Properties

Label 368.6.a.e
Level $368$
Weight $6$
Character orbit 368.a
Self dual yes
Analytic conductor $59.021$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.0212456912\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.7925.1
Defining polynomial: \( x^{3} - x^{2} - 13x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 23)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 2 \beta_1 + 7) q^{3} + (2 \beta_{2} - 3 \beta_1 - 21) q^{5} + (4 \beta_{2} - 17 \beta_1 + 87) q^{7} + (41 \beta_{2} + 25 \beta_1 + 35) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 2 \beta_1 + 7) q^{3} + (2 \beta_{2} - 3 \beta_1 - 21) q^{5} + (4 \beta_{2} - 17 \beta_1 + 87) q^{7} + (41 \beta_{2} + 25 \beta_1 + 35) q^{9} + ( - 28 \beta_{2} - 3 \beta_1 - 37) q^{11} + ( - 99 \beta_{2} + 45 \beta_1 - 324) q^{13} + ( - 30 \beta_{2} - 27 \beta_1 - 249) q^{15} + ( - 146 \beta_{2} - 57 \beta_1 - 269) q^{17} + ( - 200 \beta_{2} - 40 \beta_1 - 498) q^{19} + ( - 52 \beta_{2} + 193 \beta_1 - 475) q^{21} + 529 q^{23} + ( - 88 \beta_{2} + 86 \beta_1 - 2391) q^{25} + (559 \beta_{2} - 22 \beta_1 + 3373) q^{27} + (523 \beta_{2} - 745 \beta_1 - 704) q^{29} + ( - 479 \beta_{2} + 914 \beta_1 + 1471) q^{31} + ( - 406 \beta_{2} - 329 \beta_1 - 2431) q^{33} + (148 \beta_{2} - 160 \beta_1 - 460) q^{35} + (1662 \beta_{2} - 1117 \beta_1 + 2053) q^{37} + ( - 1017 \beta_{2} - 1494 \beta_1 - 5499) q^{39} + ( - 1847 \beta_{2} + 1171 \beta_1 - 3258) q^{41} + (1150 \beta_{2} - 2050 \beta_1 + 4510) q^{43} + ( - 1392 \beta_{2} - 66 \beta_1 - 870) q^{45} + (1021 \beta_{2} + 682 \beta_1 - 7613) q^{47} + (1428 \beta_{2} - 4306 \beta_1 - 949) q^{49} + ( - 2648 \beta_{2} - 1909 \beta_1 - 16517) q^{51} + (534 \beta_{2} + 4444 \beta_1 + 7006) q^{53} + (840 \beta_{2} + 14 \beta_1 - 130) q^{55} + ( - 3338 \beta_{2} - 2836 \beta_1 - 20486) q^{57} + ( - 1980 \beta_{2} - 934 \beta_1 - 17710) q^{59} + ( - 42 \beta_{2} + 1322 \beta_1 - 23320) q^{61} + (52 \beta_{2} + 2906 \beta_1 - 12614) q^{63} + (2250 \beta_{2} + 351 \beta_1 - 351) q^{65} + (1436 \beta_{2} + 4323 \beta_1 + 21949) q^{67} + (529 \beta_{2} + 1058 \beta_1 + 3703) q^{69} + ( - 2227 \beta_{2} + 1628 \beta_1 - 31515) q^{71} + ( - 21 \beta_{2} + 3241 \beta_1 - 26170) q^{73} + ( - 2501 \beta_{2} - 5488 \beta_1 - 15929) q^{75} + ( - 876 \beta_{2} + 532 \beta_1 + 368) q^{77} + (6662 \beta_{2} + 4778 \beta_1 - 20036) q^{79} + ( - 124 \beta_{2} + 5680 \beta_1 + 51917) q^{81} + (13758 \beta_{2} - 5867 \beta_1 - 9839) q^{83} + (4476 \beta_{2} + 508 \beta_1 + 3856) q^{85} + ( - 2623 \beta_{2} + 2554 \beta_1 - 28441) q^{87} + (8666 \beta_{2} - 5964 \beta_1 + 4304) q^{89} + ( - 6984 \beta_{2} + 14229 \beta_1 - 43587) q^{91} + (5777 \beta_{2} - 455 \beta_1 + 50366) q^{93} + (5644 \beta_{2} + 894 \beta_1 + 5298) q^{95} + (618 \beta_{2} + 3351 \beta_1 - 90313) q^{97} + ( - 4118 \beta_{2} - 8116 \beta_1 - 62360) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9} - 136 q^{11} - 1116 q^{13} - 750 q^{15} - 896 q^{17} - 1654 q^{19} - 1670 q^{21} + 1587 q^{23} - 7347 q^{25} + 10700 q^{27} - 844 q^{29} + 3020 q^{31} - 7370 q^{33} - 1072 q^{35} + 8938 q^{37} - 16020 q^{39} - 12792 q^{41} + 16730 q^{43} - 3936 q^{45} - 22500 q^{47} + 2887 q^{49} - 50290 q^{51} + 17108 q^{53} + 436 q^{55} - 61960 q^{57} - 54176 q^{59} - 71324 q^{61} - 40696 q^{63} + 846 q^{65} + 62960 q^{67} + 10580 q^{69} - 98400 q^{71} - 81772 q^{73} - 44800 q^{75} - 304 q^{77} - 58224 q^{79} + 149947 q^{81} - 9892 q^{83} + 15536 q^{85} - 90500 q^{87} + 27542 q^{89} - 151974 q^{91} + 157330 q^{93} + 20644 q^{95} - 273672 q^{97} - 183082 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 13x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} - \beta _1 + 17 ) / 2 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.57511
0.917748
3.65736
0 −9.09413 0 3.86330 0 226.379 0 −160.297 0
1.2 0 1.43100 0 −37.9865 0 43.8366 0 −240.952 0
1.3 0 27.6631 0 −23.8768 0 11.7843 0 522.249 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.6.a.e 3
4.b odd 2 1 23.6.a.a 3
12.b even 2 1 207.6.a.b 3
20.d odd 2 1 575.6.a.b 3
92.b even 2 1 529.6.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.6.a.a 3 4.b odd 2 1
207.6.a.b 3 12.b even 2 1
368.6.a.e 3 1.a even 1 1 trivial
529.6.a.a 3 92.b even 2 1
575.6.a.b 3 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 20T_{3}^{2} - 225T_{3} + 360 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(368))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 20 T^{2} - 225 T + 360 \) Copy content Toggle raw display
$5$ \( T^{3} + 58 T^{2} + 668 T - 3504 \) Copy content Toggle raw display
$7$ \( T^{3} - 282 T^{2} + 13108 T - 116944 \) Copy content Toggle raw display
$11$ \( T^{3} + 136 T^{2} - 43688 T - 840152 \) Copy content Toggle raw display
$13$ \( T^{3} + 1116 T^{2} + \cdots - 255615102 \) Copy content Toggle raw display
$17$ \( T^{3} + 896 T^{2} + \cdots + 220718408 \) Copy content Toggle raw display
$19$ \( T^{3} + 1654 T^{2} + \cdots - 460771768 \) Copy content Toggle raw display
$23$ \( (T - 529)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 844 T^{2} + \cdots - 33789223458 \) Copy content Toggle raw display
$31$ \( T^{3} - 3020 T^{2} + \cdots + 117638912880 \) Copy content Toggle raw display
$37$ \( T^{3} - 8938 T^{2} + \cdots + 1048082031344 \) Copy content Toggle raw display
$41$ \( T^{3} + 12792 T^{2} + \cdots - 1564944049486 \) Copy content Toggle raw display
$43$ \( T^{3} - 16730 T^{2} + \cdots + 95315904000 \) Copy content Toggle raw display
$47$ \( T^{3} + 22500 T^{2} + \cdots - 916008439440 \) Copy content Toggle raw display
$53$ \( T^{3} - 17108 T^{2} + \cdots + 7849670295504 \) Copy content Toggle raw display
$59$ \( T^{3} + 54176 T^{2} + \cdots + 1725012447168 \) Copy content Toggle raw display
$61$ \( T^{3} + 71324 T^{2} + \cdots + 11439907465152 \) Copy content Toggle raw display
$67$ \( T^{3} - 62960 T^{2} + \cdots + 11971711378840 \) Copy content Toggle raw display
$71$ \( T^{3} + 98400 T^{2} + \cdots + 24837760695040 \) Copy content Toggle raw display
$73$ \( T^{3} + 81772 T^{2} + \cdots + 7199078503954 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 235690469012368 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 297029282761704 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 125799322340896 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 693755159518744 \) Copy content Toggle raw display
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