Properties

Label 294.2.f.a
Level $294$
Weight $2$
Character orbit 294.f
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.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.34760181943\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i −0.866025 + 1.50000i 1.73205i 0 1.00000i −1.50000 + 2.59808i 1.50000 0.866025i
215.2 0.866025 + 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i 0.866025 1.50000i 1.73205i 0 1.00000i −1.50000 + 2.59808i 1.50000 0.866025i
227.1 −0.866025 + 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i −0.866025 1.50000i 1.73205i 0 1.00000i −1.50000 2.59808i 1.50000 + 0.866025i
227.2 0.866025 0.500000i 0.866025 1.50000i 0.500000 0.866025i 0.866025 + 1.50000i 1.73205i 0 1.00000i −1.50000 2.59808i 1.50000 + 0.866025i
\(n\): 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.d odd 6 1 inner
21.g even 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 3T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$43$ \( (T + 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$53$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$59$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$97$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
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