Properties

Label 116.4.a.a
Level $116$
Weight $4$
Character orbit 116.a
Self dual yes
Analytic conductor $6.844$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 116 = 2^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 116.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.84422156067\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + (2 \beta - 5) q^{5} + (4 \beta - 10) q^{7} - 14 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + (2 \beta - 5) q^{5} + (4 \beta - 10) q^{7} - 14 q^{9} + (\beta - 16) q^{11} + ( - 6 \beta - 27) q^{13} + (5 \beta - 26) q^{15} + ( - 22 \beta - 22) q^{17} + ( - 12 \beta - 16) q^{19} + (10 \beta - 52) q^{21} + (10 \beta - 18) q^{23} + ( - 20 \beta - 48) q^{25} + 41 \beta q^{27} - 29 q^{29} + (3 \beta + 10) q^{31} + (16 \beta - 13) q^{33} + ( - 40 \beta + 154) q^{35} + (48 \beta - 72) q^{37} + (27 \beta + 78) q^{39} + ( - 2 \beta + 48) q^{41} + (9 \beta + 120) q^{43} + ( - 28 \beta + 70) q^{45} + ( - 49 \beta + 298) q^{47} + ( - 80 \beta - 35) q^{49} + (22 \beta + 286) q^{51} + (54 \beta - 17) q^{53} + ( - 37 \beta + 106) q^{55} + (16 \beta + 156) q^{57} + ( - 70 \beta + 362) q^{59} + (102 \beta - 306) q^{61} + ( - 56 \beta + 140) q^{63} + ( - 24 \beta - 21) q^{65} + ( - 204 \beta + 264) q^{67} + (18 \beta - 130) q^{69} + (166 \beta - 52) q^{71} + (168 \beta - 436) q^{73} + (48 \beta + 260) q^{75} + ( - 74 \beta + 212) q^{77} + (165 \beta - 410) q^{79} - 155 q^{81} + (90 \beta - 114) q^{83} + (66 \beta - 462) q^{85} + 29 \beta q^{87} + ( - 382 \beta - 16) q^{89} + ( - 48 \beta - 42) q^{91} + ( - 10 \beta - 39) q^{93} + (28 \beta - 232) q^{95} + ( - 42 \beta - 948) q^{97} + ( - 14 \beta + 224) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{5} - 20 q^{7} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{5} - 20 q^{7} - 28 q^{9} - 32 q^{11} - 54 q^{13} - 52 q^{15} - 44 q^{17} - 32 q^{19} - 104 q^{21} - 36 q^{23} - 96 q^{25} - 58 q^{29} + 20 q^{31} - 26 q^{33} + 308 q^{35} - 144 q^{37} + 156 q^{39} + 96 q^{41} + 240 q^{43} + 140 q^{45} + 596 q^{47} - 70 q^{49} + 572 q^{51} - 34 q^{53} + 212 q^{55} + 312 q^{57} + 724 q^{59} - 612 q^{61} + 280 q^{63} - 42 q^{65} + 528 q^{67} - 260 q^{69} - 104 q^{71} - 872 q^{73} + 520 q^{75} + 424 q^{77} - 820 q^{79} - 310 q^{81} - 228 q^{83} - 924 q^{85} - 32 q^{89} - 84 q^{91} - 78 q^{93} - 464 q^{95} - 1896 q^{97} + 448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
2.30278
−1.30278
0 −3.60555 0 2.21110 0 4.42221 0 −14.0000 0
1.2 0 3.60555 0 −12.2111 0 −24.4222 0 −14.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(1\)

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 116.4.a.a 2
3.b odd 2 1 1044.4.a.d 2
4.b odd 2 1 464.4.a.d 2
8.b even 2 1 1856.4.a.j 2
8.d odd 2 1 1856.4.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.4.a.a 2 1.a even 1 1 trivial
464.4.a.d 2 4.b odd 2 1
1044.4.a.d 2 3.b odd 2 1
1856.4.a.j 2 8.b even 2 1
1856.4.a.k 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 13 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(116))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 13 \) Copy content Toggle raw display
$5$ \( T^{2} + 10T - 27 \) Copy content Toggle raw display
$7$ \( T^{2} + 20T - 108 \) Copy content Toggle raw display
$11$ \( T^{2} + 32T + 243 \) Copy content Toggle raw display
$13$ \( T^{2} + 54T + 261 \) Copy content Toggle raw display
$17$ \( T^{2} + 44T - 5808 \) Copy content Toggle raw display
$19$ \( T^{2} + 32T - 1616 \) Copy content Toggle raw display
$23$ \( T^{2} + 36T - 976 \) Copy content Toggle raw display
$29$ \( (T + 29)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 20T - 17 \) Copy content Toggle raw display
$37$ \( T^{2} + 144T - 24768 \) Copy content Toggle raw display
$41$ \( T^{2} - 96T + 2252 \) Copy content Toggle raw display
$43$ \( T^{2} - 240T + 13347 \) Copy content Toggle raw display
$47$ \( T^{2} - 596T + 57591 \) Copy content Toggle raw display
$53$ \( T^{2} + 34T - 37619 \) Copy content Toggle raw display
$59$ \( T^{2} - 724T + 67344 \) Copy content Toggle raw display
$61$ \( T^{2} + 612T - 41616 \) Copy content Toggle raw display
$67$ \( T^{2} - 528T - 471312 \) Copy content Toggle raw display
$71$ \( T^{2} + 104T - 355524 \) Copy content Toggle raw display
$73$ \( T^{2} + 872T - 176816 \) Copy content Toggle raw display
$79$ \( T^{2} + 820T - 185825 \) Copy content Toggle raw display
$83$ \( T^{2} + 228T - 92304 \) Copy content Toggle raw display
$89$ \( T^{2} + 32T - 1896756 \) Copy content Toggle raw display
$97$ \( T^{2} + 1896 T + 875772 \) Copy content Toggle raw display
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