LMFDB - Embedded newform 414.6.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,6,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("414.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	414.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:	$66.3989014026$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8369x^{2} - 182616x - 370980 $ x^4 - x^3 - 8369x^2 - 182616x - 370980
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$101.516$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} -38.3508 q^{5} -90.6804 q^{7} -64.0000 q^{8} +O(q^{10})$	\(q-4.00000 q^{2} +16.0000 q^{4} -38.3508 q^{5} -90.6804 q^{7} -64.0000 q^{8} +153.403 q^{10} -263.339 q^{11} +618.446 q^{13} +362.722 q^{14} +256.000 q^{16} -1691.98 q^{17} -951.063 q^{19} -613.612 q^{20} +1053.35 q^{22} -529.000 q^{23} -1654.22 q^{25} -2473.78 q^{26} -1450.89 q^{28} -744.300 q^{29} -1830.09 q^{31} -1024.00 q^{32} +6767.91 q^{34} +3477.66 q^{35} +11530.0 q^{37} +3804.25 q^{38} +2454.45 q^{40} -12287.1 q^{41} -6135.42 q^{43} -4213.42 q^{44} +2116.00 q^{46} +10359.6 q^{47} -8584.07 q^{49} +6616.88 q^{50} +9895.13 q^{52} +23877.8 q^{53} +10099.2 q^{55} +5803.54 q^{56} +2977.20 q^{58} -36664.8 q^{59} -6014.85 q^{61} +7320.36 q^{62} +4096.00 q^{64} -23717.9 q^{65} -53216.6 q^{67} -27071.6 q^{68} -13910.6 q^{70} +26211.2 q^{71} +19037.1 q^{73} -46120.1 q^{74} -15217.0 q^{76} +23879.6 q^{77} +8173.10 q^{79} -9817.79 q^{80} +49148.3 q^{82} +77689.6 q^{83} +64888.6 q^{85} +24541.7 q^{86} +16853.7 q^{88} +50234.5 q^{89} -56080.9 q^{91} -8464.00 q^{92} -41438.5 q^{94} +36474.0 q^{95} -108592. q^{97} +34336.3 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8}+O(q^{10}) $ 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8	$ 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8} + 216 q^{10} + 6 q^{11} + 1248 q^{13} - 1392 q^{14} + 1024 q^{16} + 56 q^{17} + 1530 q^{19} - 864 q^{20} - 24 q^{22} - 2116 q^{23} + 8852 q^{25} - 4992 q^{26} + 5568 q^{28} - 2292 q^{29} + 1556 q^{31} - 4096 q^{32} - 224 q^{34} - 5688 q^{35} - 9586 q^{37} - 6120 q^{38} + 3456 q^{40} + 11768 q^{41} + 13758 q^{43} + 96 q^{44} + 8464 q^{46} - 10636 q^{47} + 11360 q^{49} - 35408 q^{50} + 19968 q^{52} - 26686 q^{53} - 28900 q^{55} - 22272 q^{56} + 9168 q^{58} + 9108 q^{59} - 37878 q^{61} - 6224 q^{62} + 16384 q^{64} + 72212 q^{65} - 23302 q^{67} + 896 q^{68} + 22752 q^{70} - 31728 q^{71} - 28340 q^{73} + 38344 q^{74} + 24480 q^{76} + 121276 q^{77} - 26668 q^{79} - 13824 q^{80} - 47072 q^{82} + 119026 q^{83} - 217876 q^{85} - 55032 q^{86} - 384 q^{88} + 148236 q^{89} + 89008 q^{91} - 33856 q^{92} + 42544 q^{94} + 399292 q^{95} - 16092 q^{97} - 45440 q^{98}+O(q^{100}) $ 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8 + 216 * q^10 + 6 * q^11 + 1248 * q^13 - 1392 * q^14 + 1024 * q^16 + 56 * q^17 + 1530 * q^19 - 864 * q^20 - 24 * q^22 - 2116 * q^23 + 8852 * q^25 - 4992 * q^26 + 5568 * q^28 - 2292 * q^29 + 1556 * q^31 - 4096 * q^32 - 224 * q^34 - 5688 * q^35 - 9586 * q^37 - 6120 * q^38 + 3456 * q^40 + 11768 * q^41 + 13758 * q^43 + 96 * q^44 + 8464 * q^46 - 10636 * q^47 + 11360 * q^49 - 35408 * q^50 + 19968 * q^52 - 26686 * q^53 - 28900 * q^55 - 22272 * q^56 + 9168 * q^58 + 9108 * q^59 - 37878 * q^61 - 6224 * q^62 + 16384 * q^64 + 72212 * q^65 - 23302 * q^67 + 896 * q^68 + 22752 * q^70 - 31728 * q^71 - 28340 * q^73 + 38344 * q^74 + 24480 * q^76 + 121276 * q^77 - 26668 * q^79 - 13824 * q^80 - 47072 * q^82 + 119026 * q^83 - 217876 * q^85 - 55032 * q^86 - 384 * q^88 + 148236 * q^89 + 89008 * q^91 - 33856 * q^92 + 42544 * q^94 + 399292 * q^95 - 16092 * q^97 - 45440 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−38.3508	−0.686039	−0.343020	−	0.939328i	$-0.611450\pi$
$5$	−38.3508	−0.686039	−0.343020	+	0.939328i	$0.611450\pi$
$6$	0	0
$7$	−90.6804	−0.699469	−0.349734	−	0.936849i	$-0.613728\pi$
$7$	−90.6804	−0.699469	−0.349734	+	0.936849i	$0.613728\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	153.403	0.485103
$11$	−263.339	−0.656195	−0.328097	−	0.944644i	$-0.606407\pi$
$11$	−263.339	−0.656195	−0.328097	+	0.944644i	$0.606407\pi$
$12$	0	0
$13$	618.446	1.01495	0.507473	−	0.861668i	$-0.330580\pi$
$13$	618.446	1.01495	0.507473	+	0.861668i	$0.330580\pi$
$14$	362.722	0.494599
$15$	0	0
$16$	256.000	0.250000
$17$	−1691.98	−1.41995	−0.709973	−	0.704229i	$-0.751293\pi$
$17$	−1691.98	−1.41995	−0.709973	+	0.704229i	$0.751293\pi$
$18$	0	0
$19$	−951.063	−0.604401	−0.302201	−	0.953244i	$-0.597721\pi$
$19$	−951.063	−0.604401	−0.302201	+	0.953244i	$0.597721\pi$
$20$	−613.612	−0.343020
$21$	0	0
$22$	1053.35	0.464000
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−1654.22	−0.529350
$26$	−2473.78	−0.717675
$27$	0	0
$28$	−1450.89	−0.349734
$29$	−744.300	−0.164344	−0.0821718	−	0.996618i	$-0.526186\pi$
$29$	−744.300	−0.164344	−0.0821718	+	0.996618i	$0.526186\pi$
$30$	0	0
$31$	−1830.09	−0.342033	−0.171017	−	0.985268i	$-0.554705\pi$
$31$	−1830.09	−0.342033	−0.171017	+	0.985268i	$0.554705\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	6767.91	1.00405
$35$	3477.66	0.479863
$36$	0	0
$37$	11530.0	1.38460	0.692302	−	0.721608i	$-0.256597\pi$
$37$	11530.0	1.38460	0.692302	+	0.721608i	$0.256597\pi$
$38$	3804.25	0.427376
$39$	0	0
$40$	2454.45	0.242551
$41$	−12287.1	−1.14153	−0.570767	−	0.821112i	$-0.693354\pi$
$41$	−12287.1	−1.14153	−0.570767	+	0.821112i	$0.693354\pi$
$42$	0	0
$43$	−6135.42	−0.506026	−0.253013	−	0.967463i	$-0.581422\pi$
$43$	−6135.42	−0.506026	−0.253013	+	0.967463i	$0.581422\pi$
$44$	−4213.42	−0.328097
$45$	0	0
$46$	2116.00	0.147442
$47$	10359.6	0.684068	0.342034	−	0.939688i	$-0.388884\pi$
$47$	10359.6	0.684068	0.342034	+	0.939688i	$0.388884\pi$
$48$	0	0
$49$	−8584.07	−0.510744
$50$	6616.88	0.374307
$51$	0	0
$52$	9895.13	0.507473
$53$	23877.8	1.16763	0.583815	−	0.811887i	$-0.301559\pi$
$53$	23877.8	1.16763	0.583815	+	0.811887i	$0.301559\pi$
$54$	0	0
$55$	10099.2	0.450175
$56$	5803.54	0.247299
$57$	0	0
$58$	2977.20	0.116209
$59$	−36664.8	−1.37126	−0.685629	−	0.727951i	$-0.740473\pi$
$59$	−36664.8	−1.37126	−0.685629	+	0.727951i	$0.740473\pi$
$60$	0	0
$61$	−6014.85	−0.206966	−0.103483	−	0.994631i	$-0.532999\pi$
$61$	−6014.85	−0.206966	−0.103483	+	0.994631i	$0.532999\pi$
$62$	7320.36	0.241854
$63$	0	0
$64$	4096.00	0.125000
$65$	−23717.9	−0.696293
$66$	0	0
$67$	−53216.6	−1.44830	−0.724152	−	0.689640i	$-0.757769\pi$
$67$	−53216.6	−1.44830	−0.724152	+	0.689640i	$0.757769\pi$
$68$	−27071.6	−0.709973
$69$	0	0
$70$	−13910.6	−0.339314
$71$	26211.2	0.617079	0.308539	−	0.951212i	$-0.400160\pi$
$71$	26211.2	0.617079	0.308539	+	0.951212i	$0.400160\pi$
$72$	0	0
$73$	19037.1	0.418113	0.209056	−	0.977904i	$-0.432961\pi$
$73$	19037.1	0.418113	0.209056	+	0.977904i	$0.432961\pi$
$74$	−46120.1	−0.979063
$75$	0	0
$76$	−15217.0	−0.302201
$77$	23879.6	0.458988
$78$	0	0
$79$	8173.10	0.147339	0.0736697	−	0.997283i	$-0.476529\pi$
$79$	8173.10	0.147339	0.0736697	+	0.997283i	$0.476529\pi$
$80$	−9817.79	−0.171510
$81$	0	0
$82$	49148.3	0.807187
$83$	77689.6	1.23785	0.618924	−	0.785451i	$-0.287569\pi$
$83$	77689.6	1.23785	0.618924	+	0.785451i	$0.287569\pi$
$84$	0	0
$85$	64888.6	0.974139
$86$	24541.7	0.357814
$87$	0	0
$88$	16853.7	0.232000
$89$	50234.5	0.672245	0.336122	−	0.941818i	$-0.390884\pi$
$89$	50234.5	0.672245	0.336122	+	0.941818i	$0.390884\pi$
$90$	0	0
$91$	−56080.9	−0.709923
$92$	−8464.00	−0.104257
$93$	0	0
$94$	−41438.5	−0.483709
$95$	36474.0	0.414643
$96$	0	0
$97$	−108592.	−1.17184	−0.585918	−	0.810370i	$-0.699266\pi$
$97$	−108592.	−1.17184	−0.585918	+	0.810370i	$0.699266\pi$
$98$	34336.3	0.361150
$99$	0	0
$100$	−26467.5	−0.264675
$101$	116672.	1.13805	0.569025	−	0.822320i	$-0.307321\pi$
$101$	116672.	1.13805	0.569025	+	0.822320i	$0.307321\pi$
$102$	0	0
$103$	−3576.22	−0.0332148	−0.0166074	−	0.999862i	$-0.505287\pi$
$103$	−3576.22	−0.0332148	−0.0166074	+	0.999862i	$0.505287\pi$
$104$	−39580.5	−0.358838
$105$	0	0
$106$	−95511.3	−0.825639
$107$	8015.42	0.0676810	0.0338405	−	0.999427i	$-0.489226\pi$
$107$	8015.42	0.0676810	0.0338405	+	0.999427i	$0.489226\pi$
$108$	0	0
$109$	117068.	0.943779	0.471890	−	0.881658i	$-0.343572\pi$
$109$	117068.	0.943779	0.471890	+	0.881658i	$0.343572\pi$
$110$	−40396.9	−0.318322
$111$	0	0
$112$	−23214.2	−0.174867
$113$	115338.	0.849723	0.424862	−	0.905258i	$-0.360323\pi$
$113$	115338.	0.849723	0.424862	+	0.905258i	$0.360323\pi$
$114$	0	0
$115$	20287.6	0.143049
$116$	−11908.8	−0.0821718
$117$	0	0
$118$	146659.	0.969626
$119$	153429.	0.993208
$120$	0	0
$121$	−91703.8	−0.569409
$122$	24059.4	0.146347
$123$	0	0
$124$	−29281.4	−0.171017
$125$	183287.	1.04919
$126$	0	0
$127$	−105519.	−0.580528	−0.290264	−	0.956947i	$-0.593743\pi$
$127$	−105519.	−0.580528	−0.290264	+	0.956947i	$0.593743\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	94871.4	0.492353
$131$	174565.	0.888751	0.444375	−	0.895841i	$-0.353426\pi$
$131$	174565.	0.888751	0.444375	+	0.895841i	$0.353426\pi$
$132$	0	0
$133$	86242.8	0.422760
$134$	212866.	1.02411
$135$	0	0
$136$	108287.	0.502027
$137$	169885.	0.773308	0.386654	−	0.922225i	$-0.373631\pi$
$137$	169885.	0.773308	0.386654	+	0.922225i	$0.373631\pi$
$138$	0	0
$139$	355902.	1.56240	0.781202	−	0.624278i	$-0.214607\pi$
$139$	355902.	1.56240	0.781202	+	0.624278i	$0.214607\pi$
$140$	55642.6	0.239931
$141$	0	0
$142$	−104845.	−0.436340
$143$	−162861.	−0.666002
$144$	0	0
$145$	28544.5	0.112746
$146$	−76148.4	−0.295650
$147$	0	0
$148$	184480.	0.692302
$149$	445125.	1.64254	0.821271	−	0.570538i	$-0.193265\pi$
$149$	445125.	1.64254	0.821271	+	0.570538i	$0.193265\pi$
$150$	0	0
$151$	29755.1	0.106199	0.0530994	−	0.998589i	$-0.483090\pi$
$151$	29755.1	0.106199	0.0530994	+	0.998589i	$0.483090\pi$
$152$	60868.0	0.213688
$153$	0	0
$154$	−95518.5	−0.324553
$155$	70185.4	0.234648
$156$	0	0
$157$	451654.	1.46237	0.731184	−	0.682180i	$-0.238968\pi$
$157$	451654.	1.46237	0.731184	+	0.682180i	$0.238968\pi$
$158$	−32692.4	−0.104185
$159$	0	0
$160$	39271.2	0.121276
$161$	47969.9	0.145849
$162$	0	0
$163$	284883.	0.839843	0.419921	−	0.907560i	$-0.362058\pi$
$163$	284883.	0.839843	0.419921	+	0.907560i	$0.362058\pi$
$164$	−196593.	−0.570767
$165$	0	0
$166$	−310758.	−0.875291
$167$	238283.	0.661152	0.330576	−	0.943779i	$-0.392757\pi$
$167$	238283.	0.661152	0.330576	+	0.943779i	$0.392757\pi$
$168$	0	0
$169$	11181.9	0.0301160
$170$	−259554.	−0.688820
$171$	0	0
$172$	−98166.7	−0.253013
$173$	−560451.	−1.42371	−0.711857	−	0.702325i	$-0.752145\pi$
$173$	−560451.	−1.42371	−0.711857	+	0.702325i	$0.752145\pi$
$174$	0	0
$175$	150005.	0.370264
$176$	−67414.7	−0.164049
$177$	0	0
$178$	−200938.	−0.475349
$179$	−226758.	−0.528968	−0.264484	−	0.964390i	$-0.585202\pi$
$179$	−226758.	−0.528968	−0.264484	+	0.964390i	$0.585202\pi$
$180$	0	0
$181$	676332.	1.53449	0.767244	−	0.641355i	$-0.221627\pi$
$181$	676332.	1.53449	0.767244	+	0.641355i	$0.221627\pi$
$182$	224323.	0.501991
$183$	0	0
$184$	33856.0	0.0737210
$185$	−442185.	−0.949893
$186$	0	0
$187$	445563.	0.931761
$188$	165754.	0.342034
$189$	0	0
$190$	−145896.	−0.293197
$191$	203200.	0.403033	0.201516	−	0.979485i	$-0.435413\pi$
$191$	203200.	0.403033	0.201516	+	0.979485i	$0.435413\pi$
$192$	0	0
$193$	71112.6	0.137421	0.0687105	−	0.997637i	$-0.478112\pi$
$193$	71112.6	0.137421	0.0687105	+	0.997637i	$0.478112\pi$
$194$	434366.	0.828613
$195$	0	0
$196$	−137345.	−0.255372
$197$	−757116.	−1.38994	−0.694971	−	0.719038i	$-0.744583\pi$
$197$	−757116.	−1.38994	−0.694971	+	0.719038i	$0.744583\pi$
$198$	0	0
$199$	740992.	1.32642	0.663210	−	0.748434i	$-0.269194\pi$
$199$	740992.	1.32642	0.663210	+	0.748434i	$0.269194\pi$
$200$	105870.	0.187154
$201$	0	0
$202$	−466686.	−0.804724
$203$	67493.4	0.114953
$204$	0	0
$205$	471219.	0.783138
$206$	14304.9	0.0234864
$207$	0	0
$208$	158322.	0.253737
$209$	250452.	0.396605
$210$	0	0
$211$	8808.64	0.0136208	0.00681040	−	0.999977i	$-0.497832\pi$
$211$	8808.64	0.0136208	0.00681040	+	0.999977i	$0.497832\pi$
$212$	382045.	0.583815
$213$	0	0
$214$	−32061.7	−0.0478577
$215$	235298.	0.347154
$216$	0	0
$217$	165953.	0.239242
$218$	−468270.	−0.667353
$219$	0	0
$220$	161588.	0.225088
$221$	−1.04640e6	−1.44117
$222$	0	0
$223$	−743669.	−1.00142	−0.500711	−	0.865614i	$-0.666928\pi$
$223$	−743669.	−1.00142	−0.500711	+	0.865614i	$0.666928\pi$
$224$	92856.7	0.123650
$225$	0	0
$226$	−461353.	−0.600845
$227$	321413.	0.413999	0.206999	−	0.978341i	$-0.433630\pi$
$227$	321413.	0.413999	0.206999	+	0.978341i	$0.433630\pi$
$228$	0	0
$229$	275710.	0.347427	0.173714	−	0.984796i	$-0.444423\pi$
$229$	275710.	0.347427	0.173714	+	0.984796i	$0.444423\pi$
$230$	−81150.2	−0.101151
$231$	0	0
$232$	47635.2	0.0581043
$233$	−1.09761e6	−1.32452	−0.662258	−	0.749276i	$-0.730402\pi$
$233$	−1.09761e6	−1.32452	−0.662258	+	0.749276i	$0.730402\pi$
$234$	0	0
$235$	−397299.	−0.469297
$236$	−586636.	−0.685629
$237$	0	0
$238$	−613716.	−0.702304
$239$	237626.	0.269091	0.134546	−	0.990907i	$-0.457042\pi$
$239$	237626.	0.269091	0.134546	+	0.990907i	$0.457042\pi$
$240$	0	0
$241$	−445755.	−0.494371	−0.247186	−	0.968968i	$-0.579506\pi$
$241$	−445755.	−0.494371	−0.247186	+	0.968968i	$0.579506\pi$
$242$	366815.	0.402633
$243$	0	0
$244$	−96237.5	−0.103483
$245$	329206.	0.350390
$246$	0	0
$247$	−588181.	−0.613435
$248$	117126.	0.120927
$249$	0	0
$250$	−733147.	−0.741892
$251$	390657.	0.391391	0.195695	−	0.980665i	$-0.437304\pi$
$251$	390657.	0.391391	0.195695	+	0.980665i	$0.437304\pi$
$252$	0	0
$253$	139306.	0.136826
$254$	422078.	0.410495
$255$	0	0
$256$	65536.0	0.0625000
$257$	68901.5	0.0650722	0.0325361	−	0.999471i	$-0.489642\pi$
$257$	68901.5	0.0650722	0.0325361	+	0.999471i	$0.489642\pi$
$258$	0	0
$259$	−1.04555e6	−0.968487
$260$	−379486.	−0.348146
$261$	0	0
$262$	−698262.	−0.628442
$263$	1.20041e6	1.07014	0.535068	−	0.844809i	$-0.320286\pi$
$263$	1.20041e6	1.07014	0.535068	+	0.844809i	$0.320286\pi$
$264$	0	0
$265$	−915733.	−0.801040
$266$	−344971.	−0.298936
$267$	0	0
$268$	−851465.	−0.724152
$269$	−471457.	−0.397248	−0.198624	−	0.980076i	$-0.563647\pi$
$269$	−471457.	−0.397248	−0.198624	+	0.980076i	$0.563647\pi$
$270$	0	0
$271$	−1.28573e6	−1.06348	−0.531738	−	0.846909i	$-0.678461\pi$
$271$	−1.28573e6	−1.06348	−0.531738	+	0.846909i	$0.678461\pi$
$272$	−433146.	−0.354987
$273$	0	0
$274$	−679539.	−0.546812
$275$	435620.	0.347357
$276$	0	0
$277$	−451149.	−0.353282	−0.176641	−	0.984275i	$-0.556523\pi$
$277$	−451149.	−0.353282	−0.176641	+	0.984275i	$0.556523\pi$
$278$	−1.42361e6	−1.10479
$279$	0	0
$280$	−222570.	−0.169657
$281$	−1.33780e6	−1.01071	−0.505355	−	0.862911i	$-0.668639\pi$
$281$	−1.33780e6	−1.01071	−0.505355	+	0.862911i	$0.668639\pi$
$282$	0	0
$283$	−1.78951e6	−1.32822	−0.664108	−	0.747637i	$-0.731189\pi$
$283$	−1.78951e6	−1.32822	−0.664108	+	0.747637i	$0.731189\pi$
$284$	419379.	0.308539
$285$	0	0
$286$	651442.	0.470935
$287$	1.11420e6	0.798468
$288$	0	0
$289$	1.44293e6	1.01625
$290$	−114178.	−0.0797236
$291$	0	0
$292$	304594.	0.209056
$293$	1.31648e6	0.895872	0.447936	−	0.894066i	$-0.352159\pi$
$293$	1.31648e6	0.895872	0.447936	+	0.894066i	$0.352159\pi$
$294$	0	0
$295$	1.40612e6	0.940736
$296$	−737921.	−0.489532
$297$	0	0
$298$	−1.78050e6	−1.16145
$299$	−327158.	−0.211631
$300$	0	0
$301$	556362.	0.353949
$302$	−119020.	−0.0750938
$303$	0	0
$304$	−243472.	−0.151100
$305$	230674.	0.141987
$306$	0	0
$307$	1.02499e6	0.620690	0.310345	−	0.950624i	$-0.399555\pi$
$307$	1.02499e6	0.620690	0.310345	+	0.950624i	$0.399555\pi$
$308$	382074.	0.229494
$309$	0	0
$310$	−280741.	−0.165921
$311$	−827520.	−0.485152	−0.242576	−	0.970132i	$-0.577992\pi$
$311$	−827520.	−0.485152	−0.242576	+	0.970132i	$0.577992\pi$
$312$	0	0
$313$	2.68599e6	1.54968	0.774842	−	0.632155i	$-0.217829\pi$
$313$	2.68599e6	1.54968	0.774842	+	0.632155i	$0.217829\pi$
$314$	−1.80662e6	−1.03405
$315$	0	0
$316$	130770.	0.0736697
$317$	−1.26083e6	−0.704706	−0.352353	−	0.935867i	$-0.614618\pi$
$317$	−1.26083e6	−0.704706	−0.352353	+	0.935867i	$0.614618\pi$
$318$	0	0
$319$	196003.	0.107841
$320$	−157085.	−0.0857549
$321$	0	0
$322$	−191880.	−0.103131
$323$	1.60918e6	0.858217
$324$	0	0
$325$	−1.02304e6	−0.537262
$326$	−1.13953e6	−0.593858
$327$	0	0
$328$	786373.	0.403594
$329$	−939414.	−0.478484
$330$	0	0
$331$	−1.95121e6	−0.978891	−0.489446	−	0.872034i	$-0.662801\pi$
$331$	−1.95121e6	−0.978891	−0.489446	+	0.872034i	$0.662801\pi$
$332$	1.24303e6	0.618924
$333$	0	0
$334$	−953130.	−0.467505
$335$	2.04090e6	0.993594
$336$	0	0
$337$	−1.75741e6	−0.842944	−0.421472	−	0.906841i	$-0.638486\pi$
$337$	−1.75741e6	−0.842944	−0.421472	+	0.906841i	$0.638486\pi$
$338$	−44727.5	−0.0212952
$339$	0	0
$340$	1.03822e6	0.487070
$341$	481933.	0.224440
$342$	0	0
$343$	2.30247e6	1.05672
$344$	392667.	0.178907
$345$	0	0
$346$	2.24180e6	1.00672
$347$	−3.93398e6	−1.75392	−0.876958	−	0.480566i	$-0.840431\pi$
$347$	−3.93398e6	−1.75392	−0.876958	+	0.480566i	$0.840431\pi$
$348$	0	0
$349$	3.43702e6	1.51049	0.755245	−	0.655442i	$-0.227518\pi$
$349$	3.43702e6	1.51049	0.755245	+	0.655442i	$0.227518\pi$
$350$	−600021.	−0.261816
$351$	0	0
$352$	269659.	0.116000
$353$	−3.59456e6	−1.53536	−0.767678	−	0.640835i	$-0.778588\pi$
$353$	−3.59456e6	−1.53536	−0.767678	+	0.640835i	$0.778588\pi$
$354$	0	0
$355$	−1.00522e6	−0.423340
$356$	803753.	0.336122
$357$	0	0
$358$	907031.	0.374037
$359$	−168160.	−0.0688631	−0.0344315	−	0.999407i	$-0.510962\pi$
$359$	−168160.	−0.0688631	−0.0344315	+	0.999407i	$0.510962\pi$
$360$	0	0
$361$	−1.57158e6	−0.634699
$362$	−2.70533e6	−1.08505
$363$	0	0
$364$	−897294.	−0.354962
$365$	−730087.	−0.286842
$366$	0	0
$367$	−4.68555e6	−1.81592	−0.907958	−	0.419061i	$-0.862359\pi$
$367$	−4.68555e6	−1.81592	−0.907958	+	0.419061i	$0.862359\pi$
$368$	−135424.	−0.0521286
$369$	0	0
$370$	1.76874e6	0.671676
$371$	−2.16525e6	−0.816720
$372$	0	0
$373$	520660.	0.193768	0.0968840	−	0.995296i	$-0.469112\pi$
$373$	520660.	0.193768	0.0968840	+	0.995296i	$0.469112\pi$
$374$	−1.78225e6	−0.658855
$375$	0	0
$376$	−663015.	−0.241854
$377$	−460309.	−0.166800
$378$	0	0
$379$	1.25395e6	0.448418	0.224209	−	0.974541i	$-0.428020\pi$
$379$	1.25395e6	0.448418	0.224209	+	0.974541i	$0.428020\pi$
$380$	583584.	0.207321
$381$	0	0
$382$	−812800.	−0.284987
$383$	784912.	0.273416	0.136708	−	0.990611i	$-0.456348\pi$
$383$	784912.	0.273416	0.136708	+	0.990611i	$0.456348\pi$
$384$	0	0
$385$	−915802.	−0.314883
$386$	−284450.	−0.0971713
$387$	0	0
$388$	−1.73747e6	−0.585918
$389$	−40700.6	−0.0136372	−0.00681862	−	0.999977i	$-0.502170\pi$
$389$	−40700.6	−0.0136372	−0.00681862	+	0.999977i	$0.502170\pi$
$390$	0	0
$391$	895056.	0.296079
$392$	549380.	0.180575
$393$	0	0
$394$	3.02846e6	0.982837
$395$	−313444.	−0.101081
$396$	0	0
$397$	4.54520e6	1.44736	0.723680	−	0.690136i	$-0.242449\pi$
$397$	4.54520e6	1.44736	0.723680	+	0.690136i	$0.242449\pi$
$398$	−2.96397e6	−0.937920
$399$	0	0
$400$	−423480.	−0.132338
$401$	1.66096e6	0.515819	0.257910	−	0.966169i	$-0.416966\pi$
$401$	1.66096e6	0.515819	0.257910	+	0.966169i	$0.416966\pi$
$402$	0	0
$403$	−1.13181e6	−0.347145
$404$	1.86675e6	0.569025
$405$	0	0
$406$	−269974.	−0.0812842
$407$	−3.03630e6	−0.908570
$408$	0	0
$409$	1.41566e6	0.418458	0.209229	−	0.977867i	$-0.432905\pi$
$409$	1.41566e6	0.418458	0.209229	+	0.977867i	$0.432905\pi$
$410$	−1.88488e6	−0.553762
$411$	0	0
$412$	−57219.5	−0.0166074
$413$	3.32478e6	0.959152
$414$	0	0
$415$	−2.97945e6	−0.849213
$416$	−633288.	−0.179419
$417$	0	0
$418$	−1.00181e6	−0.280442
$419$	3.98894e6	1.11000	0.554999	−	0.831851i	$-0.312719\pi$
$419$	3.98894e6	1.11000	0.554999	+	0.831851i	$0.312719\pi$
$420$	0	0
$421$	1.37454e6	0.377965	0.188982	−	0.981980i	$-0.439481\pi$
$421$	1.37454e6	0.377965	0.188982	+	0.981980i	$0.439481\pi$
$422$	−35234.6	−0.00963137
$423$	0	0
$424$	−1.52818e6	−0.412819
$425$	2.79890e6	0.751649
$426$	0	0
$427$	545428.	0.144767
$428$	128247.	0.0338405
$429$	0	0
$430$	−941192.	−0.245475
$431$	−730160.	−0.189332	−0.0946662	−	0.995509i	$-0.530178\pi$
$431$	−730160.	−0.189332	−0.0946662	+	0.995509i	$0.530178\pi$
$432$	0	0
$433$	1.63285e6	0.418530	0.209265	−	0.977859i	$-0.432893\pi$
$433$	1.63285e6	0.418530	0.209265	+	0.977859i	$0.432893\pi$
$434$	−663813.	−0.169169
$435$	0	0
$436$	1.87308e6	0.471890
$437$	503112.	0.126026
$438$	0	0
$439$	−6.87755e6	−1.70323	−0.851614	−	0.524169i	$-0.824376\pi$
$439$	−6.87755e6	−1.70323	−0.851614	+	0.524169i	$0.824376\pi$
$440$	−646351.	−0.159161
$441$	0	0
$442$	4.18558e6	1.01906
$443$	5.05214e6	1.22311	0.611555	−	0.791202i	$-0.290544\pi$
$443$	5.05214e6	1.22311	0.611555	+	0.791202i	$0.290544\pi$
$444$	0	0
$445$	−1.92653e6	−0.461186
$446$	2.97467e6	0.708113
$447$	0	0
$448$	−371427.	−0.0874336
$449$	6.21605e6	1.45512	0.727560	−	0.686045i	$-0.240654\pi$
$449$	6.21605e6	1.45512	0.727560	+	0.686045i	$0.240654\pi$
$450$	0	0
$451$	3.23566e6	0.749069
$452$	1.84541e6	0.424862
$453$	0	0
$454$	−1.28565e6	−0.292741
$455$	2.15074e6	0.487035
$456$	0	0
$457$	−1.81089e6	−0.405604	−0.202802	−	0.979220i	$-0.565005\pi$
$457$	−1.81089e6	−0.405604	−0.202802	+	0.979220i	$0.565005\pi$
$458$	−1.10284e6	−0.245668
$459$	0	0
$460$	324601.	0.0715245
$461$	−7.18697e6	−1.57505	−0.787523	−	0.616285i	$-0.788637\pi$
$461$	−7.18697e6	−1.57505	−0.787523	+	0.616285i	$0.788637\pi$
$462$	0	0
$463$	4.35205e6	0.943498	0.471749	−	0.881733i	$-0.343623\pi$
$463$	4.35205e6	0.943498	0.471749	+	0.881733i	$0.343623\pi$
$464$	−190541.	−0.0410859
$465$	0	0
$466$	4.39043e6	0.936575
$467$	−4.68687e6	−0.994468	−0.497234	−	0.867616i	$-0.665651\pi$
$467$	−4.68687e6	−0.994468	−0.497234	+	0.867616i	$0.665651\pi$
$468$	0	0
$469$	4.82570e6	1.01304
$470$	1.58920e6	0.331843
$471$	0	0
$472$	2.34655e6	0.484813
$473$	1.61569e6	0.332052
$474$	0	0
$475$	1.57327e6	0.319940
$476$	2.45487e6	0.496604
$477$	0	0
$478$	−950506.	−0.190276
$479$	−5.58645e6	−1.11249	−0.556246	−	0.831018i	$-0.687759\pi$
$479$	−5.58645e6	−1.11249	−0.556246	+	0.831018i	$0.687759\pi$
$480$	0	0
$481$	7.13069e6	1.40530
$482$	1.78302e6	0.349573
$483$	0	0
$484$	−1.46726e6	−0.284704
$485$	4.16457e6	0.803926
$486$	0	0
$487$	−1.64830e6	−0.314929	−0.157465	−	0.987525i	$-0.550332\pi$
$487$	−1.64830e6	−0.314929	−0.157465	+	0.987525i	$0.550332\pi$
$488$	384950.	0.0731737
$489$	0	0
$490$	−1.31682e6	−0.247763
$491$	−1.32923e6	−0.248827	−0.124413	−	0.992230i	$-0.539705\pi$
$491$	−1.32923e6	−0.248827	−0.124413	+	0.992230i	$0.539705\pi$
$492$	0	0
$493$	1.25934e6	0.233359
$494$	2.35272e6	0.433764
$495$	0	0
$496$	−468503.	−0.0855083
$497$	−2.37684e6	−0.431627
$498$	0	0
$499$	1.04339e7	1.87585	0.937923	−	0.346844i	$-0.112747\pi$
$499$	1.04339e7	1.87585	0.937923	+	0.346844i	$0.112747\pi$
$500$	2.93259e6	0.524597
$501$	0	0
$502$	−1.56263e6	−0.276755
$503$	3.62876e6	0.639496	0.319748	−	0.947503i	$-0.396402\pi$
$503$	3.62876e6	0.639496	0.319748	+	0.947503i	$0.396402\pi$
$504$	0	0
$505$	−4.47444e6	−0.780748
$506$	−557224.	−0.0967506
$507$	0	0
$508$	−1.68831e6	−0.290264
$509$	1.06098e6	0.181515	0.0907576	−	0.995873i	$-0.471071\pi$
$509$	1.06098e6	0.181515	0.0907576	+	0.995873i	$0.471071\pi$
$510$	0	0
$511$	−1.72629e6	−0.292457
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−275606.	−0.0460130
$515$	137151.	0.0227866
$516$	0	0
$517$	−2.72809e6	−0.448881
$518$	4.18219e6	0.684824
$519$	0	0
$520$	1.51794e6	0.246177
$521$	1.11398e7	1.79798	0.898989	−	0.437972i	$-0.144303\pi$
$521$	1.11398e7	1.79798	0.898989	+	0.437972i	$0.144303\pi$
$522$	0	0
$523$	−1.12593e7	−1.79993	−0.899967	−	0.435958i	$-0.856410\pi$
$523$	−1.12593e7	−1.79993	−0.899967	+	0.435958i	$0.856410\pi$
$524$	2.79305e6	0.444375
$525$	0	0
$526$	−4.80162e6	−0.756700
$527$	3.09647e6	0.485669
$528$	0	0
$529$	279841.	0.0434783
$530$	3.66293e6	0.566420
$531$	0	0
$532$	1.37988e6	0.211380
$533$	−7.59889e6	−1.15860
$534$	0	0
$535$	−307397.	−0.0464318
$536$	3.40586e6	0.512053
$537$	0	0
$538$	1.88583e6	0.280897
$539$	2.26052e6	0.335147
$540$	0	0
$541$	−8.95543e6	−1.31551	−0.657754	−	0.753233i	$-0.728493\pi$
$541$	−8.95543e6	−1.31551	−0.657754	+	0.753233i	$0.728493\pi$
$542$	5.14294e6	0.751992
$543$	0	0
$544$	1.73258e6	0.251013
$545$	−4.48963e6	−0.647469
$546$	0	0
$547$	−2.21483e6	−0.316498	−0.158249	−	0.987399i	$-0.550585\pi$
$547$	−2.21483e6	−0.316498	−0.158249	+	0.987399i	$0.550585\pi$
$548$	2.71815e6	0.386654
$549$	0	0
$550$	−1.74248e6	−0.245618
$551$	707876.	0.0993295
$552$	0	0
$553$	−741139.	−0.103059
$554$	1.80460e6	0.249808
$555$	0	0
$556$	5.69443e6	0.781202
$557$	3.86142e6	0.527362	0.263681	−	0.964610i	$-0.415063\pi$
$557$	3.86142e6	0.527362	0.263681	+	0.964610i	$0.415063\pi$
$558$	0	0
$559$	−3.79442e6	−0.513589
$560$	890281.	0.119966
$561$	0	0
$562$	5.35122e6	0.714681
$563$	5.16169e6	0.686311	0.343156	−	0.939279i	$-0.388504\pi$
$563$	5.16169e6	0.686311	0.343156	+	0.939279i	$0.388504\pi$
$564$	0	0
$565$	−4.42331e6	−0.582943
$566$	7.15805e6	0.939191
$567$	0	0
$568$	−1.67751e6	−0.218170
$569$	7.51466e6	0.973035	0.486518	−	0.873671i	$-0.338267\pi$
$569$	7.51466e6	0.973035	0.486518	+	0.873671i	$0.338267\pi$
$570$	0	0
$571$	2.36883e6	0.304049	0.152024	−	0.988377i	$-0.451421\pi$
$571$	2.36883e6	0.304049	0.152024	+	0.988377i	$0.451421\pi$
$572$	−2.60577e6	−0.333001
$573$	0	0
$574$	−4.45679e6	−0.564602
$575$	875082.	0.110377
$576$	0	0
$577$	826968.	0.103407	0.0517034	−	0.998662i	$-0.483535\pi$
$577$	826968.	0.103407	0.0517034	+	0.998662i	$0.483535\pi$
$578$	−5.77171e6	−0.718596
$579$	0	0
$580$	456711.	0.0563731
$581$	−7.04492e6	−0.865836
$582$	0	0
$583$	−6.28795e6	−0.766192
$584$	−1.21837e6	−0.147825
$585$	0	0
$586$	−5.26593e6	−0.633477
$587$	−4.79086e6	−0.573876	−0.286938	−	0.957949i	$-0.592637\pi$
$587$	−4.79086e6	−0.573876	−0.286938	+	0.957949i	$0.592637\pi$
$588$	0	0
$589$	1.74053e6	0.206725
$590$	−5.62449e6	−0.665201
$591$	0	0
$592$	2.95168e6	0.346151
$593$	5.86529e6	0.684940	0.342470	−	0.939529i	$-0.388736\pi$
$593$	5.86529e6	0.684940	0.342470	+	0.939529i	$0.388736\pi$
$594$	0	0
$595$	−5.88412e6	−0.681380
$596$	7.12200e6	0.821271
$597$	0	0
$598$	1.30863e6	0.149646
$599$	−1.19832e7	−1.36460	−0.682300	−	0.731072i	$-0.739020\pi$
$599$	−1.19832e7	−1.36460	−0.682300	+	0.731072i	$0.739020\pi$
$600$	0	0
$601$	7.67163e6	0.866366	0.433183	−	0.901306i	$-0.357390\pi$
$601$	7.67163e6	0.866366	0.433183	+	0.901306i	$0.357390\pi$
$602$	−2.22545e6	−0.250280
$603$	0	0
$604$	476082.	0.0530994
$605$	3.51691e6	0.390637
$606$	0	0
$607$	1.16866e7	1.28741	0.643704	−	0.765274i	$-0.277397\pi$
$607$	1.16866e7	1.28741	0.643704	+	0.765274i	$0.277397\pi$
$608$	973889.	0.106844
$609$	0	0
$610$	−922695.	−0.100400
$611$	6.40686e6	0.694292
$612$	0	0
$613$	8.66668e6	0.931541	0.465770	−	0.884906i	$-0.345777\pi$
$613$	8.66668e6	0.931541	0.465770	+	0.884906i	$0.345777\pi$
$614$	−4.09997e6	−0.438894
$615$	0	0
$616$	−1.52830e6	−0.162277
$617$	−9.96878e6	−1.05421	−0.527107	−	0.849799i	$-0.676723\pi$
$617$	−9.96878e6	−1.05421	−0.527107	+	0.849799i	$0.676723\pi$
$618$	0	0
$619$	−8.53733e6	−0.895562	−0.447781	−	0.894143i	$-0.647786\pi$
$619$	−8.53733e6	−0.895562	−0.447781	+	0.894143i	$0.647786\pi$
$620$	1.12297e6	0.117324
$621$	0	0
$622$	3.31008e6	0.343054
$623$	−4.55529e6	−0.470214
$624$	0	0
$625$	−1.85975e6	−0.190438
$626$	−1.07439e7	−1.09579
$627$	0	0
$628$	7.22647e6	0.731184
$629$	−1.95085e7	−1.96606
$630$	0	0
$631$	−811698.	−0.0811561	−0.0405781	−	0.999176i	$-0.512920\pi$
$631$	−811698.	−0.0811561	−0.0405781	+	0.999176i	$0.512920\pi$
$632$	−523078.	−0.0520923
$633$	0	0
$634$	5.04332e6	0.498303
$635$	4.04675e6	0.398265
$636$	0	0
$637$	−5.30878e6	−0.518377
$638$	−784011.	−0.0762554
$639$	0	0
$640$	628339.	0.0606379
$641$	1.28227e7	1.23264	0.616318	−	0.787497i	$-0.288624\pi$
$641$	1.28227e7	1.23264	0.616318	+	0.787497i	$0.288624\pi$
$642$	0	0
$643$	−1.03538e7	−0.987578	−0.493789	−	0.869582i	$-0.664389\pi$
$643$	−1.03538e7	−0.987578	−0.493789	+	0.869582i	$0.664389\pi$
$644$	767519.	0.0729246
$645$	0	0
$646$	−6.43671e6	−0.606851
$647$	−1.46554e7	−1.37637	−0.688186	−	0.725534i	$-0.741593\pi$
$647$	−1.46554e7	−1.37637	−0.688186	+	0.725534i	$0.741593\pi$
$648$	0	0
$649$	9.65525e6	0.899812
$650$	4.09218e6	0.379902
$651$	0	0
$652$	4.55813e6	0.419921
$653$	−1.13745e7	−1.04388	−0.521939	−	0.852983i	$-0.674791\pi$
$653$	−1.13745e7	−1.04388	−0.521939	+	0.852983i	$0.674791\pi$
$654$	0	0
$655$	−6.69472e6	−0.609718
$656$	−3.14549e6	−0.285384
$657$	0	0
$658$	3.75766e6	0.338339
$659$	−7.89393e6	−0.708076	−0.354038	−	0.935231i	$-0.615192\pi$
$659$	−7.89393e6	−0.708076	−0.354038	+	0.935231i	$0.615192\pi$
$660$	0	0
$661$	1.06805e7	0.950795	0.475398	−	0.879771i	$-0.342304\pi$
$661$	1.06805e7	0.950795	0.475398	+	0.879771i	$0.342304\pi$
$662$	7.80485e6	0.692181
$663$	0	0
$664$	−4.97213e6	−0.437645
$665$	−3.30748e6	−0.290030
$666$	0	0
$667$	393735.	0.0342680
$668$	3.81252e6	0.330576
$669$	0	0
$670$	−8.16358e6	−0.702577
$671$	1.58394e6	0.135810
$672$	0	0
$673$	1.71584e6	0.146029	0.0730147	−	0.997331i	$-0.476738\pi$
$673$	1.71584e6	0.146029	0.0730147	+	0.997331i	$0.476738\pi$
$674$	7.02964e6	0.596051
$675$	0	0
$676$	178910.	0.0150580
$677$	2.51590e6	0.210970	0.105485	−	0.994421i	$-0.466360\pi$
$677$	2.51590e6	0.210970	0.105485	+	0.994421i	$0.466360\pi$
$678$	0	0
$679$	9.84713e6	0.819663
$680$	−4.15287e6	−0.344410
$681$	0	0
$682$	−1.92773e6	−0.158703
$683$	2.09991e7	1.72246	0.861230	−	0.508216i	$-0.169695\pi$
$683$	2.09991e7	1.72246	0.861230	+	0.508216i	$0.169695\pi$
$684$	0	0
$685$	−6.51521e6	−0.530520
$686$	−9.20989e6	−0.747212
$687$	0	0
$688$	−1.57067e6	−0.126507
$689$	1.47671e7	1.18508
$690$	0	0
$691$	−1.94678e7	−1.55104	−0.775519	−	0.631324i	$-0.782512\pi$
$691$	−1.94678e7	−1.55104	−0.775519	+	0.631324i	$0.782512\pi$
$692$	−8.96722e6	−0.711857
$693$	0	0
$694$	1.57359e7	1.24021
$695$	−1.36491e7	−1.07187
$696$	0	0
$697$	2.07895e7	1.62092
$698$	−1.37481e7	−1.06808
$699$	0	0
$700$	2.40008e6	0.185132
$701$	2.14238e7	1.64665	0.823324	−	0.567571i	$-0.192117\pi$
$701$	2.14238e7	1.64665	0.823324	+	0.567571i	$0.192117\pi$
$702$	0	0
$703$	−1.09658e7	−0.836857
$704$	−1.07863e6	−0.0820243
$705$	0	0
$706$	1.43783e7	1.08566
$707$	−1.05798e7	−0.796031
$708$	0	0
$709$	1.03212e7	0.771106	0.385553	−	0.922686i	$-0.374011\pi$
$709$	1.03212e7	0.771106	0.385553	+	0.922686i	$0.374011\pi$
$710$	4.02087e6	0.299347
$711$	0	0
$712$	−3.21501e6	−0.237674
$713$	968118.	0.0713189
$714$	0	0
$715$	6.24582e6	0.456904
$716$	−3.62812e6	−0.264484
$717$	0	0
$718$	672640.	0.0486935
$719$	−1.36680e7	−0.986013	−0.493006	−	0.870026i	$-0.664102\pi$
$719$	−1.36680e7	−0.986013	−0.493006	+	0.870026i	$0.664102\pi$
$720$	0	0
$721$	324293.	0.0232327
$722$	6.28631e6	0.448800
$723$	0	0
$724$	1.08213e7	0.767244
$725$	1.23124e6	0.0869954
$726$	0	0
$727$	9.37509e6	0.657869	0.328934	−	0.944353i	$-0.393310\pi$
$727$	9.37509e6	0.657869	0.328934	+	0.944353i	$0.393310\pi$
$728$	3.58918e6	0.250996
$729$	0	0
$730$	2.92035e6	0.202828
$731$	1.03810e7	0.718530
$732$	0	0
$733$	−9.51784e6	−0.654303	−0.327151	−	0.944972i	$-0.606089\pi$
$733$	−9.51784e6	−0.654303	−0.327151	+	0.944972i	$0.606089\pi$
$734$	1.87422e7	1.28405
$735$	0	0
$736$	541696.	0.0368605
$737$	1.40140e7	0.950370
$738$	0	0
$739$	1.82428e7	1.22880	0.614400	−	0.788995i	$-0.289398\pi$
$739$	1.82428e7	1.22880	0.614400	+	0.788995i	$0.289398\pi$
$740$	−7.07496e6	−0.474946
$741$	0	0
$742$	8.66100e6	0.577508
$743$	−9.74489e6	−0.647597	−0.323798	−	0.946126i	$-0.604960\pi$
$743$	−9.74489e6	−0.647597	−0.323798	+	0.946126i	$0.604960\pi$
$744$	0	0
$745$	−1.70709e7	−1.12685
$746$	−2.08264e6	−0.137015
$747$	0	0
$748$	7.12900e6	0.465881
$749$	−726841.	−0.0473407
$750$	0	0
$751$	9.23122e6	0.597254	0.298627	−	0.954370i	$-0.403471\pi$
$751$	9.23122e6	0.597254	0.298627	+	0.954370i	$0.403471\pi$
$752$	2.65206e6	0.171017
$753$	0	0
$754$	1.84124e6	0.117945
$755$	−1.14113e6	−0.0728565
$756$	0	0
$757$	−1.67564e7	−1.06277	−0.531387	−	0.847129i	$-0.678329\pi$
$757$	−1.67564e7	−1.06277	−0.531387	+	0.847129i	$0.678329\pi$
$758$	−5.01581e6	−0.317080
$759$	0	0
$760$	−2.33434e6	−0.146598
$761$	2.18267e7	1.36624	0.683120	−	0.730306i	$-0.260622\pi$
$761$	2.18267e7	1.36624	0.683120	+	0.730306i	$0.260622\pi$
$762$	0	0
$763$	−1.06157e7	−0.660144
$764$	3.25120e6	0.201516
$765$	0	0
$766$	−3.13965e6	−0.193334
$767$	−2.26752e7	−1.39175
$768$	0	0
$769$	7.63358e6	0.465492	0.232746	−	0.972538i	$-0.425229\pi$
$769$	7.63358e6	0.465492	0.232746	+	0.972538i	$0.425229\pi$
$770$	3.66321e6	0.222656
$771$	0	0
$772$	1.13780e6	0.0687105
$773$	−1.55360e7	−0.935168	−0.467584	−	0.883949i	$-0.654875\pi$
$773$	−1.55360e7	−0.935168	−0.467584	+	0.883949i	$0.654875\pi$
$774$	0	0
$775$	3.02737e6	0.181055
$776$	6.94986e6	0.414307
$777$	0	0
$778$	162802.	0.00964299
$779$	1.16858e7	0.689945
$780$	0	0
$781$	−6.90241e6	−0.404924
$782$	−3.58022e6	−0.209360
$783$	0	0
$784$	−2.19752e6	−0.127686
$785$	−1.73213e7	−1.00324
$786$	0	0
$787$	−2.73480e7	−1.57394	−0.786970	−	0.616992i	$-0.788351\pi$
$787$	−2.73480e7	−1.57394	−0.786970	+	0.616992i	$0.788351\pi$
$788$	−1.21138e7	−0.694971
$789$	0	0
$790$	1.25378e6	0.0714748
$791$	−1.04589e7	−0.594355
$792$	0	0
$793$	−3.71985e6	−0.210060
$794$	−1.81808e7	−1.02344
$795$	0	0
$796$	1.18559e7	0.663210
$797$	−5.37984e6	−0.300001	−0.150001	−	0.988686i	$-0.547928\pi$
$797$	−5.37984e6	−0.300001	−0.150001	+	0.988686i	$0.547928\pi$
$798$	0	0
$799$	−1.75282e7	−0.971339
$800$	1.69392e6	0.0935768
$801$	0	0
$802$	−6.64383e6	−0.364739
$803$	−5.01320e6	−0.274363
$804$	0	0
$805$	−1.83968e6	−0.100058
$806$	4.52725e6	0.245469
$807$	0	0
$808$	−7.46698e6	−0.402362
$809$	1.96864e7	1.05754	0.528769	−	0.848766i	$-0.322654\pi$
$809$	1.96864e7	1.05754	0.528769	+	0.848766i	$0.322654\pi$
$810$	0	0
$811$	−2.18016e7	−1.16395	−0.581977	−	0.813205i	$-0.697721\pi$
$811$	−2.18016e7	−1.16395	−0.581977	+	0.813205i	$0.697721\pi$
$812$	1.07989e6	0.0574766
$813$	0	0
$814$	1.21452e7	0.642456
$815$	−1.09255e7	−0.576165
$816$	0	0
$817$	5.83517e6	0.305843
$818$	−5.66265e6	−0.295894
$819$	0	0
$820$	7.53950e6	0.391569
$821$	2.45416e7	1.27071	0.635354	−	0.772221i	$-0.280854\pi$
$821$	2.45416e7	1.27071	0.635354	+	0.772221i	$0.280854\pi$
$822$	0	0
$823$	−1.37439e7	−0.707313	−0.353656	−	0.935375i	$-0.615062\pi$
$823$	−1.37439e7	−0.707313	−0.353656	+	0.935375i	$0.615062\pi$
$824$	228878.	0.0117432
$825$	0	0
$826$	−1.32991e7	−0.678223
$827$	−9.03752e6	−0.459500	−0.229750	−	0.973250i	$-0.573791\pi$
$827$	−9.03752e6	−0.459500	−0.229750	+	0.973250i	$0.573791\pi$
$828$	0	0
$829$	−7.76874e6	−0.392613	−0.196306	−	0.980543i	$-0.562895\pi$
$829$	−7.76874e6	−0.392613	−0.196306	+	0.980543i	$0.562895\pi$
$830$	1.19178e7	0.600484
$831$	0	0
$832$	2.53315e6	0.126868
$833$	1.45240e7	0.725229
$834$	0	0
$835$	−9.13832e6	−0.453576
$836$	4.00722e6	0.198302
$837$	0	0
$838$	−1.59558e7	−0.784888
$839$	4.65701e6	0.228403	0.114202	−	0.993458i	$-0.463569\pi$
$839$	4.65701e6	0.228403	0.114202	+	0.993458i	$0.463569\pi$
$840$	0	0
$841$	−1.99572e7	−0.972991
$842$	−5.49815e6	−0.267261
$843$	0	0
$844$	140938.	0.00681040
$845$	−428833.	−0.0206608
$846$	0	0
$847$	8.31574e6	0.398283
$848$	6.11272e6	0.291907
$849$	0	0
$850$	−1.11956e7	−0.531496
$851$	−6.09938e6	−0.288710
$852$	0	0
$853$	−1.99915e7	−0.940749	−0.470374	−	0.882467i	$-0.655881\pi$
$853$	−1.99915e7	−0.940749	−0.470374	+	0.882467i	$0.655881\pi$
$854$	−2.18171e6	−0.102365
$855$	0	0
$856$	−512987.	−0.0239288
$857$	−2.56405e7	−1.19254	−0.596272	−	0.802783i	$-0.703352\pi$
$857$	−2.56405e7	−1.19254	−0.596272	+	0.802783i	$0.703352\pi$
$858$	0	0
$859$	−4.01488e6	−0.185648	−0.0928239	−	0.995683i	$-0.529589\pi$
$859$	−4.01488e6	−0.185648	−0.0928239	+	0.995683i	$0.529589\pi$
$860$	3.76477e6	0.173577
$861$	0	0
$862$	2.92064e6	0.133878
$863$	1.22340e7	0.559165	0.279583	−	0.960122i	$-0.409804\pi$
$863$	1.22340e7	0.559165	0.279583	+	0.960122i	$0.409804\pi$
$864$	0	0
$865$	2.14937e7	0.976723
$866$	−6.53139e6	−0.295945
$867$	0	0
$868$	2.65525e6	0.119621
$869$	−2.15229e6	−0.0966833
$870$	0	0
$871$	−3.29116e7	−1.46995
$872$	−7.49232e6	−0.333676
$873$	0	0
$874$	−2.01245e6	−0.0891141
$875$	−1.66205e7	−0.733878
$876$	0	0
$877$	4.43307e7	1.94628	0.973141	−	0.230211i	$-0.0739416\pi$
$877$	4.43307e7	1.94628	0.973141	+	0.230211i	$0.0739416\pi$
$878$	2.75102e7	1.20436
$879$	0	0
$880$	2.58540e6	0.112544
$881$	−2.12701e7	−0.923273	−0.461637	−	0.887069i	$-0.652738\pi$
$881$	−2.12701e7	−0.923273	−0.461637	+	0.887069i	$0.652738\pi$
$882$	0	0
$883$	5.97566e6	0.257919	0.128960	−	0.991650i	$-0.458836\pi$
$883$	5.97566e6	0.257919	0.128960	+	0.991650i	$0.458836\pi$
$884$	−1.67423e7	−0.720585
$885$	0	0
$886$	−2.02085e7	−0.864870
$887$	3.34281e7	1.42660	0.713301	−	0.700858i	$-0.247199\pi$
$887$	3.34281e7	1.42660	0.713301	+	0.700858i	$0.247199\pi$
$888$	0	0
$889$	9.56854e6	0.406061
$890$	7.70613e6	0.326108
$891$	0	0
$892$	−1.18987e7	−0.500711
$893$	−9.85265e6	−0.413451
$894$	0	0
$895$	8.69633e6	0.362893
$896$	1.48571e6	0.0618249
$897$	0	0
$898$	−2.48642e7	−1.02892
$899$	1.36214e6	0.0562110
$900$	0	0
$901$	−4.04007e7	−1.65797
$902$	−1.29426e7	−0.529672
$903$	0	0
$904$	−7.38165e6	−0.300423
$905$	−2.59379e7	−1.05272
$906$	0	0
$907$	4.03668e7	1.62932	0.814660	−	0.579938i	$-0.196923\pi$
$907$	4.03668e7	1.62932	0.814660	+	0.579938i	$0.196923\pi$
$908$	5.14261e6	0.206999
$909$	0	0
$910$	−8.60298e6	−0.344386
$911$	−2.04754e7	−0.817404	−0.408702	−	0.912668i	$-0.634019\pi$
$911$	−2.04754e7	−0.817404	−0.408702	+	0.912668i	$0.634019\pi$
$912$	0	0
$913$	−2.04587e7	−0.812269
$914$	7.24357e6	0.286805
$915$	0	0
$916$	4.41136e6	0.173714
$917$	−1.58297e7	−0.621653
$918$	0	0
$919$	4.86459e7	1.90002	0.950008	−	0.312225i	$-0.101074\pi$
$919$	4.86459e7	1.90002	0.950008	+	0.312225i	$0.101074\pi$
$920$	−1.29840e6	−0.0505755
$921$	0	0
$922$	2.87479e7	1.11373
$923$	1.62102e7	0.626302
$924$	0	0
$925$	−1.90732e7	−0.732941
$926$	−1.74082e7	−0.667154
$927$	0	0
$928$	762163.	0.0290521
$929$	3.18405e7	1.21043	0.605215	−	0.796062i	$-0.293087\pi$
$929$	3.18405e7	1.21043	0.605215	+	0.796062i	$0.293087\pi$
$930$	0	0
$931$	8.16399e6	0.308694
$932$	−1.75617e7	−0.662258
$933$	0	0
$934$	1.87475e7	0.703195
$935$	−1.70877e7	−0.639225
$936$	0	0
$937$	1.15061e7	0.428135	0.214067	−	0.976819i	$-0.431329\pi$
$937$	1.15061e7	0.428135	0.214067	+	0.976819i	$0.431329\pi$
$938$	−1.93028e7	−0.716330
$939$	0	0
$940$	−6.35679e6	−0.234649
$941$	3.32759e7	1.22505	0.612527	−	0.790449i	$-0.290153\pi$
$941$	3.32759e7	1.22505	0.612527	+	0.790449i	$0.290153\pi$
$942$	0	0
$943$	6.49987e6	0.238026
$944$	−9.38618e6	−0.342814
$945$	0	0
$946$	−6.46277e6	−0.234796
$947$	−4.15829e7	−1.50674	−0.753372	−	0.657595i	$-0.771574\pi$
$947$	−4.15829e7	−1.50674	−0.753372	+	0.657595i	$0.771574\pi$
$948$	0	0
$949$	1.17734e7	0.424362
$950$	−6.29307e6	−0.226232
$951$	0	0
$952$	−9.81946e6	−0.351152
$953$	6.68573e6	0.238460	0.119230	−	0.992867i	$-0.461957\pi$
$953$	6.68573e6	0.238460	0.119230	+	0.992867i	$0.461957\pi$
$954$	0	0
$955$	−7.79288e6	−0.276496
$956$	3.80202e6	0.134546
$957$	0	0
$958$	2.23458e7	0.786651
$959$	−1.54052e7	−0.540905
$960$	0	0
$961$	−2.52799e7	−0.883013
$962$	−2.85228e7	−0.993697
$963$	0	0
$964$	−7.13207e6	−0.247186
$965$	−2.72722e6	−0.0942762
$966$	0	0
$967$	−9.17569e6	−0.315553	−0.157777	−	0.987475i	$-0.550433\pi$
$967$	−9.17569e6	−0.315553	−0.157777	+	0.987475i	$0.550433\pi$
$968$	5.86905e6	0.201316
$969$	0	0
$970$	−1.66583e7	−0.568461
$971$	3.26579e7	1.11158	0.555788	−	0.831324i	$-0.312416\pi$
$971$	3.26579e7	1.11158	0.555788	+	0.831324i	$0.312416\pi$
$972$	0	0
$973$	−3.22733e7	−1.09285
$974$	6.59319e6	0.222689
$975$	0	0
$976$	−1.53980e6	−0.0517416
$977$	−2.84132e7	−0.952321	−0.476160	−	0.879358i	$-0.657972\pi$
$977$	−2.84132e7	−0.952321	−0.476160	+	0.879358i	$0.657972\pi$
$978$	0	0
$979$	−1.32287e7	−0.441123
$980$	5.26729e6	0.175195
$981$	0	0
$982$	5.31693e6	0.175947
$983$	6.68115e6	0.220530	0.110265	−	0.993902i	$-0.464830\pi$
$983$	6.68115e6	0.220530	0.110265	+	0.993902i	$0.464830\pi$
$984$	0	0
$985$	2.90360e7	0.953555
$986$	−5.03735e6	−0.165010
$987$	0	0
$988$	−9.41089e6	−0.306717
$989$	3.24564e6	0.105514
$990$	0	0
$991$	3.99543e7	1.29235	0.646174	−	0.763190i	$-0.276368\pi$
$991$	3.99543e7	1.29235	0.646174	+	0.763190i	$0.276368\pi$
$992$	1.87401e6	0.0604635
$993$	0	0
$994$	9.50735e6	0.305206
$995$	−2.84176e7	−0.909976
$996$	0	0
$997$	5.53944e7	1.76493	0.882466	−	0.470375i	$-0.155882\pi$
$997$	5.53944e7	1.76493	0.882466	+	0.470375i	$0.155882\pi$
$998$	−4.17358e7	−1.32642
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.6.a.o.1.2		4
3.2	odd	2		138.6.a.i.1.3	✓	4
12.11	even	2		1104.6.a.m.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.6.a.i.1.3	✓	4	3.2	odd	2
414.6.a.o.1.2		4	1.1	even	1	trivial
1104.6.a.m.1.3		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} -38.3508 q^{5} -90.6804 q^{7} -64.0000 q^{8} +O(q^{10})\)	\(q-4.00000 q^{2} +16.0000 q^{4} -38.3508 q^{5} -90.6804 q^{7} -64.0000 q^{8} +153.403 q^{10} -263.339 q^{11} +618.446 q^{13} +362.722 q^{14} +256.000 q^{16} -1691.98 q^{17} -951.063 q^{19} -613.612 q^{20} +1053.35 q^{22} -529.000 q^{23} -1654.22 q^{25} -2473.78 q^{26} -1450.89 q^{28} -744.300 q^{29} -1830.09 q^{31} -1024.00 q^{32} +6767.91 q^{34} +3477.66 q^{35} +11530.0 q^{37} +3804.25 q^{38} +2454.45 q^{40} -12287.1 q^{41} -6135.42 q^{43} -4213.42 q^{44} +2116.00 q^{46} +10359.6 q^{47} -8584.07 q^{49} +6616.88 q^{50} +9895.13 q^{52} +23877.8 q^{53} +10099.2 q^{55} +5803.54 q^{56} +2977.20 q^{58} -36664.8 q^{59} -6014.85 q^{61} +7320.36 q^{62} +4096.00 q^{64} -23717.9 q^{65} -53216.6 q^{67} -27071.6 q^{68} -13910.6 q^{70} +26211.2 q^{71} +19037.1 q^{73} -46120.1 q^{74} -15217.0 q^{76} +23879.6 q^{77} +8173.10 q^{79} -9817.79 q^{80} +49148.3 q^{82} +77689.6 q^{83} +64888.6 q^{85} +24541.7 q^{86} +16853.7 q^{88} +50234.5 q^{89} -56080.9 q^{91} -8464.00 q^{92} -41438.5 q^{94} +36474.0 q^{95} -108592. q^{97} +34336.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8}+O(q^{10}) \) 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8	\( 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8} + 216 q^{10} + 6 q^{11} + 1248 q^{13} - 1392 q^{14} + 1024 q^{16} + 56 q^{17} + 1530 q^{19} - 864 q^{20} - 24 q^{22} - 2116 q^{23} + 8852 q^{25} - 4992 q^{26} + 5568 q^{28} - 2292 q^{29} + 1556 q^{31} - 4096 q^{32} - 224 q^{34} - 5688 q^{35} - 9586 q^{37} - 6120 q^{38} + 3456 q^{40} + 11768 q^{41} + 13758 q^{43} + 96 q^{44} + 8464 q^{46} - 10636 q^{47} + 11360 q^{49} - 35408 q^{50} + 19968 q^{52} - 26686 q^{53} - 28900 q^{55} - 22272 q^{56} + 9168 q^{58} + 9108 q^{59} - 37878 q^{61} - 6224 q^{62} + 16384 q^{64} + 72212 q^{65} - 23302 q^{67} + 896 q^{68} + 22752 q^{70} - 31728 q^{71} - 28340 q^{73} + 38344 q^{74} + 24480 q^{76} + 121276 q^{77} - 26668 q^{79} - 13824 q^{80} - 47072 q^{82} + 119026 q^{83} - 217876 q^{85} - 55032 q^{86} - 384 q^{88} + 148236 q^{89} + 89008 q^{91} - 33856 q^{92} + 42544 q^{94} + 399292 q^{95} - 16092 q^{97} - 45440 q^{98}+O(q^{100}) \) 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8 + 216 * q^10 + 6 * q^11 + 1248 * q^13 - 1392 * q^14 + 1024 * q^16 + 56 * q^17 + 1530 * q^19 - 864 * q^20 - 24 * q^22 - 2116 * q^23 + 8852 * q^25 - 4992 * q^26 + 5568 * q^28 - 2292 * q^29 + 1556 * q^31 - 4096 * q^32 - 224 * q^34 - 5688 * q^35 - 9586 * q^37 - 6120 * q^38 + 3456 * q^40 + 11768 * q^41 + 13758 * q^43 + 96 * q^44 + 8464 * q^46 - 10636 * q^47 + 11360 * q^49 - 35408 * q^50 + 19968 * q^52 - 26686 * q^53 - 28900 * q^55 - 22272 * q^56 + 9168 * q^58 + 9108 * q^59 - 37878 * q^61 - 6224 * q^62 + 16384 * q^64 + 72212 * q^65 - 23302 * q^67 + 896 * q^68 + 22752 * q^70 - 31728 * q^71 - 28340 * q^73 + 38344 * q^74 + 24480 * q^76 + 121276 * q^77 - 26668 * q^79 - 13824 * q^80 - 47072 * q^82 + 119026 * q^83 - 217876 * q^85 - 55032 * q^86 - 384 * q^88 + 148236 * q^89 + 89008 * q^91 - 33856 * q^92 + 42544 * q^94 + 399292 * q^95 - 16092 * q^97 - 45440 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more