Properties

Label 294.2.e.f
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: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Character values

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

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

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 1.50000 + 2.59808i 1.00000 0 −1.00000 −0.500000 0.866025i −1.50000 + 2.59808i
79.1 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.50000 2.59808i 1.00000 0 −1.00000 −0.500000 + 0.866025i −1.50000 2.59808i
\(n\): 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.f 2
3.b odd 2 1 882.2.g.b 2
4.b odd 2 1 2352.2.q.m 2
7.b odd 2 1 42.2.e.b 2
7.c even 3 1 294.2.a.a 1
7.c even 3 1 inner 294.2.e.f 2
7.d odd 6 1 42.2.e.b 2
7.d odd 6 1 294.2.a.d 1
21.c even 2 1 126.2.g.b 2
21.g even 6 1 126.2.g.b 2
21.g even 6 1 882.2.a.g 1
21.h odd 6 1 882.2.a.k 1
21.h odd 6 1 882.2.g.b 2
28.d even 2 1 336.2.q.d 2
28.f even 6 1 336.2.q.d 2
28.f even 6 1 2352.2.a.m 1
28.g odd 6 1 2352.2.a.n 1
28.g odd 6 1 2352.2.q.m 2
35.c odd 2 1 1050.2.i.e 2
35.f even 4 2 1050.2.o.b 4
35.i odd 6 1 1050.2.i.e 2
35.i odd 6 1 7350.2.a.ce 1
35.j even 6 1 7350.2.a.cw 1
35.k even 12 2 1050.2.o.b 4
56.e even 2 1 1344.2.q.j 2
56.h odd 2 1 1344.2.q.v 2
56.j odd 6 1 1344.2.q.v 2
56.j odd 6 1 9408.2.a.d 1
56.k odd 6 1 9408.2.a.bm 1
56.m even 6 1 1344.2.q.j 2
56.m even 6 1 9408.2.a.bu 1
56.p even 6 1 9408.2.a.db 1
63.i even 6 1 1134.2.h.a 2
63.k odd 6 1 1134.2.e.a 2
63.l odd 6 1 1134.2.e.a 2
63.l odd 6 1 1134.2.h.p 2
63.o even 6 1 1134.2.e.p 2
63.o even 6 1 1134.2.h.a 2
63.s even 6 1 1134.2.e.p 2
63.t odd 6 1 1134.2.h.p 2
84.h odd 2 1 1008.2.s.n 2
84.j odd 6 1 1008.2.s.n 2
84.j odd 6 1 7056.2.a.g 1
84.n even 6 1 7056.2.a.bz 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.b 2 7.b odd 2 1
42.2.e.b 2 7.d odd 6 1
126.2.g.b 2 21.c even 2 1
126.2.g.b 2 21.g even 6 1
294.2.a.a 1 7.c even 3 1
294.2.a.d 1 7.d odd 6 1
294.2.e.f 2 1.a even 1 1 trivial
294.2.e.f 2 7.c even 3 1 inner
336.2.q.d 2 28.d even 2 1
336.2.q.d 2 28.f even 6 1
882.2.a.g 1 21.g even 6 1
882.2.a.k 1 21.h odd 6 1
882.2.g.b 2 3.b odd 2 1
882.2.g.b 2 21.h odd 6 1
1008.2.s.n 2 84.h odd 2 1
1008.2.s.n 2 84.j odd 6 1
1050.2.i.e 2 35.c odd 2 1
1050.2.i.e 2 35.i odd 6 1
1050.2.o.b 4 35.f even 4 2
1050.2.o.b 4 35.k even 12 2
1134.2.e.a 2 63.k odd 6 1
1134.2.e.a 2 63.l odd 6 1
1134.2.e.p 2 63.o even 6 1
1134.2.e.p 2 63.s even 6 1
1134.2.h.a 2 63.i even 6 1
1134.2.h.a 2 63.o even 6 1
1134.2.h.p 2 63.l odd 6 1
1134.2.h.p 2 63.t odd 6 1
1344.2.q.j 2 56.e even 2 1
1344.2.q.j 2 56.m even 6 1
1344.2.q.v 2 56.h odd 2 1
1344.2.q.v 2 56.j odd 6 1
2352.2.a.m 1 28.f even 6 1
2352.2.a.n 1 28.g odd 6 1
2352.2.q.m 2 4.b odd 2 1
2352.2.q.m 2 28.g odd 6 1
7056.2.a.g 1 84.j odd 6 1
7056.2.a.bz 1 84.n even 6 1
7350.2.a.ce 1 35.i odd 6 1
7350.2.a.cw 1 35.j even 6 1
9408.2.a.d 1 56.j odd 6 1
9408.2.a.bm 1 56.k odd 6 1
9408.2.a.bu 1 56.m even 6 1
9408.2.a.db 1 56.p even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 9 T + 29 T^{2} )^{2} \)
$31$ \( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \)
$37$ \( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 10 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \)
$79$ \( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 9 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - T + 97 T^{2} )^{2} \)
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