LMFDB - Embedded newform 6040.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.03909 q^{3} +1.00000 q^{5} -1.69120 q^{7} +6.23604 q^{9} +O(q^{10})$	$q-3.03909 q^{3} +1.00000 q^{5} -1.69120 q^{7} +6.23604 q^{9} +6.10327 q^{11} -5.68607 q^{13} -3.03909 q^{15} +3.98231 q^{17} +7.23142 q^{19} +5.13970 q^{21} +0.0633183 q^{23} +1.00000 q^{25} -9.83461 q^{27} +6.97799 q^{29} -9.35112 q^{31} -18.5484 q^{33} -1.69120 q^{35} +11.3462 q^{37} +17.2804 q^{39} +10.8820 q^{41} +6.11783 q^{43} +6.23604 q^{45} -8.87522 q^{47} -4.13984 q^{49} -12.1026 q^{51} +6.52239 q^{53} +6.10327 q^{55} -21.9769 q^{57} -6.12619 q^{59} +0.903049 q^{61} -10.5464 q^{63} -5.68607 q^{65} -7.35760 q^{67} -0.192430 q^{69} -5.16720 q^{71} +5.80005 q^{73} -3.03909 q^{75} -10.3219 q^{77} +11.1781 q^{79} +11.1801 q^{81} -4.61406 q^{83} +3.98231 q^{85} -21.2067 q^{87} +7.00088 q^{89} +9.61628 q^{91} +28.4189 q^{93} +7.23142 q^{95} -7.77207 q^{97} +38.0602 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * 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
$3$	−3.03909	−1.75462	−0.877308	−	0.479927i	$-0.840663\pi$
$3$	−3.03909	−1.75462	−0.877308	+	0.479927i	$0.840663\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.69120	−0.639214	−0.319607	−	0.947550i	$-0.603551\pi$
$7$	−1.69120	−0.639214	−0.319607	+	0.947550i	$0.603551\pi$
$8$	0	0
$9$	6.23604	2.07868
$10$	0	0
$11$	6.10327	1.84021	0.920103	−	0.391678i	$-0.128105\pi$
$11$	6.10327	1.84021	0.920103	+	0.391678i	$0.128105\pi$
$12$	0	0
$13$	−5.68607	−1.57703	−0.788515	−	0.615015i	$-0.789150\pi$
$13$	−5.68607	−1.57703	−0.788515	+	0.615015i	$0.789150\pi$
$14$	0	0
$15$	−3.03909	−0.784689
$16$	0	0
$17$	3.98231	0.965853	0.482926	−	0.875661i	$-0.339574\pi$
$17$	3.98231	0.965853	0.482926	+	0.875661i	$0.339574\pi$
$18$	0	0
$19$	7.23142	1.65900	0.829501	−	0.558505i	$-0.188625\pi$
$19$	7.23142	1.65900	0.829501	+	0.558505i	$0.188625\pi$
$20$	0	0
$21$	5.13970	1.12158
$22$	0	0
$23$	0.0633183	0.0132028	0.00660139	−	0.999978i	$-0.497899\pi$
$23$	0.0633183	0.0132028	0.00660139	+	0.999978i	$0.497899\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−9.83461	−1.89267
$28$	0	0
$29$	6.97799	1.29578	0.647890	−	0.761734i	$-0.275652\pi$
$29$	6.97799	1.29578	0.647890	+	0.761734i	$0.275652\pi$
$30$	0	0
$31$	−9.35112	−1.67951	−0.839756	−	0.542964i	$-0.817302\pi$
$31$	−9.35112	−1.67951	−0.839756	+	0.542964i	$0.817302\pi$
$32$	0	0
$33$	−18.5484	−3.22885
$34$	0	0
$35$	−1.69120	−0.285865
$36$	0	0
$37$	11.3462	1.86531	0.932656	−	0.360767i	$-0.117485\pi$
$37$	11.3462	1.86531	0.932656	+	0.360767i	$0.117485\pi$
$38$	0	0
$39$	17.2804	2.76708
$40$	0	0
$41$	10.8820	1.69949	0.849743	−	0.527198i	$-0.176757\pi$
$41$	10.8820	1.69949	0.849743	+	0.527198i	$0.176757\pi$
$42$	0	0
$43$	6.11783	0.932961	0.466480	−	0.884532i	$-0.345522\pi$
$43$	6.11783	0.932961	0.466480	+	0.884532i	$0.345522\pi$
$44$	0	0
$45$	6.23604	0.929614
$46$	0	0
$47$	−8.87522	−1.29458	−0.647292	−	0.762242i	$-0.724098\pi$
$47$	−8.87522	−1.29458	−0.647292	+	0.762242i	$0.724098\pi$
$48$	0	0
$49$	−4.13984	−0.591406
$50$	0	0
$51$	−12.1026	−1.69470
$52$	0	0
$53$	6.52239	0.895920	0.447960	−	0.894054i	$-0.352151\pi$
$53$	6.52239	0.895920	0.447960	+	0.894054i	$0.352151\pi$
$54$	0	0
$55$	6.10327	0.822965
$56$	0	0
$57$	−21.9769	−2.91091
$58$	0	0
$59$	−6.12619	−0.797562	−0.398781	−	0.917046i	$-0.630567\pi$
$59$	−6.12619	−0.797562	−0.398781	+	0.917046i	$0.630567\pi$
$60$	0	0
$61$	0.903049	0.115624	0.0578118	−	0.998328i	$-0.481588\pi$
$61$	0.903049	0.115624	0.0578118	+	0.998328i	$0.481588\pi$
$62$	0	0
$63$	−10.5464	−1.32872
$64$	0	0
$65$	−5.68607	−0.705270
$66$	0	0
$67$	−7.35760	−0.898874	−0.449437	−	0.893312i	$-0.648375\pi$
$67$	−7.35760	−0.898874	−0.449437	+	0.893312i	$0.648375\pi$
$68$	0	0
$69$	−0.192430	−0.0231658
$70$	0	0
$71$	−5.16720	−0.613234	−0.306617	−	0.951833i	$-0.599197\pi$
$71$	−5.16720	−0.613234	−0.306617	+	0.951833i	$0.599197\pi$
$72$	0	0
$73$	5.80005	0.678844	0.339422	−	0.940634i	$-0.389769\pi$
$73$	5.80005	0.678844	0.339422	+	0.940634i	$0.389769\pi$
$74$	0	0
$75$	−3.03909	−0.350923
$76$	0	0
$77$	−10.3219	−1.17628
$78$	0	0
$79$	11.1781	1.25764	0.628819	−	0.777552i	$-0.283539\pi$
$79$	11.1781	1.25764	0.628819	+	0.777552i	$0.283539\pi$
$80$	0	0
$81$	11.1801	1.24223
$82$	0	0
$83$	−4.61406	−0.506459	−0.253229	−	0.967406i	$-0.581493\pi$
$83$	−4.61406	−0.506459	−0.253229	+	0.967406i	$0.581493\pi$
$84$	0	0
$85$	3.98231	0.431943
$86$	0	0
$87$	−21.2067	−2.27360
$88$	0	0
$89$	7.00088	0.742092	0.371046	−	0.928614i	$-0.378999\pi$
$89$	7.00088	0.742092	0.371046	+	0.928614i	$0.378999\pi$
$90$	0	0
$91$	9.61628	1.00806
$92$	0	0
$93$	28.4189	2.94690
$94$	0	0
$95$	7.23142	0.741928
$96$	0	0
$97$	−7.77207	−0.789134	−0.394567	−	0.918867i	$-0.629105\pi$
$97$	−7.77207	−0.789134	−0.394567	+	0.918867i	$0.629105\pi$
$98$	0	0
$99$	38.0602	3.82520
$100$	0	0
$101$	−10.3349	−1.02836	−0.514180	−	0.857682i	$-0.671904\pi$
$101$	−10.3349	−1.02836	−0.514180	+	0.857682i	$0.671904\pi$
$102$	0	0
$103$	16.1318	1.58951	0.794755	−	0.606930i	$-0.207599\pi$
$103$	16.1318	1.58951	0.794755	+	0.606930i	$0.207599\pi$
$104$	0	0
$105$	5.13970	0.501584
$106$	0	0
$107$	−18.5388	−1.79221	−0.896105	−	0.443842i	$-0.853615\pi$
$107$	−18.5388	−1.79221	−0.896105	+	0.443842i	$0.853615\pi$
$108$	0	0
$109$	−1.15483	−0.110613	−0.0553065	−	0.998469i	$-0.517614\pi$
$109$	−1.15483	−0.110613	−0.0553065	+	0.998469i	$0.517614\pi$
$110$	0	0
$111$	−34.4822	−3.27291
$112$	0	0
$113$	5.08924	0.478755	0.239378	−	0.970927i	$-0.423057\pi$
$113$	5.08924	0.478755	0.239378	+	0.970927i	$0.423057\pi$
$114$	0	0
$115$	0.0633183	0.00590447
$116$	0	0
$117$	−35.4585	−3.27814
$118$	0	0
$119$	−6.73489	−0.617387
$120$	0	0
$121$	26.2499	2.38635
$122$	0	0
$123$	−33.0714	−2.98195
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.3111	1.26991	0.634953	−	0.772551i	$-0.281020\pi$
$127$	14.3111	1.26991	0.634953	+	0.772551i	$0.281020\pi$
$128$	0	0
$129$	−18.5926	−1.63699
$130$	0	0
$131$	−8.86370	−0.774425	−0.387212	−	0.921991i	$-0.626562\pi$
$131$	−8.86370	−0.774425	−0.387212	+	0.921991i	$0.626562\pi$
$132$	0	0
$133$	−12.2298	−1.06046
$134$	0	0
$135$	−9.83461	−0.846428
$136$	0	0
$137$	−3.28569	−0.280716	−0.140358	−	0.990101i	$-0.544825\pi$
$137$	−3.28569	−0.280716	−0.140358	+	0.990101i	$0.544825\pi$
$138$	0	0
$139$	−14.1844	−1.20311	−0.601554	−	0.798832i	$-0.705452\pi$
$139$	−14.1844	−1.20311	−0.601554	+	0.798832i	$0.705452\pi$
$140$	0	0
$141$	26.9725	2.27150
$142$	0	0
$143$	−34.7036	−2.90206
$144$	0	0
$145$	6.97799	0.579490
$146$	0	0
$147$	12.5813	1.03769
$148$	0	0
$149$	−14.3784	−1.17792	−0.588961	−	0.808161i	$-0.700463\pi$
$149$	−14.3784	−1.17792	−0.588961	+	0.808161i	$0.700463\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	24.8339	2.00770
$154$	0	0
$155$	−9.35112	−0.751100
$156$	0	0
$157$	1.83920	0.146784	0.0733921	−	0.997303i	$-0.476618\pi$
$157$	1.83920	0.146784	0.0733921	+	0.997303i	$0.476618\pi$
$158$	0	0
$159$	−19.8221	−1.57200
$160$	0	0
$161$	−0.107084	−0.00843941
$162$	0	0
$163$	−13.4004	−1.04960	−0.524801	−	0.851225i	$-0.675860\pi$
$163$	−13.4004	−1.04960	−0.524801	+	0.851225i	$0.675860\pi$
$164$	0	0
$165$	−18.5484	−1.44399
$166$	0	0
$167$	−10.4752	−0.810597	−0.405299	−	0.914184i	$-0.632832\pi$
$167$	−10.4752	−0.810597	−0.405299	+	0.914184i	$0.632832\pi$
$168$	0	0
$169$	19.3313	1.48703
$170$	0	0
$171$	45.0954	3.44853
$172$	0	0
$173$	−17.0577	−1.29688	−0.648438	−	0.761268i	$-0.724577\pi$
$173$	−17.0577	−1.29688	−0.648438	+	0.761268i	$0.724577\pi$
$174$	0	0
$175$	−1.69120	−0.127843
$176$	0	0
$177$	18.6180	1.39942
$178$	0	0
$179$	26.2728	1.96372	0.981862	−	0.189597i	$-0.0607181\pi$
$179$	26.2728	1.96372	0.981862	+	0.189597i	$0.0607181\pi$
$180$	0	0
$181$	−8.93182	−0.663897	−0.331949	−	0.943297i	$-0.607706\pi$
$181$	−8.93182	−0.663897	−0.331949	+	0.943297i	$0.607706\pi$
$182$	0	0
$183$	−2.74444	−0.202875
$184$	0	0
$185$	11.3462	0.834193
$186$	0	0
$187$	24.3051	1.77737
$188$	0	0
$189$	16.6323	1.20982
$190$	0	0
$191$	5.65723	0.409343	0.204671	−	0.978831i	$-0.434387\pi$
$191$	5.65723	0.409343	0.204671	+	0.978831i	$0.434387\pi$
$192$	0	0
$193$	6.96449	0.501315	0.250657	−	0.968076i	$-0.419353\pi$
$193$	6.96449	0.501315	0.250657	+	0.968076i	$0.419353\pi$
$194$	0	0
$195$	17.2804	1.23748
$196$	0	0
$197$	2.23855	0.159490	0.0797450	−	0.996815i	$-0.474589\pi$
$197$	2.23855	0.159490	0.0797450	+	0.996815i	$0.474589\pi$
$198$	0	0
$199$	−0.0570966	−0.00404747	−0.00202374	−	0.999998i	$-0.500644\pi$
$199$	−0.0570966	−0.00404747	−0.00202374	+	0.999998i	$0.500644\pi$
$200$	0	0
$201$	22.3604	1.57718
$202$	0	0
$203$	−11.8012	−0.828280
$204$	0	0
$205$	10.8820	0.760033
$206$	0	0
$207$	0.394856	0.0274444
$208$	0	0
$209$	44.1353	3.05290
$210$	0	0
$211$	−23.2903	−1.60337	−0.801686	−	0.597746i	$-0.796063\pi$
$211$	−23.2903	−1.60337	−0.801686	+	0.597746i	$0.796063\pi$
$212$	0	0
$213$	15.7036	1.07599
$214$	0	0
$215$	6.11783	0.417233
$216$	0	0
$217$	15.8146	1.07357
$218$	0	0
$219$	−17.6268	−1.19111
$220$	0	0
$221$	−22.6437	−1.52318
$222$	0	0
$223$	13.3675	0.895152	0.447576	−	0.894246i	$-0.352287\pi$
$223$	13.3675	0.895152	0.447576	+	0.894246i	$0.352287\pi$
$224$	0	0
$225$	6.23604	0.415736
$226$	0	0
$227$	18.8659	1.25217	0.626086	−	0.779754i	$-0.284656\pi$
$227$	18.8659	1.25217	0.626086	+	0.779754i	$0.284656\pi$
$228$	0	0
$229$	−3.72392	−0.246083	−0.123042	−	0.992402i	$-0.539265\pi$
$229$	−3.72392	−0.246083	−0.123042	+	0.992402i	$0.539265\pi$
$230$	0	0
$231$	31.3690	2.06393
$232$	0	0
$233$	14.0422	0.919937	0.459968	−	0.887935i	$-0.347861\pi$
$233$	14.0422	0.919937	0.459968	+	0.887935i	$0.347861\pi$
$234$	0	0
$235$	−8.87522	−0.578955
$236$	0	0
$237$	−33.9713	−2.20667
$238$	0	0
$239$	−1.15452	−0.0746799	−0.0373400	−	0.999303i	$-0.511888\pi$
$239$	−1.15452	−0.0746799	−0.0373400	+	0.999303i	$0.511888\pi$
$240$	0	0
$241$	8.40754	0.541577	0.270789	−	0.962639i	$-0.412716\pi$
$241$	8.40754	0.541577	0.270789	+	0.962639i	$0.412716\pi$
$242$	0	0
$243$	−4.47342	−0.286970
$244$	0	0
$245$	−4.13984	−0.264485
$246$	0	0
$247$	−41.1183	−2.61630
$248$	0	0
$249$	14.0225	0.888641
$250$	0	0
$251$	−10.6007	−0.669107	−0.334554	−	0.942377i	$-0.608586\pi$
$251$	−10.6007	−0.669107	−0.334554	+	0.942377i	$0.608586\pi$
$252$	0	0
$253$	0.386449	0.0242958
$254$	0	0
$255$	−12.1026	−0.757894
$256$	0	0
$257$	26.0129	1.62264	0.811322	−	0.584600i	$-0.198749\pi$
$257$	26.0129	1.62264	0.811322	+	0.584600i	$0.198749\pi$
$258$	0	0
$259$	−19.1888	−1.19233
$260$	0	0
$261$	43.5150	2.69351
$262$	0	0
$263$	11.8770	0.732368	0.366184	−	0.930542i	$-0.380664\pi$
$263$	11.8770	0.732368	0.366184	+	0.930542i	$0.380664\pi$
$264$	0	0
$265$	6.52239	0.400667
$266$	0	0
$267$	−21.2763	−1.30209
$268$	0	0
$269$	26.2197	1.59864	0.799321	−	0.600905i	$-0.205193\pi$
$269$	26.2197	1.59864	0.799321	+	0.600905i	$0.205193\pi$
$270$	0	0
$271$	−0.308488	−0.0187393	−0.00936967	−	0.999956i	$-0.502983\pi$
$271$	−0.308488	−0.0187393	−0.00936967	+	0.999956i	$0.502983\pi$
$272$	0	0
$273$	−29.2247	−1.76876
$274$	0	0
$275$	6.10327	0.368041
$276$	0	0
$277$	21.4688	1.28993	0.644966	−	0.764211i	$-0.276871\pi$
$277$	21.4688	1.28993	0.644966	+	0.764211i	$0.276871\pi$
$278$	0	0
$279$	−58.3140	−3.49117
$280$	0	0
$281$	−12.7288	−0.759336	−0.379668	−	0.925123i	$-0.623962\pi$
$281$	−12.7288	−0.759336	−0.379668	+	0.925123i	$0.623962\pi$
$282$	0	0
$283$	12.4905	0.742485	0.371243	−	0.928536i	$-0.378932\pi$
$283$	12.4905	0.742485	0.371243	+	0.928536i	$0.378932\pi$
$284$	0	0
$285$	−21.9769	−1.30180
$286$	0	0
$287$	−18.4037	−1.08633
$288$	0	0
$289$	−1.14118	−0.0671280
$290$	0	0
$291$	23.6200	1.38463
$292$	0	0
$293$	−3.80407	−0.222236	−0.111118	−	0.993807i	$-0.535443\pi$
$293$	−3.80407	−0.222236	−0.111118	+	0.993807i	$0.535443\pi$
$294$	0	0
$295$	−6.12619	−0.356681
$296$	0	0
$297$	−60.0233	−3.48290
$298$	0	0
$299$	−0.360032	−0.0208212
$300$	0	0
$301$	−10.3465	−0.596362
$302$	0	0
$303$	31.4086	1.80438
$304$	0	0
$305$	0.903049	0.0517084
$306$	0	0
$307$	22.5648	1.28784	0.643920	−	0.765093i	$-0.277307\pi$
$307$	22.5648	1.28784	0.643920	+	0.765093i	$0.277307\pi$
$308$	0	0
$309$	−49.0258	−2.78898
$310$	0	0
$311$	−14.4258	−0.818014	−0.409007	−	0.912531i	$-0.634125\pi$
$311$	−14.4258	−0.818014	−0.409007	+	0.912531i	$0.634125\pi$
$312$	0	0
$313$	24.8730	1.40591	0.702953	−	0.711236i	$-0.251864\pi$
$313$	24.8730	1.40591	0.702953	+	0.711236i	$0.251864\pi$
$314$	0	0
$315$	−10.5464	−0.594222
$316$	0	0
$317$	0.612521	0.0344026	0.0172013	−	0.999852i	$-0.494524\pi$
$317$	0.612521	0.0344026	0.0172013	+	0.999852i	$0.494524\pi$
$318$	0	0
$319$	42.5885	2.38450
$320$	0	0
$321$	56.3409	3.14464
$322$	0	0
$323$	28.7978	1.60235
$324$	0	0
$325$	−5.68607	−0.315406
$326$	0	0
$327$	3.50964	0.194083
$328$	0	0
$329$	15.0098	0.827516
$330$	0	0
$331$	9.62320	0.528939	0.264469	−	0.964394i	$-0.414803\pi$
$331$	9.62320	0.528939	0.264469	+	0.964394i	$0.414803\pi$
$332$	0	0
$333$	70.7557	3.87739
$334$	0	0
$335$	−7.35760	−0.401989
$336$	0	0
$337$	5.06186	0.275737	0.137869	−	0.990451i	$-0.455975\pi$
$337$	5.06186	0.275737	0.137869	+	0.990451i	$0.455975\pi$
$338$	0	0
$339$	−15.4666	−0.840032
$340$	0	0
$341$	−57.0724	−3.09065
$342$	0	0
$343$	18.8397	1.01725
$344$	0	0
$345$	−0.192430	−0.0103601
$346$	0	0
$347$	−2.81360	−0.151042	−0.0755209	−	0.997144i	$-0.524062\pi$
$347$	−2.81360	−0.151042	−0.0755209	+	0.997144i	$0.524062\pi$
$348$	0	0
$349$	−23.1503	−1.23921	−0.619603	−	0.784915i	$-0.712706\pi$
$349$	−23.1503	−1.23921	−0.619603	+	0.784915i	$0.712706\pi$
$350$	0	0
$351$	55.9202	2.98480
$352$	0	0
$353$	−13.5219	−0.719696	−0.359848	−	0.933011i	$-0.617171\pi$
$353$	−13.5219	−0.719696	−0.359848	+	0.933011i	$0.617171\pi$
$354$	0	0
$355$	−5.16720	−0.274247
$356$	0	0
$357$	20.4679	1.08328
$358$	0	0
$359$	−14.3267	−0.756136	−0.378068	−	0.925778i	$-0.623411\pi$
$359$	−14.3267	−0.756136	−0.378068	+	0.925778i	$0.623411\pi$
$360$	0	0
$361$	33.2935	1.75229
$362$	0	0
$363$	−79.7757	−4.18714
$364$	0	0
$365$	5.80005	0.303588
$366$	0	0
$367$	−15.0900	−0.787691	−0.393846	−	0.919177i	$-0.628856\pi$
$367$	−15.0900	−0.787691	−0.393846	+	0.919177i	$0.628856\pi$
$368$	0	0
$369$	67.8607	3.53269
$370$	0	0
$371$	−11.0307	−0.572684
$372$	0	0
$373$	−30.3919	−1.57363	−0.786817	−	0.617186i	$-0.788273\pi$
$373$	−30.3919	−1.57363	−0.786817	+	0.617186i	$0.788273\pi$
$374$	0	0
$375$	−3.03909	−0.156938
$376$	0	0
$377$	−39.6773	−2.04348
$378$	0	0
$379$	26.5782	1.36523	0.682614	−	0.730779i	$-0.260843\pi$
$379$	26.5782	1.36523	0.682614	+	0.730779i	$0.260843\pi$
$380$	0	0
$381$	−43.4927	−2.22820
$382$	0	0
$383$	37.2704	1.90443	0.952213	−	0.305434i	$-0.0988016\pi$
$383$	37.2704	1.90443	0.952213	+	0.305434i	$0.0988016\pi$
$384$	0	0
$385$	−10.3219	−0.526051
$386$	0	0
$387$	38.1511	1.93933
$388$	0	0
$389$	3.74363	0.189810	0.0949048	−	0.995486i	$-0.469745\pi$
$389$	3.74363	0.189810	0.0949048	+	0.995486i	$0.469745\pi$
$390$	0	0
$391$	0.252154	0.0127520
$392$	0	0
$393$	26.9375	1.35882
$394$	0	0
$395$	11.1781	0.562433
$396$	0	0
$397$	−18.9299	−0.950067	−0.475033	−	0.879968i	$-0.657564\pi$
$397$	−18.9299	−0.950067	−0.475033	+	0.879968i	$0.657564\pi$
$398$	0	0
$399$	37.1674	1.86070
$400$	0	0
$401$	−3.45044	−0.172307	−0.0861534	−	0.996282i	$-0.527458\pi$
$401$	−3.45044	−0.172307	−0.0861534	+	0.996282i	$0.527458\pi$
$402$	0	0
$403$	53.1711	2.64864
$404$	0	0
$405$	11.1801	0.555543
$406$	0	0
$407$	69.2492	3.43256
$408$	0	0
$409$	33.8732	1.67492	0.837461	−	0.546497i	$-0.184039\pi$
$409$	33.8732	1.67492	0.837461	+	0.546497i	$0.184039\pi$
$410$	0	0
$411$	9.98550	0.492548
$412$	0	0
$413$	10.3606	0.509813
$414$	0	0
$415$	−4.61406	−0.226495
$416$	0	0
$417$	43.1077	2.11099
$418$	0	0
$419$	1.70403	0.0832472	0.0416236	−	0.999133i	$-0.486747\pi$
$419$	1.70403	0.0832472	0.0416236	+	0.999133i	$0.486747\pi$
$420$	0	0
$421$	21.1510	1.03084	0.515418	−	0.856939i	$-0.327637\pi$
$421$	21.1510	1.03084	0.515418	+	0.856939i	$0.327637\pi$
$422$	0	0
$423$	−55.3462	−2.69102
$424$	0	0
$425$	3.98231	0.193171
$426$	0	0
$427$	−1.52724	−0.0739082
$428$	0	0
$429$	105.467	5.09200
$430$	0	0
$431$	26.8418	1.29292	0.646461	−	0.762947i	$-0.276248\pi$
$431$	26.8418	1.29292	0.646461	+	0.762947i	$0.276248\pi$
$432$	0	0
$433$	−7.04593	−0.338606	−0.169303	−	0.985564i	$-0.554152\pi$
$433$	−7.04593	−0.338606	−0.169303	+	0.985564i	$0.554152\pi$
$434$	0	0
$435$	−21.2067	−1.01678
$436$	0	0
$437$	0.457882	0.0219035
$438$	0	0
$439$	−18.1376	−0.865659	−0.432829	−	0.901476i	$-0.642485\pi$
$439$	−18.1376	−0.865659	−0.432829	+	0.901476i	$0.642485\pi$
$440$	0	0
$441$	−25.8162	−1.22934
$442$	0	0
$443$	30.4097	1.44481	0.722403	−	0.691472i	$-0.243037\pi$
$443$	30.4097	1.44481	0.722403	+	0.691472i	$0.243037\pi$
$444$	0	0
$445$	7.00088	0.331874
$446$	0	0
$447$	43.6971	2.06680
$448$	0	0
$449$	0.685880	0.0323687	0.0161844	−	0.999869i	$-0.494848\pi$
$449$	0.685880	0.0323687	0.0161844	+	0.999869i	$0.494848\pi$
$450$	0	0
$451$	66.4159	3.12740
$452$	0	0
$453$	−3.03909	−0.142789
$454$	0	0
$455$	9.61628	0.450818
$456$	0	0
$457$	−11.1683	−0.522430	−0.261215	−	0.965281i	$-0.584123\pi$
$457$	−11.1683	−0.522430	−0.261215	+	0.965281i	$0.584123\pi$
$458$	0	0
$459$	−39.1645	−1.82804
$460$	0	0
$461$	−16.5987	−0.773080	−0.386540	−	0.922273i	$-0.626330\pi$
$461$	−16.5987	−0.773080	−0.386540	+	0.922273i	$0.626330\pi$
$462$	0	0
$463$	−23.8579	−1.10877	−0.554385	−	0.832261i	$-0.687046\pi$
$463$	−23.8579	−1.10877	−0.554385	+	0.832261i	$0.687046\pi$
$464$	0	0
$465$	28.4189	1.31789
$466$	0	0
$467$	17.5260	0.811007	0.405503	−	0.914094i	$-0.367096\pi$
$467$	17.5260	0.811007	0.405503	+	0.914094i	$0.367096\pi$
$468$	0	0
$469$	12.4432	0.574573
$470$	0	0
$471$	−5.58949	−0.257550
$472$	0	0
$473$	37.3388	1.71684
$474$	0	0
$475$	7.23142	0.331800
$476$	0	0
$477$	40.6739	1.86233
$478$	0	0
$479$	20.1437	0.920391	0.460195	−	0.887818i	$-0.347779\pi$
$479$	20.1437	0.920391	0.460195	+	0.887818i	$0.347779\pi$
$480$	0	0
$481$	−64.5155	−2.94165
$482$	0	0
$483$	0.325438	0.0148079
$484$	0	0
$485$	−7.77207	−0.352912
$486$	0	0
$487$	33.1532	1.50232	0.751158	−	0.660122i	$-0.229496\pi$
$487$	33.1532	1.50232	0.751158	+	0.660122i	$0.229496\pi$
$488$	0	0
$489$	40.7251	1.84165
$490$	0	0
$491$	−4.45704	−0.201143	−0.100572	−	0.994930i	$-0.532067\pi$
$491$	−4.45704	−0.201143	−0.100572	+	0.994930i	$0.532067\pi$
$492$	0	0
$493$	27.7885	1.25153
$494$	0	0
$495$	38.0602	1.71068
$496$	0	0
$497$	8.73878	0.391988
$498$	0	0
$499$	−24.1800	−1.08245	−0.541223	−	0.840879i	$-0.682039\pi$
$499$	−24.1800	−1.08245	−0.541223	+	0.840879i	$0.682039\pi$
$500$	0	0
$501$	31.8351	1.42229
$502$	0	0
$503$	4.88039	0.217606	0.108803	−	0.994063i	$-0.465298\pi$
$503$	4.88039	0.217606	0.108803	+	0.994063i	$0.465298\pi$
$504$	0	0
$505$	−10.3349	−0.459897
$506$	0	0
$507$	−58.7496	−2.60916
$508$	0	0
$509$	9.86943	0.437455	0.218727	−	0.975786i	$-0.429809\pi$
$509$	9.86943	0.437455	0.218727	+	0.975786i	$0.429809\pi$
$510$	0	0
$511$	−9.80905	−0.433927
$512$	0	0
$513$	−71.1182	−3.13994
$514$	0	0
$515$	16.1318	0.710850
$516$	0	0
$517$	−54.1678	−2.38230
$518$	0	0
$519$	51.8399	2.27552
$520$	0	0
$521$	8.28827	0.363116	0.181558	−	0.983380i	$-0.441886\pi$
$521$	8.28827	0.363116	0.181558	+	0.983380i	$0.441886\pi$
$522$	0	0
$523$	29.9250	1.30853	0.654264	−	0.756267i	$-0.272979\pi$
$523$	29.9250	1.30853	0.654264	+	0.756267i	$0.272979\pi$
$524$	0	0
$525$	5.13970	0.224315
$526$	0	0
$527$	−37.2391	−1.62216
$528$	0	0
$529$	−22.9960	−0.999826
$530$	0	0
$531$	−38.2032	−1.65788
$532$	0	0
$533$	−61.8759	−2.68014
$534$	0	0
$535$	−18.5388	−0.801501
$536$	0	0
$537$	−79.8454	−3.44558
$538$	0	0
$539$	−25.2666	−1.08831
$540$	0	0
$541$	10.4279	0.448329	0.224164	−	0.974551i	$-0.428035\pi$
$541$	10.4279	0.448329	0.224164	+	0.974551i	$0.428035\pi$
$542$	0	0
$543$	27.1446	1.16489
$544$	0	0
$545$	−1.15483	−0.0494676
$546$	0	0
$547$	12.8251	0.548360	0.274180	−	0.961678i	$-0.411594\pi$
$547$	12.8251	0.548360	0.274180	+	0.961678i	$0.411594\pi$
$548$	0	0
$549$	5.63145	0.240344
$550$	0	0
$551$	50.4608	2.14970
$552$	0	0
$553$	−18.9045	−0.803899
$554$	0	0
$555$	−34.4822	−1.46369
$556$	0	0
$557$	1.82518	0.0773352	0.0386676	−	0.999252i	$-0.487689\pi$
$557$	1.82518	0.0773352	0.0386676	+	0.999252i	$0.487689\pi$
$558$	0	0
$559$	−34.7864	−1.47131
$560$	0	0
$561$	−73.8654	−3.11860
$562$	0	0
$563$	−18.0785	−0.761919	−0.380959	−	0.924592i	$-0.624406\pi$
$563$	−18.0785	−0.761919	−0.380959	+	0.924592i	$0.624406\pi$
$564$	0	0
$565$	5.08924	0.214106
$566$	0	0
$567$	−18.9078	−0.794052
$568$	0	0
$569$	−33.3424	−1.39779	−0.698893	−	0.715226i	$-0.746324\pi$
$569$	−33.3424	−1.39779	−0.698893	+	0.715226i	$0.746324\pi$
$570$	0	0
$571$	15.3787	0.643578	0.321789	−	0.946811i	$-0.395716\pi$
$571$	15.3787	0.643578	0.321789	+	0.946811i	$0.395716\pi$
$572$	0	0
$573$	−17.1928	−0.718240
$574$	0	0
$575$	0.0633183	0.00264056
$576$	0	0
$577$	1.95806	0.0815150	0.0407575	−	0.999169i	$-0.487023\pi$
$577$	1.95806	0.0815150	0.0407575	+	0.999169i	$0.487023\pi$
$578$	0	0
$579$	−21.1657	−0.879616
$580$	0	0
$581$	7.80330	0.323735
$582$	0	0
$583$	39.8079	1.64868
$584$	0	0
$585$	−35.4585	−1.46603
$586$	0	0
$587$	−36.3368	−1.49978	−0.749890	−	0.661563i	$-0.769894\pi$
$587$	−36.3368	−1.49978	−0.749890	+	0.661563i	$0.769894\pi$
$588$	0	0
$589$	−67.6219	−2.78631
$590$	0	0
$591$	−6.80314	−0.279844
$592$	0	0
$593$	31.5150	1.29417	0.647084	−	0.762419i	$-0.275988\pi$
$593$	31.5150	1.29417	0.647084	+	0.762419i	$0.275988\pi$
$594$	0	0
$595$	−6.73489	−0.276104
$596$	0	0
$597$	0.173521	0.00710176
$598$	0	0
$599$	27.7027	1.13190	0.565950	−	0.824439i	$-0.308509\pi$
$599$	27.7027	1.13190	0.565950	+	0.824439i	$0.308509\pi$
$600$	0	0
$601$	34.4279	1.40434	0.702172	−	0.712007i	$-0.252214\pi$
$601$	34.4279	1.40434	0.702172	+	0.712007i	$0.252214\pi$
$602$	0	0
$603$	−45.8823	−1.86847
$604$	0	0
$605$	26.2499	1.06721
$606$	0	0
$607$	−18.8765	−0.766175	−0.383088	−	0.923712i	$-0.625139\pi$
$607$	−18.8765	−0.766175	−0.383088	+	0.923712i	$0.625139\pi$
$608$	0	0
$609$	35.8648	1.45331
$610$	0	0
$611$	50.4651	2.04160
$612$	0	0
$613$	−0.510883	−0.0206344	−0.0103172	−	0.999947i	$-0.503284\pi$
$613$	−0.510883	−0.0206344	−0.0103172	+	0.999947i	$0.503284\pi$
$614$	0	0
$615$	−33.0714	−1.33357
$616$	0	0
$617$	−21.4004	−0.861546	−0.430773	−	0.902460i	$-0.641759\pi$
$617$	−21.4004	−0.861546	−0.430773	+	0.902460i	$0.641759\pi$
$618$	0	0
$619$	0.837182	0.0336492	0.0168246	−	0.999858i	$-0.494644\pi$
$619$	0.837182	0.0336492	0.0168246	+	0.999858i	$0.494644\pi$
$620$	0	0
$621$	−0.622711	−0.0249885
$622$	0	0
$623$	−11.8399	−0.474356
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−134.131	−5.35668
$628$	0	0
$629$	45.1843	1.80162
$630$	0	0
$631$	−8.60662	−0.342624	−0.171312	−	0.985217i	$-0.554801\pi$
$631$	−8.60662	−0.342624	−0.171312	+	0.985217i	$0.554801\pi$
$632$	0	0
$633$	70.7813	2.81330
$634$	0	0
$635$	14.3111	0.567919
$636$	0	0
$637$	23.5394	0.932665
$638$	0	0
$639$	−32.2229	−1.27472
$640$	0	0
$641$	−11.8798	−0.469226	−0.234613	−	0.972089i	$-0.575382\pi$
$641$	−11.8798	−0.469226	−0.234613	+	0.972089i	$0.575382\pi$
$642$	0	0
$643$	−20.7539	−0.818452	−0.409226	−	0.912433i	$-0.634201\pi$
$643$	−20.7539	−0.818452	−0.409226	+	0.912433i	$0.634201\pi$
$644$	0	0
$645$	−18.5926	−0.732084
$646$	0	0
$647$	19.0583	0.749259	0.374630	−	0.927175i	$-0.377770\pi$
$647$	19.0583	0.749259	0.374630	+	0.927175i	$0.377770\pi$
$648$	0	0
$649$	−37.3898	−1.46768
$650$	0	0
$651$	−48.0620	−1.88370
$652$	0	0
$653$	−2.19743	−0.0859923	−0.0429961	−	0.999075i	$-0.513690\pi$
$653$	−2.19743	−0.0859923	−0.0429961	+	0.999075i	$0.513690\pi$
$654$	0	0
$655$	−8.86370	−0.346333
$656$	0	0
$657$	36.1693	1.41110
$658$	0	0
$659$	−20.6965	−0.806223	−0.403111	−	0.915151i	$-0.632071\pi$
$659$	−20.6965	−0.806223	−0.403111	+	0.915151i	$0.632071\pi$
$660$	0	0
$661$	23.6248	0.918897	0.459449	−	0.888204i	$-0.348047\pi$
$661$	23.6248	0.918897	0.459449	+	0.888204i	$0.348047\pi$
$662$	0	0
$663$	68.8161	2.67260
$664$	0	0
$665$	−12.2298	−0.474251
$666$	0	0
$667$	0.441835	0.0171079
$668$	0	0
$669$	−40.6249	−1.57065
$670$	0	0
$671$	5.51155	0.212771
$672$	0	0
$673$	32.8872	1.26771	0.633854	−	0.773453i	$-0.281472\pi$
$673$	32.8872	1.26771	0.633854	+	0.773453i	$0.281472\pi$
$674$	0	0
$675$	−9.83461	−0.378534
$676$	0	0
$677$	−11.5682	−0.444603	−0.222301	−	0.974978i	$-0.571357\pi$
$677$	−11.5682	−0.444603	−0.222301	+	0.974978i	$0.571357\pi$
$678$	0	0
$679$	13.1441	0.504426
$680$	0	0
$681$	−57.3350	−2.19708
$682$	0	0
$683$	−22.8445	−0.874121	−0.437061	−	0.899432i	$-0.643981\pi$
$683$	−22.8445	−0.874121	−0.437061	+	0.899432i	$0.643981\pi$
$684$	0	0
$685$	−3.28569	−0.125540
$686$	0	0
$687$	11.3173	0.431782
$688$	0	0
$689$	−37.0868	−1.41289
$690$	0	0
$691$	40.1144	1.52602	0.763011	−	0.646386i	$-0.223720\pi$
$691$	40.1144	1.52602	0.763011	+	0.646386i	$0.223720\pi$
$692$	0	0
$693$	−64.3675	−2.44512
$694$	0	0
$695$	−14.1844	−0.538046
$696$	0	0
$697$	43.3356	1.64145
$698$	0	0
$699$	−42.6755	−1.61414
$700$	0	0
$701$	−22.0273	−0.831961	−0.415981	−	0.909373i	$-0.636562\pi$
$701$	−22.0273	−0.831961	−0.415981	+	0.909373i	$0.636562\pi$
$702$	0	0
$703$	82.0495	3.09456
$704$	0	0
$705$	26.9725	1.01584
$706$	0	0
$707$	17.4784	0.657342
$708$	0	0
$709$	6.57973	0.247107	0.123553	−	0.992338i	$-0.460571\pi$
$709$	6.57973	0.247107	0.123553	+	0.992338i	$0.460571\pi$
$710$	0	0
$711$	69.7073	2.61423
$712$	0	0
$713$	−0.592098	−0.0221742
$714$	0	0
$715$	−34.7036	−1.29784
$716$	0	0
$717$	3.50870	0.131035
$718$	0	0
$719$	−11.0942	−0.413745	−0.206872	−	0.978368i	$-0.566328\pi$
$719$	−11.0942	−0.413745	−0.206872	+	0.978368i	$0.566328\pi$
$720$	0	0
$721$	−27.2821	−1.01604
$722$	0	0
$723$	−25.5512	−0.950261
$724$	0	0
$725$	6.97799	0.259156
$726$	0	0
$727$	−24.3252	−0.902173	−0.451087	−	0.892480i	$-0.648963\pi$
$727$	−24.3252	−0.902173	−0.451087	+	0.892480i	$0.648963\pi$
$728$	0	0
$729$	−19.9451	−0.738709
$730$	0	0
$731$	24.3631	0.901103
$732$	0	0
$733$	19.8621	0.733625	0.366813	−	0.930295i	$-0.380449\pi$
$733$	19.8621	0.733625	0.366813	+	0.930295i	$0.380449\pi$
$734$	0	0
$735$	12.5813	0.464069
$736$	0	0
$737$	−44.9054	−1.65411
$738$	0	0
$739$	6.10667	0.224638	0.112319	−	0.993672i	$-0.464172\pi$
$739$	6.10667	0.224638	0.112319	+	0.993672i	$0.464172\pi$
$740$	0	0
$741$	124.962	4.59060
$742$	0	0
$743$	35.5788	1.30526	0.652630	−	0.757677i	$-0.273666\pi$
$743$	35.5788	1.30526	0.652630	+	0.757677i	$0.273666\pi$
$744$	0	0
$745$	−14.3784	−0.526783
$746$	0	0
$747$	−28.7735	−1.05277
$748$	0	0
$749$	31.3528	1.14561
$750$	0	0
$751$	−51.3117	−1.87239	−0.936195	−	0.351482i	$-0.885678\pi$
$751$	−51.3117	−1.87239	−0.936195	+	0.351482i	$0.885678\pi$
$752$	0	0
$753$	32.2163	1.17403
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	36.1874	1.31525	0.657626	−	0.753345i	$-0.271561\pi$
$757$	36.1874	1.31525	0.657626	+	0.753345i	$0.271561\pi$
$758$	0	0
$759$	−1.17445	−0.0426299
$760$	0	0
$761$	6.41253	0.232454	0.116227	−	0.993223i	$-0.462920\pi$
$761$	6.41253	0.232454	0.116227	+	0.993223i	$0.462920\pi$
$762$	0	0
$763$	1.95305	0.0707053
$764$	0	0
$765$	24.8339	0.897871
$766$	0	0
$767$	34.8339	1.25778
$768$	0	0
$769$	−11.8535	−0.427449	−0.213724	−	0.976894i	$-0.568559\pi$
$769$	−11.8535	−0.427449	−0.213724	+	0.976894i	$0.568559\pi$
$770$	0	0
$771$	−79.0556	−2.84712
$772$	0	0
$773$	5.37570	0.193351	0.0966753	−	0.995316i	$-0.469179\pi$
$773$	5.37570	0.193351	0.0966753	+	0.995316i	$0.469179\pi$
$774$	0	0
$775$	−9.35112	−0.335902
$776$	0	0
$777$	58.3164	2.09209
$778$	0	0
$779$	78.6925	2.81945
$780$	0	0
$781$	−31.5368	−1.12848
$782$	0	0
$783$	−68.6257	−2.45248
$784$	0	0
$785$	1.83920	0.0656439
$786$	0	0
$787$	−41.3584	−1.47427	−0.737133	−	0.675747i	$-0.763821\pi$
$787$	−41.3584	−1.47427	−0.737133	+	0.675747i	$0.763821\pi$
$788$	0	0
$789$	−36.0953	−1.28503
$790$	0	0
$791$	−8.60692	−0.306027
$792$	0	0
$793$	−5.13479	−0.182342
$794$	0	0
$795$	−19.8221	−0.703018
$796$	0	0
$797$	11.1529	0.395054	0.197527	−	0.980297i	$-0.436709\pi$
$797$	11.1529	0.395054	0.197527	+	0.980297i	$0.436709\pi$
$798$	0	0
$799$	−35.3439	−1.25038
$800$	0	0
$801$	43.6578	1.54257
$802$	0	0
$803$	35.3993	1.24921
$804$	0	0
$805$	−0.107084	−0.00377422
$806$	0	0
$807$	−79.6838	−2.80500
$808$	0	0
$809$	11.8663	0.417198	0.208599	−	0.978001i	$-0.433110\pi$
$809$	11.8663	0.417198	0.208599	+	0.978001i	$0.433110\pi$
$810$	0	0
$811$	39.9538	1.40297	0.701483	−	0.712686i	$-0.252522\pi$
$811$	39.9538	1.40297	0.701483	+	0.712686i	$0.252522\pi$
$812$	0	0
$813$	0.937522	0.0328803
$814$	0	0
$815$	−13.4004	−0.469397
$816$	0	0
$817$	44.2406	1.54778
$818$	0	0
$819$	59.9675	2.09543
$820$	0	0
$821$	3.33373	0.116348	0.0581741	−	0.998306i	$-0.481472\pi$
$821$	3.33373	0.116348	0.0581741	+	0.998306i	$0.481472\pi$
$822$	0	0
$823$	−33.9228	−1.18247	−0.591237	−	0.806498i	$-0.701360\pi$
$823$	−33.9228	−1.18247	−0.591237	+	0.806498i	$0.701360\pi$
$824$	0	0
$825$	−18.5484	−0.645771
$826$	0	0
$827$	54.4994	1.89513	0.947565	−	0.319562i	$-0.103536\pi$
$827$	54.4994	1.89513	0.947565	+	0.319562i	$0.103536\pi$
$828$	0	0
$829$	19.6696	0.683155	0.341577	−	0.939854i	$-0.389039\pi$
$829$	19.6696	0.683155	0.341577	+	0.939854i	$0.389039\pi$
$830$	0	0
$831$	−65.2454	−2.26334
$832$	0	0
$833$	−16.4861	−0.571211
$834$	0	0
$835$	−10.4752	−0.362510
$836$	0	0
$837$	91.9646	3.17876
$838$	0	0
$839$	−42.7364	−1.47542	−0.737712	−	0.675115i	$-0.764094\pi$
$839$	−42.7364	−1.47542	−0.737712	+	0.675115i	$0.764094\pi$
$840$	0	0
$841$	19.6923	0.679044
$842$	0	0
$843$	38.6839	1.33234
$844$	0	0
$845$	19.3313	0.665018
$846$	0	0
$847$	−44.3939	−1.52539
$848$	0	0
$849$	−37.9598	−1.30278
$850$	0	0
$851$	0.718426	0.0246273
$852$	0	0
$853$	−48.8772	−1.67352	−0.836762	−	0.547567i	$-0.815554\pi$
$853$	−48.8772	−1.67352	−0.836762	+	0.547567i	$0.815554\pi$
$854$	0	0
$855$	45.0954	1.54223
$856$	0	0
$857$	−6.89526	−0.235537	−0.117769	−	0.993041i	$-0.537574\pi$
$857$	−6.89526	−0.235537	−0.117769	+	0.993041i	$0.537574\pi$
$858$	0	0
$859$	28.1189	0.959405	0.479702	−	0.877431i	$-0.340745\pi$
$859$	28.1189	0.959405	0.479702	+	0.877431i	$0.340745\pi$
$860$	0	0
$861$	55.9304	1.90610
$862$	0	0
$863$	31.4025	1.06895	0.534476	−	0.845183i	$-0.320509\pi$
$863$	31.4025	1.06895	0.534476	+	0.845183i	$0.320509\pi$
$864$	0	0
$865$	−17.0577	−0.579980
$866$	0	0
$867$	3.46813	0.117784
$868$	0	0
$869$	68.2231	2.31431
$870$	0	0
$871$	41.8358	1.41755
$872$	0	0
$873$	−48.4669	−1.64036
$874$	0	0
$875$	−1.69120	−0.0571730
$876$	0	0
$877$	−30.2360	−1.02100	−0.510498	−	0.859879i	$-0.670539\pi$
$877$	−30.2360	−1.02100	−0.510498	+	0.859879i	$0.670539\pi$
$878$	0	0
$879$	11.5609	0.389939
$880$	0	0
$881$	−1.30929	−0.0441112	−0.0220556	−	0.999757i	$-0.507021\pi$
$881$	−1.30929	−0.0441112	−0.0220556	+	0.999757i	$0.507021\pi$
$882$	0	0
$883$	17.5036	0.589043	0.294521	−	0.955645i	$-0.404840\pi$
$883$	17.5036	0.589043	0.294521	+	0.955645i	$0.404840\pi$
$884$	0	0
$885$	18.6180	0.625838
$886$	0	0
$887$	13.8935	0.466497	0.233249	−	0.972417i	$-0.425064\pi$
$887$	13.8935	0.466497	0.233249	+	0.972417i	$0.425064\pi$
$888$	0	0
$889$	−24.2030	−0.811741
$890$	0	0
$891$	68.2351	2.28596
$892$	0	0
$893$	−64.1804	−2.14772
$894$	0	0
$895$	26.2728	0.878204
$896$	0	0
$897$	1.09417	0.0365332
$898$	0	0
$899$	−65.2520	−2.17628
$900$	0	0
$901$	25.9742	0.865327
$902$	0	0
$903$	31.4439	1.04639
$904$	0	0
$905$	−8.93182	−0.296904
$906$	0	0
$907$	−50.7113	−1.68384	−0.841920	−	0.539602i	$-0.818575\pi$
$907$	−50.7113	−1.68384	−0.841920	+	0.539602i	$0.818575\pi$
$908$	0	0
$909$	−64.4488	−2.13763
$910$	0	0
$911$	−31.8671	−1.05580	−0.527902	−	0.849306i	$-0.677021\pi$
$911$	−31.8671	−1.05580	−0.527902	+	0.849306i	$0.677021\pi$
$912$	0	0
$913$	−28.1608	−0.931988
$914$	0	0
$915$	−2.74444	−0.0907285
$916$	0	0
$917$	14.9903	0.495023
$918$	0	0
$919$	35.5470	1.17259	0.586294	−	0.810098i	$-0.300586\pi$
$919$	35.5470	1.17259	0.586294	+	0.810098i	$0.300586\pi$
$920$	0	0
$921$	−68.5762	−2.25966
$922$	0	0
$923$	29.3810	0.967089
$924$	0	0
$925$	11.3462	0.373062
$926$	0	0
$927$	100.598	3.30408
$928$	0	0
$929$	−33.4701	−1.09812	−0.549059	−	0.835783i	$-0.685014\pi$
$929$	−33.4701	−1.09812	−0.549059	+	0.835783i	$0.685014\pi$
$930$	0	0
$931$	−29.9369	−0.981143
$932$	0	0
$933$	43.8413	1.43530
$934$	0	0
$935$	24.3051	0.794863
$936$	0	0
$937$	−30.3406	−0.991184	−0.495592	−	0.868556i	$-0.665049\pi$
$937$	−30.3406	−0.991184	−0.495592	+	0.868556i	$0.665049\pi$
$938$	0	0
$939$	−75.5912	−2.46683
$940$	0	0
$941$	−22.7386	−0.741259	−0.370629	−	0.928781i	$-0.620858\pi$
$941$	−22.7386	−0.741259	−0.370629	+	0.928781i	$0.620858\pi$
$942$	0	0
$943$	0.689031	0.0224379
$944$	0	0
$945$	16.6323	0.541049
$946$	0	0
$947$	8.12597	0.264059	0.132029	−	0.991246i	$-0.457851\pi$
$947$	8.12597	0.264059	0.132029	+	0.991246i	$0.457851\pi$
$948$	0	0
$949$	−32.9794	−1.07056
$950$	0	0
$951$	−1.86150	−0.0603634
$952$	0	0
$953$	3.37893	0.109454	0.0547271	−	0.998501i	$-0.482571\pi$
$953$	3.37893	0.109454	0.0547271	+	0.998501i	$0.482571\pi$
$954$	0	0
$955$	5.65723	0.183064
$956$	0	0
$957$	−129.430	−4.18388
$958$	0	0
$959$	5.55677	0.179437
$960$	0	0
$961$	56.4435	1.82076
$962$	0	0
$963$	−115.609	−3.72543
$964$	0	0
$965$	6.96449	0.224195
$966$	0	0
$967$	35.5648	1.14369	0.571843	−	0.820363i	$-0.306229\pi$
$967$	35.5648	1.14369	0.571843	+	0.820363i	$0.306229\pi$
$968$	0	0
$969$	−87.5190	−2.81151
$970$	0	0
$971$	−3.67141	−0.117821	−0.0589105	−	0.998263i	$-0.518763\pi$
$971$	−3.67141	−0.117821	−0.0589105	+	0.998263i	$0.518763\pi$
$972$	0	0
$973$	23.9887	0.769043
$974$	0	0
$975$	17.2804	0.553417
$976$	0	0
$977$	−31.5457	−1.00923	−0.504617	−	0.863343i	$-0.668366\pi$
$977$	−31.5457	−1.00923	−0.504617	+	0.863343i	$0.668366\pi$
$978$	0	0
$979$	42.7283	1.36560
$980$	0	0
$981$	−7.20158	−0.229929
$982$	0	0
$983$	7.33813	0.234050	0.117025	−	0.993129i	$-0.462664\pi$
$983$	7.33813	0.234050	0.117025	+	0.993129i	$0.462664\pi$
$984$	0	0
$985$	2.23855	0.0713261
$986$	0	0
$987$	−45.6160	−1.45197
$988$	0	0
$989$	0.387371	0.0123177
$990$	0	0
$991$	−38.3029	−1.21673	−0.608366	−	0.793657i	$-0.708175\pi$
$991$	−38.3029	−1.21673	−0.608366	+	0.793657i	$0.708175\pi$
$992$	0	0
$993$	−29.2457	−0.928085
$994$	0	0
$995$	−0.0570966	−0.00181008
$996$	0	0
$997$	−37.8724	−1.19943	−0.599716	−	0.800213i	$-0.704720\pi$
$997$	−37.8724	−1.19943	−0.599716	+	0.800213i	$0.704720\pi$
$998$	0	0
$999$	−111.586	−3.53042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.s.1.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.s.1.3	✓	24	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-3.03909 q^{3} +1.00000 q^{5} -1.69120 q^{7} +6.23604 q^{9} +O(q^{10})\)	\(q-3.03909 q^{3} +1.00000 q^{5} -1.69120 q^{7} +6.23604 q^{9} +6.10327 q^{11} -5.68607 q^{13} -3.03909 q^{15} +3.98231 q^{17} +7.23142 q^{19} +5.13970 q^{21} +0.0633183 q^{23} +1.00000 q^{25} -9.83461 q^{27} +6.97799 q^{29} -9.35112 q^{31} -18.5484 q^{33} -1.69120 q^{35} +11.3462 q^{37} +17.2804 q^{39} +10.8820 q^{41} +6.11783 q^{43} +6.23604 q^{45} -8.87522 q^{47} -4.13984 q^{49} -12.1026 q^{51} +6.52239 q^{53} +6.10327 q^{55} -21.9769 q^{57} -6.12619 q^{59} +0.903049 q^{61} -10.5464 q^{63} -5.68607 q^{65} -7.35760 q^{67} -0.192430 q^{69} -5.16720 q^{71} +5.80005 q^{73} -3.03909 q^{75} -10.3219 q^{77} +11.1781 q^{79} +11.1801 q^{81} -4.61406 q^{83} +3.98231 q^{85} -21.2067 q^{87} +7.00088 q^{89} +9.61628 q^{91} +28.4189 q^{93} +7.23142 q^{95} -7.77207 q^{97} +38.0602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * q^99

Introduction

L-functions

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Varieties

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Database

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