Properties

Label 1840.2.a.u
Level $1840$
Weight $2$
Character orbit 1840.a
Self dual yes
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{3} + q^{5} + ( 1 + \beta_{3} ) q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{3} + q^{5} + ( 1 + \beta_{3} ) q^{7} + ( 2 + \beta_{2} ) q^{9} + ( -1 - \beta_{1} + \beta_{3} ) q^{11} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( 1 + \beta_{2} ) q^{15} + \beta_{1} q^{17} + ( 1 + \beta_{1} - \beta_{3} ) q^{19} + ( 3 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{21} + q^{23} + q^{25} + ( 3 - \beta_{2} ) q^{27} + ( 5 - \beta_{1} + \beta_{2} ) q^{29} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( -1 - 3 \beta_{1} + \beta_{3} ) q^{33} + ( 1 + \beta_{3} ) q^{35} + ( 1 + 4 \beta_{2} - \beta_{3} ) q^{37} + ( \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{39} + ( 3 + \beta_{1} - \beta_{2} ) q^{41} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{43} + ( 2 + \beta_{2} ) q^{45} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{47} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{49} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{51} + ( 4 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -1 - \beta_{1} + \beta_{3} ) q^{55} + ( 1 + 3 \beta_{1} - \beta_{3} ) q^{57} + ( -6 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{59} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{61} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{63} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{65} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{67} + ( 1 + \beta_{2} ) q^{69} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{71} + ( -8 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{73} + ( 1 + \beta_{2} ) q^{75} + ( 4 - 2 \beta_{1} ) q^{77} + ( -1 - 3 \beta_{1} + \beta_{3} ) q^{79} -7 q^{81} + ( 4 - \beta_{1} - 2 \beta_{2} ) q^{83} + \beta_{1} q^{85} + ( 7 - 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{87} + ( -1 + \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{89} + ( 7 + \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{91} + ( -2 - 3 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{93} + ( 1 + \beta_{1} - \beta_{3} ) q^{95} + ( -5 - \beta_{1} - \beta_{3} ) q^{97} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 4q^{5} + 3q^{7} + 6q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 4q^{5} + 3q^{7} + 6q^{9} - 4q^{11} + 2q^{15} - q^{17} + 4q^{19} + 10q^{21} + 4q^{23} + 4q^{25} + 14q^{27} + 19q^{29} + q^{31} - 2q^{33} + 3q^{35} - 3q^{37} + 13q^{41} + 6q^{43} + 6q^{45} - 6q^{47} + 9q^{49} + 8q^{51} + 19q^{53} - 4q^{55} + 2q^{57} - 23q^{59} + 13q^{63} + 3q^{67} + 2q^{69} + 3q^{71} - 32q^{73} + 2q^{75} + 18q^{77} - 2q^{79} - 28q^{81} + 21q^{83} - q^{85} + 18q^{87} + 40q^{91} - 8q^{93} + 4q^{95} - 18q^{97} - 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + \beta_{1} + 7\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + \beta_{2} + 3 \beta_{1} + 5\)

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
1.32973
−0.329727
2.69353
−1.69353
0 −1.56155 0 1.00000 0 −4.06562 0 −0.561553 0
1.2 0 −1.56155 0 1.00000 0 3.50407 0 −0.561553 0
1.3 0 2.56155 0 1.00000 0 0.819031 0 3.56155 0
1.4 0 2.56155 0 1.00000 0 2.74252 0 3.56155 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 1840.2.a.u 4
4.b odd 2 1 115.2.a.c 4
5.b even 2 1 9200.2.a.cl 4
8.b even 2 1 7360.2.a.cg 4
8.d odd 2 1 7360.2.a.cj 4
12.b even 2 1 1035.2.a.o 4
20.d odd 2 1 575.2.a.h 4
20.e even 4 2 575.2.b.e 8
28.d even 2 1 5635.2.a.v 4
60.h even 2 1 5175.2.a.bx 4
92.b even 2 1 2645.2.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.a.c 4 4.b odd 2 1
575.2.a.h 4 20.d odd 2 1
575.2.b.e 8 20.e even 4 2
1035.2.a.o 4 12.b even 2 1
1840.2.a.u 4 1.a even 1 1 trivial
2645.2.a.m 4 92.b even 2 1
5175.2.a.bx 4 60.h even 2 1
5635.2.a.v 4 28.d even 2 1
7360.2.a.cg 4 8.b even 2 1
7360.2.a.cj 4 8.d odd 2 1
9200.2.a.cl 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{2} - T_{3} - 4 \)
\( T_{7}^{4} - 3 T_{7}^{3} - 14 T_{7}^{2} + 52 T_{7} - 32 \)
\( T_{11}^{4} + 4 T_{11}^{3} - 16 T_{11}^{2} - 40 T_{11} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -4 - T + T^{2} )^{2} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( -32 + 52 T - 14 T^{2} - 3 T^{3} + T^{4} \)
$11$ \( 32 - 40 T - 16 T^{2} + 4 T^{3} + T^{4} \)
$13$ \( 212 - 41 T^{2} + T^{4} \)
$17$ \( 32 - 24 T - 18 T^{2} + T^{3} + T^{4} \)
$19$ \( 32 + 40 T - 16 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( ( -1 + T )^{4} \)
$29$ \( 202 - 269 T + 117 T^{2} - 19 T^{3} + T^{4} \)
$31$ \( 2144 - 11 T - 101 T^{2} - T^{3} + T^{4} \)
$37$ \( 2008 + 16 T - 116 T^{2} + 3 T^{3} + T^{4} \)
$41$ \( -94 - 3 T + 45 T^{2} - 13 T^{3} + T^{4} \)
$43$ \( 128 + 16 T - 36 T^{2} - 6 T^{3} + T^{4} \)
$47$ \( -128 - 548 T - 83 T^{2} + 6 T^{3} + T^{4} \)
$53$ \( -8776 + 2092 T - 34 T^{2} - 19 T^{3} + T^{4} \)
$59$ \( -3136 - 560 T + 100 T^{2} + 23 T^{3} + T^{4} \)
$61$ \( -32 + 136 T - 56 T^{2} + T^{4} \)
$67$ \( 2032 + 212 T - 98 T^{2} - 3 T^{3} + T^{4} \)
$71$ \( -8 + 535 T - 149 T^{2} - 3 T^{3} + T^{4} \)
$73$ \( 1684 + 1392 T + 343 T^{2} + 32 T^{3} + T^{4} \)
$79$ \( 512 + 352 T - 140 T^{2} + 2 T^{3} + T^{4} \)
$83$ \( -1216 + 224 T + 96 T^{2} - 21 T^{3} + T^{4} \)
$89$ \( -2752 - 1496 T - 216 T^{2} + T^{4} \)
$97$ \( -1072 - 200 T + 72 T^{2} + 18 T^{3} + T^{4} \)
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