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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(2394, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 3T_{13} + 9 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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