Properties

Label 294.2.f.b
Level $294$
Weight $2$
Character orbit 294.f
Analytic conductor $2.348$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{2} - \beta_{6} q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{7} - \beta_{6} + \beta_{3}) q^{5} + (\beta_{7} - \beta_{5}) q^{6} + \beta_{4} q^{8} + ( - 3 \beta_{4} + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1) q^{2} - \beta_{6} q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{7} - \beta_{6} + \beta_{3}) q^{5} + (\beta_{7} - \beta_{5}) q^{6} + \beta_{4} q^{8} + ( - 3 \beta_{4} + 3 \beta_1) q^{9} + ( - \beta_{6} - \beta_{5}) q^{10} + ( - \beta_{6} + \beta_{3}) q^{12} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{13} + ( - 3 \beta_{4} - 3) q^{15} - \beta_{2} q^{16} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{17} + (3 \beta_{2} - 3) q^{18} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{19} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{20} + (6 \beta_{4} - 6 \beta_1) q^{23} - \beta_{5} q^{24} + (\beta_{2} - 1) q^{25} + ( - \beta_{7} + \beta_{6} - \beta_{3}) q^{26} + ( - 3 \beta_{7} + 3 \beta_{5}) q^{27} + 6 \beta_{4} q^{29} + ( - 3 \beta_{4} + 3 \beta_{2} + 3 \beta_1) q^{30} + \beta_1 q^{32} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3}) q^{34} - 3 \beta_{4} q^{36} + 2 \beta_{2} q^{37} + (\beta_{6} - \beta_{5}) q^{38} + ( - 3 \beta_{2} - 3 \beta_1 + 3) q^{39} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{40} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3}) q^{41} + 4 q^{43} + (3 \beta_{6} + 3 \beta_{5}) q^{45} + ( - 6 \beta_{2} + 6) q^{46} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{3}) q^{47} + \beta_{3} q^{48} - \beta_{4} q^{50} + ( - 6 \beta_{4} - 6 \beta_{2} + 6 \beta_1) q^{51} + (\beta_{6} + \beta_{5}) q^{52} + 6 \beta_1 q^{53} + (3 \beta_{6} - 3 \beta_{3}) q^{54} + ( - 3 \beta_{4} + 3) q^{57} - 6 \beta_{2} q^{58} + (5 \beta_{6} - 5 \beta_{5}) q^{59} + (3 \beta_{2} - 3 \beta_1 - 3) q^{60} + (5 \beta_{7} + 5 \beta_{6} - 5 \beta_{3}) q^{61} - q^{64} + (6 \beta_{4} - 6 \beta_1) q^{65} + (8 \beta_{2} - 8) q^{67} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3}) q^{68} + (6 \beta_{7} - 6 \beta_{5}) q^{69} + 3 \beta_{2} q^{72} + (4 \beta_{6} + 4 \beta_{5}) q^{73} - 2 \beta_1 q^{74} + (\beta_{6} - \beta_{3}) q^{75} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{76} + (3 \beta_{4} + 3) q^{78} + 10 \beta_{2} q^{79} + (\beta_{6} - \beta_{5}) q^{80} + ( - 9 \beta_{2} + 9) q^{81} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{3}) q^{82} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{83} - 12 q^{85} + (4 \beta_{4} - 4 \beta_1) q^{86} - 6 \beta_{5} q^{87} + ( - 3 \beta_{7} + 3 \beta_{5} - 3 \beta_{3}) q^{90} + 6 \beta_{4} q^{92} + (2 \beta_{6} + 2 \beta_{5}) q^{94} - 6 \beta_1 q^{95} - \beta_{7} q^{96} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 24 q^{15} - 4 q^{16} - 12 q^{18} - 4 q^{25} + 12 q^{30} + 8 q^{37} + 12 q^{39} + 32 q^{43} + 24 q^{46} - 24 q^{51} + 24 q^{57} - 24 q^{58} - 12 q^{60} - 8 q^{64} - 32 q^{67} + 12 q^{72} + 24 q^{78} + 40 q^{79} + 36 q^{81} - 96 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{5} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{6} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.866025 0.500000i −1.67303 0.448288i 0.500000 + 0.866025i 1.22474 2.12132i 1.22474 + 1.22474i 0 1.00000i 2.59808 + 1.50000i −2.12132 + 1.22474i
215.2 −0.866025 0.500000i 1.67303 + 0.448288i 0.500000 + 0.866025i −1.22474 + 2.12132i −1.22474 1.22474i 0 1.00000i 2.59808 + 1.50000i 2.12132 1.22474i
215.3 0.866025 + 0.500000i −0.448288 + 1.67303i 0.500000 + 0.866025i −1.22474 + 2.12132i −1.22474 + 1.22474i 0 1.00000i −2.59808 1.50000i −2.12132 + 1.22474i
215.4 0.866025 + 0.500000i 0.448288 1.67303i 0.500000 + 0.866025i 1.22474 2.12132i 1.22474 1.22474i 0 1.00000i −2.59808 1.50000i 2.12132 1.22474i
227.1 −0.866025 + 0.500000i −1.67303 + 0.448288i 0.500000 0.866025i 1.22474 + 2.12132i 1.22474 1.22474i 0 1.00000i 2.59808 1.50000i −2.12132 1.22474i
227.2 −0.866025 + 0.500000i 1.67303 0.448288i 0.500000 0.866025i −1.22474 2.12132i −1.22474 + 1.22474i 0 1.00000i 2.59808 1.50000i 2.12132 + 1.22474i
227.3 0.866025 0.500000i −0.448288 1.67303i 0.500000 0.866025i −1.22474 2.12132i −1.22474 1.22474i 0 1.00000i −2.59808 + 1.50000i −2.12132 1.22474i
227.4 0.866025 0.500000i 0.448288 + 1.67303i 0.500000 0.866025i 1.22474 + 2.12132i 1.22474 + 1.22474i 0 1.00000i −2.59808 + 1.50000i 2.12132 + 1.22474i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 227.4
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
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 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.b 8
3.b odd 2 1 inner 294.2.f.b 8
7.b odd 2 1 inner 294.2.f.b 8
7.c even 3 1 42.2.d.a 4
7.c even 3 1 inner 294.2.f.b 8
7.d odd 6 1 42.2.d.a 4
7.d odd 6 1 inner 294.2.f.b 8
21.c even 2 1 inner 294.2.f.b 8
21.g even 6 1 42.2.d.a 4
21.g even 6 1 inner 294.2.f.b 8
21.h odd 6 1 42.2.d.a 4
21.h odd 6 1 inner 294.2.f.b 8
28.f even 6 1 336.2.k.b 4
28.g odd 6 1 336.2.k.b 4
35.i odd 6 1 1050.2.b.b 4
35.j even 6 1 1050.2.b.b 4
35.k even 12 1 1050.2.d.b 4
35.k even 12 1 1050.2.d.e 4
35.l odd 12 1 1050.2.d.b 4
35.l odd 12 1 1050.2.d.e 4
56.j odd 6 1 1344.2.k.c 4
56.k odd 6 1 1344.2.k.d 4
56.m even 6 1 1344.2.k.d 4
56.p even 6 1 1344.2.k.c 4
63.g even 3 1 1134.2.m.g 8
63.h even 3 1 1134.2.m.g 8
63.i even 6 1 1134.2.m.g 8
63.j odd 6 1 1134.2.m.g 8
63.k odd 6 1 1134.2.m.g 8
63.n odd 6 1 1134.2.m.g 8
63.s even 6 1 1134.2.m.g 8
63.t odd 6 1 1134.2.m.g 8
84.j odd 6 1 336.2.k.b 4
84.n even 6 1 336.2.k.b 4
105.o odd 6 1 1050.2.b.b 4
105.p even 6 1 1050.2.b.b 4
105.w odd 12 1 1050.2.d.b 4
105.w odd 12 1 1050.2.d.e 4
105.x even 12 1 1050.2.d.b 4
105.x even 12 1 1050.2.d.e 4
168.s odd 6 1 1344.2.k.c 4
168.v even 6 1 1344.2.k.d 4
168.ba even 6 1 1344.2.k.c 4
168.be odd 6 1 1344.2.k.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.d.a 4 7.c even 3 1
42.2.d.a 4 7.d odd 6 1
42.2.d.a 4 21.g even 6 1
42.2.d.a 4 21.h odd 6 1
294.2.f.b 8 1.a even 1 1 trivial
294.2.f.b 8 3.b odd 2 1 inner
294.2.f.b 8 7.b odd 2 1 inner
294.2.f.b 8 7.c even 3 1 inner
294.2.f.b 8 7.d odd 6 1 inner
294.2.f.b 8 21.c even 2 1 inner
294.2.f.b 8 21.g even 6 1 inner
294.2.f.b 8 21.h odd 6 1 inner
336.2.k.b 4 28.f even 6 1
336.2.k.b 4 28.g odd 6 1
336.2.k.b 4 84.j odd 6 1
336.2.k.b 4 84.n even 6 1
1050.2.b.b 4 35.i odd 6 1
1050.2.b.b 4 35.j even 6 1
1050.2.b.b 4 105.o odd 6 1
1050.2.b.b 4 105.p even 6 1
1050.2.d.b 4 35.k even 12 1
1050.2.d.b 4 35.l odd 12 1
1050.2.d.b 4 105.w odd 12 1
1050.2.d.b 4 105.x even 12 1
1050.2.d.e 4 35.k even 12 1
1050.2.d.e 4 35.l odd 12 1
1050.2.d.e 4 105.w odd 12 1
1050.2.d.e 4 105.x even 12 1
1134.2.m.g 8 63.g even 3 1
1134.2.m.g 8 63.h even 3 1
1134.2.m.g 8 63.i even 6 1
1134.2.m.g 8 63.j odd 6 1
1134.2.m.g 8 63.k odd 6 1
1134.2.m.g 8 63.n odd 6 1
1134.2.m.g 8 63.s even 6 1
1134.2.m.g 8 63.t odd 6 1
1344.2.k.c 4 56.j odd 6 1
1344.2.k.c 4 56.p even 6 1
1344.2.k.c 4 168.s odd 6 1
1344.2.k.c 4 168.ba even 6 1
1344.2.k.d 4 56.k odd 6 1
1344.2.k.d 4 56.m even 6 1
1344.2.k.d 4 168.v even 6 1
1344.2.k.d 4 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 6T_{5}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$43$ \( (T - 4)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 150 T^{2} + 22500)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 150 T^{2} + 22500)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} - 96 T^{2} + 9216)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 10 T + 100)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
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