Properties

Label 725.2.p.a
Level $725$
Weight $2$
Character orbit 725.p
Analytic conductor $5.789$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(149,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.p (of order \(14\), degree \(6\), not 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: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{14})\)
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{4} - 6 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{4} - 6 q^{6} + 6 q^{9} + 14 q^{11} + 14 q^{14} + 18 q^{16} + 14 q^{19} - 14 q^{21} + 50 q^{24} - 42 q^{26} + 30 q^{29} - 42 q^{31} + 26 q^{34} - 80 q^{36} - 42 q^{39} - 84 q^{44} - 26 q^{49} + 40 q^{51} + 76 q^{54} - 42 q^{56} - 88 q^{59} - 14 q^{61} + 52 q^{64} + 42 q^{66} - 42 q^{69} - 42 q^{71} + 14 q^{76} - 98 q^{79} + 2 q^{81} - 42 q^{84} - 88 q^{86} - 14 q^{89} - 6 q^{91} - 132 q^{94} + 60 q^{96}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 −0.504621 + 2.21089i 2.55926 1.23248i −2.83146 1.36356i 0 1.43341 + 6.28018i −0.677709 1.40728i 1.61565 2.02596i 3.16036 3.96297i 0
149.2 −0.122359 + 0.536089i −1.77743 + 0.855966i 1.52952 + 0.736577i 0 −0.241390 1.05760i 1.94112 + 4.03077i −1.26771 + 1.58965i 0.556117 0.697349i 0
149.3 0.122359 0.536089i 1.77743 0.855966i 1.52952 + 0.736577i 0 −0.241390 1.05760i −1.94112 4.03077i 1.26771 1.58965i 0.556117 0.697349i 0
149.4 0.504621 2.21089i −2.55926 + 1.23248i −2.83146 1.36356i 0 1.43341 + 6.28018i 0.677709 + 1.40728i −1.61565 + 2.02596i 3.16036 3.96297i 0
274.1 −2.34472 1.12916i −0.273923 + 0.343489i 2.97573 + 3.73144i 0 1.03013 0.496082i 0.0587399 + 0.0468435i −1.60566 7.03485i 0.624612 + 2.73660i 0
274.2 −0.154074 0.0741982i 0.701005 0.879032i −1.22875 1.54080i 0 −0.173229 + 0.0834229i 2.28763 + 1.82432i 0.151100 + 0.662012i 0.386273 + 1.69237i 0
274.3 0.154074 + 0.0741982i −0.701005 + 0.879032i −1.22875 1.54080i 0 −0.173229 + 0.0834229i −2.28763 1.82432i −0.151100 0.662012i 0.386273 + 1.69237i 0
274.4 2.34472 + 1.12916i 0.273923 0.343489i 2.97573 + 3.73144i 0 1.03013 0.496082i −0.0587399 0.0468435i 1.60566 + 7.03485i 0.624612 + 2.73660i 0
299.1 −2.34472 + 1.12916i −0.273923 0.343489i 2.97573 3.73144i 0 1.03013 + 0.496082i 0.0587399 0.0468435i −1.60566 + 7.03485i 0.624612 2.73660i 0
299.2 −0.154074 + 0.0741982i 0.701005 + 0.879032i −1.22875 + 1.54080i 0 −0.173229 0.0834229i 2.28763 1.82432i 0.151100 0.662012i 0.386273 1.69237i 0
299.3 0.154074 0.0741982i −0.701005 0.879032i −1.22875 + 1.54080i 0 −0.173229 0.0834229i −2.28763 + 1.82432i −0.151100 + 0.662012i 0.386273 1.69237i 0
299.4 2.34472 1.12916i 0.273923 + 0.343489i 2.97573 3.73144i 0 1.03013 + 0.496082i −0.0587399 + 0.0468435i 1.60566 7.03485i 0.624612 2.73660i 0
324.1 −0.965958 1.21127i −0.653024 2.86109i −0.0890656 + 0.390222i 0 −2.83476 + 3.55468i −3.32768 + 0.759522i −2.23300 + 1.07536i −5.05647 + 2.43507i 0
324.2 −0.725171 0.909335i −0.219141 0.960118i 0.144024 0.631009i 0 −0.714155 + 0.895521i 1.48749 0.339509i −2.77404 + 1.33591i 1.82910 0.880850i 0
324.3 0.725171 + 0.909335i 0.219141 + 0.960118i 0.144024 0.631009i 0 −0.714155 + 0.895521i −1.48749 + 0.339509i 2.77404 1.33591i 1.82910 0.880850i 0
324.4 0.965958 + 1.21127i 0.653024 + 2.86109i −0.0890656 + 0.390222i 0 −2.83476 + 3.55468i 3.32768 0.759522i 2.23300 1.07536i −5.05647 + 2.43507i 0
399.1 −0.504621 2.21089i 2.55926 + 1.23248i −2.83146 + 1.36356i 0 1.43341 6.28018i −0.677709 + 1.40728i 1.61565 + 2.02596i 3.16036 + 3.96297i 0
399.2 −0.122359 0.536089i −1.77743 0.855966i 1.52952 0.736577i 0 −0.241390 + 1.05760i 1.94112 4.03077i −1.26771 1.58965i 0.556117 + 0.697349i 0
399.3 0.122359 + 0.536089i 1.77743 + 0.855966i 1.52952 0.736577i 0 −0.241390 + 1.05760i −1.94112 + 4.03077i 1.26771 + 1.58965i 0.556117 + 0.697349i 0
399.4 0.504621 + 2.21089i −2.55926 1.23248i −2.83146 + 1.36356i 0 1.43341 6.28018i 0.677709 1.40728i −1.61565 2.02596i 3.16036 + 3.96297i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
29.e even 14 1 inner
145.l even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.p.a 24
5.b even 2 1 inner 725.2.p.a 24
5.c odd 4 1 29.2.e.a 12
5.c odd 4 1 725.2.q.a 12
15.e even 4 1 261.2.o.a 12
20.e even 4 1 464.2.y.d 12
29.e even 14 1 inner 725.2.p.a 24
145.e even 4 1 841.2.d.m 24
145.h odd 4 1 841.2.e.i 12
145.j even 4 1 841.2.d.m 24
145.l even 14 1 inner 725.2.p.a 24
145.o even 28 1 841.2.a.k 12
145.o even 28 2 841.2.d.k 24
145.o even 28 2 841.2.d.l 24
145.o even 28 1 841.2.d.m 24
145.p odd 28 1 841.2.b.e 12
145.p odd 28 1 841.2.e.a 12
145.p odd 28 1 841.2.e.e 12
145.p odd 28 1 841.2.e.f 12
145.p odd 28 1 841.2.e.h 12
145.p odd 28 1 841.2.e.i 12
145.q odd 28 1 29.2.e.a 12
145.q odd 28 1 725.2.q.a 12
145.q odd 28 1 841.2.b.e 12
145.q odd 28 1 841.2.e.a 12
145.q odd 28 1 841.2.e.e 12
145.q odd 28 1 841.2.e.f 12
145.q odd 28 1 841.2.e.h 12
145.t even 28 1 841.2.a.k 12
145.t even 28 2 841.2.d.k 24
145.t even 28 2 841.2.d.l 24
145.t even 28 1 841.2.d.m 24
435.bc odd 28 1 7569.2.a.bp 12
435.bg even 28 1 261.2.o.a 12
435.bn odd 28 1 7569.2.a.bp 12
580.bh even 28 1 464.2.y.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.e.a 12 5.c odd 4 1
29.2.e.a 12 145.q odd 28 1
261.2.o.a 12 15.e even 4 1
261.2.o.a 12 435.bg even 28 1
464.2.y.d 12 20.e even 4 1
464.2.y.d 12 580.bh even 28 1
725.2.p.a 24 1.a even 1 1 trivial
725.2.p.a 24 5.b even 2 1 inner
725.2.p.a 24 29.e even 14 1 inner
725.2.p.a 24 145.l even 14 1 inner
725.2.q.a 12 5.c odd 4 1
725.2.q.a 12 145.q odd 28 1
841.2.a.k 12 145.o even 28 1
841.2.a.k 12 145.t even 28 1
841.2.b.e 12 145.p odd 28 1
841.2.b.e 12 145.q odd 28 1
841.2.d.k 24 145.o even 28 2
841.2.d.k 24 145.t even 28 2
841.2.d.l 24 145.o even 28 2
841.2.d.l 24 145.t even 28 2
841.2.d.m 24 145.e even 4 1
841.2.d.m 24 145.j even 4 1
841.2.d.m 24 145.o even 28 1
841.2.d.m 24 145.t even 28 1
841.2.e.a 12 145.p odd 28 1
841.2.e.a 12 145.q odd 28 1
841.2.e.e 12 145.p odd 28 1
841.2.e.e 12 145.q odd 28 1
841.2.e.f 12 145.p odd 28 1
841.2.e.f 12 145.q odd 28 1
841.2.e.h 12 145.p odd 28 1
841.2.e.h 12 145.q odd 28 1
841.2.e.i 12 145.h odd 4 1
841.2.e.i 12 145.p odd 28 1
7569.2.a.bp 12 435.bc odd 28 1
7569.2.a.bp 12 435.bn odd 28 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 3 T_{2}^{22} + 5 T_{2}^{20} + 206 T_{2}^{18} + 1620 T_{2}^{16} + 4449 T_{2}^{14} + 12992 T_{2}^{12} + 14569 T_{2}^{10} + 18003 T_{2}^{8} + 7101 T_{2}^{6} + 902 T_{2}^{4} - 36 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display