Properties

Label 1856.4.a.k
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 116)
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.

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

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 1856.4.a.k 2
4.b odd 2 1 1856.4.a.j 2
8.b even 2 1 464.4.a.d 2
8.d odd 2 1 116.4.a.a 2
24.f even 2 1 1044.4.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.4.a.a 2 8.d odd 2 1
464.4.a.d 2 8.b even 2 1
1044.4.a.d 2 24.f even 2 1
1856.4.a.j 2 4.b odd 2 1
1856.4.a.k 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{2} - 13 \)
\( T_{5}^{2} - 10 T_{5} - 27 \)
\( T_{7}^{2} - 20 T_{7} - 108 \)

Hecke characteristic polynomials

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