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Properties

Label 441.2.bb.c
Level $441$
Weight $2$
Character orbit 441.bb
Analytic conductor $3.521$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [441,2,Mod(37,441)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 32])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("441.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.bb (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 147)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + q^{2} + 3 q^{4} - 6 q^{8} + 30 q^{10} + 9 q^{11} + 42 q^{14} + 29 q^{16} + 5 q^{17} - 26 q^{19} + 5 q^{20} + q^{22} + 4 q^{23} - 56 q^{25} + 62 q^{26} + 7 q^{28} - 12 q^{29} - 36 q^{31} + 14 q^{32}+ \cdots - 119 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.65175 + 1.12614i 0 0.729391 − 1.85846i 0.400180 + 0.123439i 0 −1.90889 + 1.83197i −0.00157168 − 0.00688598i 0 −0.800006 + 0.246769i
37.2 −0.449603 + 0.306534i 0 −0.622503 + 1.58611i −1.43317 − 0.442074i 0 0.881776 − 2.49449i −0.448490 − 1.96496i 0 0.779867 − 0.240557i
37.3 1.08067 − 0.736786i 0 −0.105696 + 0.269309i −1.01467 − 0.312983i 0 0.746631 + 2.53822i 0.666286 + 2.91919i 0 −1.32712 + 0.409362i
37.4 1.84692 − 1.25921i 0 1.09483 − 2.78958i 1.49226 + 0.460300i 0 1.29066 − 2.30959i −0.495786 − 2.17218i 0 3.33570 − 1.02893i
46.1 −0.687295 + 1.75120i 0 −1.12822 − 1.04683i −2.01668 − 1.37495i 0 −0.492953 − 2.59942i −0.781251 + 0.376230i 0 3.79385 − 2.58660i
46.2 −0.293772 + 0.748519i 0 0.992125 + 0.920558i 1.11822 + 0.762387i 0 1.80149 + 1.93768i −2.42946 + 1.16997i 0 −0.899162 + 0.613038i
46.3 0.633632 − 1.61447i 0 −0.738910 − 0.685609i −2.02525 − 1.38079i 0 2.55966 − 0.669438i 1.55011 − 0.746495i 0 −3.51251 + 2.39479i
46.4 0.712776 − 1.81612i 0 −1.32415 − 1.22863i 2.50104 + 1.70518i 0 −1.59223 + 2.11301i 0.340382 − 0.163920i 0 4.87950 − 3.32679i
100.1 −2.24916 − 0.693774i 0 2.92492 + 1.99418i 0.982858 − 0.148142i 0 −2.07146 − 1.64592i −2.26005 − 2.83402i 0 −2.31338 − 0.348686i
100.2 0.393224 + 0.121294i 0 −1.51256 − 1.03125i 1.73113 − 0.260925i 0 −2.49962 + 0.867115i −0.982833 − 1.23243i 0 0.712370 + 0.107372i
100.3 0.742928 + 0.229163i 0 −1.15305 − 0.786137i −1.96019 + 0.295451i 0 1.38027 + 2.25718i −1.64597 − 2.06398i 0 −1.52399 − 0.229704i
100.4 2.06858 + 0.638073i 0 2.21941 + 1.51317i 2.32995 − 0.351184i 0 0.990624 − 2.45330i 0.926114 + 1.16131i 0 5.04378 + 0.760227i
109.1 −1.90236 + 1.76513i 0 0.353823 − 4.72145i 1.29324 − 3.29513i 0 −2.54955 + 0.706959i 4.42482 + 5.54855i 0 3.35613 + 8.55127i
109.2 −0.615612 + 0.571205i 0 −0.0967566 + 1.29113i 0.568152 − 1.44763i 0 1.56778 − 2.13121i −1.72514 − 2.16325i 0 0.477130 + 1.21571i
109.3 0.492413 − 0.456893i 0 −0.115740 + 1.54445i 0.137164 − 0.349488i 0 0.0148404 + 2.64571i 1.48629 + 1.86375i 0 −0.0921371 − 0.234762i
109.4 1.29251 − 1.19927i 0 0.0828637 − 1.10574i −1.35494 + 3.45233i 0 1.47509 − 2.19638i 0.979680 + 1.22848i 0 2.38902 + 6.08712i
163.1 −0.687295 − 1.75120i 0 −1.12822 + 1.04683i −2.01668 + 1.37495i 0 −0.492953 + 2.59942i −0.781251 − 0.376230i 0 3.79385 + 2.58660i
163.2 −0.293772 − 0.748519i 0 0.992125 − 0.920558i 1.11822 − 0.762387i 0 1.80149 − 1.93768i −2.42946 − 1.16997i 0 −0.899162 − 0.613038i
163.3 0.633632 + 1.61447i 0 −0.738910 + 0.685609i −2.02525 + 1.38079i 0 2.55966 + 0.669438i 1.55011 + 0.746495i 0 −3.51251 − 2.39479i
163.4 0.712776 + 1.81612i 0 −1.32415 + 1.22863i 2.50104 − 1.70518i 0 −1.59223 − 2.11301i 0.340382 + 0.163920i 0 4.87950 + 3.32679i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
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 441.2.bb.c 48
3.b odd 2 1 147.2.m.a ✓ 48
49.g even 21 1 inner 441.2.bb.c 48
147.n odd 42 1 147.2.m.a ✓ 48
147.n odd 42 1 7203.2.a.i 24
147.o even 42 1 7203.2.a.k 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.m.a ✓ 48 3.b odd 2 1
147.2.m.a ✓ 48 147.n odd 42 1
441.2.bb.c 48 1.a even 1 1 trivial
441.2.bb.c 48 49.g even 21 1 inner
7203.2.a.i 24 147.n odd 42 1
7203.2.a.k 24 147.o even 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 \(S_{2}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display