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Properties

Label 49.7.f.a
Level $49$
Weight $7$
Character orbit 49.f
Analytic conductor $11.273$
Analytic rank $0$
Dimension $162$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,7,Mod(6,49)] 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("49.6"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([9])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(162\)
Relative dimension: \(27\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 162 q - 5 q^{2} - 7 q^{3} - 837 q^{4} - 7 q^{5} - 791 q^{6} + 70 q^{7} - 843 q^{8} + 2990 q^{9} - 7 q^{10} + 1171 q^{11} - 13482 q^{12} - 7 q^{13} + 22533 q^{14} + 6181 q^{15} - 17901 q^{16} - 3143 q^{17}+ \cdots + 4449580 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −9.46158 + 11.8645i −18.0354 + 37.4509i −37.0023 − 162.118i −84.6381 + 175.753i −273.691 − 568.326i 152.737 − 307.116i 1398.50 + 673.484i −622.773 − 780.932i −1284.40 − 2667.09i
6.2 −8.72864 + 10.9454i −5.98668 + 12.4315i −29.3706 − 128.681i 59.0592 − 122.638i −83.8115 − 174.036i 147.205 + 309.806i 857.576 + 412.987i 335.823 + 421.108i 826.808 + 1716.88i
6.3 −8.55624 + 10.7292i 13.9693 − 29.0077i −27.6648 − 121.208i −76.3841 + 158.613i 191.703 + 398.076i −283.411 + 193.203i 745.860 + 359.187i −191.777 − 240.481i −1048.23 − 2176.67i
6.4 −8.50954 + 10.6706i 5.48095 − 11.3813i −27.2087 − 119.209i 27.2702 − 56.6272i 74.8053 + 155.335i −203.668 − 275.986i 716.582 + 345.088i 355.031 + 445.195i 372.191 + 772.862i
6.5 −7.44458 + 9.33521i 22.0427 − 45.7722i −17.4830 − 76.5981i 50.1527 − 104.143i 263.194 + 546.528i 342.985 − 3.25534i 156.718 + 75.4716i −1154.69 − 1447.93i 598.832 + 1243.49i
6.6 −5.80604 + 7.28055i −14.9167 + 30.9748i −5.05489 − 22.1469i −9.48695 + 19.6998i −138.907 − 288.443i −315.153 + 135.379i −346.368 − 166.802i −282.408 − 354.129i −88.3440 − 183.448i
6.7 −5.59155 + 7.01158i 4.21448 − 8.75145i −3.65548 − 16.0157i −5.59191 + 11.6117i 37.7960 + 78.4842i 122.008 − 320.567i −384.386 − 185.110i 395.698 + 496.190i −50.1491 − 104.136i
6.8 −5.21294 + 6.53682i −0.592083 + 1.22947i −1.31391 − 5.75662i −71.3439 + 148.147i −4.95035 − 10.2795i 322.225 + 117.558i −437.627 − 210.750i 453.363 + 568.499i −596.499 − 1238.64i
6.9 −4.75016 + 5.95651i −19.8436 + 41.2057i 1.32533 + 5.80663i 70.5425 − 146.483i −151.182 − 313.933i 221.822 − 261.618i −480.190 − 231.247i −849.618 − 1065.39i 537.440 + 1116.01i
6.10 −3.22211 + 4.04040i 9.00678 − 18.7028i 8.29851 + 36.3581i 100.083 − 207.825i 46.5458 + 96.6534i −342.293 − 22.0068i −471.630 − 227.125i 185.852 + 233.052i 517.216 + 1074.01i
6.11 −2.75625 + 3.45623i 14.4977 − 30.1049i 9.89275 + 43.3430i 0.876306 − 1.81967i 64.0898 + 133.084i −48.8736 + 339.500i −431.975 − 208.028i −241.594 − 302.950i 3.87387 + 8.04417i
6.12 −1.24733 + 1.56411i 19.3946 − 40.2732i 13.3507 + 58.4935i −90.5621 + 188.054i 38.8002 + 80.5694i −164.585 − 300.933i −223.500 − 107.632i −791.260 − 992.209i −181.176 − 376.215i
6.13 −0.577544 + 0.724218i −7.31055 + 15.1805i 14.0504 + 61.5589i −59.1331 + 122.791i −6.77183 − 14.0619i −307.715 − 151.528i −106.110 − 51.0997i 277.520 + 347.999i −54.7755 − 113.743i
6.14 −0.409124 + 0.513026i −8.59209 + 17.8417i 14.1455 + 61.9756i 41.8227 − 86.8458i −5.63800 − 11.7074i 172.114 + 296.691i −75.4192 − 36.3200i 210.023 + 263.361i 27.4434 + 56.9869i
6.15 0.970401 − 1.21684i −4.48048 + 9.30382i 13.7023 + 60.0337i 45.5444 − 94.5740i 6.97343 + 14.4805i 89.6755 − 331.070i 176.094 + 84.8023i 388.038 + 486.584i −70.8855 − 147.195i
6.16 1.02183 − 1.28133i −20.8983 + 43.3959i 13.6437 + 59.7768i −76.2794 + 158.396i 34.2499 + 71.1207i 283.852 + 192.554i 185.036 + 89.1088i −991.936 − 1243.85i 125.013 + 259.592i
6.17 1.86787 − 2.34223i 13.7924 − 28.6402i 12.2442 + 53.6454i 17.8467 − 37.0591i −41.3197 − 85.8012i 329.864 − 94.0163i 321.266 + 154.714i −175.506 − 220.078i −53.4657 − 111.023i
6.18 4.47311 − 5.60910i 5.07295 − 10.5341i 2.78804 + 12.2152i −43.9720 + 91.3088i −36.3949 − 75.5747i −131.478 + 316.801i 494.672 + 238.222i 369.292 + 463.077i 315.469 + 655.077i
6.19 4.86576 − 6.10147i −10.5310 + 21.8679i 0.689011 + 3.01876i 85.2970 − 177.121i 82.1849 + 170.659i −261.077 + 222.459i 471.770 + 227.193i 87.2214 + 109.372i −665.665 − 1382.27i
6.20 4.89633 − 6.13980i −19.6341 + 40.7707i 0.518196 + 2.27036i 9.17971 − 19.0619i 154.189 + 320.177i −229.984 − 254.472i 469.303 + 226.004i −822.228 − 1031.04i −72.0892 − 149.695i
See next 80 embeddings (of 162 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.27
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.f odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.7.f.a ✓ 162
49.f odd 14 1 inner 49.7.f.a ✓ 162
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.7.f.a ✓ 162 1.a even 1 1 trivial
49.7.f.a ✓ 162 49.f odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(49, [\chi])\).