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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Character values

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

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

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 \( T_{5}^{2} - 2T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$13$ \( (T - 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 14)^{2} \) Copy content Toggle raw display
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