Properties

Label 232.4.a.e
Level $232$
Weight $4$
Character orbit 232.a
Self dual yes
Analytic conductor $13.688$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 232.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.6884431213\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{3} + ( - \beta_{4} - \beta_{3} - 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_1 - 6) q^{7} + (2 \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{3} + ( - \beta_{4} - \beta_{3} - 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_1 - 6) q^{7} + (2 \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 9) q^{9} + ( - \beta_{5} + 3 \beta_{4} + 3 \beta_{2} + 2 \beta_1 - 1) q^{11} + (3 \beta_{5} - 6 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{13} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 - 33) q^{15} + ( - 6 \beta_{5} - 5 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 37) q^{17} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 11 \beta_{2} - 52) q^{19} + ( - 2 \beta_{5} - 5 \beta_{4} - 10 \beta_{3} + \beta_{2} - 4 \beta_1 - 45) q^{21} + (8 \beta_{5} - 5 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} - \beta_1 - 34) q^{23} + (5 \beta_{5} - \beta_{4} + 6 \beta_{3} + 6 \beta_{2} - 2 \beta_1 - 1) q^{25} + ( - 3 \beta_{5} - 3 \beta_{4} + 12 \beta_{3} + 15 \beta_{2} + 2 \beta_1 - 71) q^{27} - 29 q^{29} + ( - 4 \beta_{5} + \beta_{4} + 10 \beta_{3} - 10 \beta_{2} + 5 \beta_1 - 117) q^{31} + (9 \beta_{5} + 15 \beta_{4} + 8 \beta_{3} - 8 \beta_{2} + 6 \beta_1 + 52) q^{33} + ( - 10 \beta_{5} + 15 \beta_{4} + 15 \beta_{3} - 8 \beta_{2} + 21 \beta_1 - 76) q^{35} + ( - 2 \beta_{5} + 14 \beta_{4} + 2 \beta_{3} - 10 \beta_{2} - 6 \beta_1 - 40) q^{37} + ( - 8 \beta_{5} - 21 \beta_{4} + 6 \beta_{3} + 24 \beta_{2} - 9 \beta_1 - 219) q^{39} + ( - 14 \beta_{5} - 7 \beta_{4} + 28 \beta_{3} - 11 \beta_{2} + 8 \beta_1 - 89) q^{41} + (13 \beta_{5} + 7 \beta_{4} + 12 \beta_{3} + 9 \beta_{2} - 12 \beta_1 - 93) q^{43} + ( - 5 \beta_{5} + 13 \beta_{4} - 27 \beta_{3} + \beta_{2} - 10 \beta_1 - 18) q^{45} + (11 \beta_{5} - 12 \beta_{4} + \beta_{3} + 3 \beta_{2} - 7 \beta_1 - 165) q^{47} + (24 \beta_{5} + 10 \beta_{4} + 44 \beta_{3} + 22 \beta_{2} - 4 \beta_1 + 159) q^{49} + (22 \beta_{5} + 7 \beta_{4} - 41 \beta_{3} - 50 \beta_{2} - 25 \beta_1 - 104) q^{51} + (15 \beta_{5} - 8 \beta_{4} - 8 \beta_{3} - 11 \beta_{2} - 2 \beta_1 + 83) q^{53} + ( - 30 \beta_{5} + 9 \beta_{4} - 36 \beta_{3} - 2 \beta_{2} + 7 \beta_1 - 263) q^{55} + ( - 33 \beta_{5} - 31 \beta_{4} - 9 \beta_{3} + 39 \beta_{2} - 58) q^{57} + (6 \beta_{5} + 13 \beta_{4} - 33 \beta_{3} + 16 \beta_{2} - \beta_1 - 2) q^{59} + ( - 24 \beta_{5} + 13 \beta_{4} + 6 \beta_{3} + \beta_{2} + 38 \beta_1 + 199) q^{61} + (6 \beta_{5} - 56 \beta_{4} - 46 \beta_{3} - 24 \beta_{2} - 16 \beta_1 - 210) q^{63} + (10 \beta_{5} - 8 \beta_{4} + 37 \beta_{3} - 17 \beta_{2} + 14 \beta_1 + 196) q^{65} + ( - 10 \beta_{5} + 4 \beta_{4} + 6 \beta_{3} + 52 \beta_{2} - 4 \beta_1 + 2) q^{67} + (2 \beta_{5} + 21 \beta_{4} - 70 \beta_{3} + 19 \beta_{2} + 34 \beta_1 + 459) q^{69} + ( - 8 \beta_{5} + 24 \beta_{4} - 70 \beta_{3} + 16 \beta_{2} + 56 \beta_1 + 106) q^{71} + ( - 14 \beta_{5} - 52 \beta_{4} + 58 \beta_{3} - 40 \beta_{2} - 18 \beta_1 - 2) q^{73} + (12 \beta_{5} + 24 \beta_{4} - 50 \beta_{3} - 6 \beta_{2} + 20 \beta_1 + 314) q^{75} + (40 \beta_{5} - 77 \beta_{4} - 34 \beta_{3} - \beta_{2} - 70 \beta_1 + 7) q^{77} + ( - 9 \beta_{5} + 20 \beta_{4} - 9 \beta_{3} - 29 \beta_{2} + 21 \beta_1 - 39) q^{79} + (53 \beta_{5} + 21 \beta_{4} - 67 \beta_{3} - 65 \beta_{2} - 36 \beta_1 + 425) q^{81} + ( - 12 \beta_{5} - 55 \beta_{4} + 29 \beta_{3} - 40 \beta_{2} - 41 \beta_1 - 260) q^{83} + ( - 16 \beta_{5} + 67 \beta_{4} + 22 \beta_{3} + 67 \beta_{2} + 24 \beta_1 + 561) q^{85} + ( - 29 \beta_{3} + 29) q^{87} + ( - 12 \beta_{5} - 25 \beta_{4} + 30 \beta_{3} + 11 \beta_{2} - 44 \beta_1 - 61) q^{89} + ( - 8 \beta_{5} + 91 \beta_{4} + 53 \beta_{3} + 20 \beta_{2} + 41 \beta_1 + 442) q^{91} + ( - 22 \beta_{5} + 5 \beta_{4} - 59 \beta_{3} + 18 \beta_{2} + 24 \beta_1 + 373) q^{93} + (33 \beta_{5} + 119 \beta_{4} + 67 \beta_{3} - \beta_{2} - 22 \beta_1 + 60) q^{95} + ( - 16 \beta_{5} + 45 \beta_{4} + 124 \beta_{3} - 73 \beta_{2} - 22 \beta_1 + 27) q^{97} + ( - 15 \beta_{5} - 71 \beta_{4} + 155 \beta_{3} + 39 \beta_{2} + \cdots + 342) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9} - 19 q^{11} + 13 q^{13} - 191 q^{15} - 218 q^{17} - 290 q^{19} - 266 q^{21} - 196 q^{23} - 13 q^{25} - 437 q^{27} - 174 q^{29} - 675 q^{31} + 291 q^{33} - 466 q^{35} - 238 q^{37} - 1297 q^{39} - 464 q^{41} - 579 q^{43} - 148 q^{45} - 975 q^{47} + 914 q^{49} - 576 q^{51} + 515 q^{53} - 1605 q^{55} - 340 q^{57} - 108 q^{59} + 1158 q^{61} - 1136 q^{63} + 1239 q^{65} - 80 q^{67} + 2568 q^{69} + 438 q^{71} + 262 q^{73} + 1766 q^{75} + 194 q^{77} - 237 q^{79} + 2554 q^{81} - 1288 q^{83} + 3112 q^{85} + 145 q^{87} - 252 q^{89} + 2450 q^{91} + 2131 q^{93} + 180 q^{95} + 380 q^{97} + 2264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 9\nu^{4} + 43\nu^{3} + 447\nu^{2} + 671\nu - 113 ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 12\nu^{4} + 22\nu^{3} - 365\nu^{2} - 400\nu + 321 ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} - 3\nu^{4} + 201\nu^{3} + 555\nu^{2} + 261\nu - 159 ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 6\nu^{4} + 431\nu^{3} + 731\nu^{2} - 32\nu - 123 ) / 21 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -41\nu^{5} + 9\nu^{4} + 1595\nu^{3} + 3347\nu^{2} + 1093\nu - 1035 ) / 42 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{4} - 3\beta_{3} - \beta_{2} + \beta _1 + 1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{5} - 15\beta_{3} - 9\beta_{2} - \beta _1 + 107 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} + 17\beta_{4} - 32\beta_{3} - 16\beta_{2} + 5\beta _1 + 95 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 244\beta_{5} + 170\beta_{4} - 861\beta_{3} - 455\beta_{2} + 27\beta _1 + 4403 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1002\beta_{5} + 2736\beta_{4} - 6473\beta_{3} - 3351\beta_{2} + 729\beta _1 + 24309 ) / 8 \) 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
7.22553
0.215317
−1.32108
−4.70501
−1.90366
0.488896
0 −9.88017 0 2.64112 0 3.30580 0 70.6177 0
1.2 0 −6.36242 0 8.72860 0 −8.76166 0 13.4804 0
1.3 0 −0.116789 0 −14.4485 0 30.0998 0 −26.9864 0
1.4 0 0.586661 0 15.4201 0 −28.5777 0 −26.6558 0
1.5 0 2.88453 0 −3.33508 0 0.0162797 0 −18.6795 0
1.6 0 7.88819 0 −14.0062 0 −34.0826 0 35.2236 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(1\)

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 232.4.a.e 6
3.b odd 2 1 2088.4.a.l 6
4.b odd 2 1 464.4.a.n 6
8.b even 2 1 1856.4.a.bd 6
8.d odd 2 1 1856.4.a.bc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.a.e 6 1.a even 1 1 trivial
464.4.a.n 6 4.b odd 2 1
1856.4.a.bc 6 8.d odd 2 1
1856.4.a.bd 6 8.b even 2 1
2088.4.a.l 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 5T_{3}^{5} - 92T_{3}^{4} - 266T_{3}^{3} + 1581T_{3}^{2} - 651T_{3} - 98 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(232))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 5 T^{5} - 92 T^{4} - 266 T^{3} + \cdots - 98 \) Copy content Toggle raw display
$5$ \( T^{6} + 5 T^{5} - 356 T^{4} + \cdots - 239922 \) Copy content Toggle raw display
$7$ \( T^{6} + 38 T^{5} - 764 T^{4} + \cdots - 13824 \) Copy content Toggle raw display
$11$ \( T^{6} + 19 T^{5} + \cdots - 2827812238 \) Copy content Toggle raw display
$13$ \( T^{6} - 13 T^{5} + \cdots - 7992521554 \) Copy content Toggle raw display
$17$ \( T^{6} + 218 T^{5} + \cdots + 200391946752 \) Copy content Toggle raw display
$19$ \( T^{6} + 290 T^{5} + \cdots + 1256352636928 \) Copy content Toggle raw display
$23$ \( T^{6} + 196 T^{5} + \cdots + 4069096656896 \) Copy content Toggle raw display
$29$ \( (T + 29)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + 675 T^{5} + \cdots + 766872337410 \) Copy content Toggle raw display
$37$ \( T^{6} + 238 T^{5} + \cdots - 38445998866432 \) Copy content Toggle raw display
$41$ \( T^{6} + 464 T^{5} + \cdots + 64375103462400 \) Copy content Toggle raw display
$43$ \( T^{6} + 579 T^{5} + \cdots - 905147314238 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 154765125165962 \) Copy content Toggle raw display
$53$ \( T^{6} - 515 T^{5} + \cdots - 36699799804014 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 110093808390144 \) Copy content Toggle raw display
$61$ \( T^{6} - 1158 T^{5} + \cdots + 40\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 681378807496704 \) Copy content Toggle raw display
$71$ \( T^{6} - 438 T^{5} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{6} - 262 T^{5} + \cdots + 21\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{6} + 237 T^{5} + \cdots - 10\!\cdots\!74 \) Copy content Toggle raw display
$83$ \( T^{6} + 1288 T^{5} + \cdots + 47\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{6} + 252 T^{5} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} - 380 T^{5} + \cdots + 83\!\cdots\!28 \) Copy content Toggle raw display
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