Properties

Label 1232.4.a.s
Level $1232$
Weight $4$
Character orbit 1232.a
Self dual yes
Analytic conductor $72.690$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.6903531271\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.522072.1
Defining polynomial: \( x^{4} - x^{3} - 12x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 4) q^{3} + (2 \beta_{2} - \beta_1 + 4) q^{5} + 7 q^{7} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 4) q^{3} + (2 \beta_{2} - \beta_1 + 4) q^{5} + 7 q^{7} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 18) q^{9} + 11 q^{11} + ( - 3 \beta_{2} - 10 \beta_1 + 18) q^{13} + ( - 6 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 70) q^{15} + ( - 2 \beta_{3} - 5 \beta_{2} + 11 \beta_1 - 6) q^{17} + ( - 4 \beta_{3} + 4 \beta_{2} + 11 \beta_1 - 66) q^{19} + ( - 7 \beta_{2} - 28) q^{21} + (2 \beta_{3} - 18 \beta_{2} + 4 \beta_1 - 14) q^{23} + (14 \beta_{3} - 4 \beta_{2} - 12 \beta_1 + 17) q^{25} + ( - 18 \beta_{3} - 12 \beta_{2} - 36 \beta_1 - 86) q^{27} + ( - 6 \beta_{3} + 14 \beta_{2} - 2 \beta_1 - 88) q^{29} + (16 \beta_{3} + 9 \beta_{2} + 9 \beta_1 + 6) q^{31} + ( - 11 \beta_{2} - 44) q^{33} + (14 \beta_{2} - 7 \beta_1 + 28) q^{35} + ( - 23 \beta_{3} - 16 \beta_{2} - 9 \beta_1 + 29) q^{37} + (9 \beta_{3} + 2 \beta_{2} + 89 \beta_1 + 55) q^{39} + ( - 14 \beta_{3} - 23 \beta_{2} + 37 \beta_1 - 10) q^{41} + ( - 37 \beta_{3} - 2 \beta_{2} + 37 \beta_1 - 103) q^{43} + (18 \beta_{3} + 42 \beta_{2} + 17 \beta_1 + 210) q^{45} + ( - 8 \beta_{3} + 39 \beta_{2} + 7 \beta_1 + 18) q^{47} + 49 q^{49} + (19 \beta_{3} - 6 \beta_{2} - 73 \beta_1 + 121) q^{51} + (7 \beta_{3} - 8 \beta_{2} + 3 \beta_1 + 147) q^{53} + (22 \beta_{2} - 11 \beta_1 + 44) q^{55} + ( - 4 \beta_{3} + 64 \beta_{2} - 100 \beta_1 + 96) q^{57} + (74 \beta_{3} - 3 \beta_{2} - 50 \beta_1 + 46) q^{59} + (48 \beta_{3} + 29 \beta_{2} + 72 \beta_1 - 86) q^{61} + (21 \beta_{3} + 28 \beta_{2} + 21 \beta_1 + 126) q^{63} + (2 \beta_{3} + 54 \beta_{2} - 144 \beta_1 + 90) q^{65} + ( - 2 \beta_{3} + 62 \beta_{2} - 122 \beta_1 - 242) q^{67} + (50 \beta_{3} - 4 \beta_{2} + 22 \beta_1 + 566) q^{69} + ( - 2 \beta_{3} + 26 \beta_{2} - 94 \beta_1 - 386) q^{71} + ( - 78 \beta_{3} - 47 \beta_{2} + 139 \beta_1 + 310) q^{73} + ( - 16 \beta_{3} - 63 \beta_{2} + 108 \beta_1 + 124) q^{75} + 77 q^{77} + (71 \beta_{3} + 6 \beta_{2} + 43 \beta_1 + 277) q^{79} + ( - 9 \beta_{3} + 140 \beta_{2} + 243 \beta_1 + 314) q^{81} + (34 \beta_{3} + 76 \beta_{2} + 9 \beta_1 - 96) q^{83} + ( - 52 \beta_{3} - 2 \beta_{2} + 134 \beta_1 - 476) q^{85} + ( - 30 \beta_{3} + 122 \beta_{2} - 26 \beta_1 - 58) q^{87} + ( - 74 \beta_{3} - 48 \beta_{2} - 30 \beta_1 - 740) q^{89} + ( - 21 \beta_{2} - 70 \beta_1 + 126) q^{91} + ( - 59 \beta_{3} - 104 \beta_{2} - 99 \beta_1 - 289) q^{93} + (2 \beta_{3} - 196 \beta_{2} + 218 \beta_1 - 214) q^{95} + (104 \beta_{3} + 154 \beta_{2} - 72 \beta_1 - 102) q^{97} + (33 \beta_{3} + 44 \beta_{2} + 33 \beta_1 + 198) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35} + 84 q^{37} + 412 q^{39} + 52 q^{41} - 408 q^{43} + 826 q^{45} - 8 q^{47} + 196 q^{49} + 388 q^{51} + 624 q^{53} + 110 q^{55} + 48 q^{57} + 238 q^{59} - 162 q^{61} + 532 q^{63} - 32 q^{65} - 1340 q^{67} + 2416 q^{69} - 1788 q^{71} + 1456 q^{73} + 806 q^{75} + 308 q^{77} + 1324 q^{79} + 1444 q^{81} - 450 q^{83} - 1736 q^{85} - 588 q^{87} - 3072 q^{89} + 406 q^{91} - 1264 q^{93} - 24 q^{95} - 652 q^{97} + 836 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 12x^{2} + 5x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 12\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{3} + 2\nu^{2} + 24\nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 6\beta _1 + 3 \) 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
3.79597
−3.20317
−0.148103
0.555307
0 −10.1459 0 8.69995 0 7.00000 0 75.9402 0
1.2 0 −6.57251 0 15.5514 0 7.00000 0 16.1978 0
1.3 0 −2.77399 0 1.84418 0 7.00000 0 −19.3050 0
1.4 0 5.49244 0 −16.0955 0 7.00000 0 3.16692 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 1232.4.a.s 4
4.b odd 2 1 77.4.a.d 4
12.b even 2 1 693.4.a.l 4
20.d odd 2 1 1925.4.a.p 4
28.d even 2 1 539.4.a.g 4
44.c even 2 1 847.4.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.d 4 4.b odd 2 1
539.4.a.g 4 28.d even 2 1
693.4.a.l 4 12.b even 2 1
847.4.a.d 4 44.c even 2 1
1232.4.a.s 4 1.a even 1 1 trivial
1925.4.a.p 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3}^{4} + 14T_{3}^{3} + 6T_{3}^{2} - 436T_{3} - 1016 \) Copy content Toggle raw display
\( T_{5}^{4} - 10T_{5}^{3} - 240T_{5}^{2} + 2648T_{5} - 4016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 14 T^{3} + 6 T^{2} + \cdots - 1016 \) Copy content Toggle raw display
$5$ \( T^{4} - 10 T^{3} - 240 T^{2} + \cdots - 4016 \) Copy content Toggle raw display
$7$ \( (T - 7)^{4} \) Copy content Toggle raw display
$11$ \( (T - 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 58 T^{3} - 4926 T^{2} + \cdots - 4947656 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} - 6186 T^{2} + \cdots - 2705024 \) Copy content Toggle raw display
$19$ \( T^{4} + 258 T^{3} + \cdots - 14423904 \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} - 22392 T^{2} + \cdots - 17449856 \) Copy content Toggle raw display
$29$ \( T^{4} + 396 T^{3} + \cdots + 22336464 \) Copy content Toggle raw display
$31$ \( T^{4} - 56 T^{3} - 31890 T^{2} + \cdots - 11250248 \) Copy content Toggle raw display
$37$ \( T^{4} - 84 T^{3} - 63516 T^{2} + \cdots + 11157312 \) Copy content Toggle raw display
$41$ \( T^{4} - 52 T^{3} + \cdots - 659233664 \) Copy content Toggle raw display
$43$ \( T^{4} + 408 T^{3} + \cdots + 1210397376 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} - 121650 T^{2} + \cdots - 318931592 \) Copy content Toggle raw display
$53$ \( T^{4} - 624 T^{3} + \cdots + 403923072 \) Copy content Toggle raw display
$59$ \( T^{4} - 238 T^{3} + \cdots + 17599820728 \) Copy content Toggle raw display
$61$ \( T^{4} + 162 T^{3} + \cdots + 6668930664 \) Copy content Toggle raw display
$67$ \( T^{4} + 1340 T^{3} + \cdots - 140865466496 \) Copy content Toggle raw display
$71$ \( T^{4} + 1788 T^{3} + \cdots - 72982082688 \) Copy content Toggle raw display
$73$ \( T^{4} - 1456 T^{3} + \cdots - 322052228384 \) Copy content Toggle raw display
$79$ \( T^{4} - 1324 T^{3} + \cdots - 59537293568 \) Copy content Toggle raw display
$83$ \( T^{4} + 450 T^{3} + \cdots + 21951092064 \) Copy content Toggle raw display
$89$ \( T^{4} + 3072 T^{3} + \cdots - 109303561968 \) Copy content Toggle raw display
$97$ \( T^{4} + 652 T^{3} + \cdots + 868634650768 \) Copy content Toggle raw display
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