Properties

Label 1232.4.a.v
Level $1232$
Weight $4$
Character orbit 1232.a
Self dual yes
Analytic conductor $72.690$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 43 x^{2} - 11 x + 222\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 308)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} + \beta_{2} q^{5} + 7 q^{7} + ( -5 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} + \beta_{2} q^{5} + 7 q^{7} + ( -5 + \beta_{2} ) q^{9} -11 q^{11} + ( 24 + 2 \beta_{1} - \beta_{3} ) q^{13} + ( 11 - 11 \beta_{1} - \beta_{3} ) q^{15} + ( 13 + 11 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{17} + ( -29 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{19} + ( 7 - 7 \beta_{1} ) q^{21} + ( -55 - 13 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{23} + ( 106 - 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{25} + ( -21 + 21 \beta_{1} - \beta_{3} ) q^{27} + ( 68 - 10 \beta_{1} + 6 \beta_{2} ) q^{29} + ( -10 - 4 \beta_{1} + 11 \beta_{2} - \beta_{3} ) q^{31} + ( -11 + 11 \beta_{1} ) q^{33} + 7 \beta_{2} q^{35} + ( 98 - 42 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{37} + ( -29 - 29 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{39} + ( 91 - 15 \beta_{1} + 15 \beta_{2} + 4 \beta_{3} ) q^{41} + ( -7 - 11 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{43} + ( 231 - 3 \beta_{1} - 9 \beta_{2} - 3 \beta_{3} ) q^{45} + ( -197 + 67 \beta_{1} + \beta_{2} ) q^{47} + 49 q^{49} + ( -185 - 29 \beta_{1} - 25 \beta_{2} + 5 \beta_{3} ) q^{51} + ( 195 - 35 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{53} -11 \beta_{2} q^{55} + ( 36 - 6 \beta_{2} ) q^{57} + ( -203 + 15 \beta_{1} + 10 \beta_{2} + 8 \beta_{3} ) q^{59} + ( 272 - 66 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{61} + ( -35 + 7 \beta_{2} ) q^{63} + ( 74 - 50 \beta_{1} + 50 \beta_{2} + 8 \beta_{3} ) q^{65} + ( 259 - 51 \beta_{1} + 10 \beta_{2} - 9 \beta_{3} ) q^{67} + ( 251 + 49 \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{69} + ( 287 - 115 \beta_{1} - 22 \beta_{2} - \beta_{3} ) q^{71} + ( 413 - 53 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{73} + ( 92 - 68 \beta_{1} + 24 \beta_{2} - 5 \beta_{3} ) q^{75} -77 q^{77} + ( 33 + 29 \beta_{1} + 17 \beta_{2} - 9 \beta_{3} ) q^{79} + ( -338 - 3 \beta_{1} - 41 \beta_{2} - 3 \beta_{3} ) q^{81} + ( -89 + 47 \beta_{1} - 15 \beta_{2} + 21 \beta_{3} ) q^{83} + ( -82 + 262 \beta_{1} - 28 \beta_{2} - 4 \beta_{3} ) q^{85} + ( 344 - 124 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} ) q^{87} + ( 5 + 33 \beta_{1} - 34 \beta_{2} - 11 \beta_{3} ) q^{89} + ( 168 + 14 \beta_{1} - 7 \beta_{3} ) q^{91} + ( 184 - 110 \beta_{1} + 11 \beta_{2} - 14 \beta_{3} ) q^{93} + ( 608 + 52 \beta_{1} - 66 \beta_{2} - 16 \beta_{3} ) q^{95} + ( -83 - 179 \beta_{1} - 44 \beta_{2} + 3 \beta_{3} ) q^{97} + ( 55 - 11 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} + q^{5} + 28 q^{7} - 19 q^{9} + O(q^{10}) \) \( 4 q + 3 q^{3} + q^{5} + 28 q^{7} - 19 q^{9} - 44 q^{11} + 98 q^{13} + 33 q^{15} + 64 q^{17} - 114 q^{19} + 21 q^{21} - 231 q^{23} + 417 q^{25} - 63 q^{27} + 268 q^{29} - 33 q^{31} - 33 q^{33} + 7 q^{35} + 357 q^{37} - 140 q^{39} + 364 q^{41} - 44 q^{43} + 912 q^{45} - 720 q^{47} + 196 q^{49} - 794 q^{51} + 740 q^{53} - 11 q^{55} + 138 q^{57} - 787 q^{59} + 1020 q^{61} - 133 q^{63} + 296 q^{65} + 995 q^{67} + 1059 q^{69} + 1011 q^{71} + 1592 q^{73} + 324 q^{75} - 308 q^{77} + 178 q^{79} - 1396 q^{81} - 324 q^{83} - 94 q^{85} + 1262 q^{87} + 19 q^{89} + 686 q^{91} + 637 q^{93} + 2418 q^{95} - 555 q^{97} + 209 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 43 x^{2} - 11 x + 222\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 21 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 3 \nu^{2} - 30 \nu + 32 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 21\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 3 \beta_{2} + 36 \beta_{1} + 31\)

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
6.83284
2.21425
−2.79239
−5.25471
0 −5.83284 0 12.0220 0 7.00000 0 7.02203 0
1.2 0 −1.21425 0 −20.5256 0 7.00000 0 −25.5256 0
1.3 0 3.79239 0 −7.61779 0 7.00000 0 −12.6178 0
1.4 0 6.25471 0 17.1213 0 7.00000 0 12.1213 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.v 4
4.b odd 2 1 308.4.a.d 4
28.d even 2 1 2156.4.a.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.4.a.d 4 4.b odd 2 1
1232.4.a.v 4 1.a even 1 1 trivial
2156.4.a.f 4 28.d even 2 1

Hecke kernels

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

\( T_{3}^{4} - 3 T_{3}^{3} - 40 T_{3}^{2} + 96 T_{3} + 168 \)
\( T_{5}^{4} - T_{5}^{3} - 458 T_{5}^{2} + 1236 T_{5} + 32184 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 168 + 96 T - 40 T^{2} - 3 T^{3} + T^{4} \)
$5$ \( 32184 + 1236 T - 458 T^{2} - T^{3} + T^{4} \)
$7$ \( ( -7 + T )^{4} \)
$11$ \( ( 11 + T )^{4} \)
$13$ \( -5715264 + 266816 T - 628 T^{2} - 98 T^{3} + T^{4} \)
$17$ \( 80892704 + 710584 T - 18168 T^{2} - 64 T^{3} + T^{4} \)
$19$ \( 186624 - 19872 T - 2400 T^{2} + 114 T^{3} + T^{4} \)
$23$ \( 6309184 - 758784 T + 6184 T^{2} + 231 T^{3} + T^{4} \)
$29$ \( -62229648 + 1579504 T + 4160 T^{2} - 268 T^{3} + T^{4} \)
$31$ \( 668400408 - 2725976 T - 64680 T^{2} + 33 T^{3} + T^{4} \)
$37$ \( 2194340904 + 20514596 T - 75222 T^{2} - 357 T^{3} + T^{4} \)
$41$ \( -6396705696 + 61971400 T - 116320 T^{2} - 364 T^{3} + T^{4} \)
$43$ \( -221937664 - 7549952 T - 56352 T^{2} + 44 T^{3} + T^{4} \)
$47$ \( -3932145984 - 59864728 T + 1392 T^{2} + 720 T^{3} + T^{4} \)
$53$ \( -1987381584 + 11231184 T + 119856 T^{2} - 740 T^{3} + T^{4} \)
$59$ \( 4667157144 - 70341120 T - 41280 T^{2} + 787 T^{3} + T^{4} \)
$61$ \( 1375017408 + 58919912 T + 87672 T^{2} - 1020 T^{3} + T^{4} \)
$67$ \( 2114400384 + 89682672 T - 137780 T^{2} - 995 T^{3} + T^{4} \)
$71$ \( -156603934656 + 601638208 T - 322584 T^{2} - 1011 T^{3} + T^{4} \)
$73$ \( -8143492032 - 93000408 T + 772360 T^{2} - 1592 T^{3} + T^{4} \)
$79$ \( -14872658688 + 209727456 T - 525072 T^{2} - 178 T^{3} + T^{4} \)
$83$ \( 163701302784 - 586180416 T - 2002096 T^{2} + 324 T^{3} + T^{4} \)
$89$ \( -58144933944 - 462550788 T - 983118 T^{2} - 19 T^{3} + T^{4} \)
$97$ \( 5739946776 + 519707012 T - 2009298 T^{2} + 555 T^{3} + T^{4} \)
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