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Properties

Label 207.2.m.a
Level $207$
Weight $2$
Character orbit 207.m
Analytic conductor $1.653$
Analytic rank $0$
Dimension $440$
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] = [207,2,Mod(4,207)] 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("207.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(207, base_ring=CyclotomicField(66)) chi = DirichletCharacter(H, H._module([22, 12])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 207.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.65290332184\)
Analytic rank: \(0\)
Dimension: \(440\)
Relative dimension: \(22\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 440 q - 9 q^{2} - 20 q^{3} + 11 q^{4} - 11 q^{5} - 17 q^{6} - 9 q^{7} - 50 q^{8} - 24 q^{9} - 44 q^{10} - 17 q^{11} - 30 q^{12} - 9 q^{13} - 21 q^{14} - 34 q^{15} + 11 q^{16} - 20 q^{17} + 3 q^{18} - 36 q^{19}+ \cdots - 220 q^{99}+O(q^{100}) \) 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 −1.46746 + 2.06076i −1.47811 + 0.902882i −1.43916 − 4.15818i 2.97107 − 1.53169i 0.308441 − 4.37097i 0.452583 − 0.355916i 5.82617 + 1.71072i 1.36961 − 2.66912i −1.20348 + 8.37037i
4.2 −1.42424 + 2.00007i −0.974750 − 1.43173i −1.31767 − 3.80716i −1.71157 + 0.882374i 4.25184 + 0.0895707i 1.45139 − 1.14139i 4.77948 + 1.40338i −1.09973 + 2.79116i 0.672877 − 4.67996i
4.3 −1.30169 + 1.82796i 0.211378 + 1.71910i −0.992927 − 2.86887i −1.37586 + 0.709305i −3.41761 − 1.85134i 0.277555 − 0.218272i 2.23034 + 0.654887i −2.91064 + 0.726763i 0.494355 − 3.43832i
4.4 −1.08192 + 1.51935i 1.47170 − 0.913295i −0.483727 − 1.39764i 0.797744 − 0.411266i −0.204651 + 3.22414i 2.76230 − 2.17230i −0.932440 − 0.273789i 1.33179 − 2.68819i −0.238242 + 1.65701i
4.5 −0.976060 + 1.37068i 1.50753 + 0.852857i −0.271946 − 0.785737i 1.76457 − 0.909700i −2.64043 + 1.23390i −3.19096 + 2.50940i −1.88664 − 0.553967i 1.54527 + 2.57141i −0.475416 + 3.30659i
4.6 −0.954897 + 1.34096i −1.52649 − 0.818435i −0.232223 − 0.670964i 1.25699 − 0.648021i 2.55513 − 1.26545i −2.05983 + 1.61987i −2.03757 − 0.598284i 1.66033 + 2.49866i −0.331318 + 2.30437i
4.7 −0.665677 + 0.934812i −1.60332 + 0.655269i 0.223388 + 0.645436i −3.14339 + 1.62053i 0.454737 − 1.93500i 1.17978 − 0.927791i −2.95431 − 0.867462i 2.14125 − 2.10121i 0.577591 − 4.01723i
4.8 −0.569295 + 0.799462i 0.758988 − 1.55690i 0.339092 + 0.979743i −3.24434 + 1.67257i 0.812595 + 1.49312i −3.31555 + 2.60738i −2.85969 − 0.839681i −1.84787 − 2.36334i 0.509825 − 3.54591i
4.9 −0.381565 + 0.535833i 0.00453809 − 1.73204i 0.512611 + 1.48109i 3.01055 − 1.55205i 0.926355 + 0.663319i −0.335362 + 0.263732i −2.25153 − 0.661109i −2.99996 − 0.0157204i −0.317082 + 2.20536i
4.10 −0.229452 + 0.322220i 0.0945687 + 1.72947i 0.602958 + 1.74213i 2.96492 − 1.52852i −0.578969 − 0.366358i 3.41188 − 2.68313i −1.45879 − 0.428340i −2.98211 + 0.327107i −0.187786 + 1.30608i
4.11 −0.176547 + 0.247925i 1.62057 + 0.611351i 0.623838 + 1.80246i −1.66769 + 0.859756i −0.437676 + 0.293848i 1.42051 − 1.11710i −1.14108 − 0.335050i 2.25250 + 1.98147i 0.0812707 − 0.565250i
4.12 0.0256606 − 0.0360353i −1.16039 + 1.28588i 0.653496 + 1.88815i 0.800365 − 0.412617i 0.0165606 + 0.0748116i −3.08653 + 2.42727i 0.169702 + 0.0498289i −0.306974 − 2.98425i 0.00566909 − 0.0394294i
4.13 0.187803 − 0.263732i −1.56096 − 0.750610i 0.619851 + 1.79094i 0.277733 − 0.143181i −0.491112 + 0.270708i 0.387605 − 0.304816i 1.21004 + 0.355301i 1.87317 + 2.34334i 0.0143975 − 0.100137i
4.14 0.541462 − 0.760376i 1.52956 − 0.812681i 0.369144 + 1.06657i −0.806242 + 0.415647i 0.210254 − 1.60308i −0.783251 + 0.615955i 2.80218 + 0.822793i 1.67910 − 2.48609i −0.120501 + 0.838104i
4.15 0.680467 − 0.955583i 0.453546 − 1.67161i 0.204034 + 0.589516i 0.783941 − 0.404150i −1.28874 − 1.57088i 0.987285 − 0.776409i 2.95334 + 0.867179i −2.58859 − 1.51631i 0.147248 − 1.02413i
4.16 0.744303 − 1.04523i 0.0216039 + 1.73192i 0.115623 + 0.334070i −3.33608 + 1.71987i 1.82633 + 1.26649i 0.176203 − 0.138568i 2.89759 + 0.850810i −2.99907 + 0.0748322i −0.685401 + 4.76707i
4.17 0.865712 − 1.21572i −1.67168 + 0.453296i −0.0743883 − 0.214931i 0.376170 − 0.193929i −0.896114 + 2.42473i 3.39171 − 2.66727i 2.53832 + 0.745317i 2.58905 − 1.51553i 0.0898911 − 0.625206i
4.18 0.985191 − 1.38351i 0.806693 + 1.53273i −0.289355 − 0.836037i 1.87297 − 0.965585i 2.91528 + 0.393962i −2.05250 + 1.61410i 1.81755 + 0.533680i −1.69849 + 2.47288i 0.509343 − 3.54256i
4.19 1.30535 − 1.83310i −1.23309 − 1.21634i −1.00220 − 2.89566i 2.91364 − 1.50208i −3.83929 + 0.672636i −1.86236 + 1.46457i −2.29782 − 0.674701i 0.0410290 + 2.99972i 1.04983 − 7.30173i
4.20 1.34512 − 1.88895i −0.263222 − 1.71193i −1.10466 − 3.19172i −3.02169 + 1.55779i −3.58782 − 1.80553i 1.82903 − 1.43837i −3.06489 − 0.899932i −2.86143 + 0.901238i −1.12193 + 7.80323i
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
23.c even 11 1 inner
207.m even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.2.m.a ✓ 440
3.b odd 2 1 621.2.q.a 440
9.c even 3 1 inner 207.2.m.a ✓ 440
9.d odd 6 1 621.2.q.a 440
23.c even 11 1 inner 207.2.m.a ✓ 440
69.h odd 22 1 621.2.q.a 440
207.m even 33 1 inner 207.2.m.a ✓ 440
207.n odd 66 1 621.2.q.a 440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.2.m.a ✓ 440 1.a even 1 1 trivial
207.2.m.a ✓ 440 9.c even 3 1 inner
207.2.m.a ✓ 440 23.c even 11 1 inner
207.2.m.a ✓ 440 207.m even 33 1 inner
621.2.q.a 440 3.b odd 2 1
621.2.q.a 440 9.d odd 6 1
621.2.q.a 440 69.h odd 22 1
621.2.q.a 440 207.n odd 66 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(207, [\chi])\).