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Properties

Label 819.2.bm.f
Level $819$
Weight $2$
Character orbit 819.bm
Analytic conductor $6.540$
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] = [819,2,Mod(478,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.478");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.bm (of order \(6\), degree \(2\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.2346760387617129.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + ( - \beta_{11} - \beta_{8} + \beta_{7} + \cdots - 1) q^{4}+ \cdots + (\beta_{11} - \beta_{9} + \beta_{7} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + ( - \beta_{11} - \beta_{8} + \beta_{7} + \cdots - 1) q^{4}+ \cdots + (4 \beta_{11} + 2 \beta_{10} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{4} + 3 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{4} + 3 q^{5} - 3 q^{7} + 12 q^{10} - 12 q^{11} - 2 q^{13} - 4 q^{14} + 16 q^{16} + 34 q^{17} + 9 q^{19} + 3 q^{20} - 15 q^{22} + 6 q^{23} - 5 q^{25} + 6 q^{26} - 9 q^{28} + q^{29} + 18 q^{31} + 6 q^{35} - 19 q^{38} - q^{40} + 6 q^{41} + 11 q^{43} + 33 q^{44} + 15 q^{47} - 3 q^{49} - 18 q^{50} - 7 q^{52} + 8 q^{53} - 15 q^{55} - 27 q^{56} - 24 q^{58} + 5 q^{61} - 41 q^{62} + 2 q^{64} - 21 q^{65} + 15 q^{67} - 22 q^{68} + 3 q^{70} - 30 q^{71} + 42 q^{73} - 66 q^{74} - 45 q^{76} + 19 q^{77} - 35 q^{79} + 63 q^{80} + 5 q^{82} - 21 q^{85} + 57 q^{86} - 14 q^{88} - 7 q^{91} + 66 q^{92} + q^{94} + 4 q^{95} - 3 q^{97} + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 72 \nu^{8} - 91 \nu^{7} - 164 \nu^{6} + 313 \nu^{5} + 42 \nu^{4} + \cdots - 800 ) / 224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9 \nu^{11} + 5 \nu^{10} + 25 \nu^{9} - 32 \nu^{8} - 21 \nu^{7} + 132 \nu^{6} - 73 \nu^{5} + \cdots + 256 ) / 224 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{11} + 17 \nu^{10} + 29 \nu^{9} - 78 \nu^{8} + 21 \nu^{7} + 166 \nu^{6} - 167 \nu^{5} + \cdots + 288 ) / 224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13 \nu^{11} + 29 \nu^{10} + 5 \nu^{9} - 96 \nu^{8} + 91 \nu^{7} + 200 \nu^{6} - 289 \nu^{5} + \cdots + 544 ) / 224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8 \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 51 \nu^{8} - 42 \nu^{7} - 101 \nu^{6} + 194 \nu^{5} + \cdots - 464 ) / 112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13 \nu^{11} - 57 \nu^{10} - 5 \nu^{9} + 208 \nu^{8} - 231 \nu^{7} - 396 \nu^{6} + 821 \nu^{5} + \cdots - 1664 ) / 224 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{11} - 5 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 7 \nu^{7} - 41 \nu^{6} + 45 \nu^{5} + 35 \nu^{4} + \cdots - 88 ) / 28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 20 \nu^{8} - 44 \nu^{6} + 43 \nu^{5} + 56 \nu^{4} - 82 \nu^{3} + \cdots - 48 ) / 28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15 \nu^{11} + 20 \nu^{10} + 30 \nu^{9} - 121 \nu^{8} + 21 \nu^{7} + 269 \nu^{6} - 271 \nu^{5} + \cdots + 464 ) / 112 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17 \nu^{11} + 39 \nu^{10} + 13 \nu^{9} - 160 \nu^{8} + 133 \nu^{7} + 310 \nu^{6} - 547 \nu^{5} + \cdots + 1056 ) / 112 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} + 2\beta_{9} - 2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{3} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{11} + 2 \beta_{10} - 3 \beta_{8} + \beta_{7} - 5 \beta_{6} + 4 \beta_{5} - 7 \beta_{4} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{11} - \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} + 6 \beta_{5} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -4\beta_{10} - 2\beta_{9} - \beta_{8} + 2\beta_{5} - 4\beta_{4} + 8\beta_{3} - 2\beta_{2} + 3\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5 \beta_{11} - 9 \beta_{10} + \beta_{9} - 16 \beta_{8} + \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2 \beta_{11} - \beta_{10} - 19 \beta_{8} + \beta_{7} + 4 \beta_{6} - 15 \beta_{5} - 5 \beta_{4} + \cdots - 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(1 + \beta_{4}\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
478.1
1.32725 − 0.488273i
−1.38488 − 0.286553i
0.655911 + 1.25291i
−1.18541 + 0.771231i
0.874681 − 1.11128i
1.21245 + 0.727987i
1.21245 − 0.727987i
0.874681 + 1.11128i
−1.18541 − 0.771231i
0.655911 − 1.25291i
−1.38488 + 0.286553i
1.32725 + 0.488273i
− 2.58860i 0 −4.70085 1.39608 + 0.806027i 0 1.06153 + 2.42346i 6.99143i 0 2.08648 − 3.61389i
478.2 − 1.37905i 0 0.0982074 0.697972 + 0.402974i 0 0.0699870 − 2.64483i − 2.89354i 0 0.555723 − 0.962541i
478.3 − 0.180824i 0 1.96730 2.32670 + 1.34332i 0 −2.46263 + 0.967177i − 0.717383i 0 0.242904 − 0.420723i
478.4 0.499987i 0 1.75001 0.902810 + 0.521238i 0 2.63491 + 0.239300i 1.87496i 0 −0.260612 + 0.451393i
478.5 1.34523i 0 0.190366 −3.08979 − 1.78389i 0 −2.44601 − 1.00849i 2.94654i 0 2.39973 − 4.15646i
478.6 2.30327i 0 −3.30504 −0.733776 − 0.423646i 0 −0.357777 + 2.62145i − 3.00585i 0 0.975769 − 1.69008i
550.1 − 2.30327i 0 −3.30504 −0.733776 + 0.423646i 0 −0.357777 − 2.62145i 3.00585i 0 0.975769 + 1.69008i
550.2 − 1.34523i 0 0.190366 −3.08979 + 1.78389i 0 −2.44601 + 1.00849i − 2.94654i 0 2.39973 + 4.15646i
550.3 − 0.499987i 0 1.75001 0.902810 − 0.521238i 0 2.63491 − 0.239300i − 1.87496i 0 −0.260612 − 0.451393i
550.4 0.180824i 0 1.96730 2.32670 − 1.34332i 0 −2.46263 − 0.967177i 0.717383i 0 0.242904 + 0.420723i
550.5 1.37905i 0 0.0982074 0.697972 − 0.402974i 0 0.0699870 + 2.64483i 2.89354i 0 0.555723 + 0.962541i
550.6 2.58860i 0 −4.70085 1.39608 − 0.806027i 0 1.06153 − 2.42346i − 6.99143i 0 2.08648 + 3.61389i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 478.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.bm.f 12
3.b odd 2 1 91.2.k.b ✓ 12
7.c even 3 1 819.2.do.e 12
13.e even 6 1 819.2.do.e 12
21.c even 2 1 637.2.k.i 12
21.g even 6 1 637.2.q.i 12
21.g even 6 1 637.2.u.g 12
21.h odd 6 1 91.2.u.b yes 12
21.h odd 6 1 637.2.q.g 12
39.h odd 6 1 91.2.u.b yes 12
39.k even 12 2 1183.2.e.j 24
91.k even 6 1 inner 819.2.bm.f 12
273.u even 6 1 637.2.u.g 12
273.x odd 6 1 637.2.q.g 12
273.y even 6 1 637.2.q.i 12
273.bp odd 6 1 91.2.k.b ✓ 12
273.br even 6 1 637.2.k.i 12
273.bs odd 12 2 8281.2.a.co 12
273.bv even 12 2 8281.2.a.cp 12
273.bw even 12 2 1183.2.e.j 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.b ✓ 12 3.b odd 2 1
91.2.k.b ✓ 12 273.bp odd 6 1
91.2.u.b yes 12 21.h odd 6 1
91.2.u.b yes 12 39.h odd 6 1
637.2.k.i 12 21.c even 2 1
637.2.k.i 12 273.br even 6 1
637.2.q.g 12 21.h odd 6 1
637.2.q.g 12 273.x odd 6 1
637.2.q.i 12 21.g even 6 1
637.2.q.i 12 273.y even 6 1
637.2.u.g 12 21.g even 6 1
637.2.u.g 12 273.u even 6 1
819.2.bm.f 12 1.a even 1 1 trivial
819.2.bm.f 12 91.k even 6 1 inner
819.2.do.e 12 7.c even 3 1
819.2.do.e 12 13.e even 6 1
1183.2.e.j 24 39.k even 12 2
1183.2.e.j 24 273.bw even 12 2
8281.2.a.co 12 273.bs odd 12 2
8281.2.a.cp 12 273.bv even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{12} + 16T_{2}^{10} + 88T_{2}^{8} + 197T_{2}^{6} + 172T_{2}^{4} + 36T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{12} - 3 T_{5}^{11} - 8 T_{5}^{10} + 33 T_{5}^{9} + 81 T_{5}^{8} - 495 T_{5}^{7} + 801 T_{5}^{6} + \cdots + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 16 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + \cdots + 121 \) Copy content Toggle raw display
$7$ \( T^{12} + 3 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + 12 T^{11} + \cdots + 85849 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} - 17 T^{5} + \cdots + 19)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 9 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{6} - 3 T^{5} + \cdots + 793)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - T^{11} + \cdots + 16072081 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 241274089 \) Copy content Toggle raw display
$37$ \( T^{12} + 147 T^{10} + \cdots + 123201 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 389707081 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 418898089 \) Copy content Toggle raw display
$47$ \( T^{12} - 15 T^{11} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{12} - 8 T^{11} + \cdots + 289 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 35582408689 \) Copy content Toggle raw display
$61$ \( T^{12} - 5 T^{11} + \cdots + 3157729 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 5708255809 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 639230089 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 484396081 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 65086724641 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 402363481 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 145033849 \) Copy content Toggle raw display
$97$ \( T^{12} + 3 T^{11} + \cdots + 1681 \) Copy content Toggle raw display
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