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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,Mod(32,75)] 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("75.32"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,150,0,0,-4548] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 150 q^{3} - 4548 q^{6} + 9760 q^{7} - 191940 q^{12} - 114260 q^{13} - 1494724 q^{16} - 994800 q^{18} + 592212 q^{21} - 308260 q^{22} - 4786830 q^{27} + 12953300 q^{28} + 16124104 q^{31} - 19638120 q^{33}+ \cdots - 3987642200 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −30.5207 + 30.5207i 129.276 − 54.5047i − 1351.03i 0 −2282.07 + 5609.11i 2749.16 + 2749.16i 25607.7 + 25607.7i 13741.5 − 14092.3i 0
32.2 −24.1261 + 24.1261i −138.640 + 21.4940i − 652.142i 0 2826.28 − 3863.41i −1781.62 − 1781.62i 3381.08 + 3381.08i 18759.0 − 5959.84i 0
32.3 −24.0812 + 24.0812i 17.1863 + 139.239i − 647.806i 0 −3766.92 − 2939.18i 4153.03 + 4153.03i 3270.36 + 3270.36i −19092.3 + 4786.01i 0
32.4 −20.3831 + 20.3831i −65.1534 − 124.250i − 318.940i 0 3860.62 + 1204.57i −1024.24 − 1024.24i −3935.17 − 3935.17i −11193.1 + 16190.6i 0
32.5 −13.8737 + 13.8737i 128.756 − 55.7208i 127.039i 0 −1013.28 + 2559.39i −6797.28 − 6797.28i −8865.86 − 8865.86i 13473.4 − 14348.8i 0
32.6 −10.0225 + 10.0225i 84.4162 + 112.058i 311.097i 0 −1969.17 − 277.038i −2707.78 − 2707.78i −8249.53 − 8249.53i −5430.82 + 18919.0i 0
32.7 −6.35582 + 6.35582i −0.342183 − 140.296i 431.207i 0 893.869 + 889.519i 7866.00 + 7866.00i −5994.85 − 5994.85i −19682.8 + 96.0135i 0
32.8 −4.02674 + 4.02674i −130.186 + 52.2930i 479.571i 0 313.655 − 734.796i −17.2800 − 17.2800i −3992.80 − 3992.80i 14213.9 − 13615.7i 0
32.9 4.02674 − 4.02674i −52.2930 + 130.186i 479.571i 0 313.655 + 734.796i −17.2800 − 17.2800i 3992.80 + 3992.80i −14213.9 − 13615.7i 0
32.10 6.35582 − 6.35582i 140.296 + 0.342183i 431.207i 0 893.869 − 889.519i 7866.00 + 7866.00i 5994.85 + 5994.85i 19682.8 + 96.0135i 0
32.11 10.0225 − 10.0225i −112.058 − 84.4162i 311.097i 0 −1969.17 + 277.038i −2707.78 − 2707.78i 8249.53 + 8249.53i 5430.82 + 18919.0i 0
32.12 13.8737 − 13.8737i 55.7208 − 128.756i 127.039i 0 −1013.28 − 2559.39i −6797.28 − 6797.28i 8865.86 + 8865.86i −13473.4 − 14348.8i 0
32.13 20.3831 − 20.3831i 124.250 + 65.1534i − 318.940i 0 3860.62 − 1204.57i −1024.24 − 1024.24i 3935.17 + 3935.17i 11193.1 + 16190.6i 0
32.14 24.0812 − 24.0812i −139.239 − 17.1863i − 647.806i 0 −3766.92 + 2939.18i 4153.03 + 4153.03i −3270.36 − 3270.36i 19092.3 + 4786.01i 0
32.15 24.1261 − 24.1261i −21.4940 + 138.640i − 652.142i 0 2826.28 + 3863.41i −1781.62 − 1781.62i −3381.08 − 3381.08i −18759.0 − 5959.84i 0
32.16 30.5207 − 30.5207i 54.5047 − 129.276i − 1351.03i 0 −2282.07 − 5609.11i 2749.16 + 2749.16i −25607.7 − 25607.7i −13741.5 − 14092.3i 0
68.1 −30.5207 − 30.5207i 129.276 + 54.5047i 1351.03i 0 −2282.07 − 5609.11i 2749.16 − 2749.16i 25607.7 − 25607.7i 13741.5 + 14092.3i 0
68.2 −24.1261 − 24.1261i −138.640 − 21.4940i 652.142i 0 2826.28 + 3863.41i −1781.62 + 1781.62i 3381.08 − 3381.08i 18759.0 + 5959.84i 0
68.3 −24.0812 − 24.0812i 17.1863 − 139.239i 647.806i 0 −3766.92 + 2939.18i 4153.03 − 4153.03i 3270.36 − 3270.36i −19092.3 − 4786.01i 0
68.4 −20.3831 − 20.3831i −65.1534 + 124.250i 318.940i 0 3860.62 − 1204.57i −1024.24 + 1024.24i −3935.17 + 3935.17i −11193.1 − 16190.6i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 32.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.10.e.f 32
3.b odd 2 1 inner 75.10.e.f 32
5.b even 2 1 15.10.e.a ✓ 32
5.c odd 4 1 15.10.e.a ✓ 32
5.c odd 4 1 inner 75.10.e.f 32
15.d odd 2 1 15.10.e.a ✓ 32
15.e even 4 1 15.10.e.a ✓ 32
15.e even 4 1 inner 75.10.e.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.e.a ✓ 32 5.b even 2 1
15.10.e.a ✓ 32 5.c odd 4 1
15.10.e.a ✓ 32 15.d odd 2 1
15.10.e.a ✓ 32 15.e even 4 1
75.10.e.f 32 1.a even 1 1 trivial
75.10.e.f 32 3.b odd 2 1 inner
75.10.e.f 32 5.c odd 4 1 inner
75.10.e.f 32 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{32} + 7057841 T_{2}^{28} + 16809877421296 T_{2}^{24} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{16} - 4880 T_{7}^{15} + 11907200 T_{7}^{14} + 298591416240 T_{7}^{13} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display