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Properties

Label 425.2.m.e
Level $425$
Weight $2$
Character orbit 425.m
Analytic conductor $3.394$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,2,Mod(26,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.m (of order \(8\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
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: no (minimal twist has level 85)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{9} + 24 q^{14} + 8 q^{16} + 24 q^{19} - 32 q^{24} - 16 q^{26} - 24 q^{29} - 24 q^{31} - 8 q^{34} + 8 q^{36} + 24 q^{39} - 48 q^{41} - 72 q^{44} - 16 q^{46} - 48 q^{49} - 32 q^{54} + 24 q^{56} + 48 q^{59} + 16 q^{61} - 96 q^{66} - 32 q^{69} + 16 q^{71} + 64 q^{74} - 24 q^{76} + 72 q^{79} + 96 q^{84} - 8 q^{91} + 40 q^{94} + 96 q^{96} + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −1.48843 + 1.48843i −1.66813 − 0.690961i − 2.43082i 0 3.51133 − 1.45444i −1.02689 − 2.47912i 0.641241 + 0.641241i 0.183901 + 0.183901i 0
26.2 −0.672981 + 0.672981i 2.28645 + 0.947078i 1.09419i 0 −2.17610 + 0.901371i 0.204115 + 0.492777i −2.08233 − 2.08233i 2.20957 + 2.20957i 0
26.3 −0.352960 + 0.352960i −0.158243 − 0.0655465i 1.75084i 0 0.0789888 − 0.0327182i −1.48214 − 3.57820i −1.32390 − 1.32390i −2.10058 − 2.10058i 0
26.4 0.352960 − 0.352960i 0.158243 + 0.0655465i 1.75084i 0 0.0789888 − 0.0327182i 1.48214 + 3.57820i 1.32390 + 1.32390i −2.10058 − 2.10058i 0
26.5 0.672981 − 0.672981i −2.28645 − 0.947078i 1.09419i 0 −2.17610 + 0.901371i −0.204115 − 0.492777i 2.08233 + 2.08233i 2.20957 + 2.20957i 0
26.6 1.48843 − 1.48843i 1.66813 + 0.690961i − 2.43082i 0 3.51133 − 1.45444i 1.02689 + 2.47912i −0.641241 − 0.641241i 0.183901 + 0.183901i 0
76.1 −1.63043 + 1.63043i −0.718554 + 1.73474i − 3.31660i 0 −1.65683 − 3.99993i 1.42808 − 0.591528i 2.14663 + 2.14663i −0.371694 − 0.371694i 0
76.2 −1.23200 + 1.23200i 0.229112 − 0.553124i − 1.03565i 0 0.399184 + 0.963715i −0.958968 + 0.397218i −1.18808 − 1.18808i 1.86787 + 1.86787i 0
76.3 −0.176012 + 0.176012i −0.629010 + 1.51856i 1.93804i 0 −0.156572 − 0.377999i −1.32215 + 0.547653i −0.693142 − 0.693142i 0.210935 + 0.210935i 0
76.4 0.176012 − 0.176012i 0.629010 − 1.51856i 1.93804i 0 −0.156572 − 0.377999i 1.32215 − 0.547653i 0.693142 + 0.693142i 0.210935 + 0.210935i 0
76.5 1.23200 − 1.23200i −0.229112 + 0.553124i − 1.03565i 0 0.399184 + 0.963715i 0.958968 − 0.397218i 1.18808 + 1.18808i 1.86787 + 1.86787i 0
76.6 1.63043 − 1.63043i 0.718554 − 1.73474i − 3.31660i 0 −1.65683 − 3.99993i −1.42808 + 0.591528i −2.14663 − 2.14663i −0.371694 − 0.371694i 0
151.1 −1.63043 − 1.63043i −0.718554 − 1.73474i 3.31660i 0 −1.65683 + 3.99993i 1.42808 + 0.591528i 2.14663 − 2.14663i −0.371694 + 0.371694i 0
151.2 −1.23200 − 1.23200i 0.229112 + 0.553124i 1.03565i 0 0.399184 − 0.963715i −0.958968 − 0.397218i −1.18808 + 1.18808i 1.86787 − 1.86787i 0
151.3 −0.176012 − 0.176012i −0.629010 − 1.51856i − 1.93804i 0 −0.156572 + 0.377999i −1.32215 − 0.547653i −0.693142 + 0.693142i 0.210935 − 0.210935i 0
151.4 0.176012 + 0.176012i 0.629010 + 1.51856i − 1.93804i 0 −0.156572 + 0.377999i 1.32215 + 0.547653i 0.693142 − 0.693142i 0.210935 − 0.210935i 0
151.5 1.23200 + 1.23200i −0.229112 − 0.553124i 1.03565i 0 0.399184 − 0.963715i 0.958968 + 0.397218i 1.18808 − 1.18808i 1.86787 − 1.86787i 0
151.6 1.63043 + 1.63043i 0.718554 + 1.73474i 3.31660i 0 −1.65683 + 3.99993i −1.42808 − 0.591528i −2.14663 + 2.14663i −0.371694 + 0.371694i 0
376.1 −1.48843 − 1.48843i −1.66813 + 0.690961i 2.43082i 0 3.51133 + 1.45444i −1.02689 + 2.47912i 0.641241 − 0.641241i 0.183901 − 0.183901i 0
376.2 −0.672981 − 0.672981i 2.28645 − 0.947078i − 1.09419i 0 −2.17610 − 0.901371i 0.204115 − 0.492777i −2.08233 + 2.08233i 2.20957 − 2.20957i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.d even 8 1 inner
85.m 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.e 24
5.b even 2 1 inner 425.2.m.e 24
5.c odd 4 2 85.2.m.a ✓ 24
15.e even 4 2 765.2.bh.b 24
17.d even 8 1 inner 425.2.m.e 24
17.e odd 16 2 7225.2.a.by 24
85.k odd 8 1 85.2.m.a ✓ 24
85.m even 8 1 inner 425.2.m.e 24
85.n odd 8 1 85.2.m.a ✓ 24
85.o even 16 2 1445.2.b.i 24
85.p odd 16 2 7225.2.a.by 24
85.r even 16 2 1445.2.b.i 24
255.v even 8 1 765.2.bh.b 24
255.ba even 8 1 765.2.bh.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.m.a ✓ 24 5.c odd 4 2
85.2.m.a ✓ 24 85.k odd 8 1
85.2.m.a ✓ 24 85.n odd 8 1
425.2.m.e 24 1.a even 1 1 trivial
425.2.m.e 24 5.b even 2 1 inner
425.2.m.e 24 17.d even 8 1 inner
425.2.m.e 24 85.m even 8 1 inner
765.2.bh.b 24 15.e even 4 2
765.2.bh.b 24 255.v even 8 1
765.2.bh.b 24 255.ba even 8 1
1445.2.b.i 24 85.o even 16 2
1445.2.b.i 24 85.r even 16 2
7225.2.a.by 24 17.e odd 16 2
7225.2.a.by 24 85.p odd 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 58T_{2}^{20} + 1047T_{2}^{16} + 6000T_{2}^{12} + 4587T_{2}^{8} + 278T_{2}^{4} + 1 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display