Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{-14})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{2} q^{5} -2 q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{2} q^{5} -2 q^{9} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{11} + \beta_{3} q^{13} + \beta_{2} q^{15} + ( \beta_{1} - \beta_{2} ) q^{17} + ( \beta_{1} - \beta_{2} ) q^{19} + ( -3 + \beta_{1} + \beta_{2} ) q^{23} + ( -2 - \beta_{3} ) q^{25} -5 q^{27} -3 q^{29} -\beta_{3} q^{31} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{33} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{37} + \beta_{3} q^{39} -3 q^{41} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{43} -2 \beta_{2} q^{45} -9 q^{47} + 7 q^{49} + ( \beta_{1} - \beta_{2} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -6 + 2 \beta_{3} ) q^{55} + ( \beta_{1} - \beta_{2} ) q^{57} + 2 \beta_{3} q^{59} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{61} + ( -5 \beta_{1} + 2 \beta_{2} ) q^{65} + ( -3 + \beta_{1} + \beta_{2} ) q^{69} + 3 \beta_{3} q^{71} + \beta_{3} q^{73} + ( -2 - \beta_{3} ) q^{75} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{79} + q^{81} + ( -\beta_{1} - \beta_{2} ) q^{83} + ( -3 + \beta_{3} ) q^{85} -3 q^{87} + ( -\beta_{1} - \beta_{2} ) q^{89} -\beta_{3} q^{93} + ( -3 + \beta_{3} ) q^{95} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{97} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 8q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 8q^{9} - 12q^{23} - 8q^{25} - 20q^{27} - 12q^{29} - 12q^{41} - 36q^{47} + 28q^{49} - 24q^{55} - 12q^{69} - 8q^{75} + 4q^{81} - 12q^{85} - 12q^{87} - 12q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 2\)
\(\nu^{3}\)\(=\)\(5 \beta_{2} - 4 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1839.1
1.22474 1.87083i
1.22474 + 1.87083i
−1.22474 1.87083i
−1.22474 + 1.87083i
0 1.00000 0 −1.22474 1.87083i 0 0 0 −2.00000 0
1839.2 0 1.00000 0 −1.22474 + 1.87083i 0 0 0 −2.00000 0
1839.3 0 1.00000 0 1.22474 1.87083i 0 0 0 −2.00000 0
1839.4 0 1.00000 0 1.22474 + 1.87083i 0 0 0 −2.00000 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
20.d odd 2 1 inner
23.b odd 2 1 inner
460.g 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.m.d yes 4
4.b odd 2 1 1840.2.m.a 4
5.b even 2 1 1840.2.m.a 4
20.d odd 2 1 inner 1840.2.m.d yes 4
23.b odd 2 1 inner 1840.2.m.d yes 4
92.b even 2 1 1840.2.m.a 4
115.c odd 2 1 1840.2.m.a 4
460.g even 2 1 inner 1840.2.m.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.a 4 4.b odd 2 1
1840.2.m.a 4 5.b even 2 1
1840.2.m.a 4 92.b even 2 1
1840.2.m.a 4 115.c odd 2 1
1840.2.m.d yes 4 1.a even 1 1 trivial
1840.2.m.d yes 4 20.d odd 2 1 inner
1840.2.m.d yes 4 23.b odd 2 1 inner
1840.2.m.d yes 4 460.g even 2 1 inner

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 25 + 4 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -24 + T^{2} )^{2} \)
$13$ \( ( 21 + T^{2} )^{2} \)
$17$ \( ( -6 + T^{2} )^{2} \)
$19$ \( ( -6 + T^{2} )^{2} \)
$23$ \( ( 23 + 6 T + T^{2} )^{2} \)
$29$ \( ( 3 + T )^{4} \)
$31$ \( ( 21 + T^{2} )^{2} \)
$37$ \( ( -24 + T^{2} )^{2} \)
$41$ \( ( 3 + T )^{4} \)
$43$ \( ( 126 + T^{2} )^{2} \)
$47$ \( ( 9 + T )^{4} \)
$53$ \( ( -24 + T^{2} )^{2} \)
$59$ \( ( 84 + T^{2} )^{2} \)
$61$ \( ( 126 + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( ( 189 + T^{2} )^{2} \)
$73$ \( ( 21 + T^{2} )^{2} \)
$79$ \( ( -54 + T^{2} )^{2} \)
$83$ \( ( 14 + T^{2} )^{2} \)
$89$ \( ( 14 + T^{2} )^{2} \)
$97$ \( ( -96 + T^{2} )^{2} \)
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