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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [245,6,Mod(99,245)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("245.99"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-24,90] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.2940358542\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(0.500000 - 1.65831i\) of defining polynomial
Character \(\chi\) \(=\) 245.99
Dual form 245.6.b.a.99.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.63325i q^{2} +19.8997i q^{3} -12.0000 q^{4} +(45.0000 - 33.1662i) q^{5} -132.000 q^{6} +132.665i q^{8} -153.000 q^{9} +(220.000 + 298.496i) q^{10} +252.000 q^{11} -238.797i q^{12} +119.398i q^{13} +(660.000 + 895.489i) q^{15} -1264.00 q^{16} +689.858i q^{17} -1014.89i q^{18} +220.000 q^{19} +(-540.000 + 397.995i) q^{20} +1671.58i q^{22} +2434.40i q^{23} -2640.00 q^{24} +(925.000 - 2984.96i) q^{25} -792.000 q^{26} +1790.98i q^{27} -6930.00 q^{29} +(-5940.00 + 4377.94i) q^{30} -6752.00 q^{31} -4139.15i q^{32} +5014.74i q^{33} -4576.00 q^{34} +1836.00 q^{36} +13969.6i q^{37} +1459.31i q^{38} -2376.00 q^{39} +(4400.00 + 5969.92i) q^{40} +198.000 q^{41} -417.895i q^{43} -3024.00 q^{44} +(-6885.00 + 5074.44i) q^{45} -16148.0 q^{46} +10540.2i q^{47} -25153.3i q^{48} +(19800.0 + 6135.76i) q^{50} -13728.0 q^{51} -1432.78i q^{52} -5823.99i q^{53} -11880.0 q^{54} +(11340.0 - 8357.89i) q^{55} +4377.94i q^{57} -45968.4i q^{58} +24660.0 q^{59} +(-7920.00 - 10745.9i) q^{60} +5698.00 q^{61} -44787.7i q^{62} -12992.0 q^{64} +(3960.00 + 5372.93i) q^{65} -33264.0 q^{66} -43640.1i q^{67} -8278.30i q^{68} -48444.0 q^{69} +53352.0 q^{71} -20297.7i q^{72} -70922.7i q^{73} -92664.0 q^{74} +(59400.0 + 18407.3i) q^{75} -2640.00 q^{76} -15760.6i q^{78} +51920.0 q^{79} +(-56880.0 + 41922.1i) q^{80} -72819.0 q^{81} +1313.38i q^{82} +61841.8i q^{83} +(22880.0 + 31043.6i) q^{85} +2772.00 q^{86} -137905. i q^{87} +33431.6i q^{88} +9990.00 q^{89} +(-33660.0 - 45669.9i) q^{90} -29212.8i q^{92} -134363. i q^{93} -69916.0 q^{94} +(9900.00 - 7296.57i) q^{95} +82368.0 q^{96} +101250. i q^{97} -38556.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{4} + 90 q^{5} - 264 q^{6} - 306 q^{9} + 440 q^{10} + 504 q^{11} + 1320 q^{15} - 2528 q^{16} + 440 q^{19} - 1080 q^{20} - 5280 q^{24} + 1850 q^{25} - 1584 q^{26} - 13860 q^{29} - 11880 q^{30}+ \cdots - 77112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.63325i 1.17260i 0.810093 + 0.586302i \(0.199417\pi\)
−0.810093 + 0.586302i \(0.800583\pi\)
\(3\) 19.8997i 1.27657i 0.769800 + 0.638285i \(0.220356\pi\)
−0.769800 + 0.638285i \(0.779644\pi\)
\(4\) −12.0000 −0.375000
\(5\) 45.0000 − 33.1662i 0.804984 − 0.593296i
\(6\) −132.000 −1.49691
\(7\) 0 0
\(8\) 132.665i 0.732877i
\(9\) −153.000 −0.629630
\(10\) 220.000 + 298.496i 0.695701 + 0.943928i
\(11\) 252.000 0.627941 0.313970 − 0.949433i \(-0.398341\pi\)
0.313970 + 0.949433i \(0.398341\pi\)
\(12\) − 238.797i − 0.478714i
\(13\) 119.398i 0.195948i 0.995189 + 0.0979739i \(0.0312362\pi\)
−0.995189 + 0.0979739i \(0.968764\pi\)
\(14\) 0 0
\(15\) 660.000 + 895.489i 0.757383 + 1.02762i
\(16\) −1264.00 −1.23438
\(17\) 689.858i 0.578945i 0.957186 + 0.289473i \(0.0934799\pi\)
−0.957186 + 0.289473i \(0.906520\pi\)
\(18\) − 1014.89i − 0.738306i
\(19\) 220.000 0.139810 0.0699051 − 0.997554i \(-0.477730\pi\)
0.0699051 + 0.997554i \(0.477730\pi\)
\(20\) −540.000 + 397.995i −0.301869 + 0.222486i
\(21\) 0 0
\(22\) 1671.58i 0.736326i
\(23\) 2434.40i 0.959561i 0.877388 + 0.479781i \(0.159284\pi\)
−0.877388 + 0.479781i \(0.840716\pi\)
\(24\) −2640.00 −0.935569
\(25\) 925.000 − 2984.96i 0.296000 − 0.955188i
\(26\) −792.000 −0.229769
\(27\) 1790.98i 0.472804i
\(28\) 0 0
\(29\) −6930.00 −1.53016 −0.765082 − 0.643932i \(-0.777302\pi\)
−0.765082 + 0.643932i \(0.777302\pi\)
\(30\) −5940.00 + 4377.94i −1.20499 + 0.888111i
\(31\) −6752.00 −1.26191 −0.630955 − 0.775820i \(-0.717337\pi\)
−0.630955 + 0.775820i \(0.717337\pi\)
\(32\) − 4139.15i − 0.714556i
\(33\) 5014.74i 0.801610i
\(34\) −4576.00 −0.678873
\(35\) 0 0
\(36\) 1836.00 0.236111
\(37\) 13969.6i 1.67757i 0.544464 + 0.838785i \(0.316733\pi\)
−0.544464 + 0.838785i \(0.683267\pi\)
\(38\) 1459.31i 0.163942i
\(39\) −2376.00 −0.250141
\(40\) 4400.00 + 5969.92i 0.434813 + 0.589955i
\(41\) 198.000 0.0183952 0.00919762 − 0.999958i \(-0.497072\pi\)
0.00919762 + 0.999958i \(0.497072\pi\)
\(42\) 0 0
\(43\) − 417.895i − 0.0344664i −0.999851 − 0.0172332i \(-0.994514\pi\)
0.999851 − 0.0172332i \(-0.00548577\pi\)
\(44\) −3024.00 −0.235478
\(45\) −6885.00 + 5074.44i −0.506842 + 0.373557i
\(46\) −16148.0 −1.12519
\(47\) 10540.2i 0.695994i 0.937496 + 0.347997i \(0.113138\pi\)
−0.937496 + 0.347997i \(0.886862\pi\)
\(48\) − 25153.3i − 1.57577i
\(49\) 0 0
\(50\) 19800.0 + 6135.76i 1.12006 + 0.347091i
\(51\) −13728.0 −0.739064
\(52\) − 1432.78i − 0.0734804i
\(53\) − 5823.99i − 0.284794i −0.989810 − 0.142397i \(-0.954519\pi\)
0.989810 − 0.142397i \(-0.0454810\pi\)
\(54\) −11880.0 −0.554411
\(55\) 11340.0 − 8357.89i 0.505483 − 0.372555i
\(56\) 0 0
\(57\) 4377.94i 0.178477i
\(58\) − 45968.4i − 1.79428i
\(59\) 24660.0 0.922281 0.461140 − 0.887327i \(-0.347440\pi\)
0.461140 + 0.887327i \(0.347440\pi\)
\(60\) −7920.00 − 10745.9i −0.284019 − 0.385357i
\(61\) 5698.00 0.196064 0.0980320 − 0.995183i \(-0.468745\pi\)
0.0980320 + 0.995183i \(0.468745\pi\)
\(62\) − 44787.7i − 1.47972i
\(63\) 0 0
\(64\) −12992.0 −0.396484
\(65\) 3960.00 + 5372.93i 0.116255 + 0.157735i
\(66\) −33264.0 −0.939971
\(67\) − 43640.1i − 1.18768i −0.804583 − 0.593840i \(-0.797611\pi\)
0.804583 − 0.593840i \(-0.202389\pi\)
\(68\) − 8278.30i − 0.217104i
\(69\) −48444.0 −1.22495
\(70\) 0 0
\(71\) 53352.0 1.25604 0.628022 − 0.778196i \(-0.283865\pi\)
0.628022 + 0.778196i \(0.283865\pi\)
\(72\) − 20297.7i − 0.461441i
\(73\) − 70922.7i − 1.55768i −0.627223 − 0.778840i \(-0.715808\pi\)
0.627223 − 0.778840i \(-0.284192\pi\)
\(74\) −92664.0 −1.96712
\(75\) 59400.0 + 18407.3i 1.21936 + 0.377865i
\(76\) −2640.00 −0.0524288
\(77\) 0 0
\(78\) − 15760.6i − 0.293316i
\(79\) 51920.0 0.935981 0.467990 − 0.883734i \(-0.344978\pi\)
0.467990 + 0.883734i \(0.344978\pi\)
\(80\) −56880.0 + 41922.1i −0.993653 + 0.732350i
\(81\) −72819.0 −1.23320
\(82\) 1313.38i 0.0215703i
\(83\) 61841.8i 0.985342i 0.870216 + 0.492671i \(0.163979\pi\)
−0.870216 + 0.492671i \(0.836021\pi\)
\(84\) 0 0
\(85\) 22880.0 + 31043.6i 0.343486 + 0.466042i
\(86\) 2772.00 0.0404154
\(87\) − 137905.i − 1.95336i
\(88\) 33431.6i 0.460204i
\(89\) 9990.00 0.133687 0.0668437 − 0.997763i \(-0.478707\pi\)
0.0668437 + 0.997763i \(0.478707\pi\)
\(90\) −33660.0 − 45669.9i −0.438034 − 0.594325i
\(91\) 0 0
\(92\) − 29212.8i − 0.359836i
\(93\) − 134363.i − 1.61092i
\(94\) −69916.0 −0.816125
\(95\) 9900.00 − 7296.57i 0.112545 − 0.0829488i
\(96\) 82368.0 0.912180
\(97\) 101250.i 1.09261i 0.837586 + 0.546305i \(0.183966\pi\)
−0.837586 + 0.546305i \(0.816034\pi\)
\(98\) 0 0
\(99\) −38556.0 −0.395370
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 245.6.b.a.99.2 2
5.4 even 2 inner 245.6.b.a.99.1 2
7.6 odd 2 5.6.b.a.4.2 yes 2
21.20 even 2 45.6.b.b.19.1 2
28.27 even 2 80.6.c.a.49.2 2
35.13 even 4 25.6.a.c.1.2 2
35.27 even 4 25.6.a.c.1.1 2
35.34 odd 2 5.6.b.a.4.1 ✓ 2
56.13 odd 2 320.6.c.f.129.2 2
56.27 even 2 320.6.c.g.129.1 2
84.83 odd 2 720.6.f.f.289.1 2
105.62 odd 4 225.6.a.n.1.2 2
105.83 odd 4 225.6.a.n.1.1 2
105.104 even 2 45.6.b.b.19.2 2
140.27 odd 4 400.6.a.t.1.2 2
140.83 odd 4 400.6.a.t.1.1 2
140.139 even 2 80.6.c.a.49.1 2
280.69 odd 2 320.6.c.f.129.1 2
280.139 even 2 320.6.c.g.129.2 2
420.419 odd 2 720.6.f.f.289.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.b.a.4.1 ✓ 2 35.34 odd 2
5.6.b.a.4.2 yes 2 7.6 odd 2
25.6.a.c.1.1 2 35.27 even 4
25.6.a.c.1.2 2 35.13 even 4
45.6.b.b.19.1 2 21.20 even 2
45.6.b.b.19.2 2 105.104 even 2
80.6.c.a.49.1 2 140.139 even 2
80.6.c.a.49.2 2 28.27 even 2
225.6.a.n.1.1 2 105.83 odd 4
225.6.a.n.1.2 2 105.62 odd 4
245.6.b.a.99.1 2 5.4 even 2 inner
245.6.b.a.99.2 2 1.1 even 1 trivial
320.6.c.f.129.1 2 280.69 odd 2
320.6.c.f.129.2 2 56.13 odd 2
320.6.c.g.129.1 2 56.27 even 2
320.6.c.g.129.2 2 280.139 even 2
400.6.a.t.1.1 2 140.83 odd 4
400.6.a.t.1.2 2 140.27 odd 4
720.6.f.f.289.1 2 84.83 odd 2
720.6.f.f.289.2 2 420.419 odd 2