Properties

Label 294.3.b.e
Level $294$
Weight $3$
Character orbit 294.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.01091977219\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-11})\)
Defining polynomial: \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{3} - 2 q^{4} + (2 \beta_{3} - 2 \beta_{2} - 1) q^{5} + (2 \beta_{3} - \beta_{2} - \beta_1) q^{6} - 2 \beta_{2} q^{8} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{3} - 2 q^{4} + (2 \beta_{3} - 2 \beta_{2} - 1) q^{5} + (2 \beta_{3} - \beta_{2} - \beta_1) q^{6} - 2 \beta_{2} q^{8} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 3) q^{9} + (\beta_{2} + 2 \beta_1 + 4) q^{10} + ( - 10 \beta_{3} + \beta_{2} + 5) q^{11} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{12} + ( - 2 \beta_{2} - 4 \beta_1 + 8) q^{13} + ( - 5 \beta_{3} + 7 \beta_{2} + \beta_1 + 6) q^{15} + 4 q^{16} + ( - 2 \beta_{3} + 2 \beta_{2} + 1) q^{17} + ( - 4 \beta_{3} + 2 \beta_{2} - \beta_1 + 12) q^{18} + (3 \beta_{2} + 6 \beta_1 + 1) q^{19} + ( - 4 \beta_{3} + 4 \beta_{2} + 2) q^{20} + ( - 5 \beta_{2} - 10 \beta_1 - 2) q^{22} + ( - 10 \beta_{3} + 7 \beta_{2} + 5) q^{23} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{24} + ( - 4 \beta_{2} - 8 \beta_1 + 6) q^{25} + (8 \beta_{3} + 8 \beta_{2} - 4) q^{26} + ( - 10 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 15) q^{27} + ( - 20 \beta_{3} - 10 \beta_{2} + 10) q^{29} + ( - 2 \beta_{3} + \beta_{2} - 5 \beta_1 - 12) q^{30} + ( - 5 \beta_{2} - 10 \beta_1 + 5) q^{31} + 4 \beta_{2} q^{32} + (7 \beta_{3} - 26 \beta_{2} + 4 \beta_1 - 30) q^{33} + ( - \beta_{2} - 2 \beta_1 - 4) q^{34} + (2 \beta_{3} + 8 \beta_{2} - 4 \beta_1 - 6) q^{36} - q^{37} + ( - 12 \beta_{3} + \beta_{2} + 6) q^{38} + ( - 12 \beta_{3} - 18 \beta_{2} - 6 \beta_1 + 24) q^{39} + ( - 2 \beta_{2} - 4 \beta_1 - 8) q^{40} + ( - 12 \beta_{3} - 6 \beta_{2} + 6) q^{41} + (6 \beta_{2} + 12 \beta_1 + 26) q^{43} + (20 \beta_{3} - 2 \beta_{2} - 10) q^{44} + (13 \beta_{3} - 20 \beta_{2} - 8 \beta_1 - 21) q^{45} + ( - 5 \beta_{2} - 10 \beta_1 - 14) q^{46} + (6 \beta_{3} + 45 \beta_{2} - 3) q^{47} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{48} + (16 \beta_{3} + 6 \beta_{2} - 8) q^{50} + (5 \beta_{3} - 7 \beta_{2} - \beta_1 - 6) q^{51} + (4 \beta_{2} + 8 \beta_1 - 16) q^{52} + (6 \beta_{3} + 36 \beta_{2} - 3) q^{53} + ( - 4 \beta_{3} - 25 \beta_{2} - 10 \beta_1 - 6) q^{54} + (11 \beta_{2} + 22 \beta_1 + 59) q^{55} + (5 \beta_{3} + 14 \beta_{2} - 4 \beta_1 - 36) q^{57} + ( - 10 \beta_{2} - 20 \beta_1 + 20) q^{58} + (14 \beta_{3} + 55 \beta_{2} - 7) q^{59} + (10 \beta_{3} - 14 \beta_{2} - 2 \beta_1 - 12) q^{60} + (4 \beta_{2} + 8 \beta_1 + 53) q^{61} + (20 \beta_{3} + 5 \beta_{2} - 10) q^{62} - 8 q^{64} + 6 \beta_{2} q^{65} + ( - 8 \beta_{3} - 23 \beta_{2} + 7 \beta_1 + 60) q^{66} + (13 \beta_{2} + 26 \beta_1 - 39) q^{67} + (4 \beta_{3} - 4 \beta_{2} - 2) q^{68} + (19 \beta_{3} - 32 \beta_{2} - 2 \beta_1 - 30) q^{69} + (8 \beta_{3} - 68 \beta_{2} - 4) q^{71} + (8 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 24) q^{72} + ( - 4 \beta_{2} - 8 \beta_1 + 33) q^{73} - \beta_{2} q^{74} + ( - 14 \beta_{3} - 26 \beta_{2} - 2 \beta_1 + 48) q^{75} + ( - 6 \beta_{2} - 12 \beta_1 - 2) q^{76} + (12 \beta_{3} + 12 \beta_{2} - 12 \beta_1 + 24) q^{78} + (17 \beta_{2} + 34 \beta_1 + 13) q^{79} + (8 \beta_{3} - 8 \beta_{2} - 4) q^{80} + (35 \beta_{3} - 4 \beta_{2} + 20 \beta_1 - 42) q^{81} + ( - 6 \beta_{2} - 12 \beta_1 + 12) q^{82} + (16 \beta_{3} + 44 \beta_{2} - 8) q^{83} + (4 \beta_{2} + 8 \beta_1 + 19) q^{85} + ( - 24 \beta_{3} + 26 \beta_{2} + 12) q^{86} + ( - 10 \beta_{3} - 40 \beta_{2} + 20 \beta_1 - 60) q^{87} + (10 \beta_{2} + 20 \beta_1 + 4) q^{88} + ( - 66 \beta_{3} + 36 \beta_{2} + 33) q^{89} + (16 \beta_{3} - 8 \beta_{2} + 13 \beta_1 + 24) q^{90} + (20 \beta_{3} - 14 \beta_{2} - 10) q^{92} + ( - 15 \beta_{3} - 30 \beta_{2} + 60) q^{93} + (3 \beta_{2} + 6 \beta_1 - 90) q^{94} + (26 \beta_{3} - 35 \beta_{2} - 13) q^{95} + (8 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{96} + ( - 10 \beta_{2} - 20 \beta_1 - 72) q^{97} + ( - 29 \beta_{3} + 82 \beta_{2} + 49 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 8 q^{4} + 4 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 8 q^{4} + 4 q^{6} + 10 q^{9} + 16 q^{10} + 4 q^{12} + 32 q^{13} + 14 q^{15} + 16 q^{16} + 40 q^{18} + 4 q^{19} - 8 q^{22} - 8 q^{24} + 24 q^{25} - 80 q^{27} - 52 q^{30} + 20 q^{31} - 106 q^{33} - 16 q^{34} - 20 q^{36} - 4 q^{37} + 72 q^{39} - 32 q^{40} + 104 q^{43} - 58 q^{45} - 56 q^{46} - 8 q^{48} - 14 q^{51} - 64 q^{52} - 32 q^{54} + 236 q^{55} - 134 q^{57} + 80 q^{58} - 28 q^{60} + 212 q^{61} - 32 q^{64} + 224 q^{66} - 156 q^{67} - 82 q^{69} - 80 q^{72} + 132 q^{73} + 164 q^{75} - 8 q^{76} + 120 q^{78} + 52 q^{79} - 98 q^{81} + 48 q^{82} + 76 q^{85} - 260 q^{87} + 16 q^{88} + 128 q^{90} + 210 q^{93} - 360 q^{94} + 16 q^{96} - 288 q^{97} - 70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 8\nu + 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} - 19\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} + 22\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta_{2} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -8\beta_{3} - 8\beta_{2} + 3\beta _1 - 3 \) Copy content Toggle raw display

Character values

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

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

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
0.500000 + 0.244099i
0.500000 3.07253i
0.500000 0.244099i
0.500000 + 3.07253i
1.41421i −2.84521 0.951206i −2.00000 6.14505i −1.34521 + 4.02373i 0 2.82843i 7.19042 + 5.41276i 8.69042
197.2 1.41421i 1.84521 + 2.36542i −2.00000 0.488198i 3.34521 2.60952i 0 2.82843i −2.19042 + 8.72938i −0.690416
197.3 1.41421i −2.84521 + 0.951206i −2.00000 6.14505i −1.34521 4.02373i 0 2.82843i 7.19042 5.41276i 8.69042
197.4 1.41421i 1.84521 2.36542i −2.00000 0.488198i 3.34521 + 2.60952i 0 2.82843i −2.19042 8.72938i −0.690416
\(n\): 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.e 4
3.b odd 2 1 inner 294.3.b.e 4
7.b odd 2 1 294.3.b.i 4
7.c even 3 2 294.3.h.h 8
7.d odd 6 2 42.3.h.b 8
21.c even 2 1 294.3.b.i 4
21.g even 6 2 42.3.h.b 8
21.h odd 6 2 294.3.h.h 8
28.f even 6 2 336.3.bn.g 8
84.j odd 6 2 336.3.bn.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.h.b 8 7.d odd 6 2
42.3.h.b 8 21.g even 6 2
294.3.b.e 4 1.a even 1 1 trivial
294.3.b.e 4 3.b odd 2 1 inner
294.3.b.i 4 7.b odd 2 1
294.3.b.i 4 21.c even 2 1
294.3.h.h 8 7.c even 3 2
294.3.h.h 8 21.h odd 6 2
336.3.bn.g 8 28.f even 6 2
336.3.bn.g 8 84.j odd 6 2

Hecke kernels

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

\( T_{5}^{4} + 38T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{13}^{2} - 16T_{13} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} - 3 T^{2} + 18 T + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 38T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 554 T^{2} + 74529 \) Copy content Toggle raw display
$13$ \( (T^{2} - 16 T - 24)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 38T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 197)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 746 T^{2} + 31329 \) Copy content Toggle raw display
$29$ \( T^{4} + 2600 T^{2} + 810000 \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T - 525)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 936 T^{2} + 104976 \) Copy content Toggle raw display
$43$ \( (T^{2} - 52 T - 116)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 8298 T^{2} + \cdots + 15610401 \) Copy content Toggle raw display
$53$ \( T^{4} + 5382 T^{2} + \cdots + 6215049 \) Copy content Toggle raw display
$59$ \( T^{4} + 13178 T^{2} + \cdots + 30371121 \) Copy content Toggle raw display
$61$ \( (T^{2} - 106 T + 2457)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 78 T - 2197)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 18848 T^{2} + \cdots + 82301184 \) Copy content Toggle raw display
$73$ \( (T^{2} - 66 T + 737)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 26 T - 6189)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 9152 T^{2} + \cdots + 10036224 \) Copy content Toggle raw display
$89$ \( T^{4} + 29142 T^{2} + \cdots + 88115769 \) Copy content Toggle raw display
$97$ \( (T^{2} + 144 T + 2984)^{2} \) Copy content Toggle raw display
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