LMFDB - Embedded newform 495.6.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3 x^{9} - 246 x^{8} + 640 x^{7} + 20433 x^{6} - 44595 x^{5} - 667026 x^{4} + 1173648 x^{3} + \cdots - 30445728 $ x^10 - 3x^9 - 246x^8 + 640x^7 + 20433x^6 - 44595x^5 - 667026x^4 + 1173648x^3 + 7949136x^2 - 8226800*x - 30445728
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-8.54982$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.54982 q^{2} +41.0994 q^{4} +25.0000 q^{5} -123.368 q^{7} -77.7982 q^{8} +O(q^{10})$	\(q-8.54982 q^{2} +41.0994 q^{4} +25.0000 q^{5} -123.368 q^{7} -77.7982 q^{8} -213.745 q^{10} -121.000 q^{11} +635.596 q^{13} +1054.78 q^{14} -650.020 q^{16} -706.852 q^{17} -203.800 q^{19} +1027.48 q^{20} +1034.53 q^{22} -2058.55 q^{23} +625.000 q^{25} -5434.23 q^{26} -5070.36 q^{28} -3848.93 q^{29} +3603.03 q^{31} +8047.10 q^{32} +6043.45 q^{34} -3084.20 q^{35} +15205.8 q^{37} +1742.46 q^{38} -1944.95 q^{40} +18849.3 q^{41} +1587.74 q^{43} -4973.03 q^{44} +17600.3 q^{46} -29200.2 q^{47} -1587.29 q^{49} -5343.64 q^{50} +26122.6 q^{52} -20497.7 q^{53} -3025.00 q^{55} +9597.82 q^{56} +32907.6 q^{58} -22565.9 q^{59} -14293.8 q^{61} -30805.2 q^{62} -48000.6 q^{64} +15889.9 q^{65} +70058.4 q^{67} -29051.2 q^{68} +26369.4 q^{70} -26806.1 q^{71} +14281.0 q^{73} -130007. q^{74} -8376.07 q^{76} +14927.6 q^{77} -92109.5 q^{79} -16250.5 q^{80} -161158. q^{82} -2313.28 q^{83} -17671.3 q^{85} -13574.9 q^{86} +9413.58 q^{88} -68893.3 q^{89} -78412.3 q^{91} -84605.4 q^{92} +249657. q^{94} -5095.01 q^{95} +82688.3 q^{97} +13571.1 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8}+O(q^{10}) $ 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8	$ 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8} + 75 q^{10} - 1210 q^{11} + 932 q^{13} - 1332 q^{14} + 2701 q^{16} + 96 q^{17} + 1664 q^{19} + 4525 q^{20} - 363 q^{22} + 6288 q^{23} + 6250 q^{25} + 13380 q^{26} + 13868 q^{28} + 11208 q^{29} + 9032 q^{31} + 9801 q^{32} + 14610 q^{34} + 2900 q^{35} + 21572 q^{37} + 15870 q^{38} + 3225 q^{40} + 10800 q^{41} + 21128 q^{43} - 21901 q^{44} + 83982 q^{46} - 17400 q^{47} + 71610 q^{49} + 1875 q^{50} + 40640 q^{52} + 5004 q^{53} - 30250 q^{55} - 54012 q^{56} - 9786 q^{58} - 25272 q^{59} + 52004 q^{61} + 34740 q^{62} + 56953 q^{64} + 23300 q^{65} + 4160 q^{67} - 87978 q^{68} - 33300 q^{70} - 65232 q^{71} + 44252 q^{73} - 49842 q^{74} + 233246 q^{76} - 14036 q^{77} + 112604 q^{79} + 67525 q^{80} + 167910 q^{82} + 70032 q^{83} + 2400 q^{85} - 72978 q^{86} - 15609 q^{88} - 46848 q^{89} + 130672 q^{91} + 121302 q^{92} + 252294 q^{94} + 41600 q^{95} + 129932 q^{97} - 316137 q^{98}+O(q^{100}) $ 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8 + 75 * q^10 - 1210 * q^11 + 932 * q^13 - 1332 * q^14 + 2701 * q^16 + 96 * q^17 + 1664 * q^19 + 4525 * q^20 - 363 * q^22 + 6288 * q^23 + 6250 * q^25 + 13380 * q^26 + 13868 * q^28 + 11208 * q^29 + 9032 * q^31 + 9801 * q^32 + 14610 * q^34 + 2900 * q^35 + 21572 * q^37 + 15870 * q^38 + 3225 * q^40 + 10800 * q^41 + 21128 * q^43 - 21901 * q^44 + 83982 * q^46 - 17400 * q^47 + 71610 * q^49 + 1875 * q^50 + 40640 * q^52 + 5004 * q^53 - 30250 * q^55 - 54012 * q^56 - 9786 * q^58 - 25272 * q^59 + 52004 * q^61 + 34740 * q^62 + 56953 * q^64 + 23300 * q^65 + 4160 * q^67 - 87978 * q^68 - 33300 * q^70 - 65232 * q^71 + 44252 * q^73 - 49842 * q^74 + 233246 * q^76 - 14036 * q^77 + 112604 * q^79 + 67525 * q^80 + 167910 * q^82 + 70032 * q^83 + 2400 * q^85 - 72978 * q^86 - 15609 * q^88 - 46848 * q^89 + 130672 * q^91 + 121302 * q^92 + 252294 * q^94 + 41600 * q^95 + 129932 * q^97 - 316137 * q^98

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 $

$2$	−8.54982	−1.51141	−0.755704	−	0.654913i	$-0.772705\pi$
$2$	−8.54982	−1.51141	−0.755704	+	0.654913i	$0.772705\pi$
$3$	0	0
$3$	0	0
$4$	41.0994	1.28436
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−123.368	−0.951608	−0.475804	−	0.879551i	$-0.657843\pi$
$7$	−123.368	−0.951608	−0.475804	+	0.879551i	$0.657843\pi$
$8$	−77.7982	−0.429778
$9$	0	0
$10$	−213.745	−0.675922
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	635.596	1.04309	0.521546	−	0.853223i	$-0.325355\pi$
$13$	635.596	1.04309	0.521546	+	0.853223i	$0.325355\pi$
$14$	1054.78	1.43827
$15$	0	0
$16$	−650.020	−0.634785
$17$	−706.852	−0.593207	−0.296603	−	0.955001i	$-0.595854\pi$
$17$	−706.852	−0.593207	−0.296603	+	0.955001i	$0.595854\pi$
$18$	0	0
$19$	−203.800	−0.129515	−0.0647576	−	0.997901i	$-0.520627\pi$
$19$	−203.800	−0.129515	−0.0647576	+	0.997901i	$0.520627\pi$
$20$	1027.48	0.574382
$21$	0	0
$22$	1034.53	0.455707
$23$	−2058.55	−0.811415	−0.405707	−	0.914003i	$-0.632975\pi$
$23$	−2058.55	−0.811415	−0.405707	+	0.914003i	$0.632975\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−5434.23	−1.57654
$27$	0	0
$28$	−5070.36	−1.22220
$29$	−3848.93	−0.849855	−0.424927	−	0.905227i	$-0.639700\pi$
$29$	−3848.93	−0.849855	−0.424927	+	0.905227i	$0.639700\pi$
$30$	0	0
$31$	3603.03	0.673385	0.336693	−	0.941615i	$-0.390692\pi$
$31$	3603.03	0.673385	0.336693	+	0.941615i	$0.390692\pi$
$32$	8047.10	1.38920
$33$	0	0
$34$	6043.45	0.896578
$35$	−3084.20	−0.425572
$36$	0	0
$37$	15205.8	1.82602	0.913010	−	0.407937i	$-0.133752\pi$
$37$	15205.8	1.82602	0.913010	+	0.407937i	$0.133752\pi$
$38$	1742.46	0.195750
$39$	0	0
$40$	−1944.95	−0.192203
$41$	18849.3	1.75120	0.875599	−	0.483039i	$-0.160467\pi$
$41$	18849.3	1.75120	0.875599	+	0.483039i	$0.160467\pi$
$42$	0	0
$43$	1587.74	0.130951	0.0654754	−	0.997854i	$-0.479144\pi$
$43$	1587.74	0.130951	0.0654754	+	0.997854i	$0.479144\pi$
$44$	−4973.03	−0.387248
$45$	0	0
$46$	17600.3	1.22638
$47$	−29200.2	−1.92815	−0.964076	−	0.265625i	$-0.914422\pi$
$47$	−29200.2	−1.92815	−0.964076	+	0.265625i	$0.914422\pi$
$48$	0	0
$49$	−1587.29	−0.0944423
$50$	−5343.64	−0.302282
$51$	0	0
$52$	26122.6	1.33970
$53$	−20497.7	−1.00234	−0.501170	−	0.865349i	$-0.667097\pi$
$53$	−20497.7	−1.00234	−0.501170	+	0.865349i	$0.667097\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	9597.82	0.408980
$57$	0	0
$58$	32907.6	1.28448
$59$	−22565.9	−0.843960	−0.421980	−	0.906605i	$-0.638665\pi$
$59$	−22565.9	−0.843960	−0.421980	+	0.906605i	$0.638665\pi$
$60$	0	0
$61$	−14293.8	−0.491841	−0.245920	−	0.969290i	$-0.579090\pi$
$61$	−14293.8	−0.491841	−0.245920	+	0.969290i	$0.579090\pi$
$62$	−30805.2	−1.01776
$63$	0	0
$64$	−48000.6	−1.46486
$65$	15889.9	0.466485
$66$	0	0
$67$	70058.4	1.90666	0.953330	−	0.301931i	$-0.0976313\pi$
$67$	70058.4	1.90666	0.953330	+	0.301931i	$0.0976313\pi$
$68$	−29051.2	−0.761889
$69$	0	0
$70$	26369.4	0.643213
$71$	−26806.1	−0.631085	−0.315543	−	0.948911i	$-0.602186\pi$
$71$	−26806.1	−0.631085	−0.315543	+	0.948911i	$0.602186\pi$
$72$	0	0
$73$	14281.0	0.313654	0.156827	−	0.987626i	$-0.449873\pi$
$73$	14281.0	0.313654	0.156827	+	0.987626i	$0.449873\pi$
$74$	−130007.	−2.75986
$75$	0	0
$76$	−8376.07	−0.166344
$77$	14927.6	0.286921
$78$	0	0
$79$	−92109.5	−1.66049	−0.830246	−	0.557397i	$-0.811800\pi$
$79$	−92109.5	−1.66049	−0.830246	+	0.557397i	$0.811800\pi$
$80$	−16250.5	−0.283885
$81$	0	0
$82$	−161158.	−2.64678
$83$	−2313.28	−0.0368580	−0.0184290	−	0.999830i	$-0.505866\pi$
$83$	−2313.28	−0.0368580	−0.0184290	+	0.999830i	$0.505866\pi$
$84$	0	0
$85$	−17671.3	−0.265290
$86$	−13574.9	−0.197920
$87$	0	0
$88$	9413.58	0.129583
$89$	−68893.3	−0.921938	−0.460969	−	0.887416i	$-0.652498\pi$
$89$	−68893.3	−0.921938	−0.460969	+	0.887416i	$0.652498\pi$
$90$	0	0
$91$	−78412.3	−0.992615
$92$	−84605.4	−1.04215
$93$	0	0
$94$	249657.	2.91423
$95$	−5095.01	−0.0579210
$96$	0	0
$97$	82688.3	0.892308	0.446154	−	0.894956i	$-0.352793\pi$
$97$	82688.3	0.892308	0.446154	+	0.894956i	$0.352793\pi$
$98$	13571.1	0.142741
$99$	0	0
$100$	25687.1	0.256871
$101$	−44822.0	−0.437207	−0.218604	−	0.975814i	$-0.570150\pi$
$101$	−44822.0	−0.437207	−0.218604	+	0.975814i	$0.570150\pi$
$102$	0	0
$103$	−182739.	−1.69722	−0.848612	−	0.529016i	$-0.822561\pi$
$103$	−182739.	−1.69722	−0.848612	+	0.529016i	$0.822561\pi$
$104$	−49448.2	−0.448299
$105$	0	0
$106$	175252.	1.51495
$107$	209457.	1.76863	0.884314	−	0.466893i	$-0.154627\pi$
$107$	209457.	1.76863	0.884314	+	0.466893i	$0.154627\pi$
$108$	0	0
$109$	27535.0	0.221982	0.110991	−	0.993821i	$-0.464597\pi$
$109$	27535.0	0.221982	0.110991	+	0.993821i	$0.464597\pi$
$110$	25863.2	0.203798
$111$	0	0
$112$	80191.8	0.604067
$113$	8676.30	0.0639202	0.0319601	−	0.999489i	$-0.489825\pi$
$113$	8676.30	0.0639202	0.0319601	+	0.999489i	$0.489825\pi$
$114$	0	0
$115$	−51463.9	−0.362876
$116$	−158189.	−1.09152
$117$	0	0
$118$	192934.	1.27557
$119$	87203.0	0.564500
$120$	0	0
$121$	14641.0	0.0909091
$122$	122210.	0.743372
$123$	0	0
$124$	148082.	0.864866
$125$	15625.0	0.0894427
$126$	0	0
$127$	−28535.1	−0.156989	−0.0784946	−	0.996915i	$-0.525011\pi$
$127$	−28535.1	−0.156989	−0.0784946	+	0.996915i	$0.525011\pi$
$128$	152889.	0.824805
$129$	0	0
$130$	−135856.	−0.705050
$131$	79010.6	0.402260	0.201130	−	0.979565i	$-0.435539\pi$
$131$	79010.6	0.402260	0.201130	+	0.979565i	$0.435539\pi$
$132$	0	0
$133$	25142.5	0.123248
$134$	−598986.	−2.88174
$135$	0	0
$136$	54991.8	0.254947
$137$	265953.	1.21061	0.605304	−	0.795994i	$-0.293052\pi$
$137$	265953.	1.21061	0.605304	+	0.795994i	$0.293052\pi$
$138$	0	0
$139$	75939.2	0.333372	0.166686	−	0.986010i	$-0.446693\pi$
$139$	75939.2	0.333372	0.166686	+	0.986010i	$0.446693\pi$
$140$	−126759.	−0.546586
$141$	0	0
$142$	229187.	0.953827
$143$	−76907.1	−0.314504
$144$	0	0
$145$	−96223.2	−0.380067
$146$	−122100.	−0.474060
$147$	0	0
$148$	624950.	2.34526
$149$	−43994.5	−0.162343	−0.0811713	−	0.996700i	$-0.525866\pi$
$149$	−43994.5	−0.162343	−0.0811713	+	0.996700i	$0.525866\pi$
$150$	0	0
$151$	403186.	1.43901	0.719504	−	0.694488i	$-0.244369\pi$
$151$	403186.	1.43901	0.719504	+	0.694488i	$0.244369\pi$
$152$	15855.3	0.0556628
$153$	0	0
$154$	−127628.	−0.433654
$155$	90075.7	0.301147
$156$	0	0
$157$	166348.	0.538602	0.269301	−	0.963056i	$-0.413207\pi$
$157$	166348.	0.538602	0.269301	+	0.963056i	$0.413207\pi$
$158$	787520.	2.50968
$159$	0	0
$160$	201177.	0.621268
$161$	253960.	0.772149
$162$	0	0
$163$	407635.	1.20172	0.600859	−	0.799355i	$-0.294825\pi$
$163$	407635.	1.20172	0.600859	+	0.799355i	$0.294825\pi$
$164$	774694.	2.24916
$165$	0	0
$166$	19778.1	0.0557076
$167$	319622.	0.886842	0.443421	−	0.896314i	$-0.353765\pi$
$167$	319622.	0.886842	0.443421	+	0.896314i	$0.353765\pi$
$168$	0	0
$169$	32689.5	0.0880422
$170$	151086.	0.400962
$171$	0	0
$172$	65255.2	0.168188
$173$	−419443.	−1.06551	−0.532755	−	0.846269i	$-0.678843\pi$
$173$	−419443.	−1.06551	−0.532755	+	0.846269i	$0.678843\pi$
$174$	0	0
$175$	−77105.1	−0.190322
$176$	78652.5	0.191395
$177$	0	0
$178$	589025.	1.39342
$179$	−22741.6	−0.0530503	−0.0265252	−	0.999648i	$-0.508444\pi$
$179$	−22741.6	−0.0530503	−0.0265252	+	0.999648i	$0.508444\pi$
$180$	0	0
$181$	278350.	0.631531	0.315766	−	0.948837i	$-0.397739\pi$
$181$	278350.	0.631531	0.315766	+	0.948837i	$0.397739\pi$
$182$	670411.	1.50025
$183$	0	0
$184$	160152.	0.348728
$185$	380146.	0.816621
$186$	0	0
$187$	85529.1	0.178859
$188$	−1.20011e6	−2.47643
$189$	0	0
$190$	43561.4	0.0875423
$191$	−348379.	−0.690985	−0.345493	−	0.938421i	$-0.612288\pi$
$191$	−348379.	−0.690985	−0.345493	+	0.938421i	$0.612288\pi$
$192$	0	0
$193$	724757.	1.40055	0.700276	−	0.713872i	$-0.253060\pi$
$193$	724757.	1.40055	0.700276	+	0.713872i	$0.253060\pi$
$194$	−706970.	−1.34864
$195$	0	0
$196$	−65236.7	−0.121297
$197$	830700.	1.52503	0.762516	−	0.646970i	$-0.223964\pi$
$197$	830700.	1.52503	0.762516	+	0.646970i	$0.223964\pi$
$198$	0	0
$199$	−644919.	−1.15444	−0.577221	−	0.816588i	$-0.695863\pi$
$199$	−644919.	−1.15444	−0.577221	+	0.816588i	$0.695863\pi$
$200$	−48623.9	−0.0859557
$201$	0	0
$202$	383220.	0.660799
$203$	474835.	0.808729
$204$	0	0
$205$	471232.	0.783160
$206$	1.56239e6	2.56520
$207$	0	0
$208$	−413150.	−0.662140
$209$	24659.8	0.0390503
$210$	0	0
$211$	441463.	0.682635	0.341317	−	0.939948i	$-0.389127\pi$
$211$	441463.	0.682635	0.341317	+	0.939948i	$0.389127\pi$
$212$	−842443.	−1.28736
$213$	0	0
$214$	−1.79082e6	−2.67312
$215$	39693.5	0.0585630
$216$	0	0
$217$	−444499.	−0.640799
$218$	−235419.	−0.335506
$219$	0	0
$220$	−124326.	−0.173183
$221$	−449272.	−0.618770
$222$	0	0
$223$	−550784.	−0.741685	−0.370842	−	0.928696i	$-0.620931\pi$
$223$	−550784.	−0.741685	−0.370842	+	0.928696i	$0.620931\pi$
$224$	−992756.	−1.32197
$225$	0	0
$226$	−74180.8	−0.0966096
$227$	816796.	1.05208	0.526041	−	0.850460i	$-0.323676\pi$
$227$	816796.	1.05208	0.526041	+	0.850460i	$0.323676\pi$
$228$	0	0
$229$	1.12207e6	1.41394	0.706972	−	0.707242i	$-0.250061\pi$
$229$	1.12207e6	1.41394	0.706972	+	0.707242i	$0.250061\pi$
$230$	440007.	0.548453
$231$	0	0
$232$	299440.	0.365249
$233$	1.39917e6	1.68842	0.844210	−	0.536013i	$-0.180070\pi$
$233$	1.39917e6	1.68842	0.844210	+	0.536013i	$0.180070\pi$
$234$	0	0
$235$	−730005.	−0.862296
$236$	−927443.	−1.08394
$237$	0	0
$238$	−745570.	−0.853190
$239$	800861.	0.906906	0.453453	−	0.891280i	$-0.350192\pi$
$239$	800861.	0.906906	0.453453	+	0.891280i	$0.350192\pi$
$240$	0	0
$241$	369630.	0.409944	0.204972	−	0.978768i	$-0.434290\pi$
$241$	369630.	0.409944	0.204972	+	0.978768i	$0.434290\pi$
$242$	−125178.	−0.137401
$243$	0	0
$244$	−587468.	−0.631698
$245$	−39682.3	−0.0422359
$246$	0	0
$247$	−129535.	−0.135096
$248$	−280309.	−0.289406
$249$	0	0
$250$	−133591.	−0.135184
$251$	−610979.	−0.612127	−0.306064	−	0.952011i	$-0.599012\pi$
$251$	−610979.	−0.612127	−0.306064	+	0.952011i	$0.599012\pi$
$252$	0	0
$253$	249085.	0.244651
$254$	243970.	0.237275
$255$	0	0
$256$	228845.	0.218243
$257$	1.33469e6	1.26051	0.630256	−	0.776387i	$-0.282950\pi$
$257$	1.33469e6	1.26051	0.630256	+	0.776387i	$0.282950\pi$
$258$	0	0
$259$	−1.87591e6	−1.73766
$260$	653065.	0.599133
$261$	0	0
$262$	−675526.	−0.607979
$263$	−87498.5	−0.0780030	−0.0390015	−	0.999239i	$-0.512418\pi$
$263$	−87498.5	−0.0780030	−0.0390015	+	0.999239i	$0.512418\pi$
$264$	0	0
$265$	−512442.	−0.448260
$266$	−214964.	−0.186278
$267$	0	0
$268$	2.87936e6	2.44883
$269$	731224.	0.616126	0.308063	−	0.951366i	$-0.400319\pi$
$269$	731224.	0.616126	0.308063	+	0.951366i	$0.400319\pi$
$270$	0	0
$271$	1.77084e6	1.46473	0.732363	−	0.680914i	$-0.238417\pi$
$271$	1.77084e6	1.46473	0.732363	+	0.680914i	$0.238417\pi$
$272$	459468.	0.376559
$273$	0	0
$274$	−2.27385e6	−1.82972
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	2.02043e6	1.58214	0.791070	−	0.611725i	$-0.209524\pi$
$277$	2.02043e6	1.58214	0.791070	+	0.611725i	$0.209524\pi$
$278$	−649266.	−0.503861
$279$	0	0
$280$	239946.	0.182902
$281$	789275.	0.596297	0.298148	−	0.954520i	$-0.403631\pi$
$281$	789275.	0.596297	0.298148	+	0.954520i	$0.403631\pi$
$282$	0	0
$283$	−632675.	−0.469585	−0.234793	−	0.972045i	$-0.575441\pi$
$283$	−632675.	−0.469585	−0.234793	+	0.972045i	$0.575441\pi$
$284$	−1.10171e6	−0.810538
$285$	0	0
$286$	657542.	0.475344
$287$	−2.32540e6	−1.66645
$288$	0	0
$289$	−920218.	−0.648106
$290$	822691.	0.574436
$291$	0	0
$292$	586940.	0.402844
$293$	−2.16925e6	−1.47618	−0.738092	−	0.674700i	$-0.764273\pi$
$293$	−2.16925e6	−1.47618	−0.738092	+	0.674700i	$0.764273\pi$
$294$	0	0
$295$	−564146.	−0.377430
$296$	−1.18299e6	−0.784784
$297$	0	0
$298$	376145.	0.245366
$299$	−1.30841e6	−0.846381
$300$	0	0
$301$	−195877.	−0.124614
$302$	−3.44717e6	−2.17493
$303$	0	0
$304$	132474.	0.0822144
$305$	−357346.	−0.219958
$306$	0	0
$307$	−725573.	−0.439375	−0.219687	−	0.975570i	$-0.570504\pi$
$307$	−725573.	−0.439375	−0.219687	+	0.975570i	$0.570504\pi$
$308$	613513.	0.368508
$309$	0	0
$310$	−770131.	−0.455156
$311$	−89890.2	−0.0527001	−0.0263500	−	0.999653i	$-0.508388\pi$
$311$	−89890.2	−0.0527001	−0.0263500	+	0.999653i	$0.508388\pi$
$312$	0	0
$313$	1.64295e6	0.947904	0.473952	−	0.880551i	$-0.342827\pi$
$313$	1.64295e6	0.947904	0.473952	+	0.880551i	$0.342827\pi$
$314$	−1.42224e6	−0.814048
$315$	0	0
$316$	−3.78565e6	−2.13266
$317$	−807498.	−0.451329	−0.225664	−	0.974205i	$-0.572455\pi$
$317$	−807498.	−0.451329	−0.225664	+	0.974205i	$0.572455\pi$
$318$	0	0
$319$	465720.	0.256241
$320$	−1.20001e6	−0.655106
$321$	0	0
$322$	−2.17131e6	−1.16703
$323$	144057.	0.0768293
$324$	0	0
$325$	397248.	0.208619
$326$	−3.48521e6	−1.81629
$327$	0	0
$328$	−1.46644e6	−0.752627
$329$	3.60238e6	1.83485
$330$	0	0
$331$	−447398.	−0.224452	−0.112226	−	0.993683i	$-0.535798\pi$
$331$	−447398.	−0.224452	−0.112226	+	0.993683i	$0.535798\pi$
$332$	−95074.3	−0.0473388
$333$	0	0
$334$	−2.73271e6	−1.34038
$335$	1.75146e6	0.852684
$336$	0	0
$337$	−1.72142e6	−0.825682	−0.412841	−	0.910803i	$-0.635464\pi$
$337$	−1.72142e6	−0.825682	−0.412841	+	0.910803i	$0.635464\pi$
$338$	−279489.	−0.133068
$339$	0	0
$340$	−726279.	−0.340727
$341$	−435966.	−0.203033
$342$	0	0
$343$	2.26927e6	1.04148
$344$	−123523.	−0.0562798
$345$	0	0
$346$	3.58616e6	1.61042
$347$	3.50845e6	1.56420	0.782100	−	0.623153i	$-0.214149\pi$
$347$	3.50845e6	1.56420	0.782100	+	0.623153i	$0.214149\pi$
$348$	0	0
$349$	837754.	0.368174	0.184087	−	0.982910i	$-0.441067\pi$
$349$	837754.	0.368174	0.184087	+	0.982910i	$0.441067\pi$
$350$	659235.	0.287654
$351$	0	0
$352$	−973699.	−0.418859
$353$	−2.38882e6	−1.02034	−0.510172	−	0.860073i	$-0.670418\pi$
$353$	−2.38882e6	−1.02034	−0.510172	+	0.860073i	$0.670418\pi$
$354$	0	0
$355$	−670153.	−0.282230
$356$	−2.83147e6	−1.18410
$357$	0	0
$358$	194436.	0.0801807
$359$	−1.52336e6	−0.623832	−0.311916	−	0.950110i	$-0.600971\pi$
$359$	−1.52336e6	−0.623832	−0.311916	+	0.950110i	$0.600971\pi$
$360$	0	0
$361$	−2.43456e6	−0.983226
$362$	−2.37984e6	−0.954502
$363$	0	0
$364$	−3.22270e6	−1.27487
$365$	357025.	0.140270
$366$	0	0
$367$	1.74441e6	0.676059	0.338029	−	0.941136i	$-0.390240\pi$
$367$	1.74441e6	0.676059	0.338029	+	0.941136i	$0.390240\pi$
$368$	1.33810e6	0.515074
$369$	0	0
$370$	−3.25018e6	−1.23425
$371$	2.52876e6	0.953835
$372$	0	0
$373$	2.31250e6	0.860615	0.430308	−	0.902682i	$-0.358405\pi$
$373$	2.31250e6	0.860615	0.430308	+	0.902682i	$0.358405\pi$
$374$	−731258.	−0.270328
$375$	0	0
$376$	2.27172e6	0.828678
$377$	−2.44636e6	−0.886477
$378$	0	0
$379$	−1.34246e6	−0.480068	−0.240034	−	0.970765i	$-0.577159\pi$
$379$	−1.34246e6	−0.480068	−0.240034	+	0.970765i	$0.577159\pi$
$380$	−209402.	−0.0743912
$381$	0	0
$382$	2.97858e6	1.04436
$383$	4.11127e6	1.43212	0.716060	−	0.698039i	$-0.245944\pi$
$383$	4.11127e6	1.43212	0.716060	+	0.698039i	$0.245944\pi$
$384$	0	0
$385$	373189.	0.128315
$386$	−6.19654e6	−2.11681
$387$	0	0
$388$	3.39844e6	1.14604
$389$	3.61085e6	1.20986	0.604931	−	0.796278i	$-0.293201\pi$
$389$	3.61085e6	1.20986	0.604931	+	0.796278i	$0.293201\pi$
$390$	0	0
$391$	1.45509e6	0.481337
$392$	123488.	0.0405892
$393$	0	0
$394$	−7.10234e6	−2.30495
$395$	−2.30274e6	−0.742595
$396$	0	0
$397$	−2.93570e6	−0.934836	−0.467418	−	0.884036i	$-0.654816\pi$
$397$	−2.93570e6	−0.934836	−0.467418	+	0.884036i	$0.654816\pi$
$398$	5.51394e6	1.74483
$399$	0	0
$400$	−406263.	−0.126957
$401$	−1.80710e6	−0.561204	−0.280602	−	0.959824i	$-0.590534\pi$
$401$	−1.80710e6	−0.561204	−0.280602	+	0.959824i	$0.590534\pi$
$402$	0	0
$403$	2.29007e6	0.702403
$404$	−1.84216e6	−0.561530
$405$	0	0
$406$	−4.05976e6	−1.22232
$407$	−1.83990e6	−0.550566
$408$	0	0
$409$	−357075.	−0.105548	−0.0527741	−	0.998606i	$-0.516806\pi$
$409$	−357075.	−0.105548	−0.0527741	+	0.998606i	$0.516806\pi$
$410$	−4.02895e6	−1.18367
$411$	0	0
$412$	−7.51048e6	−2.17984
$413$	2.78391e6	0.803119
$414$	0	0
$415$	−57831.9	−0.0164834
$416$	5.11470e6	1.44906
$417$	0	0
$418$	−210837.	−0.0590210
$419$	−4.84151e6	−1.34724	−0.673621	−	0.739077i	$-0.735262\pi$
$419$	−4.84151e6	−1.34724	−0.673621	+	0.739077i	$0.735262\pi$
$420$	0	0
$421$	5.68699e6	1.56379	0.781893	−	0.623413i	$-0.214254\pi$
$421$	5.68699e6	1.56379	0.781893	+	0.623413i	$0.214254\pi$
$422$	−3.77443e6	−1.03174
$423$	0	0
$424$	1.59468e6	0.430784
$425$	−441782.	−0.118641
$426$	0	0
$427$	1.76341e6	0.468039
$428$	8.60858e6	2.27155
$429$	0	0
$430$	−339372.	−0.0885126
$431$	−4.75618e6	−1.23329	−0.616645	−	0.787242i	$-0.711508\pi$
$431$	−4.75618e6	−1.23329	−0.616645	+	0.787242i	$0.711508\pi$
$432$	0	0
$433$	2.01966e6	0.517676	0.258838	−	0.965921i	$-0.416660\pi$
$433$	2.01966e6	0.517676	0.258838	+	0.965921i	$0.416660\pi$
$434$	3.80039e6	0.968509
$435$	0	0
$436$	1.13167e6	0.285104
$437$	419534.	0.105091
$438$	0	0
$439$	−5.56198e6	−1.37742	−0.688712	−	0.725035i	$-0.741824\pi$
$439$	−5.56198e6	−1.37742	−0.688712	+	0.725035i	$0.741824\pi$
$440$	235340.	0.0579513
$441$	0	0
$442$	3.84120e6	0.935214
$443$	6.54406e6	1.58430	0.792151	−	0.610325i	$-0.208961\pi$
$443$	6.54406e6	1.58430	0.792151	+	0.610325i	$0.208961\pi$
$444$	0	0
$445$	−1.72233e6	−0.412303
$446$	4.70911e6	1.12099
$447$	0	0
$448$	5.92174e6	1.39397
$449$	1.30308e6	0.305038	0.152519	−	0.988301i	$-0.451261\pi$
$449$	1.30308e6	0.305038	0.152519	+	0.988301i	$0.451261\pi$
$450$	0	0
$451$	−2.28076e6	−0.528006
$452$	356591.	0.0820963
$453$	0	0
$454$	−6.98346e6	−1.59012
$455$	−1.96031e6	−0.443911
$456$	0	0
$457$	−3.68787e6	−0.826010	−0.413005	−	0.910729i	$-0.635521\pi$
$457$	−3.68787e6	−0.826010	−0.413005	+	0.910729i	$0.635521\pi$
$458$	−9.59352e6	−2.13705
$459$	0	0
$460$	−2.11513e6	−0.466062
$461$	4.69924e6	1.02985	0.514926	−	0.857235i	$-0.327819\pi$
$461$	4.69924e6	1.02985	0.514926	+	0.857235i	$0.327819\pi$
$462$	0	0
$463$	−5.86205e6	−1.27086	−0.635429	−	0.772159i	$-0.719177\pi$
$463$	−5.86205e6	−1.27086	−0.635429	+	0.772159i	$0.719177\pi$
$464$	2.50188e6	0.539476
$465$	0	0
$466$	−1.19626e7	−2.55189
$467$	2.90734e6	0.616884	0.308442	−	0.951243i	$-0.400192\pi$
$467$	2.90734e6	0.616884	0.308442	+	0.951243i	$0.400192\pi$
$468$	0	0
$469$	−8.64298e6	−1.81439
$470$	6.24141e6	1.30328
$471$	0	0
$472$	1.75558e6	0.362716
$473$	−192117.	−0.0394832
$474$	0	0
$475$	−127375.	−0.0259031
$476$	3.58399e6	0.725019
$477$	0	0
$478$	−6.84721e6	−1.37071
$479$	7.43086e6	1.47979	0.739895	−	0.672722i	$-0.234875\pi$
$479$	7.43086e6	1.47979	0.739895	+	0.672722i	$0.234875\pi$
$480$	0	0
$481$	9.66476e6	1.90471
$482$	−3.16027e6	−0.619593
$483$	0	0
$484$	601736.	0.116760
$485$	2.06721e6	0.399052
$486$	0	0
$487$	6.23031e6	1.19039	0.595193	−	0.803583i	$-0.297076\pi$
$487$	6.23031e6	1.19039	0.595193	+	0.803583i	$0.297076\pi$
$488$	1.11203e6	0.211382
$489$	0	0
$490$	339276.	0.0638357
$491$	−1.49022e6	−0.278963	−0.139481	−	0.990225i	$-0.544544\pi$
$491$	−1.49022e6	−0.278963	−0.139481	+	0.990225i	$0.544544\pi$
$492$	0	0
$493$	2.72062e6	0.504140
$494$	1.10750e6	0.204186
$495$	0	0
$496$	−2.34204e6	−0.427455
$497$	3.30702e6	0.600546
$498$	0	0
$499$	4.11214e6	0.739292	0.369646	−	0.929173i	$-0.379479\pi$
$499$	4.11214e6	0.739292	0.369646	+	0.929173i	$0.379479\pi$
$500$	642178.	0.114876
$501$	0	0
$502$	5.22376e6	0.925175
$503$	3.28534e6	0.578976	0.289488	−	0.957182i	$-0.406515\pi$
$503$	3.28534e6	0.578976	0.289488	+	0.957182i	$0.406515\pi$
$504$	0	0
$505$	−1.12055e6	−0.195525
$506$	−2.12963e6	−0.369767
$507$	0	0
$508$	−1.17277e6	−0.201630
$509$	5.69457e6	0.974241	0.487121	−	0.873335i	$-0.338047\pi$
$509$	5.69457e6	0.974241	0.487121	+	0.873335i	$0.338047\pi$
$510$	0	0
$511$	−1.76182e6	−0.298476
$512$	−6.84903e6	−1.15466
$513$	0	0
$514$	−1.14113e7	−1.90515
$515$	−4.56848e6	−0.759021
$516$	0	0
$517$	3.53323e6	0.581360
$518$	1.60387e7	2.62631
$519$	0	0
$520$	−1.23621e6	−0.200485
$521$	1.08789e6	0.175586	0.0877932	−	0.996139i	$-0.472019\pi$
$521$	1.08789e6	0.175586	0.0877932	+	0.996139i	$0.472019\pi$
$522$	0	0
$523$	6.92316e6	1.10675	0.553376	−	0.832932i	$-0.313339\pi$
$523$	6.92316e6	1.10675	0.553376	+	0.832932i	$0.313339\pi$
$524$	3.24729e6	0.516645
$525$	0	0
$526$	748096.	0.117894
$527$	−2.54681e6	−0.399457
$528$	0	0
$529$	−2.19870e6	−0.341606
$530$	4.38129e6	0.677504
$531$	0	0
$532$	1.03334e6	0.158294
$533$	1.19805e7	1.82666
$534$	0	0
$535$	5.23644e6	0.790954
$536$	−5.45042e6	−0.819441
$537$	0	0
$538$	−6.25183e6	−0.931218
$539$	192062.	0.0284754
$540$	0	0
$541$	8.30148e6	1.21944	0.609722	−	0.792615i	$-0.291281\pi$
$541$	8.30148e6	1.21944	0.609722	+	0.792615i	$0.291281\pi$
$542$	−1.51404e7	−2.21380
$543$	0	0
$544$	−5.68811e6	−0.824082
$545$	688374.	0.0992735
$546$	0	0
$547$	7.58757e6	1.08426	0.542131	−	0.840294i	$-0.317618\pi$
$547$	7.58757e6	1.08426	0.542131	+	0.840294i	$0.317618\pi$
$548$	1.09305e7	1.55485
$549$	0	0
$550$	646580.	0.0911414
$551$	784413.	0.110069
$552$	0	0
$553$	1.13634e7	1.58014
$554$	−1.72743e7	−2.39126
$555$	0	0
$556$	3.12105e6	0.428168
$557$	−725421.	−0.0990723	−0.0495362	−	0.998772i	$-0.515774\pi$
$557$	−725421.	−0.0990723	−0.0495362	+	0.998772i	$0.515774\pi$
$558$	0	0
$559$	1.00916e6	0.136594
$560$	2.00480e6	0.270147
$561$	0	0
$562$	−6.74816e6	−0.901248
$563$	−6.30382e6	−0.838171	−0.419085	−	0.907947i	$-0.637649\pi$
$563$	−6.30382e6	−0.838171	−0.419085	+	0.907947i	$0.637649\pi$
$564$	0	0
$565$	216907.	0.0285860
$566$	5.40926e6	0.709735
$567$	0	0
$568$	2.08547e6	0.271227
$569$	−2.18722e6	−0.283212	−0.141606	−	0.989923i	$-0.545227\pi$
$569$	−2.18722e6	−0.283212	−0.141606	+	0.989923i	$0.545227\pi$
$570$	0	0
$571$	−9.17603e6	−1.17778	−0.588891	−	0.808213i	$-0.700435\pi$
$571$	−9.17603e6	−1.17778	−0.588891	+	0.808213i	$0.700435\pi$
$572$	−3.16084e6	−0.403935
$573$	0	0
$574$	1.98818e7	2.51869
$575$	−1.28660e6	−0.162283
$576$	0	0
$577$	2.45975e6	0.307575	0.153787	−	0.988104i	$-0.450853\pi$
$577$	2.45975e6	0.307575	0.153787	+	0.988104i	$0.450853\pi$
$578$	7.86769e6	0.979553
$579$	0	0
$580$	−3.95472e6	−0.488141
$581$	285385.	0.0350744
$582$	0	0
$583$	2.48022e6	0.302217
$584$	−1.11103e6	−0.134802
$585$	0	0
$586$	1.85467e7	2.23112
$587$	3.64784e6	0.436959	0.218480	−	0.975842i	$-0.429890\pi$
$587$	3.64784e6	0.436959	0.218480	+	0.975842i	$0.429890\pi$
$588$	0	0
$589$	−734299.	−0.0872136
$590$	4.82335e6	0.570451
$591$	0	0
$592$	−9.88409e6	−1.15913
$593$	1.51169e6	0.176534	0.0882668	−	0.996097i	$-0.471867\pi$
$593$	1.51169e6	0.176534	0.0882668	+	0.996097i	$0.471867\pi$
$594$	0	0
$595$	2.18008e6	0.252452
$596$	−1.80815e6	−0.208506
$597$	0	0
$598$	1.11867e7	1.27923
$599$	−2.31883e6	−0.264059	−0.132030	−	0.991246i	$-0.542149\pi$
$599$	−2.31883e6	−0.264059	−0.132030	+	0.991246i	$0.542149\pi$
$600$	0	0
$601$	−1.00630e6	−0.113643	−0.0568215	−	0.998384i	$-0.518097\pi$
$601$	−1.00630e6	−0.113643	−0.0568215	+	0.998384i	$0.518097\pi$
$602$	1.67471e6	0.188342
$603$	0	0
$604$	1.65707e7	1.84820
$605$	366025.	0.0406558
$606$	0	0
$607$	−8.30117e6	−0.914466	−0.457233	−	0.889347i	$-0.651160\pi$
$607$	−8.30117e6	−0.914466	−0.457233	+	0.889347i	$0.651160\pi$
$608$	−1.64000e6	−0.179922
$609$	0	0
$610$	3.05524e6	0.332446
$611$	−1.85595e7	−2.01124
$612$	0	0
$613$	1.50606e7	1.61880	0.809398	−	0.587260i	$-0.199793\pi$
$613$	1.50606e7	1.61880	0.809398	+	0.587260i	$0.199793\pi$
$614$	6.20351e6	0.664075
$615$	0	0
$616$	−1.16134e6	−0.123312
$617$	−1.53471e7	−1.62298	−0.811489	−	0.584367i	$-0.801343\pi$
$617$	−1.53471e7	−1.62298	−0.811489	+	0.584367i	$0.801343\pi$
$618$	0	0
$619$	1.08830e7	1.14163	0.570813	−	0.821080i	$-0.306628\pi$
$619$	1.08830e7	1.14163	0.570813	+	0.821080i	$0.306628\pi$
$620$	3.70206e6	0.386780
$621$	0	0
$622$	768545.	0.0796514
$623$	8.49924e6	0.877323
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.40469e7	−1.43267
$627$	0	0
$628$	6.83680e6	0.691757
$629$	−1.07483e7	−1.08321
$630$	0	0
$631$	1.79709e7	1.79679	0.898393	−	0.439193i	$-0.144736\pi$
$631$	1.79709e7	1.79679	0.898393	+	0.439193i	$0.144736\pi$
$632$	7.16595e6	0.713644
$633$	0	0
$634$	6.90396e6	0.682142
$635$	−713377.	−0.0702077
$636$	0	0
$637$	−1.00888e6	−0.0985120
$638$	−3.98182e6	−0.387285
$639$	0	0
$640$	3.82223e6	0.368864
$641$	−1.60177e7	−1.53977	−0.769883	−	0.638185i	$-0.779685\pi$
$641$	−1.60177e7	−1.53977	−0.769883	+	0.638185i	$0.779685\pi$
$642$	0	0
$643$	1.32687e6	0.126561	0.0632806	−	0.997996i	$-0.479844\pi$
$643$	1.32687e6	0.126561	0.0632806	+	0.997996i	$0.479844\pi$
$644$	1.04376e7	0.991714
$645$	0	0
$646$	−1.23166e6	−0.116120
$647$	−7.17703e6	−0.674037	−0.337019	−	0.941498i	$-0.609419\pi$
$647$	−7.17703e6	−0.674037	−0.337019	+	0.941498i	$0.609419\pi$
$648$	0	0
$649$	2.73047e6	0.254463
$650$	−3.39639e6	−0.315308
$651$	0	0
$652$	1.67536e7	1.54343
$653$	−1.49977e7	−1.37639	−0.688193	−	0.725527i	$-0.741596\pi$
$653$	−1.49977e7	−1.37639	−0.688193	+	0.725527i	$0.741596\pi$
$654$	0	0
$655$	1.97526e6	0.179896
$656$	−1.22524e7	−1.11164
$657$	0	0
$658$	−3.07997e7	−2.77320
$659$	8.60760e6	0.772091	0.386046	−	0.922480i	$-0.373841\pi$
$659$	8.60760e6	0.772091	0.386046	+	0.922480i	$0.373841\pi$
$660$	0	0
$661$	−132024.	−0.0117530	−0.00587649	−	0.999983i	$-0.501871\pi$
$661$	−132024.	−0.0117530	−0.00587649	+	0.999983i	$0.501871\pi$
$662$	3.82517e6	0.339239
$663$	0	0
$664$	179969.	0.0158408
$665$	628562.	0.0551181
$666$	0	0
$667$	7.92323e6	0.689585
$668$	1.31363e7	1.13902
$669$	0	0
$670$	−1.49747e7	−1.28875
$671$	1.72955e6	0.148295
$672$	0	0
$673$	3.06430e6	0.260791	0.130396	−	0.991462i	$-0.458375\pi$
$673$	3.06430e6	0.260791	0.130396	+	0.991462i	$0.458375\pi$
$674$	1.47179e7	1.24794
$675$	0	0
$676$	1.34352e6	0.113078
$677$	5.62012e6	0.471275	0.235637	−	0.971841i	$-0.424282\pi$
$677$	5.62012e6	0.471275	0.235637	+	0.971841i	$0.424282\pi$
$678$	0	0
$679$	−1.02011e7	−0.849127
$680$	1.37479e6	0.114016
$681$	0	0
$682$	3.72743e6	0.306866
$683$	−6.11516e6	−0.501598	−0.250799	−	0.968039i	$-0.580693\pi$
$683$	−6.11516e6	−0.501598	−0.250799	+	0.968039i	$0.580693\pi$
$684$	0	0
$685$	6.64883e6	0.541400
$686$	−1.94018e7	−1.57410
$687$	0	0
$688$	−1.03206e6	−0.0831257
$689$	−1.30283e7	−1.04553
$690$	0	0
$691$	−938001.	−0.0747323	−0.0373661	−	0.999302i	$-0.511897\pi$
$691$	−938001.	−0.0747323	−0.0373661	+	0.999302i	$0.511897\pi$
$692$	−1.72389e7	−1.36849
$693$	0	0
$694$	−2.99967e7	−2.36415
$695$	1.89848e6	0.149088
$696$	0	0
$697$	−1.33237e7	−1.03882
$698$	−7.16265e6	−0.556461
$699$	0	0
$700$	−3.16897e6	−0.244441
$701$	−8.76569e6	−0.673738	−0.336869	−	0.941551i	$-0.609368\pi$
$701$	−8.76569e6	−0.673738	−0.336869	+	0.941551i	$0.609368\pi$
$702$	0	0
$703$	−3.09895e6	−0.236497
$704$	5.80807e6	0.441672
$705$	0	0
$706$	2.04240e7	1.54216
$707$	5.52961e6	0.416050
$708$	0	0
$709$	2.73257e6	0.204153	0.102077	−	0.994777i	$-0.467451\pi$
$709$	2.73257e6	0.204153	0.102077	+	0.994777i	$0.467451\pi$
$710$	5.72968e6	0.426565
$711$	0	0
$712$	5.35977e6	0.396229
$713$	−7.41703e6	−0.546394
$714$	0	0
$715$	−1.92268e6	−0.140651
$716$	−934665.	−0.0681355
$717$	0	0
$718$	1.30245e7	0.942865
$719$	−1.11913e7	−0.807341	−0.403670	−	0.914905i	$-0.632266\pi$
$719$	−1.11913e7	−0.807341	−0.403670	+	0.914905i	$0.632266\pi$
$720$	0	0
$721$	2.25442e7	1.61509
$722$	2.08151e7	1.48606
$723$	0	0
$724$	1.14400e7	0.811111
$725$	−2.40558e6	−0.169971
$726$	0	0
$727$	−2.44003e7	−1.71222	−0.856110	−	0.516794i	$-0.827125\pi$
$727$	−2.44003e7	−1.71222	−0.856110	+	0.516794i	$0.827125\pi$
$728$	6.10034e6	0.426605
$729$	0	0
$730$	−3.05250e6	−0.212006
$731$	−1.12230e6	−0.0776809
$732$	0	0
$733$	−2.24723e7	−1.54485	−0.772427	−	0.635104i	$-0.780957\pi$
$733$	−2.24723e7	−1.54485	−0.772427	+	0.635104i	$0.780957\pi$
$734$	−1.49144e7	−1.02180
$735$	0	0
$736$	−1.65654e7	−1.12722
$737$	−8.47706e6	−0.574879
$738$	0	0
$739$	2.07038e7	1.39456	0.697282	−	0.716797i	$-0.254393\pi$
$739$	2.07038e7	1.39456	0.697282	+	0.716797i	$0.254393\pi$
$740$	1.56238e7	1.04883
$741$	0	0
$742$	−2.16205e7	−1.44163
$743$	−6.17792e6	−0.410554	−0.205277	−	0.978704i	$-0.565809\pi$
$743$	−6.17792e6	−0.410554	−0.205277	+	0.978704i	$0.565809\pi$
$744$	0	0
$745$	−1.09986e6	−0.0726018
$746$	−1.97714e7	−1.30074
$747$	0	0
$748$	3.51519e6	0.229718
$749$	−2.58404e7	−1.68304
$750$	0	0
$751$	−1.29465e7	−0.837629	−0.418815	−	0.908072i	$-0.637554\pi$
$751$	−1.29465e7	−0.837629	−0.418815	+	0.908072i	$0.637554\pi$
$752$	1.89807e7	1.22396
$753$	0	0
$754$	2.09160e7	1.33983
$755$	1.00797e7	0.643544
$756$	0	0
$757$	4.20461e6	0.266678	0.133339	−	0.991071i	$-0.457430\pi$
$757$	4.20461e6	0.266678	0.133339	+	0.991071i	$0.457430\pi$
$758$	1.14778e7	0.725579
$759$	0	0
$760$	396382.	0.0248932
$761$	1.26627e7	0.792622	0.396311	−	0.918116i	$-0.370290\pi$
$761$	1.26627e7	0.792622	0.396311	+	0.918116i	$0.370290\pi$
$762$	0	0
$763$	−3.39694e6	−0.211240
$764$	−1.43182e7	−0.887471
$765$	0	0
$766$	−3.51506e7	−2.16452
$767$	−1.43428e7	−0.880328
$768$	0	0
$769$	−7.47776e6	−0.455991	−0.227995	−	0.973662i	$-0.573217\pi$
$769$	−7.47776e6	−0.455991	−0.227995	+	0.973662i	$0.573217\pi$
$770$	−3.19070e6	−0.193936
$771$	0	0
$772$	2.97871e7	1.79881
$773$	4.30971e6	0.259418	0.129709	−	0.991552i	$-0.458596\pi$
$773$	4.30971e6	0.259418	0.129709	+	0.991552i	$0.458596\pi$
$774$	0	0
$775$	2.25189e6	0.134677
$776$	−6.43300e6	−0.383495
$777$	0	0
$778$	−3.08722e7	−1.82860
$779$	−3.84149e6	−0.226807
$780$	0	0
$781$	3.24354e6	0.190279
$782$	−1.24408e7	−0.727496
$783$	0	0
$784$	1.03177e6	0.0599506
$785$	4.15870e6	0.240870
$786$	0	0
$787$	2.05712e7	1.18392	0.591961	−	0.805966i	$-0.298354\pi$
$787$	2.05712e7	1.18392	0.591961	+	0.805966i	$0.298354\pi$
$788$	3.41413e7	1.95868
$789$	0	0
$790$	1.96880e7	1.12236
$791$	−1.07038e6	−0.0608270
$792$	0	0
$793$	−9.08511e6	−0.513035
$794$	2.50997e7	1.41292
$795$	0	0
$796$	−2.65058e7	−1.48272
$797$	2.78384e7	1.55238	0.776192	−	0.630496i	$-0.217149\pi$
$797$	2.78384e7	1.55238	0.776192	+	0.630496i	$0.217149\pi$
$798$	0	0
$799$	2.06402e7	1.14379
$800$	5.02944e6	0.277840
$801$	0	0
$802$	1.54504e7	0.848209
$803$	−1.72800e6	−0.0945703
$804$	0	0
$805$	6.34900e6	0.345315
$806$	−1.95797e7	−1.06162
$807$	0	0
$808$	3.48707e6	0.187902
$809$	−3.16781e7	−1.70172	−0.850859	−	0.525393i	$-0.823918\pi$
$809$	−3.16781e7	−1.70172	−0.850859	+	0.525393i	$0.823918\pi$
$810$	0	0
$811$	−1.77228e7	−0.946194	−0.473097	−	0.881010i	$-0.656864\pi$
$811$	−1.77228e7	−0.946194	−0.473097	+	0.881010i	$0.656864\pi$
$812$	1.95154e7	1.03870
$813$	0	0
$814$	1.57308e7	0.832130
$815$	1.01909e7	0.537425
$816$	0	0
$817$	−323582.	−0.0169601
$818$	3.05292e6	0.159526
$819$	0	0
$820$	1.93674e7	1.00586
$821$	8.70974e6	0.450970	0.225485	−	0.974247i	$-0.427603\pi$
$821$	8.70974e6	0.450970	0.225485	+	0.974247i	$0.427603\pi$
$822$	0	0
$823$	−1.83219e7	−0.942909	−0.471455	−	0.881890i	$-0.656271\pi$
$823$	−1.83219e7	−0.942909	−0.471455	+	0.881890i	$0.656271\pi$
$824$	1.42168e7	0.729430
$825$	0	0
$826$	−2.38019e7	−1.21384
$827$	7.34720e6	0.373558	0.186779	−	0.982402i	$-0.440195\pi$
$827$	7.34720e6	0.373558	0.186779	+	0.982402i	$0.440195\pi$
$828$	0	0
$829$	1.29206e7	0.652974	0.326487	−	0.945202i	$-0.394135\pi$
$829$	1.29206e7	0.652974	0.326487	+	0.945202i	$0.394135\pi$
$830$	494452.	0.0249132
$831$	0	0
$832$	−3.05090e7	−1.52799
$833$	1.12198e6	0.0560238
$834$	0	0
$835$	7.99056e6	0.396608
$836$	1.01350e6	0.0501545
$837$	0	0
$838$	4.13940e7	2.03623
$839$	1.38982e7	0.681639	0.340820	−	0.940129i	$-0.389295\pi$
$839$	1.38982e7	0.681639	0.340820	+	0.940129i	$0.389295\pi$
$840$	0	0
$841$	−5.69690e6	−0.277747
$842$	−4.86227e7	−2.36352
$843$	0	0
$844$	1.81439e7	0.876746
$845$	817237.	0.0393737
$846$	0	0
$847$	−1.80623e6	−0.0865098
$848$	1.33239e7	0.636271
$849$	0	0
$850$	3.77716e6	0.179316
$851$	−3.13020e7	−1.48166
$852$	0	0
$853$	−2.34777e7	−1.10480	−0.552400	−	0.833579i	$-0.686288\pi$
$853$	−2.34777e7	−1.10480	−0.552400	+	0.833579i	$0.686288\pi$
$854$	−1.50768e7	−0.707399
$855$	0	0
$856$	−1.62954e7	−0.760118
$857$	−5.16990e6	−0.240453	−0.120226	−	0.992746i	$-0.538362\pi$
$857$	−5.16990e6	−0.240453	−0.120226	+	0.992746i	$0.538362\pi$
$858$	0	0
$859$	33444.6	0.00154648	0.000773238	−	1.00000i	$-0.499754\pi$
$859$	33444.6	0.00154648	0.000773238		1.00000i	$0.499754\pi$
$860$	1.63138e6	0.0752157
$861$	0	0
$862$	4.06645e7	1.86400
$863$	−3.83003e7	−1.75055	−0.875276	−	0.483624i	$-0.839320\pi$
$863$	−3.83003e7	−1.75055	−0.875276	+	0.483624i	$0.839320\pi$
$864$	0	0
$865$	−1.04861e7	−0.476511
$866$	−1.72677e7	−0.782419
$867$	0	0
$868$	−1.82686e7	−0.823014
$869$	1.11453e7	0.500657
$870$	0	0
$871$	4.45288e7	1.98882
$872$	−2.14217e6	−0.0954032
$873$	0	0
$874$	−3.58694e6	−0.158835
$875$	−1.92763e6	−0.0851144
$876$	0	0
$877$	2.05837e7	0.903701	0.451851	−	0.892094i	$-0.350764\pi$
$877$	2.05837e7	0.903701	0.451851	+	0.892094i	$0.350764\pi$
$878$	4.75539e7	2.08185
$879$	0	0
$880$	1.96631e6	0.0855945
$881$	3.74288e7	1.62468	0.812338	−	0.583187i	$-0.198195\pi$
$881$	3.74288e7	1.62468	0.812338	+	0.583187i	$0.198195\pi$
$882$	0	0
$883$	1.62149e7	0.699861	0.349930	−	0.936776i	$-0.386205\pi$
$883$	1.62149e7	0.699861	0.349930	+	0.936776i	$0.386205\pi$
$884$	−1.84648e7	−0.794720
$885$	0	0
$886$	−5.59506e7	−2.39453
$887$	2.32640e7	0.992832	0.496416	−	0.868085i	$-0.334649\pi$
$887$	2.32640e7	0.992832	0.496416	+	0.868085i	$0.334649\pi$
$888$	0	0
$889$	3.52032e6	0.149392
$890$	1.47256e7	0.623159
$891$	0	0
$892$	−2.26369e7	−0.952587
$893$	5.95101e6	0.249725
$894$	0	0
$895$	−568540.	−0.0237248
$896$	−1.88616e7	−0.784891
$897$	0	0
$898$	−1.11411e7	−0.461038
$899$	−1.38678e7	−0.572280
$900$	0	0
$901$	1.44888e7	0.594595
$902$	1.95001e7	0.798033
$903$	0	0
$904$	−675000.	−0.0274715
$905$	6.95875e6	0.282429
$906$	0	0
$907$	−2.69692e7	−1.08855	−0.544276	−	0.838906i	$-0.683196\pi$
$907$	−2.69692e7	−1.08855	−0.544276	+	0.838906i	$0.683196\pi$
$908$	3.35698e7	1.35125
$909$	0	0
$910$	1.67603e7	0.670931
$911$	−2.65963e7	−1.06176	−0.530879	−	0.847447i	$-0.678138\pi$
$911$	−2.65963e7	−1.06176	−0.530879	+	0.847447i	$0.678138\pi$
$912$	0	0
$913$	279906.	0.0111131
$914$	3.15307e7	1.24844
$915$	0	0
$916$	4.61165e7	1.81601
$917$	−9.74739e6	−0.382794
$918$	0	0
$919$	3.38373e7	1.32162	0.660811	−	0.750552i	$-0.270213\pi$
$919$	3.38373e7	1.32162	0.660811	+	0.750552i	$0.270213\pi$
$920$	4.00380e6	0.155956
$921$	0	0
$922$	−4.01776e7	−1.55653
$923$	−1.70379e7	−0.658280
$924$	0	0
$925$	9.50364e6	0.365204
$926$	5.01195e7	1.92079
$927$	0	0
$928$	−3.09727e7	−1.18062
$929$	1.14253e6	0.0434338	0.0217169	−	0.999764i	$-0.493087\pi$
$929$	1.14253e6	0.0434338	0.0217169	+	0.999764i	$0.493087\pi$
$930$	0	0
$931$	323490.	0.0122317
$932$	5.75050e7	2.16853
$933$	0	0
$934$	−2.48572e7	−0.932365
$935$	2.13823e6	0.0799880
$936$	0	0
$937$	−8.90840e6	−0.331475	−0.165737	−	0.986170i	$-0.553000\pi$
$937$	−8.90840e6	−0.331475	−0.165737	+	0.986170i	$0.553000\pi$
$938$	7.38959e7	2.74229
$939$	0	0
$940$	−3.00028e7	−1.10750
$941$	−1.50307e7	−0.553356	−0.276678	−	0.960963i	$-0.589234\pi$
$941$	−1.50307e7	−0.553356	−0.276678	+	0.960963i	$0.589234\pi$
$942$	0	0
$943$	−3.88023e7	−1.42095
$944$	1.46683e7	0.535733
$945$	0	0
$946$	1.64256e6	0.0596752
$947$	2.83145e7	1.02597	0.512985	−	0.858398i	$-0.328540\pi$
$947$	2.83145e7	1.02597	0.512985	+	0.858398i	$0.328540\pi$
$948$	0	0
$949$	9.07694e6	0.327170
$950$	1.08904e6	0.0391501
$951$	0	0
$952$	−6.78424e6	−0.242610
$953$	4.67607e7	1.66782	0.833910	−	0.551900i	$-0.186097\pi$
$953$	4.67607e7	1.66782	0.833910	+	0.551900i	$0.186097\pi$
$954$	0	0
$955$	−8.70948e6	−0.309018
$956$	3.29149e7	1.16479
$957$	0	0
$958$	−6.35325e7	−2.23657
$959$	−3.28101e7	−1.15202
$960$	0	0
$961$	−1.56473e7	−0.546553
$962$	−8.26320e7	−2.87879
$963$	0	0
$964$	1.51916e7	0.526514
$965$	1.81189e7	0.626346
$966$	0	0
$967$	−1.46114e7	−0.502488	−0.251244	−	0.967924i	$-0.580840\pi$
$967$	−1.46114e7	−0.502488	−0.251244	+	0.967924i	$0.580840\pi$
$968$	−1.13904e6	−0.0390708
$969$	0	0
$970$	−1.76742e7	−0.603131
$971$	1.54600e6	0.0526213	0.0263107	−	0.999654i	$-0.491624\pi$
$971$	1.54600e6	0.0526213	0.0263107	+	0.999654i	$0.491624\pi$
$972$	0	0
$973$	−9.36848e6	−0.317239
$974$	−5.32680e7	−1.79916
$975$	0	0
$976$	9.29129e6	0.312213
$977$	−1.87025e7	−0.626849	−0.313425	−	0.949613i	$-0.601476\pi$
$977$	−1.87025e7	−0.626849	−0.313425	+	0.949613i	$0.601476\pi$
$978$	0	0
$979$	8.33608e6	0.277975
$980$	−1.63092e6	−0.0542459
$981$	0	0
$982$	1.27411e7	0.421627
$983$	4.48391e7	1.48004	0.740019	−	0.672585i	$-0.234816\pi$
$983$	4.48391e7	1.48004	0.740019	+	0.672585i	$0.234816\pi$
$984$	0	0
$985$	2.07675e7	0.682015
$986$	−2.32608e7	−0.761961
$987$	0	0
$988$	−5.32380e6	−0.173512
$989$	−3.26845e6	−0.106255
$990$	0	0
$991$	−8.45032e6	−0.273331	−0.136666	−	0.990617i	$-0.543639\pi$
$991$	−8.45032e6	−0.273331	−0.136666	+	0.990617i	$0.543639\pi$
$992$	2.89939e7	0.935466
$993$	0	0
$994$	−2.82744e7	−0.907670
$995$	−1.61230e7	−0.516283
$996$	0	0
$997$	6.13864e7	1.95584	0.977922	−	0.208970i	$-0.0670112\pi$
$997$	6.13864e7	1.95584	0.977922	+	0.208970i	$0.0670112\pi$
$998$	−3.51580e7	−1.11737
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.p.1.2	yes	10
3.2	odd	2		495.6.a.o.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.6.a.o.1.9	✓	10	3.2	odd	2
495.6.a.p.1.2	yes	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-8.54982 q^{2} +41.0994 q^{4} +25.0000 q^{5} -123.368 q^{7} -77.7982 q^{8} +O(q^{10})\)	\(q-8.54982 q^{2} +41.0994 q^{4} +25.0000 q^{5} -123.368 q^{7} -77.7982 q^{8} -213.745 q^{10} -121.000 q^{11} +635.596 q^{13} +1054.78 q^{14} -650.020 q^{16} -706.852 q^{17} -203.800 q^{19} +1027.48 q^{20} +1034.53 q^{22} -2058.55 q^{23} +625.000 q^{25} -5434.23 q^{26} -5070.36 q^{28} -3848.93 q^{29} +3603.03 q^{31} +8047.10 q^{32} +6043.45 q^{34} -3084.20 q^{35} +15205.8 q^{37} +1742.46 q^{38} -1944.95 q^{40} +18849.3 q^{41} +1587.74 q^{43} -4973.03 q^{44} +17600.3 q^{46} -29200.2 q^{47} -1587.29 q^{49} -5343.64 q^{50} +26122.6 q^{52} -20497.7 q^{53} -3025.00 q^{55} +9597.82 q^{56} +32907.6 q^{58} -22565.9 q^{59} -14293.8 q^{61} -30805.2 q^{62} -48000.6 q^{64} +15889.9 q^{65} +70058.4 q^{67} -29051.2 q^{68} +26369.4 q^{70} -26806.1 q^{71} +14281.0 q^{73} -130007. q^{74} -8376.07 q^{76} +14927.6 q^{77} -92109.5 q^{79} -16250.5 q^{80} -161158. q^{82} -2313.28 q^{83} -17671.3 q^{85} -13574.9 q^{86} +9413.58 q^{88} -68893.3 q^{89} -78412.3 q^{91} -84605.4 q^{92} +249657. q^{94} -5095.01 q^{95} +82688.3 q^{97} +13571.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8}+O(q^{10}) \) 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8	\( 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8} + 75 q^{10} - 1210 q^{11} + 932 q^{13} - 1332 q^{14} + 2701 q^{16} + 96 q^{17} + 1664 q^{19} + 4525 q^{20} - 363 q^{22} + 6288 q^{23} + 6250 q^{25} + 13380 q^{26} + 13868 q^{28} + 11208 q^{29} + 9032 q^{31} + 9801 q^{32} + 14610 q^{34} + 2900 q^{35} + 21572 q^{37} + 15870 q^{38} + 3225 q^{40} + 10800 q^{41} + 21128 q^{43} - 21901 q^{44} + 83982 q^{46} - 17400 q^{47} + 71610 q^{49} + 1875 q^{50} + 40640 q^{52} + 5004 q^{53} - 30250 q^{55} - 54012 q^{56} - 9786 q^{58} - 25272 q^{59} + 52004 q^{61} + 34740 q^{62} + 56953 q^{64} + 23300 q^{65} + 4160 q^{67} - 87978 q^{68} - 33300 q^{70} - 65232 q^{71} + 44252 q^{73} - 49842 q^{74} + 233246 q^{76} - 14036 q^{77} + 112604 q^{79} + 67525 q^{80} + 167910 q^{82} + 70032 q^{83} + 2400 q^{85} - 72978 q^{86} - 15609 q^{88} - 46848 q^{89} + 130672 q^{91} + 121302 q^{92} + 252294 q^{94} + 41600 q^{95} + 129932 q^{97} - 316137 q^{98}+O(q^{100}) \) 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8 + 75 * q^10 - 1210 * q^11 + 932 * q^13 - 1332 * q^14 + 2701 * q^16 + 96 * q^17 + 1664 * q^19 + 4525 * q^20 - 363 * q^22 + 6288 * q^23 + 6250 * q^25 + 13380 * q^26 + 13868 * q^28 + 11208 * q^29 + 9032 * q^31 + 9801 * q^32 + 14610 * q^34 + 2900 * q^35 + 21572 * q^37 + 15870 * q^38 + 3225 * q^40 + 10800 * q^41 + 21128 * q^43 - 21901 * q^44 + 83982 * q^46 - 17400 * q^47 + 71610 * q^49 + 1875 * q^50 + 40640 * q^52 + 5004 * q^53 - 30250 * q^55 - 54012 * q^56 - 9786 * q^58 - 25272 * q^59 + 52004 * q^61 + 34740 * q^62 + 56953 * q^64 + 23300 * q^65 + 4160 * q^67 - 87978 * q^68 - 33300 * q^70 - 65232 * q^71 + 44252 * q^73 - 49842 * q^74 + 233246 * q^76 - 14036 * q^77 + 112604 * q^79 + 67525 * q^80 + 167910 * q^82 + 70032 * q^83 + 2400 * q^85 - 72978 * q^86 - 15609 * q^88 - 46848 * q^89 + 130672 * q^91 + 121302 * q^92 + 252294 * q^94 + 41600 * q^95 + 129932 * q^97 - 316137 * q^98

Introduction

L-functions

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