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Properties

Label 945.2.bf.d
Level $945$
Weight $2$
Character orbit 945.bf
Analytic conductor $7.546$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(109,945)]
 
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([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bf (of order \(6\), degree \(2\), 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: \(64\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4} + 2 q^{10} - 32 q^{16} + 8 q^{19} + 2 q^{25} + 12 q^{31} + 64 q^{34} - 28 q^{40} - 36 q^{46} - 44 q^{49} - 8 q^{55} - 16 q^{61} - 64 q^{64} + 76 q^{70} + 64 q^{76} + 44 q^{79} + 52 q^{85} - 84 q^{91} + 24 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −2.35894 + 1.36193i 0 2.70973 − 4.69339i −1.67986 − 1.47582i 0 0.926026 − 2.47840i 9.31418i 0 5.97267 + 1.19352i
109.2 −2.35894 + 1.36193i 0 2.70973 − 4.69339i 0.438170 + 2.19272i 0 −0.926026 + 2.47840i 9.31418i 0 −4.01995 − 4.57573i
109.3 −2.01041 + 1.16071i 0 1.69449 − 2.93494i −2.11683 − 0.720449i 0 0.251139 + 2.63381i 3.22439i 0 5.09191 − 1.00862i
109.4 −2.01041 + 1.16071i 0 1.69449 − 2.93494i −0.434486 + 2.19345i 0 −0.251139 − 2.63381i 3.22439i 0 −1.67246 − 4.91404i
109.5 −1.79330 + 1.03536i 0 1.14395 − 1.98139i 1.86420 − 1.23481i 0 −1.99306 − 1.74003i 0.596181i 0 −2.06460 + 4.14451i
109.6 −1.79330 + 1.03536i 0 1.14395 − 1.98139i 2.00148 − 0.997043i 0 1.99306 + 1.74003i 0.596181i 0 −2.55695 + 3.86025i
109.7 −1.60258 + 0.925251i 0 0.712178 − 1.23353i 0.420341 − 2.19620i 0 2.13946 − 1.55650i − 1.06523i 0 1.35841 + 3.90852i
109.8 −1.60258 + 0.925251i 0 0.712178 − 1.23353i 2.11214 + 0.734076i 0 −2.13946 + 1.55650i − 1.06523i 0 −4.06408 + 0.777842i
109.9 −1.30418 + 0.752967i 0 0.133918 − 0.231953i −2.11609 − 0.722606i 0 −2.60813 − 0.444583i − 2.60852i 0 3.30386 − 0.650940i
109.10 −1.30418 + 0.752967i 0 0.133918 − 0.231953i −0.432250 + 2.19389i 0 2.60813 + 0.444583i − 2.60852i 0 −1.08820 − 3.18669i
109.11 −0.732533 + 0.422928i 0 −0.642264 + 1.11243i −0.852190 − 2.06731i 0 0.878112 + 2.49578i − 2.77824i 0 1.49858 + 1.15396i
109.12 −0.732533 + 0.422928i 0 −0.642264 + 1.11243i 1.36425 + 1.77167i 0 −0.878112 − 2.49578i − 2.77824i 0 −1.74865 − 0.720830i
109.13 −0.511104 + 0.295086i 0 −0.825848 + 1.43041i −2.20381 + 0.378463i 0 2.64199 + 0.140966i − 2.15513i 0 1.01470 − 0.843747i
109.14 −0.511104 + 0.295086i 0 −0.825848 + 1.43041i −1.42966 + 1.71932i 0 −2.64199 − 0.140966i − 2.15513i 0 0.223359 − 1.30063i
109.15 −0.332810 + 0.192148i 0 −0.926159 + 1.60415i 1.13803 − 1.92481i 0 −1.10736 + 2.40286i − 1.48043i 0 −0.00889968 + 0.859265i
109.16 −0.332810 + 0.192148i 0 −0.926159 + 1.60415i 2.23595 − 0.0231584i 0 1.10736 − 2.40286i − 1.48043i 0 −0.739695 + 0.437340i
109.17 0.332810 − 0.192148i 0 −0.926159 + 1.60415i −2.23595 + 0.0231584i 0 1.10736 − 2.40286i 1.48043i 0 −0.739695 + 0.437340i
109.18 0.332810 − 0.192148i 0 −0.926159 + 1.60415i −1.13803 + 1.92481i 0 −1.10736 + 2.40286i 1.48043i 0 −0.00889968 + 0.859265i
109.19 0.511104 − 0.295086i 0 −0.825848 + 1.43041i 1.42966 − 1.71932i 0 −2.64199 − 0.140966i 2.15513i 0 0.223359 − 1.30063i
109.20 0.511104 − 0.295086i 0 −0.825848 + 1.43041i 2.20381 − 0.378463i 0 2.64199 + 0.140966i 2.15513i 0 1.01470 − 0.843747i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.bf.d ✓ 64
3.b odd 2 1 inner 945.2.bf.d ✓ 64
5.b even 2 1 inner 945.2.bf.d ✓ 64
7.c even 3 1 inner 945.2.bf.d ✓ 64
15.d odd 2 1 inner 945.2.bf.d ✓ 64
21.h odd 6 1 inner 945.2.bf.d ✓ 64
35.j even 6 1 inner 945.2.bf.d ✓ 64
105.o odd 6 1 inner 945.2.bf.d ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.bf.d ✓ 64 1.a even 1 1 trivial
945.2.bf.d ✓ 64 3.b odd 2 1 inner
945.2.bf.d ✓ 64 5.b even 2 1 inner
945.2.bf.d ✓ 64 7.c even 3 1 inner
945.2.bf.d ✓ 64 15.d odd 2 1 inner
945.2.bf.d ✓ 64 21.h odd 6 1 inner
945.2.bf.d ✓ 64 35.j even 6 1 inner
945.2.bf.d ✓ 64 105.o odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\):

\( T_{2}^{32} - 24 T_{2}^{30} + 348 T_{2}^{28} - 3280 T_{2}^{26} + 22862 T_{2}^{24} - 118381 T_{2}^{22} + \cdots + 2401 \) Copy content Toggle raw display
\( T_{11}^{32} + 108 T_{11}^{30} + 7197 T_{11}^{28} + 305620 T_{11}^{26} + 9547428 T_{11}^{24} + \cdots + 3063651608241 \) Copy content Toggle raw display