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Properties

Label 819.2.dl.f
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: \( x^{16} - 11x^{14} + 88x^{12} - 303x^{10} + 758x^{8} - 968x^{6} + 867x^{4} - 30x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_1) q^{2} + ( - \beta_{11} + \beta_{10}) q^{4} + (\beta_{15} - \beta_{8}) q^{5} + ( - \beta_{14} + \beta_{7} - \beta_1) q^{7} + ( - \beta_{14} - \beta_{13} + \cdots + \beta_{7}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_1) q^{2} + ( - \beta_{11} + \beta_{10}) q^{4} + (\beta_{15} - \beta_{8}) q^{5} + ( - \beta_{14} + \beta_{7} - \beta_1) q^{7} + ( - \beta_{14} - \beta_{13} + \cdots + \beta_{7}) q^{8}+ \cdots + (\beta_{15} - \beta_{13} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{4} - 4 q^{10} + 4 q^{13} - 40 q^{14} - 10 q^{16} + 8 q^{17} - 8 q^{22} + 8 q^{23} - 6 q^{25} + 4 q^{26} + 36 q^{29} + 14 q^{35} + 26 q^{38} + 6 q^{40} + 32 q^{43} - 46 q^{49} + 40 q^{52} - 36 q^{53} - 8 q^{55} - 54 q^{56} + 12 q^{61} + 80 q^{62} - 56 q^{64} - 34 q^{65} - 10 q^{68} - 18 q^{74} + 22 q^{77} + 8 q^{79} + 12 q^{82} - 98 q^{88} + 16 q^{91} - 40 q^{92} + 46 q^{94} - 38 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 11x^{14} + 88x^{12} - 303x^{10} + 758x^{8} - 968x^{6} + 867x^{4} - 30x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 84975 \nu^{14} - 847968 \nu^{12} + 6630625 \nu^{10} - 18977750 \nu^{8} + 45035320 \nu^{6} + \cdots + 141812432 ) / 47832247 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 493812 \nu^{14} - 5094945 \nu^{12} + 38532300 \nu^{10} - 110284680 \nu^{8} + 203384572 \nu^{6} + \cdots - 24499860 ) / 47832247 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 508068 \nu^{14} - 5088601 \nu^{12} + 39644700 \nu^{10} - 113468520 \nu^{8} + 257471965 \nu^{6} + \cdots + 227456047 ) / 47832247 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 516836 \nu^{14} + 6587845 \nu^{12} - 54823490 \nu^{10} + 227035283 \nu^{8} - 593353298 \nu^{6} + \cdots + 24504217 ) / 47832247 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 593043 \nu^{15} - 5936569 \nu^{13} + 46275325 \nu^{11} - 132446270 \nu^{9} + 302507285 \nu^{7} + \cdots + 464932973 \nu ) / 47832247 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 678018 \nu^{15} - 6784537 \nu^{13} + 52905950 \nu^{11} - 151424020 \nu^{9} + 347542605 \nu^{7} + \cdots + 702409899 \nu ) / 47832247 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1171830 \nu^{15} - 11879482 \nu^{13} + 91438250 \nu^{11} - 261708700 \nu^{9} + \cdots + 677910039 \nu ) / 47832247 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1684309 \nu^{15} + 18442424 \nu^{13} - 147371224 \nu^{11} + 503715002 \nu^{9} + \cdots + 1849848 \nu ) / 47832247 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1684309 \nu^{14} + 18442424 \nu^{12} - 147371224 \nu^{10} + 503715002 \nu^{8} + \cdots + 49682095 ) / 47832247 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4967952 \nu^{14} + 54479304 \nu^{12} - 435483047 \nu^{10} + 1492167256 \nu^{8} + \cdots + 147361976 ) / 47832247 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7826720 \nu^{14} + 86276344 \nu^{12} - 690441745 \nu^{10} + 2385730760 \nu^{8} + \cdots + 236620312 ) / 47832247 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16756333 \nu^{15} - 183577065 \nu^{13} + 1466942565 \nu^{11} - 5017774290 \nu^{9} + \cdots - 18413505 \nu ) / 47832247 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 24499860 \nu^{15} + 269004648 \nu^{13} - 2150892735 \nu^{11} + 7384925280 \nu^{9} + \cdots + 730072644 \nu ) / 47832247 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 24504217 \nu^{15} - 269029551 \nu^{13} + 2149783251 \nu^{11} - 7369954261 \nu^{9} + \cdots - 26984727 \nu ) / 47832247 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - 3\beta_{10} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{13} - 5\beta_{9} - \beta_{7} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + 7\beta_{11} - 16\beta_{10} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{14} + 9\beta_{13} - 30\beta_{9} - 9\beta_{6} - 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{4} + \beta_{3} + 48\beta_{2} - 99 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{8} + 56\beta_{7} - 66\beta_{6} - 195\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 66\beta_{12} - 327\beta_{11} + 661\beta_{10} - 11\beta_{5} - 66\beta_{4} + 11\beta_{3} + 327\beta_{2} - 650 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 11\beta_{15} - 382\beta_{14} - 459\beta_{13} + 1304\beta_{9} + 382\beta_{7} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 459\beta_{12} - 2222\beta_{11} + 4448\beta_{10} - 88\beta_{5} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 88\beta_{15} - 2593\beta_{14} - 3140\beta_{13} + 8804\beta_{9} - 88\beta_{8} + 3140\beta_{6} + 8804\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3140\beta_{4} - 635\beta_{3} - 15084\beta_{2} + 29464 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -635\beta_{8} - 17589\beta_{7} + 21364\beta_{6} + 59632\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 21364 \beta_{12} + 102360 \beta_{11} - 204035 \beta_{10} + 4410 \beta_{5} + 21364 \beta_{4} + \cdots + 199625 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -4410\beta_{15} + 119314\beta_{14} + 145088\beta_{13} - 404345\beta_{9} - 119314\beta_{7} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(-1\) \(-1 + \beta_{10}\)

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
−2.25575 − 1.30236i
−1.34967 − 0.779232i
−1.14630 − 0.661815i
−0.161178 − 0.0930563i
0.161178 + 0.0930563i
1.14630 + 0.661815i
1.34967 + 0.779232i
2.25575 + 1.30236i
−2.25575 + 1.30236i
−1.34967 + 0.779232i
−1.14630 + 0.661815i
−0.161178 + 0.0930563i
0.161178 − 0.0930563i
1.14630 − 0.661815i
1.34967 − 0.779232i
2.25575 − 1.30236i
−2.25575 + 1.30236i 0 2.39228 − 4.14355i 0.401974 − 0.232080i 0 2.25575 + 1.38260i 7.25298i 0 −0.604503 + 1.04703i
298.2 −1.34967 + 0.779232i 0 0.214404 − 0.371358i 1.85159 − 1.06902i 0 1.34967 + 2.27561i − 2.44865i 0 −1.66602 + 2.88564i
298.3 −1.14630 + 0.661815i 0 −0.124000 + 0.214775i −1.98394 + 1.14543i 0 1.14630 − 2.38453i − 2.97552i 0 1.51612 − 2.62600i
298.4 −0.161178 + 0.0930563i 0 −0.982681 + 1.70205i 2.28561 − 1.31960i 0 0.161178 − 2.64084i − 0.738004i 0 −0.245594 + 0.425381i
298.5 0.161178 − 0.0930563i 0 −0.982681 + 1.70205i −2.28561 + 1.31960i 0 −0.161178 + 2.64084i 0.738004i 0 −0.245594 + 0.425381i
298.6 1.14630 − 0.661815i 0 −0.124000 + 0.214775i 1.98394 − 1.14543i 0 −1.14630 + 2.38453i 2.97552i 0 1.51612 − 2.62600i
298.7 1.34967 − 0.779232i 0 0.214404 − 0.371358i −1.85159 + 1.06902i 0 −1.34967 − 2.27561i 2.44865i 0 −1.66602 + 2.88564i
298.8 2.25575 − 1.30236i 0 2.39228 − 4.14355i −0.401974 + 0.232080i 0 −2.25575 − 1.38260i − 7.25298i 0 −0.604503 + 1.04703i
415.1 −2.25575 − 1.30236i 0 2.39228 + 4.14355i 0.401974 + 0.232080i 0 2.25575 − 1.38260i − 7.25298i 0 −0.604503 − 1.04703i
415.2 −1.34967 − 0.779232i 0 0.214404 + 0.371358i 1.85159 + 1.06902i 0 1.34967 − 2.27561i 2.44865i 0 −1.66602 − 2.88564i
415.3 −1.14630 − 0.661815i 0 −0.124000 − 0.214775i −1.98394 − 1.14543i 0 1.14630 + 2.38453i 2.97552i 0 1.51612 + 2.62600i
415.4 −0.161178 − 0.0930563i 0 −0.982681 − 1.70205i 2.28561 + 1.31960i 0 0.161178 + 2.64084i 0.738004i 0 −0.245594 − 0.425381i
415.5 0.161178 + 0.0930563i 0 −0.982681 − 1.70205i −2.28561 − 1.31960i 0 −0.161178 − 2.64084i − 0.738004i 0 −0.245594 − 0.425381i
415.6 1.14630 + 0.661815i 0 −0.124000 − 0.214775i 1.98394 + 1.14543i 0 −1.14630 − 2.38453i − 2.97552i 0 1.51612 + 2.62600i
415.7 1.34967 + 0.779232i 0 0.214404 + 0.371358i −1.85159 − 1.06902i 0 −1.34967 + 2.27561i − 2.44865i 0 −1.66602 − 2.88564i
415.8 2.25575 + 1.30236i 0 2.39228 + 4.14355i −0.401974 − 0.232080i 0 −2.25575 + 1.38260i 7.25298i 0 −0.604503 − 1.04703i
\(n\): 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.f 16
3.b odd 2 1 273.2.bj.c ✓ 16
7.c even 3 1 inner 819.2.dl.f 16
13.b even 2 1 inner 819.2.dl.f 16
21.g even 6 1 1911.2.c.k 8
21.h odd 6 1 273.2.bj.c ✓ 16
21.h odd 6 1 1911.2.c.n 8
39.d odd 2 1 273.2.bj.c ✓ 16
91.r even 6 1 inner 819.2.dl.f 16
273.w odd 6 1 273.2.bj.c ✓ 16
273.w odd 6 1 1911.2.c.n 8
273.ba even 6 1 1911.2.c.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bj.c ✓ 16 3.b odd 2 1
273.2.bj.c ✓ 16 21.h odd 6 1
273.2.bj.c ✓ 16 39.d odd 2 1
273.2.bj.c ✓ 16 273.w odd 6 1
819.2.dl.f 16 1.a even 1 1 trivial
819.2.dl.f 16 7.c even 3 1 inner
819.2.dl.f 16 13.b even 2 1 inner
819.2.dl.f 16 91.r even 6 1 inner
1911.2.c.k 8 21.g even 6 1
1911.2.c.k 8 273.ba even 6 1
1911.2.c.n 8 21.h odd 6 1
1911.2.c.n 8 273.w odd 6 1

Hecke kernels

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

\( T_{2}^{16} - 11T_{2}^{14} + 88T_{2}^{12} - 303T_{2}^{10} + 758T_{2}^{8} - 968T_{2}^{6} + 867T_{2}^{4} - 30T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{19}^{16} - 32 T_{19}^{14} + 682 T_{19}^{12} - 8304 T_{19}^{10} + 73763 T_{19}^{8} - 389936 T_{19}^{6} + \cdots + 923521 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 11 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 17 T^{14} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( T^{16} + 23 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 56 T^{14} + \cdots + 10556001 \) Copy content Toggle raw display
$13$ \( (T^{8} - 2 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 4 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} - 32 T^{14} + \cdots + 923521 \) Copy content Toggle raw display
$23$ \( (T^{8} - 4 T^{7} + \cdots + 244036)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 9 T^{3} + 13 T^{2} + \cdots + 11)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} - 108 T^{14} + \cdots + 11316496 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 454371856 \) Copy content Toggle raw display
$41$ \( (T^{8} + 100 T^{6} + \cdots + 7396)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + \cdots - 3552)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1442919878656 \) Copy content Toggle raw display
$53$ \( (T^{8} + 18 T^{7} + \cdots + 5517801)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 909087685468561 \) Copy content Toggle raw display
$61$ \( (T^{8} - 6 T^{7} + \cdots + 1168561)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1387488001 \) Copy content Toggle raw display
$71$ \( (T^{8} + 332 T^{6} + \cdots + 4004001)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 2667616624656 \) Copy content Toggle raw display
$79$ \( (T^{8} - 4 T^{7} + \cdots + 199487376)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 252 T^{6} + \cdots + 26244)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} - 53 T^{14} + \cdots + 1296 \) Copy content Toggle raw display
$97$ \( (T^{8} + 63 T^{6} + \cdots + 3364)^{2} \) Copy content Toggle raw display
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