Properties

Label 1856.4.a.o
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: 3.3.148344.1
Defining polynomial: \(x^{3} - x^{2} - 70 x + 238\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 - \beta_{1} ) q^{3} + ( 7 - \beta_{2} ) q^{5} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{7} + ( 30 + 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -3 - \beta_{1} ) q^{3} + ( 7 - \beta_{2} ) q^{5} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{7} + ( 30 + 2 \beta_{1} + \beta_{2} ) q^{9} + ( 29 - 3 \beta_{1} + 2 \beta_{2} ) q^{11} + ( -41 - 2 \beta_{1} + \beta_{2} ) q^{13} + ( -19 - 5 \beta_{1} + 8 \beta_{2} ) q^{15} + ( 2 + 10 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 30 - 4 \beta_{1} + 2 \beta_{2} ) q^{19} + ( -88 + 2 \beta_{1} - 18 \beta_{2} ) q^{21} + ( -18 - 14 \beta_{2} ) q^{23} + ( 38 + 12 \beta_{1} - 15 \beta_{2} ) q^{25} + ( -107 - 3 \beta_{1} - 10 \beta_{2} ) q^{27} -29 q^{29} + ( -111 + 5 \beta_{1} + 2 \beta_{2} ) q^{31} + ( 53 - 36 \beta_{1} - 13 \beta_{2} ) q^{33} + ( -260 - 14 \beta_{1} + 10 \beta_{2} ) q^{35} + ( -64 + 8 \beta_{1} + 18 \beta_{2} ) q^{37} + ( 217 + 37 \beta_{1} - 6 \beta_{2} ) q^{39} + ( 4 + 34 \beta_{1} - 12 \beta_{2} ) q^{41} + ( -319 + 5 \beta_{1} + 6 \beta_{2} ) q^{43} + ( 92 - 2 \beta_{1} - 32 \beta_{2} ) q^{45} + ( -61 - 39 \beta_{1} - 12 \beta_{2} ) q^{47} + ( 337 + 32 \beta_{1} + 24 \beta_{2} ) q^{49} + ( -482 + 12 \beta_{1} + 6 \beta_{2} ) q^{51} + ( 377 - 10 \beta_{1} + 23 \beta_{2} ) q^{53} + ( -19 - 39 \beta_{1} + 2 \beta_{2} ) q^{55} + ( 98 - 38 \beta_{1} - 12 \beta_{2} ) q^{57} + ( 150 + 64 \beta_{1} - 26 \beta_{2} ) q^{59} + ( -130 - 6 \beta_{1} - 54 \beta_{2} ) q^{61} + ( 312 + 72 \beta_{1} + 88 \beta_{2} ) q^{63} + ( -397 - 22 \beta_{1} + 59 \beta_{2} ) q^{65} + ( -24 + 4 \beta_{1} + 16 \beta_{2} ) q^{67} + ( 82 + 46 \beta_{1} + 112 \beta_{2} ) q^{69} + ( -410 - 10 \beta_{1} + 44 \beta_{2} ) q^{71} + ( 808 + 8 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -660 + 4 \beta_{1} + 108 \beta_{2} ) q^{75} + ( 48 + 138 \beta_{1} + 30 \beta_{2} ) q^{77} + ( -195 - 149 \beta_{1} - 8 \beta_{2} ) q^{79} + ( -325 + 70 \beta_{1} + 56 \beta_{2} ) q^{81} + ( 166 - 80 \beta_{1} + 46 \beta_{2} ) q^{83} + ( 222 + 74 \beta_{1} - 68 \beta_{2} ) q^{85} + ( 87 + 29 \beta_{1} ) q^{87} + ( 668 - 6 \beta_{1} - 12 \beta_{2} ) q^{89} + ( 196 - 38 \beta_{1} - 102 \beta_{2} ) q^{91} + ( 89 + 112 \beta_{1} - 21 \beta_{2} ) q^{93} + ( -10 - 44 \beta_{1} + 6 \beta_{2} ) q^{95} + ( -116 + 22 \beta_{1} - 80 \beta_{2} ) q^{97} + ( 812 + 18 \beta_{1} + 86 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 10 q^{3} + 20 q^{5} - 8 q^{7} + 93 q^{9} + O(q^{10}) \) \( 3 q - 10 q^{3} + 20 q^{5} - 8 q^{7} + 93 q^{9} + 86 q^{11} - 124 q^{13} - 54 q^{15} + 14 q^{17} + 88 q^{19} - 280 q^{21} - 68 q^{23} + 111 q^{25} - 334 q^{27} - 87 q^{29} - 326 q^{31} + 110 q^{33} - 784 q^{35} - 166 q^{37} + 682 q^{39} + 34 q^{41} - 946 q^{43} + 242 q^{45} - 234 q^{47} + 1067 q^{49} - 1428 q^{51} + 1144 q^{53} - 94 q^{55} + 244 q^{57} + 488 q^{59} - 450 q^{61} + 1096 q^{63} - 1154 q^{65} - 52 q^{67} + 404 q^{69} - 1196 q^{71} + 2434 q^{73} - 1868 q^{75} + 312 q^{77} - 742 q^{79} - 849 q^{81} + 464 q^{83} + 672 q^{85} + 290 q^{87} + 1986 q^{89} + 448 q^{91} + 358 q^{93} - 68 q^{95} - 406 q^{97} + 2540 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 70 x + 238\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 4 \nu - 48 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 4 \beta_{1} + 48\)

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
6.07339
4.21773
−9.29111
0 −9.07339 0 −6.17957 0 34.5059 0 55.3263 0
1.2 0 −7.21773 0 20.3399 0 −22.2443 0 25.0956 0
1.3 0 6.29111 0 5.83968 0 −20.2616 0 12.5781 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.o 3
4.b odd 2 1 1856.4.a.v 3
8.b even 2 1 464.4.a.j 3
8.d odd 2 1 116.4.a.c 3
24.f even 2 1 1044.4.a.f 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.4.a.c 3 8.d odd 2 1
464.4.a.j 3 8.b even 2 1
1044.4.a.f 3 24.f even 2 1
1856.4.a.o 3 1.a even 1 1 trivial
1856.4.a.v 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{3} + 10 T_{3}^{2} - 37 T_{3} - 412 \)
\( T_{5}^{3} - 20 T_{5}^{2} - 43 T_{5} + 734 \)
\( T_{7}^{3} + 8 T_{7}^{2} - 1016 T_{7} - 15552 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -412 - 37 T + 10 T^{2} + T^{3} \)
$5$ \( 734 - 43 T - 20 T^{2} + T^{3} \)
$7$ \( -15552 - 1016 T + 8 T^{2} + T^{3} \)
$11$ \( 22716 + 1203 T - 86 T^{2} + T^{3} \)
$13$ \( 53334 + 4693 T + 124 T^{2} + T^{3} \)
$17$ \( 240296 - 7420 T - 14 T^{2} + T^{3} \)
$19$ \( 30192 + 852 T - 88 T^{2} + T^{3} \)
$23$ \( -1170336 - 33020 T + 68 T^{2} + T^{3} \)
$29$ \( ( 29 + T )^{3} \)
$31$ \( 981672 + 32835 T + 326 T^{2} + T^{3} \)
$37$ \( -7043616 - 54272 T + 166 T^{2} + T^{3} \)
$41$ \( 5243664 - 101144 T - 34 T^{2} + T^{3} \)
$43$ \( 28393308 + 289819 T + 946 T^{2} + T^{3} \)
$47$ \( -8573816 - 120045 T + 234 T^{2} + T^{3} \)
$53$ \( -8614194 + 338845 T - 1144 T^{2} + T^{3} \)
$59$ \( 71366448 - 306828 T - 488 T^{2} + T^{3} \)
$61$ \( -67933496 - 453324 T + 450 T^{2} + T^{3} \)
$67$ \( -1984128 - 46176 T + 52 T^{2} + T^{3} \)
$71$ \( -30178784 + 133964 T + 1196 T^{2} + T^{3} \)
$73$ \( -529660832 + 1969376 T - 2434 T^{2} + T^{3} \)
$79$ \( -1019501448 - 1404333 T + 742 T^{2} + T^{3} \)
$83$ \( 216436368 - 704876 T - 464 T^{2} + T^{3} \)
$89$ \( -269758448 + 1285896 T - 1986 T^{2} + T^{3} \)
$97$ \( -447258672 - 1085336 T + 406 T^{2} + T^{3} \)
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