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Properties

Label 945.2.t.c
Level $945$
Weight $2$
Character orbit 945.t
Analytic conductor $7.546$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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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: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} + 16 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} + 16 q^{5} + q^{7} - 3 q^{11} + 6 q^{13} + 15 q^{14} + 32 q^{16} + 3 q^{17} - 16 q^{20} - 21 q^{22} + 9 q^{23} - 16 q^{25} - 12 q^{26} - 31 q^{28} - 18 q^{29} - 30 q^{34} - q^{35} - q^{37} + 30 q^{38} - 6 q^{41} - 19 q^{43} - 21 q^{44} + 6 q^{46} + 30 q^{47} + 5 q^{49} + 21 q^{52} + 24 q^{53} - 30 q^{56} - 30 q^{59} + 76 q^{64} - 50 q^{67} + 3 q^{68} + 9 q^{70} + 12 q^{73} - 60 q^{74} + 54 q^{76} + 27 q^{77} + 4 q^{79} + 16 q^{80} - 24 q^{82} + 42 q^{83} - 3 q^{85} - 51 q^{86} + 42 q^{88} - 30 q^{89} - 57 q^{91} - 6 q^{92} - 42 q^{97} - 6 q^{98}+O(q^{100}) \) 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 − 2.62557i 0 −4.89362 0.500000 + 0.866025i 0 1.43974 − 2.21972i 7.59741i 0 2.27381 − 1.31279i
341.2 − 2.34793i 0 −3.51278 0.500000 + 0.866025i 0 2.22052 + 1.43851i 3.55190i 0 2.03337 − 1.17397i
341.3 − 2.21457i 0 −2.90433 0.500000 + 0.866025i 0 −0.612213 + 2.57395i 2.00271i 0 1.91788 − 1.10729i
341.4 − 1.93560i 0 −1.74654 0.500000 + 0.866025i 0 −2.29351 + 1.31902i − 0.490599i 0 1.67628 − 0.967799i
341.5 − 1.23655i 0 0.470942 0.500000 + 0.866025i 0 2.44060 − 1.02150i − 3.05545i 0 1.07088 − 0.618275i
341.6 − 1.09551i 0 0.799850 0.500000 + 0.866025i 0 −2.64482 + 0.0701258i − 3.06727i 0 0.948743 − 0.547757i
341.7 − 0.524306i 0 1.72510 0.500000 + 0.866025i 0 −1.53835 − 2.15255i − 1.95309i 0 0.454062 − 0.262153i
341.8 − 0.103991i 0 1.98919 0.500000 + 0.866025i 0 −0.107447 + 2.64357i − 0.414839i 0 0.0900587 − 0.0519954i
341.9 0.396951i 0 1.84243 0.500000 + 0.866025i 0 2.43622 + 1.03190i 1.52526i 0 −0.343770 + 0.198476i
341.10 0.465802i 0 1.78303 0.500000 + 0.866025i 0 1.73784 − 1.99497i 1.76214i 0 −0.403396 + 0.232901i
341.11 0.645959i 0 1.58274 0.500000 + 0.866025i 0 −2.64433 − 0.0866018i 2.31430i 0 −0.559417 + 0.322980i
341.12 1.57681i 0 −0.486323 0.500000 + 0.866025i 0 −2.52230 − 0.798755i 2.38678i 0 −1.36556 + 0.788404i
341.13 1.71080i 0 −0.926832 0.500000 + 0.866025i 0 1.44091 + 2.21896i 1.83597i 0 −1.48159 + 0.855399i
341.14 2.32447i 0 −3.40314 0.500000 + 0.866025i 0 −1.17052 + 2.37274i − 3.26155i 0 −2.01305 + 1.16223i
341.15 2.44435i 0 −3.97484 0.500000 + 0.866025i 0 0.510801 − 2.59597i − 4.82720i 0 −2.11687 + 1.22217i
341.16 2.51890i 0 −4.34486 0.500000 + 0.866025i 0 1.80687 − 1.93267i − 5.90648i 0 −2.18143 + 1.25945i
521.1 − 2.51890i 0 −4.34486 0.500000 − 0.866025i 0 1.80687 + 1.93267i 5.90648i 0 −2.18143 − 1.25945i
521.2 − 2.44435i 0 −3.97484 0.500000 − 0.866025i 0 0.510801 + 2.59597i 4.82720i 0 −2.11687 − 1.22217i
521.3 − 2.32447i 0 −3.40314 0.500000 − 0.866025i 0 −1.17052 − 2.37274i 3.26155i 0 −2.01305 − 1.16223i
521.4 − 1.71080i 0 −0.926832 0.500000 − 0.866025i 0 1.44091 − 2.21896i − 1.83597i 0 −1.48159 − 0.855399i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.16
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.c 32
3.b odd 2 1 315.2.t.c ✓ 32
7.d odd 6 1 945.2.be.c 32
9.c even 3 1 315.2.be.c yes 32
9.d odd 6 1 945.2.be.c 32
21.g even 6 1 315.2.be.c yes 32
63.i even 6 1 inner 945.2.t.c 32
63.t odd 6 1 315.2.t.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.t.c ✓ 32 3.b odd 2 1
315.2.t.c ✓ 32 63.t odd 6 1
315.2.be.c yes 32 9.c even 3 1
315.2.be.c yes 32 21.g even 6 1
945.2.t.c 32 1.a even 1 1 trivial
945.2.t.c 32 63.i even 6 1 inner
945.2.be.c 32 7.d odd 6 1
945.2.be.c 32 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 48 T_{2}^{30} + 1032 T_{2}^{28} + 13118 T_{2}^{26} + 109596 T_{2}^{24} + 632922 T_{2}^{22} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display