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Properties

Label 925.2.bc.e
Level $925$
Weight $2$
Character orbit 925.bc
Analytic conductor $7.386$
Analytic rank $0$
Dimension $156$
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(49,925)]
 
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, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("925.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bc (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: \(156\)
Relative dimension: \(26\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 156 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 156 q - 6 q^{4} - 12 q^{14} + 42 q^{16} + 24 q^{19} + 30 q^{21} - 30 q^{24} + 6 q^{29} - 72 q^{31} + 42 q^{34} + 216 q^{36} - 18 q^{39} - 6 q^{41} + 30 q^{44} - 72 q^{46} + 60 q^{49} - 60 q^{51} - 54 q^{54} + 240 q^{56} - 30 q^{59} - 12 q^{61} + 108 q^{64} - 12 q^{66} - 90 q^{69} + 144 q^{71} - 192 q^{74} + 42 q^{76} - 96 q^{79} + 96 q^{81} - 54 q^{84} + 216 q^{86} + 42 q^{89} - 54 q^{91} + 156 q^{94} - 252 q^{96} - 282 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.78850 − 2.13145i −0.602205 + 0.717680i −0.997051 + 5.65456i 0 2.60674 −0.350900 − 0.964090i 9.01634 − 5.20559i 0.368531 + 2.09004i 0
49.2 −1.70002 − 2.02601i 1.02436 − 1.22079i −0.867338 + 4.91892i 0 −4.21478 −0.533357 − 1.46539i 6.85940 − 3.96027i 0.0799390 + 0.453356i 0
49.3 −1.40353 − 1.67266i −1.58939 + 1.89417i −0.480599 + 2.72562i 0 5.39904 −0.115435 − 0.317156i 1.45162 − 0.838094i −0.540747 − 3.06673i 0
49.4 −1.29514 − 1.54348i 0.697445 − 0.831183i −0.357666 + 2.02842i 0 −2.18620 −0.550712 − 1.51307i 0.104199 − 0.0601595i 0.316509 + 1.79501i 0
49.5 −1.27678 − 1.52161i 1.58979 − 1.89464i −0.337829 + 1.91592i 0 −4.91272 1.42481 + 3.91462i −0.0937881 + 0.0541486i −0.541278 − 3.06974i 0
49.6 −1.23228 − 1.46857i −1.78122 + 2.12278i −0.290900 + 1.64978i 0 5.31243 1.54900 + 4.25584i −0.539202 + 0.311308i −0.812492 − 4.60787i 0
49.7 −1.04160 − 1.24133i 0.916161 − 1.09184i −0.108678 + 0.616341i 0 −2.30961 1.01845 + 2.79817i −1.92841 + 1.11337i 0.168185 + 0.953823i 0
49.8 −0.797210 − 0.950078i −0.636268 + 0.758274i 0.0801918 − 0.454791i 0 1.22766 −1.25175 − 3.43915i −2.64417 + 1.52661i 0.350801 + 1.98949i 0
49.9 −0.777601 − 0.926709i 1.92015 − 2.28834i 0.0931703 − 0.528395i 0 −3.61374 −0.899712 − 2.47194i −2.65744 + 1.53427i −1.02860 − 5.83350i 0
49.10 −0.671547 − 0.800319i −1.71214 + 2.04045i 0.157762 − 0.894712i 0 2.78280 −1.70393 − 4.68151i −2.63155 + 1.51932i −0.711068 − 4.03267i 0
49.11 −0.460177 − 0.548417i −0.215129 + 0.256381i 0.258298 − 1.46488i 0 0.239601 0.0206430 + 0.0567163i −2.16222 + 1.24836i 0.501494 + 2.84411i 0
49.12 −0.119840 − 0.142819i −0.223068 + 0.265842i 0.341261 − 1.93538i 0 0.0646998 0.671707 + 1.84550i −0.640225 + 0.369634i 0.500032 + 2.83582i 0
49.13 −0.0837501 − 0.0998095i 1.95318 − 2.32771i 0.344348 − 1.95290i 0 −0.395907 −0.426326 − 1.17132i −0.449430 + 0.259478i −1.08238 − 6.13847i 0
49.14 0.0837501 + 0.0998095i −1.95318 + 2.32771i 0.344348 − 1.95290i 0 −0.395907 0.426326 + 1.17132i 0.449430 − 0.259478i −1.08238 − 6.13847i 0
49.15 0.119840 + 0.142819i 0.223068 − 0.265842i 0.341261 − 1.93538i 0 0.0646998 −0.671707 − 1.84550i 0.640225 − 0.369634i 0.500032 + 2.83582i 0
49.16 0.460177 + 0.548417i 0.215129 − 0.256381i 0.258298 − 1.46488i 0 0.239601 −0.0206430 − 0.0567163i 2.16222 − 1.24836i 0.501494 + 2.84411i 0
49.17 0.671547 + 0.800319i 1.71214 − 2.04045i 0.157762 − 0.894712i 0 2.78280 1.70393 + 4.68151i 2.63155 − 1.51932i −0.711068 − 4.03267i 0
49.18 0.777601 + 0.926709i −1.92015 + 2.28834i 0.0931703 − 0.528395i 0 −3.61374 0.899712 + 2.47194i 2.65744 − 1.53427i −1.02860 − 5.83350i 0
49.19 0.797210 + 0.950078i 0.636268 − 0.758274i 0.0801918 − 0.454791i 0 1.22766 1.25175 + 3.43915i 2.64417 − 1.52661i 0.350801 + 1.98949i 0
49.20 1.04160 + 1.24133i −0.916161 + 1.09184i −0.108678 + 0.616341i 0 −2.30961 −1.01845 − 2.79817i 1.92841 − 1.11337i 0.168185 + 0.953823i 0
See next 80 embeddings (of 156 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.f even 9 1 inner
185.x 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.bc.e 156
5.b even 2 1 inner 925.2.bc.e 156
5.c odd 4 1 925.2.p.e ✓ 78
5.c odd 4 1 925.2.p.f yes 78
37.f even 9 1 inner 925.2.bc.e 156
185.x even 18 1 inner 925.2.bc.e 156
185.bd odd 36 1 925.2.p.e ✓ 78
185.bd odd 36 1 925.2.p.f yes 78
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
925.2.p.e ✓ 78 5.c odd 4 1
925.2.p.e ✓ 78 185.bd odd 36 1
925.2.p.f yes 78 5.c odd 4 1
925.2.p.f yes 78 185.bd odd 36 1
925.2.bc.e 156 1.a even 1 1 trivial
925.2.bc.e 156 5.b even 2 1 inner
925.2.bc.e 156 37.f even 9 1 inner
925.2.bc.e 156 185.x even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{156} + 3 T_{2}^{154} - 15 T_{2}^{152} - 1256 T_{2}^{150} - 3630 T_{2}^{148} + 20565 T_{2}^{146} + \cdots + 3486784401 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display