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Properties

Label 819.2.j.g
Level $819$
Weight $2$
Character orbit 819.j
Analytic conductor $6.540$
Analytic rank $0$
Dimension $10$
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] = [819,2,Mod(235,819)]
 
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, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.235");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.j (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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} - 8x^{7} + 41x^{6} - 40x^{5} + 59x^{4} - 10x^{3} + 18x^{2} - 4x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 273)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{4}) q^{2} + (\beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{7} - \beta_{2}) q^{5} + (\beta_{8} - \beta_{6} - \beta_{4}) q^{7} + (\beta_{9} - \beta_{8} + \beta_{6} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{4}) q^{2} + (\beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{7} - \beta_{2}) q^{5} + (\beta_{8} - \beta_{6} - \beta_{4}) q^{7} + (\beta_{9} - \beta_{8} + \beta_{6} - \beta_1) q^{8} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots - 1) q^{10}+ \cdots + (3 \beta_{9} - 3 \beta_{8} + \beta_{7} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{4} - 3 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{4} - 3 q^{5} + 4 q^{7} - 6 q^{8} + 2 q^{10} - q^{11} - 10 q^{13} - 23 q^{14} - 13 q^{17} + 7 q^{19} + 26 q^{20} - 38 q^{22} - 4 q^{23} - 16 q^{25} - 4 q^{28} + 24 q^{29} + 6 q^{31} + 21 q^{32} - 14 q^{34} + 3 q^{35} - 11 q^{37} - 14 q^{38} + 11 q^{40} + 20 q^{41} + 20 q^{43} - 29 q^{44} - q^{46} + 4 q^{47} + 22 q^{49} + 58 q^{50} + 6 q^{52} - 9 q^{53} + 24 q^{55} - 42 q^{56} - 34 q^{58} - 7 q^{59} + 23 q^{61} - 48 q^{62} - 26 q^{64} + 3 q^{65} - 25 q^{67} - 20 q^{68} + 73 q^{70} + 54 q^{71} + 18 q^{73} + 15 q^{74} - 4 q^{76} - 27 q^{77} - 8 q^{79} - 41 q^{80} + 26 q^{82} + 24 q^{83} + 20 q^{85} - 19 q^{86} + 36 q^{88} - 29 q^{89} - 4 q^{91} + 100 q^{92} - 2 q^{94} - 33 q^{95} - 26 q^{97} - 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 7x^{8} - 8x^{7} + 41x^{6} - 40x^{5} + 59x^{4} - 10x^{3} + 18x^{2} - 4x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2640 \nu^{9} + 11641 \nu^{8} + 15246 \nu^{7} + 63880 \nu^{6} + 75479 \nu^{5} + 368907 \nu^{4} + \cdots + 66894 ) / 101089 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14281 \nu^{9} + 3234 \nu^{8} - 85000 \nu^{7} + 32761 \nu^{6} - 474507 \nu^{5} + 90633 \nu^{4} + \cdots + 10560 ) / 202178 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8527 \nu^{9} - 12294 \nu^{8} + 66934 \nu^{7} - 107661 \nu^{6} + 394774 \nu^{5} - 573725 \nu^{4} + \cdots - 56648 ) / 101089 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19323 \nu^{9} + 15246 \nu^{8} + 119172 \nu^{7} + 67797 \nu^{6} + 622413 \nu^{5} + 485035 \nu^{4} + \cdots + 309726 ) / 202178 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19323 \nu^{9} - 15246 \nu^{8} - 119172 \nu^{7} - 67797 \nu^{6} - 622413 \nu^{5} - 485035 \nu^{4} + \cdots - 309726 ) / 202178 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 18327 \nu^{9} + 22689 \nu^{8} - 123529 \nu^{7} + 176861 \nu^{6} - 728569 \nu^{5} + 872266 \nu^{4} + \cdots - 53822 ) / 101089 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37803 \nu^{9} + 4356 \nu^{8} - 225894 \nu^{7} + 91577 \nu^{6} - 1251855 \nu^{5} + 268553 \nu^{4} + \cdots + 30728 ) / 202178 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20287 \nu^{9} + 24768 \nu^{8} - 134848 \nu^{7} + 190701 \nu^{6} - 795328 \nu^{5} + 952192 \nu^{4} + \cdots + 227351 ) / 101089 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 45021 \nu^{9} - 43061 \nu^{8} + 313068 \nu^{7} - 348849 \nu^{6} + 1832021 \nu^{5} - 1734081 \nu^{4} + \cdots - 173616 ) / 101089 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{4} + \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - 3\beta_{8} + 2\beta_{7} - 2\beta_{5} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{6} + \beta_{5} + 6\beta_{4} - 6\beta_{3} + 15\beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{9} + 18\beta_{8} - 10\beta_{7} - 18\beta_{4} + \beta_{3} - 10\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{9} - 36\beta_{8} + 9\beta_{7} + 33\beta_{6} - 9\beta_{5} + \beta _1 + 81 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -13\beta_{6} + 54\beta_{5} + 106\beta_{4} - 13\beta_{3} + 76\beta_{2} - 33\beta _1 - 76 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13\beta_{9} + 218\beta_{8} - 64\beta_{7} - 218\beta_{4} + 180\beta_{3} - 448\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -180\beta_{9} - 622\beta_{8} + 301\beta_{7} + 115\beta_{6} - 301\beta_{5} + 180\beta _1 + 525 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
−1.22012 + 2.11332i
0.660865 − 1.14465i
−0.281188 + 0.487032i
1.08681 − 1.88241i
0.253637 − 0.439313i
−1.22012 − 2.11332i
0.660865 + 1.14465i
−0.281188 − 0.487032i
1.08681 + 1.88241i
0.253637 + 0.439313i
−1.07479 − 1.86158i 0 −1.31033 + 2.26956i −1.86546 − 3.23108i 0 −1.88003 − 1.86158i 1.33415 0 −4.00995 + 6.94543i
235.2 −0.893230 − 1.54712i 0 −0.595718 + 1.03181i 1.71496 + 2.97040i 0 2.14626 − 1.54712i −1.44447 0 3.06371 − 5.30650i
235.3 0.0388377 + 0.0672688i 0 0.996983 − 1.72683i −1.10121 − 1.90736i 0 2.64490 + 0.0672688i 0.310233 0 0.0855372 − 0.148155i
235.4 0.660777 + 1.14450i 0 0.126747 − 0.219533i 1.01284 + 1.75429i 0 −2.38540 + 1.14450i 2.97812 0 −1.33853 + 2.31839i
235.5 1.26840 + 2.19693i 0 −2.21768 + 3.84114i −1.26113 − 2.18434i 0 1.47427 + 2.19693i −6.17804 0 3.19923 − 5.54123i
352.1 −1.07479 + 1.86158i 0 −1.31033 − 2.26956i −1.86546 + 3.23108i 0 −1.88003 + 1.86158i 1.33415 0 −4.00995 − 6.94543i
352.2 −0.893230 + 1.54712i 0 −0.595718 − 1.03181i 1.71496 − 2.97040i 0 2.14626 + 1.54712i −1.44447 0 3.06371 + 5.30650i
352.3 0.0388377 − 0.0672688i 0 0.996983 + 1.72683i −1.10121 + 1.90736i 0 2.64490 − 0.0672688i 0.310233 0 0.0855372 + 0.148155i
352.4 0.660777 − 1.14450i 0 0.126747 + 0.219533i 1.01284 − 1.75429i 0 −2.38540 − 1.14450i 2.97812 0 −1.33853 − 2.31839i
352.5 1.26840 − 2.19693i 0 −2.21768 − 3.84114i −1.26113 + 2.18434i 0 1.47427 − 2.19693i −6.17804 0 3.19923 + 5.54123i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.j.g 10
3.b odd 2 1 273.2.i.e ✓ 10
7.c even 3 1 inner 819.2.j.g 10
7.c even 3 1 5733.2.a.bq 5
7.d odd 6 1 5733.2.a.bp 5
21.g even 6 1 1911.2.a.u 5
21.h odd 6 1 273.2.i.e ✓ 10
21.h odd 6 1 1911.2.a.t 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.i.e ✓ 10 3.b odd 2 1
273.2.i.e ✓ 10 21.h odd 6 1
819.2.j.g 10 1.a even 1 1 trivial
819.2.j.g 10 7.c even 3 1 inner
1911.2.a.t 5 21.h odd 6 1
1911.2.a.u 5 21.g even 6 1
5733.2.a.bp 5 7.d odd 6 1
5733.2.a.bq 5 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 8T_{2}^{8} + 2T_{2}^{7} + 51T_{2}^{6} + 7T_{2}^{5} + 105T_{2}^{4} - 29T_{2}^{3} + 168T_{2}^{2} - 13T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 8 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 3 T^{9} + \cdots + 20736 \) Copy content Toggle raw display
$7$ \( T^{10} - 4 T^{9} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T + 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} + 13 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{10} - 7 T^{9} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( T^{10} + 4 T^{9} + \cdots + 1227664 \) Copy content Toggle raw display
$29$ \( (T^{5} - 12 T^{4} + \cdots + 251)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 6 T^{9} + \cdots + 1893376 \) Copy content Toggle raw display
$37$ \( T^{10} + 11 T^{9} + \cdots + 55591936 \) Copy content Toggle raw display
$41$ \( (T^{5} - 10 T^{4} + \cdots - 224)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 10 T^{4} + \cdots - 76)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 338265664 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 7060872841 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 5156245249 \) Copy content Toggle raw display
$61$ \( T^{10} - 23 T^{9} + \cdots + 1104601 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 127757809 \) Copy content Toggle raw display
$71$ \( (T^{5} - 27 T^{4} + \cdots - 761)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} - 18 T^{9} + \cdots + 55771024 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 344919184 \) Copy content Toggle raw display
$83$ \( (T^{5} - 12 T^{4} + \cdots - 17248)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 29 T^{9} + \cdots + 4596736 \) Copy content Toggle raw display
$97$ \( (T^{5} + 13 T^{4} + \cdots + 14308)^{2} \) Copy content Toggle raw display
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