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Properties

Label 98.7.f.a
Level $98$
Weight $7$
Character orbit 98.f
Analytic conductor $22.545$
Analytic rank $0$
Dimension $168$
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] = [98,7,Mod(13,98)] 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("98.13"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([11])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 98.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: \(22.5453001947\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 896 q^{4} + 784 q^{6} - 308 q^{7} + 9884 q^{9} - 3976 q^{11} - 2688 q^{14} - 3948 q^{15} - 28672 q^{16} + 6272 q^{17} + 11200 q^{18} - 25088 q^{20} - 19488 q^{21} + 16800 q^{22} + 81536 q^{23} + 111524 q^{25}+ \cdots - 7151816 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.25877 − 5.51503i −41.1255 + 32.7965i −28.8310 + 13.8843i −42.4452 + 33.8489i 232.641 + 185.525i 342.397 + 20.3243i 112.864 + 141.527i 453.480 − 1986.82i 240.106 + 191.478i
13.2 −1.25877 − 5.51503i −35.5602 + 28.3583i −28.8310 + 13.8843i 23.9598 − 19.1073i 201.159 + 160.419i −243.746 − 241.323i 112.864 + 141.527i 298.117 − 1306.13i −135.537 − 108.087i
13.3 −1.25877 − 5.51503i −19.8702 + 15.8459i −28.8310 + 13.8843i −117.184 + 93.4508i 112.403 + 89.6382i −193.658 + 283.100i 112.864 + 141.527i −18.4875 + 80.9988i 662.890 + 528.637i
13.4 −1.25877 − 5.51503i −19.3295 + 15.4147i −28.8310 + 13.8843i 123.980 − 98.8710i 109.344 + 87.1990i −276.450 + 203.037i 112.864 + 141.527i −26.2034 + 114.805i −701.339 − 559.299i
13.5 −1.25877 − 5.51503i −18.7998 + 14.9923i −28.8310 + 13.8843i 186.857 − 149.013i 106.347 + 84.8093i 85.2034 − 332.249i 112.864 + 141.527i −33.5561 + 147.019i −1057.02 − 842.946i
13.6 −1.25877 − 5.51503i −15.6831 + 12.5068i −28.8310 + 13.8843i −21.3191 + 17.0014i 88.7170 + 70.7494i 342.451 − 19.3944i 112.864 + 141.527i −72.6797 + 318.430i 120.599 + 96.1745i
13.7 −1.25877 − 5.51503i −3.69702 + 2.94827i −28.8310 + 13.8843i −119.499 + 95.2971i 20.9135 + 16.6779i 130.832 − 317.068i 112.864 + 141.527i −157.242 + 688.923i 675.987 + 539.082i
13.8 −1.25877 − 5.51503i 0.352082 − 0.280776i −28.8310 + 13.8843i 110.012 − 87.7315i −1.99168 − 1.58831i 246.875 + 238.121i 112.864 + 141.527i −162.173 + 710.525i −622.321 − 496.284i
13.9 −1.25877 − 5.51503i 6.90144 − 5.50372i −28.8310 + 13.8843i 0.562721 − 0.448755i −39.0405 − 31.1337i −248.979 − 235.921i 112.864 + 141.527i −144.879 + 634.755i −3.18323 − 2.53854i
13.10 −1.25877 − 5.51503i 16.7165 − 13.3310i −28.8310 + 13.8843i −124.897 + 99.6020i −94.5630 − 75.4115i −81.3447 + 333.215i 112.864 + 141.527i −60.4905 + 265.026i 706.524 + 563.434i
13.11 −1.25877 − 5.51503i 22.5591 − 17.9903i −28.8310 + 13.8843i 29.8758 − 23.8251i −127.614 − 101.768i 117.501 + 322.246i 112.864 + 141.527i 23.0449 − 100.966i −169.003 − 134.775i
13.12 −1.25877 − 5.51503i 26.3407 − 21.0060i −28.8310 + 13.8843i 149.526 − 119.243i −149.006 − 118.828i −311.552 + 143.473i 112.864 + 141.527i 90.3627 − 395.905i −845.848 − 674.541i
13.13 −1.25877 − 5.51503i 32.0016 − 25.5204i −28.8310 + 13.8843i 57.0904 − 45.5281i −181.028 − 144.365i 248.903 − 236.001i 112.864 + 141.527i 210.591 − 922.662i −322.952 − 257.546i
13.14 −1.25877 − 5.51503i 37.0915 − 29.5795i −28.8310 + 13.8843i −127.979 + 102.060i −209.821 − 167.327i −315.608 − 134.316i 112.864 + 141.527i 338.615 − 1483.57i 723.958 + 577.337i
13.15 1.25877 + 5.51503i −38.1358 + 30.4123i −28.8310 + 13.8843i 57.9755 − 46.2340i −215.729 − 172.038i −248.598 + 236.322i −112.864 − 141.527i 367.214 − 1608.87i 327.959 + 261.539i
13.16 1.25877 + 5.51503i −35.4077 + 28.2367i −28.8310 + 13.8843i −109.600 + 87.4034i −200.296 − 159.731i 257.319 − 226.795i −112.864 − 141.527i 294.175 − 1288.87i −619.994 − 494.428i
13.17 1.25877 + 5.51503i −22.8730 + 18.2406i −28.8310 + 13.8843i 89.6887 − 71.5244i −129.389 − 103.184i 221.241 + 262.110i −112.864 − 141.527i 28.2365 − 123.712i 507.356 + 404.603i
13.18 1.25877 + 5.51503i −21.6124 + 17.2353i −28.8310 + 13.8843i 42.5270 − 33.9142i −122.258 − 97.4977i 112.962 − 323.865i −112.864 − 141.527i 7.82228 − 34.2716i 240.569 + 191.847i
13.19 1.25877 + 5.51503i −18.2125 + 14.5240i −28.8310 + 13.8843i −99.2196 + 79.1250i −103.025 − 82.1600i −339.471 + 49.0745i −112.864 − 141.527i −41.4689 + 181.687i −561.271 − 447.598i
13.20 1.25877 + 5.51503i −2.94208 + 2.34623i −28.8310 + 13.8843i −115.607 + 92.1932i −16.6429 − 13.2723i −342.828 + 10.8523i −112.864 − 141.527i −159.067 + 696.917i −653.970 − 521.524i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 13.28
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 98.7.f.a ✓ 168
49.f odd 14 1 inner 98.7.f.a ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.7.f.a ✓ 168 1.a even 1 1 trivial
98.7.f.a ✓ 168 49.f odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(98, [\chi])\).