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Properties

Label 798.2.bo.b
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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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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 9 x^{10} - 10 x^{9} + 63 x^{8} - 60 x^{7} + 185 x^{6} - 180 x^{5} + 390 x^{4} - 260 x^{3} + \cdots + 1 \) 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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{8} q^{3} + ( - \beta_{11} + \beta_{6}) q^{4} + (\beta_{10} + \beta_{8} + \beta_{7} + \cdots - 1) q^{5}+ \cdots - \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{8} q^{3} + ( - \beta_{11} + \beta_{6}) q^{4} + (\beta_{10} + \beta_{8} + \beta_{7} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{11} - \beta_{10} + 2 \beta_{8} + \cdots + 2) q^{99}+O(q^{100}) \) 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 \( 12 q - 9 q^{5} - 6 q^{7} + 6 q^{8} - 9 q^{10} + 3 q^{11} + 6 q^{12} - 9 q^{13} + 9 q^{15} + 9 q^{17} - 12 q^{18} - 9 q^{19} - 12 q^{20} + 9 q^{22} + 9 q^{23} + 15 q^{25} + 9 q^{26} + 6 q^{27} - 3 q^{29} + 6 q^{30} - 18 q^{31} - 18 q^{33} + 48 q^{37} + 3 q^{38} - 18 q^{39} - 9 q^{40} + 9 q^{41} - 12 q^{43} - 9 q^{44} + 6 q^{45} - 21 q^{46} - 12 q^{47} - 6 q^{49} + 9 q^{51} + 9 q^{52} + 9 q^{53} + 45 q^{55} - 12 q^{56} - 6 q^{57} - 36 q^{59} + 39 q^{61} + 12 q^{62} - 6 q^{64} + 12 q^{65} - 9 q^{66} - 27 q^{67} + 12 q^{68} - 21 q^{69} - 9 q^{71} + 51 q^{73} - 30 q^{74} + 9 q^{76} - 6 q^{77} + 9 q^{79} - 9 q^{80} - 9 q^{82} + 12 q^{83} + 6 q^{84} - 27 q^{85} - 6 q^{86} - 3 q^{88} - 6 q^{89} + 9 q^{90} + 9 q^{91} + 9 q^{92} + 3 q^{93} + 6 q^{94} + 12 q^{95} + 12 q^{96} - 57 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 9 x^{10} - 10 x^{9} + 63 x^{8} - 60 x^{7} + 185 x^{6} - 180 x^{5} + 390 x^{4} - 260 x^{3} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 75654 \nu^{11} - 84690 \nu^{10} - 665625 \nu^{9} + 37620 \nu^{8} - 3953625 \nu^{7} + \cdots - 1386970 ) / 22028973 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 363344 \nu^{11} + 1717712 \nu^{10} - 5604865 \nu^{9} + 12604187 \nu^{8} - 65786640 \nu^{7} + \cdots - 73061634 ) / 66086919 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 430566 \nu^{11} - 5052362 \nu^{10} - 15016744 \nu^{9} - 46464466 \nu^{8} - 65461855 \nu^{7} + \cdots + 27762044 ) / 66086919 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 776161 \nu^{11} - 3946097 \nu^{10} + 845412 \nu^{9} - 43881342 \nu^{8} + 43697353 \nu^{7} + \cdots - 4590362 ) / 66086919 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2148278 \nu^{11} + 2717593 \nu^{10} + 23987491 \nu^{9} + 3568498 \nu^{8} + 142818165 \nu^{7} + \cdots - 27398700 ) / 66086919 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2219717 \nu^{11} - 4757792 \nu^{10} - 23509254 \nu^{9} - 17254566 \nu^{8} - 112678193 \nu^{7} + \cdots - 2560250 ) / 66086919 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2560250 \nu^{11} - 2219717 \nu^{10} + 18284458 \nu^{9} - 49111754 \nu^{8} + 144041184 \nu^{7} + \cdots - 53087160 ) / 66086919 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3493754 \nu^{11} + 706892 \nu^{10} + 25896690 \nu^{9} - 36404691 \nu^{8} + 163147745 \nu^{7} + \cdots - 6738640 ) / 66086919 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1386970 \nu^{11} + 75654 \nu^{10} - 12398040 \nu^{9} + 14535325 \nu^{8} - 87416730 \nu^{7} + \cdots + 1613550 ) / 22028973 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4590362 \nu^{11} - 776161 \nu^{10} - 37367161 \nu^{9} + 45058208 \nu^{8} - 245311464 \nu^{7} + \cdots - 71757732 ) / 66086919 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + 2\beta_{10} - \beta_{9} - \beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + \beta_{9} + 2\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 4\beta_{2} - 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{11} - 9 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + 7 \beta_{7} - 6 \beta_{6} - 5 \beta_{5} + \cdots + 3 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{11} + 18 \beta_{10} - 2 \beta_{9} - 11 \beta_{8} - 2 \beta_{7} + 31 \beta_{6} + 12 \beta_{5} + \cdots - 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 55 \beta_{11} + 50 \beta_{9} + 55 \beta_{8} - 23 \beta_{7} - 59 \beta_{6} - 23 \beta_{5} - 27 \beta_{4} + \cdots + 56 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 96 \beta_{11} - 141 \beta_{10} - 82 \beta_{9} - 59 \beta_{8} + 110 \beta_{7} - 141 \beta_{6} + \cdots + 133 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 161 \beta_{11} + 404 \beta_{10} - 164 \beta_{9} - 274 \beta_{8} - 164 \beta_{7} + 805 \beta_{6} + \cdots - 404 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1225 \beta_{11} + 920 \beta_{9} + 1225 \beta_{8} - 644 \beta_{7} - 1058 \beta_{6} - 644 \beta_{5} + \cdots + 1092 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2208 \beta_{11} - 3072 \beta_{10} - 1702 \beta_{9} - 1214 \beta_{8} + 2807 \beta_{7} + \cdots + 2288 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3393 \beta_{11} + 8488 \beta_{10} - 2397 \beta_{9} - 6211 \beta_{8} - 2397 \beta_{7} + 17016 \beta_{6} + \cdots - 8488 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(1\) \(\beta_{6} - \beta_{11}\) \(1\)

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
0.684125 − 1.18494i
−0.857773 + 1.48571i
−1.39778 − 2.42102i
0.631733 + 1.09419i
−1.39778 + 2.42102i
0.631733 − 1.09419i
0.970755 + 1.68140i
−0.0310621 − 0.0538012i
0.684125 + 1.18494i
−0.857773 − 1.48571i
0.970755 − 1.68140i
−0.0310621 + 0.0538012i
0.939693 − 0.342020i −0.173648 + 0.984808i 0.766044 − 0.642788i −3.25388 − 2.73033i 0.173648 + 0.984808i −0.500000 − 0.866025i 0.500000 − 0.866025i −0.939693 − 0.342020i −3.99147 − 1.45278i
43.2 0.939693 − 0.342020i −0.173648 + 0.984808i 0.766044 − 0.642788i −0.891552 − 0.748101i 0.173648 + 0.984808i −0.500000 − 0.866025i 0.500000 − 0.866025i −0.939693 − 0.342020i −1.09365 − 0.398056i
85.1 −0.173648 − 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i −2.01362 − 0.732898i 0.766044 + 0.642788i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.173648 − 0.984808i −0.372102 + 2.11030i
85.2 −0.173648 − 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i 1.80061 + 0.655369i 0.766044 + 0.642788i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.173648 − 0.984808i 0.332739 − 1.88706i
169.1 −0.173648 + 0.984808i −0.766044 − 0.642788i −0.939693 − 0.342020i −2.01362 + 0.732898i 0.766044 − 0.642788i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.173648 + 0.984808i −0.372102 − 2.11030i
169.2 −0.173648 + 0.984808i −0.766044 − 0.642788i −0.939693 − 0.342020i 1.80061 − 0.655369i 0.766044 − 0.642788i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.173648 + 0.984808i 0.332739 + 1.88706i
253.1 −0.766044 + 0.642788i 0.939693 + 0.342020i 0.173648 − 0.984808i −0.244743 − 1.38801i −0.939693 + 0.342020i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.766044 + 0.642788i 1.07968 + 0.905958i
253.2 −0.766044 + 0.642788i 0.939693 + 0.342020i 0.173648 − 0.984808i 0.103184 + 0.585186i −0.939693 + 0.342020i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.766044 + 0.642788i −0.455194 − 0.381953i
631.1 0.939693 + 0.342020i −0.173648 − 0.984808i 0.766044 + 0.642788i −3.25388 + 2.73033i 0.173648 − 0.984808i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.939693 + 0.342020i −3.99147 + 1.45278i
631.2 0.939693 + 0.342020i −0.173648 − 0.984808i 0.766044 + 0.642788i −0.891552 + 0.748101i 0.173648 − 0.984808i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.939693 + 0.342020i −1.09365 + 0.398056i
757.1 −0.766044 − 0.642788i 0.939693 − 0.342020i 0.173648 + 0.984808i −0.244743 + 1.38801i −0.939693 − 0.342020i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.766044 − 0.642788i 1.07968 − 0.905958i
757.2 −0.766044 − 0.642788i 0.939693 − 0.342020i 0.173648 + 0.984808i 0.103184 − 0.585186i −0.939693 − 0.342020i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.766044 − 0.642788i −0.455194 + 0.381953i
\(n\): 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.b ✓ 12
19.e even 9 1 inner 798.2.bo.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bo.b ✓ 12 1.a even 1 1 trivial
798.2.bo.b ✓ 12 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 9 T_{5}^{11} + 33 T_{5}^{10} + 28 T_{5}^{9} - 87 T_{5}^{8} - 201 T_{5}^{7} - 61 T_{5}^{6} + \cdots + 289 \) 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
$5$ \( T^{12} + 9 T^{11} + \cdots + 289 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{11} + \cdots + 273529 \) Copy content Toggle raw display
$13$ \( T^{12} + 9 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} - 9 T^{11} + \cdots + 130321 \) Copy content Toggle raw display
$19$ \( T^{12} + 9 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 9 T^{11} + \cdots + 37222201 \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} + \cdots + 1819801 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 281467729 \) Copy content Toggle raw display
$37$ \( (T^{6} - 24 T^{5} + \cdots + 64081)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 9 T^{11} + \cdots + 58262689 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 327863449 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 374461201 \) Copy content Toggle raw display
$53$ \( T^{12} - 9 T^{11} + \cdots + 361 \) Copy content Toggle raw display
$59$ \( T^{12} + 36 T^{11} + \cdots + 62995969 \) Copy content Toggle raw display
$61$ \( T^{12} - 39 T^{11} + \cdots + 2621161 \) Copy content Toggle raw display
$67$ \( T^{12} + 27 T^{11} + \cdots + 908209 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 982007569 \) Copy content Toggle raw display
$73$ \( T^{12} - 51 T^{11} + \cdots + 10556001 \) Copy content Toggle raw display
$79$ \( T^{12} - 9 T^{11} + \cdots + 39677401 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 656948161 \) Copy content Toggle raw display
$89$ \( T^{12} + 6 T^{11} + \cdots + 12187081 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 206870689 \) Copy content Toggle raw display
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