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Properties

Label 819.2.y.f
Level $819$
Weight $2$
Character orbit 819.y
Analytic conductor $6.540$
Analytic rank $0$
Dimension $12$
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(307,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.y (of order \(4\), 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 \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 60x^{8} - 8x^{7} + 80x^{5} + 320x^{4} + 160x^{3} + 32x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + ( - \beta_{11} - \beta_{6} + 3 \beta_{4}) q^{4} + (\beta_{4} - \beta_1 - 1) q^{5} + ( - \beta_{10} - \beta_{9} + \beta_{7} + \cdots - 1) q^{7}+ \cdots + (\beta_{11} + \beta_{9} + \beta_{7} + \cdots + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + ( - \beta_{11} - \beta_{6} + 3 \beta_{4}) q^{4} + (\beta_{4} - \beta_1 - 1) q^{5} + ( - \beta_{10} - \beta_{9} + \beta_{7} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{11} - \beta_{9} + 3 \beta_{7} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{5} - 4 q^{7} + 4 q^{11} - 36 q^{16} + 8 q^{17} - 8 q^{20} + 32 q^{22} - 4 q^{26} + 12 q^{28} + 8 q^{29} + 24 q^{31} - 20 q^{32} + 20 q^{35} - 4 q^{37} - 40 q^{38} + 20 q^{41} - 8 q^{44} + 20 q^{46} - 32 q^{47} - 20 q^{50} - 56 q^{52} + 16 q^{53} + 20 q^{56} - 8 q^{59} + 16 q^{65} - 32 q^{67} - 20 q^{70} + 12 q^{71} - 32 q^{73} + 64 q^{74} + 12 q^{77} + 24 q^{79} + 4 q^{80} - 32 q^{85} - 4 q^{89} + 32 q^{91} - 112 q^{92} - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 60x^{8} - 8x^{7} + 80x^{5} + 320x^{4} + 160x^{3} + 32x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12309 \nu^{11} + 5381 \nu^{10} - 140579 \nu^{9} + 28834 \nu^{8} - 745616 \nu^{7} + \cdots - 956416 ) / 54445816 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 170557 \nu^{11} - 60738 \nu^{10} + 1270592 \nu^{9} - 400085 \nu^{8} + 10325938 \nu^{7} + \cdots + 13274488 ) / 54445816 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 556438 \nu^{11} - 583343 \nu^{10} + 119552 \nu^{9} - 24618 \nu^{8} - 33375518 \nu^{7} + \cdots + 9887728 ) / 54445816 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 583343 \nu^{11} + 119552 \nu^{10} - 24618 \nu^{9} + 10762 \nu^{8} - 35281738 \nu^{7} + \cdots + 8903008 ) / 54445816 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2412741 \nu^{11} - 3033606 \nu^{10} + 2016198 \nu^{9} - 400372 \nu^{8} + 144220708 \nu^{7} + \cdots - 22588976 ) / 108891632 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3446659 \nu^{11} - 676828 \nu^{10} - 1302018 \nu^{9} + 2509892 \nu^{8} + 205726644 \nu^{7} + \cdots + 62241040 ) / 108891632 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1750029 \nu^{11} + 358656 \nu^{10} - 73854 \nu^{9} + 32286 \nu^{8} - 105845214 \nu^{7} + \cdots + 26709024 ) / 54445816 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5569761 \nu^{11} - 1112876 \nu^{10} - 1166686 \nu^{9} + 239104 \nu^{8} + 334136424 \nu^{7} + \cdots - 316463600 ) / 108891632 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3299703 \nu^{11} + 676169 \nu^{10} - 135251 \nu^{9} - 134391 \nu^{8} - 196850238 \nu^{7} + \cdots + 58170888 ) / 54445816 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8041693 \nu^{11} - 1112876 \nu^{10} - 1166686 \nu^{9} + 239104 \nu^{8} - 482550816 \nu^{7} + \cdots + 119102928 ) / 108891632 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{11} - \beta_{10} - \beta_{8} - \beta_{5} - 4\beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{8} - 6\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{10} - 14\beta_{9} - 8\beta_{8} - 2\beta_{7} - 6\beta_{5} - 8\beta_{3} + 8\beta_{2} - 6\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{11} + 2\beta_{9} - 4\beta_{4} - 20\beta_{2} - 40\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 100 \beta_{11} + 60 \beta_{10} + 60 \beta_{8} + 20 \beta_{6} + 36 \beta_{5} + 164 \beta_{4} + \cdots - 36 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 28\beta_{11} - 8\beta_{10} - 28\beta_{9} - 168\beta_{8} + 276\beta_{5} - 56\beta_{4} - 56 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 444 \beta_{10} + 728 \beta_{9} + 452 \beta_{8} + 160 \beta_{7} + 220 \beta_{5} + 444 \beta_{3} + \cdots + 1176 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 296 \beta_{11} - 296 \beta_{9} - 8 \beta_{7} + 8 \beta_{6} + 576 \beta_{4} - 144 \beta_{3} + \cdots - 576 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5352 \beta_{11} - 3272 \beta_{10} - 3432 \beta_{8} - 1208 \beta_{6} - 1344 \beta_{5} - 8608 \beta_{4} + \cdots + 1344 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2808 \beta_{11} + 1760 \beta_{10} + 2808 \beta_{9} + 10720 \beta_{8} + 160 \beta_{7} + 160 \beta_{6} + \cdots + 5312 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.27310 − 1.27310i
−1.96818 + 1.96818i
0.236276 − 0.236276i
1.85068 − 1.85068i
−0.528642 + 0.528642i
−0.863233 + 0.863233i
1.27310 + 1.27310i
−1.96818 − 1.96818i
0.236276 + 0.236276i
1.85068 + 1.85068i
−0.528642 − 0.528642i
−0.863233 − 0.863233i
−1.75588 − 1.75588i 0 4.16620i −2.27310 + 2.27310i 0 −2.08970 + 1.62270i 3.80357 − 3.80357i 0 7.98257
307.2 −1.71968 − 1.71968i 0 3.91457i 0.968182 − 0.968182i 0 0.407631 − 2.61416i 3.29244 − 3.29244i 0 −3.32992
307.3 −0.695639 − 0.695639i 0 − 1.03217i −1.23628 + 1.23628i 0 1.20338 − 2.35624i −2.10930 + 2.10930i 0 1.72000
307.4 0.786556 + 0.786556i 0 − 0.762660i −2.85068 + 2.85068i 0 −1.83993 − 1.90123i 2.17299 − 2.17299i 0 −4.48443
307.5 1.43819 + 1.43819i 0 2.13679i −0.471358 + 0.471358i 0 2.56584 + 0.645342i −0.196726 + 0.196726i 0 −1.35580
307.6 1.94644 + 1.94644i 0 5.57728i −0.136767 + 0.136767i 0 −2.24723 − 1.39641i −6.96297 + 6.96297i 0 −0.532416
811.1 −1.75588 + 1.75588i 0 − 4.16620i −2.27310 − 2.27310i 0 −2.08970 − 1.62270i 3.80357 + 3.80357i 0 7.98257
811.2 −1.71968 + 1.71968i 0 − 3.91457i 0.968182 + 0.968182i 0 0.407631 + 2.61416i 3.29244 + 3.29244i 0 −3.32992
811.3 −0.695639 + 0.695639i 0 1.03217i −1.23628 − 1.23628i 0 1.20338 + 2.35624i −2.10930 − 2.10930i 0 1.72000
811.4 0.786556 − 0.786556i 0 0.762660i −2.85068 − 2.85068i 0 −1.83993 + 1.90123i 2.17299 + 2.17299i 0 −4.48443
811.5 1.43819 − 1.43819i 0 − 2.13679i −0.471358 − 0.471358i 0 2.56584 − 0.645342i −0.196726 − 0.196726i 0 −1.35580
811.6 1.94644 − 1.94644i 0 − 5.57728i −0.136767 − 0.136767i 0 −2.24723 + 1.39641i −6.96297 − 6.96297i 0 −0.532416
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.y.f 12
3.b odd 2 1 273.2.p.f yes 12
7.b odd 2 1 819.2.y.g 12
13.d odd 4 1 819.2.y.g 12
21.c even 2 1 273.2.p.e ✓ 12
39.f even 4 1 273.2.p.e ✓ 12
91.i even 4 1 inner 819.2.y.f 12
273.o odd 4 1 273.2.p.f yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.p.e ✓ 12 21.c even 2 1
273.2.p.e ✓ 12 39.f even 4 1
273.2.p.f yes 12 3.b odd 2 1
273.2.p.f yes 12 273.o odd 4 1
819.2.y.f 12 1.a even 1 1 trivial
819.2.y.f 12 91.i even 4 1 inner
819.2.y.g 12 7.b odd 2 1
819.2.y.g 12 13.d odd 4 1

Hecke kernels

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

\( T_{2}^{12} + 75T_{2}^{8} + 4T_{2}^{7} - 52T_{2}^{5} + 1247T_{2}^{4} - 188T_{2}^{3} + 8T_{2}^{2} + 148T_{2} + 1369 \) Copy content Toggle raw display
\( T_{5}^{12} + 12 T_{5}^{11} + 72 T_{5}^{10} + 240 T_{5}^{9} + 480 T_{5}^{8} + 520 T_{5}^{7} + 480 T_{5}^{6} + \cdots + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 75 T^{8} + \cdots + 1369 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 12 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{12} + 4 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 1936 \) Copy content Toggle raw display
$13$ \( T^{12} - 10 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} - 4 T^{5} + \cdots - 696)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 64 T^{9} + \cdots + 295936 \) Copy content Toggle raw display
$23$ \( T^{12} + 92 T^{10} + \cdots + 10816 \) Copy content Toggle raw display
$29$ \( (T^{6} - 4 T^{5} + \cdots - 352)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 24 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 17676234304 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 112444816 \) Copy content Toggle raw display
$43$ \( T^{12} + 272 T^{10} + \cdots + 75759616 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 6961566096 \) Copy content Toggle raw display
$53$ \( (T^{6} - 8 T^{5} + \cdots - 192)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 123032464 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11283538176 \) Copy content Toggle raw display
$67$ \( T^{12} + 32 T^{11} + \cdots + 2359296 \) Copy content Toggle raw display
$71$ \( T^{12} - 12 T^{11} + \cdots + 47169424 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 127872038464 \) Copy content Toggle raw display
$79$ \( (T^{6} - 12 T^{5} + \cdots + 49424)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 2417098896 \) Copy content Toggle raw display
$89$ \( T^{12} + 4 T^{11} + \cdots + 169744 \) Copy content Toggle raw display
$97$ \( T^{12} - 208 T^{9} + \cdots + 3655744 \) Copy content Toggle raw display
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