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Properties

Label 702.2.t.a
Level $702$
Weight $2$
Character orbit 702.t
Analytic conductor $5.605$
Analytic rank $0$
Dimension $28$
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] = [702,2,Mod(181,702)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(702, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) 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("702.181"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 702.t (of order \(6\), degree \(2\), not 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: \(5.60549822189\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 234)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{4} + 2 q^{13} - 8 q^{14} - 14 q^{16} - 16 q^{17} + 8 q^{23} + 14 q^{25} - 8 q^{26} + 16 q^{29} + 68 q^{35} - 4 q^{43} + 10 q^{49} - 2 q^{52} + 120 q^{53} + 8 q^{56} + 28 q^{61} - 68 q^{62}+ \cdots - 8 q^{92}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −3.32246 + 1.91822i 0 −2.91802 − 1.68472i − 1.00000i 0 3.83645
181.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −2.40542 + 1.38877i 0 0.759011 + 0.438215i − 1.00000i 0 2.77754
181.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.515076 + 0.297379i 0 −1.45217 − 0.838409i − 1.00000i 0 0.594758
181.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.419378 − 0.242128i 0 4.37722 + 2.52719i − 1.00000i 0 −0.484256
181.5 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.548308 − 0.316566i 0 2.15366 + 1.24342i − 1.00000i 0 −0.633132
181.6 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.53992 − 1.46642i 0 1.90832 + 1.10177i − 1.00000i 0 −2.93284
181.7 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.73536 − 1.57926i 0 −1.36392 − 0.787458i − 1.00000i 0 −3.15852
181.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.73536 + 1.57926i 0 1.36392 + 0.787458i 1.00000i 0 −3.15852
181.9 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.53992 + 1.46642i 0 −1.90832 − 1.10177i 1.00000i 0 −2.93284
181.10 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.548308 + 0.316566i 0 −2.15366 − 1.24342i 1.00000i 0 −0.633132
181.11 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.419378 + 0.242128i 0 −4.37722 − 2.52719i 1.00000i 0 −0.484256
181.12 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.515076 − 0.297379i 0 1.45217 + 0.838409i 1.00000i 0 0.594758
181.13 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.40542 − 1.38877i 0 −0.759011 − 0.438215i 1.00000i 0 2.77754
181.14 0.866025 + 0.500000i 0 0.500000 + 0.866025i 3.32246 − 1.91822i 0 2.91802 + 1.68472i 1.00000i 0 3.83645
415.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −3.32246 − 1.91822i 0 −2.91802 + 1.68472i 1.00000i 0 3.83645
415.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −2.40542 − 1.38877i 0 0.759011 − 0.438215i 1.00000i 0 2.77754
415.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.515076 − 0.297379i 0 −1.45217 + 0.838409i 1.00000i 0 0.594758
415.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.419378 + 0.242128i 0 4.37722 − 2.52719i 1.00000i 0 −0.484256
415.5 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.548308 + 0.316566i 0 2.15366 − 1.24342i 1.00000i 0 −0.633132
415.6 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 2.53992 + 1.46642i 0 1.90832 − 1.10177i 1.00000i 0 −2.93284
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
13.b even 2 1 inner
117.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 702.2.t.a 28
3.b odd 2 1 234.2.t.a ✓ 28
9.c even 3 1 inner 702.2.t.a 28
9.c even 3 1 2106.2.b.d 14
9.d odd 6 1 234.2.t.a ✓ 28
9.d odd 6 1 2106.2.b.c 14
13.b even 2 1 inner 702.2.t.a 28
39.d odd 2 1 234.2.t.a ✓ 28
117.n odd 6 1 234.2.t.a ✓ 28
117.n odd 6 1 2106.2.b.c 14
117.t even 6 1 inner 702.2.t.a 28
117.t even 6 1 2106.2.b.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.2.t.a ✓ 28 3.b odd 2 1
234.2.t.a ✓ 28 9.d odd 6 1
234.2.t.a ✓ 28 39.d odd 2 1
234.2.t.a ✓ 28 117.n odd 6 1
702.2.t.a 28 1.a even 1 1 trivial
702.2.t.a 28 9.c even 3 1 inner
702.2.t.a 28 13.b even 2 1 inner
702.2.t.a 28 117.t even 6 1 inner
2106.2.b.c 14 9.d odd 6 1
2106.2.b.c 14 117.n odd 6 1
2106.2.b.d 14 9.c even 3 1
2106.2.b.d 14 117.t even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(702, [\chi])\).