Properties

Label 725.2.e.c
Level $725$
Weight $2$
Character orbit 725.e
Analytic conductor $5.789$
Analytic rank $0$
Dimension $26$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(157,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.e (of order \(4\), degree \(2\), 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 4 q^{3} - 22 q^{4} + 4 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 4 q^{3} - 22 q^{4} + 4 q^{7} + 10 q^{9} - 8 q^{11} + 8 q^{12} + 14 q^{13} + 4 q^{14} + 6 q^{16} + 16 q^{21} - 8 q^{22} + 4 q^{23} + 6 q^{26} + 4 q^{27} - 8 q^{28} + 8 q^{31} + 32 q^{34} - 22 q^{36} - 16 q^{37} - 8 q^{38} + 16 q^{39} - 6 q^{41} - 4 q^{42} - 12 q^{43} - 32 q^{46} + 36 q^{47} - 4 q^{48} - 26 q^{52} - 14 q^{53} - 32 q^{56} - 12 q^{57} - 58 q^{58} + 18 q^{61} + 28 q^{62} + 60 q^{63} - 30 q^{64} + 20 q^{66} + 32 q^{67} - 12 q^{69} + 20 q^{76} - 56 q^{78} - 4 q^{79} - 86 q^{81} + 58 q^{82} + 60 q^{83} - 76 q^{84} + 12 q^{87} + 68 q^{88} + 46 q^{89} - 28 q^{92} - 8 q^{93} + 8 q^{97} - 34 q^{98} - 36 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 2.26693i 2.65401 −3.13899 0 6.01647i 2.59753 + 2.59753i 2.58201i 4.04378 0
157.2 2.23019i −1.25170 −2.97373 0 2.79153i −0.483409 0.483409i 2.17160i −1.43324 0
157.3 1.41066i −2.58872 0.0100302 0 3.65181i 0.510628 + 0.510628i 2.83547i 3.70148 0
157.4 1.26373i 0.913274 0.402981 0 1.15413i −2.03055 2.03055i 3.03672i −2.16593 0
157.5 0.895351i 2.11245 1.19835 0 1.89138i −1.38465 1.38465i 2.86364i 1.46244 0
157.6 0.222351i −1.02589 1.95056 0 0.228107i 2.35964 + 2.35964i 0.878412i −1.94756 0
157.7 0.342532i 2.64611 1.88267 0 0.906377i 1.55474 + 1.55474i 1.32994i 4.00189 0
157.8 0.839004i −0.711801 1.29607 0 0.597203i −1.13987 1.13987i 2.76542i −2.49334 0
157.9 1.36192i 0.228160 0.145179 0 0.310736i −3.45046 3.45046i 2.92156i −2.94794 0
157.10 1.77873i 1.38965 −1.16389 0 2.47181i 3.41296 + 3.41296i 1.48721i −1.06888 0
157.11 1.82099i −2.59340 −1.31599 0 4.72256i 0.820621 + 0.820621i 1.24557i 3.72575 0
157.12 2.41122i 1.85973 −3.81399 0 4.48422i 0.291676 + 0.291676i 4.37394i 0.458586 0
157.13 2.73482i −1.63186 −5.47925 0 4.46285i −1.05887 1.05887i 9.51511i −0.337028 0
568.1 2.73482i −1.63186 −5.47925 0 4.46285i −1.05887 + 1.05887i 9.51511i −0.337028 0
568.2 2.41122i 1.85973 −3.81399 0 4.48422i 0.291676 0.291676i 4.37394i 0.458586 0
568.3 1.82099i −2.59340 −1.31599 0 4.72256i 0.820621 0.820621i 1.24557i 3.72575 0
568.4 1.77873i 1.38965 −1.16389 0 2.47181i 3.41296 3.41296i 1.48721i −1.06888 0
568.5 1.36192i 0.228160 0.145179 0 0.310736i −3.45046 + 3.45046i 2.92156i −2.94794 0
568.6 0.839004i −0.711801 1.29607 0 0.597203i −1.13987 + 1.13987i 2.76542i −2.49334 0
568.7 0.342532i 2.64611 1.88267 0 0.906377i 1.55474 1.55474i 1.32994i 4.00189 0
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.e.c 26
5.b even 2 1 145.2.e.a 26
5.c odd 4 1 145.2.j.a yes 26
5.c odd 4 1 725.2.j.c 26
29.c odd 4 1 725.2.j.c 26
145.e even 4 1 inner 725.2.e.c 26
145.f odd 4 1 145.2.j.a yes 26
145.j even 4 1 145.2.e.a 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.e.a 26 5.b even 2 1
145.2.e.a 26 145.j even 4 1
145.2.j.a yes 26 5.c odd 4 1
145.2.j.a yes 26 145.f odd 4 1
725.2.e.c 26 1.a even 1 1 trivial
725.2.e.c 26 145.e even 4 1 inner
725.2.j.c 26 5.c odd 4 1
725.2.j.c 26 29.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} + 37 T_{2}^{24} + 598 T_{2}^{22} + 5562 T_{2}^{20} + 33015 T_{2}^{18} + 131099 T_{2}^{16} + 354976 T_{2}^{14} + 655536 T_{2}^{12} + 811627 T_{2}^{10} + 648415 T_{2}^{8} + 311274 T_{2}^{6} + 78422 T_{2}^{4} + \cdots + 225 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display