Properties

Label 294.3.h.b
Level $294$
Weight $3$
Character orbit 294.h
Analytic conductor $8.011$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 294.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.01091977219\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{3} + 2 \beta_{2} q^{4} + 6 \beta_1 q^{5} + ( - \beta_{3} - 4) q^{6} + 2 \beta_{3} q^{8} + ( - 7 \beta_{2} + 4 \beta_1 + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{3} + 2 \beta_{2} q^{4} + 6 \beta_1 q^{5} + ( - \beta_{3} - 4) q^{6} + 2 \beta_{3} q^{8} + ( - 7 \beta_{2} + 4 \beta_1 + 7) q^{9} + 12 \beta_{2} q^{10} + ( - 2 \beta_{2} - 4 \beta_1 + 2) q^{12} + q^{13} + ( - 6 \beta_{3} - 24) q^{15} + (4 \beta_{2} - 4) q^{16} + ( - 6 \beta_{3} + 6 \beta_1) q^{17} + ( - 7 \beta_{3} + 8 \beta_{2} + 7 \beta_1) q^{18} + (31 \beta_{2} - 31) q^{19} + 12 \beta_{3} q^{20} + 6 \beta_1 q^{23} + ( - 2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{24} + 47 \beta_{2} q^{25} + \beta_1 q^{26} + (10 \beta_{3} - 23) q^{27} + 12 \beta_{3} q^{29} + ( - 12 \beta_{2} - 24 \beta_1 + 12) q^{30} - 7 \beta_{2} q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + 12 q^{34} + (8 \beta_{3} + 14) q^{36} + ( - \beta_{2} + 1) q^{37} + (31 \beta_{3} - 31 \beta_1) q^{38} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{39} + (24 \beta_{2} - 24) q^{40} - 24 \beta_{3} q^{41} - 31 q^{43} + ( - 42 \beta_{3} + 48 \beta_{2} + 42 \beta_1) q^{45} + 12 \beta_{2} q^{46} + 30 \beta_1 q^{47} + ( - 8 \beta_{3} + 4) q^{48} + 47 \beta_{3} q^{50} + (24 \beta_{2} - 6 \beta_1 - 24) q^{51} + 2 \beta_{2} q^{52} + ( - 18 \beta_{3} + 18 \beta_1) q^{53} + (20 \beta_{2} - 23 \beta_1 - 20) q^{54} + ( - 62 \beta_{3} + 31) q^{57} + (24 \beta_{2} - 24) q^{58} + ( - 6 \beta_{3} + 6 \beta_1) q^{59} + ( - 12 \beta_{3} - 48 \beta_{2} + 12 \beta_1) q^{60} + ( - 50 \beta_{2} + 50) q^{61} - 7 \beta_{3} q^{62} - 8 q^{64} + 6 \beta_1 q^{65} - 65 \beta_{2} q^{67} + 12 \beta_1 q^{68} + ( - 6 \beta_{3} - 24) q^{69} - 42 \beta_{3} q^{71} + (16 \beta_{2} + 14 \beta_1 - 16) q^{72} - 97 \beta_{2} q^{73} + ( - \beta_{3} + \beta_1) q^{74} + ( - 47 \beta_{2} - 94 \beta_1 + 47) q^{75} - 62 q^{76} + ( - \beta_{3} - 4) q^{78} + ( - 103 \beta_{2} + 103) q^{79} + (24 \beta_{3} - 24 \beta_1) q^{80} + ( - 56 \beta_{3} - 17 \beta_{2} + 56 \beta_1) q^{81} + ( - 48 \beta_{2} + 48) q^{82} - 30 \beta_{3} q^{83} + 72 q^{85} - 31 \beta_1 q^{86} + ( - 12 \beta_{3} - 48 \beta_{2} + 12 \beta_1) q^{87} + 84 \beta_1 q^{89} + (48 \beta_{3} + 84) q^{90} + 12 \beta_{3} q^{92} + (7 \beta_{2} + 14 \beta_1 - 7) q^{93} + 60 \beta_{2} q^{94} + (186 \beta_{3} - 186 \beta_1) q^{95} + ( - 16 \beta_{2} + 4 \beta_1 + 16) q^{96} + 166 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{4} - 16 q^{6} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 4 q^{4} - 16 q^{6} + 14 q^{9} + 24 q^{10} + 4 q^{12} + 4 q^{13} - 96 q^{15} - 8 q^{16} + 16 q^{18} - 62 q^{19} - 16 q^{24} + 94 q^{25} - 92 q^{27} + 24 q^{30} - 14 q^{31} + 48 q^{34} + 56 q^{36} + 2 q^{37} - 2 q^{39} - 48 q^{40} - 124 q^{43} + 96 q^{45} + 24 q^{46} + 16 q^{48} - 48 q^{51} + 4 q^{52} - 40 q^{54} + 124 q^{57} - 48 q^{58} - 96 q^{60} + 100 q^{61} - 32 q^{64} - 130 q^{67} - 96 q^{69} - 32 q^{72} - 194 q^{73} + 94 q^{75} - 248 q^{76} - 16 q^{78} + 206 q^{79} - 34 q^{81} + 96 q^{82} + 288 q^{85} - 96 q^{87} + 336 q^{90} - 14 q^{93} + 120 q^{94} + 32 q^{96} + 664 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 1.94949 + 2.28024i 1.00000 1.73205i −7.34847 + 4.24264i −4.00000 1.41421i 0 2.82843i −1.39898 + 8.89060i 6.00000 10.3923i
263.2 1.22474 0.707107i −2.94949 0.548188i 1.00000 1.73205i 7.34847 4.24264i −4.00000 + 1.41421i 0 2.82843i 8.39898 + 3.23375i 6.00000 10.3923i
275.1 −1.22474 0.707107i 1.94949 2.28024i 1.00000 + 1.73205i −7.34847 4.24264i −4.00000 + 1.41421i 0 2.82843i −1.39898 8.89060i 6.00000 + 10.3923i
275.2 1.22474 + 0.707107i −2.94949 + 0.548188i 1.00000 + 1.73205i 7.34847 + 4.24264i −4.00000 1.41421i 0 2.82843i 8.39898 3.23375i 6.00000 + 10.3923i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 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 294.3.h.b 4
3.b odd 2 1 inner 294.3.h.b 4
7.b odd 2 1 42.3.h.a 4
7.c even 3 1 294.3.b.c 2
7.c even 3 1 inner 294.3.h.b 4
7.d odd 6 1 42.3.h.a 4
7.d odd 6 1 294.3.b.b 2
21.c even 2 1 42.3.h.a 4
21.g even 6 1 42.3.h.a 4
21.g even 6 1 294.3.b.b 2
21.h odd 6 1 294.3.b.c 2
21.h odd 6 1 inner 294.3.h.b 4
28.d even 2 1 336.3.bn.c 4
28.f even 6 1 336.3.bn.c 4
84.h odd 2 1 336.3.bn.c 4
84.j odd 6 1 336.3.bn.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.h.a 4 7.b odd 2 1
42.3.h.a 4 7.d odd 6 1
42.3.h.a 4 21.c even 2 1
42.3.h.a 4 21.g even 6 1
294.3.b.b 2 7.d odd 6 1
294.3.b.b 2 21.g even 6 1
294.3.b.c 2 7.c even 3 1
294.3.b.c 2 21.h odd 6 1
294.3.h.b 4 1.a even 1 1 trivial
294.3.h.b 4 3.b odd 2 1 inner
294.3.h.b 4 7.c even 3 1 inner
294.3.h.b 4 21.h odd 6 1 inner
336.3.bn.c 4 28.d even 2 1
336.3.bn.c 4 28.f even 6 1
336.3.bn.c 4 84.h odd 2 1
336.3.bn.c 4 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(294, [\chi])\):

\( T_{5}^{4} - 72T_{5}^{2} + 5184 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} - 5 T^{2} + 18 T + 81 \) Copy content Toggle raw display
$5$ \( T^{4} - 72T^{2} + 5184 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 72T^{2} + 5184 \) Copy content Toggle raw display
$19$ \( (T^{2} + 31 T + 961)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 72T^{2} + 5184 \) Copy content Toggle raw display
$29$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1152)^{2} \) Copy content Toggle raw display
$43$ \( (T + 31)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 1800 T^{2} + \cdots + 3240000 \) Copy content Toggle raw display
$53$ \( T^{4} - 648 T^{2} + 419904 \) Copy content Toggle raw display
$59$ \( T^{4} - 72T^{2} + 5184 \) Copy content Toggle raw display
$61$ \( (T^{2} - 50 T + 2500)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 65 T + 4225)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 3528)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 97 T + 9409)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 103 T + 10609)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1800)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 14112 T^{2} + \cdots + 199148544 \) Copy content Toggle raw display
$97$ \( (T - 166)^{4} \) Copy content Toggle raw display
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