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Properties

Label 798.2.bo.a
Level $798$
Weight $2$
Character orbit 798.bo
Analytic conductor $6.372$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.15035401757601.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 15 x^{10} - 19 x^{9} + 15 x^{8} - 9 x^{7} + 2 x^{6} + 9 x^{5} + 15 x^{4} + 19 x^{3} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{11} q^{3} + \beta_{6} q^{4} + ( - \beta_{10} + \beta_{7} - \beta_{6} + \cdots - 1) q^{5}+ \cdots + (\beta_{7} - \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_{11} q^{3} + \beta_{6} q^{4} + ( - \beta_{10} + \beta_{7} - \beta_{6} + \cdots - 1) q^{5}+ \cdots + (\beta_{11} - \beta_{10} + \beta_{8} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 9 q^{5} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 9 q^{5} - 6 q^{7} - 6 q^{8} + 9 q^{10} - 3 q^{11} + 6 q^{12} + 9 q^{13} + 9 q^{15} + 9 q^{17} + 12 q^{18} + 3 q^{19} - 9 q^{22} + 3 q^{23} + 3 q^{25} + 9 q^{26} + 6 q^{27} + 15 q^{29} + 18 q^{31} - 18 q^{33} + 12 q^{37} + 3 q^{38} + 18 q^{39} + 9 q^{40} + 9 q^{41} - 6 q^{43} - 9 q^{44} - 9 q^{46} - 12 q^{47} - 6 q^{49} + 12 q^{50} + 9 q^{51} - 9 q^{52} + 9 q^{53} - 9 q^{55} + 12 q^{56} - 12 q^{57} - 24 q^{58} + 6 q^{59} + 33 q^{61} + 6 q^{62} - 6 q^{64} + 9 q^{66} + 9 q^{67} + 9 q^{69} - 39 q^{71} - 33 q^{73} - 30 q^{74} + 24 q^{75} + 15 q^{76} + 6 q^{77} - 33 q^{79} - 9 q^{80} + 9 q^{82} + 12 q^{83} + 6 q^{84} + 9 q^{85} + 12 q^{86} - 12 q^{87} - 3 q^{88} - 6 q^{89} - 9 q^{90} - 9 q^{91} + 3 q^{92} + 3 q^{93} + 6 q^{94} - 30 q^{95} - 12 q^{96} - 15 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 15 x^{10} - 19 x^{9} + 15 x^{8} - 9 x^{7} + 2 x^{6} + 9 x^{5} + 15 x^{4} + 19 x^{3} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 9 \nu^{8} - 6 \nu^{7} + 3 \nu^{6} + \nu^{5} - 8 \nu^{4} + \cdots - 63 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 7 \nu^{10} + 22 \nu^{9} - 41 \nu^{8} + 56 \nu^{7} - 65 \nu^{6} + 67 \nu^{5} - 58 \nu^{4} + \cdots + 1 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 7 \nu^{10} + 22 \nu^{9} - 41 \nu^{8} + 56 \nu^{7} - 65 \nu^{6} + 67 \nu^{5} - 58 \nu^{4} + \cdots + 1 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5 \nu^{11} - 34 \nu^{10} + 102 \nu^{9} - 175 \nu^{8} + 209 \nu^{7} - 198 \nu^{6} + 143 \nu^{5} + \cdots + 6 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6 \nu^{11} + 41 \nu^{10} - 124 \nu^{9} + 216 \nu^{8} - 265 \nu^{7} + 263 \nu^{6} - 210 \nu^{5} + \cdots - 7 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19 \nu^{11} - 105 \nu^{10} + 222 \nu^{9} - 167 \nu^{8} - 56 \nu^{7} + 241 \nu^{6} - 355 \nu^{5} + \cdots + 115 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13 \nu^{11} - 80 \nu^{10} + 206 \nu^{9} - 271 \nu^{8} + 219 \nu^{7} - 132 \nu^{6} + 35 \nu^{5} + \cdots + 36 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15 \nu^{11} - 98 \nu^{10} + 276 \nu^{9} - 423 \nu^{8} + 421 \nu^{7} - 306 \nu^{6} + 135 \nu^{5} + \cdots + 24 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24 \nu^{11} + 159 \nu^{10} - 458 \nu^{9} + 732 \nu^{8} - 783 \nu^{7} + 637 \nu^{6} - 354 \nu^{5} + \cdots - 35 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 71 \nu^{11} + 451 \nu^{10} - 1218 \nu^{9} + 1739 \nu^{8} - 1566 \nu^{7} + 1021 \nu^{6} - 341 \nu^{5} + \cdots - 149 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{4} + 3\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{9} + 3\beta_{6} + 5\beta_{5} - \beta_{4} + 5\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + 5\beta_{10} + 7\beta_{9} + \beta_{8} + 5\beta_{6} + 13\beta_{5} - \beta_{4} + 6\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{11} + 13 \beta_{10} + 25 \beta_{9} + 9 \beta_{8} - \beta_{7} + 7 \beta_{6} + 24 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 25 \beta_{11} + 24 \beta_{10} + 62 \beta_{9} + 41 \beta_{8} - 9 \beta_{7} + 9 \beta_{6} + 30 \beta_{5} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 62 \beta_{11} + 30 \beta_{10} + 116 \beta_{9} + 128 \beta_{8} - 41 \beta_{7} + 11 \beta_{6} - 116 \beta_{3} + \cdots - 72 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 116 \beta_{11} + 146 \beta_{9} + 306 \beta_{8} - 128 \beta_{7} + 12 \beta_{6} - 146 \beta_{5} + \cdots - 291 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 146 \beta_{11} - 146 \beta_{10} + 568 \beta_{8} - 306 \beta_{7} - 568 \beta_{5} + 84 \beta_{4} + \cdots - 894 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 568 \beta_{10} - 714 \beta_{9} + 714 \beta_{8} - 568 \beta_{7} - 96 \beta_{6} - 1538 \beta_{5} + \cdots - 2202 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(1\) \(\beta_{6}\) \(1\)

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
−0.146093 − 0.828535i
0.206400 + 1.17056i
1.55180 + 1.30212i
−0.378156 − 0.317311i
1.55180 − 1.30212i
−0.378156 + 0.317311i
2.17250 − 0.790725i
−0.406455 + 0.147937i
−0.146093 + 0.828535i
0.206400 − 1.17056i
2.17250 + 0.790725i
−0.406455 − 0.147937i
−0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 − 0.642788i −2.08418 − 1.74883i −0.173648 − 0.984808i −0.500000 − 0.866025i −0.500000 + 0.866025i −0.939693 − 0.342020i 2.55662 + 0.930535i
43.2 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 − 0.642788i −0.529162 − 0.444020i −0.173648 − 0.984808i −0.500000 − 0.866025i −0.500000 + 0.866025i −0.939693 − 0.342020i 0.649114 + 0.236258i
85.1 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i −2.22992 − 0.811625i −0.766044 − 0.642788i −0.500000 + 0.866025i −0.500000 − 0.866025i 0.173648 − 0.984808i 0.412073 − 2.33698i
85.2 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i 0.137525 + 0.0500551i −0.766044 − 0.642788i −0.500000 + 0.866025i −0.500000 − 0.866025i 0.173648 − 0.984808i −0.0254136 + 0.144128i
169.1 0.173648 − 0.984808i −0.766044 − 0.642788i −0.939693 − 0.342020i −2.22992 + 0.811625i −0.766044 + 0.642788i −0.500000 − 0.866025i −0.500000 + 0.866025i 0.173648 + 0.984808i 0.412073 + 2.33698i
169.2 0.173648 − 0.984808i −0.766044 − 0.642788i −0.939693 − 0.342020i 0.137525 − 0.0500551i −0.766044 + 0.642788i −0.500000 − 0.866025i −0.500000 + 0.866025i 0.173648 + 0.984808i −0.0254136 − 0.144128i
253.1 0.766044 − 0.642788i 0.939693 + 0.342020i 0.173648 − 0.984808i −0.135417 − 0.767989i 0.939693 − 0.342020i −0.500000 + 0.866025i −0.500000 − 0.866025i 0.766044 + 0.642788i −0.597389 − 0.501269i
253.2 0.766044 − 0.642788i 0.939693 + 0.342020i 0.173648 − 0.984808i 0.341154 + 1.93478i 0.939693 − 0.342020i −0.500000 + 0.866025i −0.500000 − 0.866025i 0.766044 + 0.642788i 1.50499 + 1.26284i
631.1 −0.939693 − 0.342020i −0.173648 − 0.984808i 0.766044 + 0.642788i −2.08418 + 1.74883i −0.173648 + 0.984808i −0.500000 + 0.866025i −0.500000 − 0.866025i −0.939693 + 0.342020i 2.55662 − 0.930535i
631.2 −0.939693 − 0.342020i −0.173648 − 0.984808i 0.766044 + 0.642788i −0.529162 + 0.444020i −0.173648 + 0.984808i −0.500000 + 0.866025i −0.500000 − 0.866025i −0.939693 + 0.342020i 0.649114 − 0.236258i
757.1 0.766044 + 0.642788i 0.939693 − 0.342020i 0.173648 + 0.984808i −0.135417 + 0.767989i 0.939693 + 0.342020i −0.500000 − 0.866025i −0.500000 + 0.866025i 0.766044 − 0.642788i −0.597389 + 0.501269i
757.2 0.766044 + 0.642788i 0.939693 − 0.342020i 0.173648 + 0.984808i 0.341154 − 1.93478i 0.939693 + 0.342020i −0.500000 − 0.866025i −0.500000 + 0.866025i 0.766044 − 0.642788i 1.50499 − 1.26284i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.2
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.a ✓ 12
19.e even 9 1 inner 798.2.bo.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bo.a ✓ 12 1.a even 1 1 trivial
798.2.bo.a ✓ 12 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 9 T_{5}^{11} + 39 T_{5}^{10} + 108 T_{5}^{9} + 231 T_{5}^{8} + 405 T_{5}^{7} + 521 T_{5}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 9 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 3 T^{11} + \cdots + 32041 \) Copy content Toggle raw display
$13$ \( T^{12} - 9 T^{11} + \cdots + 38809 \) Copy content Toggle raw display
$17$ \( T^{12} - 9 T^{11} + \cdots + 6561 \) Copy content Toggle raw display
$19$ \( T^{12} - 3 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 3 T^{11} + \cdots + 289 \) Copy content Toggle raw display
$29$ \( T^{12} - 15 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{12} - 18 T^{11} + \cdots + 88943761 \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} + \cdots - 19907)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 9 T^{11} + \cdots + 89661961 \) Copy content Toggle raw display
$43$ \( T^{12} + 6 T^{11} + \cdots + 130321 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 579990889 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 150724729 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 8913436921 \) Copy content Toggle raw display
$61$ \( T^{12} - 33 T^{11} + \cdots + 237169 \) Copy content Toggle raw display
$67$ \( T^{12} - 9 T^{11} + \cdots + 187489 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1666517329 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1174638529 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 15412477609 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 110960273449 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 3562537969 \) Copy content Toggle raw display
$97$ \( T^{12} + 15 T^{11} + \cdots + 38809 \) Copy content Toggle raw display
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