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Properties

Label 804.2.i.d
Level $804$
Weight $2$
Character orbit 804.i
Analytic conductor $6.420$
Analytic rank $0$
Dimension $8$
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] = [804,2,Mod(37,804)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("804.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 804.i (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: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 11x^{6} + 4x^{5} + 91x^{4} - 6x^{3} + 129x^{2} + 36x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{6} q^{5} + (\beta_{6} - \beta_1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_{6} q^{5} + (\beta_{6} - \beta_1) q^{7} + q^{9} + 2 \beta_{2} q^{11} + (\beta_{4} + \beta_{3}) q^{13} - \beta_{6} q^{15} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{17} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \cdots - 1) q^{19}+ \cdots + 2 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} + 2 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} + 2 q^{5} + q^{7} + 8 q^{9} + 8 q^{11} - 2 q^{15} + 4 q^{17} - 5 q^{19} - q^{21} + q^{23} + 2 q^{25} - 8 q^{27} - 2 q^{29} + 9 q^{31} - 8 q^{33} + 21 q^{35} - 5 q^{37} + 11 q^{41} + 6 q^{43} + 2 q^{45} + 11 q^{47} + 7 q^{49} - 4 q^{51} + 40 q^{53} + 2 q^{55} + 5 q^{57} + 28 q^{59} + 20 q^{61} + q^{63} - 4 q^{65} - 33 q^{67} - q^{69} + 11 q^{71} - 7 q^{73} - 2 q^{75} - 2 q^{77} + 12 q^{79} + 8 q^{81} - 4 q^{83} + 5 q^{85} + 2 q^{87} - 16 q^{89} - 8 q^{91} - 9 q^{93} - 18 q^{95} + 20 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 11x^{6} + 4x^{5} + 91x^{4} - 6x^{3} + 129x^{2} + 36x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 47\nu^{7} - 199\nu^{6} + 597\nu^{5} - 1256\nu^{4} + 2985\nu^{3} - 14726\nu^{2} + 3555\nu - 1044 ) / 18060 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20\nu^{7} + 37\nu^{6} + 190\nu^{5} + 170\nu^{4} + 2756\nu^{3} + 330\nu^{2} + 360\nu - 15411 ) / 4515 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83\nu^{7} - 915\nu^{6} + 1541\nu^{5} - 7572\nu^{4} + 481\nu^{3} - 58078\nu^{2} - 3021\nu - 11988 ) / 18060 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 67\nu^{7} + 139\nu^{6} - 417\nu^{5} + 3128\nu^{4} - 2085\nu^{3} + 10286\nu^{2} - 62907\nu + 13344 ) / 9030 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -38\nu^{7} + 20\nu^{6} - 361\nu^{5} - 323\nu^{4} - 3611\nu^{3} - 627\nu^{2} - 684\nu - 1692 ) / 4515 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 577\nu^{7} - 573\nu^{6} + 6535\nu^{5} + 2346\nu^{4} + 52541\nu^{3} + 340\nu^{2} + 74499\nu + 20844 ) / 9030 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{4} - 5\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - 9\beta_{6} - \beta_{5} - \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} - 11\beta_{4} - 11\beta_{3} + 41\beta_{2} - 16\beta _1 - 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{7} + 91\beta_{6} - 18\beta_{4} + 56\beta_{2} - 91\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 18\beta_{7} + 212\beta_{6} + 18\beta_{5} + 113\beta_{3} + 397 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 113\beta_{5} + 248\beta_{4} + 248\beta_{3} - 798\beta_{2} + 966\beta _1 + 798 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.30177 − 2.25473i
−0.531167 − 0.920008i
0.641129 + 1.11047i
1.69181 + 2.93030i
−1.30177 + 2.25473i
−0.531167 + 0.920008i
0.641129 − 1.11047i
1.69181 − 2.93030i
0 −1.00000 0 −2.60354 0 −1.30177 + 2.25473i 0 1.00000 0
37.2 0 −1.00000 0 −1.06233 0 −0.531167 + 0.920008i 0 1.00000 0
37.3 0 −1.00000 0 1.28226 0 0.641129 − 1.11047i 0 1.00000 0
37.4 0 −1.00000 0 3.38361 0 1.69181 − 2.93030i 0 1.00000 0
565.1 0 −1.00000 0 −2.60354 0 −1.30177 − 2.25473i 0 1.00000 0
565.2 0 −1.00000 0 −1.06233 0 −0.531167 − 0.920008i 0 1.00000 0
565.3 0 −1.00000 0 1.28226 0 0.641129 + 1.11047i 0 1.00000 0
565.4 0 −1.00000 0 3.38361 0 1.69181 + 2.93030i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 804.2.i.d ✓ 8
3.b odd 2 1 2412.2.l.e 8
67.c even 3 1 inner 804.2.i.d ✓ 8
201.g odd 6 1 2412.2.l.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.i.d ✓ 8 1.a even 1 1 trivial
804.2.i.d ✓ 8 67.c even 3 1 inner
2412.2.l.e 8 3.b odd 2 1
2412.2.l.e 8 201.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\):

\( T_{5}^{4} - T_{5}^{3} - 10T_{5}^{2} + 3T_{5} + 12 \) Copy content Toggle raw display
\( T_{7}^{8} - T_{7}^{7} + 11T_{7}^{6} + 4T_{7}^{5} + 91T_{7}^{4} - 6T_{7}^{3} + 129T_{7}^{2} + 36T_{7} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} - 10 T^{2} + \cdots + 12)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 29 T^{6} + \cdots + 30625 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + \cdots + 22500 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} + \cdots + 19600 \) Copy content Toggle raw display
$23$ \( T^{8} - T^{7} + \cdots + 51076 \) Copy content Toggle raw display
$29$ \( T^{8} + 2 T^{7} + \cdots + 1188100 \) Copy content Toggle raw display
$31$ \( T^{8} - 9 T^{7} + \cdots + 2166784 \) Copy content Toggle raw display
$37$ \( T^{8} + 5 T^{7} + \cdots + 2992900 \) Copy content Toggle raw display
$41$ \( T^{8} - 11 T^{7} + \cdots + 2876416 \) Copy content Toggle raw display
$43$ \( (T^{4} - 3 T^{3} + \cdots + 347)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 11 T^{7} + \cdots + 2876416 \) Copy content Toggle raw display
$53$ \( (T^{4} - 20 T^{3} + \cdots + 90)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 14 T^{3} + \cdots - 2208)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 20 T^{7} + \cdots + 286794225 \) Copy content Toggle raw display
$67$ \( T^{8} + 33 T^{7} + \cdots + 20151121 \) Copy content Toggle raw display
$71$ \( T^{8} - 11 T^{7} + \cdots + 199148544 \) Copy content Toggle raw display
$73$ \( T^{8} + 7 T^{7} + \cdots + 149769 \) Copy content Toggle raw display
$79$ \( T^{8} - 12 T^{7} + \cdots + 24059025 \) Copy content Toggle raw display
$83$ \( T^{8} + 4 T^{7} + \cdots + 20736 \) Copy content Toggle raw display
$89$ \( (T^{4} + 8 T^{3} + \cdots - 200)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 20 T^{7} + \cdots + 2152089 \) Copy content Toggle raw display
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