Properties

Label 1840.2.e.c
Level $1840$
Weight $2$
Character orbit 1840.e
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 + 2 \beta_{2} ) q^{5} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 + 2 \beta_{2} ) q^{5} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{7} + ( 2 + \beta_{2} ) q^{9} + ( 4 - \beta_{2} ) q^{11} + ( -\beta_{1} + \beta_{3} ) q^{13} + ( -\beta_{1} + 2 \beta_{3} ) q^{15} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{17} + ( -5 - 3 \beta_{2} ) q^{19} -3 q^{21} -\beta_{3} q^{23} + 5 q^{25} + ( 4 \beta_{1} + \beta_{3} ) q^{27} -6 \beta_{2} q^{29} + ( 7 + 3 \beta_{2} ) q^{31} + ( 5 \beta_{1} - \beta_{3} ) q^{33} + ( 3 \beta_{1} + 9 \beta_{3} ) q^{35} -6 \beta_{1} q^{37} + ( 1 - 2 \beta_{2} ) q^{39} + ( -4 + \beta_{2} ) q^{41} + ( 2 \beta_{1} - 8 \beta_{3} ) q^{43} + ( 4 + 3 \beta_{2} ) q^{45} + ( -6 \beta_{1} - 8 \beta_{3} ) q^{47} + ( -11 - 9 \beta_{2} ) q^{49} + ( -3 - \beta_{2} ) q^{51} -2 \beta_{3} q^{53} + ( 2 + 9 \beta_{2} ) q^{55} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{57} -6 q^{59} + ( -2 - 3 \beta_{2} ) q^{61} + ( 6 \beta_{1} + 9 \beta_{3} ) q^{63} + ( 3 \beta_{1} - \beta_{3} ) q^{65} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{67} + \beta_{2} q^{69} + ( -2 - \beta_{2} ) q^{71} + ( 4 \beta_{1} - 10 \beta_{3} ) q^{73} + 5 \beta_{1} q^{75} + ( 12 \beta_{1} + 9 \beta_{3} ) q^{77} + ( 2 + 6 \beta_{2} ) q^{79} + ( 2 + 6 \beta_{2} ) q^{81} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{83} + ( 5 \beta_{1} + 10 \beta_{3} ) q^{85} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{87} + ( 6 + 6 \beta_{2} ) q^{89} -3 \beta_{2} q^{91} + ( 4 \beta_{1} + 3 \beta_{3} ) q^{93} + ( -11 - 7 \beta_{2} ) q^{95} + ( 13 \beta_{1} + 8 \beta_{3} ) q^{97} + ( 7 + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{9} + O(q^{10}) \) \( 4q + 6q^{9} + 18q^{11} - 14q^{19} - 12q^{21} + 20q^{25} + 12q^{29} + 22q^{31} + 8q^{39} - 18q^{41} + 10q^{45} - 26q^{49} - 10q^{51} - 10q^{55} - 24q^{59} - 2q^{61} - 2q^{69} - 6q^{71} - 4q^{79} - 4q^{81} + 12q^{89} + 6q^{91} - 30q^{95} + 22q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
369.1
1.61803i
0.618034i
0.618034i
1.61803i
0 1.61803i 0 −2.23607 0 1.85410i 0 0.381966 0
369.2 0 0.618034i 0 2.23607 0 4.85410i 0 2.61803 0
369.3 0 0.618034i 0 2.23607 0 4.85410i 0 2.61803 0
369.4 0 1.61803i 0 −2.23607 0 1.85410i 0 0.381966 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.e.c 4
4.b odd 2 1 230.2.b.a 4
5.b even 2 1 inner 1840.2.e.c 4
5.c odd 4 1 9200.2.a.bo 2
5.c odd 4 1 9200.2.a.by 2
12.b even 2 1 2070.2.d.c 4
20.d odd 2 1 230.2.b.a 4
20.e even 4 1 1150.2.a.l 2
20.e even 4 1 1150.2.a.n 2
60.h even 2 1 2070.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.b.a 4 4.b odd 2 1
230.2.b.a 4 20.d odd 2 1
1150.2.a.l 2 20.e even 4 1
1150.2.a.n 2 20.e even 4 1
1840.2.e.c 4 1.a even 1 1 trivial
1840.2.e.c 4 5.b even 2 1 inner
2070.2.d.c 4 12.b even 2 1
2070.2.d.c 4 60.h even 2 1
9200.2.a.bo 2 5.c odd 4 1
9200.2.a.by 2 5.c odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 3 T_{3}^{2} + 1 \)
\( T_{7}^{4} + 27 T_{7}^{2} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + 3 T^{2} + T^{4} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( 81 + 27 T^{2} + T^{4} \)
$11$ \( ( 19 - 9 T + T^{2} )^{2} \)
$13$ \( 1 + 7 T^{2} + T^{4} \)
$17$ \( 25 + 35 T^{2} + T^{4} \)
$19$ \( ( 1 + 7 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( -36 - 6 T + T^{2} )^{2} \)
$31$ \( ( 19 - 11 T + T^{2} )^{2} \)
$37$ \( 1296 + 108 T^{2} + T^{4} \)
$41$ \( ( 19 + 9 T + T^{2} )^{2} \)
$43$ \( 5776 + 172 T^{2} + T^{4} \)
$47$ \( 400 + 140 T^{2} + T^{4} \)
$53$ \( ( 4 + T^{2} )^{2} \)
$59$ \( ( 6 + T )^{4} \)
$61$ \( ( -11 + T + T^{2} )^{2} \)
$67$ \( 16 + 28 T^{2} + T^{4} \)
$71$ \( ( 1 + 3 T + T^{2} )^{2} \)
$73$ \( 15376 + 328 T^{2} + T^{4} \)
$79$ \( ( -44 + 2 T + T^{2} )^{2} \)
$83$ \( 1936 + 92 T^{2} + T^{4} \)
$89$ \( ( -36 - 6 T + T^{2} )^{2} \)
$97$ \( 43681 + 427 T^{2} + T^{4} \)
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