↑

Properties

Label 925.2.o.d
Level $925$
Weight $2$
Character orbit 925.o
Analytic conductor $7.386$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [925,2,Mod(174,925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(925, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("925.174");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.o (of order \(6\), degree \(2\), 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: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 22 q^{4} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 22 q^{4} + 20 q^{9} - 40 q^{14} - 26 q^{16} - 10 q^{19} - 12 q^{21} + 42 q^{24} - 40 q^{26} - 12 q^{29} + 76 q^{31} + 10 q^{34} - 4 q^{36} - 28 q^{39} - 26 q^{41} + 30 q^{44} - 26 q^{46} + 52 q^{49} - 92 q^{51} + 74 q^{54} + 14 q^{59} + 32 q^{61} - 40 q^{64} + 164 q^{66} - 42 q^{69} - 4 q^{71} - 96 q^{74} + 34 q^{76} + 42 q^{79} - 32 q^{81} - 152 q^{84} - 32 q^{86} - 58 q^{89} + 64 q^{91} + 26 q^{94} + 52 q^{96} + 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
174.1 −2.29020 + 1.32225i 1.01350 + 0.585145i 2.49668 − 4.32437i 0 −3.09483 −0.0480259 − 0.0277278i 7.91592i −0.815210 − 1.41199i 0
174.2 −2.18842 + 1.26349i −2.76175 − 1.59450i 2.19280 − 3.79803i 0 8.05850 2.85005 + 1.64548i 6.02833i 3.58484 + 6.20912i 0
174.3 −2.06750 + 1.19367i 0.547621 + 0.316169i 1.84971 − 3.20380i 0 −1.50961 −4.45286 − 2.57086i 4.05713i −1.30007 − 2.25180i 0
174.4 −2.05767 + 1.18799i −0.0643028 − 0.0371252i 1.82266 − 3.15694i 0 0.176418 3.20181 + 1.84857i 3.90926i −1.49724 − 2.59330i 0
174.5 −1.50474 + 0.868763i −1.10706 − 0.639160i 0.509498 − 0.882476i 0 2.22111 −1.52138 − 0.878368i − 1.70452i −0.682950 − 1.18290i 0
174.6 −1.23169 + 0.711119i 2.52555 + 1.45812i 0.0113813 − 0.0197130i 0 −4.14760 4.52117 + 2.61030i − 2.81210i 2.75225 + 4.76704i 0
174.7 −1.20392 + 0.695082i −2.14707 − 1.23961i −0.0337208 + 0.0584061i 0 3.44653 −0.117956 − 0.0681017i − 2.87408i 1.57328 + 2.72500i 0
174.8 −1.06320 + 0.613840i 0.819784 + 0.473302i −0.246400 + 0.426778i 0 −1.16213 1.74824 + 1.00935i − 3.06036i −1.05197 − 1.82207i 0
174.9 −0.978390 + 0.564874i 2.45341 + 1.41648i −0.361836 + 0.626718i 0 −3.20052 −0.203143 − 0.117284i − 3.07706i 2.51281 + 4.35232i 0
174.10 −0.479496 + 0.276837i 1.80447 + 1.04181i −0.846722 + 1.46657i 0 −1.15365 −3.61105 − 2.08484i − 2.04497i 0.670751 + 1.16177i 0
174.11 −0.348412 + 0.201156i 0.227139 + 0.131139i −0.919073 + 1.59188i 0 −0.105517 1.99551 + 1.15211i − 1.54413i −1.46561 − 2.53850i 0
174.12 −0.193739 + 0.111855i −1.82447 − 1.05336i −0.974977 + 1.68871i 0 0.471293 1.18597 + 0.684718i − 0.883645i 0.719118 + 1.24555i 0
174.13 0.193739 − 0.111855i 1.82447 + 1.05336i −0.974977 + 1.68871i 0 0.471293 −1.18597 − 0.684718i 0.883645i 0.719118 + 1.24555i 0
174.14 0.348412 − 0.201156i −0.227139 − 0.131139i −0.919073 + 1.59188i 0 −0.105517 −1.99551 − 1.15211i 1.54413i −1.46561 − 2.53850i 0
174.15 0.479496 − 0.276837i −1.80447 − 1.04181i −0.846722 + 1.46657i 0 −1.15365 3.61105 + 2.08484i 2.04497i 0.670751 + 1.16177i 0
174.16 0.978390 − 0.564874i −2.45341 − 1.41648i −0.361836 + 0.626718i 0 −3.20052 0.203143 + 0.117284i 3.07706i 2.51281 + 4.35232i 0
174.17 1.06320 − 0.613840i −0.819784 − 0.473302i −0.246400 + 0.426778i 0 −1.16213 −1.74824 − 1.00935i 3.06036i −1.05197 − 1.82207i 0
174.18 1.20392 − 0.695082i 2.14707 + 1.23961i −0.0337208 + 0.0584061i 0 3.44653 0.117956 + 0.0681017i 2.87408i 1.57328 + 2.72500i 0
174.19 1.23169 − 0.711119i −2.52555 − 1.45812i 0.0113813 − 0.0197130i 0 −4.14760 −4.52117 − 2.61030i 2.81210i 2.75225 + 4.76704i 0
174.20 1.50474 − 0.868763i 1.10706 + 0.639160i 0.509498 − 0.882476i 0 2.22111 1.52138 + 0.878368i 1.70452i −0.682950 − 1.18290i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 174.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.o.d 48
5.b even 2 1 inner 925.2.o.d 48
5.c odd 4 1 925.2.e.d ✓ 24
5.c odd 4 1 925.2.e.e yes 24
37.c even 3 1 inner 925.2.o.d 48
185.n even 6 1 inner 925.2.o.d 48
185.s odd 12 1 925.2.e.d ✓ 24
185.s odd 12 1 925.2.e.e yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
925.2.e.d ✓ 24 5.c odd 4 1
925.2.e.d ✓ 24 185.s odd 12 1
925.2.e.e yes 24 5.c odd 4 1
925.2.e.e yes 24 185.s odd 12 1
925.2.o.d 48 1.a even 1 1 trivial
925.2.o.d 48 5.b even 2 1 inner
925.2.o.d 48 37.c even 3 1 inner
925.2.o.d 48 185.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 35 T_{2}^{46} + 700 T_{2}^{44} - 9531 T_{2}^{42} + 97803 T_{2}^{40} - 785338 T_{2}^{38} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display