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Properties

Label 147.2.m.a
Level $147$
Weight $2$
Character orbit 147.m
Analytic conductor $1.174$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [147,2,Mod(4,147)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("147.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.m (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - q^{2} - 4 q^{3} + 3 q^{4} - 2 q^{6} + 6 q^{8} + 4 q^{9} + 30 q^{10} - 9 q^{11} - 3 q^{12} - 42 q^{14} - 7 q^{15} + 29 q^{16} - 5 q^{17} + 6 q^{18} - 26 q^{19} - 5 q^{20} + 7 q^{21} + q^{22} - 4 q^{23}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.84692 − 1.25921i 0.733052 + 0.680173i 1.09483 + 2.78958i −1.49226 + 0.460300i −0.497409 − 2.17929i 1.29066 + 2.30959i 0.495786 − 2.17218i 0.0747301 + 0.997204i 3.33570 + 1.02893i
4.2 −1.08067 − 0.736786i 0.733052 + 0.680173i −0.105696 − 0.269309i 1.01467 − 0.312983i −0.291043 − 1.27514i 0.746631 − 2.53822i −0.666286 + 2.91919i 0.0747301 + 0.997204i −1.32712 − 0.409362i
4.3 0.449603 + 0.306534i 0.733052 + 0.680173i −0.622503 − 1.58611i 1.43317 − 0.442074i 0.121086 + 0.530513i 0.881776 + 2.49449i 0.448490 − 1.96496i 0.0747301 + 0.997204i 0.779867 + 0.240557i
4.4 1.65175 + 1.12614i 0.733052 + 0.680173i 0.729391 + 1.85846i −0.400180 + 0.123439i 0.444845 + 1.94899i −1.90889 − 1.83197i 0.00157168 − 0.00688598i 0.0747301 + 0.997204i −0.800006 − 0.246769i
16.1 −0.712776 − 1.81612i −0.0747301 − 0.997204i −1.32415 + 1.22863i −2.50104 + 1.70518i −1.75778 + 0.846502i −1.59223 − 2.11301i −0.340382 − 0.163920i −0.988831 + 0.149042i 4.87950 + 3.32679i
16.2 −0.633632 − 1.61447i −0.0747301 − 0.997204i −0.738910 + 0.685609i 2.02525 − 1.38079i −1.56260 + 0.752509i 2.55966 + 0.669438i −1.55011 − 0.746495i −0.988831 + 0.149042i −3.51251 − 2.39479i
16.3 0.293772 + 0.748519i −0.0747301 − 0.997204i 0.992125 − 0.920558i −1.11822 + 0.762387i 0.724472 − 0.348887i 1.80149 − 1.93768i 2.42946 + 1.16997i −0.988831 + 0.149042i −0.899162 − 0.613038i
16.4 0.687295 + 1.75120i −0.0747301 − 0.997204i −1.12822 + 1.04683i 2.01668 − 1.37495i 1.69494 − 0.816240i −0.492953 + 2.59942i 0.781251 + 0.376230i −0.988831 + 0.149042i 3.79385 + 2.58660i
25.1 −2.06858 + 0.638073i −0.365341 − 0.930874i 2.21941 − 1.51317i −2.32995 − 0.351184i 1.34970 + 1.69247i 0.990624 + 2.45330i −0.926114 + 1.16131i −0.733052 + 0.680173i 5.04378 − 0.760227i
25.2 −0.742928 + 0.229163i −0.365341 − 0.930874i −1.15305 + 0.786137i 1.96019 + 0.295451i 0.484744 + 0.607850i 1.38027 − 2.25718i 1.64597 − 2.06398i −0.733052 + 0.680173i −1.52399 + 0.229704i
25.3 −0.393224 + 0.121294i −0.365341 − 0.930874i −1.51256 + 1.03125i −1.73113 − 0.260925i 0.256570 + 0.321729i −2.49962 − 0.867115i 0.982833 − 1.23243i −0.733052 + 0.680173i 0.712370 − 0.107372i
25.4 2.24916 − 0.693774i −0.365341 − 0.930874i 2.92492 − 1.99418i −0.982858 − 0.148142i −1.46753 − 1.84022i −2.07146 + 1.64592i 2.26005 − 2.83402i −0.733052 + 0.680173i −2.31338 + 0.348686i
37.1 −1.84692 + 1.25921i 0.733052 − 0.680173i 1.09483 − 2.78958i −1.49226 − 0.460300i −0.497409 + 2.17929i 1.29066 − 2.30959i 0.495786 + 2.17218i 0.0747301 − 0.997204i 3.33570 − 1.02893i
37.2 −1.08067 + 0.736786i 0.733052 − 0.680173i −0.105696 + 0.269309i 1.01467 + 0.312983i −0.291043 + 1.27514i 0.746631 + 2.53822i −0.666286 − 2.91919i 0.0747301 − 0.997204i −1.32712 + 0.409362i
37.3 0.449603 − 0.306534i 0.733052 − 0.680173i −0.622503 + 1.58611i 1.43317 + 0.442074i 0.121086 − 0.530513i 0.881776 − 2.49449i 0.448490 + 1.96496i 0.0747301 − 0.997204i 0.779867 − 0.240557i
37.4 1.65175 − 1.12614i 0.733052 − 0.680173i 0.729391 − 1.85846i −0.400180 − 0.123439i 0.444845 − 1.94899i −1.90889 + 1.83197i 0.00157168 + 0.00688598i 0.0747301 − 0.997204i −0.800006 + 0.246769i
46.1 −0.712776 + 1.81612i −0.0747301 + 0.997204i −1.32415 − 1.22863i −2.50104 − 1.70518i −1.75778 − 0.846502i −1.59223 + 2.11301i −0.340382 + 0.163920i −0.988831 − 0.149042i 4.87950 − 3.32679i
46.2 −0.633632 + 1.61447i −0.0747301 + 0.997204i −0.738910 − 0.685609i 2.02525 + 1.38079i −1.56260 − 0.752509i 2.55966 − 0.669438i −1.55011 + 0.746495i −0.988831 − 0.149042i −3.51251 + 2.39479i
46.3 0.293772 − 0.748519i −0.0747301 + 0.997204i 0.992125 + 0.920558i −1.11822 − 0.762387i 0.724472 + 0.348887i 1.80149 + 1.93768i 2.42946 − 1.16997i −0.988831 − 0.149042i −0.899162 + 0.613038i
46.4 0.687295 − 1.75120i −0.0747301 + 0.997204i −1.12822 − 1.04683i 2.01668 + 1.37495i 1.69494 + 0.816240i −0.492953 − 2.59942i 0.781251 − 0.376230i −0.988831 − 0.149042i 3.79385 − 2.58660i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.2.m.a ✓ 48
3.b odd 2 1 441.2.bb.c 48
49.g even 21 1 inner 147.2.m.a ✓ 48
49.g even 21 1 7203.2.a.i 24
49.h odd 42 1 7203.2.a.k 24
147.n odd 42 1 441.2.bb.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.m.a ✓ 48 1.a even 1 1 trivial
147.2.m.a ✓ 48 49.g even 21 1 inner
441.2.bb.c 48 3.b odd 2 1
441.2.bb.c 48 147.n odd 42 1
7203.2.a.i 24 49.g even 21 1
7203.2.a.k 24 49.h odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + T_{2}^{47} - 5 T_{2}^{46} - 10 T_{2}^{45} - 12 T_{2}^{44} + 63 T_{2}^{43} + 317 T_{2}^{42} + \cdots + 1681 \) acting on \(S_{2}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display