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Properties

Label 722.2.i.b
Level $722$
Weight $2$
Character orbit 722.i
Analytic conductor $5.765$
Analytic rank $0$
Dimension $612$
Inner twists $2$

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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] = [722,2,Mod(7,722)] 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("722.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(722, base_ring=CyclotomicField(114)) chi = DirichletCharacter(H, H._module([50])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.i (of order \(57\), degree \(36\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [612] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(612\)
Relative dimension: \(17\) over \(\Q(\zeta_{57})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{57}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 612 q + 17 q^{2} - 19 q^{3} + 17 q^{4} + q^{5} - 2 q^{7} - 34 q^{8} + 4 q^{9} + q^{10} - 28 q^{11} + 4 q^{13} - 18 q^{14} - 52 q^{15} + 17 q^{16} + 18 q^{17} - 46 q^{18} - 12 q^{19} - 2 q^{20} - 65 q^{21}+ \cdots - 116 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.191711 − 0.981451i −3.10452 + 0.343608i −0.926494 + 0.376309i 0.978297 − 0.951702i 0.932404 + 2.98106i −3.55508 + 2.76703i 0.546948 + 0.837166i 6.59258 − 1.47743i −1.12160 − 0.777700i
7.2 −0.191711 − 0.981451i −2.98587 + 0.330476i −0.926494 + 0.376309i 2.65519 − 2.58301i 0.896768 + 2.86713i 3.52766 − 2.74569i 0.546948 + 0.837166i 5.87879 − 1.31747i −3.04413 − 2.11075i
7.3 −0.191711 − 0.981451i −2.14479 + 0.237385i −0.926494 + 0.376309i −1.67672 + 1.63114i 0.644161 + 2.05950i 2.08346 − 1.62162i 0.546948 + 0.837166i 1.61637 − 0.362238i 1.92233 + 1.33292i
7.4 −0.191711 − 0.981451i −2.06111 + 0.228124i −0.926494 + 0.376309i −2.51756 + 2.44912i 0.619030 + 1.97915i −2.55923 + 1.99193i 0.546948 + 0.837166i 1.26876 − 0.284335i 2.88634 + 2.00134i
7.5 −0.191711 − 0.981451i −2.00265 + 0.221654i −0.926494 + 0.376309i 0.168453 − 0.163874i 0.601472 + 1.92301i 0.945236 − 0.735707i 0.546948 + 0.837166i 1.03410 − 0.231747i −0.193129 − 0.133913i
7.6 −0.191711 − 0.981451i −1.01174 + 0.111979i −0.926494 + 0.376309i 1.78089 − 1.73247i 0.303863 + 0.971503i 0.733877 − 0.571199i 0.546948 + 0.837166i −1.91632 + 0.429457i −2.04176 − 1.41572i
7.7 −0.191711 − 0.981451i −0.938145 + 0.103834i −0.926494 + 0.376309i −2.13625 + 2.07818i 0.281760 + 0.900838i 3.84605 − 2.99350i 0.546948 + 0.837166i −2.05805 + 0.461221i 2.44917 + 1.69822i
7.8 −0.191711 − 0.981451i −0.569788 + 0.0630641i −0.926494 + 0.376309i 0.634360 − 0.617115i 0.171129 + 0.547129i −3.33569 + 2.59628i 0.546948 + 0.837166i −2.60671 + 0.584177i −0.727282 − 0.504286i
7.9 −0.191711 − 0.981451i −0.239790 + 0.0265399i −0.926494 + 0.376309i 2.82465 − 2.74786i 0.0720180 + 0.230254i −1.19053 + 0.926625i 0.546948 + 0.837166i −2.87059 + 0.643315i −3.23841 − 2.24546i
7.10 −0.191711 − 0.981451i −0.158755 + 0.0175709i −0.926494 + 0.376309i 0.872040 − 0.848334i 0.0476800 + 0.152441i −1.10423 + 0.859458i 0.546948 + 0.837166i −2.90249 + 0.650464i −0.999778 − 0.693231i
7.11 −0.191711 − 0.981451i 0.617860 − 0.0683847i −0.926494 + 0.376309i −0.614082 + 0.597388i −0.185567 − 0.593289i 1.07269 − 0.834910i 0.546948 + 0.837166i −2.55031 + 0.571539i 0.704034 + 0.488166i
7.12 −0.191711 − 0.981451i 1.23882 − 0.137113i −0.926494 + 0.376309i −1.33225 + 1.29603i −0.372065 − 1.18956i −1.17328 + 0.913200i 0.546948 + 0.837166i −1.41151 + 0.316326i 1.52740 + 1.05907i
7.13 −0.191711 − 0.981451i 1.92323 − 0.212863i −0.926494 + 0.376309i −2.69730 + 2.62397i −0.577618 − 1.84675i 2.10128 − 1.63549i 0.546948 + 0.837166i 0.726103 − 0.162724i 3.09240 + 2.14423i
7.14 −0.191711 − 0.981451i 2.40435 − 0.266113i −0.926494 + 0.376309i 0.204013 − 0.198466i −0.722116 − 2.30873i 0.434702 − 0.338342i 0.546948 + 0.837166i 2.78268 − 0.623612i −0.233897 − 0.162180i
7.15 −0.191711 − 0.981451i 2.40915 − 0.266645i −0.926494 + 0.376309i 2.50090 − 2.43291i −0.723560 − 2.31335i −0.591047 + 0.460031i 0.546948 + 0.837166i 2.80553 − 0.628735i −2.86723 − 1.98809i
7.16 −0.191711 − 0.981451i 3.06305 − 0.339018i −0.926494 + 0.376309i 1.04887 − 1.02036i −0.919949 − 2.94124i 2.94199 − 2.28984i 0.546948 + 0.837166i 6.33994 − 1.42081i −1.20251 − 0.833801i
7.17 −0.191711 − 0.981451i 3.37734 − 0.373804i −0.926494 + 0.376309i −1.97671 + 1.92298i −1.01434 − 3.24304i −3.88410 + 3.02312i 0.546948 + 0.837166i 8.33933 − 1.86889i 2.26626 + 1.57139i
11.1 −0.592235 − 0.805765i −0.447948 − 3.23037i −0.298515 + 0.954405i 2.48156 + 1.72067i −2.33763 + 2.27408i −1.96145 + 2.13070i 0.945817 − 0.324699i −7.34782 + 2.07777i −0.0832070 − 3.01860i
11.2 −0.592235 − 0.805765i −0.413326 − 2.98069i −0.298515 + 0.954405i −2.01439 − 1.39674i −2.15695 + 2.09831i 3.01918 − 3.27970i 0.945817 − 0.324699i −5.82688 + 1.64769i 0.0675427 + 2.45032i
11.3 −0.592235 − 0.805765i −0.381900 − 2.75406i −0.298515 + 0.954405i −3.38886 − 2.34978i −1.99295 + 1.93878i −2.94059 + 3.19433i 0.945817 − 0.324699i −4.55222 + 1.28724i 0.113629 + 4.12225i
See next 80 embeddings (of 612 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.17
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
361.i even 57 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.i.b ✓ 612
361.i even 57 1 inner 722.2.i.b ✓ 612
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.i.b ✓ 612 1.a even 1 1 trivial
722.2.i.b ✓ 612 361.i even 57 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{612} + 19 T_{3}^{611} + 153 T_{3}^{610} + 568 T_{3}^{609} - 299 T_{3}^{608} + \cdots + 15\!\cdots\!09 \) acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\). Copy content Toggle raw display