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Properties

Label 725.2.p.b
Level $725$
Weight $2$
Character orbit 725.p
Analytic conductor $5.789$
Analytic rank $0$
Dimension $48$
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: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{14})\)
Twist minimal: no (minimal twist has level 145)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{6} + 8 q^{9} - 28 q^{11} + 84 q^{14} - 20 q^{16} + 98 q^{21} - 76 q^{24} - 14 q^{29} + 14 q^{31} - 40 q^{34} - 56 q^{36} - 14 q^{39} + 42 q^{44} + 4 q^{49} + 12 q^{51} - 214 q^{54} - 84 q^{56} + 132 q^{59} + 112 q^{61} - 66 q^{64} - 140 q^{66} + 56 q^{69} + 106 q^{71} - 66 q^{74} - 84 q^{76} + 112 q^{79} - 58 q^{81} - 28 q^{84} + 60 q^{86} - 28 q^{89} - 62 q^{91} + 76 q^{94} - 2 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 −0.461174 + 2.02054i −2.96300 + 1.42691i −2.06795 − 0.995870i 0 −1.51666 − 6.64490i −1.85178 − 3.84525i 0.381511 − 0.478400i 4.87285 − 6.11036i 0
149.2 −0.416060 + 1.82288i 0.896481 − 0.431723i −1.34784 − 0.649083i 0 0.413987 + 1.81380i −0.393550 − 0.817215i −0.587568 + 0.736786i −1.25318 + 1.57143i 0
149.3 −0.217216 + 0.951685i −2.24150 + 1.07945i 0.943415 + 0.454325i 0 −0.540406 − 2.36767i −0.892230 − 1.85273i −1.85455 + 2.32553i 1.98863 − 2.49366i 0
149.4 −0.0209948 + 0.0919844i −0.209175 + 0.100733i 1.79392 + 0.863905i 0 −0.00487430 − 0.0213557i 1.49626 + 3.10702i −0.234781 + 0.294407i −1.83686 + 2.30335i 0
149.5 0.0209948 − 0.0919844i 0.209175 − 0.100733i 1.79392 + 0.863905i 0 −0.00487430 − 0.0213557i −1.49626 − 3.10702i 0.234781 − 0.294407i −1.83686 + 2.30335i 0
149.6 0.217216 − 0.951685i 2.24150 − 1.07945i 0.943415 + 0.454325i 0 −0.540406 − 2.36767i 0.892230 + 1.85273i 1.85455 − 2.32553i 1.98863 − 2.49366i 0
149.7 0.416060 − 1.82288i −0.896481 + 0.431723i −1.34784 − 0.649083i 0 0.413987 + 1.81380i 0.393550 + 0.817215i 0.587568 − 0.736786i −1.25318 + 1.57143i 0
149.8 0.461174 − 2.02054i 2.96300 − 1.42691i −2.06795 − 0.995870i 0 −1.51666 − 6.64490i 1.85178 + 3.84525i −0.381511 + 0.478400i 4.87285 − 6.11036i 0
274.1 −2.22082 − 1.06949i 0.731567 − 0.917356i 2.54126 + 3.18664i 0 −2.60579 + 1.25488i 0.412875 + 0.329257i −1.13861 − 4.98858i 0.361211 + 1.58257i 0
274.2 −1.20749 − 0.581499i −0.772561 + 0.968761i −0.127077 − 0.159349i 0 1.49620 − 0.720531i 3.02093 + 2.40911i 0.657236 + 2.87954i 0.325916 + 1.42793i 0
274.3 −1.16693 − 0.561962i −1.90343 + 2.38683i −0.201065 − 0.252127i 0 3.56247 − 1.71559i −2.58438 − 2.06098i 0.669356 + 2.93264i −1.40633 − 6.16153i 0
274.4 −0.852581 − 0.410581i 0.646721 − 0.810963i −0.688663 − 0.863556i 0 −0.884348 + 0.425880i 0.979248 + 0.780924i 0.653721 + 2.86414i 0.428151 + 1.87585i 0
274.5 0.852581 + 0.410581i −0.646721 + 0.810963i −0.688663 − 0.863556i 0 −0.884348 + 0.425880i −0.979248 − 0.780924i −0.653721 − 2.86414i 0.428151 + 1.87585i 0
274.6 1.16693 + 0.561962i 1.90343 − 2.38683i −0.201065 − 0.252127i 0 3.56247 − 1.71559i 2.58438 + 2.06098i −0.669356 − 2.93264i −1.40633 − 6.16153i 0
274.7 1.20749 + 0.581499i 0.772561 − 0.968761i −0.127077 − 0.159349i 0 1.49620 − 0.720531i −3.02093 − 2.40911i −0.657236 − 2.87954i 0.325916 + 1.42793i 0
274.8 2.22082 + 1.06949i −0.731567 + 0.917356i 2.54126 + 3.18664i 0 −2.60579 + 1.25488i −0.412875 − 0.329257i 1.13861 + 4.98858i 0.361211 + 1.58257i 0
299.1 −2.22082 + 1.06949i 0.731567 + 0.917356i 2.54126 − 3.18664i 0 −2.60579 − 1.25488i 0.412875 − 0.329257i −1.13861 + 4.98858i 0.361211 − 1.58257i 0
299.2 −1.20749 + 0.581499i −0.772561 − 0.968761i −0.127077 + 0.159349i 0 1.49620 + 0.720531i 3.02093 − 2.40911i 0.657236 − 2.87954i 0.325916 − 1.42793i 0
299.3 −1.16693 + 0.561962i −1.90343 − 2.38683i −0.201065 + 0.252127i 0 3.56247 + 1.71559i −2.58438 + 2.06098i 0.669356 − 2.93264i −1.40633 + 6.16153i 0
299.4 −0.852581 + 0.410581i 0.646721 + 0.810963i −0.688663 + 0.863556i 0 −0.884348 − 0.425880i 0.979248 − 0.780924i 0.653721 − 2.86414i 0.428151 − 1.87585i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.8
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.b 48
5.b even 2 1 inner 725.2.p.b 48
5.c odd 4 1 145.2.m.a ✓ 24
5.c odd 4 1 725.2.q.b 24
29.e even 14 1 inner 725.2.p.b 48
145.l even 14 1 inner 725.2.p.b 48
145.o even 28 1 4205.2.a.y 24
145.q odd 28 1 145.2.m.a ✓ 24
145.q odd 28 1 725.2.q.b 24
145.t even 28 1 4205.2.a.y 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.m.a ✓ 24 5.c odd 4 1
145.2.m.a ✓ 24 145.q odd 28 1
725.2.p.b 48 1.a even 1 1 trivial
725.2.p.b 48 5.b even 2 1 inner
725.2.p.b 48 29.e even 14 1 inner
725.2.p.b 48 145.l even 14 1 inner
725.2.q.b 24 5.c odd 4 1
725.2.q.b 24 145.q odd 28 1
4205.2.a.y 24 145.o even 28 1
4205.2.a.y 24 145.t even 28 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 8 T_{2}^{46} + 74 T_{2}^{44} + 449 T_{2}^{42} + 2715 T_{2}^{40} + 18044 T_{2}^{38} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display