Properties

Label 294.3.h.d
Level $294$
Weight $3$
Character orbit 294.h
Analytic conductor $8.011$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - 2 \beta_{4} + \beta_{3}) q^{5} + (\beta_{7} + \beta_{3} - 2) q^{6} - 2 \beta_{6} q^{8} + ( - 2 \beta_{3} - 5 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - 2 \beta_{4} + \beta_{3}) q^{5} + (\beta_{7} + \beta_{3} - 2) q^{6} - 2 \beta_{6} q^{8} + ( - 2 \beta_{3} - 5 \beta_{2}) q^{9} + ( - 2 \beta_{5} - 4 \beta_{2} - 2 \beta_1 + 4) q^{10} + (2 \beta_{7} + 5 \beta_{6} - 5 \beta_{4}) q^{11} + ( - 2 \beta_{5} + 2 \beta_{4}) q^{12} + ( - 4 \beta_1 + 10) q^{13} + (2 \beta_{7} - 7 \beta_{6} + 2 \beta_{3} + 2 \beta_1 - 4) q^{15} - 4 \beta_{2} q^{16} + (5 \beta_{7} + 2 \beta_{6} - 2 \beta_{4}) q^{17} + ( - 5 \beta_{6} + 4 \beta_{5} + 5 \beta_{4} + 4 \beta_1) q^{18} + 16 \beta_{2} q^{19} + (2 \beta_{7} - 4 \beta_{6} + 2 \beta_{3}) q^{20} + (4 \beta_1 + 10) q^{22} + ( - \beta_{4} - 10 \beta_{3}) q^{23} + (2 \beta_{7} + 4 \beta_{2} - 4) q^{24} + ( - 8 \beta_{5} + 3 \beta_{2} - 8 \beta_1 - 3) q^{25} + ( - 10 \beta_{4} + 4 \beta_{3}) q^{26} + (19 \beta_{6} + \beta_1) q^{27} + (2 \beta_{7} + 20 \beta_{6} + 2 \beta_{3}) q^{29} + ( - 4 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 14 \beta_{2}) q^{30} + (10 \beta_{5} - 32 \beta_{2} + 10 \beta_1 + 32) q^{31} + ( - 4 \beta_{6} + 4 \beta_{4}) q^{32} + (4 \beta_{5} + 14 \beta_{4} + 5 \beta_{3} - 10 \beta_{2}) q^{33} + (10 \beta_1 + 4) q^{34} + ( - 4 \beta_{7} - 4 \beta_{3} - 10) q^{36} - 20 \beta_{2} q^{37} + (16 \beta_{6} - 16 \beta_{4}) q^{38} + (4 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} + 10 \beta_{4} - 28 \beta_{2} + \cdots + 28) q^{39}+ \cdots + ( - 10 \beta_{7} - 25 \beta_{6} - 10 \beta_{3} + 20 \beta_1 + 56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 16 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 16 q^{6} - 20 q^{9} + 16 q^{10} + 80 q^{13} - 32 q^{15} - 16 q^{16} + 64 q^{19} + 80 q^{22} - 16 q^{24} - 12 q^{25} - 56 q^{30} + 128 q^{31} - 40 q^{33} + 32 q^{34} - 80 q^{36} - 80 q^{37} + 112 q^{39} - 32 q^{40} + 160 q^{43} - 112 q^{45} + 8 q^{46} - 16 q^{51} + 80 q^{52} + 152 q^{54} - 64 q^{55} + 160 q^{58} - 32 q^{60} + 56 q^{61} - 64 q^{64} - 112 q^{66} - 240 q^{67} - 16 q^{69} - 120 q^{73} - 224 q^{75} + 256 q^{76} - 160 q^{78} + 128 q^{79} + 124 q^{81} - 240 q^{82} - 496 q^{85} + 160 q^{87} + 80 q^{88} - 160 q^{90} + 280 q^{93} - 48 q^{94} + 32 q^{96} - 720 q^{97} + 448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 148 ) / 55 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 533\nu ) / 165 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 203\nu ) / 165 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23\nu^{6} + 220\nu^{4} - 1265\nu^{2} + 1656 ) / 495 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 55\nu^{5} - 341\nu^{3} + 81\nu ) / 297 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 79\nu^{7} - 605\nu^{5} + 4345\nu^{3} - 5688\nu ) / 1485 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 4\beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} + 11\beta_{6} + 5\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{5} - 23\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 31\beta_{7} + 79\beta_{6} - 79\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -55\beta _1 - 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -533\beta_{4} - 203\beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

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

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
2.23256 1.28897i
−1.00781 + 0.581861i
−2.23256 + 1.28897i
1.00781 0.581861i
2.23256 + 1.28897i
−1.00781 0.581861i
−2.23256 1.28897i
1.00781 + 0.581861i
−1.22474 + 0.707107i −0.0981308 + 2.99839i 1.00000 1.73205i 0.790881 0.456615i −2.00000 3.74166i 0 2.82843i −8.98074 0.588470i −0.645751 + 1.11847i
263.2 −1.22474 + 0.707107i 2.54762 1.58418i 1.00000 1.73205i −5.68986 + 3.28504i −2.00000 + 3.74166i 0 2.82843i 3.98074 8.07178i 4.64575 8.04668i
263.3 1.22474 0.707107i −2.54762 + 1.58418i 1.00000 1.73205i −0.790881 + 0.456615i −2.00000 + 3.74166i 0 2.82843i 3.98074 8.07178i −0.645751 + 1.11847i
263.4 1.22474 0.707107i 0.0981308 2.99839i 1.00000 1.73205i 5.68986 3.28504i −2.00000 3.74166i 0 2.82843i −8.98074 0.588470i 4.64575 8.04668i
275.1 −1.22474 0.707107i −0.0981308 2.99839i 1.00000 + 1.73205i 0.790881 + 0.456615i −2.00000 + 3.74166i 0 2.82843i −8.98074 + 0.588470i −0.645751 1.11847i
275.2 −1.22474 0.707107i 2.54762 + 1.58418i 1.00000 + 1.73205i −5.68986 3.28504i −2.00000 3.74166i 0 2.82843i 3.98074 + 8.07178i 4.64575 + 8.04668i
275.3 1.22474 + 0.707107i −2.54762 1.58418i 1.00000 + 1.73205i −0.790881 0.456615i −2.00000 3.74166i 0 2.82843i 3.98074 + 8.07178i −0.645751 1.11847i
275.4 1.22474 + 0.707107i 0.0981308 + 2.99839i 1.00000 + 1.73205i 5.68986 + 3.28504i −2.00000 + 3.74166i 0 2.82843i −8.98074 + 0.588470i 4.64575 + 8.04668i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 275.4
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.d 8
3.b odd 2 1 inner 294.3.h.d 8
7.b odd 2 1 294.3.h.g 8
7.c even 3 1 42.3.b.a 4
7.c even 3 1 inner 294.3.h.d 8
7.d odd 6 1 294.3.b.h 4
7.d odd 6 1 294.3.h.g 8
21.c even 2 1 294.3.h.g 8
21.g even 6 1 294.3.b.h 4
21.g even 6 1 294.3.h.g 8
21.h odd 6 1 42.3.b.a 4
21.h odd 6 1 inner 294.3.h.d 8
28.g odd 6 1 336.3.d.b 4
35.j even 6 1 1050.3.e.a 4
35.l odd 12 2 1050.3.c.a 8
56.k odd 6 1 1344.3.d.e 4
56.p even 6 1 1344.3.d.c 4
63.g even 3 1 1134.3.q.a 8
63.h even 3 1 1134.3.q.a 8
63.j odd 6 1 1134.3.q.a 8
63.n odd 6 1 1134.3.q.a 8
84.n even 6 1 336.3.d.b 4
105.o odd 6 1 1050.3.e.a 4
105.x even 12 2 1050.3.c.a 8
168.s odd 6 1 1344.3.d.c 4
168.v even 6 1 1344.3.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.b.a 4 7.c even 3 1
42.3.b.a 4 21.h odd 6 1
294.3.b.h 4 7.d odd 6 1
294.3.b.h 4 21.g even 6 1
294.3.h.d 8 1.a even 1 1 trivial
294.3.h.d 8 3.b odd 2 1 inner
294.3.h.d 8 7.c even 3 1 inner
294.3.h.d 8 21.h odd 6 1 inner
294.3.h.g 8 7.b odd 2 1
294.3.h.g 8 7.d odd 6 1
294.3.h.g 8 21.c even 2 1
294.3.h.g 8 21.g even 6 1
336.3.d.b 4 28.g odd 6 1
336.3.d.b 4 84.n even 6 1
1050.3.c.a 8 35.l odd 12 2
1050.3.c.a 8 105.x even 12 2
1050.3.e.a 4 35.j even 6 1
1050.3.e.a 4 105.o odd 6 1
1134.3.q.a 8 63.g even 3 1
1134.3.q.a 8 63.h even 3 1
1134.3.q.a 8 63.j odd 6 1
1134.3.q.a 8 63.n odd 6 1
1344.3.d.c 4 56.p even 6 1
1344.3.d.c 4 168.s odd 6 1
1344.3.d.e 4 56.k odd 6 1
1344.3.d.e 4 168.v even 6 1

Hecke kernels

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

\( T_{5}^{8} - 44T_{5}^{6} + 1900T_{5}^{4} - 1584T_{5}^{2} + 1296 \) Copy content Toggle raw display
\( T_{13}^{2} - 20T_{13} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 10 T^{6} + 19 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} - 44 T^{6} + 1900 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 212 T^{6} + 44908 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( (T^{2} - 20 T - 12)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} - 716 T^{6} + \cdots + 13680577296 \) Copy content Toggle raw display
$19$ \( (T^{2} - 16 T + 256)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 2804 T^{6} + \cdots + 3819694995216 \) Copy content Toggle raw display
$29$ \( (T^{4} + 1712 T^{2} + 553536)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 64 T^{3} + 3772 T^{2} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 20 T + 400)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 5868 T^{2} + 443556)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 40 T - 608)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 2592 T^{2} + 6718464)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 3392 T^{6} + \cdots + 84934656 \) Copy content Toggle raw display
$61$ \( (T^{4} - 28 T^{3} + 3388 T^{2} + \cdots + 6780816)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 120 T^{3} + 10912 T^{2} + \cdots + 12166144)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 7988 T^{2} + 2232036)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 60 T^{3} + 4492 T^{2} + \cdots + 795664)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 64 T^{3} + 5872 T^{2} + \cdots + 3154176)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 21200 T^{2} + 360000)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 3852 T^{6} + \cdots + 7852750780176 \) Copy content Toggle raw display
$97$ \( (T^{2} + 180 T + 7652)^{4} \) Copy content Toggle raw display
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