Properties

Label 1216.4.a.v
Level $1216$
Weight $4$
Character orbit 1216.a
Self dual yes
Analytic conductor $71.746$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.7463225670\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.35529.1
Defining polynomial: \(x^{3} - x^{2} - 52 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{2} ) q^{3} + ( -3 - \beta_{2} ) q^{5} + ( -15 + \beta_{1} + 2 \beta_{2} ) q^{7} + ( 19 + 3 \beta_{1} - 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} ) q^{3} + ( -3 - \beta_{2} ) q^{5} + ( -15 + \beta_{1} + 2 \beta_{2} ) q^{7} + ( 19 + 3 \beta_{1} - 3 \beta_{2} ) q^{9} + ( 29 - 8 \beta_{1} - 3 \beta_{2} ) q^{11} + ( 2 + 5 \beta_{1} - 7 \beta_{2} ) q^{13} + ( -30 + 2 \beta_{2} ) q^{15} + ( 23 + 13 \beta_{1} - 6 \beta_{2} ) q^{17} + 19 q^{19} + ( 76 - 15 \beta_{1} - 23 \beta_{2} ) q^{21} + ( -38 + 11 \beta_{1} + 9 \beta_{2} ) q^{23} + ( -68 - 6 \beta_{1} + 3 \beta_{2} ) q^{25} + ( -42 + 19 \beta_{1} + 13 \beta_{2} ) q^{27} + ( 24 + 21 \beta_{1} + 23 \beta_{2} ) q^{29} + ( 44 - 16 \beta_{1} + 12 \beta_{2} ) q^{31} + ( -218 - 10 \beta_{1} + 28 \beta_{2} ) q^{33} + ( -33 + 6 \beta_{1} + 17 \beta_{2} ) q^{35} + ( 150 + 16 \beta_{1} + 36 \beta_{2} ) q^{37} + ( -130 + 53 \beta_{1} + 47 \beta_{2} ) q^{39} + ( -58 - 14 \beta_{1} - 26 \beta_{2} ) q^{41} + ( -11 + 22 \beta_{1} + 27 \beta_{2} ) q^{43} + ( 141 - 36 \beta_{1} - 13 \beta_{2} ) q^{45} + ( -59 + 14 \beta_{1} - 41 \beta_{2} ) q^{47} + ( 36 - 39 \beta_{1} - 76 \beta_{2} ) q^{49} + ( 28 + 119 \beta_{1} + 79 \beta_{2} ) q^{51} + ( -46 - 59 \beta_{1} - 31 \beta_{2} ) q^{53} + ( -87 + 30 \beta_{1} - 45 \beta_{2} ) q^{55} + ( 19 \beta_{1} + 19 \beta_{2} ) q^{57} + ( 452 + 25 \beta_{1} + 37 \beta_{2} ) q^{59} + ( 109 + 78 \beta_{1} + 13 \beta_{2} ) q^{61} + ( -525 + 28 \beta_{1} + 107 \beta_{2} ) q^{63} + ( 420 - 72 \beta_{1} + 8 \beta_{2} ) q^{65} + ( 322 - 23 \beta_{1} - 61 \beta_{2} ) q^{67} + ( 446 + \beta_{1} - 61 \beta_{2} ) q^{69} + ( -334 - 50 \beta_{1} + 88 \beta_{2} ) q^{71} + ( -249 + 167 \beta_{1} + 84 \beta_{2} ) q^{73} + ( -6 - 113 \beta_{1} - 95 \beta_{2} ) q^{75} + ( -653 + 104 \beta_{1} + 127 \beta_{2} ) q^{77} + ( 444 + 70 \beta_{1} + 10 \beta_{2} ) q^{79} + ( 181 - 48 \beta_{1} + 12 \beta_{2} ) q^{81} + ( 592 + 26 \beta_{1} + 54 \beta_{2} ) q^{83} + ( 453 - 114 \beta_{1} + 3 \beta_{2} ) q^{85} + ( 1026 + 81 \beta_{1} - 49 \beta_{2} ) q^{87} + ( 632 + 70 \beta_{1} + 68 \beta_{2} ) q^{89} + ( -586 + 35 \beta_{1} + 165 \beta_{2} ) q^{91} + ( 104 - 88 \beta_{1} - 48 \beta_{2} ) q^{93} + ( -57 - 19 \beta_{2} ) q^{95} + ( -16 + 96 \beta_{1} + 54 \beta_{2} ) q^{97} + ( -103 - 146 \beta_{1} - 297 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} - 9q^{5} - 44q^{7} + 60q^{9} + O(q^{10}) \) \( 3q + q^{3} - 9q^{5} - 44q^{7} + 60q^{9} + 79q^{11} + 11q^{13} - 90q^{15} + 82q^{17} + 57q^{19} + 213q^{21} - 103q^{23} - 210q^{25} - 107q^{27} + 93q^{29} + 116q^{31} - 664q^{33} - 93q^{35} + 466q^{37} - 337q^{39} - 188q^{41} - 11q^{43} + 387q^{45} - 163q^{47} + 69q^{49} + 203q^{51} - 197q^{53} - 231q^{55} + 19q^{57} + 1381q^{59} + 405q^{61} - 1547q^{63} + 1188q^{65} + 943q^{67} + 1339q^{69} - 1052q^{71} - 580q^{73} - 131q^{75} - 1855q^{77} + 1402q^{79} + 495q^{81} + 1802q^{83} + 1245q^{85} + 3159q^{87} + 1966q^{89} - 1723q^{91} + 224q^{93} - 171q^{95} + 48q^{97} - 455q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 52 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 3 \nu - 34 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(4 \beta_{2} + 3 \beta_{1} + 34\)

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
0.0768183
−6.76918
7.69236
0 −8.47932 0 5.55614 0 −32.0355 0 44.8989 0
1.2 0 1.26314 0 −11.0323 0 −5.70454 0 −25.4045 0
1.3 0 8.21618 0 −3.52382 0 −6.26000 0 40.5056 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 1216.4.a.v 3
4.b odd 2 1 1216.4.a.t 3
8.b even 2 1 304.4.a.h 3
8.d odd 2 1 76.4.a.b 3
24.f even 2 1 684.4.a.i 3
40.e odd 2 1 1900.4.a.c 3
40.k even 4 2 1900.4.c.c 6
152.b even 2 1 1444.4.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.a.b 3 8.d odd 2 1
304.4.a.h 3 8.b even 2 1
684.4.a.i 3 24.f even 2 1
1216.4.a.t 3 4.b odd 2 1
1216.4.a.v 3 1.a even 1 1 trivial
1444.4.a.e 3 152.b even 2 1
1900.4.a.c 3 40.e odd 2 1
1900.4.c.c 6 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\):

\( T_{3}^{3} - T_{3}^{2} - 70 T_{3} + 88 \)
\( T_{5}^{3} + 9 T_{5}^{2} - 42 T_{5} - 216 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 88 - 70 T - T^{2} + T^{3} \)
$5$ \( -216 - 42 T + 9 T^{2} + T^{3} \)
$7$ \( 1144 + 419 T + 44 T^{2} + T^{3} \)
$11$ \( 108888 - 666 T - 79 T^{2} + T^{3} \)
$13$ \( 201816 - 6434 T - 11 T^{2} + T^{3} \)
$17$ \( 1022082 - 13065 T - 82 T^{2} + T^{3} \)
$19$ \( ( -19 + T )^{3} \)
$23$ \( -235392 - 3336 T + 103 T^{2} + T^{3} \)
$29$ \( 2252268 - 32064 T - 93 T^{2} + T^{3} \)
$31$ \( -1084416 - 28640 T - 116 T^{2} + T^{3} \)
$37$ \( 15144328 - 1060 T - 466 T^{2} + T^{3} \)
$41$ \( -5041152 - 26556 T + 188 T^{2} + T^{3} \)
$43$ \( 2359248 - 45296 T + 11 T^{2} + T^{3} \)
$47$ \( 3850752 - 146664 T + 163 T^{2} + T^{3} \)
$53$ \( 11566104 - 142266 T + 197 T^{2} + T^{3} \)
$59$ \( -52865928 + 555726 T - 1381 T^{2} + T^{3} \)
$61$ \( 847124 - 223668 T - 405 T^{2} + T^{3} \)
$67$ \( 1169904 + 83536 T - 943 T^{2} + T^{3} \)
$71$ \( -521792928 - 520668 T + 1052 T^{2} + T^{3} \)
$73$ \( -726527962 - 1118827 T + 580 T^{2} + T^{3} \)
$79$ \( -18139904 + 427568 T - 1402 T^{2} + T^{3} \)
$83$ \( -91984992 + 917424 T - 1802 T^{2} + T^{3} \)
$89$ \( -47198784 + 955656 T - 1966 T^{2} + T^{3} \)
$97$ \( -82010336 - 418356 T - 48 T^{2} + T^{3} \)
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