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Properties

Label 819.2.o.j
Level $819$
Weight $2$
Character orbit 819.o
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(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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 13x^{14} + 116x^{12} + 557x^{10} + 1944x^{8} + 3316x^{6} + 3985x^{4} + 462x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_1) q^{2} + ( - \beta_{11} - \beta_{8} - \beta_{3} - 1) q^{4} + (\beta_{14} - \beta_{7}) q^{5} + (\beta_{8} + 1) q^{7} + ( - \beta_{13} - \beta_{9}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_1) q^{2} + ( - \beta_{11} - \beta_{8} - \beta_{3} - 1) q^{4} + (\beta_{14} - \beta_{7}) q^{5} + (\beta_{8} + 1) q^{7} + ( - \beta_{13} - \beta_{9}) q^{8} + (\beta_{12} + \beta_{11} + \cdots - \beta_{2}) q^{10}+ \cdots - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 10 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 10 q^{4} + 8 q^{7} - 8 q^{10} - 10 q^{13} - 2 q^{16} + 10 q^{19} + 14 q^{22} - 4 q^{25} + 10 q^{28} - 4 q^{31} - 36 q^{34} + 14 q^{37} + 40 q^{40} - 18 q^{43} + 30 q^{46} - 8 q^{49} - 8 q^{52} + 12 q^{55} + 46 q^{58} + 20 q^{61} - 88 q^{64} + 18 q^{67} - 16 q^{70} - 44 q^{73} - 16 q^{76} + 28 q^{79} - 50 q^{82} + 32 q^{85} + 18 q^{88} - 8 q^{91} - 6 q^{94} + 42 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 13x^{14} + 116x^{12} + 557x^{10} + 1944x^{8} + 3316x^{6} + 3985x^{4} + 462x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1601330 \nu^{14} + 37131273 \nu^{12} + 410551573 \nu^{10} + 2522415677 \nu^{8} + \cdots + 2573444587 ) / 4891522027 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 280741 \nu^{14} + 3248688 \nu^{12} + 28058209 \nu^{10} + 116300932 \nu^{8} + 379663364 \nu^{6} + \cdots - 2011728946 ) / 698788861 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 280741 \nu^{15} - 3248688 \nu^{13} - 28058209 \nu^{11} - 116300932 \nu^{9} + \cdots + 3409306668 \nu ) / 698788861 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 722814 \nu^{14} + 9419656 \nu^{12} + 72240486 \nu^{10} + 299435928 \nu^{8} + 742289677 \nu^{6} + \cdots - 322701276 ) / 698788861 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1283501 \nu^{14} - 14984381 \nu^{12} - 128277449 \nu^{10} - 531708452 \nu^{8} + \cdots + 2301085931 ) / 698788861 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1564242 \nu^{15} + 18233069 \nu^{13} + 156335658 \nu^{11} + 648009384 \nu^{9} + \cdots - 5710392599 \nu ) / 698788861 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12091091 \nu^{14} + 155218996 \nu^{12} + 1379825740 \nu^{10} + 6538330224 \nu^{8} + \cdots + 473682412 ) / 4891522027 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12091091 \nu^{15} + 155218996 \nu^{13} + 1379825740 \nu^{11} + 6538330224 \nu^{9} + \cdots + 473682412 \nu ) / 4891522027 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2567797 \nu^{15} + 30901413 \nu^{13} + 256634353 \nu^{11} + 1063746244 \nu^{9} + \cdots - 8743611682 \nu ) / 698788861 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 36273273 \nu^{14} - 465656988 \nu^{12} - 4139477220 \nu^{10} - 19614990672 \nu^{8} + \cdots - 1421047236 ) / 4891522027 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 36573242 \nu^{14} + 484431088 \nu^{12} + 4340858182 \nu^{10} + 21268681752 \nu^{8} + \cdots + 17906037002 ) / 4891522027 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 60455455 \nu^{15} - 776094980 \nu^{13} - 6899128700 \nu^{11} - 32691651120 \nu^{9} + \cdots - 2368412060 \nu ) / 4891522027 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 95063510 \nu^{15} - 1237785252 \nu^{13} - 11043579419 \nu^{11} - 53146226348 \nu^{9} + \cdots - 44139595738 \nu ) / 4891522027 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 161038596 \nu^{15} - 2089361584 \nu^{13} - 18573466960 \nu^{11} - 88454080105 \nu^{9} + \cdots - 6376112848 \nu ) / 4891522027 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 3\beta_{8} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + 5\beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} - 6\beta_{11} - 14\beta_{8} - 6\beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - 7\beta_{13} - 26\beta_{9} + 7\beta_{4} - 26\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} + \beta_{5} + 34\beta_{3} + 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{10} - 9\beta_{7} - 41\beta_{4} + 139\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 51\beta_{12} + 190\beta_{11} + 385\beta_{8} - 51\beta_{6} - 13\beta_{5} - 13\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 13\beta_{15} - 64\beta_{14} + 228\beta_{13} + 752\beta_{9} + 64\beta_{7} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -305\beta_{12} - 1057\beta_{11} - 2092\beta_{8} - 1057\beta_{3} + 116\beta_{2} - 2092 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -116\beta_{15} + 421\beta_{14} - 1246\beta_{13} - 116\beta_{10} - 4090\beta_{9} + 1246\beta_{4} - 4090\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1783\beta_{6} + 885\beta_{5} + 5873\beta_{3} + 11445 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 885\beta_{10} - 2668\beta_{7} - 6771\beta_{4} + 22306\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 10324\beta_{12} + 32630\beta_{11} + 62815\beta_{8} - 10324\beta_{6} - 6208\beta_{5} - 6208\beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 6208\beta_{15} - 16532\beta_{14} + 36746\beta_{13} + 121867\beta_{9} + 16532\beta_{7} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(-1 - \beta_{8}\) \(1\)

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
1.18435 + 2.05136i
1.14313 + 1.97996i
0.715117 + 1.23862i
0.170795 + 0.295826i
−0.170795 − 0.295826i
−0.715117 − 1.23862i
−1.14313 − 1.97996i
−1.18435 − 2.05136i
1.18435 − 2.05136i
1.14313 − 1.97996i
0.715117 − 1.23862i
0.170795 − 0.295826i
−0.170795 + 0.295826i
−0.715117 + 1.23862i
−1.14313 + 1.97996i
−1.18435 + 2.05136i
−1.18435 + 2.05136i 0 −1.80539 − 3.12703i 2.85526 0 0.500000 + 0.866025i 3.81548 0 −3.38164 + 5.85717i
568.2 −1.14313 + 1.97996i 0 −1.61348 − 2.79463i −0.611802 0 0.500000 + 0.866025i 2.80515 0 0.699368 − 1.21134i
568.3 −0.715117 + 1.23862i 0 −0.0227851 − 0.0394650i −1.62279 0 0.500000 + 0.866025i −2.79529 0 1.16049 − 2.01002i
568.4 −0.170795 + 0.295826i 0 0.941658 + 1.63100i 2.79996 0 0.500000 + 0.866025i −1.32650 0 −0.478219 + 0.828299i
568.5 0.170795 − 0.295826i 0 0.941658 + 1.63100i −2.79996 0 0.500000 + 0.866025i 1.32650 0 −0.478219 + 0.828299i
568.6 0.715117 − 1.23862i 0 −0.0227851 − 0.0394650i 1.62279 0 0.500000 + 0.866025i 2.79529 0 1.16049 − 2.01002i
568.7 1.14313 − 1.97996i 0 −1.61348 − 2.79463i 0.611802 0 0.500000 + 0.866025i −2.80515 0 0.699368 − 1.21134i
568.8 1.18435 − 2.05136i 0 −1.80539 − 3.12703i −2.85526 0 0.500000 + 0.866025i −3.81548 0 −3.38164 + 5.85717i
757.1 −1.18435 − 2.05136i 0 −1.80539 + 3.12703i 2.85526 0 0.500000 − 0.866025i 3.81548 0 −3.38164 − 5.85717i
757.2 −1.14313 − 1.97996i 0 −1.61348 + 2.79463i −0.611802 0 0.500000 − 0.866025i 2.80515 0 0.699368 + 1.21134i
757.3 −0.715117 − 1.23862i 0 −0.0227851 + 0.0394650i −1.62279 0 0.500000 − 0.866025i −2.79529 0 1.16049 + 2.01002i
757.4 −0.170795 − 0.295826i 0 0.941658 − 1.63100i 2.79996 0 0.500000 − 0.866025i −1.32650 0 −0.478219 − 0.828299i
757.5 0.170795 + 0.295826i 0 0.941658 − 1.63100i −2.79996 0 0.500000 − 0.866025i 1.32650 0 −0.478219 − 0.828299i
757.6 0.715117 + 1.23862i 0 −0.0227851 + 0.0394650i 1.62279 0 0.500000 − 0.866025i 2.79529 0 1.16049 + 2.01002i
757.7 1.14313 + 1.97996i 0 −1.61348 + 2.79463i 0.611802 0 0.500000 − 0.866025i −2.80515 0 0.699368 + 1.21134i
757.8 1.18435 + 2.05136i 0 −1.80539 + 3.12703i −2.85526 0 0.500000 − 0.866025i −3.81548 0 −3.38164 − 5.85717i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 568.8
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.j ✓ 16
3.b odd 2 1 inner 819.2.o.j ✓ 16
13.c even 3 1 inner 819.2.o.j ✓ 16
39.i odd 6 1 inner 819.2.o.j ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.o.j ✓ 16 1.a even 1 1 trivial
819.2.o.j ✓ 16 3.b odd 2 1 inner
819.2.o.j ✓ 16 13.c even 3 1 inner
819.2.o.j ✓ 16 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 \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{16} + 13T_{2}^{14} + 116T_{2}^{12} + 557T_{2}^{10} + 1944T_{2}^{8} + 3316T_{2}^{6} + 3985T_{2}^{4} + 462T_{2}^{2} + 49 \) Copy content Toggle raw display
\( T_{11}^{16} + 88 T_{11}^{14} + 5171 T_{11}^{12} + 167142 T_{11}^{10} + 3900789 T_{11}^{8} + \cdots + 12350321424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 13 T^{14} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 19 T^{6} + \cdots + 63)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 12350321424 \) Copy content Toggle raw display
$13$ \( (T^{8} + 5 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 18291751009 \) Copy content Toggle raw display
$19$ \( (T^{8} - 5 T^{7} + \cdots + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 71 T^{14} + \cdots + 82301184 \) Copy content Toggle raw display
$29$ \( T^{16} + 108 T^{14} + \cdots + 19140625 \) Copy content Toggle raw display
$31$ \( (T^{4} + T^{3} - 76 T^{2} + \cdots + 420)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} - 7 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 1316818944 \) Copy content Toggle raw display
$43$ \( (T^{8} + 9 T^{7} + \cdots + 145924)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 427 T^{6} + \cdots + 68228188)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 158 T^{6} + \cdots + 5103)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 191 T^{14} + \cdots + 784 \) Copy content Toggle raw display
$61$ \( (T^{8} - 10 T^{7} + \cdots + 3243601)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 9 T^{7} + \cdots + 109746576)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 1469342224 \) Copy content Toggle raw display
$73$ \( (T^{4} + 11 T^{3} + \cdots - 163)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 7 T^{3} + \cdots + 1470)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 224 T^{6} + \cdots + 6300)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} - 21 T^{7} + \cdots + 5466244)^{2} \) Copy content Toggle raw display
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