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Properties

Label 798.2.bu.a
Level $798$
Weight $2$
Character orbit 798.bu
Analytic conductor $6.372$
Analytic rank $0$
Dimension $162$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [798,2,Mod(53,798)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(798, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 12, 11])) 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("798.53"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bu (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [162,0,0,0,0,0,0,-81] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(162\)
Relative dimension: \(27\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 162 q - 81 q^{8} + 15 q^{13} + 6 q^{14} + 27 q^{15} - 6 q^{17} - 18 q^{18} + 9 q^{19} - 6 q^{22} - 6 q^{25} - 9 q^{27} + 9 q^{28} + 6 q^{29} - 24 q^{33} + 6 q^{34} + 24 q^{35} + 18 q^{37} - 9 q^{39} - 9 q^{43}+ \cdots - 57 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 0.173648 + 0.984808i −1.71587 + 0.236219i −0.939693 + 0.342020i 1.34976 − 3.70842i −0.530587 − 1.64878i −2.42362 + 1.06115i −0.500000 − 0.866025i 2.88840 − 0.810640i 3.88647 + 0.685289i
53.2 0.173648 + 0.984808i −1.69691 + 0.347119i −0.939693 + 0.342020i 0.179971 − 0.494467i −0.636511 − 1.61086i 1.75103 + 1.98341i −0.500000 − 0.866025i 2.75902 − 1.17806i 0.518206 + 0.0913737i
53.3 0.173648 + 0.984808i −1.67343 + 0.446800i −0.939693 + 0.342020i −1.36048 + 3.73789i −0.730601 − 1.57042i −2.59662 − 0.507500i −0.500000 − 0.866025i 2.60074 − 1.49538i −3.91734 − 0.690734i
53.4 0.173648 + 0.984808i −1.60498 − 0.651180i −0.939693 + 0.342020i −0.122408 + 0.336313i 0.362585 − 1.69367i −2.54737 + 0.714792i −0.500000 − 0.866025i 2.15193 + 2.09026i −0.352459 − 0.0621481i
53.5 0.173648 + 0.984808i −1.59418 + 0.677189i −0.939693 + 0.342020i −0.859500 + 2.36146i −0.943728 − 1.45237i 1.57496 − 2.12591i −0.500000 − 0.866025i 2.08283 − 2.15913i −2.47483 − 0.436380i
53.6 0.173648 + 0.984808i −1.57618 − 0.718095i −0.939693 + 0.342020i 0.379776 − 1.04342i 0.433485 − 1.67693i 2.43049 − 1.04534i −0.500000 − 0.866025i 1.96868 + 2.26369i 1.09352 + 0.192817i
53.7 0.173648 + 0.984808i −1.15251 − 1.29295i −0.939693 + 0.342020i −1.20215 + 3.30288i 1.07318 − 1.35952i 0.891053 + 2.49119i −0.500000 − 0.866025i −0.343437 + 2.98028i −3.46145 − 0.610347i
53.8 0.173648 + 0.984808i −1.05722 + 1.37197i −0.939693 + 0.342020i 0.564194 − 1.55011i −1.53471 − 0.802916i −0.631754 − 2.56922i −0.500000 − 0.866025i −0.764586 − 2.90093i 1.62453 + 0.286449i
53.9 0.173648 + 0.984808i −0.928867 − 1.46192i −0.939693 + 0.342020i −0.507545 + 1.39447i 1.27841 − 1.16861i 0.546045 − 2.58879i −0.500000 − 0.866025i −1.27441 + 2.71586i −1.46142 − 0.257688i
53.10 0.173648 + 0.984808i −0.633326 − 1.61211i −0.939693 + 0.342020i 1.36633 − 3.75396i 1.47764 − 0.903645i −0.864472 − 2.50054i −0.500000 − 0.866025i −2.19780 + 2.04198i 3.93419 + 0.693704i
53.11 0.173648 + 0.984808i −0.372558 + 1.69151i −0.939693 + 0.342020i 0.146667 − 0.402963i −1.73050 − 0.0731705i 2.63845 + 0.196355i −0.500000 − 0.866025i −2.72240 − 1.26037i 0.422310 + 0.0744646i
53.12 0.173648 + 0.984808i −0.349299 + 1.69646i −0.939693 + 0.342020i 1.38704 − 3.81087i −1.73135 − 0.0494041i 2.01440 + 1.71528i −0.500000 − 0.866025i −2.75598 − 1.18515i 3.99383 + 0.704221i
53.13 0.173648 + 0.984808i −0.111227 + 1.72848i −0.939693 + 0.342020i −0.359967 + 0.989002i −1.72153 + 0.190609i −2.21348 − 1.44931i −0.500000 − 0.866025i −2.97526 − 0.384507i −1.03648 − 0.182760i
53.14 0.173648 + 0.984808i −0.00509466 − 1.73204i −0.939693 + 0.342020i −0.359410 + 0.987471i 1.70485 − 0.305783i −2.62362 − 0.341477i −0.500000 − 0.866025i −2.99995 + 0.0176484i −1.03488 − 0.182477i
53.15 0.173648 + 0.984808i 0.246871 − 1.71437i −0.939693 + 0.342020i 0.436242 − 1.19856i 1.73119 − 0.0545760i −0.993792 + 2.45201i −0.500000 − 0.866025i −2.87811 − 0.846456i 1.25611 + 0.221486i
53.16 0.173648 + 0.984808i 0.345131 − 1.69732i −0.939693 + 0.342020i −0.571310 + 1.56966i 1.73146 + 0.0451522i 2.51710 − 0.815004i −0.500000 − 0.866025i −2.76177 − 1.17159i −1.64502 − 0.290062i
53.17 0.173648 + 0.984808i 0.600751 + 1.62453i −0.939693 + 0.342020i −1.35069 + 3.71100i −1.49553 + 0.873721i 0.939767 + 2.47322i −0.500000 − 0.866025i −2.27820 + 1.95188i −3.88916 − 0.685764i
53.18 0.173648 + 0.984808i 0.816756 + 1.52739i −0.939693 + 0.342020i 0.0712889 − 0.195865i −1.36235 + 1.06958i −2.34853 + 1.21837i −0.500000 − 0.866025i −1.66582 + 2.49501i 0.205268 + 0.0361943i
53.19 0.173648 + 0.984808i 1.00105 − 1.41347i −0.939693 + 0.342020i 1.13790 − 3.12636i 1.56582 + 0.740400i 2.35336 − 1.20900i −0.500000 − 0.866025i −0.995780 − 2.82992i 3.27646 + 0.577729i
53.20 0.173648 + 0.984808i 1.04076 − 1.38449i −0.939693 + 0.342020i −1.09478 + 3.00789i 1.54419 + 0.784533i −1.00516 − 2.44738i −0.500000 − 0.866025i −0.833641 − 2.88185i −3.15230 − 0.555835i
See next 80 embeddings (of 162 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.27
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
399.br even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.bu.a ✓ 162
3.b odd 2 1 798.2.bu.b yes 162
7.c even 3 1 798.2.cc.a yes 162
19.f odd 18 1 798.2.cc.b yes 162
21.h odd 6 1 798.2.cc.b yes 162
57.j even 18 1 798.2.cc.a yes 162
133.be odd 18 1 798.2.bu.b yes 162
399.br even 18 1 inner 798.2.bu.a ✓ 162
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bu.a ✓ 162 1.a even 1 1 trivial
798.2.bu.a ✓ 162 399.br even 18 1 inner
798.2.bu.b yes 162 3.b odd 2 1
798.2.bu.b yes 162 133.be odd 18 1
798.2.cc.a yes 162 7.c even 3 1
798.2.cc.a yes 162 57.j even 18 1
798.2.cc.b yes 162 19.f odd 18 1
798.2.cc.b yes 162 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{162} + 3 T_{5}^{160} + 18 T_{5}^{159} + 27 T_{5}^{158} - 12 T_{5}^{157} - 13220 T_{5}^{156} + \cdots + 56\!\cdots\!68 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display