↑

Properties

Label 945.2.t.b
Level $945$
Weight $2$
Character orbit 945.t
Analytic conductor $7.546$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(341,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.341");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.t (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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 315)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 30 q^{4} - 15 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 30 q^{4} - 15 q^{5} - 3 q^{7} + 3 q^{10} - 9 q^{11} - 12 q^{13} + 27 q^{14} + 42 q^{16} + 3 q^{17} + 15 q^{20} + 15 q^{22} - 15 q^{25} - 24 q^{26} + 27 q^{28} + 48 q^{34} + 6 q^{35} - 3 q^{37} + 30 q^{38} + 3 q^{40} + 18 q^{41} + 12 q^{43} - 15 q^{44} + 9 q^{46} + 60 q^{47} - 15 q^{49} - 3 q^{50} - 33 q^{52} + 30 q^{53} - 42 q^{56} - 30 q^{59} - 12 q^{62} - 138 q^{64} + 12 q^{67} + 21 q^{68} - 18 q^{70} + 6 q^{73} - 54 q^{74} - 54 q^{76} + 9 q^{77} + 24 q^{79} - 21 q^{80} + 6 q^{82} + 6 q^{83} + 3 q^{85} + 60 q^{86} - 48 q^{88} + 3 q^{89} + 15 q^{91} + 3 q^{92} + 36 q^{97} - 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1 − 2.80758i 0 −5.88252 −0.500000 − 0.866025i 0 1.63885 + 2.07705i 10.9005i 0 −2.43144 + 1.40379i
341.2 − 2.34202i 0 −3.48504 −0.500000 − 0.866025i 0 −2.07734 + 1.63849i 3.47799i 0 −2.02825 + 1.17101i
341.3 − 1.82056i 0 −1.31444 −0.500000 − 0.866025i 0 −1.92980 − 1.80994i − 1.24811i 0 −1.57665 + 0.910280i
341.4 − 1.34690i 0 0.185856 −0.500000 − 0.866025i 0 1.86235 + 1.87927i − 2.94413i 0 −1.16645 + 0.673451i
341.5 − 1.34681i 0 0.186101 −0.500000 − 0.866025i 0 −0.829875 + 2.51223i − 2.94426i 0 −1.16637 + 0.673405i
341.6 − 0.929301i 0 1.13640 −0.500000 − 0.866025i 0 0.0655497 − 2.64494i − 2.91466i 0 −0.804799 + 0.464651i
341.7 − 0.0264028i 0 1.99930 −0.500000 − 0.866025i 0 2.64251 − 0.130871i − 0.105593i 0 −0.0228655 + 0.0132014i
341.8 0.509846i 0 1.74006 −0.500000 − 0.866025i 0 −2.47153 + 0.944226i 1.90685i 0 0.441540 − 0.254923i
341.9 0.692853i 0 1.51995 −0.500000 − 0.866025i 0 −0.669425 + 2.55966i 2.43881i 0 0.600029 − 0.346427i
341.10 0.917882i 0 1.15749 −0.500000 − 0.866025i 0 0.697139 − 2.55225i 2.89821i 0 0.794909 − 0.458941i
341.11 1.23569i 0 0.473066 −0.500000 − 0.866025i 0 2.47654 + 0.930987i 3.05595i 0 1.07014 − 0.617846i
341.12 1.75842i 0 −1.09206 −0.500000 − 0.866025i 0 −1.42571 − 2.22875i 1.59655i 0 1.52284 − 0.879212i
341.13 2.16248i 0 −2.67633 −0.500000 − 0.866025i 0 −0.841895 − 2.50823i − 1.46255i 0 1.87276 − 1.08124i
341.14 2.34636i 0 −3.50542 −0.500000 − 0.866025i 0 2.00080 + 1.73112i − 3.53226i 0 2.03201 − 1.17318i
341.15 2.72808i 0 −5.44243 −0.500000 − 0.866025i 0 −2.63818 + 0.200022i − 9.39122i 0 2.36259 − 1.36404i
521.1 − 2.72808i 0 −5.44243 −0.500000 + 0.866025i 0 −2.63818 − 0.200022i 9.39122i 0 2.36259 + 1.36404i
521.2 − 2.34636i 0 −3.50542 −0.500000 + 0.866025i 0 2.00080 − 1.73112i 3.53226i 0 2.03201 + 1.17318i
521.3 − 2.16248i 0 −2.67633 −0.500000 + 0.866025i 0 −0.841895 + 2.50823i 1.46255i 0 1.87276 + 1.08124i
521.4 − 1.75842i 0 −1.09206 −0.500000 + 0.866025i 0 −1.42571 + 2.22875i − 1.59655i 0 1.52284 + 0.879212i
521.5 − 1.23569i 0 0.473066 −0.500000 + 0.866025i 0 2.47654 − 0.930987i − 3.05595i 0 1.07014 + 0.617846i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.t.b 30
3.b odd 2 1 315.2.t.b ✓ 30
7.d odd 6 1 945.2.be.b 30
9.c even 3 1 315.2.be.b yes 30
9.d odd 6 1 945.2.be.b 30
21.g even 6 1 315.2.be.b yes 30
63.i even 6 1 inner 945.2.t.b 30
63.t odd 6 1 315.2.t.b ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.t.b ✓ 30 3.b odd 2 1
315.2.t.b ✓ 30 63.t odd 6 1
315.2.be.b yes 30 9.c even 3 1
315.2.be.b yes 30 21.g even 6 1
945.2.t.b 30 1.a even 1 1 trivial
945.2.t.b 30 63.i even 6 1 inner
945.2.be.b 30 7.d odd 6 1
945.2.be.b 30 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} + 45 T_{2}^{28} + 897 T_{2}^{26} + 10463 T_{2}^{24} + 79503 T_{2}^{22} + 414717 T_{2}^{20} + \cdots + 27 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display