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: \(x^{4} - x^{3} - 12 x^{2} + 5 x + 1\)
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.

\(f(q)\) \(=\) \( q + ( -4 - \beta_{2} ) q^{3} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{5} + 7 q^{7} + ( 18 + 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -4 - \beta_{2} ) q^{3} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{5} + 7 q^{7} + ( 18 + 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{9} + 11 q^{11} + ( 18 - 10 \beta_{1} - 3 \beta_{2} ) q^{13} + ( -70 + 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{15} + ( -6 + 11 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -66 + 11 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{19} + ( -28 - 7 \beta_{2} ) q^{21} + ( -14 + 4 \beta_{1} - 18 \beta_{2} + 2 \beta_{3} ) q^{23} + ( 17 - 12 \beta_{1} - 4 \beta_{2} + 14 \beta_{3} ) q^{25} + ( -86 - 36 \beta_{1} - 12 \beta_{2} - 18 \beta_{3} ) q^{27} + ( -88 - 2 \beta_{1} + 14 \beta_{2} - 6 \beta_{3} ) q^{29} + ( 6 + 9 \beta_{1} + 9 \beta_{2} + 16 \beta_{3} ) q^{31} + ( -44 - 11 \beta_{2} ) q^{33} + ( 28 - 7 \beta_{1} + 14 \beta_{2} ) q^{35} + ( 29 - 9 \beta_{1} - 16 \beta_{2} - 23 \beta_{3} ) q^{37} + ( 55 + 89 \beta_{1} + 2 \beta_{2} + 9 \beta_{3} ) q^{39} + ( -10 + 37 \beta_{1} - 23 \beta_{2} - 14 \beta_{3} ) q^{41} + ( -103 + 37 \beta_{1} - 2 \beta_{2} - 37 \beta_{3} ) q^{43} + ( 210 + 17 \beta_{1} + 42 \beta_{2} + 18 \beta_{3} ) q^{45} + ( 18 + 7 \beta_{1} + 39 \beta_{2} - 8 \beta_{3} ) q^{47} + 49 q^{49} + ( 121 - 73 \beta_{1} - 6 \beta_{2} + 19 \beta_{3} ) q^{51} + ( 147 + 3 \beta_{1} - 8 \beta_{2} + 7 \beta_{3} ) q^{53} + ( 44 - 11 \beta_{1} + 22 \beta_{2} ) q^{55} + ( 96 - 100 \beta_{1} + 64 \beta_{2} - 4 \beta_{3} ) q^{57} + ( 46 - 50 \beta_{1} - 3 \beta_{2} + 74 \beta_{3} ) q^{59} + ( -86 + 72 \beta_{1} + 29 \beta_{2} + 48 \beta_{3} ) q^{61} + ( 126 + 21 \beta_{1} + 28 \beta_{2} + 21 \beta_{3} ) q^{63} + ( 90 - 144 \beta_{1} + 54 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -242 - 122 \beta_{1} + 62 \beta_{2} - 2 \beta_{3} ) q^{67} + ( 566 + 22 \beta_{1} - 4 \beta_{2} + 50 \beta_{3} ) q^{69} + ( -386 - 94 \beta_{1} + 26 \beta_{2} - 2 \beta_{3} ) q^{71} + ( 310 + 139 \beta_{1} - 47 \beta_{2} - 78 \beta_{3} ) q^{73} + ( 124 + 108 \beta_{1} - 63 \beta_{2} - 16 \beta_{3} ) q^{75} + 77 q^{77} + ( 277 + 43 \beta_{1} + 6 \beta_{2} + 71 \beta_{3} ) q^{79} + ( 314 + 243 \beta_{1} + 140 \beta_{2} - 9 \beta_{3} ) q^{81} + ( -96 + 9 \beta_{1} + 76 \beta_{2} + 34 \beta_{3} ) q^{83} + ( -476 + 134 \beta_{1} - 2 \beta_{2} - 52 \beta_{3} ) q^{85} + ( -58 - 26 \beta_{1} + 122 \beta_{2} - 30 \beta_{3} ) q^{87} + ( -740 - 30 \beta_{1} - 48 \beta_{2} - 74 \beta_{3} ) q^{89} + ( 126 - 70 \beta_{1} - 21 \beta_{2} ) q^{91} + ( -289 - 99 \beta_{1} - 104 \beta_{2} - 59 \beta_{3} ) q^{93} + ( -214 + 218 \beta_{1} - 196 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -102 - 72 \beta_{1} + 154 \beta_{2} + 104 \beta_{3} ) q^{97} + ( 198 + 33 \beta_{1} + 44 \beta_{2} + 33 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + O(q^{10}) \) \( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35} + 84 q^{37} + 412 q^{39} + 52 q^{41} - 408 q^{43} + 826 q^{45} - 8 q^{47} + 196 q^{49} + 388 q^{51} + 624 q^{53} + 110 q^{55} + 48 q^{57} + 238 q^{59} - 162 q^{61} + 532 q^{63} - 32 q^{65} - 1340 q^{67} + 2416 q^{69} - 1788 q^{71} + 1456 q^{73} + 806 q^{75} + 308 q^{77} + 1324 q^{79} + 1444 q^{81} - 450 q^{83} - 1736 q^{85} - 588 q^{87} - 3072 q^{89} + 406 q^{91} - 1264 q^{93} - 24 q^{95} - 652 q^{97} + 836 q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 12 \nu - 3 \)
\(\beta_{3}\)\(=\)\( -2 \nu^{3} + 2 \nu^{2} + 24 \nu - 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} + 13\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 6 \beta_{1} + 3\)

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( -1016 - 436 T + 6 T^{2} + 14 T^{3} + T^{4} \)
$5$ \( -4016 + 2648 T - 240 T^{2} - 10 T^{3} + T^{4} \)
$7$ \( ( -7 + T )^{4} \)
$11$ \( ( -11 + T )^{4} \)
$13$ \( -4947656 + 342428 T - 4926 T^{2} - 58 T^{3} + T^{4} \)
$17$ \( -2705024 + 257456 T - 6186 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( -14423904 - 161712 T + 16188 T^{2} + 258 T^{3} + T^{4} \)
$23$ \( -17449856 - 1550464 T - 22392 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( 22336464 + 1623888 T + 40872 T^{2} + 396 T^{3} + T^{4} \)
$31$ \( -11250248 + 1490896 T - 31890 T^{2} - 56 T^{3} + T^{4} \)
$37$ \( 11157312 + 647520 T - 63516 T^{2} - 84 T^{3} + T^{4} \)
$41$ \( -659233664 + 14752640 T - 83706 T^{2} - 52 T^{3} + T^{4} \)
$43$ \( 1210397376 - 31826832 T - 90804 T^{2} + 408 T^{3} + T^{4} \)
$47$ \( -318931592 + 12296144 T - 121650 T^{2} + 8 T^{3} + T^{4} \)
$53$ \( 403923072 - 12426816 T + 135948 T^{2} - 624 T^{3} + T^{4} \)
$59$ \( 17599820728 + 150381524 T - 501426 T^{2} - 238 T^{3} + T^{4} \)
$61$ \( 6668930664 - 166886028 T - 587718 T^{2} + 162 T^{3} + T^{4} \)
$67$ \( -140865466496 - 658322080 T - 172104 T^{2} + 1340 T^{3} + T^{4} \)
$71$ \( -72982082688 - 109888032 T + 761784 T^{2} + 1788 T^{3} + T^{4} \)
$73$ \( -322052228384 + 1133869160 T - 388842 T^{2} - 1456 T^{3} + T^{4} \)
$79$ \( -59537293568 + 346623968 T + 71820 T^{2} - 1324 T^{3} + T^{4} \)
$83$ \( 21951092064 - 140617200 T - 383316 T^{2} + 450 T^{3} + T^{4} \)
$89$ \( -109303561968 + 704256000 T + 2868408 T^{2} + 3072 T^{3} + T^{4} \)
$97$ \( 868634650768 - 628130672 T - 1896360 T^{2} + 652 T^{3} + T^{4} \)
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