LMFDB - Embedded newform 495.6.a.d.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.76300$ 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)

$f(q)$	$=$	$q-7.52601 q^{2} +24.6408 q^{4} +25.0000 q^{5} -234.126 q^{7} +55.3856 q^{8} +O(q^{10})$	\(q-7.52601 q^{2} +24.6408 q^{4} +25.0000 q^{5} -234.126 q^{7} +55.3856 q^{8} -188.150 q^{10} -121.000 q^{11} +236.184 q^{13} +1762.04 q^{14} -1205.34 q^{16} +608.015 q^{17} -1799.81 q^{19} +616.019 q^{20} +910.647 q^{22} +4773.40 q^{23} +625.000 q^{25} -1777.52 q^{26} -5769.05 q^{28} -2804.99 q^{29} +10258.3 q^{31} +7299.04 q^{32} -4575.93 q^{34} -5853.16 q^{35} -7629.84 q^{37} +13545.4 q^{38} +1384.64 q^{40} -8355.48 q^{41} -11967.9 q^{43} -2981.53 q^{44} -35924.7 q^{46} -8385.48 q^{47} +38008.1 q^{49} -4703.75 q^{50} +5819.76 q^{52} -3453.00 q^{53} -3025.00 q^{55} -12967.2 q^{56} +21110.4 q^{58} +51032.4 q^{59} +9988.24 q^{61} -77204.1 q^{62} -16361.8 q^{64} +5904.60 q^{65} -36567.6 q^{67} +14982.0 q^{68} +44050.9 q^{70} +30522.3 q^{71} +83879.5 q^{73} +57422.2 q^{74} -44348.7 q^{76} +28329.3 q^{77} +103505. q^{79} -30133.4 q^{80} +62883.4 q^{82} +4341.54 q^{83} +15200.4 q^{85} +90070.3 q^{86} -6701.65 q^{88} +45464.1 q^{89} -55296.9 q^{91} +117620. q^{92} +63109.2 q^{94} -44995.2 q^{95} -183170. q^{97} -286049. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8	$ 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8} + 50 q^{10} - 363 q^{11} + 450 q^{13} + 1504 q^{14} - 1360 q^{16} + 334 q^{17} - 4036 q^{19} + 700 q^{20} - 242 q^{22} + 7060 q^{23} + 1875 q^{25} - 2932 q^{26} - 8320 q^{28} - 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3644 q^{34} - 5800 q^{35} + 2250 q^{37} + 12632 q^{38} - 600 q^{40} - 10654 q^{41} - 35528 q^{43} - 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 7667 q^{49} + 1250 q^{50} - 14520 q^{52} + 12826 q^{53} - 9075 q^{55} - 17088 q^{56} - 17196 q^{58} + 81876 q^{59} - 62298 q^{61} - 109184 q^{62} - 72256 q^{64} + 11250 q^{65} - 46148 q^{67} + 35832 q^{68} + 37600 q^{70} + 64724 q^{71} + 810 q^{73} + 44796 q^{74} + 44656 q^{76} + 28072 q^{77} + 43876 q^{79} - 34000 q^{80} + 56060 q^{82} + 101024 q^{83} + 8350 q^{85} - 24128 q^{86} + 2904 q^{88} - 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 74552 q^{94} - 100900 q^{95} - 319746 q^{97} - 431134 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8 + 50 * q^10 - 363 * q^11 + 450 * q^13 + 1504 * q^14 - 1360 * q^16 + 334 * q^17 - 4036 * q^19 + 700 * q^20 - 242 * q^22 + 7060 * q^23 + 1875 * q^25 - 2932 * q^26 - 8320 * q^28 - 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3644 * q^34 - 5800 * q^35 + 2250 * q^37 + 12632 * q^38 - 600 * q^40 - 10654 * q^41 - 35528 * q^43 - 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 7667 * q^49 + 1250 * q^50 - 14520 * q^52 + 12826 * q^53 - 9075 * q^55 - 17088 * q^56 - 17196 * q^58 + 81876 * q^59 - 62298 * q^61 - 109184 * q^62 - 72256 * q^64 + 11250 * q^65 - 46148 * q^67 + 35832 * q^68 + 37600 * q^70 + 64724 * q^71 + 810 * q^73 + 44796 * q^74 + 44656 * q^76 + 28072 * q^77 + 43876 * q^79 - 34000 * q^80 + 56060 * q^82 + 101024 * q^83 + 8350 * q^85 - 24128 * q^86 + 2904 * q^88 - 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 74552 * q^94 - 100900 * q^95 - 319746 * q^97 - 431134 * 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 $

$2$	−7.52601	−1.33042	−0.665211	−	0.746655i	$-0.731659\pi$
$2$	−7.52601	−1.33042	−0.665211	+	0.746655i	$0.731659\pi$
$3$	0	0
$3$	0	0
$4$	24.6408	0.770024
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−234.126	−1.80595	−0.902973	−	0.429696i	$-0.858621\pi$
$7$	−234.126	−1.80595	−0.902973	+	0.429696i	$0.858621\pi$
$8$	55.3856	0.305965
$9$	0	0
$10$	−188.150	−0.594983
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	236.184	0.387608	0.193804	−	0.981040i	$-0.437917\pi$
$13$	236.184	0.387608	0.193804	+	0.981040i	$0.437917\pi$
$14$	1762.04	2.40267
$15$	0	0
$16$	−1205.34	−1.17709
$17$	608.015	0.510261	0.255130	−	0.966907i	$-0.417882\pi$
$17$	608.015	0.510261	0.255130	+	0.966907i	$0.417882\pi$
$18$	0	0
$19$	−1799.81	−1.14378	−0.571890	−	0.820331i	$-0.693789\pi$
$19$	−1799.81	−1.14378	−0.571890	+	0.820331i	$0.693789\pi$
$20$	616.019	0.344365
$21$	0	0
$22$	910.647	0.401137
$23$	4773.40	1.88152	0.940759	−	0.339075i	$-0.110114\pi$
$23$	4773.40	1.88152	0.940759	+	0.339075i	$0.110114\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−1777.52	−0.515682
$27$	0	0
$28$	−5769.05	−1.39062
$29$	−2804.99	−0.619351	−0.309676	−	0.950842i	$-0.600220\pi$
$29$	−2804.99	−0.619351	−0.309676	+	0.950842i	$0.600220\pi$
$30$	0	0
$31$	10258.3	1.91722	0.958610	−	0.284724i	$-0.0919019\pi$
$31$	10258.3	1.91722	0.958610	+	0.284724i	$0.0919019\pi$
$32$	7299.04	1.26006
$33$	0	0
$34$	−4575.93	−0.678862
$35$	−5853.16	−0.807644
$36$	0	0
$37$	−7629.84	−0.916244	−0.458122	−	0.888889i	$-0.651478\pi$
$37$	−7629.84	−0.916244	−0.458122	+	0.888889i	$0.651478\pi$
$38$	13545.4	1.52171
$39$	0	0
$40$	1384.64	0.136832
$41$	−8355.48	−0.776268	−0.388134	−	0.921603i	$-0.626880\pi$
$41$	−8355.48	−0.776268	−0.388134	+	0.921603i	$0.626880\pi$
$42$	0	0
$43$	−11967.9	−0.987065	−0.493533	−	0.869727i	$-0.664295\pi$
$43$	−11967.9	−0.987065	−0.493533	+	0.869727i	$0.664295\pi$
$44$	−2981.53	−0.232171
$45$	0	0
$46$	−35924.7	−2.50321
$47$	−8385.48	−0.553711	−0.276856	−	0.960912i	$-0.589292\pi$
$47$	−8385.48	−0.553711	−0.276856	+	0.960912i	$0.589292\pi$
$48$	0	0
$49$	38008.1	2.26144
$50$	−4703.75	−0.266085
$51$	0	0
$52$	5819.76	0.298467
$53$	−3453.00	−0.168852	−0.0844261	−	0.996430i	$-0.526906\pi$
$53$	−3453.00	−0.168852	−0.0844261	+	0.996430i	$0.526906\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	−12967.2	−0.552556
$57$	0	0
$58$	21110.4	0.823999
$59$	51032.4	1.90860	0.954302	−	0.298842i	$-0.0966005\pi$
$59$	51032.4	1.90860	0.954302	+	0.298842i	$0.0966005\pi$
$60$	0	0
$61$	9988.24	0.343688	0.171844	−	0.985124i	$-0.445028\pi$
$61$	9988.24	0.343688	0.171844	+	0.985124i	$0.445028\pi$
$62$	−77204.1	−2.55071
$63$	0	0
$64$	−16361.8	−0.499323
$65$	5904.60	0.173343
$66$	0	0
$67$	−36567.6	−0.995198	−0.497599	−	0.867407i	$-0.665785\pi$
$67$	−36567.6	−0.995198	−0.497599	+	0.867407i	$0.665785\pi$
$68$	14982.0	0.392913
$69$	0	0
$70$	44050.9	1.07451
$71$	30522.3	0.718573	0.359286	−	0.933227i	$-0.383020\pi$
$71$	30522.3	0.718573	0.359286	+	0.933227i	$0.383020\pi$
$72$	0	0
$73$	83879.5	1.84225	0.921125	−	0.389267i	$-0.127272\pi$
$73$	83879.5	1.84225	0.921125	+	0.389267i	$0.127272\pi$
$74$	57422.2	1.21899
$75$	0	0
$76$	−44348.7	−0.880738
$77$	28329.3	0.544513
$78$	0	0
$79$	103505.	1.86592	0.932961	−	0.359978i	$-0.117216\pi$
$79$	103505.	1.86592	0.932961	+	0.359978i	$0.117216\pi$
$80$	−30133.4	−0.526409
$81$	0	0
$82$	62883.4	1.03276
$83$	4341.54	0.0691749	0.0345875	−	0.999402i	$-0.488988\pi$
$83$	4341.54	0.0691749	0.0345875	+	0.999402i	$0.488988\pi$
$84$	0	0
$85$	15200.4	0.228196
$86$	90070.3	1.31321
$87$	0	0
$88$	−6701.65	−0.0922519
$89$	45464.1	0.608406	0.304203	−	0.952607i	$-0.401610\pi$
$89$	45464.1	0.608406	0.304203	+	0.952607i	$0.401610\pi$
$90$	0	0
$91$	−55296.9	−0.699999
$92$	117620.	1.44881
$93$	0	0
$94$	63109.2	0.736670
$95$	−44995.2	−0.511514
$96$	0	0
$97$	−183170.	−1.97663	−0.988314	−	0.152432i	$-0.951289\pi$
$97$	−183170.	−1.97663	−0.988314	+	0.152432i	$0.951289\pi$
$98$	−286049.	−3.00868
$99$	0	0
$100$	15400.5	0.154005
$101$	28833.3	0.281249	0.140625	−	0.990063i	$-0.455089\pi$
$101$	28833.3	0.281249	0.140625	+	0.990063i	$0.455089\pi$
$102$	0	0
$103$	−59444.6	−0.552102	−0.276051	−	0.961143i	$-0.589026\pi$
$103$	−59444.6	−0.552102	−0.276051	+	0.961143i	$0.589026\pi$
$104$	13081.2	0.118594
$105$	0	0
$106$	25987.3	0.224645
$107$	−13840.6	−0.116868	−0.0584342	−	0.998291i	$-0.518611\pi$
$107$	−13840.6	−0.116868	−0.0584342	+	0.998291i	$0.518611\pi$
$108$	0	0
$109$	−13278.1	−0.107046	−0.0535231	−	0.998567i	$-0.517045\pi$
$109$	−13278.1	−0.107046	−0.0535231	+	0.998567i	$0.517045\pi$
$110$	22766.2	0.179394
$111$	0	0
$112$	282201.	2.12576
$113$	−130086.	−0.958370	−0.479185	−	0.877714i	$-0.659068\pi$
$113$	−130086.	−0.958370	−0.479185	+	0.877714i	$0.659068\pi$
$114$	0	0
$115$	119335.	0.841441
$116$	−69117.2	−0.476915
$117$	0	0
$118$	−384070.	−2.53925
$119$	−142352.	−0.921504
$120$	0	0
$121$	14641.0	0.0909091
$122$	−75171.5	−0.457250
$123$	0	0
$124$	252773.	1.47631
$125$	15625.0	0.0894427
$126$	0	0
$127$	−274904.	−1.51242	−0.756209	−	0.654330i	$-0.772951\pi$
$127$	−274904.	−1.51242	−0.756209	+	0.654330i	$0.772951\pi$
$128$	−110430.	−0.595748
$129$	0	0
$130$	−44438.1	−0.230620
$131$	119374.	0.607761	0.303881	−	0.952710i	$-0.401718\pi$
$131$	119374.	0.607761	0.303881	+	0.952710i	$0.401718\pi$
$132$	0	0
$133$	421382.	2.06560
$134$	275208.	1.32403
$135$	0	0
$136$	33675.3	0.156122
$137$	132278.	0.602123	0.301061	−	0.953605i	$-0.402659\pi$
$137$	132278.	0.602123	0.301061	+	0.953605i	$0.402659\pi$
$138$	0	0
$139$	−102326.	−0.449209	−0.224604	−	0.974450i	$-0.572109\pi$
$139$	−102326.	−0.449209	−0.224604	+	0.974450i	$0.572109\pi$
$140$	−144226.	−0.621905
$141$	0	0
$142$	−229711.	−0.956005
$143$	−28578.3	−0.116868
$144$	0	0
$145$	−70124.8	−0.276982
$146$	−631278.	−2.45097
$147$	0	0
$148$	−188005.	−0.705530
$149$	−371696.	−1.37158	−0.685791	−	0.727798i	$-0.740544\pi$
$149$	−371696.	−1.37158	−0.685791	+	0.727798i	$0.740544\pi$
$150$	0	0
$151$	−111725.	−0.398757	−0.199379	−	0.979923i	$-0.563892\pi$
$151$	−111725.	−0.398757	−0.199379	+	0.979923i	$0.563892\pi$
$152$	−99683.4	−0.349956
$153$	0	0
$154$	−213206.	−0.724433
$155$	256458.	0.857406
$156$	0	0
$157$	−244726.	−0.792375	−0.396187	−	0.918170i	$-0.629667\pi$
$157$	−244726.	−0.792375	−0.396187	+	0.918170i	$0.629667\pi$
$158$	−778979.	−2.48246
$159$	0	0
$160$	182476.	0.563515
$161$	−1.11758e6	−3.39792
$162$	0	0
$163$	435623.	1.28423	0.642114	−	0.766610i	$-0.278058\pi$
$163$	435623.	1.28423	0.642114	+	0.766610i	$0.278058\pi$
$164$	−205885.	−0.597745
$165$	0	0
$166$	−32674.5	−0.0920319
$167$	−288486.	−0.800450	−0.400225	−	0.916417i	$-0.631068\pi$
$167$	−288486.	−0.800450	−0.400225	+	0.916417i	$0.631068\pi$
$168$	0	0
$169$	−315510.	−0.849760
$170$	−114398.	−0.303597
$171$	0	0
$172$	−294898.	−0.760064
$173$	−539633.	−1.37083	−0.685414	−	0.728154i	$-0.740379\pi$
$173$	−539633.	−1.37083	−0.685414	+	0.728154i	$0.740379\pi$
$174$	0	0
$175$	−146329.	−0.361189
$176$	145846.	0.354905
$177$	0	0
$178$	−342163.	−0.809438
$179$	378874.	0.883816	0.441908	−	0.897060i	$-0.354302\pi$
$179$	378874.	0.883816	0.441908	+	0.897060i	$0.354302\pi$
$180$	0	0
$181$	−621633.	−1.41038	−0.705192	−	0.709016i	$-0.749139\pi$
$181$	−621633.	−1.41038	−0.705192	+	0.709016i	$0.749139\pi$
$182$	416165.	0.931294
$183$	0	0
$184$	264378.	0.575679
$185$	−190746.	−0.409757
$186$	0	0
$187$	−73569.8	−0.153849
$188$	−206625.	−0.426371
$189$	0	0
$190$	338634.	0.680529
$191$	−8509.61	−0.0168782	−0.00843911	−	0.999964i	$-0.502686\pi$
$191$	−8509.61	−0.0168782	−0.00843911	+	0.999964i	$0.502686\pi$
$192$	0	0
$193$	−564588.	−1.09103	−0.545517	−	0.838100i	$-0.683667\pi$
$193$	−564588.	−1.09103	−0.545517	+	0.838100i	$0.683667\pi$
$194$	1.37854e6	2.62975
$195$	0	0
$196$	936549.	1.74137
$197$	432994.	0.794908	0.397454	−	0.917622i	$-0.369894\pi$
$197$	432994.	0.794908	0.397454	+	0.917622i	$0.369894\pi$
$198$	0	0
$199$	−614871.	−1.10066	−0.550328	−	0.834949i	$-0.685497\pi$
$199$	−614871.	−1.10066	−0.550328	+	0.834949i	$0.685497\pi$
$200$	34616.0	0.0611930
$201$	0	0
$202$	−217000.	−0.374180
$203$	656723.	1.11852
$204$	0	0
$205$	−208887.	−0.347158
$206$	447380.	0.734528
$207$	0	0
$208$	−284682.	−0.456248
$209$	217777.	0.344862
$210$	0	0
$211$	72348.2	0.111872	0.0559360	−	0.998434i	$-0.482186\pi$
$211$	72348.2	0.111872	0.0559360	+	0.998434i	$0.482186\pi$
$212$	−85084.6	−0.130020
$213$	0	0
$214$	104165.	0.155484
$215$	−299197.	−0.441429
$216$	0	0
$217$	−2.40174e6	−3.46240
$218$	99931.4	0.142417
$219$	0	0
$220$	−74538.3	−0.103830
$221$	143604.	0.197781
$222$	0	0
$223$	370371.	0.498740	0.249370	−	0.968408i	$-0.419776\pi$
$223$	370371.	0.498740	0.249370	+	0.968408i	$0.419776\pi$
$224$	−1.70890e6	−2.27560
$225$	0	0
$226$	979025.	1.27504
$227$	−325481.	−0.419239	−0.209619	−	0.977783i	$-0.567222\pi$
$227$	−325481.	−0.419239	−0.209619	+	0.977783i	$0.567222\pi$
$228$	0	0
$229$	−57345.0	−0.0722614	−0.0361307	−	0.999347i	$-0.511503\pi$
$229$	−57345.0	−0.0722614	−0.0361307	+	0.999347i	$0.511503\pi$
$230$	−898116.	−1.11947
$231$	0	0
$232$	−155356.	−0.189500
$233$	833891.	1.00628	0.503141	−	0.864205i	$-0.332178\pi$
$233$	833891.	1.00628	0.503141	+	0.864205i	$0.332178\pi$
$234$	0	0
$235$	−209637.	−0.247627
$236$	1.25748e6	1.46967
$237$	0	0
$238$	1.07134e6	1.22599
$239$	813986.	0.921769	0.460885	−	0.887460i	$-0.347532\pi$
$239$	813986.	0.921769	0.460885	+	0.887460i	$0.347532\pi$
$240$	0	0
$241$	−1.18456e6	−1.31376	−0.656878	−	0.753997i	$-0.728123\pi$
$241$	−1.18456e6	−1.31376	−0.656878	+	0.753997i	$0.728123\pi$
$242$	−110188.	−0.120948
$243$	0	0
$244$	246118.	0.264648
$245$	950202.	1.01135
$246$	0	0
$247$	−425086.	−0.443338
$248$	568163.	0.586602
$249$	0	0
$250$	−117594.	−0.118997
$251$	601749.	0.602880	0.301440	−	0.953485i	$-0.402533\pi$
$251$	601749.	0.602880	0.301440	+	0.953485i	$0.402533\pi$
$252$	0	0
$253$	−577582.	−0.567299
$254$	2.06893e6	2.01216
$255$	0	0
$256$	1.35468e6	1.29192
$257$	−904570.	−0.854298	−0.427149	−	0.904181i	$-0.640482\pi$
$257$	−904570.	−0.854298	−0.427149	+	0.904181i	$0.640482\pi$
$258$	0	0
$259$	1.78634e6	1.65469
$260$	145494.	0.133479
$261$	0	0
$262$	−898413.	−0.808580
$263$	−1.47174e6	−1.31202	−0.656011	−	0.754751i	$-0.727758\pi$
$263$	−1.47174e6	−1.31202	−0.656011	+	0.754751i	$0.727758\pi$
$264$	0	0
$265$	−86325.0	−0.0755130
$266$	−3.17133e6	−2.74813
$267$	0	0
$268$	−901054.	−0.766327
$269$	−897427.	−0.756168	−0.378084	−	0.925771i	$-0.623417\pi$
$269$	−897427.	−0.756168	−0.378084	+	0.925771i	$0.623417\pi$
$270$	0	0
$271$	134225.	0.111022	0.0555111	−	0.998458i	$-0.482321\pi$
$271$	134225.	0.111022	0.0555111	+	0.998458i	$0.482321\pi$
$272$	−732863.	−0.600621
$273$	0	0
$274$	−995523.	−0.801078
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	−2.01306e6	−1.57637	−0.788184	−	0.615439i	$-0.788979\pi$
$277$	−2.01306e6	−1.57637	−0.788184	+	0.615439i	$0.788979\pi$
$278$	770105.	0.597637
$279$	0	0
$280$	−324180.	−0.247111
$281$	−1.45204e6	−1.09702	−0.548509	−	0.836145i	$-0.684804\pi$
$281$	−1.45204e6	−1.09702	−0.548509	+	0.836145i	$0.684804\pi$
$282$	0	0
$283$	−2.24416e6	−1.66566	−0.832832	−	0.553525i	$-0.813282\pi$
$283$	−2.24416e6	−1.66566	−0.832832	+	0.553525i	$0.813282\pi$
$284$	752092.	0.553318
$285$	0	0
$286$	215080.	0.155484
$287$	1.95624e6	1.40190
$288$	0	0
$289$	−1.05017e6	−0.739634
$290$	527760.	0.368503
$291$	0	0
$292$	2.06686e6	1.41858
$293$	−148693.	−0.101187	−0.0505933	−	0.998719i	$-0.516111\pi$
$293$	−148693.	−0.101187	−0.0505933	+	0.998719i	$0.516111\pi$
$294$	0	0
$295$	1.27581e6	0.853554
$296$	−422583.	−0.280338
$297$	0	0
$298$	2.79739e6	1.82478
$299$	1.12740e6	0.729291
$300$	0	0
$301$	2.80199e6	1.78259
$302$	840845.	0.530516
$303$	0	0
$304$	2.16938e6	1.34633
$305$	249706.	0.153702
$306$	0	0
$307$	2.02304e6	1.22507	0.612533	−	0.790445i	$-0.290151\pi$
$307$	2.02304e6	1.22507	0.612533	+	0.790445i	$0.290151\pi$
$308$	698055.	0.419289
$309$	0	0
$310$	−1.93010e6	−1.14071
$311$	−1.87652e6	−1.10015	−0.550074	−	0.835116i	$-0.685401\pi$
$311$	−1.87652e6	−1.10015	−0.550074	+	0.835116i	$0.685401\pi$
$312$	0	0
$313$	−89494.6	−0.0516340	−0.0258170	−	0.999667i	$-0.508219\pi$
$313$	−89494.6	−0.0516340	−0.0258170	+	0.999667i	$0.508219\pi$
$314$	1.84181e6	1.05419
$315$	0	0
$316$	2.55044e6	1.43680
$317$	−260115.	−0.145384	−0.0726920	−	0.997354i	$-0.523159\pi$
$317$	−260115.	−0.145384	−0.0726920	+	0.997354i	$0.523159\pi$
$318$	0	0
$319$	339404.	0.186741
$320$	−409045.	−0.223304
$321$	0	0
$322$	8.41090e6	4.52067
$323$	−1.09431e6	−0.583626
$324$	0	0
$325$	147615.	0.0775215
$326$	−3.27850e6	−1.70856
$327$	0	0
$328$	−462773.	−0.237511
$329$	1.96326e6	0.999973
$330$	0	0
$331$	2.19339e6	1.10039	0.550193	−	0.835037i	$-0.314554\pi$
$331$	2.19339e6	1.10039	0.550193	+	0.835037i	$0.314554\pi$
$332$	106979.	0.0532664
$333$	0	0
$334$	2.17115e6	1.06494
$335$	−914190.	−0.445066
$336$	0	0
$337$	933037.	0.447532	0.223766	−	0.974643i	$-0.428165\pi$
$337$	933037.	0.447532	0.223766	+	0.974643i	$0.428165\pi$
$338$	2.37453e6	1.13054
$339$	0	0
$340$	374549.	0.175716
$341$	−1.24126e6	−0.578063
$342$	0	0
$343$	−4.96373e6	−2.27810
$344$	−662848.	−0.302007
$345$	0	0
$346$	4.06128e6	1.82378
$347$	1.72984e6	0.771229	0.385615	−	0.922660i	$-0.373989\pi$
$347$	1.72984e6	0.771229	0.385615	+	0.922660i	$0.373989\pi$
$348$	0	0
$349$	1.01770e6	0.447258	0.223629	−	0.974674i	$-0.428210\pi$
$349$	1.01770e6	0.447258	0.223629	+	0.974674i	$0.428210\pi$
$350$	1.10127e6	0.480534
$351$	0	0
$352$	−883183.	−0.379922
$353$	3.29128e6	1.40581	0.702907	−	0.711282i	$-0.251885\pi$
$353$	3.29128e6	1.40581	0.702907	+	0.711282i	$0.251885\pi$
$354$	0	0
$355$	763056.	0.321355
$356$	1.12027e6	0.468488
$357$	0	0
$358$	−2.85140e6	−1.17585
$359$	−2.32128e6	−0.950584	−0.475292	−	0.879828i	$-0.657658\pi$
$359$	−2.32128e6	−0.950584	−0.475292	+	0.879828i	$0.657658\pi$
$360$	0	0
$361$	763211.	0.308231
$362$	4.67841e6	1.87641
$363$	0	0
$364$	−1.36256e6	−0.539016
$365$	2.09699e6	0.823879
$366$	0	0
$367$	−1.81399e6	−0.703024	−0.351512	−	0.936183i	$-0.614332\pi$
$367$	−1.81399e6	−0.703024	−0.351512	+	0.936183i	$0.614332\pi$
$368$	−5.75356e6	−2.21471
$369$	0	0
$370$	1.43556e6	0.545149
$371$	808438.	0.304938
$372$	0	0
$373$	−160496.	−0.0597298	−0.0298649	−	0.999554i	$-0.509508\pi$
$373$	−160496.	−0.0597298	−0.0298649	+	0.999554i	$0.509508\pi$
$374$	553687.	0.204685
$375$	0	0
$376$	−464435.	−0.169416
$377$	−662495.	−0.240065
$378$	0	0
$379$	−622442.	−0.222587	−0.111294	−	0.993788i	$-0.535499\pi$
$379$	−622442.	−0.222587	−0.111294	+	0.993788i	$0.535499\pi$
$380$	−1.10872e6	−0.393878
$381$	0	0
$382$	64043.4	0.0224552
$383$	−1.49075e6	−0.519289	−0.259644	−	0.965704i	$-0.583605\pi$
$383$	−1.49075e6	−0.519289	−0.259644	+	0.965704i	$0.583605\pi$
$384$	0	0
$385$	708232.	0.243514
$386$	4.24909e6	1.45154
$387$	0	0
$388$	−4.51345e6	−1.52205
$389$	1.66085e6	0.556490	0.278245	−	0.960510i	$-0.410247\pi$
$389$	1.66085e6	0.556490	0.278245	+	0.960510i	$0.410247\pi$
$390$	0	0
$391$	2.90230e6	0.960065
$392$	2.10510e6	0.691923
$393$	0	0
$394$	−3.25872e6	−1.05756
$395$	2.58762e6	0.834465
$396$	0	0
$397$	2.46357e6	0.784493	0.392247	−	0.919860i	$-0.371698\pi$
$397$	2.46357e6	0.784493	0.392247	+	0.919860i	$0.371698\pi$
$398$	4.62752e6	1.46434
$399$	0	0
$400$	−753336.	−0.235417
$401$	664739.	0.206438	0.103219	−	0.994659i	$-0.467086\pi$
$401$	664739.	0.206438	0.103219	+	0.994659i	$0.467086\pi$
$402$	0	0
$403$	2.42285e6	0.743129
$404$	710475.	0.216569
$405$	0	0
$406$	−4.94250e6	−1.48810
$407$	923210.	0.276258
$408$	0	0
$409$	−111122.	−0.0328468	−0.0164234	−	0.999865i	$-0.505228\pi$
$409$	−111122.	−0.0328468	−0.0164234	+	0.999865i	$0.505228\pi$
$410$	1.57208e6	0.461866
$411$	0	0
$412$	−1.46476e6	−0.425132
$413$	−1.19480e7	−3.44684
$414$	0	0
$415$	108539.	0.0309360
$416$	1.72392e6	0.488408
$417$	0	0
$418$	−1.63899e6	−0.458813
$419$	5.34446e6	1.48720	0.743598	−	0.668627i	$-0.233118\pi$
$419$	5.34446e6	1.48720	0.743598	+	0.668627i	$0.233118\pi$
$420$	0	0
$421$	−483590.	−0.132976	−0.0664878	−	0.997787i	$-0.521179\pi$
$421$	−483590.	−0.132976	−0.0664878	+	0.997787i	$0.521179\pi$
$422$	−544493.	−0.148837
$423$	0	0
$424$	−191246.	−0.0516629
$425$	380010.	0.102052
$426$	0	0
$427$	−2.33851e6	−0.620682
$428$	−341044.	−0.0899915
$429$	0	0
$430$	2.25176e6	0.587287
$431$	241542.	0.0626325	0.0313162	−	0.999510i	$-0.490030\pi$
$431$	241542.	0.0626325	0.0313162	+	0.999510i	$0.490030\pi$
$432$	0	0
$433$	−1.03998e6	−0.266567	−0.133284	−	0.991078i	$-0.542552\pi$
$433$	−1.03998e6	−0.266567	−0.133284	+	0.991078i	$0.542552\pi$
$434$	1.80755e7	4.60645
$435$	0	0
$436$	−327184.	−0.0824281
$437$	−8.59121e6	−2.15204
$438$	0	0
$439$	−3.32986e6	−0.824642	−0.412321	−	0.911039i	$-0.635282\pi$
$439$	−3.32986e6	−0.824642	−0.412321	+	0.911039i	$0.635282\pi$
$440$	−167541.	−0.0412563
$441$	0	0
$442$	−1.08076e6	−0.263132
$443$	−5.12280e6	−1.24022	−0.620109	−	0.784515i	$-0.712912\pi$
$443$	−5.12280e6	−1.24022	−0.620109	+	0.784515i	$0.712912\pi$
$444$	0	0
$445$	1.13660e6	0.272088
$446$	−2.78741e6	−0.663535
$447$	0	0
$448$	3.83073e6	0.901750
$449$	−5.82328e6	−1.36318	−0.681588	−	0.731736i	$-0.738710\pi$
$449$	−5.82328e6	−1.36318	−0.681588	+	0.731736i	$0.738710\pi$
$450$	0	0
$451$	1.01101e6	0.234054
$452$	−3.20541e6	−0.737968
$453$	0	0
$454$	2.44957e6	0.557765
$455$	−1.38242e6	−0.313049
$456$	0	0
$457$	−5.32427e6	−1.19253	−0.596266	−	0.802787i	$-0.703349\pi$
$457$	−5.32427e6	−1.19253	−0.596266	+	0.802787i	$0.703349\pi$
$458$	431579.	0.0961382
$459$	0	0
$460$	2.94051e6	0.647930
$461$	−673694.	−0.147642	−0.0738211	−	0.997272i	$-0.523519\pi$
$461$	−673694.	−0.147642	−0.0738211	+	0.997272i	$0.523519\pi$
$462$	0	0
$463$	−1.92230e6	−0.416742	−0.208371	−	0.978050i	$-0.566816\pi$
$463$	−1.92230e6	−0.416742	−0.208371	+	0.978050i	$0.566816\pi$
$464$	3.38096e6	0.729030
$465$	0	0
$466$	−6.27587e6	−1.33878
$467$	7.94668e6	1.68614	0.843070	−	0.537804i	$-0.180746\pi$
$467$	7.94668e6	1.68614	0.843070	+	0.537804i	$0.180746\pi$
$468$	0	0
$469$	8.56144e6	1.79727
$470$	1.57773e6	0.329449
$471$	0	0
$472$	2.82646e6	0.583966
$473$	1.44811e6	0.297611
$474$	0	0
$475$	−1.12488e6	−0.228756
$476$	−3.50767e6	−0.709580
$477$	0	0
$478$	−6.12606e6	−1.22634
$479$	1.80345e6	0.359141	0.179570	−	0.983745i	$-0.442529\pi$
$479$	1.80345e6	0.359141	0.179570	+	0.983745i	$0.442529\pi$
$480$	0	0
$481$	−1.80205e6	−0.355143
$482$	8.91501e6	1.74785
$483$	0	0
$484$	360766.	0.0700022
$485$	−4.57925e6	−0.883975
$486$	0	0
$487$	3.04965e6	0.582678	0.291339	−	0.956620i	$-0.405899\pi$
$487$	3.04965e6	0.582678	0.291339	+	0.956620i	$0.405899\pi$
$488$	553204.	0.105156
$489$	0	0
$490$	−7.15123e6	−1.34552
$491$	7.53319e6	1.41018	0.705091	−	0.709117i	$-0.250906\pi$
$491$	7.53319e6	1.41018	0.705091	+	0.709117i	$0.250906\pi$
$492$	0	0
$493$	−1.70548e6	−0.316031
$494$	3.19920e6	0.589826
$495$	0	0
$496$	−1.23647e7	−2.25673
$497$	−7.14606e6	−1.29770
$498$	0	0
$499$	6.01840e6	1.08201	0.541003	−	0.841021i	$-0.318045\pi$
$499$	6.01840e6	1.08201	0.541003	+	0.841021i	$0.318045\pi$
$500$	385012.	0.0688731
$501$	0	0
$502$	−4.52876e6	−0.802085
$503$	−8.91752e6	−1.57153	−0.785767	−	0.618522i	$-0.787732\pi$
$503$	−8.91752e6	−1.57153	−0.785767	+	0.618522i	$0.787732\pi$
$504$	0	0
$505$	720833.	0.125778
$506$	4.34688e6	0.754748
$507$	0	0
$508$	−6.77385e6	−1.16460
$509$	−9.71294e6	−1.66171	−0.830857	−	0.556486i	$-0.812149\pi$
$509$	−9.71294e6	−1.66171	−0.830857	+	0.556486i	$0.812149\pi$
$510$	0	0
$511$	−1.96384e7	−3.32701
$512$	−6.66153e6	−1.12305
$513$	0	0
$514$	6.80780e6	1.13658
$515$	−1.48611e6	−0.246907
$516$	0	0
$517$	1.01464e6	0.166950
$518$	−1.34440e7	−2.20143
$519$	0	0
$520$	327030.	0.0530370
$521$	591081.	0.0954009	0.0477004	−	0.998862i	$-0.484811\pi$
$521$	591081.	0.0954009	0.0477004	+	0.998862i	$0.484811\pi$
$522$	0	0
$523$	2.83152e6	0.452653	0.226327	−	0.974051i	$-0.427328\pi$
$523$	2.83152e6	0.452653	0.226327	+	0.974051i	$0.427328\pi$
$524$	2.94148e6	0.467991
$525$	0	0
$526$	1.10763e7	1.74554
$527$	6.23721e6	0.978282
$528$	0	0
$529$	1.63490e7	2.54011
$530$	649682.	0.100464
$531$	0	0
$532$	1.03832e7	1.59057
$533$	−1.97343e6	−0.300887
$534$	0	0
$535$	−346016.	−0.0522651
$536$	−2.02532e6	−0.304496
$537$	0	0
$538$	6.75404e6	1.00602
$539$	−4.59898e6	−0.681851
$540$	0	0
$541$	9.93914e6	1.46001	0.730004	−	0.683442i	$-0.239518\pi$
$541$	9.93914e6	1.46001	0.730004	+	0.683442i	$0.239518\pi$
$542$	−1.01018e6	−0.147706
$543$	0	0
$544$	4.43793e6	0.642958
$545$	−331953.	−0.0478725
$546$	0	0
$547$	−4.34846e6	−0.621394	−0.310697	−	0.950509i	$-0.600563\pi$
$547$	−4.34846e6	−0.621394	−0.310697	+	0.950509i	$0.600563\pi$
$548$	3.25942e6	0.463649
$549$	0	0
$550$	569154.	0.0802275
$551$	5.04845e6	0.708401
$552$	0	0
$553$	−2.42332e7	−3.36976
$554$	1.51503e7	2.09724
$555$	0	0
$556$	−2.52139e6	−0.345902
$557$	−1.07234e7	−1.46452	−0.732261	−	0.681024i	$-0.761535\pi$
$557$	−1.07234e7	−1.46452	−0.732261	+	0.681024i	$0.761535\pi$
$558$	0	0
$559$	−2.82662e6	−0.382594
$560$	7.05503e6	0.950667
$561$	0	0
$562$	1.09281e7	1.45950
$563$	−1.67966e6	−0.223332	−0.111666	−	0.993746i	$-0.535619\pi$
$563$	−1.67966e6	−0.223332	−0.111666	+	0.993746i	$0.535619\pi$
$564$	0	0
$565$	−3.25214e6	−0.428596
$566$	1.68896e7	2.21604
$567$	0	0
$568$	1.69049e6	0.219858
$569$	−1.04510e7	−1.35325	−0.676626	−	0.736327i	$-0.736559\pi$
$569$	−1.04510e7	−1.35325	−0.676626	+	0.736327i	$0.736559\pi$
$570$	0	0
$571$	670155.	0.0860171	0.0430085	−	0.999075i	$-0.486306\pi$
$571$	670155.	0.0860171	0.0430085	+	0.999075i	$0.486306\pi$
$572$	−704191.	−0.0899913
$573$	0	0
$574$	−1.47226e7	−1.86512
$575$	2.98338e6	0.376304
$576$	0	0
$577$	−7.87543e6	−0.984769	−0.492385	−	0.870378i	$-0.663875\pi$
$577$	−7.87543e6	−0.984769	−0.492385	+	0.870378i	$0.663875\pi$
$578$	7.90362e6	0.984026
$579$	0	0
$580$	−1.72793e6	−0.213283
$581$	−1.01647e6	−0.124926
$582$	0	0
$583$	417813.	0.0509109
$584$	4.64571e6	0.563664
$585$	0	0
$586$	1.11907e6	0.134621
$587$	9.13414e6	1.09414	0.547069	−	0.837087i	$-0.315743\pi$
$587$	9.13414e6	1.09414	0.547069	+	0.837087i	$0.315743\pi$
$588$	0	0
$589$	−1.84630e7	−2.19288
$590$	−9.60176e6	−1.13559
$591$	0	0
$592$	9.19652e6	1.07850
$593$	−1.25096e7	−1.46085	−0.730425	−	0.682993i	$-0.760678\pi$
$593$	−1.25096e7	−1.46085	−0.730425	+	0.682993i	$0.760678\pi$
$594$	0	0
$595$	−3.55881e6	−0.412109
$596$	−9.15887e6	−1.05615
$597$	0	0
$598$	−8.48484e6	−0.970265
$599$	−1.59560e7	−1.81700	−0.908502	−	0.417881i	$-0.862773\pi$
$599$	−1.59560e7	−1.81700	−0.908502	+	0.417881i	$0.862773\pi$
$600$	0	0
$601$	763693.	0.0862448	0.0431224	−	0.999070i	$-0.486269\pi$
$601$	763693.	0.0862448	0.0431224	+	0.999070i	$0.486269\pi$
$602$	−2.10878e7	−2.37159
$603$	0	0
$604$	−2.75300e6	−0.307053
$605$	366025.	0.0406558
$606$	0	0
$607$	−8.51753e6	−0.938301	−0.469150	−	0.883118i	$-0.655440\pi$
$607$	−8.51753e6	−0.938301	−0.469150	+	0.883118i	$0.655440\pi$
$608$	−1.31369e7	−1.44123
$609$	0	0
$610$	−1.87929e6	−0.204488
$611$	−1.98052e6	−0.214623
$612$	0	0
$613$	9.58182e6	1.02990	0.514952	−	0.857219i	$-0.327810\pi$
$613$	9.58182e6	1.02990	0.514952	+	0.857219i	$0.327810\pi$
$614$	−1.52254e7	−1.62985
$615$	0	0
$616$	1.56903e6	0.166602
$617$	1.12974e7	1.19472	0.597362	−	0.801972i	$-0.296216\pi$
$617$	1.12974e7	1.19472	0.597362	+	0.801972i	$0.296216\pi$
$618$	0	0
$619$	−4.12945e6	−0.433177	−0.216588	−	0.976263i	$-0.569493\pi$
$619$	−4.12945e6	−0.433177	−0.216588	+	0.976263i	$0.569493\pi$
$620$	6.31932e6	0.660224
$621$	0	0
$622$	1.41227e7	1.46366
$623$	−1.06443e7	−1.09875
$624$	0	0
$625$	390625.	0.0400000
$626$	673537.	0.0686951
$627$	0	0
$628$	−6.03023e6	−0.610148
$629$	−4.63906e6	−0.467523
$630$	0	0
$631$	1.29753e7	1.29731	0.648656	−	0.761082i	$-0.275332\pi$
$631$	1.29753e7	1.29731	0.648656	+	0.761082i	$0.275332\pi$
$632$	5.73268e6	0.570907
$633$	0	0
$634$	1.95762e6	0.193422
$635$	−6.87260e6	−0.676374
$636$	0	0
$637$	8.97691e6	0.876553
$638$	−2.55436e6	−0.248445
$639$	0	0
$640$	−2.76075e6	−0.266427
$641$	130086.	0.0125051	0.00625255	−	0.999980i	$-0.498010\pi$
$641$	130086.	0.0125051	0.00625255	+	0.999980i	$0.498010\pi$
$642$	0	0
$643$	−6.47507e6	−0.617614	−0.308807	−	0.951125i	$-0.599930\pi$
$643$	−6.47507e6	−0.617614	−0.308807	+	0.951125i	$0.599930\pi$
$644$	−2.75380e7	−2.61648
$645$	0	0
$646$	8.23579e6	0.776469
$647$	5.42540e6	0.509531	0.254766	−	0.967003i	$-0.418002\pi$
$647$	5.42540e6	0.509531	0.254766	+	0.967003i	$0.418002\pi$
$648$	0	0
$649$	−6.17492e6	−0.575466
$650$	−1.11095e6	−0.103136
$651$	0	0
$652$	1.07341e7	0.988886
$653$	−9.15697e6	−0.840367	−0.420183	−	0.907439i	$-0.638034\pi$
$653$	−9.15697e6	−0.840367	−0.420183	+	0.907439i	$0.638034\pi$
$654$	0	0
$655$	2.98436e6	0.271799
$656$	1.00712e7	0.913735
$657$	0	0
$658$	−1.47755e7	−1.33039
$659$	4.88080e6	0.437802	0.218901	−	0.975747i	$-0.429753\pi$
$659$	4.88080e6	0.437802	0.218901	+	0.975747i	$0.429753\pi$
$660$	0	0
$661$	1.75690e7	1.56402	0.782010	−	0.623266i	$-0.214194\pi$
$661$	1.75690e7	1.56402	0.782010	+	0.623266i	$0.214194\pi$
$662$	−1.65074e7	−1.46398
$663$	0	0
$664$	240459.	0.0211651
$665$	1.05346e7	0.923767
$666$	0	0
$667$	−1.33894e7	−1.16532
$668$	−7.10853e6	−0.616366
$669$	0	0
$670$	6.88020e6	0.592126
$671$	−1.20858e6	−0.103626
$672$	0	0
$673$	1.60675e7	1.36745	0.683725	−	0.729740i	$-0.260359\pi$
$673$	1.60675e7	1.36745	0.683725	+	0.729740i	$0.260359\pi$
$674$	−7.02205e6	−0.595407
$675$	0	0
$676$	−7.77441e6	−0.654336
$677$	1.51931e7	1.27401	0.637007	−	0.770858i	$-0.280172\pi$
$677$	1.51931e7	1.27401	0.637007	+	0.770858i	$0.280172\pi$
$678$	0	0
$679$	4.28849e7	3.56968
$680$	841882.	0.0698198
$681$	0	0
$682$	9.34170e6	0.769068
$683$	9.75866e6	0.800457	0.400229	−	0.916415i	$-0.368931\pi$
$683$	9.75866e6	0.800457	0.400229	+	0.916415i	$0.368931\pi$
$684$	0	0
$685$	3.30694e6	0.269278
$686$	3.73571e7	3.03084
$687$	0	0
$688$	1.44253e7	1.16186
$689$	−815544.	−0.0654484
$690$	0	0
$691$	−1.74840e7	−1.39298	−0.696490	−	0.717567i	$-0.745256\pi$
$691$	−1.74840e7	−1.39298	−0.696490	+	0.717567i	$0.745256\pi$
$692$	−1.32970e7	−1.05557
$693$	0	0
$694$	−1.30188e7	−1.02606
$695$	−2.55815e6	−0.200892
$696$	0	0
$697$	−5.08026e6	−0.396099
$698$	−7.65925e6	−0.595042
$699$	0	0
$700$	−3.60566e6	−0.278125
$701$	−4.31053e6	−0.331311	−0.165656	−	0.986184i	$-0.552974\pi$
$701$	−4.31053e6	−0.331311	−0.165656	+	0.986184i	$0.552974\pi$
$702$	0	0
$703$	1.37322e7	1.04798
$704$	1.97978e6	0.150551
$705$	0	0
$706$	−2.47702e7	−1.87033
$707$	−6.75064e6	−0.507921
$708$	0	0
$709$	−1.44521e6	−0.107973	−0.0539866	−	0.998542i	$-0.517193\pi$
$709$	−1.44521e6	−0.107973	−0.0539866	+	0.998542i	$0.517193\pi$
$710$	−5.74277e6	−0.427539
$711$	0	0
$712$	2.51806e6	0.186151
$713$	4.89671e7	3.60728
$714$	0	0
$715$	−714457.	−0.0522650
$716$	9.33574e6	0.680559
$717$	0	0
$718$	1.74699e7	1.26468
$719$	4.22567e6	0.304841	0.152420	−	0.988316i	$-0.451293\pi$
$719$	4.22567e6	0.304841	0.152420	+	0.988316i	$0.451293\pi$
$720$	0	0
$721$	1.39175e7	0.997066
$722$	−5.74393e6	−0.410078
$723$	0	0
$724$	−1.53175e7	−1.08603
$725$	−1.75312e6	−0.123870
$726$	0	0
$727$	1.62930e7	1.14331	0.571657	−	0.820493i	$-0.306301\pi$
$727$	1.62930e7	1.14331	0.571657	+	0.820493i	$0.306301\pi$
$728$	−3.06265e6	−0.214175
$729$	0	0
$730$	−1.57819e7	−1.09611
$731$	−7.27665e6	−0.503661
$732$	0	0
$733$	−3.00376e6	−0.206493	−0.103247	−	0.994656i	$-0.532923\pi$
$733$	−3.00376e6	−0.206493	−0.103247	+	0.994656i	$0.532923\pi$
$734$	1.36521e7	0.935319
$735$	0	0
$736$	3.48412e7	2.37082
$737$	4.42468e6	0.300064
$738$	0	0
$739$	5.58694e6	0.376325	0.188163	−	0.982138i	$-0.439747\pi$
$739$	5.58694e6	0.376325	0.188163	+	0.982138i	$0.439747\pi$
$740$	−4.70013e6	−0.315522
$741$	0	0
$742$	−6.08431e6	−0.405697
$743$	8.33428e6	0.553855	0.276927	−	0.960891i	$-0.410684\pi$
$743$	8.33428e6	0.553855	0.276927	+	0.960891i	$0.410684\pi$
$744$	0	0
$745$	−9.29240e6	−0.613390
$746$	1.20789e6	0.0794659
$747$	0	0
$748$	−1.81282e6	−0.118468
$749$	3.24046e6	0.211058
$750$	0	0
$751$	−1.55555e7	−1.00643	−0.503215	−	0.864162i	$-0.667849\pi$
$751$	−1.55555e7	−1.00643	−0.503215	+	0.864162i	$0.667849\pi$
$752$	1.01073e7	0.651766
$753$	0	0
$754$	4.98594e6	0.319388
$755$	−2.79313e6	−0.178330
$756$	0	0
$757$	2.42999e7	1.54122	0.770611	−	0.637306i	$-0.219951\pi$
$757$	2.42999e7	1.54122	0.770611	+	0.637306i	$0.219951\pi$
$758$	4.68450e6	0.296135
$759$	0	0
$760$	−2.49209e6	−0.156505
$761$	−1.84687e7	−1.15605	−0.578023	−	0.816021i	$-0.696175\pi$
$761$	−1.84687e7	−1.15605	−0.578023	+	0.816021i	$0.696175\pi$
$762$	0	0
$763$	3.10876e6	0.193320
$764$	−209684.	−0.0129966
$765$	0	0
$766$	1.12194e7	0.690873
$767$	1.20530e7	0.739790
$768$	0	0
$769$	−1.51725e6	−0.0925210	−0.0462605	−	0.998929i	$-0.514730\pi$
$769$	−1.51725e6	−0.0925210	−0.0462605	+	0.998929i	$0.514730\pi$
$770$	−5.33016e6	−0.323976
$771$	0	0
$772$	−1.39119e7	−0.840122
$773$	−3.28907e6	−0.197981	−0.0989907	−	0.995088i	$-0.531561\pi$
$773$	−3.28907e6	−0.197981	−0.0989907	+	0.995088i	$0.531561\pi$
$774$	0	0
$775$	6.41145e6	0.383444
$776$	−1.01450e7	−0.604779
$777$	0	0
$778$	−1.24996e7	−0.740367
$779$	1.50383e7	0.887879
$780$	0	0
$781$	−3.69319e6	−0.216658
$782$	−2.18427e7	−1.27729
$783$	0	0
$784$	−4.58126e7	−2.66192
$785$	−6.11815e6	−0.354361
$786$	0	0
$787$	−1.46769e7	−0.844692	−0.422346	−	0.906435i	$-0.638793\pi$
$787$	−1.46769e7	−0.844692	−0.422346	+	0.906435i	$0.638793\pi$
$788$	1.06693e7	0.612098
$789$	0	0
$790$	−1.94745e7	−1.11019
$791$	3.04564e7	1.73076
$792$	0	0
$793$	2.35906e6	0.133216
$794$	−1.85409e7	−1.04371
$795$	0	0
$796$	−1.51509e7	−0.847531
$797$	−4.95422e6	−0.276267	−0.138134	−	0.990414i	$-0.544110\pi$
$797$	−4.95422e6	−0.276267	−0.138134	+	0.990414i	$0.544110\pi$
$798$	0	0
$799$	−5.09850e6	−0.282537
$800$	4.56190e6	0.252012
$801$	0	0
$802$	−5.00283e6	−0.274650
$803$	−1.01494e7	−0.555459
$804$	0	0
$805$	−2.79395e7	−1.51960
$806$	−1.82344e7	−0.988675
$807$	0	0
$808$	1.59695e6	0.0860524
$809$	−1.90776e7	−1.02483	−0.512414	−	0.858738i	$-0.671249\pi$
$809$	−1.90776e7	−1.02483	−0.512414	+	0.858738i	$0.671249\pi$
$810$	0	0
$811$	2.54611e7	1.35933	0.679665	−	0.733523i	$-0.262125\pi$
$811$	2.54611e7	1.35933	0.679665	+	0.733523i	$0.262125\pi$
$812$	1.61822e7	0.861284
$813$	0	0
$814$	−6.94809e6	−0.367540
$815$	1.08906e7	0.574324
$816$	0	0
$817$	2.15399e7	1.12899
$818$	836308.	0.0437002
$819$	0	0
$820$	−5.14714e6	−0.267320
$821$	3.75556e7	1.94454	0.972269	−	0.233867i	$-0.0751380\pi$
$821$	3.75556e7	1.94454	0.972269	+	0.233867i	$0.0751380\pi$
$822$	0	0
$823$	−3.42815e7	−1.76425	−0.882126	−	0.471014i	$-0.843888\pi$
$823$	−3.42815e7	−1.76425	−0.882126	+	0.471014i	$0.843888\pi$
$824$	−3.29237e6	−0.168924
$825$	0	0
$826$	8.99209e7	4.58575
$827$	−2.92800e7	−1.48870	−0.744351	−	0.667789i	$-0.767241\pi$
$827$	−2.92800e7	−1.48870	−0.744351	+	0.667789i	$0.767241\pi$
$828$	0	0
$829$	−7.40301e6	−0.374130	−0.187065	−	0.982348i	$-0.559897\pi$
$829$	−7.40301e6	−0.374130	−0.187065	+	0.982348i	$0.559897\pi$
$830$	−816862.	−0.0411579
$831$	0	0
$832$	−3.86440e6	−0.193541
$833$	2.31095e7	1.15393
$834$	0	0
$835$	−7.21216e6	−0.357972
$836$	5.36619e6	0.265552
$837$	0	0
$838$	−4.02224e7	−1.97860
$839$	3.44573e7	1.68996	0.844981	−	0.534797i	$-0.179612\pi$
$839$	3.44573e7	1.68996	0.844981	+	0.534797i	$0.179612\pi$
$840$	0	0
$841$	−1.26432e7	−0.616404
$842$	3.63950e6	0.176914
$843$	0	0
$844$	1.78271e6	0.0861441
$845$	−7.88775e6	−0.380024
$846$	0	0
$847$	−3.42784e6	−0.164177
$848$	4.16203e6	0.198754
$849$	0	0
$850$	−2.85995e6	−0.135772
$851$	−3.64203e7	−1.72393
$852$	0	0
$853$	−1.70125e7	−0.800561	−0.400281	−	0.916393i	$-0.631087\pi$
$853$	−1.70125e7	−0.800561	−0.400281	+	0.916393i	$0.631087\pi$
$854$	1.75996e7	0.825769
$855$	0	0
$856$	−766572.	−0.0357576
$857$	−2.11650e7	−0.984387	−0.492194	−	0.870486i	$-0.663805\pi$
$857$	−2.11650e7	−0.984387	−0.492194	+	0.870486i	$0.663805\pi$
$858$	0	0
$859$	−3.17802e7	−1.46952	−0.734758	−	0.678330i	$-0.762704\pi$
$859$	−3.17802e7	−1.46952	−0.734758	+	0.678330i	$0.762704\pi$
$860$	−7.37244e6	−0.339911
$861$	0	0
$862$	−1.81785e6	−0.0833277
$863$	3.53199e7	1.61433	0.807166	−	0.590325i	$-0.201000\pi$
$863$	3.53199e7	1.61433	0.807166	+	0.590325i	$0.201000\pi$
$864$	0	0
$865$	−1.34908e7	−0.613053
$866$	7.82692e6	0.354647
$867$	0	0
$868$	−5.91807e7	−2.66613
$869$	−1.25241e7	−0.562597
$870$	0	0
$871$	−8.63669e6	−0.385746
$872$	−735417.	−0.0327524
$873$	0	0
$874$	6.46575e7	2.86313
$875$	−3.65822e6	−0.161529
$876$	0	0
$877$	256700.	0.0112701	0.00563503	−	0.999984i	$-0.498206\pi$
$877$	256700.	0.0112701	0.00563503	+	0.999984i	$0.498206\pi$
$878$	2.50606e7	1.09712
$879$	0	0
$880$	3.64614e6	0.158718
$881$	−3.37169e7	−1.46355	−0.731776	−	0.681545i	$-0.761308\pi$
$881$	−3.37169e7	−1.46355	−0.731776	+	0.681545i	$0.761308\pi$
$882$	0	0
$883$	−2.45722e7	−1.06058	−0.530289	−	0.847817i	$-0.677917\pi$
$883$	−2.45722e7	−1.06058	−0.530289	+	0.847817i	$0.677917\pi$
$884$	3.53850e6	0.152296
$885$	0	0
$886$	3.85542e7	1.65001
$887$	−220190.	−0.00939697	−0.00469849	−	0.999989i	$-0.501496\pi$
$887$	−220190.	−0.00939697	−0.00469849	+	0.999989i	$0.501496\pi$
$888$	0	0
$889$	6.43623e7	2.73135
$890$	−8.55408e6	−0.361991
$891$	0	0
$892$	9.12622e6	0.384042
$893$	1.50923e7	0.633323
$894$	0	0
$895$	9.47184e6	0.395254
$896$	2.58546e7	1.07589
$897$	0	0
$898$	4.38260e7	1.81360
$899$	−2.87745e7	−1.18743
$900$	0	0
$901$	−2.09948e6	−0.0861587
$902$	−7.60889e6	−0.311390
$903$	0	0
$904$	−7.20487e6	−0.293228
$905$	−1.55408e7	−0.630743
$906$	0	0
$907$	−6.02862e6	−0.243332	−0.121666	−	0.992571i	$-0.538824\pi$
$907$	−6.02862e6	−0.243332	−0.121666	+	0.992571i	$0.538824\pi$
$908$	−8.02011e6	−0.322824
$909$	0	0
$910$	1.04041e7	0.416487
$911$	−1.68076e7	−0.670979	−0.335490	−	0.942044i	$-0.608902\pi$
$911$	−1.68076e7	−0.670979	−0.335490	+	0.942044i	$0.608902\pi$
$912$	0	0
$913$	−525327.	−0.0208570
$914$	4.00705e7	1.58657
$915$	0	0
$916$	−1.41302e6	−0.0556430
$917$	−2.79487e7	−1.09758
$918$	0	0
$919$	4.57189e7	1.78569	0.892846	−	0.450362i	$-0.148705\pi$
$919$	4.57189e7	1.78569	0.892846	+	0.450362i	$0.148705\pi$
$920$	6.60944e6	0.257451
$921$	0	0
$922$	5.07022e6	0.196426
$923$	7.20887e6	0.278524
$924$	0	0
$925$	−4.76865e6	−0.183249
$926$	1.44672e7	0.554443
$927$	0	0
$928$	−2.04737e7	−0.780418
$929$	2.84053e7	1.07984	0.539921	−	0.841716i	$-0.318454\pi$
$929$	2.84053e7	1.07984	0.539921	+	0.841716i	$0.318454\pi$
$930$	0	0
$931$	−6.84073e7	−2.58659
$932$	2.05477e7	0.774861
$933$	0	0
$934$	−5.98068e7	−2.24328
$935$	−1.83925e6	−0.0688035
$936$	0	0
$937$	−8.99373e6	−0.334650	−0.167325	−	0.985902i	$-0.553513\pi$
$937$	−8.99373e6	−0.334650	−0.167325	+	0.985902i	$0.553513\pi$
$938$	−6.44334e7	−2.39113
$939$	0	0
$940$	−5.16562e6	−0.190679
$941$	2.96562e7	1.09180	0.545898	−	0.837852i	$-0.316189\pi$
$941$	2.96562e7	1.09180	0.545898	+	0.837852i	$0.316189\pi$
$942$	0	0
$943$	−3.98841e7	−1.46056
$944$	−6.15113e7	−2.24659
$945$	0	0
$946$	−1.08985e7	−0.395949
$947$	1.62357e7	0.588297	0.294148	−	0.955760i	$-0.404964\pi$
$947$	1.62357e7	0.588297	0.294148	+	0.955760i	$0.404964\pi$
$948$	0	0
$949$	1.98110e7	0.714070
$950$	8.46586e6	0.304342
$951$	0	0
$952$	−7.88427e6	−0.281948
$953$	−2.13990e7	−0.763241	−0.381621	−	0.924319i	$-0.624634\pi$
$953$	−2.13990e7	−0.763241	−0.381621	+	0.924319i	$0.624634\pi$
$954$	0	0
$955$	−212740.	−0.00754817
$956$	2.00572e7	0.709785
$957$	0	0
$958$	−1.35728e7	−0.477809
$959$	−3.09697e7	−1.08740
$960$	0	0
$961$	7.66039e7	2.67573
$962$	1.35622e7	0.472490
$963$	0	0
$964$	−2.91885e7	−1.01162
$965$	−1.41147e7	−0.487925
$966$	0	0
$967$	−6.31736e6	−0.217255	−0.108627	−	0.994083i	$-0.534646\pi$
$967$	−6.31736e6	−0.217255	−0.108627	+	0.994083i	$0.534646\pi$
$968$	810900.	0.0278150
$969$	0	0
$970$	3.44635e7	1.17606
$971$	−4.14293e7	−1.41013	−0.705065	−	0.709142i	$-0.749082\pi$
$971$	−4.14293e7	−1.41013	−0.705065	+	0.709142i	$0.749082\pi$
$972$	0	0
$973$	2.39572e7	0.811247
$974$	−2.29517e7	−0.775207
$975$	0	0
$976$	−1.20392e7	−0.404550
$977$	333761.	0.0111866	0.00559331	−	0.999984i	$-0.498220\pi$
$977$	333761.	0.0111866	0.00559331	+	0.999984i	$0.498220\pi$
$978$	0	0
$979$	−5.50116e6	−0.183441
$980$	2.34137e7	0.778763
$981$	0	0
$982$	−5.66948e7	−1.87614
$983$	3.43832e7	1.13491	0.567457	−	0.823403i	$-0.307927\pi$
$983$	3.43832e7	1.13491	0.567457	+	0.823403i	$0.307927\pi$
$984$	0	0
$985$	1.08249e7	0.355493
$986$	1.28354e7	0.420454
$987$	0	0
$988$	−1.04745e7	−0.341381
$989$	−5.71275e7	−1.85718
$990$	0	0
$991$	4.26710e6	0.138022	0.0690111	−	0.997616i	$-0.478016\pi$
$991$	4.26710e6	0.138022	0.0690111	+	0.997616i	$0.478016\pi$
$992$	7.48758e7	2.41581
$993$	0	0
$994$	5.37813e7	1.72649
$995$	−1.53718e7	−0.492228
$996$	0	0
$997$	2.61381e7	0.832790	0.416395	−	0.909184i	$-0.363293\pi$
$997$	2.61381e7	0.832790	0.416395	+	0.909184i	$0.363293\pi$
$998$	−4.52945e7	−1.43953
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.d.1.1		3
3.2	odd	2		165.6.a.b.1.3	✓	3
15.14	odd	2		825.6.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.b.1.3	✓	3	3.2	odd	2
495.6.a.d.1.1		3	1.1	even	1	trivial
825.6.a.i.1.1		3	15.14	odd	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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-7.52601 q^{2} +24.6408 q^{4} +25.0000 q^{5} -234.126 q^{7} +55.3856 q^{8} +O(q^{10})\)	\(q-7.52601 q^{2} +24.6408 q^{4} +25.0000 q^{5} -234.126 q^{7} +55.3856 q^{8} -188.150 q^{10} -121.000 q^{11} +236.184 q^{13} +1762.04 q^{14} -1205.34 q^{16} +608.015 q^{17} -1799.81 q^{19} +616.019 q^{20} +910.647 q^{22} +4773.40 q^{23} +625.000 q^{25} -1777.52 q^{26} -5769.05 q^{28} -2804.99 q^{29} +10258.3 q^{31} +7299.04 q^{32} -4575.93 q^{34} -5853.16 q^{35} -7629.84 q^{37} +13545.4 q^{38} +1384.64 q^{40} -8355.48 q^{41} -11967.9 q^{43} -2981.53 q^{44} -35924.7 q^{46} -8385.48 q^{47} +38008.1 q^{49} -4703.75 q^{50} +5819.76 q^{52} -3453.00 q^{53} -3025.00 q^{55} -12967.2 q^{56} +21110.4 q^{58} +51032.4 q^{59} +9988.24 q^{61} -77204.1 q^{62} -16361.8 q^{64} +5904.60 q^{65} -36567.6 q^{67} +14982.0 q^{68} +44050.9 q^{70} +30522.3 q^{71} +83879.5 q^{73} +57422.2 q^{74} -44348.7 q^{76} +28329.3 q^{77} +103505. q^{79} -30133.4 q^{80} +62883.4 q^{82} +4341.54 q^{83} +15200.4 q^{85} +90070.3 q^{86} -6701.65 q^{88} +45464.1 q^{89} -55296.9 q^{91} +117620. q^{92} +63109.2 q^{94} -44995.2 q^{95} -183170. q^{97} -286049. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8	\( 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8} + 50 q^{10} - 363 q^{11} + 450 q^{13} + 1504 q^{14} - 1360 q^{16} + 334 q^{17} - 4036 q^{19} + 700 q^{20} - 242 q^{22} + 7060 q^{23} + 1875 q^{25} - 2932 q^{26} - 8320 q^{28} - 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3644 q^{34} - 5800 q^{35} + 2250 q^{37} + 12632 q^{38} - 600 q^{40} - 10654 q^{41} - 35528 q^{43} - 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 7667 q^{49} + 1250 q^{50} - 14520 q^{52} + 12826 q^{53} - 9075 q^{55} - 17088 q^{56} - 17196 q^{58} + 81876 q^{59} - 62298 q^{61} - 109184 q^{62} - 72256 q^{64} + 11250 q^{65} - 46148 q^{67} + 35832 q^{68} + 37600 q^{70} + 64724 q^{71} + 810 q^{73} + 44796 q^{74} + 44656 q^{76} + 28072 q^{77} + 43876 q^{79} - 34000 q^{80} + 56060 q^{82} + 101024 q^{83} + 8350 q^{85} - 24128 q^{86} + 2904 q^{88} - 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 74552 q^{94} - 100900 q^{95} - 319746 q^{97} - 431134 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8 + 50 * q^10 - 363 * q^11 + 450 * q^13 + 1504 * q^14 - 1360 * q^16 + 334 * q^17 - 4036 * q^19 + 700 * q^20 - 242 * q^22 + 7060 * q^23 + 1875 * q^25 - 2932 * q^26 - 8320 * q^28 - 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3644 * q^34 - 5800 * q^35 + 2250 * q^37 + 12632 * q^38 - 600 * q^40 - 10654 * q^41 - 35528 * q^43 - 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 7667 * q^49 + 1250 * q^50 - 14520 * q^52 + 12826 * q^53 - 9075 * q^55 - 17088 * q^56 - 17196 * q^58 + 81876 * q^59 - 62298 * q^61 - 109184 * q^62 - 72256 * q^64 + 11250 * q^65 - 46148 * q^67 + 35832 * q^68 + 37600 * q^70 + 64724 * q^71 + 810 * q^73 + 44796 * q^74 + 44656 * q^76 + 28072 * q^77 + 43876 * q^79 - 34000 * q^80 + 56060 * q^82 + 101024 * q^83 + 8350 * q^85 - 24128 * q^86 + 2904 * q^88 - 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 74552 * q^94 - 100900 * q^95 - 319746 * q^97 - 431134 * q^98

Introduction

L-functions

Modular forms

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Database

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