Properties

Label 1840.2.m.b
Level $1840$
Weight $2$
Character orbit 1840.m
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
CM discriminant -115
Inner twists $8$

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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: \(\Q(\sqrt{5}, \sqrt{-23})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} -\beta_{2} q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{5} -\beta_{2} q^{7} -3 q^{9} + 3 \beta_{1} q^{17} + \beta_{2} q^{23} + 5 q^{25} - q^{29} + \beta_{3} q^{31} + \beta_{3} q^{35} -5 \beta_{1} q^{37} + 7 q^{41} -2 \beta_{2} q^{43} -3 \beta_{1} q^{45} -16 q^{49} -\beta_{1} q^{53} + \beta_{3} q^{59} + 3 \beta_{2} q^{63} -\beta_{2} q^{67} + \beta_{3} q^{71} + 9 q^{81} + 3 \beta_{2} q^{83} + 15 q^{85} -2 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} + 20q^{25} - 4q^{29} + 28q^{41} - 64q^{49} + 36q^{81} + 60q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu \)\()/7\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 16 \nu \)\()/7\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 9\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{2} + 8 \beta_{1}\)

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
1.11803 + 2.39792i
1.11803 2.39792i
−1.11803 + 2.39792i
−1.11803 2.39792i
0 0 0 −2.23607 0 4.79583i 0 −3.00000 0
1839.2 0 0 0 −2.23607 0 4.79583i 0 −3.00000 0
1839.3 0 0 0 2.23607 0 4.79583i 0 −3.00000 0
1839.4 0 0 0 2.23607 0 4.79583i 0 −3.00000 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
115.c odd 2 1 CM by \(\Q(\sqrt{-115}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
23.b odd 2 1 inner
92.b even 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.b 4
4.b odd 2 1 inner 1840.2.m.b 4
5.b even 2 1 inner 1840.2.m.b 4
20.d odd 2 1 inner 1840.2.m.b 4
23.b odd 2 1 inner 1840.2.m.b 4
92.b even 2 1 inner 1840.2.m.b 4
115.c odd 2 1 CM 1840.2.m.b 4
460.g even 2 1 inner 1840.2.m.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.b 4 1.a even 1 1 trivial
1840.2.m.b 4 4.b odd 2 1 inner
1840.2.m.b 4 5.b even 2 1 inner
1840.2.m.b 4 20.d odd 2 1 inner
1840.2.m.b 4 23.b odd 2 1 inner
1840.2.m.b 4 92.b even 2 1 inner
1840.2.m.b 4 115.c odd 2 1 CM
1840.2.m.b 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 \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3} \)
\( T_{7}^{2} + 23 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( ( 23 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -45 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( 23 + T^{2} )^{2} \)
$29$ \( ( 1 + T )^{4} \)
$31$ \( ( 115 + T^{2} )^{2} \)
$37$ \( ( -125 + T^{2} )^{2} \)
$41$ \( ( -7 + T )^{4} \)
$43$ \( ( 92 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( -5 + T^{2} )^{2} \)
$59$ \( ( 115 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( 23 + T^{2} )^{2} \)
$71$ \( ( 115 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 207 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( -20 + T^{2} )^{2} \)
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