Properties

Label 75.10.e.g
Level $75$
Weight $10$
Character orbit 75.e
Analytic conductor $38.628$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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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 = [48,0,0,0,0,10788] 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: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 10788 q^{6} - 3683496 q^{16} + 2202288 q^{21} - 50381304 q^{31} + 225716076 q^{36} - 336695184 q^{46} + 673842108 q^{51} + 69620856 q^{61} - 445930020 q^{66} + 4632842472 q^{76} - 4102516332 q^{81}+ \cdots - 14562582156 q^{96}+O(q^{100}) \) 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 −29.6409 + 29.6409i −121.086 − 70.8607i − 1245.17i 0 5689.47 − 1488.72i −7257.95 − 7257.95i 21731.7 + 21731.7i 9640.52 + 17160.4i 0
32.2 −29.6409 + 29.6409i −70.8607 − 121.086i − 1245.17i 0 5689.47 + 1488.72i 7257.95 + 7257.95i 21731.7 + 21731.7i −9640.52 + 17160.4i 0
32.3 −27.2604 + 27.2604i 84.7063 + 111.838i − 974.263i 0 −5357.90 − 739.636i 4326.79 + 4326.79i 12601.5 + 12601.5i −5332.69 + 18946.8i 0
32.4 −27.2604 + 27.2604i 111.838 + 84.7063i − 974.263i 0 −5357.90 + 739.636i −4326.79 − 4326.79i 12601.5 + 12601.5i 5332.69 + 18946.8i 0
32.5 −20.2018 + 20.2018i −114.253 + 81.4208i − 304.223i 0 663.260 − 3952.95i −1648.69 − 1648.69i −4197.47 − 4197.47i 6424.31 − 18605.1i 0
32.6 −20.2018 + 20.2018i 81.4208 − 114.253i − 304.223i 0 663.260 + 3952.95i 1648.69 + 1648.69i −4197.47 − 4197.47i −6424.31 − 18605.1i 0
32.7 −15.0332 + 15.0332i −65.5116 + 124.061i 60.0064i 0 −880.190 − 2849.89i −4276.91 − 4276.91i −8599.08 − 8599.08i −11099.5 − 16254.9i 0
32.8 −15.0332 + 15.0332i 124.061 − 65.5116i 60.0064i 0 −880.190 + 2849.89i 4276.91 + 4276.91i −8599.08 − 8599.08i 11099.5 − 16254.9i 0
32.9 −11.0913 + 11.0913i −138.195 − 24.1892i 265.967i 0 1801.05 − 1264.47i 7391.28 + 7391.28i −8628.65 − 8628.65i 18512.8 + 6685.66i 0
32.10 −11.0913 + 11.0913i −24.1892 − 138.195i 265.967i 0 1801.05 + 1264.47i −7391.28 − 7391.28i −8628.65 − 8628.65i −18512.8 + 6685.66i 0
32.11 −3.26503 + 3.26503i 38.9302 + 134.787i 490.679i 0 −567.190 − 312.974i 2739.43 + 2739.43i −3273.77 − 3273.77i −16651.9 + 10494.6i 0
32.12 −3.26503 + 3.26503i 134.787 + 38.9302i 490.679i 0 −567.190 + 312.974i −2739.43 − 2739.43i −3273.77 − 3273.77i 16651.9 + 10494.6i 0
32.13 3.26503 − 3.26503i −134.787 − 38.9302i 490.679i 0 −567.190 + 312.974i 2739.43 + 2739.43i 3273.77 + 3273.77i 16651.9 + 10494.6i 0
32.14 3.26503 − 3.26503i −38.9302 − 134.787i 490.679i 0 −567.190 − 312.974i −2739.43 − 2739.43i 3273.77 + 3273.77i −16651.9 + 10494.6i 0
32.15 11.0913 − 11.0913i 24.1892 + 138.195i 265.967i 0 1801.05 + 1264.47i 7391.28 + 7391.28i 8628.65 + 8628.65i −18512.8 + 6685.66i 0
32.16 11.0913 − 11.0913i 138.195 + 24.1892i 265.967i 0 1801.05 − 1264.47i −7391.28 − 7391.28i 8628.65 + 8628.65i 18512.8 + 6685.66i 0
32.17 15.0332 − 15.0332i −124.061 + 65.5116i 60.0064i 0 −880.190 + 2849.89i −4276.91 − 4276.91i 8599.08 + 8599.08i 11099.5 − 16254.9i 0
32.18 15.0332 − 15.0332i 65.5116 − 124.061i 60.0064i 0 −880.190 − 2849.89i 4276.91 + 4276.91i 8599.08 + 8599.08i −11099.5 − 16254.9i 0
32.19 20.2018 − 20.2018i −81.4208 + 114.253i − 304.223i 0 663.260 + 3952.95i −1648.69 − 1648.69i 4197.47 + 4197.47i −6424.31 − 18605.1i 0
32.20 20.2018 − 20.2018i 114.253 − 81.4208i − 304.223i 0 663.260 − 3952.95i 1648.69 + 1648.69i 4197.47 + 4197.47i 6424.31 − 18605.1i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 32.24
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.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.10.e.g ✓ 48
3.b odd 2 1 inner 75.10.e.g ✓ 48
5.b even 2 1 inner 75.10.e.g ✓ 48
5.c odd 4 2 inner 75.10.e.g ✓ 48
15.d odd 2 1 inner 75.10.e.g ✓ 48
15.e even 4 2 inner 75.10.e.g ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.10.e.g ✓ 48 1.a even 1 1 trivial
75.10.e.g ✓ 48 3.b odd 2 1 inner
75.10.e.g ✓ 48 5.b even 2 1 inner
75.10.e.g ✓ 48 5.c odd 4 2 inner
75.10.e.g ✓ 48 15.d odd 2 1 inner
75.10.e.g ✓ 48 15.e even 4 2 inner

Hecke kernels

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

\( T_{2}^{24} + 6228117 T_{2}^{20} + 11943561467988 T_{2}^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{24} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display