Properties

Label 2394.2.o.a
Level $2394$
Weight $2$
Character orbit 2394.o
Analytic conductor $19.116$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.o (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} - q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} - q^{7} + q^{8} -3 \zeta_{6} q^{10} -4 q^{11} -3 \zeta_{6} q^{13} + ( 1 - \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -2 + 2 \zeta_{6} ) q^{17} + ( -3 + 5 \zeta_{6} ) q^{19} + 3 q^{20} + ( 4 - 4 \zeta_{6} ) q^{22} + \zeta_{6} q^{23} -4 \zeta_{6} q^{25} + 3 q^{26} + \zeta_{6} q^{28} -6 q^{31} -\zeta_{6} q^{32} -2 \zeta_{6} q^{34} + ( 3 - 3 \zeta_{6} ) q^{35} + 4 q^{37} + ( -2 - 3 \zeta_{6} ) q^{38} + ( -3 + 3 \zeta_{6} ) q^{40} + ( 4 - 4 \zeta_{6} ) q^{41} + ( 6 - 6 \zeta_{6} ) q^{43} + 4 \zeta_{6} q^{44} - q^{46} -2 \zeta_{6} q^{47} + q^{49} + 4 q^{50} + ( -3 + 3 \zeta_{6} ) q^{52} -4 \zeta_{6} q^{53} + ( 12 - 12 \zeta_{6} ) q^{55} - q^{56} + ( 13 - 13 \zeta_{6} ) q^{59} -5 \zeta_{6} q^{61} + ( 6 - 6 \zeta_{6} ) q^{62} + q^{64} + 9 q^{65} + 12 \zeta_{6} q^{67} + 2 q^{68} + 3 \zeta_{6} q^{70} + ( -7 + 7 \zeta_{6} ) q^{71} + ( 2 - 2 \zeta_{6} ) q^{73} + ( -4 + 4 \zeta_{6} ) q^{74} + ( 5 - 2 \zeta_{6} ) q^{76} + 4 q^{77} + ( 8 - 8 \zeta_{6} ) q^{79} -3 \zeta_{6} q^{80} + 4 \zeta_{6} q^{82} -17 q^{83} -6 \zeta_{6} q^{85} + 6 \zeta_{6} q^{86} -4 q^{88} + 4 \zeta_{6} q^{89} + 3 \zeta_{6} q^{91} + ( 1 - \zeta_{6} ) q^{92} + 2 q^{94} + ( -6 - 9 \zeta_{6} ) q^{95} + ( 10 - 10 \zeta_{6} ) q^{97} + ( -1 + \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 3 q^{5} - 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q - q^{2} - q^{4} - 3 q^{5} - 2 q^{7} + 2 q^{8} - 3 q^{10} - 8 q^{11} - 3 q^{13} + q^{14} - q^{16} - 2 q^{17} - q^{19} + 6 q^{20} + 4 q^{22} + q^{23} - 4 q^{25} + 6 q^{26} + q^{28} - 12 q^{31} - q^{32} - 2 q^{34} + 3 q^{35} + 8 q^{37} - 7 q^{38} - 3 q^{40} + 4 q^{41} + 6 q^{43} + 4 q^{44} - 2 q^{46} - 2 q^{47} + 2 q^{49} + 8 q^{50} - 3 q^{52} - 4 q^{53} + 12 q^{55} - 2 q^{56} + 13 q^{59} - 5 q^{61} + 6 q^{62} + 2 q^{64} + 18 q^{65} + 12 q^{67} + 4 q^{68} + 3 q^{70} - 7 q^{71} + 2 q^{73} - 4 q^{74} + 8 q^{76} + 8 q^{77} + 8 q^{79} - 3 q^{80} + 4 q^{82} - 34 q^{83} - 6 q^{85} + 6 q^{86} - 8 q^{88} + 4 q^{89} + 3 q^{91} + q^{92} + 4 q^{94} - 21 q^{95} + 10 q^{97} - q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −1.50000 + 2.59808i 0 −1.00000 1.00000 0 −1.50000 2.59808i
1261.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.50000 2.59808i 0 −1.00000 1.00000 0 −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 2394.2.o.a 2
3.b odd 2 1 798.2.k.i 2
19.c even 3 1 inner 2394.2.o.a 2
57.h odd 6 1 798.2.k.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.k.i 2 3.b odd 2 1
798.2.k.i 2 57.h odd 6 1
2394.2.o.a 2 1.a even 1 1 trivial
2394.2.o.a 2 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2394, [\chi])\):

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 3 T_{13} + 9 \)
\( T_{17}^{2} + 2 T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 9 + 3 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 9 + 3 T + T^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 19 + T + T^{2} \)
$23$ \( 1 - T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 6 + T )^{2} \)
$37$ \( ( -4 + T )^{2} \)
$41$ \( 16 - 4 T + T^{2} \)
$43$ \( 36 - 6 T + T^{2} \)
$47$ \( 4 + 2 T + T^{2} \)
$53$ \( 16 + 4 T + T^{2} \)
$59$ \( 169 - 13 T + T^{2} \)
$61$ \( 25 + 5 T + T^{2} \)
$67$ \( 144 - 12 T + T^{2} \)
$71$ \( 49 + 7 T + T^{2} \)
$73$ \( 4 - 2 T + T^{2} \)
$79$ \( 64 - 8 T + T^{2} \)
$83$ \( ( 17 + T )^{2} \)
$89$ \( 16 - 4 T + T^{2} \)
$97$ \( 100 - 10 T + T^{2} \)
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