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Properties

Label 3703.2.a.s
Level $3703$
Weight $2$
Character orbit 3703.a
Self dual yes
Analytic conductor $29.569$
Analytic rank $0$
Dimension $35$
CM no
Inner twists $1$

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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] = [3703,2,Mod(1,3703)] 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("3703.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3703, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3703 = 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3703.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [35,9,9,35,-2,23,35] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.5686038685\)
Analytic rank: \(0\)
Dimension: \(35\)
Twist minimal: no (minimal twist has level 161)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 35 q + 9 q^{2} + 9 q^{3} + 35 q^{4} - 2 q^{5} + 23 q^{6} + 35 q^{7} + 39 q^{8} + 50 q^{9} - 4 q^{10} + 16 q^{11} + 34 q^{12} + 15 q^{13} + 9 q^{14} + 5 q^{15} + 35 q^{16} - 2 q^{17} + 38 q^{18} - 4 q^{19}+ \cdots + 130 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.52959 −1.34102 4.39881 0.292635 3.39222 1.00000 −6.06798 −1.20167 −0.740246
1.2 −2.25407 3.06146 3.08081 −0.837888 −6.90074 1.00000 −2.43621 6.37255 1.88865
1.3 −2.21078 0.702853 2.88757 −2.24685 −1.55386 1.00000 −1.96222 −2.50600 4.96729
1.4 −2.18773 −2.44630 2.78616 1.84072 5.35183 1.00000 −1.71989 2.98436 −4.02700
1.5 −1.97519 1.23186 1.90139 −1.74201 −2.43317 1.00000 0.194769 −1.48252 3.44081
1.6 −1.71369 −2.44888 0.936720 3.71582 4.19661 1.00000 1.82213 2.99700 −6.36775
1.7 −1.64398 −0.597140 0.702666 −1.87885 0.981685 1.00000 2.13279 −2.64342 3.08879
1.8 −1.64096 0.758984 0.692758 −4.13680 −1.24546 1.00000 2.14514 −2.42394 6.78833
1.9 −1.28255 −1.77698 −0.355076 −0.249342 2.27906 1.00000 3.02049 0.157650 0.319793
1.10 −1.17022 −2.06028 −0.630575 3.39473 2.41099 1.00000 3.07836 1.24476 −3.97260
1.11 −1.13945 3.22849 −0.701658 3.04321 −3.67870 1.00000 3.07840 7.42315 −3.46758
1.12 −0.723027 0.158657 −1.47723 2.33642 −0.114713 1.00000 2.51413 −2.97483 −1.68930
1.13 −0.668433 1.91962 −1.55320 2.57475 −1.28314 1.00000 2.37507 0.684939 −1.72105
1.14 −0.535301 2.23159 −1.71345 0.746222 −1.19457 1.00000 1.98782 1.97998 −0.399454
1.15 −0.255701 −1.58354 −1.93462 −0.906911 0.404914 1.00000 1.00609 −0.492387 0.231898
1.16 −0.237583 −2.56203 −1.94355 −3.46405 0.608695 1.00000 0.936923 3.56398 0.823001
1.17 −0.186718 3.29789 −1.96514 −3.57308 −0.615776 1.00000 0.740362 7.87608 0.667158
1.18 0.224769 −3.18096 −1.94948 −2.82377 −0.714981 1.00000 −0.887719 7.11853 −0.634695
1.19 0.558777 −0.457582 −1.68777 0.855963 −0.255687 1.00000 −2.06064 −2.79062 0.478293
1.20 0.837998 1.49104 −1.29776 3.97461 1.24949 1.00000 −2.76352 −0.776813 3.33071
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)
\(23\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3703.2.a.s 35
23.b odd 2 1 3703.2.a.t 35
23.c even 11 2 161.2.i.b ✓ 70
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.i.b ✓ 70 23.c even 11 2
3703.2.a.s 35 1.a even 1 1 trivial
3703.2.a.t 35 23.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3703))\):

\( T_{2}^{35} - 9 T_{2}^{34} - 12 T_{2}^{33} + 326 T_{2}^{32} - 381 T_{2}^{31} - 5149 T_{2}^{30} + \cdots + 1607 \) Copy content Toggle raw display
\( T_{5}^{35} + 2 T_{5}^{34} - 109 T_{5}^{33} - 217 T_{5}^{32} + 5328 T_{5}^{31} + 10622 T_{5}^{30} + \cdots - 22076416 \) Copy content Toggle raw display