Properties

Label 294.3.b.b
Level $294$
Weight $3$
Character orbit 294.b
Analytic conductor $8.011$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 294.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.01091977219\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) 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: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - 2 \beta - 1) q^{3} - 2 q^{4} - 6 \beta q^{5} + ( - \beta + 4) q^{6} - 2 \beta q^{8} + (4 \beta - 7) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + ( - 2 \beta - 1) q^{3} - 2 q^{4} - 6 \beta q^{5} + ( - \beta + 4) q^{6} - 2 \beta q^{8} + (4 \beta - 7) q^{9} + 12 q^{10} + (4 \beta + 2) q^{12} - q^{13} + (6 \beta - 24) q^{15} + 4 q^{16} + 6 \beta q^{17} + ( - 7 \beta - 8) q^{18} - 31 q^{19} + 12 \beta q^{20} + 6 \beta q^{23} + (2 \beta - 8) q^{24} - 47 q^{25} - \beta q^{26} + (10 \beta + 23) q^{27} - 12 \beta q^{29} + ( - 24 \beta - 12) q^{30} - 7 q^{31} + 4 \beta q^{32} - 12 q^{34} + ( - 8 \beta + 14) q^{36} - q^{37} - 31 \beta q^{38} + (2 \beta + 1) q^{39} - 24 q^{40} - 24 \beta q^{41} - 31 q^{43} + (42 \beta + 48) q^{45} - 12 q^{46} - 30 \beta q^{47} + ( - 8 \beta - 4) q^{48} - 47 \beta q^{50} + ( - 6 \beta + 24) q^{51} + 2 q^{52} - 18 \beta q^{53} + (23 \beta - 20) q^{54} + (62 \beta + 31) q^{57} + 24 q^{58} + 6 \beta q^{59} + ( - 12 \beta + 48) q^{60} + 50 q^{61} - 7 \beta q^{62} - 8 q^{64} + 6 \beta q^{65} + 65 q^{67} - 12 \beta q^{68} + ( - 6 \beta + 24) q^{69} + 42 \beta q^{71} + (14 \beta + 16) q^{72} - 97 q^{73} - \beta q^{74} + (94 \beta + 47) q^{75} + 62 q^{76} + (\beta - 4) q^{78} - 103 q^{79} - 24 \beta q^{80} + ( - 56 \beta + 17) q^{81} + 48 q^{82} - 30 \beta q^{83} + 72 q^{85} - 31 \beta q^{86} + (12 \beta - 48) q^{87} - 84 \beta q^{89} + (48 \beta - 84) q^{90} - 12 \beta q^{92} + (14 \beta + 7) q^{93} + 60 q^{94} + 186 \beta q^{95} + ( - 4 \beta + 16) q^{96} - 166 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{4} + 8 q^{6} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{4} + 8 q^{6} - 14 q^{9} + 24 q^{10} + 4 q^{12} - 2 q^{13} - 48 q^{15} + 8 q^{16} - 16 q^{18} - 62 q^{19} - 16 q^{24} - 94 q^{25} + 46 q^{27} - 24 q^{30} - 14 q^{31} - 24 q^{34} + 28 q^{36} - 2 q^{37} + 2 q^{39} - 48 q^{40} - 62 q^{43} + 96 q^{45} - 24 q^{46} - 8 q^{48} + 48 q^{51} + 4 q^{52} - 40 q^{54} + 62 q^{57} + 48 q^{58} + 96 q^{60} + 100 q^{61} - 16 q^{64} + 130 q^{67} + 48 q^{69} + 32 q^{72} - 194 q^{73} + 94 q^{75} + 124 q^{76} - 8 q^{78} - 206 q^{79} + 34 q^{81} + 96 q^{82} + 144 q^{85} - 96 q^{87} - 168 q^{90} + 14 q^{93} + 120 q^{94} + 32 q^{96} - 332 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
1.41421i
1.41421i
1.41421i −1.00000 + 2.82843i −2.00000 8.48528i 4.00000 + 1.41421i 0 2.82843i −7.00000 5.65685i 12.0000
197.2 1.41421i −1.00000 2.82843i −2.00000 8.48528i 4.00000 1.41421i 0 2.82843i −7.00000 + 5.65685i 12.0000
\(n\): 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

Twists

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 72 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 72 \) Copy content Toggle raw display
$19$ \( (T + 31)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 72 \) Copy content Toggle raw display
$29$ \( T^{2} + 288 \) Copy content Toggle raw display
$31$ \( (T + 7)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 1152 \) Copy content Toggle raw display
$43$ \( (T + 31)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 1800 \) Copy content Toggle raw display
$53$ \( T^{2} + 648 \) Copy content Toggle raw display
$59$ \( T^{2} + 72 \) Copy content Toggle raw display
$61$ \( (T - 50)^{2} \) Copy content Toggle raw display
$67$ \( (T - 65)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 3528 \) Copy content Toggle raw display
$73$ \( (T + 97)^{2} \) Copy content Toggle raw display
$79$ \( (T + 103)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1800 \) Copy content Toggle raw display
$89$ \( T^{2} + 14112 \) Copy content Toggle raw display
$97$ \( (T + 166)^{2} \) Copy content Toggle raw display
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