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Properties

Label 714.2.bh.b
Level $714$
Weight $2$
Character orbit 714.bh
Analytic conductor $5.701$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [714,2,Mod(25,714)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(714, base_ring=CyclotomicField(24)) chi = DirichletCharacter(H, H._module([0, 16, 15])) 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("714.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.bh (of order \(24\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,0,0,8,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 8 q^{5} + 8 q^{10} + 16 q^{14} + 16 q^{15} + 48 q^{16} + 8 q^{17} + 48 q^{18} - 8 q^{19} + 16 q^{22} + 24 q^{25} + 32 q^{28} - 32 q^{29} - 16 q^{33} - 48 q^{37} - 16 q^{39} + 128 q^{41} + 16 q^{42}+ \cdots + 160 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.965926 − 0.258819i −0.608761 − 0.793353i 0.866025 + 0.500000i −0.337776 − 2.56567i 0.382683 + 0.923880i −0.00794442 + 2.64574i −0.707107 − 0.707107i −0.258819 + 0.965926i −0.337776 + 2.56567i
25.2 −0.965926 − 0.258819i −0.608761 − 0.793353i 0.866025 + 0.500000i −0.135385 − 1.02835i 0.382683 + 0.923880i 0.105025 − 2.64367i −0.707107 − 0.707107i −0.258819 + 0.965926i −0.135385 + 1.02835i
25.3 −0.965926 − 0.258819i −0.608761 − 0.793353i 0.866025 + 0.500000i −0.118819 − 0.902520i 0.382683 + 0.923880i 2.16960 + 1.51421i −0.707107 − 0.707107i −0.258819 + 0.965926i −0.118819 + 0.902520i
25.4 −0.965926 − 0.258819i −0.608761 − 0.793353i 0.866025 + 0.500000i 0.128065 + 0.972754i 0.382683 + 0.923880i −2.23332 − 1.41855i −0.707107 − 0.707107i −0.258819 + 0.965926i 0.128065 − 0.972754i
25.5 −0.965926 − 0.258819i −0.608761 − 0.793353i 0.866025 + 0.500000i 0.430273 + 3.26824i 0.382683 + 0.923880i 0.971259 + 2.46103i −0.707107 − 0.707107i −0.258819 + 0.965926i 0.430273 − 3.26824i
25.6 −0.965926 − 0.258819i −0.608761 − 0.793353i 0.866025 + 0.500000i 0.459415 + 3.48960i 0.382683 + 0.923880i 1.51635 − 2.16810i −0.707107 − 0.707107i −0.258819 + 0.965926i 0.459415 − 3.48960i
25.7 −0.965926 − 0.258819i 0.608761 + 0.793353i 0.866025 + 0.500000i −0.416216 − 3.16147i −0.382683 − 0.923880i 0.299906 + 2.62870i −0.707107 − 0.707107i −0.258819 + 0.965926i −0.416216 + 3.16147i
25.8 −0.965926 − 0.258819i 0.608761 + 0.793353i 0.866025 + 0.500000i −0.107430 − 0.816011i −0.382683 − 0.923880i 2.49739 − 0.873515i −0.707107 − 0.707107i −0.258819 + 0.965926i −0.107430 + 0.816011i
25.9 −0.965926 − 0.258819i 0.608761 + 0.793353i 0.866025 + 0.500000i −0.102499 − 0.778556i −0.382683 − 0.923880i 2.60629 + 0.455261i −0.707107 − 0.707107i −0.258819 + 0.965926i −0.102499 + 0.778556i
25.10 −0.965926 − 0.258819i 0.608761 + 0.793353i 0.866025 + 0.500000i −0.0294272 − 0.223522i −0.382683 − 0.923880i −2.00544 − 1.72575i −0.707107 − 0.707107i −0.258819 + 0.965926i −0.0294272 + 0.223522i
25.11 −0.965926 − 0.258819i 0.608761 + 0.793353i 0.866025 + 0.500000i 0.305655 + 2.32168i −0.382683 − 0.923880i −1.22034 + 2.34750i −0.707107 − 0.707107i −0.258819 + 0.965926i 0.305655 − 2.32168i
25.12 −0.965926 − 0.258819i 0.608761 + 0.793353i 0.866025 + 0.500000i 0.406505 + 3.08771i −0.382683 − 0.923880i −2.26996 − 1.35915i −0.707107 − 0.707107i −0.258819 + 0.965926i 0.406505 − 3.08771i
121.1 0.965926 − 0.258819i −0.793353 − 0.608761i 0.866025 − 0.500000i −3.71943 − 0.489672i −0.923880 − 0.382683i 2.41756 + 1.07490i 0.707107 − 0.707107i 0.258819 + 0.965926i −3.71943 + 0.489672i
121.2 0.965926 − 0.258819i −0.793353 − 0.608761i 0.866025 − 0.500000i −1.87389 − 0.246703i −0.923880 − 0.382683i −2.13946 + 1.55651i 0.707107 − 0.707107i 0.258819 + 0.965926i −1.87389 + 0.246703i
121.3 0.965926 − 0.258819i −0.793353 − 0.608761i 0.866025 − 0.500000i −0.0113440 − 0.00149347i −0.923880 − 0.382683i 0.142386 − 2.64192i 0.707107 − 0.707107i 0.258819 + 0.965926i −0.0113440 + 0.00149347i
121.4 0.965926 − 0.258819i −0.793353 − 0.608761i 0.866025 − 0.500000i 0.145332 + 0.0191333i −0.923880 − 0.382683i −1.41829 + 2.23349i 0.707107 − 0.707107i 0.258819 + 0.965926i 0.145332 − 0.0191333i
121.5 0.965926 − 0.258819i −0.793353 − 0.608761i 0.866025 − 0.500000i 1.58279 + 0.208378i −0.923880 − 0.382683i 2.57043 − 0.626823i 0.707107 − 0.707107i 0.258819 + 0.965926i 1.58279 − 0.208378i
121.6 0.965926 − 0.258819i −0.793353 − 0.608761i 0.866025 − 0.500000i 3.23325 + 0.425665i −0.923880 − 0.382683i 0.135863 + 2.64226i 0.707107 − 0.707107i 0.258819 + 0.965926i 3.23325 − 0.425665i
121.7 0.965926 − 0.258819i 0.793353 + 0.608761i 0.866025 − 0.500000i −3.39033 − 0.446345i 0.923880 + 0.382683i 1.71000 − 2.01888i 0.707107 − 0.707107i 0.258819 + 0.965926i −3.39033 + 0.446345i
121.8 0.965926 − 0.258819i 0.793353 + 0.608761i 0.866025 − 0.500000i −3.03089 − 0.399024i 0.923880 + 0.382683i 0.328120 + 2.62533i 0.707107 − 0.707107i 0.258819 + 0.965926i −3.03089 + 0.399024i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
17.d even 8 1 inner
119.q even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.bh.b ✓ 96
7.c even 3 1 inner 714.2.bh.b ✓ 96
17.d even 8 1 inner 714.2.bh.b ✓ 96
119.q even 24 1 inner 714.2.bh.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.bh.b ✓ 96 1.a even 1 1 trivial
714.2.bh.b ✓ 96 7.c even 3 1 inner
714.2.bh.b ✓ 96 17.d even 8 1 inner
714.2.bh.b ✓ 96 119.q even 24 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{96} - 8 T_{5}^{95} + 20 T_{5}^{94} + 64 T_{5}^{93} - 568 T_{5}^{92} + 1488 T_{5}^{91} + \cdots + 15527402881 \) acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\). Copy content Toggle raw display