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Properties

Label 31.7.h.a
Level $31$
Weight $7$
Character orbit 31.h
Analytic conductor $7.132$
Analytic rank $0$
Dimension $120$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [31,7,Mod(3,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 31.h (of order \(30\), degree \(8\), 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: \(7.13167659222\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(15\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 6 q^{2} - 7 q^{3} - 1034 q^{4} + 68 q^{5} + 1431 q^{6} - 1753 q^{7} + 811 q^{8} - 3488 q^{9} + 1158 q^{10} + 633 q^{11} + 12777 q^{12} - 10627 q^{13} + 16244 q^{14} + 8395 q^{15} - 4178 q^{16} + 1713 q^{17}+ \cdots - 3201573 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −10.9086 + 7.92554i 24.5923 + 2.58476i 36.4056 − 112.045i −35.9360 + 62.2430i −288.752 + 166.711i −396.026 + 84.1779i 224.214 + 690.060i −114.969 − 24.4373i −101.299 − 963.794i
3.2 −10.5954 + 7.69802i −9.40640 − 0.988652i 33.2262 − 102.260i 3.18721 − 5.52041i 107.275 − 61.9354i 559.643 − 118.956i 176.139 + 542.099i −625.567 − 132.968i 8.72642 + 83.0263i
3.3 −9.78414 + 7.10859i −41.2325 − 4.33371i 25.4202 − 78.2354i 84.9544 − 147.145i 434.231 − 250.704i −520.971 + 110.736i 68.2472 + 210.043i 968.269 + 205.812i 214.791 + 2043.60i
3.4 −6.15108 + 4.46902i 31.4465 + 3.30516i −1.91345 + 5.88901i 118.800 − 205.767i −208.201 + 120.205i −18.5129 + 3.93503i −164.917 − 507.561i 264.889 + 56.3040i 188.831 + 1796.61i
3.5 −5.64430 + 4.10083i −33.7620 − 3.54853i −4.73571 + 14.5750i −94.8796 + 164.336i 205.115 − 118.423i −72.5032 + 15.4110i −171.019 − 526.344i 414.214 + 88.0438i −138.385 − 1316.65i
3.6 −4.30092 + 3.12480i 44.0319 + 4.62794i −11.0436 + 33.9886i −60.2226 + 104.309i −203.839 + 117.686i 223.276 − 47.4587i −163.850 − 504.277i 1204.32 + 255.986i −66.9310 − 636.806i
3.7 −3.40745 + 2.47566i −4.50848 − 0.473860i −14.2952 + 43.9963i 3.09445 − 5.35975i 16.5355 − 9.54679i 75.1395 − 15.9714i −143.507 − 441.670i −692.968 − 147.295i 2.72471 + 25.9239i
3.8 0.956037 − 0.694602i −3.64397 − 0.382996i −19.3456 + 59.5395i 44.8637 − 77.7063i −3.74980 + 2.16495i −354.332 + 75.3156i 46.2323 + 142.288i −699.938 − 148.776i −11.0835 − 105.453i
3.9 1.79320 − 1.30284i −45.9055 − 4.82486i −18.2589 + 56.1951i 51.8358 − 89.7823i −88.6038 + 51.1554i 331.482 − 70.4587i 84.3075 + 259.472i 1370.96 + 291.407i −24.0196 − 228.531i
3.10 3.63195 − 2.63877i 19.6548 + 2.06580i −13.5491 + 41.6999i −95.3245 + 165.107i 76.8365 − 44.3616i −314.282 + 66.8028i 149.613 + 460.460i −331.025 − 70.3616i 89.4647 + 851.199i
3.11 5.47146 − 3.97525i 20.3573 + 2.13963i −5.64283 + 17.3668i 21.9247 − 37.9746i 119.889 − 69.2182i 611.595 − 129.999i 171.917 + 529.107i −303.230 − 64.4535i −30.9987 − 294.932i
3.12 7.18633 − 5.22117i 49.0045 + 5.15058i 4.60558 − 14.1745i 67.5410 − 116.984i 379.054 − 218.847i −500.379 + 106.359i 134.765 + 414.765i 1661.84 + 353.235i −125.424 − 1193.33i
3.13 8.05615 − 5.85314i −31.0823 − 3.26689i 10.8653 − 33.4399i −68.8559 + 119.262i −269.526 + 155.611i −37.8568 + 8.04670i 88.7433 + 273.124i 242.369 + 51.5172i 143.343 + 1363.81i
3.14 10.4803 − 7.61435i −16.5290 − 1.73726i 32.0803 − 98.7331i 74.3205 − 128.727i −186.456 + 107.650i −175.897 + 37.3880i −159.380 − 490.522i −442.881 − 94.1374i −201.274 − 1914.99i
3.15 11.9075 − 8.65128i 25.2718 + 2.65618i 47.1660 − 145.162i −50.0274 + 86.6500i 323.903 − 187.005i 188.008 − 39.9623i −403.123 − 1240.68i −81.4594 − 17.3147i 153.934 + 1464.58i
11.1 −4.91477 − 15.1261i −28.9908 + 26.1034i −152.867 + 111.064i −58.9849 − 102.165i 537.326 + 310.225i −38.2952 − 364.354i 1607.79 + 1168.12i 82.8760 − 788.513i −1255.46 + 1394.33i
11.2 −4.12294 − 12.6891i 34.5075 − 31.0707i −92.2379 + 67.0148i −33.4493 − 57.9359i −536.533 − 309.767i 0.258431 + 2.45881i 539.834 + 392.212i 149.179 − 1419.34i −597.246 + 663.309i
11.3 −3.87999 − 11.9414i 0.287429 − 0.258802i −75.7653 + 55.0467i 76.8496 + 133.107i −4.20568 − 2.42815i 36.1992 + 344.412i 301.194 + 218.830i −76.1856 + 724.858i 1291.31 − 1434.15i
11.4 −2.84211 − 8.74713i 5.74195 − 5.17008i −16.6576 + 12.1024i −21.0413 − 36.4446i −61.5426 − 35.5317i −50.4756 − 480.244i −323.004 − 234.676i −69.9609 + 665.634i −258.983 + 287.630i
11.5 −2.31599 − 7.12789i −16.2945 + 14.6716i 6.33409 − 4.60199i −92.6733 − 160.515i 142.315 + 82.1659i 60.8261 + 578.722i −435.526 − 316.428i −25.9475 + 246.874i −929.502 + 1032.32i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.h odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.7.h.a ✓ 120
31.h odd 30 1 inner 31.7.h.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.7.h.a ✓ 120 1.a even 1 1 trivial
31.7.h.a ✓ 120 31.h odd 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(31, [\chi])\).