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Properties

Label 925.2.ba.a
Level $925$
Weight $2$
Character orbit 925.ba
Analytic conductor $7.386$
Analytic rank $0$
Dimension $36$
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] = [925,2,Mod(99,925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(925, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("925.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.ba (of order \(18\), 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: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 37)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{4} + 6 q^{9} + 18 q^{11} - 18 q^{14} - 6 q^{16} - 12 q^{19} - 24 q^{21} - 90 q^{24} - 30 q^{26} + 36 q^{29} + 66 q^{34} - 144 q^{36} + 12 q^{39} - 48 q^{41} + 36 q^{46} - 42 q^{49} + 162 q^{51} - 90 q^{54} + 162 q^{56} + 12 q^{59} + 84 q^{61} - 12 q^{64} - 162 q^{66} + 72 q^{69} - 18 q^{71} - 30 q^{74} - 138 q^{76} + 12 q^{79} - 90 q^{81} + 48 q^{84} - 84 q^{86} - 36 q^{89} - 6 q^{91} - 36 q^{94} + 30 q^{96} - 120 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −1.93010 + 1.61954i 0.812851 − 0.968719i 0.755055 − 4.28213i 0 3.18617i −1.09182 − 2.99976i 2.95820 + 5.12375i 0.243256 + 1.37957i 0
99.2 −1.77542 + 1.48976i −1.98304 + 2.36330i 0.585455 − 3.32028i 0 − 7.15011i −0.373914 − 1.02732i 1.58933 + 2.75280i −1.13178 − 6.41863i 0
99.3 −0.306129 + 0.256873i 0.0397456 − 0.0473670i −0.319565 + 1.81234i 0 0.0247100i 1.52073 + 4.17818i −0.767337 − 1.32907i 0.520281 + 2.95066i 0
99.4 0.306129 − 0.256873i −0.0397456 + 0.0473670i −0.319565 + 1.81234i 0 0.0247100i −1.52073 − 4.17818i 0.767337 + 1.32907i 0.520281 + 2.95066i 0
99.5 1.77542 − 1.48976i 1.98304 − 2.36330i 0.585455 − 3.32028i 0 − 7.15011i 0.373914 + 1.02732i −1.58933 − 2.75280i −1.13178 − 6.41863i 0
99.6 1.93010 − 1.61954i −0.812851 + 0.968719i 0.755055 − 4.28213i 0 3.18617i 1.09182 + 2.99976i −2.95820 − 5.12375i 0.243256 + 1.37957i 0
299.1 −1.93010 − 1.61954i 0.812851 + 0.968719i 0.755055 + 4.28213i 0 − 3.18617i −1.09182 + 2.99976i 2.95820 − 5.12375i 0.243256 − 1.37957i 0
299.2 −1.77542 − 1.48976i −1.98304 − 2.36330i 0.585455 + 3.32028i 0 7.15011i −0.373914 + 1.02732i 1.58933 − 2.75280i −1.13178 + 6.41863i 0
299.3 −0.306129 − 0.256873i 0.0397456 + 0.0473670i −0.319565 − 1.81234i 0 − 0.0247100i 1.52073 − 4.17818i −0.767337 + 1.32907i 0.520281 − 2.95066i 0
299.4 0.306129 + 0.256873i −0.0397456 − 0.0473670i −0.319565 − 1.81234i 0 − 0.0247100i −1.52073 + 4.17818i 0.767337 − 1.32907i 0.520281 − 2.95066i 0
299.5 1.77542 + 1.48976i 1.98304 + 2.36330i 0.585455 + 3.32028i 0 7.15011i 0.373914 − 1.02732i −1.58933 + 2.75280i −1.13178 + 6.41863i 0
299.6 1.93010 + 1.61954i −0.812851 − 0.968719i 0.755055 + 4.28213i 0 − 3.18617i 1.09182 − 2.99976i −2.95820 + 5.12375i 0.243256 − 1.37957i 0
324.1 −0.456785 − 2.59056i −2.00969 − 0.354362i −4.62296 + 1.68262i 0 5.36808i 1.33057 − 1.58572i 3.84010 + 6.65125i 1.09419 + 0.398252i 0
324.2 −0.197069 − 1.11764i 2.96139 + 0.522172i 0.669111 − 0.243536i 0 − 3.41266i −1.05349 + 1.25550i −1.53892 − 2.66549i 5.67807 + 2.06665i 0
324.3 −0.0885970 − 0.502458i 1.13369 + 0.199899i 1.63477 − 0.595008i 0 − 0.587340i 0.354616 − 0.422615i −0.954012 − 1.65240i −1.57379 − 0.572814i 0
324.4 0.0885970 + 0.502458i −1.13369 − 0.199899i 1.63477 − 0.595008i 0 − 0.587340i −0.354616 + 0.422615i 0.954012 + 1.65240i −1.57379 − 0.572814i 0
324.5 0.197069 + 1.11764i −2.96139 − 0.522172i 0.669111 − 0.243536i 0 − 3.41266i 1.05349 − 1.25550i 1.53892 + 2.66549i 5.67807 + 2.06665i 0
324.6 0.456785 + 2.59056i 2.00969 + 0.354362i −4.62296 + 1.68262i 0 5.36808i −1.33057 + 1.58572i −3.84010 − 6.65125i 1.09419 + 0.398252i 0
374.1 −0.456785 + 2.59056i −2.00969 + 0.354362i −4.62296 − 1.68262i 0 − 5.36808i 1.33057 + 1.58572i 3.84010 − 6.65125i 1.09419 − 0.398252i 0
374.2 −0.197069 + 1.11764i 2.96139 − 0.522172i 0.669111 + 0.243536i 0 3.41266i −1.05349 − 1.25550i −1.53892 + 2.66549i 5.67807 − 2.06665i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.h even 18 1 inner
185.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.ba.a 36
5.b even 2 1 inner 925.2.ba.a 36
5.c odd 4 1 37.2.h.a ✓ 18
5.c odd 4 1 925.2.bb.a 18
15.e even 4 1 333.2.bl.d 18
20.e even 4 1 592.2.bq.d 18
37.h even 18 1 inner 925.2.ba.a 36
185.v even 18 1 inner 925.2.ba.a 36
185.y odd 36 1 37.2.h.a ✓ 18
185.y odd 36 1 925.2.bb.a 18
185.y odd 36 1 1369.2.b.g 18
185.z even 36 1 1369.2.a.m 18
185.bc even 36 1 1369.2.a.m 18
185.bd odd 36 1 1369.2.b.g 18
555.cf even 36 1 333.2.bl.d 18
740.ce even 36 1 592.2.bq.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.h.a ✓ 18 5.c odd 4 1
37.2.h.a ✓ 18 185.y odd 36 1
333.2.bl.d 18 15.e even 4 1
333.2.bl.d 18 555.cf even 36 1
592.2.bq.d 18 20.e even 4 1
592.2.bq.d 18 740.ce even 36 1
925.2.ba.a 36 1.a even 1 1 trivial
925.2.ba.a 36 5.b even 2 1 inner
925.2.ba.a 36 37.h even 18 1 inner
925.2.ba.a 36 185.v even 18 1 inner
925.2.bb.a 18 5.c odd 4 1
925.2.bb.a 18 185.y odd 36 1
1369.2.a.m 18 185.z even 36 1
1369.2.a.m 18 185.bc even 36 1
1369.2.b.g 18 185.y odd 36 1
1369.2.b.g 18 185.bd odd 36 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 3 T_{2}^{34} + 24 T_{2}^{32} + 263 T_{2}^{30} - 1197 T_{2}^{28} + 6093 T_{2}^{26} + \cdots + 59049 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display