Properties

Label 1216.4.a.i
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: \(\Q(\sqrt{93}) \)
Defining polynomial: \(x^{2} - x - 23\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{93}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( 2 + 2 \beta ) q^{5} + ( -22 - \beta ) q^{7} -26 q^{9} +O(q^{10})\) \( q - q^{3} + ( 2 + 2 \beta ) q^{5} + ( -22 - \beta ) q^{7} -26 q^{9} + ( 20 - 2 \beta ) q^{11} + ( 10 + 3 \beta ) q^{13} + ( -2 - 2 \beta ) q^{15} + ( 23 - 8 \beta ) q^{17} + 19 q^{19} + ( 22 + \beta ) q^{21} + ( 110 - 5 \beta ) q^{23} + ( 251 + 8 \beta ) q^{25} + 53 q^{27} + ( 42 + 19 \beta ) q^{29} + ( 42 - 18 \beta ) q^{31} + ( -20 + 2 \beta ) q^{33} + ( -230 - 46 \beta ) q^{35} + ( -152 + 6 \beta ) q^{37} + ( -10 - 3 \beta ) q^{39} + ( 84 - 20 \beta ) q^{41} + ( -186 - 2 \beta ) q^{43} + ( -52 - 52 \beta ) q^{45} + ( -292 - 24 \beta ) q^{47} + ( 234 + 44 \beta ) q^{49} + ( -23 + 8 \beta ) q^{51} + ( -242 - 33 \beta ) q^{53} + ( -332 + 36 \beta ) q^{55} -19 q^{57} + ( -537 + 4 \beta ) q^{59} + ( -44 - 14 \beta ) q^{61} + ( 572 + 26 \beta ) q^{63} + ( 578 + 26 \beta ) q^{65} + ( 715 + 28 \beta ) q^{67} + ( -110 + 5 \beta ) q^{69} + ( -924 + 2 \beta ) q^{71} + ( -647 - 8 \beta ) q^{73} + ( -251 - 8 \beta ) q^{75} + ( -254 + 24 \beta ) q^{77} + ( -416 - 78 \beta ) q^{79} + 649 q^{81} + ( 264 + 90 \beta ) q^{83} + ( -1442 + 30 \beta ) q^{85} + ( -42 - 19 \beta ) q^{87} + ( 922 - 74 \beta ) q^{89} + ( -499 - 76 \beta ) q^{91} + ( -42 + 18 \beta ) q^{93} + ( 38 + 38 \beta ) q^{95} + ( -182 - 52 \beta ) q^{97} + ( -520 + 52 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{5} - 44q^{7} - 52q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{5} - 44q^{7} - 52q^{9} + 40q^{11} + 20q^{13} - 4q^{15} + 46q^{17} + 38q^{19} + 44q^{21} + 220q^{23} + 502q^{25} + 106q^{27} + 84q^{29} + 84q^{31} - 40q^{33} - 460q^{35} - 304q^{37} - 20q^{39} + 168q^{41} - 372q^{43} - 104q^{45} - 584q^{47} + 468q^{49} - 46q^{51} - 484q^{53} - 664q^{55} - 38q^{57} - 1074q^{59} - 88q^{61} + 1144q^{63} + 1156q^{65} + 1430q^{67} - 220q^{69} - 1848q^{71} - 1294q^{73} - 502q^{75} - 508q^{77} - 832q^{79} + 1298q^{81} + 528q^{83} - 2884q^{85} - 84q^{87} + 1844q^{89} - 998q^{91} - 84q^{93} + 76q^{95} - 364q^{97} - 1040q^{99} + O(q^{100}) \)

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
−4.32183
5.32183
0 −1.00000 0 −17.2873 0 −12.3563 0 −26.0000 0
1.2 0 −1.00000 0 21.2873 0 −31.6437 0 −26.0000 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.i 2
4.b odd 2 1 1216.4.a.n 2
8.b even 2 1 608.4.a.d yes 2
8.d odd 2 1 608.4.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.c 2 8.d odd 2 1
608.4.a.d yes 2 8.b even 2 1
1216.4.a.i 2 1.a even 1 1 trivial
1216.4.a.n 2 4.b odd 2 1

Hecke kernels

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

\( T_{3} + 1 \)
\( T_{5}^{2} - 4 T_{5} - 368 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -368 - 4 T + T^{2} \)
$7$ \( 391 + 44 T + T^{2} \)
$11$ \( 28 - 40 T + T^{2} \)
$13$ \( -737 - 20 T + T^{2} \)
$17$ \( -5423 - 46 T + T^{2} \)
$19$ \( ( -19 + T )^{2} \)
$23$ \( 9775 - 220 T + T^{2} \)
$29$ \( -31809 - 84 T + T^{2} \)
$31$ \( -28368 - 84 T + T^{2} \)
$37$ \( 19756 + 304 T + T^{2} \)
$41$ \( -30144 - 168 T + T^{2} \)
$43$ \( 34224 + 372 T + T^{2} \)
$47$ \( 31696 + 584 T + T^{2} \)
$53$ \( -42713 + 484 T + T^{2} \)
$59$ \( 286881 + 1074 T + T^{2} \)
$61$ \( -16292 + 88 T + T^{2} \)
$67$ \( 438313 - 1430 T + T^{2} \)
$71$ \( 853404 + 1848 T + T^{2} \)
$73$ \( 412657 + 1294 T + T^{2} \)
$79$ \( -392756 + 832 T + T^{2} \)
$83$ \( -683604 - 528 T + T^{2} \)
$89$ \( 340816 - 1844 T + T^{2} \)
$97$ \( -218348 + 364 T + T^{2} \)
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