Properties

Label 804.2.i.b
Level 804
Weight 2
Character orbit 804.i
Analytic conductor 6.420
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

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

Self dual: no
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + 2 q^{5} -5 \zeta_{6} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + 2 q^{5} -5 \zeta_{6} q^{7} + q^{9} -5 \zeta_{6} q^{11} + ( -3 + 3 \zeta_{6} ) q^{13} -2 q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} + ( -7 + 7 \zeta_{6} ) q^{19} + 5 \zeta_{6} q^{21} + ( -3 + 3 \zeta_{6} ) q^{23} - q^{25} - q^{27} + \zeta_{6} q^{29} -5 \zeta_{6} q^{31} + 5 \zeta_{6} q^{33} -10 \zeta_{6} q^{35} + ( 5 - 5 \zeta_{6} ) q^{37} + ( 3 - 3 \zeta_{6} ) q^{39} -11 \zeta_{6} q^{41} + 4 q^{43} + 2 q^{45} + 3 \zeta_{6} q^{47} + ( -18 + 18 \zeta_{6} ) q^{49} + ( 3 - 3 \zeta_{6} ) q^{51} -6 q^{53} -10 \zeta_{6} q^{55} + ( 7 - 7 \zeta_{6} ) q^{57} -8 q^{59} + ( 1 - \zeta_{6} ) q^{61} -5 \zeta_{6} q^{63} + ( -6 + 6 \zeta_{6} ) q^{65} + ( 7 + 2 \zeta_{6} ) q^{67} + ( 3 - 3 \zeta_{6} ) q^{69} -\zeta_{6} q^{71} + ( 9 - 9 \zeta_{6} ) q^{73} + q^{75} + ( -25 + 25 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + q^{81} + ( 9 - 9 \zeta_{6} ) q^{83} + ( -6 + 6 \zeta_{6} ) q^{85} -\zeta_{6} q^{87} -18 q^{89} + 15 q^{91} + 5 \zeta_{6} q^{93} + ( -14 + 14 \zeta_{6} ) q^{95} + ( 17 - 17 \zeta_{6} ) q^{97} -5 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{5} - 5q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{5} - 5q^{7} + 2q^{9} - 5q^{11} - 3q^{13} - 4q^{15} - 3q^{17} - 7q^{19} + 5q^{21} - 3q^{23} - 2q^{25} - 2q^{27} + q^{29} - 5q^{31} + 5q^{33} - 10q^{35} + 5q^{37} + 3q^{39} - 11q^{41} + 8q^{43} + 4q^{45} + 3q^{47} - 18q^{49} + 3q^{51} - 12q^{53} - 10q^{55} + 7q^{57} - 16q^{59} + q^{61} - 5q^{63} - 6q^{65} + 16q^{67} + 3q^{69} - q^{71} + 9q^{73} + 2q^{75} - 25q^{77} - q^{79} + 2q^{81} + 9q^{83} - 6q^{85} - q^{87} - 36q^{89} + 30q^{91} + 5q^{93} - 14q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 0 2.00000 0 −2.50000 + 4.33013i 0 1.00000 0
565.1 0 −1.00000 0 2.00000 0 −2.50000 4.33013i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
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.b 2
3.b odd 2 1 2412.2.l.a 2
67.c even 3 1 inner 804.2.i.b 2
201.g odd 6 1 2412.2.l.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.i.b 2 1.a even 1 1 trivial
804.2.i.b 2 67.c even 3 1 inner
2412.2.l.a 2 3.b odd 2 1
2412.2.l.a 2 201.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 \(S_{2}^{\mathrm{new}}(804, [\chi])\):

\( T_{5} - 2 \)
\( T_{7}^{2} + 5 T_{7} + 25 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 - 2 T + 5 T^{2} )^{2} \)
$7$ \( ( 1 + T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \)
$11$ \( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \)
$13$ \( 1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4} \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \)
$31$ \( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \)
$37$ \( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 11 T + 80 T^{2} + 451 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 + 8 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \)
$67$ \( 1 - 16 T + 67 T^{2} \)
$71$ \( 1 + T - 70 T^{2} + 71 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 9 T + 8 T^{2} - 657 T^{3} + 5329 T^{4} \)
$79$ \( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 18 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4} \)
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