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Properties

Label 294.3.g.b
Level $294$
Weight $3$
Character orbit 294.g
Analytic conductor $8.011$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 294.g (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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 2) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{5} + ( - \beta_{3} - 2 \beta_1) q^{6} + 2 \beta_{3} q^{8} + (3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 2) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{5} + ( - \beta_{3} - 2 \beta_1) q^{6} + 2 \beta_{3} q^{8} + (3 \beta_{2} + 3) q^{9} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{10} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1) q^{11} + ( - 2 \beta_{2} + 2) q^{12} + ( - 2 \beta_{3} - 16 \beta_{2} - 4 \beta_1 - 8) q^{13} + (3 \beta_{3} - 6) q^{15} + ( - 4 \beta_{2} - 4) q^{16} + ( - 11 \beta_{3} - 2 \beta_{2} + 11 \beta_1 - 4) q^{17} + (3 \beta_{3} + 3 \beta_1) q^{18} + ( - 16 \beta_{3} - 2 \beta_{2} - 8 \beta_1 + 2) q^{19} + (2 \beta_{3} + 8 \beta_{2} + 4 \beta_1 + 4) q^{20} + ( - 6 \beta_{3} - 6) q^{22} + ( - 6 \beta_{2} - 9 \beta_1 - 6) q^{23} + ( - 2 \beta_{3} + 2 \beta_1) q^{24} + ( - 12 \beta_{3} + 7 \beta_{2} - 12 \beta_1) q^{25} + ( - 16 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 4) q^{26} + ( - 6 \beta_{2} - 3) q^{27} + 30 q^{29} + ( - 6 \beta_{2} - 6 \beta_1 - 6) q^{30} + ( - 12 \beta_{3} - 12 \beta_{2} + 12 \beta_1 - 24) q^{31} + ( - 4 \beta_{3} - 4 \beta_1) q^{32} + ( - 6 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 6) q^{33} + ( - 2 \beta_{3} + 44 \beta_{2} - 4 \beta_1 + 22) q^{34} - 6 q^{36} + (20 \beta_{2} + 36 \beta_1 + 20) q^{37} + ( - 2 \beta_{3} + 16 \beta_{2} + 2 \beta_1 + 32) q^{38} + (6 \beta_{3} + 24 \beta_{2} + 6 \beta_1) q^{39} + (8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 4) q^{40} + ( - 7 \beta_{3} - 28 \beta_{2} - 14 \beta_1 - 14) q^{41} + ( - 30 \beta_{3} - 32) q^{43} + (12 \beta_{2} - 6 \beta_1 + 12) q^{44} + ( - 3 \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 12) q^{45} + ( - 6 \beta_{3} - 18 \beta_{2} - 6 \beta_1) q^{46} + ( - 8 \beta_{3} + 28 \beta_{2} - 4 \beta_1 - 28) q^{47} + (8 \beta_{2} + 4) q^{48} + (7 \beta_{3} + 24) q^{50} + (6 \beta_{2} - 33 \beta_1 + 6) q^{51} + ( - 4 \beta_{3} + 16 \beta_{2} + 4 \beta_1 + 32) q^{52} + (12 \beta_{3} - 54 \beta_{2} + 12 \beta_1) q^{53} + ( - 6 \beta_{3} - 3 \beta_1) q^{54} + ( - 12 \beta_{2} - 6) q^{55} + (24 \beta_{3} - 6) q^{57} + 30 \beta_1 q^{58} + (20 \beta_{3} - 28 \beta_{2} - 20 \beta_1 - 56) q^{59} + ( - 6 \beta_{3} - 12 \beta_{2} - 6 \beta_1) q^{60} + (8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 4) q^{61} + ( - 12 \beta_{3} + 48 \beta_{2} - 24 \beta_1 + 24) q^{62} + 8 q^{64} + ( - 60 \beta_{2} - 36 \beta_1 - 60) q^{65} + (6 \beta_{3} + 6 \beta_{2} - 6 \beta_1 + 12) q^{66} + (12 \beta_{3} + 44 \beta_{2} + 12 \beta_1) q^{67} + (44 \beta_{3} - 4 \beta_{2} + 22 \beta_1 + 4) q^{68} + (9 \beta_{3} + 12 \beta_{2} + 18 \beta_1 + 6) q^{69} + (57 \beta_{3} - 30) q^{71} - 6 \beta_1 q^{72} + (26 \beta_{3} + 4 \beta_{2} - 26 \beta_1 + 8) q^{73} + (20 \beta_{3} + 72 \beta_{2} + 20 \beta_1) q^{74} + (24 \beta_{3} - 7 \beta_{2} + 12 \beta_1 + 7) q^{75} + (16 \beta_{3} + 8 \beta_{2} + 32 \beta_1 + 4) q^{76} + (24 \beta_{3} - 12) q^{78} + ( - 32 \beta_{2} + 72 \beta_1 - 32) q^{79} + (4 \beta_{3} - 8 \beta_{2} - 4 \beta_1 - 16) q^{80} + 9 \beta_{2} q^{81} + ( - 28 \beta_{3} - 14 \beta_{2} - 14 \beta_1 + 14) q^{82} + (20 \beta_{3} - 64 \beta_{2} + 40 \beta_1 - 32) q^{83} + ( - 60 \beta_{3} + 54) q^{85} + (60 \beta_{2} - 32 \beta_1 + 60) q^{86} + ( - 30 \beta_{2} - 60) q^{87} + (12 \beta_{3} - 12 \beta_{2} + 12 \beta_1) q^{88} + (42 \beta_{3} - 54 \beta_{2} + 21 \beta_1 + 54) q^{89} + (6 \beta_{3} + 12 \beta_{2} + 12 \beta_1 + 6) q^{90} + ( - 18 \beta_{3} + 12) q^{92} + (36 \beta_{2} - 36 \beta_1 + 36) q^{93} + (28 \beta_{3} + 8 \beta_{2} - 28 \beta_1 + 16) q^{94} + ( - 54 \beta_{3} - 60 \beta_{2} - 54 \beta_1) q^{95} + (8 \beta_{3} + 4 \beta_1) q^{96} + (10 \beta_{3} - 88 \beta_{2} + 20 \beta_1 - 44) q^{97} + (9 \beta_{3} + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} - 4 q^{4} + 12 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} - 4 q^{4} + 12 q^{5} + 6 q^{9} + 12 q^{10} + 12 q^{11} + 12 q^{12} - 24 q^{15} - 8 q^{16} - 12 q^{17} + 12 q^{19} - 24 q^{22} - 12 q^{23} - 14 q^{25} + 24 q^{26} + 120 q^{29} - 12 q^{30} - 72 q^{31} - 36 q^{33} - 24 q^{36} + 40 q^{37} + 96 q^{38} - 48 q^{39} - 24 q^{40} - 128 q^{43} + 24 q^{44} + 36 q^{45} + 36 q^{46} - 168 q^{47} + 96 q^{50} + 12 q^{51} + 96 q^{52} + 108 q^{53} - 24 q^{57} - 168 q^{59} + 24 q^{60} - 24 q^{61} + 32 q^{64} - 120 q^{65} + 36 q^{66} - 88 q^{67} + 24 q^{68} - 120 q^{71} + 24 q^{73} - 144 q^{74} + 42 q^{75} - 48 q^{78} - 64 q^{79} - 48 q^{80} - 18 q^{81} + 84 q^{82} + 216 q^{85} + 120 q^{86} - 180 q^{87} + 24 q^{88} + 324 q^{89} + 48 q^{92} + 72 q^{93} + 48 q^{94} + 120 q^{95} + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) 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

Display \(\beta_i\) in terms of \(\nu^j\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.707107 + 1.22474i
0.707107 − 1.22474i
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 − 1.73205i 0.878680 + 0.507306i − 2.44949i 0 2.82843 1.50000 − 2.59808i −1.24264 + 0.717439i
19.2 0.707107 − 1.22474i −1.50000 + 0.866025i −1.00000 − 1.73205i 5.12132 + 2.95680i 2.44949i 0 −2.82843 1.50000 − 2.59808i 7.24264 − 4.18154i
31.1 −0.707107 − 1.22474i −1.50000 − 0.866025i −1.00000 + 1.73205i 0.878680 − 0.507306i 2.44949i 0 2.82843 1.50000 + 2.59808i −1.24264 − 0.717439i
31.2 0.707107 + 1.22474i −1.50000 − 0.866025i −1.00000 + 1.73205i 5.12132 − 2.95680i − 2.44949i 0 −2.82843 1.50000 + 2.59808i 7.24264 + 4.18154i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d 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.g.b 4
3.b odd 2 1 882.3.n.a 4
7.b odd 2 1 294.3.g.c 4
7.c even 3 1 42.3.c.a ✓ 4
7.c even 3 1 294.3.g.c 4
7.d odd 6 1 42.3.c.a ✓ 4
7.d odd 6 1 inner 294.3.g.b 4
21.c even 2 1 882.3.n.d 4
21.g even 6 1 126.3.c.b 4
21.g even 6 1 882.3.n.a 4
21.h odd 6 1 126.3.c.b 4
21.h odd 6 1 882.3.n.d 4
28.f even 6 1 336.3.f.c 4
28.g odd 6 1 336.3.f.c 4
35.i odd 6 1 1050.3.f.a 4
35.j even 6 1 1050.3.f.a 4
35.k even 12 2 1050.3.h.a 8
35.l odd 12 2 1050.3.h.a 8
56.j odd 6 1 1344.3.f.f 4
56.k odd 6 1 1344.3.f.e 4
56.m even 6 1 1344.3.f.e 4
56.p even 6 1 1344.3.f.f 4
84.j odd 6 1 1008.3.f.g 4
84.n even 6 1 1008.3.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a ✓ 4 7.c even 3 1
42.3.c.a ✓ 4 7.d odd 6 1
126.3.c.b 4 21.g even 6 1
126.3.c.b 4 21.h odd 6 1
294.3.g.b 4 1.a even 1 1 trivial
294.3.g.b 4 7.d odd 6 1 inner
294.3.g.c 4 7.b odd 2 1
294.3.g.c 4 7.c even 3 1
336.3.f.c 4 28.f even 6 1
336.3.f.c 4 28.g odd 6 1
882.3.n.a 4 3.b odd 2 1
882.3.n.a 4 21.g even 6 1
882.3.n.d 4 21.c even 2 1
882.3.n.d 4 21.h odd 6 1
1008.3.f.g 4 84.j odd 6 1
1008.3.f.g 4 84.n even 6 1
1050.3.f.a 4 35.i odd 6 1
1050.3.f.a 4 35.j even 6 1
1050.3.h.a 8 35.k even 12 2
1050.3.h.a 8 35.l odd 12 2
1344.3.f.e 4 56.k odd 6 1
1344.3.f.e 4 56.m even 6 1
1344.3.f.f 4 56.j odd 6 1
1344.3.f.f 4 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 12T_{5}^{3} + 54T_{5}^{2} - 72T_{5} + 36 \) acting on \(S_{3}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 12 T^{3} + 54 T^{2} - 72 T + 36 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 12 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$13$ \( T^{4} + 432 T^{2} + 28224 \) Copy content Toggle raw display
$17$ \( T^{4} + 12 T^{3} - 666 T^{2} + \cdots + 509796 \) Copy content Toggle raw display
$19$ \( T^{4} - 12 T^{3} - 324 T^{2} + \cdots + 138384 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + 270 T^{2} + \cdots + 15876 \) Copy content Toggle raw display
$29$ \( (T - 30)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 72 T^{3} + 1296 T^{2} + \cdots + 186624 \) Copy content Toggle raw display
$37$ \( T^{4} - 40 T^{3} + 3792 T^{2} + \cdots + 4804864 \) Copy content Toggle raw display
$41$ \( T^{4} + 1764 T^{2} + 86436 \) Copy content Toggle raw display
$43$ \( (T^{2} + 64 T - 776)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 168 T^{3} + 11664 T^{2} + \cdots + 5089536 \) Copy content Toggle raw display
$53$ \( T^{4} - 108 T^{3} + 9036 T^{2} + \cdots + 6906384 \) Copy content Toggle raw display
$59$ \( T^{4} + 168 T^{3} + 9360 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$61$ \( T^{4} + 24 T^{3} + 144 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$67$ \( T^{4} + 88 T^{3} + 6096 T^{2} + \cdots + 2715904 \) Copy content Toggle raw display
$71$ \( (T^{2} + 60 T - 5598)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} - 3816 T^{2} + \cdots + 16064064 \) Copy content Toggle raw display
$79$ \( T^{4} + 64 T^{3} + 13440 T^{2} + \cdots + 87310336 \) Copy content Toggle raw display
$83$ \( T^{4} + 10944 T^{2} + \cdots + 451584 \) Copy content Toggle raw display
$89$ \( T^{4} - 324 T^{3} + \cdots + 37234404 \) Copy content Toggle raw display
$97$ \( T^{4} + 12816 T^{2} + \cdots + 27123264 \) Copy content Toggle raw display
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