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Properties

Label 819.2.dl.g
Level $819$
Weight $2$
Character orbit 819.dl
Analytic conductor $6.540$
Analytic rank $0$
Dimension $16$
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(298,819)]
 
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, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.298");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.dl (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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 11x^{14} + 85x^{12} - 310x^{10} + 807x^{8} - 1196x^{6} + 1273x^{4} - 688x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + ( - \beta_{12} + \beta_{9} + \cdots + \beta_{2}) q^{4}+ \cdots + (\beta_{15} + \beta_{14} + \cdots - 3 \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + ( - \beta_{12} + \beta_{9} + \cdots + \beta_{2}) q^{4}+ \cdots + (3 \beta_{15} + 3 \beta_{14} + \cdots - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 16 q^{10} + 6 q^{13} + 10 q^{14} - 28 q^{16} + 2 q^{17} + 60 q^{22} - 24 q^{23} + 10 q^{25} - 14 q^{26} - 24 q^{29} + 30 q^{35} + 26 q^{40} - 76 q^{43} + 2 q^{49} - 10 q^{53} - 16 q^{55} - 72 q^{56} + 26 q^{61} + 104 q^{62} - 84 q^{64} + 32 q^{65} + 12 q^{68} + 54 q^{74} + 10 q^{77} - 10 q^{79} - 48 q^{82} + 68 q^{88} - 57 q^{91} - 16 q^{92} - 48 q^{94} - 18 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 11x^{14} + 85x^{12} - 310x^{10} + 807x^{8} - 1196x^{6} + 1273x^{4} - 688x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 93708 \nu^{14} - 905544 \nu^{12} + 6775609 \nu^{10} - 19993643 \nu^{8} + 48900157 \nu^{6} + \cdots + 122770133 ) / 53934263 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2439541 \nu^{14} + 25335623 \nu^{12} - 192872281 \nu^{10} + 647847966 \nu^{8} + \cdots + 1373874032 ) / 862948208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2051459 \nu^{15} - 31489585 \nu^{13} + 239720495 \nu^{11} - 1065436466 \nu^{9} + \cdots - 635028480 \nu ) / 1725896416 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5499477 \nu^{14} - 79437735 \nu^{12} + 653830489 \nu^{10} - 3074557294 \nu^{8} + \cdots - 7631273072 ) / 862948208 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7318623 \nu^{14} - 76006869 \nu^{12} + 578616843 \nu^{10} - 1943543898 \nu^{8} + \cdots - 1532777472 ) / 862948208 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2439541 \nu^{15} + 25335623 \nu^{13} - 192872281 \nu^{11} + 647847966 \nu^{9} + \cdots + 510925824 \nu ) / 862948208 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6901309 \nu^{15} + 58008863 \nu^{13} - 407614369 \nu^{11} + 844736062 \nu^{9} + \cdots + 1111287280 \nu ) / 1725896416 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7983216 \nu^{15} + 6901309 \nu^{14} + 78057212 \nu^{13} - 58008863 \nu^{12} + \cdots - 1111287280 ) / 1725896416 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 34417935 \nu^{15} + 8205836 \nu^{14} + 390591909 \nu^{13} - 125958340 \nu^{12} + \cdots - 2540113920 ) / 6903585664 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 34417935 \nu^{15} + 8205836 \nu^{14} - 390591909 \nu^{13} - 125958340 \nu^{12} + \cdots - 2540113920 ) / 6903585664 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7983216 \nu^{15} - 6901309 \nu^{14} + 78057212 \nu^{13} + 58008863 \nu^{12} + \cdots + 1111287280 ) / 1725896416 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1183968 \nu^{14} - 11648434 \nu^{12} + 85607464 \nu^{10} - 252612728 \nu^{8} + 551180115 \nu^{6} + \cdots + 141925895 ) / 53934263 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5499477 \nu^{15} - 79437735 \nu^{13} + 653830489 \nu^{11} - 3074557294 \nu^{9} + \cdots - 7631273072 \nu ) / 862948208 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1183968 \nu^{15} - 11648434 \nu^{13} + 85607464 \nu^{11} - 252612728 \nu^{9} + \cdots + 141925895 \nu ) / 53934263 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - 3\beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + 2\beta_{12} - 2\beta_{11} + 2\beta_{10} + 2\beta_{9} - 4\beta_{7} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + \beta_{9} - 8\beta_{6} + \beta_{5} - 14\beta_{3} + 8\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{14} - 16\beta_{11} + 16\beta_{10} - 10\beta_{8} - 22\beta_{7} - 22\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{13} - 10\beta_{12} + 10\beta_{11} + 10\beta_{10} + 10\beta_{9} + 54\beta_{2} - 79 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -63\beta_{15} - 108\beta_{12} - 108\beta_{9} - 74\beta_{8} + 74\beta_{4} - 133\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -63\beta_{13} + 74\beta_{11} + 74\beta_{10} + 349\beta_{6} - 63\beta_{5} + 478\beta_{3} - 478 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 412 \beta_{15} - 412 \beta_{14} - 698 \beta_{12} + 698 \beta_{11} - 698 \beta_{10} + \cdots + 497 \beta_{4} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 497\beta_{12} - 497\beta_{9} + 2223\beta_{6} - 412\beta_{5} + 2968\beta_{3} - 2223\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -2635\beta_{14} + 4446\beta_{11} - 4446\beta_{10} + 3217\beta_{8} + 5191\beta_{7} + 5191\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2635\beta_{13} + 3217\beta_{12} - 3217\beta_{11} - 3217\beta_{10} - 3217\beta_{9} - 14083\beta_{2} + 18613 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 16718\beta_{15} + 28166\beta_{12} + 28166\beta_{9} + 20517\beta_{8} - 20517\beta_{4} + 32696\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 16718\beta_{13} - 20517\beta_{11} - 20517\beta_{10} - 89028\beta_{6} + 16718\beta_{5} - 117185\beta_{3} + 117185 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 105746 \beta_{15} + 105746 \beta_{14} + 178056 \beta_{12} - 178056 \beta_{11} + 178056 \beta_{10} + \cdots - 130062 \beta_{4} \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1
−1.39394 − 0.804793i
1.02312 + 0.590698i
0.725066 + 0.418617i
−2.17587 − 1.25624i
2.17587 + 1.25624i
−0.725066 − 0.418617i
−1.02312 − 0.590698i
1.39394 + 0.804793i
−1.39394 + 0.804793i
1.02312 − 0.590698i
0.725066 − 0.418617i
−2.17587 + 1.25624i
2.17587 − 1.25624i
−0.725066 + 0.418617i
−1.02312 + 0.590698i
1.39394 − 0.804793i
−2.35955 + 1.36229i 0 2.71165 − 4.69671i 1.84188 − 1.06341i 0 −1.72383 + 2.00709i 9.32701i 0 −2.89733 + 5.01833i
298.2 −1.69305 + 0.977485i 0 0.910952 − 1.57782i 0.130553 − 0.0753750i 0 −0.807080 − 2.51965i − 0.348171i 0 −0.147356 + 0.255228i
298.3 −0.558156 + 0.322252i 0 −0.792308 + 1.37232i 3.70788 − 2.14075i 0 2.12926 + 1.57043i − 2.31030i 0 −1.37972 + 2.38974i
298.4 −0.504542 + 0.291297i 0 −0.830292 + 1.43811i −1.26176 + 0.728479i 0 2.46845 − 0.952230i − 2.13264i 0 0.424408 − 0.735096i
298.5 0.504542 − 0.291297i 0 −0.830292 + 1.43811i 1.26176 − 0.728479i 0 −2.46845 + 0.952230i 2.13264i 0 0.424408 − 0.735096i
298.6 0.558156 − 0.322252i 0 −0.792308 + 1.37232i −3.70788 + 2.14075i 0 −2.12926 − 1.57043i 2.31030i 0 −1.37972 + 2.38974i
298.7 1.69305 − 0.977485i 0 0.910952 − 1.57782i −0.130553 + 0.0753750i 0 0.807080 + 2.51965i 0.348171i 0 −0.147356 + 0.255228i
298.8 2.35955 − 1.36229i 0 2.71165 − 4.69671i −1.84188 + 1.06341i 0 1.72383 − 2.00709i − 9.32701i 0 −2.89733 + 5.01833i
415.1 −2.35955 − 1.36229i 0 2.71165 + 4.69671i 1.84188 + 1.06341i 0 −1.72383 − 2.00709i − 9.32701i 0 −2.89733 − 5.01833i
415.2 −1.69305 − 0.977485i 0 0.910952 + 1.57782i 0.130553 + 0.0753750i 0 −0.807080 + 2.51965i 0.348171i 0 −0.147356 − 0.255228i
415.3 −0.558156 − 0.322252i 0 −0.792308 − 1.37232i 3.70788 + 2.14075i 0 2.12926 − 1.57043i 2.31030i 0 −1.37972 − 2.38974i
415.4 −0.504542 − 0.291297i 0 −0.830292 − 1.43811i −1.26176 − 0.728479i 0 2.46845 + 0.952230i 2.13264i 0 0.424408 + 0.735096i
415.5 0.504542 + 0.291297i 0 −0.830292 − 1.43811i 1.26176 + 0.728479i 0 −2.46845 − 0.952230i − 2.13264i 0 0.424408 + 0.735096i
415.6 0.558156 + 0.322252i 0 −0.792308 − 1.37232i −3.70788 − 2.14075i 0 −2.12926 + 1.57043i − 2.31030i 0 −1.37972 − 2.38974i
415.7 1.69305 + 0.977485i 0 0.910952 + 1.57782i −0.130553 − 0.0753750i 0 0.807080 − 2.51965i − 0.348171i 0 −0.147356 − 0.255228i
415.8 2.35955 + 1.36229i 0 2.71165 + 4.69671i −1.84188 − 1.06341i 0 1.72383 + 2.00709i 9.32701i 0 −2.89733 − 5.01833i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 298.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r 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.dl.g 16
3.b odd 2 1 273.2.bj.d ✓ 16
7.c even 3 1 inner 819.2.dl.g 16
13.b even 2 1 inner 819.2.dl.g 16
21.g even 6 1 1911.2.c.m 8
21.h odd 6 1 273.2.bj.d ✓ 16
21.h odd 6 1 1911.2.c.j 8
39.d odd 2 1 273.2.bj.d ✓ 16
91.r even 6 1 inner 819.2.dl.g 16
273.w odd 6 1 273.2.bj.d ✓ 16
273.w odd 6 1 1911.2.c.j 8
273.ba even 6 1 1911.2.c.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bj.d ✓ 16 3.b odd 2 1
273.2.bj.d ✓ 16 21.h odd 6 1
273.2.bj.d ✓ 16 39.d odd 2 1
273.2.bj.d ✓ 16 273.w odd 6 1
819.2.dl.g 16 1.a even 1 1 trivial
819.2.dl.g 16 7.c even 3 1 inner
819.2.dl.g 16 13.b even 2 1 inner
819.2.dl.g 16 91.r even 6 1 inner
1911.2.c.j 8 21.h odd 6 1
1911.2.c.j 8 273.w odd 6 1
1911.2.c.m 8 21.g even 6 1
1911.2.c.m 8 273.ba even 6 1

Hecke kernels

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

\( T_{2}^{16} - 12T_{2}^{14} + 107T_{2}^{12} - 398T_{2}^{10} + 1089T_{2}^{8} - 755T_{2}^{6} + 381T_{2}^{4} - 92T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{19}^{16} - 112 T_{19}^{14} + 8490 T_{19}^{12} - 340528 T_{19}^{10} + 9830787 T_{19}^{8} + \cdots + 61013446081 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 12 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 25 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{16} - T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 29 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{8} - 3 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - T^{7} + 23 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 61013446081 \) Copy content Toggle raw display
$23$ \( (T^{8} + 12 T^{7} + \cdots + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} + \cdots - 104)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} - 131 T^{14} + \cdots + 35153041 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 5323914784321 \) Copy content Toggle raw display
$41$ \( (T^{8} + 212 T^{6} + \cdots + 649636)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 19 T^{3} + \cdots + 169)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 959512576 \) Copy content Toggle raw display
$53$ \( (T^{8} + 5 T^{7} + \cdots + 44249104)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 6146560000 \) Copy content Toggle raw display
$61$ \( (T^{8} - 13 T^{7} + \cdots + 44169316)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{8} + 173 T^{6} + \cdots + 33124)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} - 83 T^{14} + \cdots + 25411681 \) Copy content Toggle raw display
$79$ \( (T^{8} + 5 T^{7} + \cdots + 116014441)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 232 T^{6} + \cdots + 4218916)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} - 493 T^{14} + \cdots + 9834496 \) Copy content Toggle raw display
$97$ \( (T^{8} + 67 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
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