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: \(x^{8} - x^{4} + 1\)
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 primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{2}\)

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 \( T_{5}^{4} + 6 T_{5}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( 1 - 9 T^{4} + 81 T^{8} \)
$5$ \( ( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} )^{2} \)
$7$ 1
$11$ \( ( 1 + 11 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 - 20 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 32 T^{2} + 663 T^{4} + 11552 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 22 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 + 31 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 58 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 4 T + 43 T^{2} )^{8} \)
$47$ \( ( 1 - 70 T^{2} + 2691 T^{4} - 154630 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 32 T^{2} - 2457 T^{4} + 111392 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 28 T^{2} - 2937 T^{4} - 104188 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 71 T^{2} )^{8} \)
$73$ \( ( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} )^{2}( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 160 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 - 89 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 - 170 T^{2} + 9409 T^{4} )^{4} \)
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