Properties

Label 1216.4.a.r
Level $1216$
Weight $4$
Character orbit 1216.a
Self dual yes
Analytic conductor $71.746$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.7057.1
Defining polynomial: \(x^{3} - x^{2} - 22 x + 32\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{3} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{5} + ( 2 + \beta_{1} + \beta_{2} ) q^{7} + ( 6 - 4 \beta_{1} + 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{3} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{5} + ( 2 + \beta_{1} + \beta_{2} ) q^{7} + ( 6 - 4 \beta_{1} + 3 \beta_{2} ) q^{9} + ( -36 - 2 \beta_{1} + 5 \beta_{2} ) q^{11} + ( -10 + 9 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 42 - 2 \beta_{1} - 4 \beta_{2} ) q^{15} + ( 3 + 8 \beta_{1} + 2 \beta_{2} ) q^{17} + 19 q^{19} + ( -38 + 3 \beta_{1} ) q^{21} + ( 110 + 5 \beta_{1} - 14 \beta_{2} ) q^{23} + ( 59 + 20 \beta_{1} - 15 \beta_{2} ) q^{25} + ( -59 + 16 \beta_{1} - \beta_{2} ) q^{27} + ( 46 - 11 \beta_{1} ) q^{29} + ( 138 - 6 \beta_{1} + 6 \beta_{2} ) q^{31} + ( -116 + 22 \beta_{1} + 18 \beta_{2} ) q^{33} + ( -114 - 10 \beta_{1} + 9 \beta_{2} ) q^{35} + ( -40 + 54 \beta_{1} + 18 \beta_{2} ) q^{37} + ( 38 - 17 \beta_{1} + 50 \beta_{2} ) q^{39} + ( -128 + 14 \beta_{2} ) q^{41} + ( -150 - 38 \beta_{1} + 19 \beta_{2} ) q^{43} + ( 148 + 40 \beta_{1} - 15 \beta_{2} ) q^{45} + ( -84 - 48 \beta_{1} + 53 \beta_{2} ) q^{47} + ( -266 + 4 \beta_{1} - 4 \beta_{2} ) q^{49} + ( -99 + 25 \beta_{2} ) q^{51} + ( 82 - 67 \beta_{1} + 62 \beta_{2} ) q^{53} + ( 12 + 112 \beta_{1} + 49 \beta_{2} ) q^{55} + ( -19 - 19 \beta_{2} ) q^{57} + ( 67 + 12 \beta_{1} - 13 \beta_{2} ) q^{59} + ( 188 + 34 \beta_{1} + 45 \beta_{2} ) q^{61} + ( -28 - 30 \beta_{1} + 23 \beta_{2} ) q^{63} + ( -530 - 78 \beta_{1} + 54 \beta_{2} ) q^{65} + ( 63 + 68 \beta_{1} + 57 \beta_{2} ) q^{67} + ( 318 - 61 \beta_{1} - 62 \beta_{2} ) q^{69} + ( -312 + 98 \beta_{1} - 18 \beta_{2} ) q^{71} + ( -175 - 64 \beta_{1} - 104 \beta_{2} ) q^{73} + ( 341 - 80 \beta_{1} + 51 \beta_{2} ) q^{75} + ( 34 - 68 \beta_{1} - 31 \beta_{2} ) q^{77} + ( 148 - 150 \beta_{1} + 8 \beta_{2} ) q^{79} + ( -135 + 88 \beta_{1} + 44 \beta_{2} ) q^{81} + ( 204 + 58 \beta_{1} + 168 \beta_{2} ) q^{83} + ( -646 - 78 \beta_{1} + 55 \beta_{2} ) q^{85} + ( -2 + 11 \beta_{1} - 90 \beta_{2} ) q^{87} + ( -442 - 2 \beta_{1} - 30 \beta_{2} ) q^{89} + ( 241 + 52 \beta_{1} - 53 \beta_{2} ) q^{91} + ( -306 + 30 \beta_{1} - 174 \beta_{2} ) q^{93} + ( -38 - 38 \beta_{1} - 19 \beta_{2} ) q^{95} + ( -258 - 88 \beta_{1} + 226 \beta_{2} ) q^{97} + ( 424 + 104 \beta_{1} + 33 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 4q^{3} - 7q^{5} + 7q^{7} + 21q^{9} + O(q^{10}) \) \( 3q - 4q^{3} - 7q^{5} + 7q^{7} + 21q^{9} - 103q^{11} - 32q^{13} + 122q^{15} + 11q^{17} + 57q^{19} - 114q^{21} + 316q^{23} + 162q^{25} - 178q^{27} + 138q^{29} + 420q^{31} - 330q^{33} - 333q^{35} - 102q^{37} + 164q^{39} - 370q^{41} - 431q^{43} + 429q^{45} - 199q^{47} - 802q^{49} - 272q^{51} + 308q^{53} + 85q^{55} - 76q^{57} + 188q^{59} + 609q^{61} - 61q^{63} - 1536q^{65} + 246q^{67} + 892q^{69} - 954q^{71} - 629q^{73} + 1074q^{75} + 71q^{77} + 452q^{79} - 361q^{81} + 780q^{83} - 1883q^{85} - 96q^{87} - 1356q^{89} + 670q^{91} - 1092q^{93} - 133q^{95} - 548q^{97} + 1305q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 22 x + 32\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 3 \nu - 16 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 14 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{2} + \beta_{1} + 29\)\()/2\)

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
−4.86867
4.36181
1.50686
0 −8.28632 0 −2.38427 0 5.83529 0 41.6631 0
1.2 0 −1.33180 0 −18.4427 0 10.3872 0 −25.2263 0
1.3 0 5.61812 0 13.8269 0 −9.22252 0 4.56325 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.r 3
4.b odd 2 1 1216.4.a.w 3
8.b even 2 1 152.4.a.c 3
8.d odd 2 1 304.4.a.g 3
24.h odd 2 1 1368.4.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.a.c 3 8.b even 2 1
304.4.a.g 3 8.d odd 2 1
1216.4.a.r 3 1.a even 1 1 trivial
1216.4.a.w 3 4.b odd 2 1
1368.4.a.d 3 24.h odd 2 1

Hecke kernels

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

\( T_{3}^{3} + 4 T_{3}^{2} - 43 T_{3} - 62 \)
\( T_{5}^{3} + 7 T_{5}^{2} - 244 T_{5} - 608 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -62 - 43 T + 4 T^{2} + T^{3} \)
$5$ \( -608 - 244 T + 7 T^{2} + T^{3} \)
$7$ \( 559 - 89 T - 7 T^{2} + T^{3} \)
$11$ \( -22156 + 2212 T + 103 T^{2} + T^{3} \)
$13$ \( -131420 - 3677 T + 32 T^{2} + T^{3} \)
$17$ \( -32173 - 3417 T - 11 T^{2} + T^{3} \)
$19$ \( ( -19 + T )^{3} \)
$23$ \( 242336 + 23147 T - 316 T^{2} + T^{3} \)
$29$ \( 345766 + 419 T - 138 T^{2} + T^{3} \)
$31$ \( -2336256 + 55584 T - 420 T^{2} + T^{3} \)
$37$ \( -15565400 - 162852 T + 102 T^{2} + T^{3} \)
$41$ \( 707456 + 36160 T + 370 T^{2} + T^{3} \)
$43$ \( -5420480 - 20508 T + 431 T^{2} + T^{3} \)
$47$ \( -45306112 - 215112 T + 199 T^{2} + T^{3} \)
$53$ \( -6630640 - 340901 T - 308 T^{2} + T^{3} \)
$59$ \( 1077242 - 2195 T - 188 T^{2} + T^{3} \)
$61$ \( 50618548 - 43132 T - 609 T^{2} + T^{3} \)
$67$ \( 96246632 - 394447 T - 246 T^{2} + T^{3} \)
$71$ \( -237273448 - 168772 T + 954 T^{2} + T^{3} \)
$73$ \( -417051529 - 644845 T + 629 T^{2} + T^{3} \)
$79$ \( 601611824 - 1027892 T - 452 T^{2} + T^{3} \)
$83$ \( 1048786960 - 1404148 T - 780 T^{2} + T^{3} \)
$89$ \( 71672320 + 568736 T + 1356 T^{2} + T^{3} \)
$97$ \( -2035482752 - 2588924 T + 548 T^{2} + T^{3} \)
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