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Properties

Label 378.2.w.a
Level $378$
Weight $2$
Character orbit 378.w
Analytic conductor $3.018$
Analytic rank $0$
Dimension $72$
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] = [378,2,Mod(25,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([10, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.w (of order \(9\), degree \(6\), 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: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 3 q^{6} - 3 q^{7} - 36 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 3 q^{6} - 3 q^{7} - 36 q^{8} - 12 q^{9} + 12 q^{10} - 6 q^{11} - 12 q^{13} - 3 q^{14} - 6 q^{15} + 24 q^{17} + 36 q^{19} - 12 q^{21} - 6 q^{22} + 12 q^{23} - 30 q^{25} - 18 q^{26} - 6 q^{27} + 3 q^{29} + 3 q^{30} + 9 q^{31} - 18 q^{33} + 9 q^{34} + 9 q^{35} + 9 q^{36} + 3 q^{39} - 6 q^{41} + 3 q^{42} + 24 q^{43} - 30 q^{45} - 9 q^{47} - 6 q^{48} + 51 q^{49} + 6 q^{50} + 12 q^{51} + 6 q^{52} - 15 q^{53} - 27 q^{54} + 72 q^{55} + 6 q^{56} - 63 q^{57} + 3 q^{58} + 15 q^{59} - 3 q^{60} - 18 q^{61} - 24 q^{62} - 48 q^{63} - 36 q^{64} - 18 q^{65} - 36 q^{66} + 66 q^{67} - 18 q^{68} - 21 q^{69} - 6 q^{70} + 12 q^{71} - 12 q^{72} - 66 q^{73} + 9 q^{74} + 15 q^{75} - 15 q^{77} + 30 q^{78} + 9 q^{79} - 6 q^{80} - 33 q^{82} - 18 q^{83} - 12 q^{84} + 21 q^{85} - 12 q^{86} - 48 q^{87} + 12 q^{88} + 72 q^{89} + 69 q^{90} + 12 q^{91} + 30 q^{92} + 60 q^{93} - 36 q^{94} + 93 q^{95} - 48 q^{97} + 6 q^{98} - 54 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.939693 + 0.342020i −1.70460 + 0.307144i 0.766044 − 0.642788i −2.61773 − 0.952774i 1.49675 − 0.871628i −1.47144 − 2.19883i −0.500000 + 0.866025i 2.81133 − 1.04711i 2.78573
25.2 −0.939693 + 0.342020i −1.58205 − 0.705062i 0.766044 − 0.642788i 1.56427 + 0.569347i 1.72779 + 0.121449i 0.319985 − 2.62633i −0.500000 + 0.866025i 2.00577 + 2.23089i −1.66466
25.3 −0.939693 + 0.342020i −1.25749 + 1.19110i 0.766044 − 0.642788i 1.98579 + 0.722770i 0.774279 − 1.54935i 2.63247 + 0.264792i −0.500000 + 0.866025i 0.162584 − 2.99559i −2.11324
25.4 −0.939693 + 0.342020i −1.03071 − 1.39199i 0.766044 − 0.642788i −3.69961 − 1.34655i 1.44464 + 0.955517i 0.644767 + 2.56598i −0.500000 + 0.866025i −0.875266 + 2.86948i 3.93704
25.5 −0.939693 + 0.342020i −0.649954 − 1.60548i 0.766044 − 0.642788i 3.27017 + 1.19024i 1.15986 + 1.28636i 1.64949 + 2.06862i −0.500000 + 0.866025i −2.15512 + 2.08697i −3.48004
25.6 −0.939693 + 0.342020i −0.195246 − 1.72101i 0.766044 − 0.642788i 0.182581 + 0.0664541i 0.772092 + 1.55044i −2.63939 − 0.183356i −0.500000 + 0.866025i −2.92376 + 0.672041i −0.194299
25.7 −0.939693 + 0.342020i −0.135903 + 1.72671i 0.766044 − 0.642788i −1.85969 − 0.676871i −0.462863 − 1.66906i 2.63192 + 0.270209i −0.500000 + 0.866025i −2.96306 − 0.469329i 1.97904
25.8 −0.939693 + 0.342020i 0.807797 + 1.53214i 0.766044 − 0.642788i 1.59297 + 0.579794i −1.28310 − 1.16346i −2.02988 + 1.69693i −0.500000 + 0.866025i −1.69493 + 2.47532i −1.69520
25.9 −0.939693 + 0.342020i 1.06159 − 1.36859i 0.766044 − 0.642788i −1.03287 − 0.375933i −0.529482 + 1.64914i 2.42642 − 1.05474i −0.500000 + 0.866025i −0.746062 − 2.90575i 1.09915
25.10 −0.939693 + 0.342020i 1.14375 + 1.30071i 0.766044 − 0.642788i 1.59112 + 0.579119i −1.51964 − 0.831077i 0.142417 − 2.64192i −0.500000 + 0.866025i −0.383668 + 2.97537i −1.69323
25.11 −0.939693 + 0.342020i 1.65621 − 0.506908i 0.766044 − 0.642788i 0.501605 + 0.182569i −1.38296 + 1.04280i −1.88202 + 1.85957i −0.500000 + 0.866025i 2.48609 − 1.67910i −0.533797
25.12 −0.939693 + 0.342020i 1.71296 + 0.256429i 0.766044 − 0.642788i −3.35800 − 1.22221i −1.69736 + 0.344903i −2.15868 − 1.52974i −0.500000 + 0.866025i 2.86849 + 0.878508i 3.57351
121.1 −0.939693 − 0.342020i −1.70460 − 0.307144i 0.766044 + 0.642788i −2.61773 + 0.952774i 1.49675 + 0.871628i −1.47144 + 2.19883i −0.500000 − 0.866025i 2.81133 + 1.04711i 2.78573
121.2 −0.939693 − 0.342020i −1.58205 + 0.705062i 0.766044 + 0.642788i 1.56427 − 0.569347i 1.72779 − 0.121449i 0.319985 + 2.62633i −0.500000 − 0.866025i 2.00577 − 2.23089i −1.66466
121.3 −0.939693 − 0.342020i −1.25749 − 1.19110i 0.766044 + 0.642788i 1.98579 − 0.722770i 0.774279 + 1.54935i 2.63247 − 0.264792i −0.500000 − 0.866025i 0.162584 + 2.99559i −2.11324
121.4 −0.939693 − 0.342020i −1.03071 + 1.39199i 0.766044 + 0.642788i −3.69961 + 1.34655i 1.44464 − 0.955517i 0.644767 − 2.56598i −0.500000 − 0.866025i −0.875266 − 2.86948i 3.93704
121.5 −0.939693 − 0.342020i −0.649954 + 1.60548i 0.766044 + 0.642788i 3.27017 − 1.19024i 1.15986 − 1.28636i 1.64949 − 2.06862i −0.500000 − 0.866025i −2.15512 − 2.08697i −3.48004
121.6 −0.939693 − 0.342020i −0.195246 + 1.72101i 0.766044 + 0.642788i 0.182581 − 0.0664541i 0.772092 − 1.55044i −2.63939 + 0.183356i −0.500000 − 0.866025i −2.92376 − 0.672041i −0.194299
121.7 −0.939693 − 0.342020i −0.135903 − 1.72671i 0.766044 + 0.642788i −1.85969 + 0.676871i −0.462863 + 1.66906i 2.63192 − 0.270209i −0.500000 − 0.866025i −2.96306 + 0.469329i 1.97904
121.8 −0.939693 − 0.342020i 0.807797 − 1.53214i 0.766044 + 0.642788i 1.59297 − 0.579794i −1.28310 + 1.16346i −2.02988 − 1.69693i −0.500000 − 0.866025i −1.69493 − 2.47532i −1.69520
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.w even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.w.a yes 72
7.c even 3 1 378.2.v.b ✓ 72
27.e even 9 1 378.2.v.b ✓ 72
189.w even 9 1 inner 378.2.w.a yes 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.v.b ✓ 72 7.c even 3 1
378.2.v.b ✓ 72 27.e even 9 1
378.2.w.a yes 72 1.a even 1 1 trivial
378.2.w.a yes 72 189.w even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} + 15 T_{5}^{70} + 41 T_{5}^{69} + 24 T_{5}^{68} + 87 T_{5}^{67} + 4671 T_{5}^{66} - 4338 T_{5}^{65} + 53712 T_{5}^{64} + 95076 T_{5}^{63} - 17937 T_{5}^{62} - 229095 T_{5}^{61} + 14772387 T_{5}^{60} + \cdots + 18\!\cdots\!89 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display