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Properties

Label 798.2.bp.a
Level $798$
Weight $2$
Character orbit 798.bp
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(289,798)]
 
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, 6, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bp (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: \( x^{12} + 9x^{10} - 14x^{9} + 69x^{8} - 72x^{7} + 151x^{6} - 78x^{5} + 180x^{4} - 66x^{3} + 117x^{2} + 27x + 9 \) 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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + \beta_{10} q^{3} - \beta_{3} q^{4} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_{2}) q^{5}+ \cdots - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} + \beta_{10} q^{3} - \beta_{3} q^{4} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - 2 \beta_{10} - 2 \beta_{9} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{5} + 12 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{5} + 12 q^{7} + 6 q^{8} + 3 q^{10} + 18 q^{11} - 6 q^{12} - 9 q^{13} - 9 q^{14} - 3 q^{15} + 6 q^{18} - 3 q^{19} + 18 q^{20} + 9 q^{21} + 18 q^{22} + 6 q^{23} - 21 q^{25} + 3 q^{26} - 6 q^{27} + 15 q^{28} - 3 q^{29} - 18 q^{30} - 18 q^{33} - 18 q^{34} + 12 q^{35} + 6 q^{37} - 3 q^{38} - 3 q^{39} + 3 q^{40} - 9 q^{41} - 15 q^{42} + 51 q^{43} + 18 q^{45} + 9 q^{46} - 30 q^{47} + 24 q^{49} + 21 q^{50} + 18 q^{51} - 9 q^{52} - 6 q^{53} + 9 q^{55} + 15 q^{56} + 3 q^{57} - 6 q^{59} - 3 q^{60} - 36 q^{61} + 18 q^{62} + 15 q^{63} - 6 q^{64} + 6 q^{65} - 15 q^{67} - 6 q^{68} - 9 q^{69} + 42 q^{70} - 3 q^{71} - 15 q^{73} + 27 q^{74} - 21 q^{75} - 9 q^{76} + 27 q^{77} + 9 q^{78} + 63 q^{79} - 3 q^{80} - 9 q^{82} - 24 q^{83} - 15 q^{84} - 60 q^{85} + 48 q^{86} + 36 q^{88} + 21 q^{89} + 3 q^{90} - 3 q^{91} - 12 q^{92} - 18 q^{93} + 6 q^{94} + 9 q^{95} + 6 q^{96} - 45 q^{97} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 9x^{10} - 14x^{9} + 69x^{8} - 72x^{7} + 151x^{6} - 78x^{5} + 180x^{4} - 66x^{3} + 117x^{2} + 27x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22300 \nu^{11} - 39972 \nu^{10} - 207135 \nu^{9} - 16552 \nu^{8} - 1068399 \nu^{7} + \cdots - 1430838 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 54202 \nu^{11} - 56620 \nu^{10} - 362478 \nu^{9} + 660840 \nu^{8} - 1457487 \nu^{7} + \cdots + 6547776 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 71196 \nu^{11} - 245497 \nu^{10} + 494337 \nu^{9} - 3002842 \nu^{8} + 7566829 \nu^{7} + \cdots + 2219844 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 74027 \nu^{11} + 333687 \nu^{10} + 850507 \nu^{9} + 1595916 \nu^{8} + 1206871 \nu^{7} + \cdots + 966006 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 107334 \nu^{11} + 74027 \nu^{10} - 632319 \nu^{9} + 2353183 \nu^{8} - 5810130 \nu^{7} + \cdots - 3471528 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 125253 \nu^{11} - 151332 \nu^{10} + 682852 \nu^{9} - 3373952 \nu^{8} + 7397665 \nu^{7} + \cdots - 2772270 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 144076 \nu^{11} + 324620 \nu^{10} + 1890631 \nu^{9} + 1006066 \nu^{8} + 9884477 \nu^{7} + \cdots + 1707330 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 158982 \nu^{11} + 22300 \nu^{10} - 1390866 \nu^{9} + 2432883 \nu^{8} - 10953206 \nu^{7} + \cdots + 1548036 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 200696 \nu^{11} - 199280 \nu^{10} - 1988619 \nu^{9} + 1276385 \nu^{8} - 12069988 \nu^{7} + \cdots - 1219512 ) / 5269059 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 352831 \nu^{11} + 72400 \nu^{10} + 2638417 \nu^{9} - 5007488 \nu^{8} + 18310912 \nu^{7} + \cdots + 4112292 ) / 5269059 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{6} - \beta_{5} + \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 5 \beta_{10} + 2 \beta_{8} + \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{11} - 10 \beta_{10} - 11 \beta_{9} - 7 \beta_{8} - 9 \beta_{7} - 19 \beta_{6} + \cdots + 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{11} - 31 \beta_{10} + 26 \beta_{9} - 10 \beta_{8} + 10 \beta_{7} + 25 \beta_{6} - 14 \beta_{5} + \cdots - 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 25 \beta_{11} + 182 \beta_{10} + 83 \beta_{8} + 58 \beta_{7} + 90 \beta_{6} + 83 \beta_{5} + 99 \beta_{4} + \cdots + 92 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 246 \beta_{11} - 262 \beta_{10} - 269 \beta_{9} - 154 \beta_{8} - 246 \beta_{7} - 559 \beta_{6} + \cdots + 233 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 530 \beta_{11} - 895 \beta_{10} + 866 \beta_{9} - 262 \beta_{8} + 262 \beta_{7} + 868 \beta_{6} + \cdots - 866 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 868 \beta_{11} + 5477 \beta_{10} + 2432 \beta_{8} + 1564 \beta_{7} + 2859 \beta_{6} + 2432 \beta_{5} + \cdots + 2660 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7665 \beta_{11} - 8347 \beta_{10} - 8372 \beta_{9} - 5047 \beta_{8} - 7665 \beta_{7} + \cdots + 6656 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 15458 \beta_{11} - 27715 \beta_{10} + 26024 \beta_{9} - 8347 \beta_{8} + 8347 \beta_{7} + 25741 \beta_{6} + \cdots - 26024 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(-1 + \beta_{9}\) \(\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−1.56099 − 2.70372i
0.621301 + 1.07612i
0.896524 − 1.55282i
−0.130479 + 0.225997i
0.735449 + 1.27384i
−0.561801 − 0.973068i
−1.56099 + 2.70372i
0.621301 − 1.07612i
0.896524 + 1.55282i
−0.130479 − 0.225997i
0.735449 − 1.27384i
−0.561801 + 0.973068i
−0.766044 + 0.642788i 0.766044 − 0.642788i 0.173648 − 0.984808i −0.483291 − 2.74088i −0.173648 + 0.984808i 2.62797 − 0.306212i 0.500000 + 0.866025i 0.173648 − 0.984808i 2.13203 + 1.78898i
289.2 −0.766044 + 0.642788i 0.766044 − 0.642788i 0.173648 − 0.984808i 0.677884 + 3.84447i −0.173648 + 0.984808i 2.36475 + 1.18656i 0.500000 + 0.866025i 0.173648 − 0.984808i −2.99047 − 2.50930i
529.1 −0.173648 + 0.984808i 0.173648 − 0.984808i −0.939693 − 0.342020i −2.46455 + 0.897023i 0.939693 + 0.342020i −2.43265 + 1.04030i 0.500000 − 0.866025i −0.939693 − 0.342020i −0.455430 − 2.58287i
529.2 −0.173648 + 0.984808i 0.173648 − 0.984808i −0.939693 − 0.342020i −1.79422 + 0.653043i 0.939693 + 0.342020i 1.19482 + 2.36059i 0.500000 − 0.866025i −0.939693 − 0.342020i −0.331559 − 1.88036i
541.1 0.939693 + 0.342020i −0.939693 − 0.342020i 0.766044 + 0.642788i −0.585552 + 0.491337i −0.766044 − 0.642788i −0.399965 − 2.61534i 0.500000 + 0.866025i 0.766044 + 0.642788i −0.718286 + 0.261435i
541.2 0.939693 + 0.342020i −0.939693 − 0.342020i 0.766044 + 0.642788i 3.14973 − 2.64294i −0.766044 − 0.642788i 2.64506 − 0.0602612i 0.500000 + 0.866025i 0.766044 + 0.642788i 3.86372 − 1.40628i
613.1 −0.766044 − 0.642788i 0.766044 + 0.642788i 0.173648 + 0.984808i −0.483291 + 2.74088i −0.173648 − 0.984808i 2.62797 + 0.306212i 0.500000 − 0.866025i 0.173648 + 0.984808i 2.13203 − 1.78898i
613.2 −0.766044 − 0.642788i 0.766044 + 0.642788i 0.173648 + 0.984808i 0.677884 − 3.84447i −0.173648 − 0.984808i 2.36475 − 1.18656i 0.500000 − 0.866025i 0.173648 + 0.984808i −2.99047 + 2.50930i
709.1 −0.173648 − 0.984808i 0.173648 + 0.984808i −0.939693 + 0.342020i −2.46455 − 0.897023i 0.939693 − 0.342020i −2.43265 − 1.04030i 0.500000 + 0.866025i −0.939693 + 0.342020i −0.455430 + 2.58287i
709.2 −0.173648 − 0.984808i 0.173648 + 0.984808i −0.939693 + 0.342020i −1.79422 − 0.653043i 0.939693 − 0.342020i 1.19482 − 2.36059i 0.500000 + 0.866025i −0.939693 + 0.342020i −0.331559 + 1.88036i
739.1 0.939693 − 0.342020i −0.939693 + 0.342020i 0.766044 − 0.642788i −0.585552 − 0.491337i −0.766044 + 0.642788i −0.399965 + 2.61534i 0.500000 − 0.866025i 0.766044 − 0.642788i −0.718286 − 0.261435i
739.2 0.939693 − 0.342020i −0.939693 + 0.342020i 0.766044 − 0.642788i 3.14973 + 2.64294i −0.766044 + 0.642788i 2.64506 + 0.0602612i 0.500000 − 0.866025i 0.766044 − 0.642788i 3.86372 + 1.40628i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.u 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.bp.a ✓ 12
7.c even 3 1 798.2.bq.c yes 12
19.e even 9 1 798.2.bq.c yes 12
133.u even 9 1 inner 798.2.bp.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bp.a ✓ 12 1.a even 1 1 trivial
798.2.bp.a ✓ 12 133.u even 9 1 inner
798.2.bq.c yes 12 7.c even 3 1
798.2.bq.c yes 12 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 3 T_{5}^{11} + 15 T_{5}^{10} + 80 T_{5}^{9} + 255 T_{5}^{8} + 1131 T_{5}^{7} + 5437 T_{5}^{6} + \cdots + 29241 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{11} + \cdots + 29241 \) Copy content Toggle raw display
$7$ \( T^{12} - 12 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} - 9 T^{5} + 63 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 9 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} - 9 T^{10} + \cdots + 29241 \) Copy content Toggle raw display
$19$ \( T^{12} + 3 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 813447441 \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} + \cdots + 5103081 \) Copy content Toggle raw display
$31$ \( (T^{6} - 87 T^{4} + \cdots + 2339)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 579990889 \) Copy content Toggle raw display
$41$ \( T^{12} + 9 T^{11} + \cdots + 4782969 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 4893981849 \) Copy content Toggle raw display
$47$ \( T^{12} + 30 T^{11} + \cdots + 227529 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 443902761 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 12621848409 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 245266921 \) Copy content Toggle raw display
$67$ \( T^{12} + 15 T^{11} + \cdots + 1369 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 8621679609 \) Copy content Toggle raw display
$73$ \( T^{12} + 15 T^{11} + \cdots + 494209 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 54101364409 \) Copy content Toggle raw display
$83$ \( T^{12} + 24 T^{11} + \cdots + 11390625 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1749163329 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 663011001 \) Copy content Toggle raw display
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