Properties

Label 294.2.d.a
Level $294$
Weight $2$
Character orbit 294.d
Analytic conductor $2.348$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.34760181943\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
1.00000i −1.73205 −1.00000 −1.73205 1.73205i 0 1.00000i 3.00000 1.73205i
293.2 1.00000i 1.73205 −1.00000 1.73205 1.73205i 0 1.00000i 3.00000 1.73205i
293.3 1.00000i −1.73205 −1.00000 −1.73205 1.73205i 0 1.00000i 3.00000 1.73205i
293.4 1.00000i 1.73205 −1.00000 1.73205 1.73205i 0 1.00000i 3.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.2.d.a 4
3.b odd 2 1 inner 294.2.d.a 4
4.b odd 2 1 2352.2.k.e 4
7.b odd 2 1 inner 294.2.d.a 4
7.c even 3 1 42.2.f.a 4
7.c even 3 1 294.2.f.a 4
7.d odd 6 1 42.2.f.a 4
7.d odd 6 1 294.2.f.a 4
12.b even 2 1 2352.2.k.e 4
21.c even 2 1 inner 294.2.d.a 4
21.g even 6 1 42.2.f.a 4
21.g even 6 1 294.2.f.a 4
21.h odd 6 1 42.2.f.a 4
21.h odd 6 1 294.2.f.a 4
28.d even 2 1 2352.2.k.e 4
28.f even 6 1 336.2.bc.e 4
28.g odd 6 1 336.2.bc.e 4
35.i odd 6 1 1050.2.s.b 4
35.j even 6 1 1050.2.s.b 4
35.k even 12 1 1050.2.u.a 4
35.k even 12 1 1050.2.u.d 4
35.l odd 12 1 1050.2.u.a 4
35.l odd 12 1 1050.2.u.d 4
63.g even 3 1 1134.2.t.d 4
63.h even 3 1 1134.2.l.c 4
63.i even 6 1 1134.2.l.c 4
63.j odd 6 1 1134.2.l.c 4
63.k odd 6 1 1134.2.t.d 4
63.n odd 6 1 1134.2.t.d 4
63.s even 6 1 1134.2.t.d 4
63.t odd 6 1 1134.2.l.c 4
84.h odd 2 1 2352.2.k.e 4
84.j odd 6 1 336.2.bc.e 4
84.n even 6 1 336.2.bc.e 4
105.o odd 6 1 1050.2.s.b 4
105.p even 6 1 1050.2.s.b 4
105.w odd 12 1 1050.2.u.a 4
105.w odd 12 1 1050.2.u.d 4
105.x even 12 1 1050.2.u.a 4
105.x even 12 1 1050.2.u.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.f.a 4 7.c even 3 1
42.2.f.a 4 7.d odd 6 1
42.2.f.a 4 21.g even 6 1
42.2.f.a 4 21.h odd 6 1
294.2.d.a 4 1.a even 1 1 trivial
294.2.d.a 4 3.b odd 2 1 inner
294.2.d.a 4 7.b odd 2 1 inner
294.2.d.a 4 21.c even 2 1 inner
294.2.f.a 4 7.c even 3 1
294.2.f.a 4 7.d odd 6 1
294.2.f.a 4 21.g even 6 1
294.2.f.a 4 21.h odd 6 1
336.2.bc.e 4 28.f even 6 1
336.2.bc.e 4 28.g odd 6 1
336.2.bc.e 4 84.j odd 6 1
336.2.bc.e 4 84.n even 6 1
1050.2.s.b 4 35.i odd 6 1
1050.2.s.b 4 35.j even 6 1
1050.2.s.b 4 105.o odd 6 1
1050.2.s.b 4 105.p even 6 1
1050.2.u.a 4 35.k even 12 1
1050.2.u.a 4 35.l odd 12 1
1050.2.u.a 4 105.w odd 12 1
1050.2.u.a 4 105.x even 12 1
1050.2.u.d 4 35.k even 12 1
1050.2.u.d 4 35.l odd 12 1
1050.2.u.d 4 105.w odd 12 1
1050.2.u.d 4 105.x even 12 1
1134.2.l.c 4 63.h even 3 1
1134.2.l.c 4 63.i even 6 1
1134.2.l.c 4 63.j odd 6 1
1134.2.l.c 4 63.t odd 6 1
1134.2.t.d 4 63.g even 3 1
1134.2.t.d 4 63.k odd 6 1
1134.2.t.d 4 63.n odd 6 1
1134.2.t.d 4 63.s even 6 1
2352.2.k.e 4 4.b odd 2 1
2352.2.k.e 4 12.b even 2 1
2352.2.k.e 4 28.d even 2 1
2352.2.k.e 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$43$ \( (T + 8)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T - 2)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
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