Properties

Label 1856.4.a.q
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.4481.1
Defining polynomial: \(x^{3} - 17 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 232)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -4 - \beta_{1} - \beta_{2} ) q^{5} + ( -13 + \beta_{1} - \beta_{2} ) q^{7} + ( 20 + 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -4 - \beta_{1} - \beta_{2} ) q^{5} + ( -13 + \beta_{1} - \beta_{2} ) q^{7} + ( 20 + 2 \beta_{2} ) q^{9} + ( -21 + 3 \beta_{1} + 2 \beta_{2} ) q^{11} + ( -10 + 5 \beta_{1} - \beta_{2} ) q^{13} + ( -28 - 14 \beta_{1} + \beta_{2} ) q^{15} + ( 81 - 3 \beta_{1} - \beta_{2} ) q^{17} + ( -104 + 6 \beta_{1} ) q^{19} + ( 73 - 21 \beta_{1} + 5 \beta_{2} ) q^{21} + ( -9 + 5 \beta_{1} - 3 \beta_{2} ) q^{23} + ( 144 + 12 \beta_{1} - 4 \beta_{2} ) q^{25} + ( -21 + 11 \beta_{1} - 6 \beta_{2} ) q^{27} + 29 q^{29} + ( 84 + 16 \beta_{1} + 3 \beta_{2} ) q^{31} + 131 q^{33} + ( 241 - 9 \beta_{1} + 5 \beta_{2} ) q^{35} + ( -68 - 24 \beta_{1} - 4 \beta_{2} ) q^{37} + ( 254 - 14 \beta_{1} + 13 \beta_{2} ) q^{39} + ( -159 + 11 \beta_{1} - 7 \beta_{2} ) q^{41} + ( -31 - 59 \beta_{1} + 4 \beta_{2} ) q^{43} + ( -522 - 6 \beta_{1} - 4 \beta_{2} ) q^{45} + ( 126 + 28 \beta_{1} + \beta_{2} ) q^{47} + ( 135 - 58 \beta_{1} + 22 \beta_{2} ) q^{49} + ( -205 + 69 \beta_{1} - 3 \beta_{2} ) q^{51} + ( -332 - 47 \beta_{1} + 11 \beta_{2} ) q^{53} + ( -454 - 10 \beta_{1} + 37 \beta_{2} ) q^{55} + ( 380 - 98 \beta_{1} + 12 \beta_{2} ) q^{57} + ( 133 - 17 \beta_{1} + 35 \beta_{2} ) q^{59} + ( 183 + 35 \beta_{1} + 25 \beta_{2} ) q^{61} + ( -758 + 70 \beta_{1} - 30 \beta_{2} ) q^{63} + ( 101 - 72 \beta_{1} + 2 \beta_{2} ) q^{65} + ( 266 - 6 \beta_{1} - 22 \beta_{2} ) q^{67} + ( 281 - 31 \beta_{1} + 19 \beta_{2} ) q^{69} + ( -266 - 74 \beta_{1} - 16 \beta_{2} ) q^{71} + ( 538 - 42 \beta_{1} - 6 \beta_{2} ) q^{73} + ( 464 + 120 \beta_{1} + 36 \beta_{2} ) q^{75} + ( -45 - 31 \beta_{1} + 23 \beta_{2} ) q^{77} + ( 144 + 118 \beta_{1} + 5 \beta_{2} ) q^{79} + ( 71 - 64 \beta_{1} - 14 \beta_{2} ) q^{81} + ( -497 + 53 \beta_{1} + 29 \beta_{2} ) q^{83} + ( -7 - 43 \beta_{1} - 89 \beta_{2} ) q^{85} + ( -29 + 29 \beta_{1} ) q^{87} + ( -819 + 59 \beta_{1} + 17 \beta_{2} ) q^{89} + ( 679 - 135 \beta_{1} + 35 \beta_{2} ) q^{91} + ( 610 + 127 \beta_{1} + 23 \beta_{2} ) q^{93} + ( 224 + 14 \beta_{1} + 104 \beta_{2} ) q^{95} + ( -33 + 9 \beta_{1} + 71 \beta_{2} ) q^{97} + ( 436 + 50 \beta_{1} - 54 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9} - 65 q^{11} - 29 q^{13} - 85 q^{15} + 244 q^{17} - 312 q^{19} + 214 q^{21} - 24 q^{23} + 436 q^{25} - 57 q^{27} + 87 q^{29} + 249 q^{31} + 393 q^{33} + 718 q^{35} - 200 q^{37} + 749 q^{39} - 470 q^{41} - 97 q^{43} - 1562 q^{45} + 377 q^{47} + 383 q^{49} - 612 q^{51} - 1007 q^{53} - 1399 q^{55} + 1128 q^{57} + 364 q^{59} + 524 q^{61} - 2244 q^{63} + 301 q^{65} + 820 q^{67} + 824 q^{69} - 782 q^{71} + 1620 q^{73} + 1356 q^{75} - 158 q^{77} + 427 q^{79} + 227 q^{81} - 1520 q^{83} + 68 q^{85} - 87 q^{87} - 2474 q^{89} + 2002 q^{91} + 1807 q^{93} + 568 q^{95} - 170 q^{97} + 1362 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 17 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 23 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 23\)\()/2\)

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
−3.86388
−0.476971
4.34085
0 −8.72775 0 −10.8591 0 −35.3146 0 49.1737 0
1.2 0 −1.95394 0 18.5450 0 7.63711 0 −23.1821 0
1.3 0 7.68170 0 −18.6859 0 −10.3225 0 32.0084 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.q 3
4.b odd 2 1 1856.4.a.t 3
8.b even 2 1 232.4.a.a 3
8.d odd 2 1 464.4.a.h 3
24.h odd 2 1 2088.4.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.a.a 3 8.b even 2 1
464.4.a.h 3 8.d odd 2 1
1856.4.a.q 3 1.a even 1 1 trivial
1856.4.a.t 3 4.b odd 2 1
2088.4.a.a 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{3} + 3 T_{3}^{2} - 65 T_{3} - 131 \)
\( T_{5}^{3} + 11 T_{5}^{2} - 345 T_{5} - 3763 \)
\( T_{7}^{3} + 38 T_{7}^{2} + 16 T_{7} - 2784 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -131 - 65 T + 3 T^{2} + T^{3} \)
$5$ \( -3763 - 345 T + 11 T^{2} + T^{3} \)
$7$ \( -2784 + 16 T + 38 T^{2} + T^{3} \)
$11$ \( -17161 - 393 T + 65 T^{2} + T^{3} \)
$13$ \( 11819 - 1977 T + 29 T^{2} + T^{3} \)
$17$ \( -462496 + 18996 T - 244 T^{2} + T^{3} \)
$19$ \( 856448 + 30000 T + 312 T^{2} + T^{3} \)
$23$ \( 76432 - 5324 T + 24 T^{2} + T^{3} \)
$29$ \( ( -29 + T )^{3} \)
$31$ \( -3931 + 1963 T - 249 T^{2} + T^{3} \)
$37$ \( 732672 - 27712 T + 200 T^{2} + T^{3} \)
$41$ \( 670464 + 44816 T + 470 T^{2} + T^{3} \)
$43$ \( -15326777 - 248729 T + 97 T^{2} + T^{3} \)
$47$ \( 2208837 - 5173 T - 377 T^{2} + T^{3} \)
$53$ \( -67774943 + 123887 T + 1007 T^{2} + T^{3} \)
$59$ \( 91904896 - 437020 T - 364 T^{2} + T^{3} \)
$61$ \( 68889888 - 180108 T - 524 T^{2} + T^{3} \)
$67$ \( 521984 + 54016 T - 820 T^{2} + T^{3} \)
$71$ \( 10922248 - 212644 T + 782 T^{2} + T^{3} \)
$73$ \( -75323392 + 752064 T - 1620 T^{2} + T^{3} \)
$79$ \( -63730809 - 871389 T - 427 T^{2} + T^{3} \)
$83$ \( -77896912 + 340084 T + 1520 T^{2} + T^{3} \)
$89$ \( 261171936 + 1740368 T + 2474 T^{2} + T^{3} \)
$97$ \( 692649728 - 1771632 T + 170 T^{2} + T^{3} \)
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