Properties

Label 1216.4.a.p
Level $1216$
Weight $4$
Character orbit 1216.a
Self dual yes
Analytic conductor $71.746$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
Defining polynomial: \(x^{2} - x - 18\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 5 - \beta ) q^{3} + ( 6 - 3 \beta ) q^{5} + ( 11 - 4 \beta ) q^{7} + ( 16 - 9 \beta ) q^{9} +O(q^{10})\) \( q + ( 5 - \beta ) q^{3} + ( 6 - 3 \beta ) q^{5} + ( 11 - 4 \beta ) q^{7} + ( 16 - 9 \beta ) q^{9} + ( -8 - \beta ) q^{11} + ( -15 + 13 \beta ) q^{13} + ( 84 - 18 \beta ) q^{15} + ( -41 + 2 \beta ) q^{17} + 19 q^{19} + ( 127 - 27 \beta ) q^{21} + ( -43 + 13 \beta ) q^{23} + ( 73 - 27 \beta ) q^{25} + ( 107 - 25 \beta ) q^{27} + ( 9 - 21 \beta ) q^{29} + ( -84 - 44 \beta ) q^{31} + ( -22 + 4 \beta ) q^{33} + ( 282 - 45 \beta ) q^{35} + ( -82 - 28 \beta ) q^{37} + ( -309 + 67 \beta ) q^{39} + ( -20 - 10 \beta ) q^{41} + ( 342 - 7 \beta ) q^{43} + ( 582 - 75 \beta ) q^{45} + ( 230 - 71 \beta ) q^{47} + ( 66 - 72 \beta ) q^{49} + ( -241 + 49 \beta ) q^{51} + ( 601 + 17 \beta ) q^{53} + ( 6 + 21 \beta ) q^{55} + ( 95 - 19 \beta ) q^{57} + ( -131 - 25 \beta ) q^{59} + ( -212 + 111 \beta ) q^{61} + ( 824 - 127 \beta ) q^{63} + ( -792 + 84 \beta ) q^{65} + ( 573 + 77 \beta ) q^{67} + ( -449 + 95 \beta ) q^{69} + ( -158 + 116 \beta ) q^{71} + ( 97 + 184 \beta ) q^{73} + ( 851 - 181 \beta ) q^{75} + ( -16 + 25 \beta ) q^{77} + ( -646 - 58 \beta ) q^{79} + ( 553 + 36 \beta ) q^{81} + ( -238 - 194 \beta ) q^{83} + ( -354 + 129 \beta ) q^{85} + ( 423 - 93 \beta ) q^{87} + ( -212 + 188 \beta ) q^{89} + ( -1101 + 151 \beta ) q^{91} + ( 372 - 92 \beta ) q^{93} + ( 114 - 57 \beta ) q^{95} + ( 698 - 102 \beta ) q^{97} + ( 34 + 65 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 9q^{3} + 9q^{5} + 18q^{7} + 23q^{9} + O(q^{10}) \) \( 2q + 9q^{3} + 9q^{5} + 18q^{7} + 23q^{9} - 17q^{11} - 17q^{13} + 150q^{15} - 80q^{17} + 38q^{19} + 227q^{21} - 73q^{23} + 119q^{25} + 189q^{27} - 3q^{29} - 212q^{31} - 40q^{33} + 519q^{35} - 192q^{37} - 551q^{39} - 50q^{41} + 677q^{43} + 1089q^{45} + 389q^{47} + 60q^{49} - 433q^{51} + 1219q^{53} + 33q^{55} + 171q^{57} - 287q^{59} - 313q^{61} + 1521q^{63} - 1500q^{65} + 1223q^{67} - 803q^{69} - 200q^{71} + 378q^{73} + 1521q^{75} - 7q^{77} - 1350q^{79} + 1142q^{81} - 670q^{83} - 579q^{85} + 753q^{87} - 236q^{89} - 2051q^{91} + 652q^{93} + 171q^{95} + 1294q^{97} + 133q^{99} + O(q^{100}) \)

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
4.77200
−3.77200
0 0.227998 0 −8.31601 0 −8.08801 0 −26.9480 0
1.2 0 8.77200 0 17.3160 0 26.0880 0 49.9480 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.p 2
4.b odd 2 1 1216.4.a.g 2
8.b even 2 1 304.4.a.c 2
8.d odd 2 1 38.4.a.c 2
24.f even 2 1 342.4.a.h 2
40.e odd 2 1 950.4.a.e 2
40.k even 4 2 950.4.b.i 4
56.e even 2 1 1862.4.a.e 2
152.b even 2 1 722.4.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.c 2 8.d odd 2 1
304.4.a.c 2 8.b even 2 1
342.4.a.h 2 24.f even 2 1
722.4.a.f 2 152.b even 2 1
950.4.a.e 2 40.e odd 2 1
950.4.b.i 4 40.k even 4 2
1216.4.a.g 2 4.b odd 2 1
1216.4.a.p 2 1.a even 1 1 trivial
1862.4.a.e 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\):

\( T_{3}^{2} - 9 T_{3} + 2 \)
\( T_{5}^{2} - 9 T_{5} - 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2 - 9 T + T^{2} \)
$5$ \( -144 - 9 T + T^{2} \)
$7$ \( -211 - 18 T + T^{2} \)
$11$ \( 54 + 17 T + T^{2} \)
$13$ \( -3012 + 17 T + T^{2} \)
$17$ \( 1527 + 80 T + T^{2} \)
$19$ \( ( -19 + T )^{2} \)
$23$ \( -1752 + 73 T + T^{2} \)
$29$ \( -8046 + 3 T + T^{2} \)
$31$ \( -24096 + 212 T + T^{2} \)
$37$ \( -5092 + 192 T + T^{2} \)
$41$ \( -1200 + 50 T + T^{2} \)
$43$ \( 113688 - 677 T + T^{2} \)
$47$ \( -54168 - 389 T + T^{2} \)
$53$ \( 366216 - 1219 T + T^{2} \)
$59$ \( 9186 + 287 T + T^{2} \)
$61$ \( -200366 + 313 T + T^{2} \)
$67$ \( 265728 - 1223 T + T^{2} \)
$71$ \( -235572 + 200 T + T^{2} \)
$73$ \( -582151 - 378 T + T^{2} \)
$79$ \( 394232 + 1350 T + T^{2} \)
$83$ \( -574632 + 670 T + T^{2} \)
$89$ \( -631104 + 236 T + T^{2} \)
$97$ \( 228736 - 1294 T + T^{2} \)
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