Properties

Label 1216.4.a.bc
Level $1216$
Weight $4$
Character orbit 1216.a
Self dual yes
Analytic conductor $71.746$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 20 x^{3} + 24 x^{2} + 2 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} ) q^{3} + ( -5 + \beta_{1} ) q^{5} + ( 5 - \beta_{1} - \beta_{2} - 3 \beta_{4} ) q^{7} + ( 29 - 2 \beta_{1} + 3 \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{3} ) q^{3} + ( -5 + \beta_{1} ) q^{5} + ( 5 - \beta_{1} - \beta_{2} - 3 \beta_{4} ) q^{7} + ( 29 - 2 \beta_{1} + 3 \beta_{2} + \beta_{4} ) q^{9} + ( -12 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{11} + ( -40 - \beta_{2} + 5 \beta_{4} ) q^{13} + ( -9 + 2 \beta_{1} - 7 \beta_{2} - 9 \beta_{3} + \beta_{4} ) q^{15} + ( -14 - \beta_{1} + 4 \beta_{2} - \beta_{3} + 5 \beta_{4} ) q^{17} + 19 q^{19} + ( 7 + 4 \beta_{1} + 16 \beta_{2} - \beta_{3} - 8 \beta_{4} ) q^{21} + ( 8 - 6 \beta_{1} + 13 \beta_{2} - 15 \beta_{4} ) q^{23} + ( -1 - 13 \beta_{1} - \beta_{2} + \beta_{3} - 10 \beta_{4} ) q^{25} + ( -22 \beta_{1} + 3 \beta_{2} + 24 \beta_{3} - 13 \beta_{4} ) q^{27} + ( -1 - 2 \beta_{1} - 4 \beta_{2} - 13 \beta_{3} + 6 \beta_{4} ) q^{29} + ( -147 - 11 \beta_{2} - 9 \beta_{3} + 17 \beta_{4} ) q^{31} + ( 47 + 22 \beta_{1} - 13 \beta_{2} - 33 \beta_{3} + 27 \beta_{4} ) q^{33} + ( -64 + 21 \beta_{1} + 9 \beta_{2} + 15 \beta_{3} - 2 \beta_{4} ) q^{35} + ( -41 - 12 \beta_{1} - 7 \beta_{2} - 21 \beta_{3} + 19 \beta_{4} ) q^{37} + ( -26 + 6 \beta_{1} - 7 \beta_{2} - 34 \beta_{3} + 25 \beta_{4} ) q^{39} + ( 13 + 24 \beta_{1} + 9 \beta_{2} - 15 \beta_{3} - \beta_{4} ) q^{41} + ( 76 - 17 \beta_{1} - 9 \beta_{2} + \beta_{3} + 12 \beta_{4} ) q^{43} + ( -347 + 37 \beta_{1} - 22 \beta_{2} - 34 \beta_{3} + 32 \beta_{4} ) q^{45} + ( -7 - \beta_{1} + 28 \beta_{2} - 10 \beta_{3} + 32 \beta_{4} ) q^{47} + ( 203 + 5 \beta_{1} - 20 \beta_{2} - 33 \beta_{3} - 9 \beta_{4} ) q^{49} + ( -71 - 24 \beta_{1} - 18 \beta_{2} + 17 \beta_{3} - 2 \beta_{4} ) q^{51} + ( -197 + 4 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 20 \beta_{4} ) q^{53} + ( 291 - 33 \beta_{1} + 14 \beta_{2} + 20 \beta_{3} - 32 \beta_{4} ) q^{55} + ( 19 + 19 \beta_{3} ) q^{57} + ( 96 + 8 \beta_{1} - 45 \beta_{2} - 54 \beta_{3} + 61 \beta_{4} ) q^{59} + ( -297 - 23 \beta_{1} - 30 \beta_{2} - 28 \beta_{3} ) q^{61} + ( -279 - 59 \beta_{1} - 36 \beta_{2} + 40 \beta_{3} - 28 \beta_{4} ) q^{63} + ( 84 - 48 \beta_{1} + 8 \beta_{2} + 12 \beta_{4} ) q^{65} + ( -183 + 22 \beta_{1} + 42 \beta_{2} - 29 \beta_{3} - 28 \beta_{4} ) q^{67} + ( -50 - 90 \beta_{1} + 33 \beta_{2} + 54 \beta_{3} - 131 \beta_{4} ) q^{69} + ( 316 + 46 \beta_{1} + 32 \beta_{2} + 22 \beta_{3} - 16 \beta_{4} ) q^{71} + ( 528 + 19 \beta_{1} - 20 \beta_{2} - 67 \beta_{3} - 17 \beta_{4} ) q^{73} + ( 90 - 22 \beta_{1} + 117 \beta_{2} + 26 \beta_{3} - 47 \beta_{4} ) q^{75} + ( -27 + 33 \beta_{1} + 52 \beta_{2} - 32 \beta_{3} - 42 \beta_{4} ) q^{77} + ( -107 + 24 \beta_{1} - 13 \beta_{2} + 43 \beta_{3} - 77 \beta_{4} ) q^{79} + ( 587 - 56 \beta_{1} + 162 \beta_{2} + 50 \beta_{3} - 92 \beta_{4} ) q^{81} + ( 110 - 16 \beta_{1} + 16 \beta_{2} - 42 \beta_{3} - 90 \beta_{4} ) q^{83} + ( -111 - 25 \beta_{1} - 24 \beta_{2} - 42 \beta_{3} + 36 \beta_{4} ) q^{85} + ( -680 + 46 \beta_{1} - 25 \beta_{2} + 16 \beta_{3} + 29 \beta_{4} ) q^{87} + ( 207 + 6 \beta_{1} - 59 \beta_{2} - 79 \beta_{3} - 63 \beta_{4} ) q^{89} + ( -662 - 14 \beta_{1} + 61 \beta_{2} - 34 \beta_{3} + 117 \beta_{4} ) q^{91} + ( -564 + 84 \beta_{1} - 28 \beta_{2} - 148 \beta_{3} + 114 \beta_{4} ) q^{93} + ( -95 + 19 \beta_{1} ) q^{95} + ( 104 + 50 \beta_{1} - 96 \beta_{2} - 16 \beta_{3} + 34 \beta_{4} ) q^{97} + ( -1426 + 107 \beta_{1} - 187 \beta_{2} - 33 \beta_{3} + 108 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 3q^{3} - 27q^{5} + 20q^{7} + 154q^{9} + O(q^{10}) \) \( 5q + 3q^{3} - 27q^{5} + 20q^{7} + 154q^{9} - 67q^{11} - 191q^{13} - 36q^{15} - 52q^{17} + 95q^{19} + 29q^{21} + 35q^{23} - 2q^{25} - 27q^{27} + 33q^{29} - 694q^{31} + 298q^{33} - 387q^{35} - 108q^{37} - 31q^{39} + 54q^{41} + 427q^{43} - 1699q^{45} + 79q^{47} + 1033q^{49} - 363q^{51} - 1025q^{53} + 1431q^{55} + 57q^{57} + 649q^{59} - 1413q^{61} - 1449q^{63} + 548q^{65} - 915q^{67} - 407q^{69} + 1444q^{71} + 2682q^{73} + 465q^{75} - 169q^{77} - 836q^{79} + 2925q^{81} + 502q^{83} - 373q^{85} - 3491q^{87} + 996q^{89} - 2919q^{91} - 2492q^{93} - 513q^{95} + 424q^{97} - 7249q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 20 x^{3} + 24 x^{2} + 2 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{4} + \nu^{3} + 18 \nu^{2} - 24 \nu + 14 \)\()/2\)
\(\beta_{2}\)\(=\)\( -\nu^{4} + 20 \nu^{2} - 2 \nu - 8 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 20 \nu^{2} + 6 \nu + 7 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 20 \nu^{2} + 24 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} - 2 \beta_{1} + 16\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{4} + 11 \beta_{3} + 9 \beta_{2} - 1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-20 \beta_{4} - \beta_{3} - 3 \beta_{2} - 40 \beta_{1} + 303\)\()/2\)

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
1.14402
−4.53955
−0.389032
0.457551
4.32701
0 −9.59869 0 −0.0570048 0 −24.6071 0 65.1348 0
1.2 0 −6.71806 0 −17.1669 0 31.2505 0 18.1323 0
1.3 0 2.66179 0 7.98961 0 27.0741 0 −19.9149 0
1.4 0 6.60208 0 −1.58045 0 −19.8743 0 16.5875 0
1.5 0 10.0529 0 −16.1852 0 6.15678 0 74.0603 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.bc 5
4.b odd 2 1 1216.4.a.z 5
8.b even 2 1 608.4.a.f 5
8.d odd 2 1 608.4.a.g yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.f 5 8.b even 2 1
608.4.a.g yes 5 8.d odd 2 1
1216.4.a.z 5 4.b odd 2 1
1216.4.a.bc 5 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{5} - 3 T_{3}^{4} - 140 T_{3}^{3} + 384 T_{3}^{2} + 4256 T_{3} - 11392 \)
\( T_{5}^{5} + 27 T_{5}^{4} + 53 T_{5}^{3} - 2199 T_{5}^{2} - 3634 T_{5} - 200 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( -11392 + 4256 T + 384 T^{2} - 140 T^{3} - 3 T^{4} + T^{5} \)
$5$ \( -200 - 3634 T - 2199 T^{2} + 53 T^{3} + 27 T^{4} + T^{5} \)
$7$ \( -2547508 + 357677 T + 16864 T^{2} - 1174 T^{3} - 20 T^{4} + T^{5} \)
$11$ \( 42711800 + 289554 T - 121799 T^{2} - 1415 T^{3} + 67 T^{4} + T^{5} \)
$13$ \( -38142080 + 967712 T + 297464 T^{2} + 12278 T^{3} + 191 T^{4} + T^{5} \)
$17$ \( -9857070 + 6431579 T - 162966 T^{2} - 6732 T^{3} + 52 T^{4} + T^{5} \)
$19$ \( ( -19 + T )^{5} \)
$23$ \( -5659603968 + 148521472 T + 2681760 T^{2} - 40628 T^{3} - 35 T^{4} + T^{5} \)
$29$ \( 1528050752 + 170497248 T + 75888 T^{2} - 28662 T^{3} - 33 T^{4} + T^{5} \)
$31$ \( 1546708480 - 444588576 T + 6330600 T^{2} + 144340 T^{3} + 694 T^{4} + T^{5} \)
$37$ \( 264959360 + 1528208256 T - 6146080 T^{2} - 140268 T^{3} + 108 T^{4} + T^{5} \)
$41$ \( 397858568960 + 10082529344 T - 1154208 T^{2} - 211000 T^{3} - 54 T^{4} + T^{5} \)
$43$ \( -331567864112 - 293319520 T + 28568807 T^{2} - 47021 T^{3} - 427 T^{4} + T^{5} \)
$47$ \( -905506740224 + 13775924460 T - 2808805 T^{2} - 344889 T^{3} - 79 T^{4} + T^{5} \)
$53$ \( -210846255280 + 1095222032 T + 51700440 T^{2} + 369616 T^{3} + 1025 T^{4} + T^{5} \)
$59$ \( -34775624292320 + 71067723688 T + 413524396 T^{2} - 712054 T^{3} - 649 T^{4} + T^{5} \)
$61$ \( -29987676575940 - 194167345288 T - 313293389 T^{2} + 366675 T^{3} + 1413 T^{4} + T^{5} \)
$67$ \( 24348069855968 - 51920795608 T - 534239380 T^{2} - 436030 T^{3} + 915 T^{4} + T^{5} \)
$71$ \( 35176103924480 - 316715514256 T + 781980880 T^{2} - 50720 T^{3} - 1444 T^{4} + T^{5} \)
$73$ \( 71728412586510 - 495511081027 T + 98929572 T^{2} + 2006794 T^{3} - 2682 T^{4} + T^{5} \)
$79$ \( -31064499767680 - 350770092176 T - 1115391248 T^{2} - 905520 T^{3} + 836 T^{4} + T^{5} \)
$83$ \( -8567040270336 - 104179182592 T + 680570368 T^{2} - 871208 T^{3} - 502 T^{4} + T^{5} \)
$89$ \( -13823629765888 + 776942848112 T + 1127620016 T^{2} - 1823056 T^{3} - 996 T^{4} + T^{5} \)
$97$ \( -461524239069440 + 747255069888 T + 1109159696 T^{2} - 1955148 T^{3} - 424 T^{4} + T^{5} \)
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