Properties

Label 798.2.k.h
Level $798$
Weight $2$
Character orbit 798.k
Analytic conductor $6.372$
Analytic rank $0$
Dimension $2$
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.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

Character values

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

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
463.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.50000 + 2.59808i 0.500000 0.866025i 1.00000 −1.00000 −0.500000 + 0.866025i −1.50000 + 2.59808i
505.1 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.50000 2.59808i 0.500000 + 0.866025i 1.00000 −1.00000 −0.500000 0.866025i −1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.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.k.h 2
3.b odd 2 1 2394.2.o.b 2
19.c even 3 1 inner 798.2.k.h 2
57.h odd 6 1 2394.2.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.k.h 2 1.a even 1 1 trivial
798.2.k.h 2 19.c even 3 1 inner
2394.2.o.b 2 3.b odd 2 1
2394.2.o.b 2 57.h odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{11} + 6 \) Copy content Toggle raw display
\( T_{13}^{2} + 5T_{13} + 25 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( (T - 15)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
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