Properties

Label 232.4.a.d
Level $232$
Weight $4$
Character orbit 232.a
Self dual yes
Analytic conductor $13.688$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + (\beta_{3} + 2) q^{5} + (\beta_{4} + \beta_{2} + 6) q^{7} + (\beta_{4} - \beta_{3} - 3 \beta_1 + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + (\beta_{3} + 2) q^{5} + (\beta_{4} + \beta_{2} + 6) q^{7} + (\beta_{4} - \beta_{3} - 3 \beta_1 + 6) q^{9} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 8) q^{11} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - 5 \beta_1 + 7) q^{13} + ( - \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 18) q^{15} + (\beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 17) q^{17} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 31) q^{19} + ( - \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 11 \beta_1 + 17) q^{21} + (5 \beta_{4} - 8 \beta_{3} + 9 \beta_{2} + 6 \beta_1 + 64) q^{23} + (5 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 6 \beta_1 + 27) q^{25} + (2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 8 \beta_1 + 68) q^{27} + 29 q^{29} + ( - 3 \beta_{4} - 3 \beta_{2} + 13 \beta_1 + 85) q^{31} + (\beta_{4} + 5 \beta_{3} - 7 \beta_{2} + 6 \beta_1 + 20) q^{33} + (7 \beta_{4} + 8 \beta_{3} + 5 \beta_{2} + 26 \beta_1 + 112) q^{35} + ( - 12 \beta_{4} + 14 \beta_{3} - 2 \beta_{2} + 6 \beta_1) q^{37} + (11 \beta_{4} - 20 \beta_{3} + 13 \beta_{2} - 9 \beta_1 + 135) q^{39} + ( - 5 \beta_{4} - 2 \beta_{3} + 18 \beta_{2} + 17 \beta_1 + 15) q^{41} + ( - 22 \beta_{4} + 30 \beta_{3} - 27 \beta_{2} + 14 \beta_1 + 74) q^{43} + ( - 6 \beta_{4} + 16 \beta_{3} - 15 \beta_{2} - 9 \beta_1 - 25) q^{45} + ( - 11 \beta_{4} - 6 \beta_{3} - 24 \beta_{2} - 18 \beta_1 + 144) q^{47} + (12 \beta_{4} + 8 \beta_{3} + 24 \beta_{2} + 36 \beta_1 - 35) q^{49} + (9 \beta_{4} - 24 \beta_{3} + 23 \beta_{2} - 40 \beta_1 + 126) q^{51} + (6 \beta_{4} - 15 \beta_{3} - 3 \beta_{2} - 9 \beta_1 - 21) q^{53} + (\beta_{4} - 11 \beta_{2} - 11 \beta_1 + 149) q^{55} + ( - 2 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} - 7 \beta_1 - 113) q^{57} + (33 \beta_{4} - 8 \beta_{3} + 7 \beta_{2} - 12 \beta_1 + 174) q^{59} + (17 \beta_{4} + 4 \beta_{3} - 20 \beta_{2} - 27 \beta_1 - 125) q^{61} + ( - 12 \beta_{4} - 8 \beta_{3} - 2 \beta_{2} - 42 \beta_1 + 130) q^{63} + ( - 9 \beta_{4} - 11 \beta_{3} - 22 \beta_{2} + 29 \beta_1 - 143) q^{65} + ( - 28 \beta_{4} + 18 \beta_{2} - 50 \beta_1 + 30) q^{67} + (\beta_{4} - 58 \beta_{3} + 42 \beta_{2} - 93 \beta_1 - 193) q^{69} + ( - 8 \beta_{4} + 4 \beta_{3} + 36 \beta_{2} - 2 \beta_1 + 210) q^{71} + (30 \beta_{4} - 2 \beta_{3} + 26 \beta_{2} + 72 \beta_1 - 242) q^{73} + ( - 22 \beta_{4} + 48 \beta_{3} - 36 \beta_{2} - 34 \beta_1 - 78) q^{75} + (19 \beta_{4} - 4 \beta_{3} - 8 \beta_{2} - \beta_1 - 185) q^{77} + ( - 5 \beta_{4} + 10 \beta_{3} - 2 \beta_{2} + 42) q^{79} + ( - 34 \beta_{4} + 16 \beta_{3} + 9 \beta_{2} + 15 \beta_1 - 358) q^{81} + (33 \beta_{4} + 40 \beta_{3} - 19 \beta_{2} + 46 \beta_1 + 68) q^{83} + ( - 13 \beta_{4} + 14 \beta_{3} - 8 \beta_{2} + 31 \beta_1 - 341) q^{85} + ( - 29 \beta_1 + 29) q^{87} + ( - 13 \beta_{4} - 26 \beta_{3} - 36 \beta_{2} + 71 \beta_1 - 367) q^{89} + (19 \beta_{4} - 56 \beta_{3} - 5 \beta_{2} - 96 \beta_1 - 214) q^{91} + ( - 10 \beta_{4} + 25 \beta_{3} + 6 \beta_{2} - 44 \beta_1 - 364) q^{93} + (16 \beta_{4} + 6 \beta_{3} + 29 \beta_{2} + 47 \beta_1 + 189) q^{95} + (15 \beta_{4} - 6 \beta_{3} - 24 \beta_{2} + 27 \beta_1 - 515) q^{97} + (34 \beta_{4} + 14 \beta_{3} - 13 \beta_{2} - 3 \beta_1 - 313) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{3} + 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 4 q^{3} + 10 q^{5} + 32 q^{7} + 29 q^{9} + 36 q^{11} + 26 q^{13} + 88 q^{15} + 82 q^{17} + 156 q^{19} + 72 q^{21} + 336 q^{23} + 151 q^{25} + 352 q^{27} + 145 q^{29} + 432 q^{31} + 108 q^{33} + 600 q^{35} - 18 q^{37} + 688 q^{39} + 82 q^{41} + 340 q^{43} - 146 q^{45} + 680 q^{47} - 115 q^{49} + 608 q^{51} - 102 q^{53} + 736 q^{55} - 576 q^{57} + 924 q^{59} - 618 q^{61} + 584 q^{63} - 704 q^{65} + 44 q^{67} - 1056 q^{69} + 1032 q^{71} - 1078 q^{73} - 468 q^{75} - 888 q^{77} + 200 q^{79} - 1843 q^{81} + 452 q^{83} - 1700 q^{85} + 116 q^{87} - 1790 q^{89} - 1128 q^{91} - 1884 q^{93} + 1024 q^{95} - 2518 q^{97} - 1500 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} - 5\nu^{3} + 23\nu^{2} + 64\nu - 90 ) / 19 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} - 5\nu^{3} + 23\nu^{2} + 140\nu - 109 ) / 19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{4} - 6\nu^{3} + 134\nu^{2} - 60\nu - 203 ) / 19 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{4} - 7\nu^{3} + 283\nu^{2} - 184\nu - 829 ) / 19 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - 2\beta_{3} + \beta _1 + 27 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} + 4\beta_{3} + 10\beta_{2} - 21\beta _1 - 43 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14\beta_{4} - 33\beta_{3} - 9\beta_{2} + 29\beta _1 + 344 \) 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
−6.05843
2.92646
1.52813
−1.69859
4.30242
0 −5.90002 0 −14.9988 0 −6.82211 0 7.81028 0
1.2 0 −4.03215 0 15.2584 0 33.3511 0 −10.7418 0
1.3 0 −1.01127 0 0.397400 0 −14.4223 0 −25.9773 0
1.4 0 7.11424 0 16.3850 0 5.74609 0 23.6124 0
1.5 0 7.82921 0 −7.04208 0 14.1473 0 34.2965 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.d 5
3.b odd 2 1 2088.4.a.f 5
4.b odd 2 1 464.4.a.m 5
8.b even 2 1 1856.4.a.z 5
8.d odd 2 1 1856.4.a.ba 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.a.d 5 1.a even 1 1 trivial
464.4.a.m 5 4.b odd 2 1
1856.4.a.z 5 8.b even 2 1
1856.4.a.ba 5 8.d odd 2 1
2088.4.a.f 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 4T_{3}^{4} - 74T_{3}^{3} + 128T_{3}^{2} + 1525T_{3} + 1340 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(232))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 4 T^{4} - 74 T^{3} + \cdots + 1340 \) Copy content Toggle raw display
$5$ \( T^{5} - 10 T^{4} - 338 T^{3} + \cdots - 10494 \) Copy content Toggle raw display
$7$ \( T^{5} - 32 T^{4} - 288 T^{3} + \cdots - 266752 \) Copy content Toggle raw display
$11$ \( T^{5} - 36 T^{4} - 1346 T^{3} + \cdots + 1741860 \) Copy content Toggle raw display
$13$ \( T^{5} - 26 T^{4} + \cdots + 105404410 \) Copy content Toggle raw display
$17$ \( T^{5} - 82 T^{4} - 2776 T^{3} + \cdots - 49184 \) Copy content Toggle raw display
$19$ \( T^{5} - 156 T^{4} + 4216 T^{3} + \cdots - 1820736 \) Copy content Toggle raw display
$23$ \( T^{5} - 336 T^{4} + \cdots + 7489438848 \) Copy content Toggle raw display
$29$ \( (T - 29)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} - 432 T^{4} + \cdots + 445071048 \) Copy content Toggle raw display
$37$ \( T^{5} + 18 T^{4} + \cdots - 15294686720 \) Copy content Toggle raw display
$41$ \( T^{5} - 82 T^{4} + \cdots - 731491061376 \) Copy content Toggle raw display
$43$ \( T^{5} - 340 T^{4} + \cdots - 571309913052 \) Copy content Toggle raw display
$47$ \( T^{5} - 680 T^{4} + \cdots + 2559413417896 \) Copy content Toggle raw display
$53$ \( T^{5} + 102 T^{4} + \cdots + 103910584482 \) Copy content Toggle raw display
$59$ \( T^{5} - 924 T^{4} + \cdots + 16799541984192 \) Copy content Toggle raw display
$61$ \( T^{5} + 618 T^{4} + \cdots + 2366067286944 \) Copy content Toggle raw display
$67$ \( T^{5} - 44 T^{4} + \cdots + 29804817076224 \) Copy content Toggle raw display
$71$ \( T^{5} - 1032 T^{4} + \cdots - 50004302698368 \) Copy content Toggle raw display
$73$ \( T^{5} + 1078 T^{4} + \cdots - 4755790305792 \) Copy content Toggle raw display
$79$ \( T^{5} - 200 T^{4} + \cdots - 13404287016 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 181825403631808 \) Copy content Toggle raw display
$89$ \( T^{5} + 1790 T^{4} + \cdots - 79946879886976 \) Copy content Toggle raw display
$97$ \( T^{5} + 2518 T^{4} + \cdots - 41073987679360 \) Copy content Toggle raw display
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