Properties

Label 294.2.e.c
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: \(\Q(\sqrt{-3}) \)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} - q^{6} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} - q^{6} + q^{8} -\zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{10} + ( 4 - 4 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} + 6 q^{13} + 2 q^{15} -\zeta_{6} q^{16} + ( -2 + 2 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} + 4 \zeta_{6} q^{19} -2 q^{20} -4 q^{22} -8 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} + ( 1 - \zeta_{6} ) q^{25} -6 \zeta_{6} q^{26} - q^{27} -2 q^{29} -2 \zeta_{6} q^{30} + ( -1 + \zeta_{6} ) q^{32} -4 \zeta_{6} q^{33} + 2 q^{34} + q^{36} + 10 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} + ( 6 - 6 \zeta_{6} ) q^{39} + 2 \zeta_{6} q^{40} -6 q^{41} -4 q^{43} + 4 \zeta_{6} q^{44} + ( 2 - 2 \zeta_{6} ) q^{45} + ( -8 + 8 \zeta_{6} ) q^{46} - q^{48} - q^{50} + 2 \zeta_{6} q^{51} + ( -6 + 6 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + \zeta_{6} q^{54} + 8 q^{55} + 4 q^{57} + 2 \zeta_{6} q^{58} + ( -4 + 4 \zeta_{6} ) q^{59} + ( -2 + 2 \zeta_{6} ) q^{60} -6 \zeta_{6} q^{61} + q^{64} + 12 \zeta_{6} q^{65} + ( -4 + 4 \zeta_{6} ) q^{66} + ( -4 + 4 \zeta_{6} ) q^{67} -2 \zeta_{6} q^{68} -8 q^{69} + 8 q^{71} -\zeta_{6} q^{72} + ( -10 + 10 \zeta_{6} ) q^{73} + ( 10 - 10 \zeta_{6} ) q^{74} -\zeta_{6} q^{75} -4 q^{76} -6 q^{78} + ( 2 - 2 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 6 \zeta_{6} q^{82} -4 q^{83} -4 q^{85} + 4 \zeta_{6} q^{86} + ( -2 + 2 \zeta_{6} ) q^{87} + ( 4 - 4 \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} -2 q^{90} + 8 q^{92} + ( -8 + 8 \zeta_{6} ) q^{95} + \zeta_{6} q^{96} -14 q^{97} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{3} - q^{4} + 2q^{5} - 2q^{6} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{3} - q^{4} + 2q^{5} - 2q^{6} + 2q^{8} - q^{9} + 2q^{10} + 4q^{11} + q^{12} + 12q^{13} + 4q^{15} - q^{16} - 2q^{17} - q^{18} + 4q^{19} - 4q^{20} - 8q^{22} - 8q^{23} + q^{24} + q^{25} - 6q^{26} - 2q^{27} - 4q^{29} - 2q^{30} - q^{32} - 4q^{33} + 4q^{34} + 2q^{36} + 10q^{37} + 4q^{38} + 6q^{39} + 2q^{40} - 12q^{41} - 8q^{43} + 4q^{44} + 2q^{45} - 8q^{46} - 2q^{48} - 2q^{50} + 2q^{51} - 6q^{52} - 6q^{53} + q^{54} + 16q^{55} + 8q^{57} + 2q^{58} - 4q^{59} - 2q^{60} - 6q^{61} + 2q^{64} + 12q^{65} - 4q^{66} - 4q^{67} - 2q^{68} - 16q^{69} + 16q^{71} - q^{72} - 10q^{73} + 10q^{74} - q^{75} - 8q^{76} - 12q^{78} + 2q^{80} - q^{81} + 6q^{82} - 8q^{83} - 8q^{85} + 4q^{86} - 2q^{87} + 4q^{88} + 6q^{89} - 4q^{90} + 16q^{92} - 8q^{95} + q^{96} - 28q^{97} - 8q^{99} + O(q^{100}) \)

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 1.00000 + 1.73205i −1.00000 0 1.00000 −0.500000 0.866025i 1.00000 1.73205i
79.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 1.73205i −1.00000 0 1.00000 −0.500000 + 0.866025i 1.00000 + 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
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.c 2
3.b odd 2 1 882.2.g.h 2
4.b odd 2 1 2352.2.q.i 2
7.b odd 2 1 294.2.e.a 2
7.c even 3 1 42.2.a.a 1
7.c even 3 1 inner 294.2.e.c 2
7.d odd 6 1 294.2.a.g 1
7.d odd 6 1 294.2.e.a 2
21.c even 2 1 882.2.g.j 2
21.g even 6 1 882.2.a.b 1
21.g even 6 1 882.2.g.j 2
21.h odd 6 1 126.2.a.a 1
21.h odd 6 1 882.2.g.h 2
28.d even 2 1 2352.2.q.n 2
28.f even 6 1 2352.2.a.l 1
28.f even 6 1 2352.2.q.n 2
28.g odd 6 1 336.2.a.d 1
28.g odd 6 1 2352.2.q.i 2
35.i odd 6 1 7350.2.a.f 1
35.j even 6 1 1050.2.a.i 1
35.l odd 12 2 1050.2.g.a 2
56.j odd 6 1 9408.2.a.n 1
56.k odd 6 1 1344.2.a.i 1
56.m even 6 1 9408.2.a.bw 1
56.p even 6 1 1344.2.a.q 1
63.g even 3 1 1134.2.f.g 2
63.h even 3 1 1134.2.f.g 2
63.j odd 6 1 1134.2.f.j 2
63.n odd 6 1 1134.2.f.j 2
77.h odd 6 1 5082.2.a.d 1
84.j odd 6 1 7056.2.a.k 1
84.n even 6 1 1008.2.a.j 1
91.r even 6 1 7098.2.a.f 1
105.o odd 6 1 3150.2.a.bo 1
105.x even 12 2 3150.2.g.r 2
112.u odd 12 2 5376.2.c.e 2
112.w even 12 2 5376.2.c.bc 2
140.p odd 6 1 8400.2.a.k 1
168.s odd 6 1 4032.2.a.e 1
168.v even 6 1 4032.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 7.c even 3 1
126.2.a.a 1 21.h odd 6 1
294.2.a.g 1 7.d odd 6 1
294.2.e.a 2 7.b odd 2 1
294.2.e.a 2 7.d odd 6 1
294.2.e.c 2 1.a even 1 1 trivial
294.2.e.c 2 7.c even 3 1 inner
336.2.a.d 1 28.g odd 6 1
882.2.a.b 1 21.g even 6 1
882.2.g.h 2 3.b odd 2 1
882.2.g.h 2 21.h odd 6 1
882.2.g.j 2 21.c even 2 1
882.2.g.j 2 21.g even 6 1
1008.2.a.j 1 84.n even 6 1
1050.2.a.i 1 35.j even 6 1
1050.2.g.a 2 35.l odd 12 2
1134.2.f.g 2 63.g even 3 1
1134.2.f.g 2 63.h even 3 1
1134.2.f.j 2 63.j odd 6 1
1134.2.f.j 2 63.n odd 6 1
1344.2.a.i 1 56.k odd 6 1
1344.2.a.q 1 56.p even 6 1
2352.2.a.l 1 28.f even 6 1
2352.2.q.i 2 4.b odd 2 1
2352.2.q.i 2 28.g odd 6 1
2352.2.q.n 2 28.d even 2 1
2352.2.q.n 2 28.f even 6 1
3150.2.a.bo 1 105.o odd 6 1
3150.2.g.r 2 105.x even 12 2
4032.2.a.e 1 168.s odd 6 1
4032.2.a.m 1 168.v even 6 1
5082.2.a.d 1 77.h odd 6 1
5376.2.c.e 2 112.u odd 12 2
5376.2.c.bc 2 112.w even 12 2
7056.2.a.k 1 84.j odd 6 1
7098.2.a.f 1 91.r even 6 1
7350.2.a.f 1 35.i odd 6 1
8400.2.a.k 1 140.p odd 6 1
9408.2.a.n 1 56.j odd 6 1
9408.2.a.bw 1 56.m even 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \)
$19$ \( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 2 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 31 T^{2} + 961 T^{4} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} ) \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 8 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
$79$ \( 1 - 79 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 + 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 14 T + 97 T^{2} )^{2} \)
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