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Properties

Label 375.2.i.d
Level $375$
Weight $2$
Character orbit 375.i
Analytic conductor $2.994$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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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] = [375,2,Mod(49,375)] 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("375.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(375, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.i (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.99439007580\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{4} + 6 q^{9} - 8 q^{11} - 12 q^{14} + 32 q^{16} - 14 q^{19} - 6 q^{21} - 12 q^{24} - 112 q^{26} + 2 q^{29} + 26 q^{31} + 50 q^{34} - 4 q^{39} + 16 q^{41} - 66 q^{44} - 44 q^{46} + 56 q^{49}+ \cdots - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.26995 + 1.74793i 0.951057 − 0.309017i −0.824463 − 2.53744i 0 −0.667650 + 2.05481i 3.16056i 1.37265 + 0.446002i 0.809017 − 0.587785i 0
49.2 −1.18666 + 1.63330i −0.951057 + 0.309017i −0.641469 − 1.97424i 0 0.623865 − 1.92006i 1.01887i 0.145612 + 0.0473123i 0.809017 − 0.587785i 0
49.3 −0.0832830 + 0.114629i −0.951057 + 0.309017i 0.611830 + 1.88302i 0 0.0437845 − 0.134755i − 0.858311i −0.536314 − 0.174259i 0.809017 − 0.587785i 0
49.4 0.0832830 − 0.114629i 0.951057 − 0.309017i 0.611830 + 1.88302i 0 0.0437845 − 0.134755i 0.858311i 0.536314 + 0.174259i 0.809017 − 0.587785i 0
49.5 1.18666 − 1.63330i 0.951057 − 0.309017i −0.641469 − 1.97424i 0 0.623865 − 1.92006i − 1.01887i −0.145612 − 0.0473123i 0.809017 − 0.587785i 0
49.6 1.26995 − 1.74793i −0.951057 + 0.309017i −0.824463 − 2.53744i 0 −0.667650 + 2.05481i − 3.16056i −1.37265 − 0.446002i 0.809017 − 0.587785i 0
199.1 −1.26995 − 1.74793i 0.951057 + 0.309017i −0.824463 + 2.53744i 0 −0.667650 − 2.05481i − 3.16056i 1.37265 − 0.446002i 0.809017 + 0.587785i 0
199.2 −1.18666 − 1.63330i −0.951057 − 0.309017i −0.641469 + 1.97424i 0 0.623865 + 1.92006i − 1.01887i 0.145612 − 0.0473123i 0.809017 + 0.587785i 0
199.3 −0.0832830 − 0.114629i −0.951057 − 0.309017i 0.611830 − 1.88302i 0 0.0437845 + 0.134755i 0.858311i −0.536314 + 0.174259i 0.809017 + 0.587785i 0
199.4 0.0832830 + 0.114629i 0.951057 + 0.309017i 0.611830 − 1.88302i 0 0.0437845 + 0.134755i − 0.858311i 0.536314 − 0.174259i 0.809017 + 0.587785i 0
199.5 1.18666 + 1.63330i 0.951057 + 0.309017i −0.641469 + 1.97424i 0 0.623865 + 1.92006i 1.01887i −0.145612 + 0.0473123i 0.809017 + 0.587785i 0
199.6 1.26995 + 1.74793i −0.951057 − 0.309017i −0.824463 + 2.53744i 0 −0.667650 − 2.05481i 3.16056i −1.37265 + 0.446002i 0.809017 + 0.587785i 0
274.1 −2.55553 − 0.830342i 0.587785 − 0.809017i 4.22323 + 3.06835i 0 −2.17386 + 1.57940i 1.68704i −5.08599 − 7.00026i −0.309017 − 0.951057i 0
274.2 −2.32085 − 0.754089i −0.587785 + 0.809017i 3.19965 + 2.32468i 0 1.97423 − 1.43436i 3.44028i −2.80415 − 3.85959i −0.309017 − 0.951057i 0
274.3 −0.234682 − 0.0762527i −0.587785 + 0.809017i −1.56877 − 1.13978i 0 0.199632 − 0.145041i 1.24676i 0.571334 + 0.786373i −0.309017 − 0.951057i 0
274.4 0.234682 + 0.0762527i 0.587785 − 0.809017i −1.56877 − 1.13978i 0 0.199632 − 0.145041i − 1.24676i −0.571334 − 0.786373i −0.309017 − 0.951057i 0
274.5 2.32085 + 0.754089i 0.587785 − 0.809017i 3.19965 + 2.32468i 0 1.97423 − 1.43436i − 3.44028i 2.80415 + 3.85959i −0.309017 − 0.951057i 0
274.6 2.55553 + 0.830342i −0.587785 + 0.809017i 4.22323 + 3.06835i 0 −2.17386 + 1.57940i − 1.68704i 5.08599 + 7.00026i −0.309017 − 0.951057i 0
349.1 −2.55553 + 0.830342i 0.587785 + 0.809017i 4.22323 − 3.06835i 0 −2.17386 − 1.57940i − 1.68704i −5.08599 + 7.00026i −0.309017 + 0.951057i 0
349.2 −2.32085 + 0.754089i −0.587785 − 0.809017i 3.19965 − 2.32468i 0 1.97423 + 1.43436i − 3.44028i −2.80415 + 3.85959i −0.309017 + 0.951057i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.i.d 24
5.b even 2 1 inner 375.2.i.d 24
5.c odd 4 1 75.2.g.c ✓ 12
5.c odd 4 1 375.2.g.c 12
15.e even 4 1 225.2.h.d 12
25.d even 5 1 inner 375.2.i.d 24
25.d even 5 1 1875.2.b.f 12
25.e even 10 1 inner 375.2.i.d 24
25.e even 10 1 1875.2.b.f 12
25.f odd 20 1 75.2.g.c ✓ 12
25.f odd 20 1 375.2.g.c 12
25.f odd 20 1 1875.2.a.j 6
25.f odd 20 1 1875.2.a.k 6
75.l even 20 1 225.2.h.d 12
75.l even 20 1 5625.2.a.p 6
75.l even 20 1 5625.2.a.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.c ✓ 12 5.c odd 4 1
75.2.g.c ✓ 12 25.f odd 20 1
225.2.h.d 12 15.e even 4 1
225.2.h.d 12 75.l even 20 1
375.2.g.c 12 5.c odd 4 1
375.2.g.c 12 25.f odd 20 1
375.2.i.d 24 1.a even 1 1 trivial
375.2.i.d 24 5.b even 2 1 inner
375.2.i.d 24 25.d even 5 1 inner
375.2.i.d 24 25.e even 10 1 inner
1875.2.a.j 6 25.f odd 20 1
1875.2.a.k 6 25.f odd 20 1
1875.2.b.f 12 25.d even 5 1
1875.2.b.f 12 25.e even 10 1
5625.2.a.p 6 75.l even 20 1
5625.2.a.q 6 75.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 16 T_{2}^{22} + 132 T_{2}^{20} - 717 T_{2}^{18} + 4268 T_{2}^{16} - 19372 T_{2}^{14} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display