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: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{5} - q^{6} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{5} - q^{6} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + q^{8} + \beta_{2} q^{9} + (3 \beta_{2} - \beta_1 + 3) q^{10} + (\beta_{2} + 1) q^{11} - \beta_{2} q^{12} - 2 \beta_{3} q^{13} + ( - \beta_{3} - 2 \beta_1 + 1) q^{14} + (\beta_{3} + 3) q^{15} + \beta_{2} q^{16} + 3 \beta_1 q^{17} + ( - \beta_{2} - 1) q^{18} + \beta_{2} q^{19} + ( - \beta_{3} - 3) q^{20} + (\beta_{3} - \beta_{2} - \beta_1) q^{21} - q^{22} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{23} + (\beta_{2} + 1) q^{24} + ( - 6 \beta_{2} + 6 \beta_1 - 6) q^{25} + (2 \beta_{3} + 2 \beta_1) q^{26} - q^{27} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{28} + (2 \beta_{3} + 5) q^{29} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{30} + (5 \beta_{2} + 4 \beta_1 + 5) q^{31} + ( - \beta_{2} - 1) q^{32} + \beta_{2} q^{33} + 3 \beta_{3} q^{34} + (2 \beta_{3} - 4 \beta_{2} + 6 \beta_1 - 5) q^{35} + q^{36} + (3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{37} + ( - \beta_{2} - 1) q^{38} + 2 \beta_1 q^{39} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{40} + ( - \beta_{3} - 2) q^{41} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{42} + ( - 3 \beta_{3} + 2) q^{43} - \beta_{2} q^{44} + (3 \beta_{2} - \beta_1 + 3) q^{45} + ( - 2 \beta_{2} - \beta_1 - 2) q^{46} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{47} - q^{48} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{49} + (6 \beta_{3} + 6) q^{50} + (3 \beta_{3} + 3 \beta_1) q^{51} - 2 \beta_1 q^{52} + (\beta_{2} - 2 \beta_1 + 1) q^{53} - \beta_{2} q^{54} + (\beta_{3} + 3) q^{55} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{56} - q^{57} + ( - 2 \beta_{3} + 5 \beta_{2} - 2 \beta_1) q^{58} + (\beta_{2} - 5 \beta_1 + 1) q^{59} + ( - 3 \beta_{2} + \beta_1 - 3) q^{60} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{61} + (4 \beta_{3} - 5) q^{62} + ( - \beta_{3} - 2 \beta_1 + 1) q^{63} + q^{64} + ( - 6 \beta_{3} + 4 \beta_{2} - 6 \beta_1) q^{65} + ( - \beta_{2} - 1) q^{66} + (2 \beta_{2} + 2) q^{67} + ( - 3 \beta_{3} - 3 \beta_1) q^{68} + (\beta_{3} - 2) q^{69} + (4 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{70} + (7 \beta_{3} + 2) q^{71} + \beta_{2} q^{72} + (12 \beta_{2} + 2 \beta_1 + 12) q^{73} + (4 \beta_{2} - 3 \beta_1 + 4) q^{74} + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{75} + q^{76} + (\beta_{3} - \beta_{2} - \beta_1) q^{77} + 2 \beta_{3} q^{78} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{79} + (3 \beta_{2} - \beta_1 + 3) q^{80} + ( - \beta_{2} - 1) q^{81} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{82} + (10 \beta_{3} - 3) q^{83} + (\beta_{3} + 2 \beta_1 - 1) q^{84} + ( - 9 \beta_{3} - 6) q^{85} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{86} + (5 \beta_{2} - 2 \beta_1 + 5) q^{87} + (\beta_{2} + 1) q^{88} + ( - 3 \beta_{3} + 8 \beta_{2} - 3 \beta_1) q^{89} + ( - \beta_{3} - 3) q^{90} + (4 \beta_{2} - 2 \beta_1 - 4) q^{91} + ( - \beta_{3} + 2) q^{92} + (4 \beta_{3} + 5 \beta_{2} + 4 \beta_1) q^{93} + ( - 6 \beta_{2} + 2 \beta_1 - 6) q^{94} + (3 \beta_{2} - \beta_1 + 3) q^{95} - \beta_{2} q^{96} + ( - \beta_{3} - 7) q^{97} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 5) q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 6 q^{5} - 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 6 q^{5} - 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9} + 6 q^{10} + 2 q^{11} + 2 q^{12} + 4 q^{14} + 12 q^{15} - 2 q^{16} - 2 q^{18} - 2 q^{19} - 12 q^{20} + 2 q^{21} - 4 q^{22} - 4 q^{23} + 2 q^{24} - 12 q^{25} - 4 q^{27} - 2 q^{28} + 20 q^{29} - 6 q^{30} + 10 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{35} + 4 q^{36} + 8 q^{37} - 2 q^{38} + 6 q^{40} - 8 q^{41} + 2 q^{42} + 8 q^{43} + 2 q^{44} + 6 q^{45} - 4 q^{46} - 12 q^{47} - 4 q^{48} + 10 q^{49} + 24 q^{50} + 2 q^{53} + 2 q^{54} + 12 q^{55} - 2 q^{56} - 4 q^{57} - 10 q^{58} + 2 q^{59} - 6 q^{60} + 4 q^{61} - 20 q^{62} + 4 q^{63} + 4 q^{64} - 8 q^{65} - 2 q^{66} + 4 q^{67} - 8 q^{69} + 18 q^{70} + 8 q^{71} - 2 q^{72} + 24 q^{73} + 8 q^{74} + 12 q^{75} + 4 q^{76} + 2 q^{77} + 6 q^{79} + 6 q^{80} - 2 q^{81} + 4 q^{82} - 12 q^{83} - 4 q^{84} - 24 q^{85} - 4 q^{86} + 10 q^{87} + 2 q^{88} - 16 q^{89} - 12 q^{90} - 24 q^{91} + 8 q^{92} - 10 q^{93} - 12 q^{94} + 6 q^{95} + 2 q^{96} - 28 q^{97} + 10 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(798, [\chi])\):

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

Hecke characteristic polynomials

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