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Properties

Label 945.2.l.c
Level $945$
Weight $2$
Character orbit 945.l
Analytic conductor $7.546$
Analytic rank $0$
Dimension $36$
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(46,945)]
 
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([4, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.l (of order \(3\), 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: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 44 q^{4} + 18 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 44 q^{4} + 18 q^{5} - q^{7} - q^{11} + 2 q^{13} - 9 q^{14} + 60 q^{16} + 5 q^{17} - 2 q^{19} + 22 q^{20} - 19 q^{22} + 3 q^{23} - 18 q^{25} + 4 q^{26} + 5 q^{28} + 8 q^{29} + 20 q^{32} + 10 q^{34} + q^{35} - 15 q^{37} + 22 q^{38} + 4 q^{41} - 29 q^{43} + 7 q^{44} - 24 q^{46} - 46 q^{47} - 7 q^{49} - 7 q^{52} - 2 q^{55} + 12 q^{56} - 20 q^{58} - 10 q^{59} + 6 q^{61} + 12 q^{62} + 128 q^{64} + 4 q^{65} + 70 q^{67} + 17 q^{68} - 3 q^{70} - 24 q^{71} - 10 q^{73} - 22 q^{74} + 10 q^{76} - 35 q^{77} + 56 q^{79} + 30 q^{80} - 8 q^{82} + 22 q^{83} - 5 q^{85} - 19 q^{86} - 50 q^{88} + 4 q^{89} + 7 q^{91} + 50 q^{92} + 4 q^{94} - 4 q^{95} + 16 q^{97} - 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −2.75701 0 5.60109 0.500000 − 0.866025i 0 −2.40845 − 1.09515i −9.92824 0 −1.37850 + 2.38764i
46.2 −2.58565 0 4.68556 0.500000 − 0.866025i 0 2.41754 + 1.07494i −6.94392 0 −1.29282 + 2.23923i
46.3 −2.08660 0 2.35391 0.500000 − 0.866025i 0 −0.122839 − 2.64290i −0.738471 0 −1.04330 + 1.80705i
46.4 −1.89985 0 1.60945 0.500000 − 0.866025i 0 1.72052 − 2.00992i 0.741992 0 −0.949927 + 1.64532i
46.5 −1.85256 0 1.43197 0.500000 − 0.866025i 0 0.370146 + 2.61973i 1.05231 0 −0.926279 + 1.60436i
46.6 −1.50060 0 0.251799 0.500000 − 0.866025i 0 0.0793460 + 2.64456i 2.62335 0 −0.750300 + 1.29956i
46.7 −0.831231 0 −1.30905 0.500000 − 0.866025i 0 −2.57526 + 0.606656i 2.75059 0 −0.415616 + 0.719867i
46.8 −0.699049 0 −1.51133 0.500000 − 0.866025i 0 2.62064 + 0.363625i 2.45459 0 −0.349525 + 0.605394i
46.9 −0.0255806 0 −1.99935 0.500000 − 0.866025i 0 −2.37158 + 1.17286i 0.102306 0 −0.0127903 + 0.0221535i
46.10 0.259663 0 −1.93258 0.500000 − 0.866025i 0 −0.593390 − 2.57835i −1.02114 0 0.129832 − 0.224875i
46.11 0.390993 0 −1.84712 0.500000 − 0.866025i 0 2.26118 − 1.37370i −1.50420 0 0.195497 − 0.338610i
46.12 1.17766 0 −0.613115 0.500000 − 0.866025i 0 −1.48383 + 2.19049i −3.07736 0 0.588830 − 1.01988i
46.13 1.42579 0 0.0328702 0.500000 − 0.866025i 0 −1.98240 + 1.75217i −2.80471 0 0.712894 − 1.23477i
46.14 1.58441 0 0.510363 0.500000 − 0.866025i 0 −1.17061 − 2.37269i −2.36020 0 0.792206 − 1.37214i
46.15 1.69039 0 0.857411 0.500000 − 0.866025i 0 2.40616 + 1.10017i −1.93142 0 0.845194 − 1.46392i
46.16 2.35938 0 3.56668 0.500000 − 0.866025i 0 2.33019 − 1.25307i 3.69640 0 1.17969 − 2.04328i
46.17 2.65219 0 5.03414 0.500000 − 0.866025i 0 −1.76710 − 1.96910i 8.04712 0 1.32610 − 2.29687i
46.18 2.69765 0 5.27730 0.500000 − 0.866025i 0 −0.230272 + 2.63571i 8.84100 0 1.34882 − 2.33623i
226.1 −2.75701 0 5.60109 0.500000 + 0.866025i 0 −2.40845 + 1.09515i −9.92824 0 −1.37850 − 2.38764i
226.2 −2.58565 0 4.68556 0.500000 + 0.866025i 0 2.41754 − 1.07494i −6.94392 0 −1.29282 − 2.23923i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.l.c 36
3.b odd 2 1 315.2.l.c yes 36
7.c even 3 1 945.2.k.c 36
9.c even 3 1 945.2.k.c 36
9.d odd 6 1 315.2.k.c ✓ 36
21.h odd 6 1 315.2.k.c ✓ 36
63.h even 3 1 inner 945.2.l.c 36
63.j odd 6 1 315.2.l.c yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.k.c ✓ 36 9.d odd 6 1
315.2.k.c ✓ 36 21.h odd 6 1
315.2.l.c yes 36 3.b odd 2 1
315.2.l.c yes 36 63.j odd 6 1
945.2.k.c 36 7.c even 3 1
945.2.k.c 36 9.c even 3 1
945.2.l.c 36 1.a even 1 1 trivial
945.2.l.c 36 63.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 29 T_{2}^{16} + 344 T_{2}^{14} - 2 T_{2}^{13} - 2159 T_{2}^{12} + 42 T_{2}^{11} + 7749 T_{2}^{10} + \cdots - 9 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display