Properties

Label 1216.4.a.l
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: \(\Q(\sqrt{177}) \)
Defining polynomial: \(x^{2} - x - 44\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( -6 + 2 \beta ) q^{5} + ( -28 - \beta ) q^{7} + ( 17 + \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{3} + ( -6 + 2 \beta ) q^{5} + ( -28 - \beta ) q^{7} + ( 17 + \beta ) q^{9} + ( 4 + 2 \beta ) q^{11} + ( -10 + 7 \beta ) q^{13} + ( 88 - 4 \beta ) q^{15} + ( -18 - 15 \beta ) q^{17} -19 q^{19} + ( -44 - 29 \beta ) q^{21} + ( 84 - 13 \beta ) q^{23} + ( 87 - 20 \beta ) q^{25} + ( 44 - 9 \beta ) q^{27} + ( 54 - 29 \beta ) q^{29} + 16 \beta q^{31} + ( 88 + 6 \beta ) q^{33} + ( 80 - 52 \beta ) q^{35} + ( -198 + 16 \beta ) q^{37} + ( 308 - 3 \beta ) q^{39} + ( -398 + 6 \beta ) q^{41} + ( 124 + 48 \beta ) q^{43} + ( -14 + 30 \beta ) q^{45} + ( 120 - 40 \beta ) q^{47} + ( 485 + 57 \beta ) q^{49} + ( -660 - 33 \beta ) q^{51} + ( -194 - 9 \beta ) q^{53} + 152 q^{55} -19 \beta q^{57} + ( 136 - 71 \beta ) q^{59} + ( 362 - 44 \beta ) q^{61} + ( -520 - 46 \beta ) q^{63} + ( 676 - 48 \beta ) q^{65} + ( -448 - 43 \beta ) q^{67} + ( -572 + 71 \beta ) q^{69} + ( -192 - 22 \beta ) q^{71} + ( 62 - \beta ) q^{73} + ( -880 + 67 \beta ) q^{75} + ( -200 - 62 \beta ) q^{77} + ( -24 - 58 \beta ) q^{79} + ( -855 + 8 \beta ) q^{81} + ( 1116 - 6 \beta ) q^{83} + ( -1212 + 24 \beta ) q^{85} + ( -1276 + 25 \beta ) q^{87} + ( -430 - 10 \beta ) q^{89} + ( -28 - 193 \beta ) q^{91} + ( 704 + 16 \beta ) q^{93} + ( 114 - 38 \beta ) q^{95} + ( -894 - 76 \beta ) q^{97} + ( 156 + 40 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 10q^{5} - 57q^{7} + 35q^{9} + O(q^{10}) \) \( 2q + q^{3} - 10q^{5} - 57q^{7} + 35q^{9} + 10q^{11} - 13q^{13} + 172q^{15} - 51q^{17} - 38q^{19} - 117q^{21} + 155q^{23} + 154q^{25} + 79q^{27} + 79q^{29} + 16q^{31} + 182q^{33} + 108q^{35} - 380q^{37} + 613q^{39} - 790q^{41} + 296q^{43} + 2q^{45} + 200q^{47} + 1027q^{49} - 1353q^{51} - 397q^{53} + 304q^{55} - 19q^{57} + 201q^{59} + 680q^{61} - 1086q^{63} + 1304q^{65} - 939q^{67} - 1073q^{69} - 406q^{71} + 123q^{73} - 1693q^{75} - 462q^{77} - 106q^{79} - 1702q^{81} + 2226q^{83} - 2400q^{85} - 2527q^{87} - 870q^{89} - 249q^{91} + 1424q^{93} + 190q^{95} - 1864q^{97} + 352q^{99} + O(q^{100}) \)

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
−6.15207
7.15207
0 −6.15207 0 −18.3041 0 −21.8479 0 10.8479 0
1.2 0 7.15207 0 8.30413 0 −35.1521 0 24.1521 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.l 2
4.b odd 2 1 1216.4.a.j 2
8.b even 2 1 304.4.a.d 2
8.d odd 2 1 38.4.a.b 2
24.f even 2 1 342.4.a.k 2
40.e odd 2 1 950.4.a.h 2
40.k even 4 2 950.4.b.g 4
56.e even 2 1 1862.4.a.b 2
152.b even 2 1 722.4.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.b 2 8.d odd 2 1
304.4.a.d 2 8.b even 2 1
342.4.a.k 2 24.f even 2 1
722.4.a.i 2 152.b even 2 1
950.4.a.h 2 40.e odd 2 1
950.4.b.g 4 40.k even 4 2
1216.4.a.j 2 4.b odd 2 1
1216.4.a.l 2 1.a even 1 1 trivial
1862.4.a.b 2 56.e even 2 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} - 44 \)
\( T_{5}^{2} + 10 T_{5} - 152 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -44 - T + T^{2} \)
$5$ \( -152 + 10 T + T^{2} \)
$7$ \( 768 + 57 T + T^{2} \)
$11$ \( -152 - 10 T + T^{2} \)
$13$ \( -2126 + 13 T + T^{2} \)
$17$ \( -9306 + 51 T + T^{2} \)
$19$ \( ( 19 + T )^{2} \)
$23$ \( -1472 - 155 T + T^{2} \)
$29$ \( -35654 - 79 T + T^{2} \)
$31$ \( -11264 - 16 T + T^{2} \)
$37$ \( 24772 + 380 T + T^{2} \)
$41$ \( 154432 + 790 T + T^{2} \)
$43$ \( -80048 - 296 T + T^{2} \)
$47$ \( -60800 - 200 T + T^{2} \)
$53$ \( 35818 + 397 T + T^{2} \)
$59$ \( -212964 - 201 T + T^{2} \)
$61$ \( 29932 - 680 T + T^{2} \)
$67$ \( 138612 + 939 T + T^{2} \)
$71$ \( 19792 + 406 T + T^{2} \)
$73$ \( 3738 - 123 T + T^{2} \)
$79$ \( -146048 + 106 T + T^{2} \)
$83$ \( 1237176 - 2226 T + T^{2} \)
$89$ \( 184800 + 870 T + T^{2} \)
$97$ \( 613036 + 1864 T + T^{2} \)
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