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Properties

Label 425.2.m.d
Level $425$
Weight $2$
Character orbit 425.m
Analytic conductor $3.394$
Analytic rank $0$
Dimension $24$
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] = [425,2,Mod(26,425)] 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("425.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.m (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,-8,0,0,12,0,4,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(12)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{6} + 12 q^{9} + 4 q^{11} + 12 q^{12} - 24 q^{14} - 24 q^{16} - 4 q^{17} + 40 q^{18} - 20 q^{19} - 16 q^{22} - 8 q^{23} + 16 q^{24} + 16 q^{26} + 12 q^{27} - 48 q^{28} + 4 q^{29} + 24 q^{31}+ \cdots - 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −1.71892 + 1.71892i 0.679288 + 0.281370i − 3.90934i 0 −1.65129 + 0.683987i 0.222643 + 0.537508i 3.28199 + 3.28199i −1.73906 − 1.73906i 0
26.2 −0.813283 + 0.813283i −1.89910 − 0.786634i 0.677141i 0 2.18426 − 0.904752i −0.143683 − 0.346882i −2.17727 − 2.17727i 0.866476 + 0.866476i 0
26.3 −0.392996 + 0.392996i 2.31283 + 0.958007i 1.69111i 0 −1.28543 + 0.532441i 1.54446 + 3.72866i −1.45059 − 1.45059i 2.31010 + 2.31010i 0
26.4 0.982785 − 0.982785i −0.102388 − 0.0424107i 0.0682683i 0 −0.142306 + 0.0589452i 0.656642 + 1.58527i 2.03266 + 2.03266i −2.11264 − 2.11264i 0
26.5 1.52941 − 1.52941i 2.56367 + 1.06191i − 2.67820i 0 5.54499 − 2.29681i −1.20249 − 2.90308i −1.03724 − 1.03724i 3.32343 + 3.32343i 0
26.6 1.82721 − 1.82721i −2.84719 − 1.17935i − 4.67741i 0 −7.35734 + 3.04751i −1.07757 − 2.60148i −4.89219 − 4.89219i 4.59433 + 4.59433i 0
76.1 −1.91681 + 1.91681i −0.405091 + 0.977976i − 5.34834i 0 −1.09811 − 2.65108i −1.31353 + 0.544082i 6.41814 + 6.41814i 1.32898 + 1.32898i 0
76.2 −0.921271 + 0.921271i 0.128082 − 0.309218i 0.302521i 0 0.166875 + 0.402872i 3.60742 − 1.49424i −2.12124 − 2.12124i 2.04211 + 2.04211i 0
76.3 −0.639117 + 0.639117i −1.10274 + 2.66226i 1.18306i 0 −0.996713 − 2.40628i −0.671652 + 0.278207i −2.03435 − 2.03435i −3.75027 − 3.75027i 0
76.4 −0.187572 + 0.187572i 0.692349 − 1.67148i 1.92963i 0 0.183657 + 0.443388i −4.55230 + 1.88562i −0.737090 − 0.737090i −0.193173 − 0.193173i 0
76.5 0.917051 − 0.917051i 0.703348 − 1.69803i 0.318036i 0 −0.912176 − 2.20219i 3.23987 − 1.34200i 2.12576 + 2.12576i −0.267297 − 0.267297i 0
76.6 1.33351 − 1.33351i −0.723051 + 1.74560i − 1.55648i 0 1.36358 + 3.29196i −0.309803 + 0.128325i 0.591431 + 0.591431i −0.402996 − 0.402996i 0
151.1 −1.91681 − 1.91681i −0.405091 − 0.977976i 5.34834i 0 −1.09811 + 2.65108i −1.31353 − 0.544082i 6.41814 − 6.41814i 1.32898 − 1.32898i 0
151.2 −0.921271 − 0.921271i 0.128082 + 0.309218i − 0.302521i 0 0.166875 − 0.402872i 3.60742 + 1.49424i −2.12124 + 2.12124i 2.04211 − 2.04211i 0
151.3 −0.639117 − 0.639117i −1.10274 − 2.66226i − 1.18306i 0 −0.996713 + 2.40628i −0.671652 − 0.278207i −2.03435 + 2.03435i −3.75027 + 3.75027i 0
151.4 −0.187572 − 0.187572i 0.692349 + 1.67148i − 1.92963i 0 0.183657 − 0.443388i −4.55230 − 1.88562i −0.737090 + 0.737090i −0.193173 + 0.193173i 0
151.5 0.917051 + 0.917051i 0.703348 + 1.69803i − 0.318036i 0 −0.912176 + 2.20219i 3.23987 + 1.34200i 2.12576 − 2.12576i −0.267297 + 0.267297i 0
151.6 1.33351 + 1.33351i −0.723051 − 1.74560i 1.55648i 0 1.36358 − 3.29196i −0.309803 − 0.128325i 0.591431 − 0.591431i −0.402996 + 0.402996i 0
376.1 −1.71892 − 1.71892i 0.679288 − 0.281370i 3.90934i 0 −1.65129 − 0.683987i 0.222643 − 0.537508i 3.28199 − 3.28199i −1.73906 + 1.73906i 0
376.2 −0.813283 − 0.813283i −1.89910 + 0.786634i − 0.677141i 0 2.18426 + 0.904752i −0.143683 + 0.346882i −2.17727 + 2.17727i 0.866476 − 0.866476i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.m.d yes 24
5.b even 2 1 425.2.m.c ✓ 24
5.c odd 4 1 425.2.n.d 24
5.c odd 4 1 425.2.n.e 24
17.d even 8 1 inner 425.2.m.d yes 24
17.e odd 16 2 7225.2.a.cb 24
85.k odd 8 1 425.2.n.d 24
85.m even 8 1 425.2.m.c ✓ 24
85.n odd 8 1 425.2.n.e 24
85.p odd 16 2 7225.2.a.bx 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.m.c ✓ 24 5.b even 2 1
425.2.m.c ✓ 24 85.m even 8 1
425.2.m.d yes 24 1.a even 1 1 trivial
425.2.m.d yes 24 17.d even 8 1 inner
425.2.n.d 24 5.c odd 4 1
425.2.n.d 24 85.k odd 8 1
425.2.n.e 24 5.c odd 4 1
425.2.n.e 24 85.n odd 8 1
7225.2.a.bx 24 85.p odd 16 2
7225.2.a.cb 24 17.e odd 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 90 T_{2}^{20} - 12 T_{2}^{19} + 100 T_{2}^{17} + 2351 T_{2}^{16} - 392 T_{2}^{15} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display