Properties

Label 294.4.e.n
Level $294$
Weight $4$
Character orbit 294.e
Analytic conductor $17.347$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.3465615417\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + 2 \beta_{2} ) q^{2} + 3 \beta_{2} q^{3} + 4 \beta_{2} q^{4} + ( 6 + \beta_{1} + 6 \beta_{2} ) q^{5} -6 q^{6} -8 q^{8} + ( -9 - 9 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 2 + 2 \beta_{2} ) q^{2} + 3 \beta_{2} q^{3} + 4 \beta_{2} q^{4} + ( 6 + \beta_{1} + 6 \beta_{2} ) q^{5} -6 q^{6} -8 q^{8} + ( -9 - 9 \beta_{2} ) q^{9} + ( 2 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{10} + ( 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{11} + ( -12 - 12 \beta_{2} ) q^{12} + ( -24 - 3 \beta_{3} ) q^{13} + ( -18 + 3 \beta_{3} ) q^{15} + ( -16 - 16 \beta_{2} ) q^{16} + ( -\beta_{1} - 42 \beta_{2} - \beta_{3} ) q^{17} -18 \beta_{2} q^{18} + ( -36 + 2 \beta_{1} - 36 \beta_{2} ) q^{19} + ( -24 + 4 \beta_{3} ) q^{20} + ( 4 + 12 \beta_{3} ) q^{22} + ( 154 + 6 \beta_{1} + 154 \beta_{2} ) q^{23} -24 \beta_{2} q^{24} + ( 12 \beta_{1} + 9 \beta_{2} + 12 \beta_{3} ) q^{25} + ( -48 + 6 \beta_{1} - 48 \beta_{2} ) q^{26} + 27 q^{27} + ( -40 + 18 \beta_{3} ) q^{29} + ( -36 - 6 \beta_{1} - 36 \beta_{2} ) q^{30} + ( -6 \beta_{1} - 192 \beta_{2} - 6 \beta_{3} ) q^{31} -32 \beta_{2} q^{32} + ( 6 - 18 \beta_{1} + 6 \beta_{2} ) q^{33} + ( 84 - 2 \beta_{3} ) q^{34} + 36 q^{36} + ( -268 - 12 \beta_{1} - 268 \beta_{2} ) q^{37} + ( 4 \beta_{1} - 72 \beta_{2} + 4 \beta_{3} ) q^{38} + ( 9 \beta_{1} - 72 \beta_{2} + 9 \beta_{3} ) q^{39} + ( -48 - 8 \beta_{1} - 48 \beta_{2} ) q^{40} + ( -378 - 5 \beta_{3} ) q^{41} + ( 200 + 24 \beta_{3} ) q^{43} + ( 8 - 24 \beta_{1} + 8 \beta_{2} ) q^{44} + ( -9 \beta_{1} - 54 \beta_{2} - 9 \beta_{3} ) q^{45} + ( 12 \beta_{1} + 308 \beta_{2} + 12 \beta_{3} ) q^{46} + ( 156 + 10 \beta_{1} + 156 \beta_{2} ) q^{47} + 48 q^{48} + ( -18 + 24 \beta_{3} ) q^{50} + ( 126 + 3 \beta_{1} + 126 \beta_{2} ) q^{51} + ( 12 \beta_{1} - 96 \beta_{2} + 12 \beta_{3} ) q^{52} + ( 24 \beta_{1} - 26 \beta_{2} + 24 \beta_{3} ) q^{53} + ( 54 + 54 \beta_{2} ) q^{54} + ( -576 + 34 \beta_{3} ) q^{55} + ( 108 + 6 \beta_{3} ) q^{57} + ( -80 - 36 \beta_{1} - 80 \beta_{2} ) q^{58} + ( 2 \beta_{1} - 432 \beta_{2} + 2 \beta_{3} ) q^{59} + ( -12 \beta_{1} - 72 \beta_{2} - 12 \beta_{3} ) q^{60} + ( 708 + 13 \beta_{1} + 708 \beta_{2} ) q^{61} + ( 384 - 12 \beta_{3} ) q^{62} + 64 q^{64} + ( 150 - 6 \beta_{1} + 150 \beta_{2} ) q^{65} + ( -36 \beta_{1} + 12 \beta_{2} - 36 \beta_{3} ) q^{66} + ( -24 \beta_{1} + 72 \beta_{2} - 24 \beta_{3} ) q^{67} + ( 168 + 4 \beta_{1} + 168 \beta_{2} ) q^{68} + ( -462 + 18 \beta_{3} ) q^{69} + ( -762 - 30 \beta_{3} ) q^{71} + ( 72 + 72 \beta_{2} ) q^{72} + ( 83 \beta_{1} - 372 \beta_{2} + 83 \beta_{3} ) q^{73} + ( -24 \beta_{1} - 536 \beta_{2} - 24 \beta_{3} ) q^{74} + ( -27 - 36 \beta_{1} - 27 \beta_{2} ) q^{75} + ( 144 + 8 \beta_{3} ) q^{76} + ( 144 + 18 \beta_{3} ) q^{78} + ( -488 + 84 \beta_{1} - 488 \beta_{2} ) q^{79} + ( -16 \beta_{1} - 96 \beta_{2} - 16 \beta_{3} ) q^{80} + 81 \beta_{2} q^{81} + ( -756 + 10 \beta_{1} - 756 \beta_{2} ) q^{82} + ( 156 - 136 \beta_{3} ) q^{83} + ( 350 - 48 \beta_{3} ) q^{85} + ( 400 - 48 \beta_{1} + 400 \beta_{2} ) q^{86} + ( -54 \beta_{1} - 120 \beta_{2} - 54 \beta_{3} ) q^{87} + ( -48 \beta_{1} + 16 \beta_{2} - 48 \beta_{3} ) q^{88} + ( 54 + 29 \beta_{1} + 54 \beta_{2} ) q^{89} + ( 108 - 18 \beta_{3} ) q^{90} + ( -616 + 24 \beta_{3} ) q^{92} + ( 576 + 18 \beta_{1} + 576 \beta_{2} ) q^{93} + ( 20 \beta_{1} + 312 \beta_{2} + 20 \beta_{3} ) q^{94} + ( -24 \beta_{1} - 20 \beta_{2} - 24 \beta_{3} ) q^{95} + ( 96 + 96 \beta_{2} ) q^{96} + ( 372 + 125 \beta_{3} ) q^{97} + ( -18 - 54 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 6q^{3} - 8q^{4} + 12q^{5} - 24q^{6} - 32q^{8} - 18q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 6q^{3} - 8q^{4} + 12q^{5} - 24q^{6} - 32q^{8} - 18q^{9} - 24q^{10} + 4q^{11} - 24q^{12} - 96q^{13} - 72q^{15} - 32q^{16} + 84q^{17} + 36q^{18} - 72q^{19} - 96q^{20} + 16q^{22} + 308q^{23} + 48q^{24} - 18q^{25} - 96q^{26} + 108q^{27} - 160q^{29} - 72q^{30} + 384q^{31} + 64q^{32} + 12q^{33} + 336q^{34} + 144q^{36} - 536q^{37} + 144q^{38} + 144q^{39} - 96q^{40} - 1512q^{41} + 800q^{43} + 16q^{44} + 108q^{45} - 616q^{46} + 312q^{47} + 192q^{48} - 72q^{50} + 252q^{51} + 192q^{52} + 52q^{53} + 108q^{54} - 2304q^{55} + 432q^{57} - 160q^{58} + 864q^{59} + 144q^{60} + 1416q^{61} + 1536q^{62} + 256q^{64} + 300q^{65} - 24q^{66} - 144q^{67} + 336q^{68} - 1848q^{69} - 3048q^{71} + 144q^{72} + 744q^{73} + 1072q^{74} - 54q^{75} + 576q^{76} + 576q^{78} - 976q^{79} + 192q^{80} - 162q^{81} - 1512q^{82} + 624q^{83} + 1400q^{85} + 800q^{86} + 240q^{87} - 32q^{88} + 108q^{89} + 432q^{90} - 2464q^{92} + 1152q^{93} - 624q^{94} + 40q^{95} + 192q^{96} + 1488q^{97} - 72q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 7 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( 7 \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/7\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)\(/7\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −1.94975 3.37706i −6.00000 0 −8.00000 −4.50000 7.79423i 3.89949 6.75412i
67.2 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 7.94975 + 13.7694i −6.00000 0 −8.00000 −4.50000 7.79423i −15.8995 + 27.5387i
79.1 1.00000 1.73205i −1.50000 2.59808i −2.00000 3.46410i −1.94975 + 3.37706i −6.00000 0 −8.00000 −4.50000 + 7.79423i 3.89949 + 6.75412i
79.2 1.00000 1.73205i −1.50000 2.59808i −2.00000 3.46410i 7.94975 13.7694i −6.00000 0 −8.00000 −4.50000 + 7.79423i −15.8995 27.5387i
\(n\): 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 294.4.e.n 4
3.b odd 2 1 882.4.g.y 4
7.b odd 2 1 294.4.e.o 4
7.c even 3 1 294.4.a.k yes 2
7.c even 3 1 inner 294.4.e.n 4
7.d odd 6 1 294.4.a.j 2
7.d odd 6 1 294.4.e.o 4
21.c even 2 1 882.4.g.bd 4
21.g even 6 1 882.4.a.bc 2
21.g even 6 1 882.4.g.bd 4
21.h odd 6 1 882.4.a.bi 2
21.h odd 6 1 882.4.g.y 4
28.f even 6 1 2352.4.a.cd 2
28.g odd 6 1 2352.4.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.j 2 7.d odd 6 1
294.4.a.k yes 2 7.c even 3 1
294.4.e.n 4 1.a even 1 1 trivial
294.4.e.n 4 7.c even 3 1 inner
294.4.e.o 4 7.b odd 2 1
294.4.e.o 4 7.d odd 6 1
882.4.a.bc 2 21.g even 6 1
882.4.a.bi 2 21.h odd 6 1
882.4.g.y 4 3.b odd 2 1
882.4.g.y 4 21.h odd 6 1
882.4.g.bd 4 21.c even 2 1
882.4.g.bd 4 21.g even 6 1
2352.4.a.bn 2 28.g odd 6 1
2352.4.a.cd 2 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(294, [\chi])\):

\( T_{5}^{4} - 12 T_{5}^{3} + 206 T_{5}^{2} + 744 T_{5} + 3844 \)
\( T_{11}^{4} - 4 T_{11}^{3} + 3540 T_{11}^{2} + 14096 T_{11} + 12418576 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T + T^{2} )^{2} \)
$3$ \( ( 9 + 3 T + T^{2} )^{2} \)
$5$ \( 3844 + 744 T + 206 T^{2} - 12 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 12418576 + 14096 T + 3540 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( ( -306 + 48 T + T^{2} )^{2} \)
$17$ \( 2775556 - 139944 T + 5390 T^{2} - 84 T^{3} + T^{4} \)
$19$ \( 817216 + 65088 T + 4280 T^{2} + 72 T^{3} + T^{4} \)
$23$ \( 407555344 - 6217904 T + 74676 T^{2} - 308 T^{3} + T^{4} \)
$29$ \( ( -30152 + 80 T + T^{2} )^{2} \)
$31$ \( 1111288896 - 12801024 T + 114120 T^{2} - 384 T^{3} + T^{4} \)
$37$ \( 3330674944 + 30933632 T + 229584 T^{2} + 536 T^{3} + T^{4} \)
$41$ \( ( 140434 + 756 T + T^{2} )^{2} \)
$43$ \( ( -16448 - 400 T + T^{2} )^{2} \)
$47$ \( 211295296 - 4535232 T + 82808 T^{2} - 312 T^{3} + T^{4} \)
$53$ \( 3110515984 + 2900144 T + 58476 T^{2} - 52 T^{3} + T^{4} \)
$59$ \( 34682357824 - 160904448 T + 560264 T^{2} - 864 T^{3} + T^{4} \)
$61$ \( 234936028804 - 686338032 T + 1520354 T^{2} - 1416 T^{3} + T^{4} \)
$67$ \( 2627997696 - 7382016 T + 72000 T^{2} + 144 T^{3} + T^{4} \)
$71$ \( ( 492444 + 1524 T + T^{2} )^{2} \)
$73$ \( 288087680644 + 399333072 T + 1090274 T^{2} - 744 T^{3} + T^{4} \)
$79$ \( 205520782336 - 442463744 T + 1405920 T^{2} + 976 T^{3} + T^{4} \)
$83$ \( ( -1788272 - 312 T + T^{2} )^{2} \)
$89$ \( 6320568004 + 8586216 T + 91166 T^{2} - 108 T^{3} + T^{4} \)
$97$ \( ( -1392866 - 744 T + T^{2} )^{2} \)
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