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Properties

Label 819.2.o.i
Level $819$
Weight $2$
Character orbit 819.o
Analytic conductor $6.540$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(568,819)]
 
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, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.568");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.o (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 8x^{10} + 51x^{8} - 98x^{6} + 145x^{4} - 39x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + (\beta_{4} - \beta_{3} - 1) q^{4} + ( - \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{3} - 1) q^{7} + ( - \beta_{10} - 2 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + (\beta_{4} - \beta_{3} - 1) q^{4} + ( - \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{3} - 1) q^{7} + ( - \beta_{10} - 2 \beta_{2}) q^{8} + ( - \beta_{8} - \beta_{5} + \beta_{4} - 3 \beta_{3}) q^{10} + ( - \beta_{9} - \beta_{7}) q^{11} + (\beta_{8} + \beta_{4} + 1) q^{13} - \beta_{2} q^{14} + ( - \beta_{8} + 3 \beta_{5} - 3 \beta_{4} + 4 \beta_{3}) q^{16} + (\beta_{7} - \beta_{2}) q^{17} + ( - 2 \beta_{8} + \beta_{3} - 2 \beta_1 + 1) q^{19} + ( - \beta_{11} - \beta_{10} + \beta_{9} + 5 \beta_{7} - \beta_{6} - 5 \beta_{2}) q^{20} + ( - \beta_{8} - \beta_{4} + 3 \beta_{3} - \beta_1 + 3) q^{22} + ( - \beta_{11} - \beta_{9} - \beta_{7}) q^{23} + (\beta_1 + 4) q^{25} + ( - \beta_{10} + \beta_{9} - \beta_{6} - 2 \beta_{2}) q^{26} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{28} + ( - \beta_{11} + 2 \beta_{9} + 2 \beta_{7}) q^{29} + ( - \beta_{5} - \beta_1 + 5) q^{31} + (\beta_{11} + \beta_{10} - \beta_{9} - 8 \beta_{7} + \beta_{6} + 8 \beta_{2}) q^{32} + (\beta_{5} - 3) q^{34} + ( - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{2}) q^{35} + ( - 3 \beta_{8} + \beta_{5} - \beta_{4} + 2 \beta_{3}) q^{37} + (2 \beta_{6} - \beta_{2}) q^{38} + (6 \beta_{5} + 4 \beta_1 - 9) q^{40} - 2 \beta_{7} q^{41} + ( - 2 \beta_{8} - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 2) q^{43} + (\beta_{10} - \beta_{6} + 3 \beta_{2}) q^{44} + (2 \beta_{4} + 3 \beta_{3} + 3) q^{46} + ( - \beta_{10} + \beta_{6} + 2 \beta_{2}) q^{47} + \beta_{3} q^{49} + ( - \beta_{9} + 5 \beta_{7}) q^{50} + ( - \beta_{8} + 3 \beta_{5} - 3 \beta_{4} + 8 \beta_{3} - \beta_1 + 2) q^{52} + (\beta_{10} + 3 \beta_{2}) q^{53} + (\beta_{8} + 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3}) q^{55} + (\beta_{11} + \beta_{10} - 2 \beta_{7} + 2 \beta_{2}) q^{56} + (3 \beta_{8} + 5 \beta_{4} - 6 \beta_{3} + 3 \beta_1 - 6) q^{58} + (\beta_{11} + \beta_{10} - 3 \beta_{9} + \beta_{7} + 3 \beta_{6} - \beta_{2}) q^{59} + (2 \beta_{8} - 2 \beta_{4} + \beta_{3} + 2 \beta_1 + 1) q^{61} + ( - \beta_{11} + \beta_{9} + 7 \beta_{7}) q^{62} + ( - 5 \beta_{5} + 16) q^{64} + ( - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{7} - 3 \beta_{6} - 3 \beta_{2}) q^{65} + ( - 3 \beta_{8} + 2 \beta_{5} - 2 \beta_{4} + 8 \beta_{3}) q^{67} + (\beta_{11} - 4 \beta_{7}) q^{68} + (\beta_{5} - \beta_1 - 3) q^{70} + (3 \beta_{9} - 2 \beta_{7} - 3 \beta_{6} + 2 \beta_{2}) q^{71} + ( - 3 \beta_{5} - 2 \beta_1 - 1) q^{73} + (\beta_{11} + \beta_{10} - 3 \beta_{9} - 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{2}) q^{74} + ( - 2 \beta_{8} + \beta_{5} - \beta_{4} + 5 \beta_{3}) q^{76} + (\beta_{6} + \beta_{2}) q^{77} + ( - 3 \beta_{5} - 5 \beta_1 + 5) q^{79} + (4 \beta_{11} - 2 \beta_{9} - 13 \beta_{7}) q^{80} + ( - 2 \beta_{4} + 6 \beta_{3} + 6) q^{82} + ( - 3 \beta_{6} - 3 \beta_{2}) q^{83} + ( - \beta_{8} + \beta_{4} - 3 \beta_{3} - \beta_1 - 3) q^{85} + (\beta_{10} + 2 \beta_{6} - \beta_{2}) q^{86} + ( - 2 \beta_{8} - 4 \beta_{5} + 4 \beta_{4} - 3 \beta_{3}) q^{88} + (\beta_{11} + 2 \beta_{9} + 5 \beta_{7}) q^{89} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{91} + ( - 2 \beta_{6} - 5 \beta_{2}) q^{92} + (\beta_{5} - \beta_{4} - 6 \beta_{3}) q^{94} + (2 \beta_{11} + 2 \beta_{10} + \beta_{9} + 5 \beta_{7} - \beta_{6} - 5 \beta_{2}) q^{95} + ( - \beta_{8} - \beta_{4} - 11 \beta_{3} - \beta_1 - 11) q^{97} + ( - \beta_{7} + \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{4} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{4} - 6 q^{7} + 22 q^{10} + 8 q^{13} - 28 q^{16} + 2 q^{19} + 18 q^{22} + 52 q^{25} - 8 q^{28} + 60 q^{31} - 40 q^{34} - 8 q^{37} - 116 q^{40} - 14 q^{43} + 14 q^{46} - 6 q^{49} - 32 q^{52} + 12 q^{55} - 40 q^{58} + 14 q^{61} + 212 q^{64} - 46 q^{67} - 44 q^{70} - 8 q^{73} - 28 q^{76} + 52 q^{79} + 40 q^{82} - 22 q^{85} + 30 q^{88} - 4 q^{91} + 34 q^{94} - 66 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 8x^{10} + 51x^{8} - 98x^{6} + 145x^{4} - 39x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 64\nu^{10} - 408\nu^{8} + 2601\nu^{6} - 1160\nu^{4} + 312\nu^{2} + 19260 ) / 7083 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 178\nu^{11} - 1725\nu^{9} + 11292\nu^{7} - 30968\nu^{5} + 43956\nu^{3} - 21690\nu ) / 7083 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -221\nu^{10} + 1704\nu^{8} - 10863\nu^{6} + 19057\nu^{4} - 30885\nu^{2} + 1224 ) / 7083 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -31\nu^{10} + 296\nu^{8} - 1887\nu^{6} + 4792\nu^{4} - 5365\nu^{2} + 1443 ) / 787 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 280\nu^{10} - 1785\nu^{8} + 10494\nu^{6} - 5075\nu^{4} + 1365\nu^{2} + 9891 ) / 7083 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -133\nu^{11} + 1143\nu^{9} - 7385\nu^{7} + 16675\nu^{5} - 24947\nu^{3} + 12360\nu ) / 2361 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -174\nu^{11} + 1306\nu^{9} - 8129\nu^{7} + 12401\nu^{5} - 15211\nu^{3} - 2487\nu ) / 2361 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -221\nu^{10} + 1704\nu^{8} - 10863\nu^{6} + 19057\nu^{4} - 28524\nu^{2} + 1224 ) / 2361 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 743\nu^{11} - 5622\nu^{9} + 35250\nu^{7} - 56260\nu^{5} + 76518\nu^{3} + 13320\nu ) / 7083 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -385\nu^{11} + 3143\nu^{9} - 20135\nu^{7} + 41311\nu^{5} - 63558\nu^{3} + 31554\nu ) / 2361 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -1691\nu^{11} + 12846\nu^{9} - 81303\nu^{7} + 133648\nu^{5} - 193287\nu^{3} - 34767\nu ) / 7083 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{7} + \beta_{6} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} + \beta_{10} + 10\beta_{9} + 8\beta_{7} - 5\beta_{6} - 4\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{8} - \beta_{4} - 15\beta_{3} + 6\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{11} + 14\beta_{10} + 29\beta_{9} + 19\beta_{7} - 58\beta_{6} - 38\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{5} + 35\beta _1 - 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -43\beta_{11} + 43\beta_{10} - 170\beta_{9} - 103\beta_{7} - 170\beta_{6} - 103\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -205\beta_{8} - 51\beta_{5} + 51\beta_{4} + 486\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -512\beta_{11} - 256\beta_{10} - 1996\beta_{9} - 1178\beta_{7} + 998\beta_{6} + 589\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1203\beta_{8} + 307\beta_{4} + 2841\beta_{3} - 1203\beta _1 + 2841 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -1510\beta_{11} - 3020\beta_{10} - 5861\beta_{9} - 3430\beta_{7} + 11722\beta_{6} + 6860\beta_{2} ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(-1 - \beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
568.1
−1.17800 + 0.680121i
0.455008 − 0.262699i
−2.09887 + 1.21179i
2.09887 − 1.21179i
−0.455008 + 0.262699i
1.17800 − 0.680121i
−1.17800 − 0.680121i
0.455008 + 0.262699i
−2.09887 − 1.21179i
2.09887 + 1.21179i
−0.455008 − 0.262699i
1.17800 + 0.680121i
−1.38547 + 2.39970i 0 −2.83905 − 4.91737i −3.18587 0 −0.500000 − 0.866025i 10.1918 0 4.41392 − 7.64513i
568.2 −0.628502 + 1.08860i 0 0.209969 + 0.363678i −3.42403 0 −0.500000 − 0.866025i −3.04187 0 2.15201 − 3.72739i
568.3 −0.430653 + 0.745913i 0 0.629076 + 1.08959i 2.47514 0 −0.500000 − 0.866025i −2.80627 0 −1.06593 + 1.84624i
568.4 0.430653 − 0.745913i 0 0.629076 + 1.08959i −2.47514 0 −0.500000 − 0.866025i 2.80627 0 −1.06593 + 1.84624i
568.5 0.628502 − 1.08860i 0 0.209969 + 0.363678i 3.42403 0 −0.500000 − 0.866025i 3.04187 0 2.15201 − 3.72739i
568.6 1.38547 − 2.39970i 0 −2.83905 − 4.91737i 3.18587 0 −0.500000 − 0.866025i −10.1918 0 4.41392 − 7.64513i
757.1 −1.38547 − 2.39970i 0 −2.83905 + 4.91737i −3.18587 0 −0.500000 + 0.866025i 10.1918 0 4.41392 + 7.64513i
757.2 −0.628502 − 1.08860i 0 0.209969 − 0.363678i −3.42403 0 −0.500000 + 0.866025i −3.04187 0 2.15201 + 3.72739i
757.3 −0.430653 − 0.745913i 0 0.629076 − 1.08959i 2.47514 0 −0.500000 + 0.866025i −2.80627 0 −1.06593 − 1.84624i
757.4 0.430653 + 0.745913i 0 0.629076 − 1.08959i −2.47514 0 −0.500000 + 0.866025i 2.80627 0 −1.06593 − 1.84624i
757.5 0.628502 + 1.08860i 0 0.209969 − 0.363678i 3.42403 0 −0.500000 + 0.866025i 3.04187 0 2.15201 + 3.72739i
757.6 1.38547 + 2.39970i 0 −2.83905 + 4.91737i 3.18587 0 −0.500000 + 0.866025i −10.1918 0 4.41392 + 7.64513i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 568.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.c even 3 1 inner
39.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.o.i ✓ 12
3.b odd 2 1 inner 819.2.o.i ✓ 12
13.c even 3 1 inner 819.2.o.i ✓ 12
39.i odd 6 1 inner 819.2.o.i ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.o.i ✓ 12 1.a even 1 1 trivial
819.2.o.i ✓ 12 3.b odd 2 1 inner
819.2.o.i ✓ 12 13.c even 3 1 inner
819.2.o.i ✓ 12 39.i odd 6 1 inner

Hecke kernels

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

\( T_{2}^{12} + 10T_{2}^{10} + 81T_{2}^{8} + 172T_{2}^{6} + 271T_{2}^{4} + 171T_{2}^{2} + 81 \) Copy content Toggle raw display
\( T_{11}^{12} + 24T_{11}^{10} + 459T_{11}^{8} + 2646T_{11}^{6} + 11745T_{11}^{4} + 9477T_{11}^{2} + 6561 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 10 T^{10} + 81 T^{8} + 172 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 28 T^{4} + 253 T^{2} - 729)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 24 T^{10} + 459 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( (T^{6} - 4 T^{5} + 14 T^{4} - 85 T^{3} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 10 T^{10} + 81 T^{8} + 172 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( (T^{6} - T^{5} + 34 T^{4} + 111 T^{3} + \cdots + 1521)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 103 T^{10} + \cdots + 74805201 \) Copy content Toggle raw display
$29$ \( T^{12} + 157 T^{10} + \cdots + 5082121521 \) Copy content Toggle raw display
$31$ \( (T^{3} - 15 T^{2} + 60 T - 41)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 4 T^{5} + 77 T^{4} + 166 T^{3} + \cdots + 42025)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 40 T^{10} + 1296 T^{8} + \cdots + 331776 \) Copy content Toggle raw display
$43$ \( (T^{6} + 7 T^{5} + 65 T^{4} - 122 T^{3} + \cdots + 25)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 127 T^{4} + 3232 T^{2} + \cdots - 729)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 175 T^{4} + 1183 T^{2} + \cdots - 2025)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 199 T^{10} + 35739 T^{8} + \cdots + 2313441 \) Copy content Toggle raw display
$61$ \( (T^{6} - 7 T^{5} + 154 T^{4} + \cdots + 178929)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 23 T^{5} + 439 T^{4} + \cdots + 257049)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 196 T^{10} + \cdots + 66074188401 \) Copy content Toggle raw display
$73$ \( (T^{3} + 2 T^{2} - 115 T - 233)^{4} \) Copy content Toggle raw display
$79$ \( (T^{3} - 13 T^{2} - 166 T + 2155)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 216 T^{4} + 9477 T^{2} + \cdots - 59049)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 367 T^{10} + \cdots + 903687890625 \) Copy content Toggle raw display
$97$ \( (T^{6} + 33 T^{5} + 741 T^{4} + \cdots + 1380625)^{2} \) Copy content Toggle raw display
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