Properties

Label 1216.4.a.k
Level $1216$
Weight $4$
Character orbit 1216.a
Self dual yes
Analytic conductor $71.746$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + ( 3 - \beta ) q^{5} + ( 5 - 4 \beta ) q^{7} + ( -13 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} + ( 3 - \beta ) q^{5} + ( 5 - 4 \beta ) q^{7} + ( -13 + \beta ) q^{9} + ( -11 + 7 \beta ) q^{11} + ( 26 + 7 \beta ) q^{13} + ( 14 - 2 \beta ) q^{15} + ( -47 - 10 \beta ) q^{17} -19 q^{19} + ( 56 - \beta ) q^{21} + ( -6 + 33 \beta ) q^{23} + ( -102 - 5 \beta ) q^{25} + ( -14 + 39 \beta ) q^{27} + ( 64 + 9 \beta ) q^{29} + ( -16 + 28 \beta ) q^{31} + ( -98 + 4 \beta ) q^{33} + ( 71 - 13 \beta ) q^{35} + ( 90 - 28 \beta ) q^{37} + ( -98 - 33 \beta ) q^{39} + ( -150 + 90 \beta ) q^{41} + ( 45 - 23 \beta ) q^{43} + ( -53 + 15 \beta ) q^{45} + ( -159 + 45 \beta ) q^{47} + ( -94 - 24 \beta ) q^{49} + ( 140 + 57 \beta ) q^{51} + ( -106 + 3 \beta ) q^{53} + ( -131 + 25 \beta ) q^{55} + 19 \beta q^{57} + ( 368 + 63 \beta ) q^{59} + ( -101 + 53 \beta ) q^{61} + ( -121 + 53 \beta ) q^{63} + ( -20 - 12 \beta ) q^{65} + ( 98 + 5 \beta ) q^{67} + ( -462 - 27 \beta ) q^{69} + ( -446 + 100 \beta ) q^{71} + ( -87 - 72 \beta ) q^{73} + ( 70 + 107 \beta ) q^{75} + ( -447 + 51 \beta ) q^{77} + ( 184 - 114 \beta ) q^{79} + ( -195 - 52 \beta ) q^{81} + ( 264 - 154 \beta ) q^{83} + ( -1 + 27 \beta ) q^{85} + ( -126 - 73 \beta ) q^{87} + ( -184 - 196 \beta ) q^{89} + ( -262 - 97 \beta ) q^{91} + ( -392 - 12 \beta ) q^{93} + ( -57 + 19 \beta ) q^{95} + ( -276 + 374 \beta ) q^{97} + ( 241 - 95 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 5q^{5} + 6q^{7} - 25q^{9} + O(q^{10}) \) \( 2q - q^{3} + 5q^{5} + 6q^{7} - 25q^{9} - 15q^{11} + 59q^{13} + 26q^{15} - 104q^{17} - 38q^{19} + 111q^{21} + 21q^{23} - 209q^{25} + 11q^{27} + 137q^{29} - 4q^{31} - 192q^{33} + 129q^{35} + 152q^{37} - 229q^{39} - 210q^{41} + 67q^{43} - 91q^{45} - 273q^{47} - 212q^{49} + 337q^{51} - 209q^{53} - 237q^{55} + 19q^{57} + 799q^{59} - 149q^{61} - 189q^{63} - 52q^{65} + 201q^{67} - 951q^{69} - 792q^{71} - 246q^{73} + 247q^{75} - 843q^{77} + 254q^{79} - 442q^{81} + 374q^{83} + 25q^{85} - 325q^{87} - 564q^{89} - 621q^{91} - 796q^{93} - 95q^{95} - 178q^{97} + 387q^{99} + O(q^{100}) \)

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
4.27492
−3.27492
0 −4.27492 0 −1.27492 0 −12.0997 0 −8.72508 0
1.2 0 3.27492 0 6.27492 0 18.0997 0 −16.2749 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.k 2
4.b odd 2 1 1216.4.a.m 2
8.b even 2 1 304.4.a.e 2
8.d odd 2 1 152.4.a.a 2
24.f even 2 1 1368.4.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.a.a 2 8.d odd 2 1
304.4.a.e 2 8.b even 2 1
1216.4.a.k 2 1.a even 1 1 trivial
1216.4.a.m 2 4.b odd 2 1
1368.4.a.a 2 24.f even 2 1

Hecke kernels

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

\( T_{3}^{2} + T_{3} - 14 \)
\( T_{5}^{2} - 5 T_{5} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -14 + T + T^{2} \)
$5$ \( -8 - 5 T + T^{2} \)
$7$ \( -219 - 6 T + T^{2} \)
$11$ \( -642 + 15 T + T^{2} \)
$13$ \( 172 - 59 T + T^{2} \)
$17$ \( 1279 + 104 T + T^{2} \)
$19$ \( ( 19 + T )^{2} \)
$23$ \( -15408 - 21 T + T^{2} \)
$29$ \( 3538 - 137 T + T^{2} \)
$31$ \( -11168 + 4 T + T^{2} \)
$37$ \( -5396 - 152 T + T^{2} \)
$41$ \( -104400 + 210 T + T^{2} \)
$43$ \( -6416 - 67 T + T^{2} \)
$47$ \( -10224 + 273 T + T^{2} \)
$53$ \( 10792 + 209 T + T^{2} \)
$59$ \( 103042 - 799 T + T^{2} \)
$61$ \( -34478 + 149 T + T^{2} \)
$67$ \( 9744 - 201 T + T^{2} \)
$71$ \( 14316 + 792 T + T^{2} \)
$73$ \( -58743 + 246 T + T^{2} \)
$79$ \( -169064 - 254 T + T^{2} \)
$83$ \( -302984 - 374 T + T^{2} \)
$89$ \( -467904 + 564 T + T^{2} \)
$97$ \( -1985312 + 178 T + T^{2} \)
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