Properties

Label 294.2.e.d
Level $294$
Weight $2$
Character orbit 294.e
Analytic conductor $2.348$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.34760181943\)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{4} + 4 \zeta_{6} q^{5} - q^{6} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{4} + 4 \zeta_{6} q^{5} - q^{6} - q^{8} - \zeta_{6} q^{9} + (4 \zeta_{6} - 4) q^{10} + ( - 4 \zeta_{6} + 4) q^{11} - \zeta_{6} q^{12} - 4 q^{13} - 4 q^{15} - \zeta_{6} q^{16} + ( - \zeta_{6} + 1) q^{18} + 4 \zeta_{6} q^{19} - 4 q^{20} + 4 q^{22} + ( - \zeta_{6} + 1) q^{24} + (11 \zeta_{6} - 11) q^{25} - 4 \zeta_{6} q^{26} + q^{27} + 2 q^{29} - 4 \zeta_{6} q^{30} + ( - 8 \zeta_{6} + 8) q^{31} + ( - \zeta_{6} + 1) q^{32} + 4 \zeta_{6} q^{33} + q^{36} + 6 \zeta_{6} q^{37} + (4 \zeta_{6} - 4) q^{38} + ( - 4 \zeta_{6} + 4) q^{39} - 4 \zeta_{6} q^{40} + 4 q^{43} + 4 \zeta_{6} q^{44} + ( - 4 \zeta_{6} + 4) q^{45} - 8 \zeta_{6} q^{47} + q^{48} - 11 q^{50} + ( - 4 \zeta_{6} + 4) q^{52} + ( - 10 \zeta_{6} + 10) q^{53} + \zeta_{6} q^{54} + 16 q^{55} - 4 q^{57} + 2 \zeta_{6} q^{58} + ( - 4 \zeta_{6} + 4) q^{59} + ( - 4 \zeta_{6} + 4) q^{60} - 4 \zeta_{6} q^{61} + 8 q^{62} + q^{64} - 16 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{66} + (4 \zeta_{6} - 4) q^{67} + 8 q^{71} + \zeta_{6} q^{72} + (16 \zeta_{6} - 16) q^{73} + (6 \zeta_{6} - 6) q^{74} - 11 \zeta_{6} q^{75} - 4 q^{76} + 4 q^{78} + 8 \zeta_{6} q^{79} + ( - 4 \zeta_{6} + 4) q^{80} + (\zeta_{6} - 1) q^{81} + 12 q^{83} + 4 \zeta_{6} q^{86} + (2 \zeta_{6} - 2) q^{87} + (4 \zeta_{6} - 4) q^{88} + 8 \zeta_{6} q^{89} + 4 q^{90} + 8 \zeta_{6} q^{93} + ( - 8 \zeta_{6} + 8) q^{94} + (16 \zeta_{6} - 16) q^{95} + \zeta_{6} q^{96} - 8 q^{97} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{8} - q^{9} - 4 q^{10} + 4 q^{11} - q^{12} - 8 q^{13} - 8 q^{15} - q^{16} + q^{18} + 4 q^{19} - 8 q^{20} + 8 q^{22} + q^{24} - 11 q^{25} - 4 q^{26} + 2 q^{27} + 4 q^{29} - 4 q^{30} + 8 q^{31} + q^{32} + 4 q^{33} + 2 q^{36} + 6 q^{37} - 4 q^{38} + 4 q^{39} - 4 q^{40} + 8 q^{43} + 4 q^{44} + 4 q^{45} - 8 q^{47} + 2 q^{48} - 22 q^{50} + 4 q^{52} + 10 q^{53} + q^{54} + 32 q^{55} - 8 q^{57} + 2 q^{58} + 4 q^{59} + 4 q^{60} - 4 q^{61} + 16 q^{62} + 2 q^{64} - 16 q^{65} - 4 q^{66} - 4 q^{67} + 16 q^{71} + q^{72} - 16 q^{73} - 6 q^{74} - 11 q^{75} - 8 q^{76} + 8 q^{78} + 8 q^{79} + 4 q^{80} - q^{81} + 24 q^{83} + 4 q^{86} - 2 q^{87} - 4 q^{88} + 8 q^{89} + 8 q^{90} + 8 q^{93} + 8 q^{94} - 16 q^{95} + q^{96} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 2.00000 + 3.46410i −1.00000 0 −1.00000 −0.500000 0.866025i −2.00000 + 3.46410i
79.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 2.00000 3.46410i −1.00000 0 −1.00000 −0.500000 + 0.866025i −2.00000 3.46410i
\(n\): 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 294.2.e.d 2
3.b odd 2 1 882.2.g.a 2
4.b odd 2 1 2352.2.q.y 2
7.b odd 2 1 294.2.e.e 2
7.c even 3 1 294.2.a.c yes 1
7.c even 3 1 inner 294.2.e.d 2
7.d odd 6 1 294.2.a.b 1
7.d odd 6 1 294.2.e.e 2
21.c even 2 1 882.2.g.f 2
21.g even 6 1 882.2.a.f 1
21.g even 6 1 882.2.g.f 2
21.h odd 6 1 882.2.a.l 1
21.h odd 6 1 882.2.g.a 2
28.d even 2 1 2352.2.q.a 2
28.f even 6 1 2352.2.a.y 1
28.f even 6 1 2352.2.q.a 2
28.g odd 6 1 2352.2.a.b 1
28.g odd 6 1 2352.2.q.y 2
35.i odd 6 1 7350.2.a.cj 1
35.j even 6 1 7350.2.a.br 1
56.j odd 6 1 9408.2.a.br 1
56.k odd 6 1 9408.2.a.de 1
56.m even 6 1 9408.2.a.b 1
56.p even 6 1 9408.2.a.bo 1
84.j odd 6 1 7056.2.a.a 1
84.n even 6 1 7056.2.a.ca 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.a.b 1 7.d odd 6 1
294.2.a.c yes 1 7.c even 3 1
294.2.e.d 2 1.a even 1 1 trivial
294.2.e.d 2 7.c even 3 1 inner
294.2.e.e 2 7.b odd 2 1
294.2.e.e 2 7.d odd 6 1
882.2.a.f 1 21.g even 6 1
882.2.a.l 1 21.h odd 6 1
882.2.g.a 2 3.b odd 2 1
882.2.g.a 2 21.h odd 6 1
882.2.g.f 2 21.c even 2 1
882.2.g.f 2 21.g even 6 1
2352.2.a.b 1 28.g odd 6 1
2352.2.a.y 1 28.f even 6 1
2352.2.q.a 2 28.d even 2 1
2352.2.q.a 2 28.f even 6 1
2352.2.q.y 2 4.b odd 2 1
2352.2.q.y 2 28.g odd 6 1
7056.2.a.a 1 84.j odd 6 1
7056.2.a.ca 1 84.n even 6 1
7350.2.a.br 1 35.j even 6 1
7350.2.a.cj 1 35.i odd 6 1
9408.2.a.b 1 56.m even 6 1
9408.2.a.bo 1 56.p even 6 1
9408.2.a.br 1 56.j odd 6 1
9408.2.a.de 1 56.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 4T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\). 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
$5$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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