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Properties

Label 62.7.h.a
Level $62$
Weight $7$
Character orbit 62.h
Analytic conductor $14.263$
Analytic rank $0$
Dimension $128$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [62,7,Mod(3,62)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(62, 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("62.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 62 = 2 \cdot 31 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 62.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: \(14.2633531844\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 1024 q^{4} - 144 q^{5} + 1544 q^{7} - 3248 q^{9} - 1344 q^{10} - 1280 q^{11} + 9900 q^{13} - 9120 q^{14} - 9520 q^{15} - 32768 q^{16} - 4840 q^{17} + 11200 q^{18} - 11984 q^{19} - 4608 q^{20} - 21804 q^{21}+ \cdots + 6466596 q^{99}+O(q^{100}) \) 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 −4.57649 + 3.32502i −49.3410 − 5.18595i 9.88854 − 30.4338i −32.6988 + 56.6359i 243.052 − 140.326i 539.656 − 114.707i 55.9381 + 172.160i 1694.57 + 360.193i −38.6697 − 367.918i
3.2 −4.57649 + 3.32502i −34.4182 − 3.61750i 9.88854 − 30.4338i 57.5189 − 99.6257i 169.543 − 97.8856i −34.7996 + 7.39689i 55.9381 + 172.160i 458.458 + 97.4482i 68.0221 + 647.187i
3.3 −4.57649 + 3.32502i −29.1393 − 3.06266i 9.88854 − 30.4338i −73.7968 + 127.820i 143.539 − 82.8723i −456.043 + 96.9349i 55.9381 + 172.160i 126.647 + 26.9197i −87.2724 − 830.341i
3.4 −4.57649 + 3.32502i −10.2557 − 1.07792i 9.88854 − 30.4338i −14.8730 + 25.7608i 50.5191 − 29.1672i −100.802 + 21.4262i 55.9381 + 172.160i −609.053 − 129.458i −17.5889 − 167.347i
3.5 −4.57649 + 3.32502i 3.37776 + 0.355017i 9.88854 − 30.4338i 102.360 − 177.293i −16.6387 + 9.60639i 183.105 − 38.9202i 55.9381 + 172.160i −701.786 − 149.169i 121.052 + 1151.73i
3.6 −4.57649 + 3.32502i 15.5982 + 1.63944i 9.88854 − 30.4338i −119.326 + 206.678i −76.8363 + 44.3615i 318.522 − 67.7039i 55.9381 + 172.160i −472.453 − 100.423i −141.115 − 1342.62i
3.7 −4.57649 + 3.32502i 30.4379 + 3.19915i 9.88854 − 30.4338i 20.2023 − 34.9914i −149.936 + 86.5656i −476.650 + 101.315i 55.9381 + 172.160i 203.162 + 43.1834i 23.8913 + 227.311i
3.8 −4.57649 + 3.32502i 43.4732 + 4.56922i 9.88854 − 30.4338i 13.1919 − 22.8491i −214.147 + 123.638i 265.399 − 56.4124i 55.9381 + 172.160i 1155.97 + 245.709i 15.6008 + 148.432i
3.9 4.57649 − 3.32502i −51.4736 − 5.41010i 9.88854 − 30.4338i −40.0035 + 69.2882i −253.557 + 146.391i −492.186 + 104.617i −55.9381 − 172.160i 1907.19 + 405.387i 47.3084 + 450.109i
3.10 4.57649 − 3.32502i −35.0871 − 3.68781i 9.88854 − 30.4338i 95.1532 − 164.810i −172.838 + 99.7880i 259.315 − 55.1191i −55.9381 − 172.160i 504.437 + 107.221i −112.529 − 1070.64i
3.11 4.57649 − 3.32502i −23.6344 − 2.48408i 9.88854 − 30.4338i −24.7800 + 42.9202i −116.422 + 67.2164i 139.041 − 29.5540i −55.9381 − 172.160i −160.655 − 34.1482i 29.3049 + 278.818i
3.12 4.57649 − 3.32502i −15.5310 − 1.63237i 9.88854 − 30.4338i −72.8842 + 126.239i −76.5050 + 44.1702i 528.294 − 112.292i −55.9381 − 172.160i −474.523 − 100.863i 86.1932 + 820.074i
3.13 4.57649 − 3.32502i 2.34617 + 0.246592i 9.88854 − 30.4338i 27.8406 − 48.2213i 11.5571 − 6.67251i −563.950 + 119.871i −55.9381 − 172.160i −707.626 − 150.411i −32.9244 − 313.255i
3.14 4.57649 − 3.32502i 14.9143 + 1.56756i 9.88854 − 30.4338i −87.6873 + 151.879i 73.4674 − 42.4164i −289.524 + 61.5402i −55.9381 − 172.160i −493.090 − 104.809i 103.699 + 986.634i
3.15 4.57649 − 3.32502i 27.1876 + 2.85753i 9.88854 − 30.4338i 69.0328 − 119.568i 133.925 − 77.3216i 200.192 − 42.5522i −55.9381 − 172.160i 17.9280 + 3.81071i −81.6385 − 776.738i
3.16 4.57649 − 3.32502i 51.0110 + 5.36147i 9.88854 − 30.4338i −50.8015 + 87.9908i 251.278 − 145.076i −56.9648 + 12.1082i −55.9381 − 172.160i 1860.31 + 395.420i 60.0781 + 571.605i
11.1 −1.74806 − 5.37999i −34.0546 + 30.6629i −25.8885 + 18.8091i 110.734 + 191.796i 224.496 + 129.613i 27.6421 + 262.997i 146.448 + 106.400i 143.302 − 1363.42i 838.293 − 931.019i
11.2 −1.74806 − 5.37999i −22.3809 + 20.1518i −25.8885 + 18.8091i −53.7040 − 93.0180i 147.540 + 85.1822i 46.4818 + 442.245i 146.448 + 106.400i 18.6062 − 177.026i −406.558 + 451.528i
11.3 −1.74806 − 5.37999i −12.7083 + 11.4426i −25.8885 + 18.8091i 34.4516 + 59.6719i 83.7757 + 48.3679i −13.5804 − 129.208i 146.448 + 106.400i −45.6338 + 434.176i 260.811 − 289.660i
11.4 −1.74806 − 5.37999i 2.16828 − 1.95233i −25.8885 + 18.8091i −19.4805 − 33.7413i −14.2938 − 8.25253i 21.4015 + 203.622i 146.448 + 106.400i −75.3114 + 716.540i −147.474 + 163.787i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.16
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 62.7.h.a ✓ 128
31.h odd 30 1 inner 62.7.h.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.7.h.a ✓ 128 1.a even 1 1 trivial
62.7.h.a ✓ 128 31.h odd 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(62, [\chi])\).