Properties

Label 798.2.j.h
Level $798$
Weight $2$
Character orbit 798.j
Analytic conductor $6.372$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{2} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{5} - q^{6} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{7} + q^{8} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{2} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{5} - q^{6} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{7} + q^{8} + \beta_{2} q^{9} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{10} + ( 1 + \beta_{2} ) q^{11} -\beta_{2} q^{12} -2 \beta_{3} q^{13} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{14} + ( 3 + \beta_{3} ) q^{15} + \beta_{2} q^{16} + 3 \beta_{1} q^{17} + ( -1 - \beta_{2} ) q^{18} + \beta_{2} q^{19} + ( -3 - \beta_{3} ) q^{20} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{21} - q^{22} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{23} + ( 1 + \beta_{2} ) q^{24} + ( -6 + 6 \beta_{1} - 6 \beta_{2} ) q^{25} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{26} - q^{27} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{28} + ( 5 + 2 \beta_{3} ) q^{29} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{30} + ( 5 + 4 \beta_{1} + 5 \beta_{2} ) q^{31} + ( -1 - \beta_{2} ) q^{32} + \beta_{2} q^{33} + 3 \beta_{3} q^{34} + ( -5 + 6 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{35} + q^{36} + ( 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{37} + ( -1 - \beta_{2} ) q^{38} + 2 \beta_{1} q^{39} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{40} + ( -2 - \beta_{3} ) q^{41} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{42} + ( 2 - 3 \beta_{3} ) q^{43} -\beta_{2} q^{44} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{45} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{46} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{47} - q^{48} + ( 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{49} + ( 6 + 6 \beta_{3} ) q^{50} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{51} -2 \beta_{1} q^{52} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{53} -\beta_{2} q^{54} + ( 3 + \beta_{3} ) q^{55} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{56} - q^{57} + ( -2 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{58} + ( 1 - 5 \beta_{1} + \beta_{2} ) q^{59} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{60} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -5 + 4 \beta_{3} ) q^{62} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{63} + q^{64} + ( -6 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{65} + ( -1 - \beta_{2} ) q^{66} + ( 2 + 2 \beta_{2} ) q^{67} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{68} + ( -2 + \beta_{3} ) q^{69} + ( 4 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{70} + ( 2 + 7 \beta_{3} ) q^{71} + \beta_{2} q^{72} + ( 12 + 2 \beta_{1} + 12 \beta_{2} ) q^{73} + ( 4 - 3 \beta_{1} + 4 \beta_{2} ) q^{74} + ( 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{75} + q^{76} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{77} + 2 \beta_{3} q^{78} + ( 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{79} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{80} + ( -1 - \beta_{2} ) q^{81} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{82} + ( -3 + 10 \beta_{3} ) q^{83} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{84} + ( -6 - 9 \beta_{3} ) q^{85} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{86} + ( 5 - 2 \beta_{1} + 5 \beta_{2} ) q^{87} + ( 1 + \beta_{2} ) q^{88} + ( -3 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} ) q^{89} + ( -3 - \beta_{3} ) q^{90} + ( -4 - 2 \beta_{1} + 4 \beta_{2} ) q^{91} + ( 2 - \beta_{3} ) q^{92} + ( 4 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{93} + ( -6 + 2 \beta_{1} - 6 \beta_{2} ) q^{94} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{95} -\beta_{2} q^{96} + ( -7 - \beta_{3} ) q^{97} + ( 5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{98} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 2q^{3} - 2q^{4} + 6q^{5} - 4q^{6} - 2q^{7} + 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 2q^{3} - 2q^{4} + 6q^{5} - 4q^{6} - 2q^{7} + 4q^{8} - 2q^{9} + 6q^{10} + 2q^{11} + 2q^{12} + 4q^{14} + 12q^{15} - 2q^{16} - 2q^{18} - 2q^{19} - 12q^{20} + 2q^{21} - 4q^{22} - 4q^{23} + 2q^{24} - 12q^{25} - 4q^{27} - 2q^{28} + 20q^{29} - 6q^{30} + 10q^{31} - 2q^{32} - 2q^{33} - 12q^{35} + 4q^{36} + 8q^{37} - 2q^{38} + 6q^{40} - 8q^{41} + 2q^{42} + 8q^{43} + 2q^{44} + 6q^{45} - 4q^{46} - 12q^{47} - 4q^{48} + 10q^{49} + 24q^{50} + 2q^{53} + 2q^{54} + 12q^{55} - 2q^{56} - 4q^{57} - 10q^{58} + 2q^{59} - 6q^{60} + 4q^{61} - 20q^{62} + 4q^{63} + 4q^{64} - 8q^{65} - 2q^{66} + 4q^{67} - 8q^{69} + 18q^{70} + 8q^{71} - 2q^{72} + 24q^{73} + 8q^{74} + 12q^{75} + 4q^{76} + 2q^{77} + 6q^{79} + 6q^{80} - 2q^{81} + 4q^{82} - 12q^{83} - 4q^{84} - 24q^{85} - 4q^{86} + 10q^{87} + 2q^{88} - 16q^{89} - 12q^{90} - 24q^{91} + 8q^{92} - 10q^{93} - 12q^{94} + 6q^{95} + 2q^{96} - 28q^{97} + 10q^{98} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
457.1
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.792893 1.37333i −1.00000 −2.62132 + 0.358719i 1.00000 −0.500000 + 0.866025i 0.792893 + 1.37333i
457.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 2.20711 3.82282i −1.00000 1.62132 2.09077i 1.00000 −0.500000 + 0.866025i 2.20711 + 3.82282i
571.1 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.792893 + 1.37333i −1.00000 −2.62132 0.358719i 1.00000 −0.500000 0.866025i 0.792893 1.37333i
571.2 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 2.20711 + 3.82282i −1.00000 1.62132 + 2.09077i 1.00000 −0.500000 0.866025i 2.20711 3.82282i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.j.h 4
7.c even 3 1 inner 798.2.j.h 4
7.c even 3 1 5586.2.a.bg 2
7.d odd 6 1 5586.2.a.br 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.h 4 1.a even 1 1 trivial
798.2.j.h 4 7.c even 3 1 inner
5586.2.a.bg 2 7.c even 3 1
5586.2.a.br 2 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\):

\( T_{5}^{4} - 6 T_{5}^{3} + 29 T_{5}^{2} - 42 T_{5} + 49 \)
\( T_{11}^{2} - T_{11} + 1 \)
\( T_{13}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( 49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4} \)
$7$ \( 49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( ( -8 + T^{2} )^{2} \)
$17$ \( 324 + 18 T^{2} + T^{4} \)
$19$ \( ( 1 + T + T^{2} )^{2} \)
$23$ \( 4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4} \)
$29$ \( ( 17 - 10 T + T^{2} )^{2} \)
$31$ \( 49 + 70 T + 107 T^{2} - 10 T^{3} + T^{4} \)
$37$ \( 4 + 16 T + 66 T^{2} - 8 T^{3} + T^{4} \)
$41$ \( ( 2 + 4 T + T^{2} )^{2} \)
$43$ \( ( -14 - 4 T + T^{2} )^{2} \)
$47$ \( 784 + 336 T + 116 T^{2} + 12 T^{3} + T^{4} \)
$53$ \( 49 + 14 T + 11 T^{2} - 2 T^{3} + T^{4} \)
$59$ \( 2401 + 98 T + 53 T^{2} - 2 T^{3} + T^{4} \)
$61$ \( 16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4} \)
$67$ \( ( 4 - 2 T + T^{2} )^{2} \)
$71$ \( ( -94 - 4 T + T^{2} )^{2} \)
$73$ \( 18496 - 3264 T + 440 T^{2} - 24 T^{3} + T^{4} \)
$79$ \( 1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4} \)
$83$ \( ( -191 + 6 T + T^{2} )^{2} \)
$89$ \( 2116 + 736 T + 210 T^{2} + 16 T^{3} + T^{4} \)
$97$ \( ( 47 + 14 T + T^{2} )^{2} \)
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