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Properties

Label 798.2.bo.e
Level $798$
Weight $2$
Character orbit 798.bo
Analytic conductor $6.372$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [798,2,Mod(43,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bo (of order \(9\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 42x^{16} + 639x^{14} + 4594x^{12} + 16845x^{10} + 31167x^{8} + 26608x^{6} + 8322x^{4} + 396x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + \beta_{12} q^{3} + ( - \beta_{11} + \beta_{10}) q^{4} + (\beta_{12} - \beta_{11} + \cdots + \beta_{3}) q^{5}+ \cdots - \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} + \beta_{12} q^{3} + ( - \beta_{11} + \beta_{10}) q^{4} + (\beta_{12} - \beta_{11} + \cdots + \beta_{3}) q^{5}+ \cdots + ( - \beta_{15} - \beta_{13} + \cdots + \beta_{10}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 6 q^{5} + 9 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 6 q^{5} + 9 q^{7} + 9 q^{8} + 6 q^{10} - 3 q^{11} + 9 q^{12} - 3 q^{13} + 6 q^{15} - 18 q^{18} + 9 q^{19} + 12 q^{20} + 9 q^{23} + 12 q^{25} - 9 q^{26} + 9 q^{27} - 3 q^{29} - 6 q^{30} + 3 q^{31} + 9 q^{33} - 12 q^{35} - 36 q^{37} - 3 q^{38} + 18 q^{39} + 6 q^{40} + 12 q^{41} - 12 q^{43} - 6 q^{45} + 12 q^{46} + 54 q^{47} - 9 q^{49} + 15 q^{50} - 3 q^{52} - 33 q^{53} + 18 q^{55} + 18 q^{56} - 3 q^{57} + 12 q^{58} - 12 q^{60} - 27 q^{61} - 18 q^{62} - 9 q^{64} + 36 q^{65} + 9 q^{66} + 27 q^{67} - 6 q^{68} + 12 q^{69} + 12 q^{70} - 21 q^{71} - 75 q^{73} - 15 q^{74} - 30 q^{75} + 21 q^{76} - 6 q^{77} + 6 q^{78} + 3 q^{79} - 6 q^{80} - 12 q^{82} + 6 q^{83} - 9 q^{84} + 6 q^{85} + 12 q^{86} - 6 q^{87} + 3 q^{88} + 60 q^{89} + 6 q^{90} + 3 q^{91} + 9 q^{92} + 9 q^{93} - 6 q^{94} + 24 q^{95} + 18 q^{96} + 39 q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 42x^{16} + 639x^{14} + 4594x^{12} + 16845x^{10} + 31167x^{8} + 26608x^{6} + 8322x^{4} + 396x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1503756289 \nu^{16} - 62939735896 \nu^{14} - 954932237107 \nu^{12} - 6890812412418 \nu^{10} + \cdots - 255083930847 ) / 1228390661501 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4442566848 \nu^{16} + 179783187865 \nu^{14} + 2559904115007 \nu^{12} + 16342577502897 \nu^{10} + \cdots - 4307332527503 ) / 1228390661501 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7982923547 \nu^{17} + 8327899466 \nu^{16} - 335263265548 \nu^{15} + 343596382786 \nu^{14} + \cdots + 292360402032 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7982923547 \nu^{17} + 8327899466 \nu^{16} + 335263265548 \nu^{15} + 343596382786 \nu^{14} + \cdots + 292360402032 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10344976450 \nu^{17} + 5889015948 \nu^{16} + 434376178548 \nu^{15} + 243131305714 \nu^{14} + \cdots + 136955263437 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 18327899997 \nu^{17} + 2438883518 \nu^{16} + 769639444096 \nu^{15} + 100465077072 \nu^{14} + \cdots + 155405138595 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26310823544 \nu^{17} - 1503756289 \nu^{16} + 1104902709644 \nu^{15} - 62939735896 \nu^{14} + \cdots - 255083930847 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 143910 \nu^{17} + 6060247 \nu^{15} + 92617044 \nu^{13} + 670780398 \nu^{11} + 2489607372 \nu^{9} + \cdots + 4489453 ) / 8978906 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 51801712865 \nu^{17} + 18327899997 \nu^{16} - 2173233056812 \nu^{15} + \cdots + 3137086552282 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 51801712865 \nu^{17} - 18327899997 \nu^{16} - 2173233056812 \nu^{15} + \cdots - 3137086552282 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 85027976949 \nu^{17} - 10344976450 \nu^{16} + 3569671275569 \nu^{15} - 434376178548 \nu^{14} + \cdots - 3114678059414 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 85027976949 \nu^{17} + 10344976450 \nu^{16} + 3569671275569 \nu^{15} + 434376178548 \nu^{14} + \cdots + 3114678059414 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 164983694422 \nu^{17} + 48372176955 \nu^{16} - 6930550776709 \nu^{15} + \cdots + 9588951394834 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 164983694422 \nu^{17} + 48372176955 \nu^{16} + 6930550776709 \nu^{15} + 2030392387377 \nu^{14} + \cdots + 9588951394834 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 191484180933 \nu^{17} + 4442566848 \nu^{16} - 8041247519659 \nu^{15} + \cdots - 4307332527503 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 241921104786 \nu^{17} + 45899303263 \nu^{16} - 10154552209673 \nu^{15} + \cdots + 10128077928206 ) / 2456781323002 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 241921104786 \nu^{17} - 45899303263 \nu^{16} - 10154552209673 \nu^{15} + \cdots - 10128077928206 ) / 2456781323002 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{17} + \beta_{16} - 2 \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{6} + \beta_{5} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{17} - 3 \beta_{16} - 6 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} - 9 \beta_{12} - 9 \beta_{11} + \cdots + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{17} - 12 \beta_{16} - 4 \beta_{14} - 4 \beta_{13} + 33 \beta_{12} - 33 \beta_{11} - 17 \beta_{10} + \cdots + 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 87 \beta_{17} + 87 \beta_{16} + 126 \beta_{15} - 102 \beta_{14} + 102 \beta_{13} + 264 \beta_{12} + \cdots - 174 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 177 \beta_{17} + 177 \beta_{16} + 83 \beta_{14} + 83 \beta_{13} - 533 \beta_{12} + 533 \beta_{11} + \cdots - 733 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1890 \beta_{17} - 1890 \beta_{16} - 2532 \beta_{15} + 2289 \beta_{14} - 2289 \beta_{13} - 5820 \beta_{12} + \cdots + 3807 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2923 \beta_{17} - 2923 \beta_{16} - 1500 \beta_{14} - 1500 \beta_{13} + 9049 \beta_{12} - 9049 \beta_{11} + \cdots + 12266 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 37425 \beta_{17} + 37425 \beta_{16} + 49182 \beta_{15} - 45729 \beta_{14} + 45729 \beta_{13} + \cdots - 75792 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 50977 \beta_{17} + 50977 \beta_{16} + 26841 \beta_{14} + 26841 \beta_{13} - 159012 \beta_{12} + \cdots - 214629 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 713181 \beta_{17} - 713181 \beta_{16} - 932640 \beta_{15} + 873771 \beta_{14} - 873771 \beta_{13} + \cdots + 1448778 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 912761 \beta_{17} - 912761 \beta_{16} - 484185 \beta_{14} - 484185 \beta_{13} + 2851724 \beta_{12} + \cdots + 3844238 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 13349973 \beta_{17} + 13349973 \beta_{16} + 17435532 \beta_{15} - 16369842 \beta_{14} + 16369842 \beta_{13} + \cdots - 27163296 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 16556402 \beta_{17} + 16556402 \beta_{16} + 8801571 \beta_{14} + 8801571 \beta_{13} - 51730815 \beta_{12} + \cdots - 69708307 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 247701144 \beta_{17} - 247701144 \beta_{16} - 323394036 \beta_{15} + 303813759 \beta_{14} + \cdots + 504411708 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 302270574 \beta_{17} - 302270574 \beta_{16} - 160792246 \beta_{14} - 160792246 \beta_{13} + \cdots + 1272266832 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 4575360387 \beta_{17} + 4575360387 \beta_{16} + 5972934090 \beta_{15} - 5612300016 \beta_{14} + \cdots - 9320988402 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(1\) \(\beta_{10} - \beta_{11}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
− 2.03294i
− 0.215134i
1.56404i
1.12853i
− 0.740171i
− 2.35797i
− 1.12853i
0.740171i
2.35797i
− 3.09864i
0.0967669i
4.28745i
2.03294i
0.215134i
− 1.56404i
3.09864i
− 0.0967669i
− 4.28745i
0.939693 − 0.342020i −0.173648 + 0.984808i 0.766044 − 0.642788i −2.03524 − 1.70777i 0.173648 + 0.984808i 0.500000 + 0.866025i 0.500000 − 0.866025i −0.939693 − 0.342020i −2.49660 − 0.908687i
43.2 0.939693 − 0.342020i −0.173648 + 0.984808i 0.766044 − 0.642788i 0.707492 + 0.593656i 0.173648 + 0.984808i 0.500000 + 0.866025i 0.500000 − 0.866025i −0.939693 − 0.342020i 0.867868 + 0.315878i
43.3 0.939693 − 0.342020i −0.173648 + 0.984808i 0.766044 − 0.642788i 3.39193 + 2.84617i 0.173648 + 0.984808i 0.500000 + 0.866025i 0.500000 − 0.866025i −0.939693 − 0.342020i 4.16082 + 1.51441i
85.1 −0.173648 − 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i −3.74270 − 1.36223i 0.766044 + 0.642788i 0.500000 − 0.866025i 0.500000 + 0.866025i 0.173648 − 0.984808i −0.691622 + 3.92238i
85.2 −0.173648 − 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i −1.48523 − 0.540578i 0.766044 + 0.642788i 0.500000 − 0.866025i 0.500000 + 0.866025i 0.173648 − 0.984808i −0.274459 + 1.55653i
85.3 −0.173648 − 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i 0.469150 + 0.170757i 0.766044 + 0.642788i 0.500000 − 0.866025i 0.500000 + 0.866025i 0.173648 − 0.984808i 0.0866954 − 0.491674i
169.1 −0.173648 + 0.984808i −0.766044 − 0.642788i −0.939693 − 0.342020i −3.74270 + 1.36223i 0.766044 − 0.642788i 0.500000 + 0.866025i 0.500000 − 0.866025i 0.173648 + 0.984808i −0.691622 − 3.92238i
169.2 −0.173648 + 0.984808i −0.766044 − 0.642788i −0.939693 − 0.342020i −1.48523 + 0.540578i 0.766044 − 0.642788i 0.500000 + 0.866025i 0.500000 − 0.866025i 0.173648 + 0.984808i −0.274459 − 1.55653i
169.3 −0.173648 + 0.984808i −0.766044 − 0.642788i −0.939693 − 0.342020i 0.469150 − 0.170757i 0.766044 − 0.642788i 0.500000 + 0.866025i 0.500000 − 0.866025i 0.173648 + 0.984808i 0.0866954 + 0.491674i
253.1 −0.766044 + 0.642788i 0.939693 + 0.342020i 0.173648 − 0.984808i −0.520768 − 2.95342i −0.939693 + 0.342020i 0.500000 − 0.866025i 0.500000 + 0.866025i 0.766044 + 0.642788i 2.29735 + 1.92771i
253.2 −0.766044 + 0.642788i 0.939693 + 0.342020i 0.173648 − 0.984808i −0.141209 − 0.800839i −0.939693 + 0.342020i 0.500000 − 0.866025i 0.500000 + 0.866025i 0.766044 + 0.642788i 0.622942 + 0.522710i
253.3 −0.766044 + 0.642788i 0.939693 + 0.342020i 0.173648 − 0.984808i 0.356570 + 2.02221i −0.939693 + 0.342020i 0.500000 − 0.866025i 0.500000 + 0.866025i 0.766044 + 0.642788i −1.57300 − 1.31990i
631.1 0.939693 + 0.342020i −0.173648 − 0.984808i 0.766044 + 0.642788i −2.03524 + 1.70777i 0.173648 − 0.984808i 0.500000 − 0.866025i 0.500000 + 0.866025i −0.939693 + 0.342020i −2.49660 + 0.908687i
631.2 0.939693 + 0.342020i −0.173648 − 0.984808i 0.766044 + 0.642788i 0.707492 − 0.593656i 0.173648 − 0.984808i 0.500000 − 0.866025i 0.500000 + 0.866025i −0.939693 + 0.342020i 0.867868 − 0.315878i
631.3 0.939693 + 0.342020i −0.173648 − 0.984808i 0.766044 + 0.642788i 3.39193 − 2.84617i 0.173648 − 0.984808i 0.500000 − 0.866025i 0.500000 + 0.866025i −0.939693 + 0.342020i 4.16082 − 1.51441i
757.1 −0.766044 − 0.642788i 0.939693 − 0.342020i 0.173648 + 0.984808i −0.520768 + 2.95342i −0.939693 − 0.342020i 0.500000 + 0.866025i 0.500000 − 0.866025i 0.766044 − 0.642788i 2.29735 − 1.92771i
757.2 −0.766044 − 0.642788i 0.939693 − 0.342020i 0.173648 + 0.984808i −0.141209 + 0.800839i −0.939693 − 0.342020i 0.500000 + 0.866025i 0.500000 − 0.866025i 0.766044 − 0.642788i 0.622942 − 0.522710i
757.3 −0.766044 − 0.642788i 0.939693 − 0.342020i 0.173648 + 0.984808i 0.356570 − 2.02221i −0.939693 − 0.342020i 0.500000 + 0.866025i 0.500000 − 0.866025i 0.766044 − 0.642788i −1.57300 + 1.31990i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.bo.e ✓ 18
19.e even 9 1 inner 798.2.bo.e ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bo.e ✓ 18 1.a even 1 1 trivial
798.2.bo.e ✓ 18 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 6 T_{5}^{17} + 12 T_{5}^{16} + 39 T_{5}^{15} + 342 T_{5}^{14} + 1872 T_{5}^{13} + \cdots + 29241 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} + 6 T^{17} + \cdots + 29241 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{18} + 3 T^{17} + \cdots + 14969161 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 20936932416 \) Copy content Toggle raw display
$17$ \( T^{18} + 66 T^{16} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 694480609 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 19249208561664 \) Copy content Toggle raw display
$31$ \( T^{18} - 3 T^{17} + \cdots + 9 \) Copy content Toggle raw display
$37$ \( (T^{9} + 18 T^{8} + \cdots + 628371)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 334780209 \) Copy content Toggle raw display
$43$ \( T^{18} + 12 T^{17} + \cdots + 331776 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 34271282972224 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 334944617536 \) Copy content Toggle raw display
$59$ \( T^{18} + 90 T^{16} + \cdots + 31629376 \) Copy content Toggle raw display
$61$ \( T^{18} + 27 T^{17} + \cdots + 5607424 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 245305511221824 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 134351025998121 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 965355339548224 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 80\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 6462391321 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 36\!\cdots\!16 \) Copy content Toggle raw display
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