3 q^{75} + 33 q^{77} - 5 q^{79} + 3 q^{81} - 8 q^{83} + q^{85} + 18 q^{87} + 11 q^{89} - q^{91} - 6 q^{93} + 6 q^{95} - 25 q^{97} - q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9 - q^11 + 3 * q^13 - 3 * q^15 - q^17 - 6 * q^19 - q^21 + 7 * q^23 + 3 * q^25 + 3 * q^27 + 18 * q^29 - 6 * q^31 - q^33 + q^35 + 13 * q^37 + 3 * q^39 + q^41 - 3 * q^45 + 18 * q^47 + 12 * q^49 - q^51 + 11 * q^53 + q^55 - 6 * q^57 + 8 * q^59 + 9 * q^61 - q^63 - 3 * q^65 - 4 * q^67 + 7 * q^69 + 11 * q^71 - 6 * q^73 + 3 * q^75 + 33 * q^77 - 5 * q^79 + 3 * q^81 - 8 * q^83 + q^85 + 18 * q^87 + 11 * q^89 - q^91 - 6 * q^93 + 6 * q^95 - 25 * q^97 - 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.71982	1.02800	0.513998	−	0.857791i	$-0.328164\pi$
$7$	2.71982	1.02800	0.513998	+	0.857791i	$0.328164\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.71982	0.820058	0.410029	−	0.912073i	$-0.365519\pi$
$11$	2.71982	0.820058	0.410029	+	0.912073i	$0.365519\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.83709	−0.688095	−0.344048	−	0.938952i	$-0.611798\pi$
$17$	−2.83709	−0.688095	−0.344048	+	0.938952i	$0.611798\pi$
$18$	0	0
$19$	3.55691	0.816012	0.408006	−	0.912979i	$-0.366224\pi$
$19$	3.55691	0.816012	0.408006	+	0.912979i	$0.366224\pi$
$20$	0	0
$21$	2.71982	0.593514
$22$	0	0
$23$	4.83709	1.00860	0.504302	−	0.863528i	$-0.331750\pi$
$23$	4.83709	1.00860	0.504302	+	0.863528i	$0.331750\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−7.55691	−1.35726	−0.678631	−	0.734479i	$-0.737426\pi$
$31$	−7.55691	−1.35726	−0.678631	+	0.734479i	$0.737426\pi$
$32$	0	0
$33$	2.71982	0.473461
$34$	0	0
$35$	−2.71982	−0.459734
$36$	0	0
$37$	−4.27674	−0.703091	−0.351546	−	0.936171i	$-0.614344\pi$
$37$	−4.27674	−0.703091	−0.351546	+	0.936171i	$0.614344\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	2.83709	0.443079	0.221540	−	0.975151i	$-0.428892\pi$
$41$	2.83709	0.443079	0.221540	+	0.975151i	$0.428892\pi$
$42$	0	0
$43$	−11.1138	−1.69484	−0.847421	−	0.530921i	$-0.821846\pi$
$43$	−11.1138	−1.69484	−0.847421	+	0.530921i	$0.821846\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	11.5569	1.68575	0.842875	−	0.538110i	$-0.180862\pi$
$47$	11.5569	1.68575	0.842875	+	0.538110i	$0.180862\pi$
$48$	0	0
$49$	0.397442	0.0567775
$50$	0	0
$51$	−2.83709	−0.397272
$52$	0	0
$53$	1.16291	0.159738	0.0798690	−	0.996805i	$-0.474550\pi$
$53$	1.16291	0.159738	0.0798690	+	0.996805i	$0.474550\pi$
$54$	0	0
$55$	−2.71982	−0.366741
$56$	0	0
$57$	3.55691	0.471125
$58$	0	0
$59$	2.11727	0.275645	0.137822	−	0.990457i	$-0.455990\pi$
$59$	2.11727	0.275645	0.137822	+	0.990457i	$0.455990\pi$
$60$	0	0
$61$	6.60256	0.845371	0.422685	−	0.906276i	$-0.361088\pi$
$61$	6.60256	0.845371	0.422685	+	0.906276i	$0.361088\pi$
$62$	0	0
$63$	2.71982	0.342666
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−1.88273	−0.230013	−0.115006	−	0.993365i	$-0.536689\pi$
$67$	−1.88273	−0.230013	−0.115006	+	0.993365i	$0.536689\pi$
$68$	0	0
$69$	4.83709	0.582317
$70$	0	0
$71$	6.71982	0.797496	0.398748	−	0.917060i	$-0.369445\pi$
$71$	6.71982	0.797496	0.398748	+	0.917060i	$0.369445\pi$
$72$	0	0
$73$	9.11383	1.06669	0.533346	−	0.845897i	$-0.320934\pi$
$73$	9.11383	1.06669	0.533346	+	0.845897i	$0.320934\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	7.39744	0.843017
$78$	0	0
$79$	−10.2767	−1.15622	−0.578112	−	0.815958i	$-0.696210\pi$
$79$	−10.2767	−1.15622	−0.578112	+	0.815958i	$0.696210\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.11727	−0.232400	−0.116200	−	0.993226i	$-0.537071\pi$
$83$	−2.11727	−0.232400	−0.116200	+	0.993226i	$0.537071\pi$
$84$	0	0
$85$	2.83709	0.307726
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	1.16291	0.123268	0.0616341	−	0.998099i	$-0.480369\pi$
$89$	1.16291	0.123268	0.0616341	+	0.998099i	$0.480369\pi$
$90$	0	0
$91$	2.71982	0.285115
$92$	0	0
$93$	−7.55691	−0.783616
$94$	0	0
$95$	−3.55691	−0.364932
$96$	0	0
$97$	−10.8371	−1.10034	−0.550170	−	0.835053i	$-0.685437\pi$
$97$	−10.8371	−1.10034	−0.550170	+	0.835053i	$0.685437\pi$
$98$	0	0
$99$	2.71982	0.273353
$100$	0	0
$101$	7.67418	0.763610	0.381805	−	0.924243i	$-0.375303\pi$
$101$	7.67418	0.763610	0.381805	+	0.924243i	$0.375303\pi$
$102$	0	0
$103$	−3.76547	−0.371023	−0.185511	−	0.982642i	$-0.559394\pi$
$103$	−3.76547	−0.371023	−0.185511	+	0.982642i	$0.559394\pi$
$104$	0	0
$105$	−2.71982	−0.265428
$106$	0	0
$107$	12.6026	1.21834	0.609168	−	0.793041i	$-0.291504\pi$
$107$	12.6026	1.21834	0.609168	+	0.793041i	$0.291504\pi$
$108$	0	0
$109$	11.4396	1.09572	0.547860	−	0.836570i	$-0.315443\pi$
$109$	11.4396	1.09572	0.547860	+	0.836570i	$0.315443\pi$
$110$	0	0
$111$	−4.27674	−0.405930
$112$	0	0
$113$	−13.1138	−1.23365	−0.616823	−	0.787102i	$-0.711580\pi$
$113$	−13.1138	−1.23365	−0.616823	+	0.787102i	$0.711580\pi$
$114$	0	0
$115$	−4.83709	−0.451061
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−7.71639	−0.707360
$120$	0	0
$121$	−3.60256	−0.327505
$122$	0	0
$123$	2.83709	0.255812
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.4396	1.19258	0.596288	−	0.802771i	$-0.296642\pi$
$127$	13.4396	1.19258	0.596288	+	0.802771i	$0.296642\pi$
$128$	0	0
$129$	−11.1138	−0.978518
$130$	0	0
$131$	9.43965	0.824746	0.412373	−	0.911015i	$-0.364700\pi$
$131$	9.43965	0.824746	0.412373	+	0.911015i	$0.364700\pi$
$132$	0	0
$133$	9.67418	0.838858
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	1.76547	0.150834	0.0754170	−	0.997152i	$-0.475971\pi$
$137$	1.76547	0.150834	0.0754170	+	0.997152i	$0.475971\pi$
$138$	0	0
$139$	6.27674	0.532386	0.266193	−	0.963920i	$-0.414234\pi$
$139$	6.27674	0.532386	0.266193	+	0.963920i	$0.414234\pi$
$140$	0	0
$141$	11.5569	0.973268
$142$	0	0
$143$	2.71982	0.227443
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0.397442	0.0327805
$148$	0	0
$149$	−20.8302	−1.70648	−0.853239	−	0.521520i	$-0.825365\pi$
$149$	−20.8302	−1.70648	−0.853239	+	0.521520i	$0.825365\pi$
$150$	0	0
$151$	−4.99656	−0.406614	−0.203307	−	0.979115i	$-0.565169\pi$
$151$	−4.99656	−0.406614	−0.203307	+	0.979115i	$0.565169\pi$
$152$	0	0
$153$	−2.83709	−0.229365
$154$	0	0
$155$	7.55691	0.606986
$156$	0	0
$157$	8.87930	0.708645	0.354322	−	0.935123i	$-0.384712\pi$
$157$	8.87930	0.708645	0.354322	+	0.935123i	$0.384712\pi$
$158$	0	0
$159$	1.16291	0.0922247
$160$	0	0
$161$	13.1560	1.03684
$162$	0	0
$163$	13.8337	1.08354	0.541768	−	0.840528i	$-0.317755\pi$
$163$	13.8337	1.08354	0.541768	+	0.840528i	$0.317755\pi$
$164$	0	0
$165$	−2.71982	−0.211738
$166$	0	0
$167$	9.88273	0.764749	0.382374	−	0.924007i	$-0.375106\pi$
$167$	9.88273	0.764749	0.382374	+	0.924007i	$0.375106\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.55691	0.272004
$172$	0	0
$173$	13.1138	0.997026	0.498513	−	0.866882i	$-0.333880\pi$
$173$	13.1138	0.997026	0.498513	+	0.866882i	$0.333880\pi$
$174$	0	0
$175$	2.71982	0.205599
$176$	0	0
$177$	2.11727	0.159143
$178$	0	0
$179$	−8.55348	−0.639317	−0.319658	−	0.947533i	$-0.603568\pi$
$179$	−8.55348	−0.639317	−0.319658	+	0.947533i	$0.603568\pi$
$180$	0	0
$181$	3.72326	0.276748	0.138374	−	0.990380i	$-0.455812\pi$
$181$	3.72326	0.276748	0.138374	+	0.990380i	$0.455812\pi$
$182$	0	0
$183$	6.60256	0.488075
$184$	0	0
$185$	4.27674	0.314432
$186$	0	0
$187$	−7.71639	−0.564278
$188$	0	0
$189$	2.71982	0.197838
$190$	0	0
$191$	4.23453	0.306400	0.153200	−	0.988195i	$-0.451042\pi$
$191$	4.23453	0.306400	0.153200	+	0.988195i	$0.451042\pi$
$192$	0	0
$193$	−23.3906	−1.68369	−0.841845	−	0.539719i	$-0.818530\pi$
$193$	−23.3906	−1.68369	−0.841845	+	0.539719i	$0.818530\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	14.5535	1.03689	0.518446	−	0.855110i	$-0.326511\pi$
$197$	14.5535	1.03689	0.518446	+	0.855110i	$0.326511\pi$
$198$	0	0
$199$	15.1138	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$199$	15.1138	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$200$	0	0
$201$	−1.88273	−0.132798
$202$	0	0
$203$	16.3189	1.14537
$204$	0	0
$205$	−2.83709	−0.198151
$206$	0	0
$207$	4.83709	0.336201
$208$	0	0
$209$	9.67418	0.669177
$210$	0	0
$211$	18.2277	1.25484	0.627422	−	0.778680i	$-0.284110\pi$
$211$	18.2277	1.25484	0.627422	+	0.778680i	$0.284110\pi$
$212$	0	0
$213$	6.71982	0.460435
$214$	0	0
$215$	11.1138	0.757957
$216$	0	0
$217$	−20.5535	−1.39526
$218$	0	0
$219$	9.11383	0.615855
$220$	0	0
$221$	−2.83709	−0.190843
$222$	0	0
$223$	−10.1173	−0.677502	−0.338751	−	0.940876i	$-0.610004\pi$
$223$	−10.1173	−0.677502	−0.338751	+	0.940876i	$0.610004\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−11.3224	−0.751493	−0.375746	−	0.926723i	$-0.622614\pi$
$227$	−11.3224	−0.751493	−0.375746	+	0.926723i	$0.622614\pi$
$228$	0	0
$229$	−6.23453	−0.411990	−0.205995	−	0.978553i	$-0.566043\pi$
$229$	−6.23453	−0.411990	−0.205995	+	0.978553i	$0.566043\pi$
$230$	0	0
$231$	7.39744	0.486716
$232$	0	0
$233$	6.83709	0.447913	0.223956	−	0.974599i	$-0.428103\pi$
$233$	6.83709	0.447913	0.223956	+	0.974599i	$0.428103\pi$
$234$	0	0
$235$	−11.5569	−0.753890
$236$	0	0
$237$	−10.2767	−0.667546
$238$	0	0
$239$	−1.28018	−0.0828077	−0.0414039	−	0.999142i	$-0.513183\pi$
$239$	−1.28018	−0.0828077	−0.0414039	+	0.999142i	$0.513183\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.397442	−0.0253917
$246$	0	0
$247$	3.55691	0.226321
$248$	0	0
$249$	−2.11727	−0.134176
$250$	0	0
$251$	−18.2277	−1.15052	−0.575260	−	0.817971i	$-0.695099\pi$
$251$	−18.2277	−1.15052	−0.575260	+	0.817971i	$0.695099\pi$
$252$	0	0
$253$	13.1560	0.827113
$254$	0	0
$255$	2.83709	0.177665
$256$	0	0
$257$	1.11383	0.0694787	0.0347394	−	0.999396i	$-0.488940\pi$
$257$	1.11383	0.0694787	0.0347394	+	0.999396i	$0.488940\pi$
$258$	0	0
$259$	−11.6320	−0.722776
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	−1.16291	−0.0714370
$266$	0	0
$267$	1.16291	0.0711689
$268$	0	0
$269$	15.6742	0.955672	0.477836	−	0.878449i	$-0.341421\pi$
$269$	15.6742	0.955672	0.477836	+	0.878449i	$0.341421\pi$
$270$	0	0
$271$	−0.443086	−0.0269155	−0.0134578	−	0.999909i	$-0.504284\pi$
$271$	−0.443086	−0.0269155	−0.0134578	+	0.999909i	$0.504284\pi$
$272$	0	0
$273$	2.71982	0.164611
$274$	0	0
$275$	2.71982	0.164012
$276$	0	0
$277$	−4.87930	−0.293168	−0.146584	−	0.989198i	$-0.546828\pi$
$277$	−4.87930	−0.293168	−0.146584	+	0.989198i	$0.546828\pi$
$278$	0	0
$279$	−7.55691	−0.452421
$280$	0	0
$281$	−9.11383	−0.543685	−0.271843	−	0.962342i	$-0.587633\pi$
$281$	−9.11383	−0.543685	−0.271843	+	0.962342i	$0.587633\pi$
$282$	0	0
$283$	−33.3415	−1.98195	−0.990973	−	0.134063i	$-0.957198\pi$
$283$	−33.3415	−1.98195	−0.990973	+	0.134063i	$0.957198\pi$
$284$	0	0
$285$	−3.55691	−0.210693
$286$	0	0
$287$	7.71639	0.455484
$288$	0	0
$289$	−8.95092	−0.526525
$290$	0	0
$291$	−10.8371	−0.635281
$292$	0	0
$293$	−29.4328	−1.71948	−0.859740	−	0.510731i	$-0.829375\pi$
$293$	−29.4328	−1.71948	−0.859740	+	0.510731i	$0.829375\pi$
$294$	0	0
$295$	−2.11727	−0.123272
$296$	0	0
$297$	2.71982	0.157820
$298$	0	0
$299$	4.83709	0.279736
$300$	0	0
$301$	−30.2277	−1.74229
$302$	0	0
$303$	7.67418	0.440870
$304$	0	0
$305$	−6.60256	−0.378061
$306$	0	0
$307$	21.8337	1.24611	0.623056	−	0.782177i	$-0.285891\pi$
$307$	21.8337	1.24611	0.623056	+	0.782177i	$0.285891\pi$
$308$	0	0
$309$	−3.76547	−0.214210
$310$	0	0
$311$	−25.1070	−1.42368	−0.711842	−	0.702339i	$-0.752139\pi$
$311$	−25.1070	−1.42368	−0.711842	+	0.702339i	$0.752139\pi$
$312$	0	0
$313$	8.22766	0.465055	0.232527	−	0.972590i	$-0.425300\pi$
$313$	8.22766	0.465055	0.232527	+	0.972590i	$0.425300\pi$
$314$	0	0
$315$	−2.71982	−0.153245
$316$	0	0
$317$	27.6742	1.55434	0.777168	−	0.629293i	$-0.216655\pi$
$317$	27.6742	1.55434	0.777168	+	0.629293i	$0.216655\pi$
$318$	0	0
$319$	16.3189	0.913685
$320$	0	0
$321$	12.6026	0.703406
$322$	0	0
$323$	−10.0913	−0.561494
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	11.4396	0.632614
$328$	0	0
$329$	31.4328	1.73294
$330$	0	0
$331$	13.2311	0.727247	0.363623	−	0.931546i	$-0.381540\pi$
$331$	13.2311	0.727247	0.363623	+	0.931546i	$0.381540\pi$
$332$	0	0
$333$	−4.27674	−0.234364
$334$	0	0
$335$	1.88273	0.102865
$336$	0	0
$337$	−4.32582	−0.235642	−0.117821	−	0.993035i	$-0.537591\pi$
$337$	−4.32582	−0.235642	−0.117821	+	0.993035i	$0.537591\pi$
$338$	0	0
$339$	−13.1138	−0.712245
$340$	0	0
$341$	−20.5535	−1.11303
$342$	0	0
$343$	−17.9578	−0.969630
$344$	0	0
$345$	−4.83709	−0.260420
$346$	0	0
$347$	−6.27674	−0.336953	−0.168476	−	0.985706i	$-0.553885\pi$
$347$	−6.27674	−0.336953	−0.168476	+	0.985706i	$0.553885\pi$
$348$	0	0
$349$	17.6673	0.945709	0.472855	−	0.881140i	$-0.343224\pi$
$349$	17.6673	0.945709	0.472855	+	0.881140i	$0.343224\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−13.7655	−0.732662	−0.366331	−	0.930485i	$-0.619386\pi$
$353$	−13.7655	−0.732662	−0.366331	+	0.930485i	$0.619386\pi$
$354$	0	0
$355$	−6.71982	−0.356651
$356$	0	0
$357$	−7.71639	−0.408394
$358$	0	0
$359$	−0.996562	−0.0525965	−0.0262983	−	0.999654i	$-0.508372\pi$
$359$	−0.996562	−0.0525965	−0.0262983	+	0.999654i	$0.508372\pi$
$360$	0	0
$361$	−6.34836	−0.334124
$362$	0	0
$363$	−3.60256	−0.189085
$364$	0	0
$365$	−9.11383	−0.477040
$366$	0	0
$367$	−14.2277	−0.742678	−0.371339	−	0.928497i	$-0.621101\pi$
$367$	−14.2277	−0.742678	−0.371339	+	0.928497i	$0.621101\pi$
$368$	0	0
$369$	2.83709	0.147693
$370$	0	0
$371$	3.16291	0.164210
$372$	0	0
$373$	15.6742	0.811578	0.405789	−	0.913967i	$-0.366997\pi$
$373$	15.6742	0.811578	0.405789	+	0.913967i	$0.366997\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	−26.2017	−1.34589	−0.672945	−	0.739693i	$-0.734971\pi$
$379$	−26.2017	−1.34589	−0.672945	+	0.739693i	$0.734971\pi$
$380$	0	0
$381$	13.4396	0.688534
$382$	0	0
$383$	−22.4362	−1.14644	−0.573218	−	0.819403i	$-0.694305\pi$
$383$	−22.4362	−1.14644	−0.573218	+	0.819403i	$0.694305\pi$
$384$	0	0
$385$	−7.39744	−0.377009
$386$	0	0
$387$	−11.1138	−0.564948
$388$	0	0
$389$	31.6742	1.60594	0.802972	−	0.596016i	$-0.203251\pi$
$389$	31.6742	1.60594	0.802972	+	0.596016i	$0.203251\pi$
$390$	0	0
$391$	−13.7233	−0.694015
$392$	0	0
$393$	9.43965	0.476167
$394$	0	0
$395$	10.2767	0.517079
$396$	0	0
$397$	17.9509	0.900931	0.450465	−	0.892794i	$-0.351258\pi$
$397$	17.9509	0.900931	0.450465	+	0.892794i	$0.351258\pi$
$398$	0	0
$399$	9.67418	0.484315
$400$	0	0
$401$	13.5829	0.678297	0.339149	−	0.940733i	$-0.389861\pi$
$401$	13.5829	0.678297	0.339149	+	0.940733i	$0.389861\pi$
$402$	0	0
$403$	−7.55691	−0.376437
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−11.6320	−0.576576
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	1.76547	0.0870841
$412$	0	0
$413$	5.75859	0.283362
$414$	0	0
$415$	2.11727	0.103933
$416$	0	0
$417$	6.27674	0.307373
$418$	0	0
$419$	−12.3189	−0.601820	−0.300910	−	0.953653i	$-0.597290\pi$
$419$	−12.3189	−0.601820	−0.300910	+	0.953653i	$0.597290\pi$
$420$	0	0
$421$	22.7880	1.11062	0.555310	−	0.831644i	$-0.312600\pi$
$421$	22.7880	1.11062	0.555310	+	0.831644i	$0.312600\pi$
$422$	0	0
$423$	11.5569	0.561916
$424$	0	0
$425$	−2.83709	−0.137619
$426$	0	0
$427$	17.9578	0.869039
$428$	0	0
$429$	2.71982	0.131314
$430$	0	0
$431$	8.99656	0.433349	0.216675	−	0.976244i	$-0.430479\pi$
$431$	8.99656	0.433349	0.216675	+	0.976244i	$0.430479\pi$
$432$	0	0
$433$	−20.3258	−0.976797	−0.488398	−	0.872621i	$-0.662419\pi$
$433$	−20.3258	−0.976797	−0.488398	+	0.872621i	$0.662419\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	17.2051	0.823032
$438$	0	0
$439$	25.3906	1.21183	0.605913	−	0.795531i	$-0.292808\pi$
$439$	25.3906	1.21183	0.605913	+	0.795531i	$0.292808\pi$
$440$	0	0
$441$	0.397442	0.0189258
$442$	0	0
$443$	−10.9284	−0.519223	−0.259611	−	0.965713i	$-0.583594\pi$
$443$	−10.9284	−0.519223	−0.259611	+	0.965713i	$0.583594\pi$
$444$	0	0
$445$	−1.16291	−0.0551272
$446$	0	0
$447$	−20.8302	−0.985235
$448$	0	0
$449$	2.83709	0.133891	0.0669453	−	0.997757i	$-0.478675\pi$
$449$	2.83709	0.133891	0.0669453	+	0.997757i	$0.478675\pi$
$450$	0	0
$451$	7.71639	0.363350
$452$	0	0
$453$	−4.99656	−0.234759
$454$	0	0
$455$	−2.71982	−0.127507
$456$	0	0
$457$	−13.7164	−0.641625	−0.320813	−	0.947143i	$-0.603956\pi$
$457$	−13.7164	−0.641625	−0.320813	+	0.947143i	$0.603956\pi$
$458$	0	0
$459$	−2.83709	−0.132424
$460$	0	0
$461$	−19.6251	−0.914032	−0.457016	−	0.889458i	$-0.651082\pi$
$461$	−19.6251	−0.914032	−0.457016	+	0.889458i	$0.651082\pi$
$462$	0	0
$463$	−27.0388	−1.25660	−0.628299	−	0.777972i	$-0.716249\pi$
$463$	−27.0388	−1.25660	−0.628299	+	0.777972i	$0.716249\pi$
$464$	0	0
$465$	7.55691	0.350444
$466$	0	0
$467$	−28.9215	−1.33833	−0.669164	−	0.743115i	$-0.733348\pi$
$467$	−28.9215	−1.33833	−0.669164	+	0.743115i	$0.733348\pi$
$468$	0	0
$469$	−5.12070	−0.236452
$470$	0	0
$471$	8.87930	0.409136
$472$	0	0
$473$	−30.2277	−1.38987
$474$	0	0
$475$	3.55691	0.163202
$476$	0	0
$477$	1.16291	0.0532460
$478$	0	0
$479$	12.1595	0.555580	0.277790	−	0.960642i	$-0.410398\pi$
$479$	12.1595	0.555580	0.277790	+	0.960642i	$0.410398\pi$
$480$	0	0
$481$	−4.27674	−0.195002
$482$	0	0
$483$	13.1560	0.598620
$484$	0	0
$485$	10.8371	0.492087
$486$	0	0
$487$	0.159472	0.00722636	0.00361318	−	0.999993i	$-0.498850\pi$
$487$	0.159472	0.00722636	0.00361318	+	0.999993i	$0.498850\pi$
$488$	0	0
$489$	13.8337	0.625579
$490$	0	0
$491$	42.2277	1.90571	0.952854	−	0.303430i	$-0.0981318\pi$
$491$	42.2277	1.90571	0.952854	+	0.303430i	$0.0981318\pi$
$492$	0	0
$493$	−17.0225	−0.766657
$494$	0	0
$495$	−2.71982	−0.122247
$496$	0	0
$497$	18.2767	0.819824
$498$	0	0
$499$	−7.79145	−0.348793	−0.174397	−	0.984676i	$-0.555797\pi$
$499$	−7.79145	−0.348793	−0.174397	+	0.984676i	$0.555797\pi$
$500$	0	0
$501$	9.88273	0.441528
$502$	0	0
$503$	−27.3484	−1.21940	−0.609702	−	0.792631i	$-0.708711\pi$
$503$	−27.3484	−1.21940	−0.609702	+	0.792631i	$0.708711\pi$
$504$	0	0
$505$	−7.67418	−0.341497
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−33.4819	−1.48406	−0.742029	−	0.670368i	$-0.766136\pi$
$509$	−33.4819	−1.48406	−0.742029	+	0.670368i	$0.766136\pi$
$510$	0	0
$511$	24.7880	1.09656
$512$	0	0
$513$	3.55691	0.157042
$514$	0	0
$515$	3.76547	0.165926
$516$	0	0
$517$	31.4328	1.38241
$518$	0	0
$519$	13.1138	0.575633
$520$	0	0
$521$	−17.3484	−0.760045	−0.380023	−	0.924977i	$-0.624084\pi$
$521$	−17.3484	−0.760045	−0.380023	+	0.924977i	$0.624084\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	2.71982	0.118703
$526$	0	0
$527$	21.4396	0.933926
$528$	0	0
$529$	0.397442	0.0172801
$530$	0	0
$531$	2.11727	0.0918815
$532$	0	0
$533$	2.83709	0.122888
$534$	0	0
$535$	−12.6026	−0.544856
$536$	0	0
$537$	−8.55348	−0.369110
$538$	0	0
$539$	1.08097	0.0465608
$540$	0	0
$541$	−32.6448	−1.40351	−0.701754	−	0.712419i	$-0.747599\pi$
$541$	−32.6448	−1.40351	−0.701754	+	0.712419i	$0.747599\pi$
$542$	0	0
$543$	3.72326	0.159780
$544$	0	0
$545$	−11.4396	−0.490021
$546$	0	0
$547$	−34.2277	−1.46347	−0.731734	−	0.681590i	$-0.761289\pi$
$547$	−34.2277	−1.46347	−0.731734	+	0.681590i	$0.761289\pi$
$548$	0	0
$549$	6.60256	0.281790
$550$	0	0
$551$	21.3415	0.909178
$552$	0	0
$553$	−27.9509	−1.18859
$554$	0	0
$555$	4.27674	0.181537
$556$	0	0
$557$	6.65164	0.281839	0.140919	−	0.990021i	$-0.454994\pi$
$557$	6.65164	0.281839	0.140919	+	0.990021i	$0.454994\pi$
$558$	0	0
$559$	−11.1138	−0.470065
$560$	0	0
$561$	−7.71639	−0.325786
$562$	0	0
$563$	−40.2699	−1.69717	−0.848586	−	0.529057i	$-0.822546\pi$
$563$	−40.2699	−1.69717	−0.848586	+	0.529057i	$0.822546\pi$
$564$	0	0
$565$	13.1138	0.551703
$566$	0	0
$567$	2.71982	0.114222
$568$	0	0
$569$	−13.4328	−0.563131	−0.281566	−	0.959542i	$-0.590854\pi$
$569$	−13.4328	−0.563131	−0.281566	+	0.959542i	$0.590854\pi$
$570$	0	0
$571$	35.7164	1.49468	0.747342	−	0.664439i	$-0.231330\pi$
$571$	35.7164	1.49468	0.747342	+	0.664439i	$0.231330\pi$
$572$	0	0
$573$	4.23453	0.176900
$574$	0	0
$575$	4.83709	0.201721
$576$	0	0
$577$	−13.7164	−0.571021	−0.285510	−	0.958376i	$-0.592163\pi$
$577$	−13.7164	−0.571021	−0.285510	+	0.958376i	$0.592163\pi$
$578$	0	0
$579$	−23.3906	−0.972079
$580$	0	0
$581$	−5.75859	−0.238907
$582$	0	0
$583$	3.16291	0.130994
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−30.6707	−1.26592	−0.632959	−	0.774186i	$-0.718160\pi$
$587$	−30.6707	−1.26592	−0.632959	+	0.774186i	$0.718160\pi$
$588$	0	0
$589$	−26.8793	−1.10754
$590$	0	0
$591$	14.5535	0.598650
$592$	0	0
$593$	−45.6673	−1.87533	−0.937666	−	0.347538i	$-0.887018\pi$
$593$	−45.6673	−1.87533	−0.937666	+	0.347538i	$0.887018\pi$
$594$	0	0
$595$	7.71639	0.316341
$596$	0	0
$597$	15.1138	0.618568
$598$	0	0
$599$	40.2208	1.64338	0.821688	−	0.569937i	$-0.193032\pi$
$599$	40.2208	1.64338	0.821688	+	0.569937i	$0.193032\pi$
$600$	0	0
$601$	17.3974	0.709656	0.354828	−	0.934932i	$-0.384539\pi$
$601$	17.3974	0.709656	0.354828	+	0.934932i	$0.384539\pi$
$602$	0	0
$603$	−1.88273	−0.0766708
$604$	0	0
$605$	3.60256	0.146465
$606$	0	0
$607$	14.2277	0.577483	0.288741	−	0.957407i	$-0.406763\pi$
$607$	14.2277	0.577483	0.288741	+	0.957407i	$0.406763\pi$
$608$	0	0
$609$	16.3189	0.661277
$610$	0	0
$611$	11.5569	0.467543
$612$	0	0
$613$	−40.8302	−1.64912	−0.824558	−	0.565777i	$-0.808576\pi$
$613$	−40.8302	−1.64912	−0.824558	+	0.565777i	$0.808576\pi$
$614$	0	0
$615$	−2.83709	−0.114403
$616$	0	0
$617$	5.11383	0.205875	0.102937	−	0.994688i	$-0.467176\pi$
$617$	5.11383	0.205875	0.102937	+	0.994688i	$0.467176\pi$
$618$	0	0
$619$	−11.5569	−0.464512	−0.232256	−	0.972655i	$-0.574611\pi$
$619$	−11.5569	−0.464512	−0.232256	+	0.972655i	$0.574611\pi$
$620$	0	0
$621$	4.83709	0.194106
$622$	0	0
$623$	3.16291	0.126719
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	9.67418	0.386350
$628$	0	0
$629$	12.1335	0.483794
$630$	0	0
$631$	−35.2242	−1.40225	−0.701127	−	0.713036i	$-0.747319\pi$
$631$	−35.2242	−1.40225	−0.701127	+	0.713036i	$0.747319\pi$
$632$	0	0
$633$	18.2277	0.724484
$634$	0	0
$635$	−13.4396	−0.533336
$636$	0	0
$637$	0.397442	0.0157472
$638$	0	0
$639$	6.71982	0.265832
$640$	0	0
$641$	−21.9018	−0.865071	−0.432535	−	0.901617i	$-0.642381\pi$
$641$	−21.9018	−0.865071	−0.432535	+	0.901617i	$0.642381\pi$
$642$	0	0
$643$	−7.50783	−0.296080	−0.148040	−	0.988981i	$-0.547296\pi$
$643$	−7.50783	−0.296080	−0.148040	+	0.988981i	$0.547296\pi$
$644$	0	0
$645$	11.1138	0.437607
$646$	0	0
$647$	14.0422	0.552056	0.276028	−	0.961150i	$-0.410982\pi$
$647$	14.0422	0.552056	0.276028	+	0.961150i	$0.410982\pi$
$648$	0	0
$649$	5.75859	0.226044
$650$	0	0
$651$	−20.5535	−0.805554
$652$	0	0
$653$	7.99312	0.312795	0.156398	−	0.987694i	$-0.450012\pi$
$653$	7.99312	0.312795	0.156398	+	0.987694i	$0.450012\pi$
$654$	0	0
$655$	−9.43965	−0.368838
$656$	0	0
$657$	9.11383	0.355564
$658$	0	0
$659$	25.3415	0.987164	0.493582	−	0.869699i	$-0.335687\pi$
$659$	25.3415	0.987164	0.493582	+	0.869699i	$0.335687\pi$
$660$	0	0
$661$	27.4396	1.06728	0.533639	−	0.845712i	$-0.320824\pi$
$661$	27.4396	1.06728	0.533639	+	0.845712i	$0.320824\pi$
$662$	0	0
$663$	−2.83709	−0.110183
$664$	0	0
$665$	−9.67418	−0.375149
$666$	0	0
$667$	29.0225	1.12376
$668$	0	0
$669$	−10.1173	−0.391156
$670$	0	0
$671$	17.9578	0.693253
$672$	0	0
$673$	27.1070	1.04490	0.522448	−	0.852671i	$-0.325019\pi$
$673$	27.1070	1.04490	0.522448	+	0.852671i	$0.325019\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−36.5957	−1.40649	−0.703243	−	0.710949i	$-0.748265\pi$
$677$	−36.5957	−1.40649	−0.703243	+	0.710949i	$0.748265\pi$
$678$	0	0
$679$	−29.4750	−1.13115
$680$	0	0
$681$	−11.3224	−0.433875
$682$	0	0
$683$	13.4656	0.515248	0.257624	−	0.966245i	$-0.417060\pi$
$683$	13.4656	0.515248	0.257624	+	0.966245i	$0.417060\pi$
$684$	0	0
$685$	−1.76547	−0.0674550
$686$	0	0
$687$	−6.23453	−0.237862
$688$	0	0
$689$	1.16291	0.0443033
$690$	0	0
$691$	−29.5500	−1.12414	−0.562068	−	0.827091i	$-0.689994\pi$
$691$	−29.5500	−1.12414	−0.562068	+	0.827091i	$0.689994\pi$
$692$	0	0
$693$	7.39744	0.281006
$694$	0	0
$695$	−6.27674	−0.238090
$696$	0	0
$697$	−8.04908	−0.304881
$698$	0	0
$699$	6.83709	0.258603
$700$	0	0
$701$	43.6604	1.64903	0.824516	−	0.565839i	$-0.191448\pi$
$701$	43.6604	1.64903	0.824516	+	0.565839i	$0.191448\pi$
$702$	0	0
$703$	−15.2120	−0.573731
$704$	0	0
$705$	−11.5569	−0.435259
$706$	0	0
$707$	20.8724	0.784988
$708$	0	0
$709$	−26.7880	−1.00604	−0.503022	−	0.864273i	$-0.667779\pi$
$709$	−26.7880	−1.00604	−0.503022	+	0.864273i	$0.667779\pi$
$710$	0	0
$711$	−10.2767	−0.385408
$712$	0	0
$713$	−36.5535	−1.36894
$714$	0	0
$715$	−2.71982	−0.101716
$716$	0	0
$717$	−1.28018	−0.0478091
$718$	0	0
$719$	34.8793	1.30078	0.650389	−	0.759601i	$-0.274606\pi$
$719$	34.8793	1.30078	0.650389	+	0.759601i	$0.274606\pi$
$720$	0	0
$721$	−10.2414	−0.381410
$722$	0	0
$723$	−6.00000	−0.223142
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	−37.4396	−1.38856	−0.694280	−	0.719705i	$-0.744277\pi$
$727$	−37.4396	−1.38856	−0.694280	+	0.719705i	$0.744277\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	31.5309	1.16621
$732$	0	0
$733$	−47.1560	−1.74175	−0.870874	−	0.491506i	$-0.836446\pi$
$733$	−47.1560	−1.74175	−0.870874	+	0.491506i	$0.836446\pi$
$734$	0	0
$735$	−0.397442	−0.0146599
$736$	0	0
$737$	−5.12070	−0.188624
$738$	0	0
$739$	31.7914	1.16947	0.584734	−	0.811225i	$-0.301199\pi$
$739$	31.7914	1.16947	0.584734	+	0.811225i	$0.301199\pi$
$740$	0	0
$741$	3.55691	0.130667
$742$	0	0
$743$	−16.6776	−0.611842	−0.305921	−	0.952057i	$-0.598964\pi$
$743$	−16.6776	−0.611842	−0.305921	+	0.952057i	$0.598964\pi$
$744$	0	0
$745$	20.8302	0.763160
$746$	0	0
$747$	−2.11727	−0.0774667
$748$	0	0
$749$	34.2767	1.25244
$750$	0	0
$751$	−16.1855	−0.590616	−0.295308	−	0.955402i	$-0.595422\pi$
$751$	−16.1855	−0.590616	−0.295308	+	0.955402i	$0.595422\pi$
$752$	0	0
$753$	−18.2277	−0.664253
$754$	0	0
$755$	4.99656	0.181844
$756$	0	0
$757$	12.3258	0.447990	0.223995	−	0.974590i	$-0.428090\pi$
$757$	12.3258	0.447990	0.223995	+	0.974590i	$0.428090\pi$
$758$	0	0
$759$	13.1560	0.477534
$760$	0	0
$761$	−0.00687569	−0.000249244 0	−0.000124622	−	1.00000i	$-0.500040\pi$
$761$	−0.00687569	−0.000249244 0	−0.000124622		1.00000i	$0.500040\pi$
$762$	0	0
$763$	31.1138	1.12640
$764$	0	0
$765$	2.83709	0.102575
$766$	0	0
$767$	2.11727	0.0764501
$768$	0	0
$769$	−20.3258	−0.732968	−0.366484	−	0.930424i	$-0.619439\pi$
$769$	−20.3258	−0.732968	−0.366484	+	0.930424i	$0.619439\pi$
$770$	0	0
$771$	1.11383	0.0401136
$772$	0	0
$773$	9.90184	0.356144	0.178072	−	0.984017i	$-0.443014\pi$
$773$	9.90184	0.356144	0.178072	+	0.984017i	$0.443014\pi$
$774$	0	0
$775$	−7.55691	−0.271452
$776$	0	0
$777$	−11.6320	−0.417295
$778$	0	0
$779$	10.0913	0.361558
$780$	0	0
$781$	18.2767	0.653993
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	−8.87930	−0.316916
$786$	0	0
$787$	−36.3449	−1.29556	−0.647778	−	0.761829i	$-0.724302\pi$
$787$	−36.3449	−1.29556	−0.647778	+	0.761829i	$0.724302\pi$
$788$	0	0
$789$	−8.00000	−0.284808
$790$	0	0
$791$	−35.6673	−1.26818
$792$	0	0
$793$	6.60256	0.234464
$794$	0	0
$795$	−1.16291	−0.0412442
$796$	0	0
$797$	18.8371	0.667244	0.333622	−	0.942707i	$-0.391729\pi$
$797$	18.8371	0.667244	0.333622	+	0.942707i	$0.391729\pi$
$798$	0	0
$799$	−32.7880	−1.15996
$800$	0	0
$801$	1.16291	0.0410894
$802$	0	0
$803$	24.7880	0.874750
$804$	0	0
$805$	−13.1560	−0.463689
$806$	0	0
$807$	15.6742	0.551757
$808$	0	0
$809$	32.2277	1.13306	0.566532	−	0.824040i	$-0.308285\pi$
$809$	32.2277	1.13306	0.566532	+	0.824040i	$0.308285\pi$
$810$	0	0
$811$	23.0034	0.807760	0.403880	−	0.914812i	$-0.367661\pi$
$811$	23.0034	0.807760	0.403880	+	0.914812i	$0.367661\pi$
$812$	0	0
$813$	−0.443086	−0.0155397
$814$	0	0
$815$	−13.8337	−0.484572
$816$	0	0
$817$	−39.5309	−1.38301
$818$	0	0
$819$	2.71982	0.0950383
$820$	0	0
$821$	49.9372	1.74282	0.871410	−	0.490556i	$-0.163206\pi$
$821$	49.9372	1.74282	0.871410	+	0.490556i	$0.163206\pi$
$822$	0	0
$823$	−28.2345	−0.984194	−0.492097	−	0.870540i	$-0.663769\pi$
$823$	−28.2345	−0.984194	−0.492097	+	0.870540i	$0.663769\pi$
$824$	0	0
$825$	2.71982	0.0946921
$826$	0	0
$827$	9.55004	0.332087	0.166044	−	0.986118i	$-0.446901\pi$
$827$	9.55004	0.332087	0.166044	+	0.986118i	$0.446901\pi$
$828$	0	0
$829$	−37.9862	−1.31932	−0.659658	−	0.751565i	$-0.729299\pi$
$829$	−37.9862	−1.31932	−0.659658	+	0.751565i	$0.729299\pi$
$830$	0	0
$831$	−4.87930	−0.169261
$832$	0	0
$833$	−1.12758	−0.0390683
$834$	0	0
$835$	−9.88273	−0.342006
$836$	0	0
$837$	−7.55691	−0.261205
$838$	0	0
$839$	4.72670	0.163184	0.0815919	−	0.996666i	$-0.474000\pi$
$839$	4.72670	0.163184	0.0815919	+	0.996666i	$0.474000\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−9.11383	−0.313897
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−9.79832	−0.336674
$848$	0	0
$849$	−33.3415	−1.14428
$850$	0	0
$851$	−20.6870	−0.709140
$852$	0	0
$853$	−48.3611	−1.65585	−0.827927	−	0.560836i	$-0.810480\pi$
$853$	−48.3611	−1.65585	−0.827927	+	0.560836i	$0.810480\pi$
$854$	0	0
$855$	−3.55691	−0.121644
$856$	0	0
$857$	6.83709	0.233551	0.116775	−	0.993158i	$-0.462744\pi$
$857$	6.83709	0.233551	0.116775	+	0.993158i	$0.462744\pi$
$858$	0	0
$859$	−12.6026	−0.429994	−0.214997	−	0.976615i	$-0.568974\pi$
$859$	−12.6026	−0.429994	−0.214997	+	0.976615i	$0.568974\pi$
$860$	0	0
$861$	7.71639	0.262974
$862$	0	0
$863$	−8.20855	−0.279422	−0.139711	−	0.990192i	$-0.544617\pi$
$863$	−8.20855	−0.279422	−0.139711	+	0.990192i	$0.544617\pi$
$864$	0	0
$865$	−13.1138	−0.445884
$866$	0	0
$867$	−8.95092	−0.303989
$868$	0	0
$869$	−27.9509	−0.948170
$870$	0	0
$871$	−1.88273	−0.0637940
$872$	0	0
$873$	−10.8371	−0.366780
$874$	0	0
$875$	−2.71982	−0.0919468
$876$	0	0
$877$	13.5309	0.456907	0.228454	−	0.973555i	$-0.426633\pi$
$877$	13.5309	0.456907	0.228454	+	0.973555i	$0.426633\pi$
$878$	0	0
$879$	−29.4328	−0.992743
$880$	0	0
$881$	−9.34836	−0.314954	−0.157477	−	0.987523i	$-0.550336\pi$
$881$	−9.34836	−0.314954	−0.157477	+	0.987523i	$0.550336\pi$
$882$	0	0
$883$	55.1001	1.85427	0.927133	−	0.374733i	$-0.122266\pi$
$883$	55.1001	1.85427	0.927133	+	0.374733i	$0.122266\pi$
$884$	0	0
$885$	−2.11727	−0.0711711
$886$	0	0
$887$	−0.133492	−0.00448223	−0.00224112	−	0.999997i	$-0.500713\pi$
$887$	−0.133492	−0.00448223	−0.00224112	+	0.999997i	$0.500713\pi$
$888$	0	0
$889$	36.5535	1.22596
$890$	0	0
$891$	2.71982	0.0911175
$892$	0	0
$893$	41.1070	1.37559
$894$	0	0
$895$	8.55348	0.285911
$896$	0	0
$897$	4.83709	0.161506
$898$	0	0
$899$	−45.3415	−1.51222
$900$	0	0
$901$	−3.29928	−0.109915
$902$	0	0
$903$	−30.2277	−1.00591
$904$	0	0
$905$	−3.72326	−0.123765
$906$	0	0
$907$	58.5466	1.94401	0.972004	−	0.234964i	$-0.0754973\pi$
$907$	58.5466	1.94401	0.972004	+	0.234964i	$0.0754973\pi$
$908$	0	0
$909$	7.67418	0.254537
$910$	0	0
$911$	−50.4622	−1.67189	−0.835943	−	0.548816i	$-0.815079\pi$
$911$	−50.4622	−1.67189	−0.835943	+	0.548816i	$0.815079\pi$
$912$	0	0
$913$	−5.75859	−0.190582
$914$	0	0
$915$	−6.60256	−0.218274
$916$	0	0
$917$	25.6742	0.847836
$918$	0	0
$919$	−56.9735	−1.87938	−0.939691	−	0.342026i	$-0.888887\pi$
$919$	−56.9735	−1.87938	−0.939691	+	0.342026i	$0.888887\pi$
$920$	0	0
$921$	21.8337	0.719443
$922$	0	0
$923$	6.71982	0.221186
$924$	0	0
$925$	−4.27674	−0.140618
$926$	0	0
$927$	−3.76547	−0.123674
$928$	0	0
$929$	−36.5957	−1.20067	−0.600333	−	0.799750i	$-0.704965\pi$
$929$	−36.5957	−1.20067	−0.600333	+	0.799750i	$0.704965\pi$
$930$	0	0
$931$	1.41367	0.0463311
$932$	0	0
$933$	−25.1070	−0.821965
$934$	0	0
$935$	7.71639	0.252353
$936$	0	0
$937$	−47.1070	−1.53892	−0.769459	−	0.638697i	$-0.779474\pi$
$937$	−47.1070	−1.53892	−0.769459	+	0.638697i	$0.779474\pi$
$938$	0	0
$939$	8.22766	0.268499
$940$	0	0
$941$	−36.3611	−1.18534	−0.592670	−	0.805446i	$-0.701926\pi$
$941$	−36.3611	−1.18534	−0.592670	+	0.805446i	$0.701926\pi$
$942$	0	0
$943$	13.7233	0.446891
$944$	0	0
$945$	−2.71982	−0.0884759
$946$	0	0
$947$	−44.5795	−1.44864	−0.724319	−	0.689465i	$-0.757846\pi$
$947$	−44.5795	−1.44864	−0.724319	+	0.689465i	$0.757846\pi$
$948$	0	0
$949$	9.11383	0.295847
$950$	0	0
$951$	27.6742	0.897397
$952$	0	0
$953$	19.8596	0.643317	0.321658	−	0.946856i	$-0.395760\pi$
$953$	19.8596	0.643317	0.321658	+	0.946856i	$0.395760\pi$
$954$	0	0
$955$	−4.23453	−0.137026
$956$	0	0
$957$	16.3189	0.527517
$958$	0	0
$959$	4.80176	0.155057
$960$	0	0
$961$	26.1070	0.842160
$962$	0	0
$963$	12.6026	0.406112
$964$	0	0
$965$	23.3906	0.752969
$966$	0	0
$967$	−47.4068	−1.52450	−0.762250	−	0.647283i	$-0.775905\pi$
$967$	−47.4068	−1.52450	−0.762250	+	0.647283i	$0.775905\pi$
$968$	0	0
$969$	−10.0913	−0.324179
$970$	0	0
$971$	10.6448	0.341607	0.170803	−	0.985305i	$-0.445364\pi$
$971$	10.6448	0.341607	0.170803	+	0.985305i	$0.445364\pi$
$972$	0	0
$973$	17.0716	0.547291
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−1.21199	−0.0387750	−0.0193875	−	0.999812i	$-0.506172\pi$
$977$	−1.21199	−0.0387750	−0.0193875	+	0.999812i	$0.506172\pi$
$978$	0	0
$979$	3.16291	0.101087
$980$	0	0
$981$	11.4396	0.365240
$982$	0	0
$983$	−51.8759	−1.65458	−0.827291	−	0.561773i	$-0.810119\pi$
$983$	−51.8759	−1.65458	−0.827291	+	0.561773i	$0.810119\pi$
$984$	0	0
$985$	−14.5535	−0.463712
$986$	0	0
$987$	31.4328	1.00052
$988$	0	0
$989$	−53.7586	−1.70942
$990$	0	0
$991$	21.6251	0.686944	0.343472	−	0.939163i	$-0.388397\pi$
$991$	21.6251	0.686944	0.343472	+	0.939163i	$0.388397\pi$
$992$	0	0
$993$	13.2311	0.419876
$994$	0	0
$995$	−15.1138	−0.479141
$996$	0	0
$997$	23.2051	0.734913	0.367457	−	0.930041i	$-0.380229\pi$
$997$	23.2051	0.734913	0.367457	+	0.930041i	$0.380229\pi$
$998$	0	0
$999$	−4.27674	−0.135310

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.a.bj.1.3		3
3.2	odd	2		9360.2.a.dd.1.3		3
4.3	odd	2		195.2.a.e.1.3	✓	3
12.11	even	2		585.2.a.n.1.1		3
20.3	even	4		975.2.c.i.274.1		6
20.7	even	4		975.2.c.i.274.6		6
20.19	odd	2		975.2.a.o.1.1		3
28.27	even	2		9555.2.a.bq.1.3		3
52.51	odd	2		2535.2.a.bc.1.1		3
60.23	odd	4		2925.2.c.w.2224.6		6
60.47	odd	4		2925.2.c.w.2224.1		6
60.59	even	2		2925.2.a.bh.1.3		3
156.155	even	2		7605.2.a.bx.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.e.1.3	✓	3	4.3	odd	2
585.2.a.n.1.1		3	12.11	even	2
975.2.a.o.1.1		3	20.19	odd	2
975.2.c.i.274.1		6	20.3	even	4
975.2.c.i.274.6		6	20.7	even	4
2535.2.a.bc.1.1		3	52.51	odd	2
2925.2.a.bh.1.3		3	60.59	even	2
2925.2.c.w.2224.1		6	60.47	odd	4
2925.2.c.w.2224.6		6	60.23	odd	4
3120.2.a.bj.1.3		3	1.1	even	1	trivial
7605.2.a.bx.1.3		3	156.155	even	2
9360.2.a.dd.1.3		3	3.2	odd	2
9555.2.a.bq.1.3		3	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +2.71982 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +2.71982 q^{7} +1.00000 q^{9} +2.71982 q^{11} +1.00000 q^{13} -1.00000 q^{15} -2.83709 q^{17} +3.55691 q^{19} +2.71982 q^{21} +4.83709 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -7.55691 q^{31} +2.71982 q^{33} -2.71982 q^{35} -4.27674 q^{37} +1.00000 q^{39} +2.83709 q^{41} -11.1138 q^{43} -1.00000 q^{45} +11.5569 q^{47} +0.397442 q^{49} -2.83709 q^{51} +1.16291 q^{53} -2.71982 q^{55} +3.55691 q^{57} +2.11727 q^{59} +6.60256 q^{61} +2.71982 q^{63} -1.00000 q^{65} -1.88273 q^{67} +4.83709 q^{69} +6.71982 q^{71} +9.11383 q^{73} +1.00000 q^{75} +7.39744 q^{77} -10.2767 q^{79} +1.00000 q^{81} -2.11727 q^{83} +2.83709 q^{85} +6.00000 q^{87} +1.16291 q^{89} +2.71982 q^{91} -7.55691 q^{93} -3.55691 q^{95} -10.8371 q^{97} +2.71982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - 3 q^{15} - q^{17} - 6 q^{19} - q^{21} + 7 q^{23} + 3 q^{25} + 3 q^{27} + 18 q^{29} - 6 q^{31} - q^{33} + q^{35} + 13 q^{37} + 3 q^{39} + q^{41} - 3 q^{45} + 18 q^{47} + 12 q^{49} - q^{51} + 11 q^{53} + q^{55} - 6 q^{57} + 8 q^{59} + 9 q^{61} - q^{63} - 3 q^{65} - 4 q^{67} + 7 q^{69} + 11 q^{71} - 6 q^{73} + 3 q^{75} + 33 q^{77} - 5 q^{79} + 3 q^{81} - 8 q^{83} + q^{85} + 18 q^{87} + 11 q^{89} - q^{91} - 6 q^{93} + 6 q^{95} - 25 q^{97} - q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9 - q^11 + 3 * q^13 - 3 * q^15 - q^17 - 6 * q^19 - q^21 + 7 * q^23 + 3 * q^25 + 3 * q^27 + 18 * q^29 - 6 * q^31 - q^33 + q^35 + 13 * q^37 + 3 * q^39 + q^41 - 3 * q^45 + 18 * q^47 + 12 * q^49 - q^51 + 11 * q^53 + q^55 - 6 * q^57 + 8 * q^59 + 9 * q^61 - q^63 - 3 * q^65 - 4 * q^67 + 7 * q^69 + 11 * q^71 - 6 * q^73 + 3 * q^75 + 33 * q^77 - 5 * q^79 + 3 * q^81 - 8 * q^83 + q^85 + 18 * q^87 + 11 * q^89 - q^91 - 6 * q^93 + 6 * q^95 - 25 * q^97 - q^99

Introduction

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