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Properties

Label 444.2.c.d
Level $444$
Weight $2$
Character orbit 444.c
Analytic conductor $3.545$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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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] = [444,2,Mod(371,444)] 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("444.371"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,0] 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: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{4} - 3 q^{6} - 2 q^{9} - 2 q^{10} - 11 q^{12} + 34 q^{16} + 21 q^{18} + 4 q^{21} - 8 q^{22} - 21 q^{24} - 32 q^{25} + 22 q^{28} + 28 q^{30} - 8 q^{33} - 42 q^{34} - 31 q^{36} - 36 q^{37} + 42 q^{40}+ \cdots + 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
371.1 −1.40958 − 0.114392i −0.186475 + 1.72198i 1.97383 + 0.322488i − 1.23599i 0.459831 − 2.40594i 1.14796i −2.74538 − 0.680362i −2.93045 − 0.642213i −0.141387 + 1.74222i
371.2 −1.40958 + 0.114392i −0.186475 − 1.72198i 1.97383 − 0.322488i 1.23599i 0.459831 + 2.40594i − 1.14796i −2.74538 + 0.680362i −2.93045 + 0.642213i −0.141387 − 1.74222i
371.3 −1.40586 − 0.153479i 1.09775 − 1.33976i 1.95289 + 0.431540i 0.486732i −1.74891 + 1.71503i − 2.92279i −2.67926 − 0.906413i −0.589899 − 2.94143i 0.0747031 − 0.684277i
371.4 −1.40586 + 0.153479i 1.09775 + 1.33976i 1.95289 − 0.431540i − 0.486732i −1.74891 − 1.71503i 2.92279i −2.67926 + 0.906413i −0.589899 + 2.94143i 0.0747031 + 0.684277i
371.5 −1.32526 − 0.493635i 1.36684 + 1.06383i 1.51265 + 1.30839i 4.43172i −1.28628 − 2.08458i − 0.340530i −1.35879 − 2.48066i 0.736517 + 2.90819i 2.18765 − 5.87320i
371.6 −1.32526 + 0.493635i 1.36684 − 1.06383i 1.51265 − 1.30839i − 4.43172i −1.28628 + 2.08458i 0.340530i −1.35879 + 2.48066i 0.736517 − 2.90819i 2.18765 + 5.87320i
371.7 −1.22055 − 0.714322i −1.25718 + 1.19143i 0.979488 + 1.74373i 0.781284i 2.38551 − 0.556170i − 2.09732i 0.0500714 − 2.82798i 0.160994 − 2.99568i 0.558088 − 0.953596i
371.8 −1.22055 + 0.714322i −1.25718 − 1.19143i 0.979488 − 1.74373i − 0.781284i 2.38551 + 0.556170i 2.09732i 0.0500714 + 2.82798i 0.160994 + 2.99568i 0.558088 + 0.953596i
371.9 −0.907099 − 1.08498i −1.67776 − 0.430274i −0.354342 + 1.96836i − 1.79178i 1.05506 + 2.21062i 1.23053i 2.45704 − 1.40105i 2.62973 + 1.44379i −1.94403 + 1.62532i
371.10 −0.907099 + 1.08498i −1.67776 + 0.430274i −0.354342 − 1.96836i 1.79178i 1.05506 − 2.21062i − 1.23053i 2.45704 + 1.40105i 2.62973 − 1.44379i −1.94403 − 1.62532i
371.11 −0.587205 − 1.28654i 1.09126 + 1.34505i −1.31038 + 1.51093i 0.966829i 1.08967 − 2.19377i − 3.52282i 2.71333 + 0.798633i −0.618304 + 2.93559i 1.24387 − 0.567727i
371.12 −0.587205 + 1.28654i 1.09126 − 1.34505i −1.31038 − 1.51093i − 0.966829i 1.08967 + 2.19377i 3.52282i 2.71333 − 0.798633i −0.618304 − 2.93559i 1.24387 + 0.567727i
371.13 −0.572407 − 1.29319i 0.253239 − 1.71344i −1.34470 + 1.48047i − 2.40717i −2.36076 + 0.653297i 1.45151i 2.68425 + 0.891530i −2.87174 − 0.867817i −3.11294 + 1.37788i
371.14 −0.572407 + 1.29319i 0.253239 + 1.71344i −1.34470 − 1.48047i 2.40717i −2.36076 − 0.653297i − 1.45151i 2.68425 − 0.891530i −2.87174 + 0.867817i −3.11294 − 1.37788i
371.15 −0.154417 − 1.40576i −1.68294 + 0.409533i −1.95231 + 0.434145i 3.46237i 0.835578 + 2.30257i − 3.74077i 0.911773 + 2.67744i 2.66457 − 1.37844i 4.86726 − 0.534649i
371.16 −0.154417 + 1.40576i −1.68294 − 0.409533i −1.95231 − 0.434145i − 3.46237i 0.835578 − 2.30257i 3.74077i 0.911773 − 2.67744i 2.66457 + 1.37844i 4.86726 + 0.534649i
371.17 −0.146420 − 1.40661i 1.28814 + 1.15789i −1.95712 + 0.411913i − 3.00951i 1.44009 − 1.98145i 4.53873i 0.865964 + 2.69260i 0.318591 + 2.98304i −4.23321 + 0.440652i
371.18 −0.146420 + 1.40661i 1.28814 − 1.15789i −1.95712 − 0.411913i 3.00951i 1.44009 + 1.98145i − 4.53873i 0.865964 − 2.69260i 0.318591 − 2.98304i −4.23321 − 0.440652i
371.19 0.146420 − 1.40661i −1.28814 − 1.15789i −1.95712 − 0.411913i − 3.00951i −1.81731 + 1.64237i − 4.53873i −0.865964 + 2.69260i 0.318591 + 2.98304i −4.23321 − 0.440652i
371.20 0.146420 + 1.40661i −1.28814 + 1.15789i −1.95712 + 0.411913i 3.00951i −1.81731 − 1.64237i 4.53873i −0.865964 − 2.69260i 0.318591 − 2.98304i −4.23321 + 0.440652i
See all 36 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 371.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 444.2.c.d ✓ 36
3.b odd 2 1 inner 444.2.c.d ✓ 36
4.b odd 2 1 inner 444.2.c.d ✓ 36
12.b even 2 1 inner 444.2.c.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.c.d ✓ 36 1.a even 1 1 trivial
444.2.c.d ✓ 36 3.b odd 2 1 inner
444.2.c.d ✓ 36 4.b odd 2 1 inner
444.2.c.d ✓ 36 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\):

\( T_{5}^{18} + 53 T_{5}^{16} + 1075 T_{5}^{14} + 10774 T_{5}^{12} + 57426 T_{5}^{10} + 164560 T_{5}^{8} + \cdots + 8192 \) Copy content Toggle raw display
\( T_{11}^{18} - 104 T_{11}^{16} + 4195 T_{11}^{14} - 83541 T_{11}^{12} + 875652 T_{11}^{10} + \cdots - 819200 \) Copy content Toggle raw display