Properties

Label 1840.2.m.e
Level $1840$
Weight $2$
Character orbit 1840.m
Analytic conductor $14.692$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.14166950625.1
Defining polynomial: \(x^{8} - 9 x^{6} + 32 x^{4} - 441 x^{2} + 2401\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + \beta_{7} q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{5} + \beta_{7} q^{7} -3 q^{9} + ( -\beta_{1} + \beta_{4} ) q^{17} + ( \beta_{3} + \beta_{7} ) q^{23} + 5 q^{25} -\beta_{5} q^{29} + ( -2 \beta_{2} + \beta_{6} ) q^{31} + ( 2 \beta_{2} + \beta_{6} ) q^{35} + ( 3 \beta_{1} + \beta_{4} ) q^{37} + ( -4 - \beta_{5} ) q^{41} + ( -2 \beta_{3} - 2 \beta_{7} ) q^{43} -3 \beta_{1} q^{45} + ( -3 - \beta_{5} ) q^{49} + ( -\beta_{1} - 3 \beta_{4} ) q^{53} + ( -6 \beta_{2} + \beta_{6} ) q^{59} -3 \beta_{7} q^{63} + ( 4 \beta_{3} - 3 \beta_{7} ) q^{67} + ( 6 \beta_{2} + \beta_{6} ) q^{71} + 9 q^{81} + ( -4 \beta_{3} + \beta_{7} ) q^{83} + ( -8 - \beta_{5} ) q^{85} -2 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} + 40q^{25} + 4q^{29} - 28q^{41} - 20q^{49} + 72q^{81} - 60q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 9 x^{6} + 32 x^{4} - 441 x^{2} + 2401\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 9 \nu^{7} - 32 \nu^{5} - 496 \nu^{3} - 2401 \nu \)\()/5488\)
\(\beta_{2}\)\(=\)\((\)\( -9 \nu^{6} + 32 \nu^{4} - 288 \nu^{2} + 3185 \)\()/784\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 9 \nu^{5} + 32 \nu^{3} - 98 \nu \)\()/343\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 9 \nu^{5} - 32 \nu^{3} + 784 \nu \)\()/343\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 9 \nu^{4} + 17 \nu^{2} + 196 \)\()/49\)
\(\beta_{6}\)\(=\)\((\)\( -29 \nu^{6} + 16 \nu^{4} - 144 \nu^{2} + 8869 \)\()/784\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{5} + 18 \nu^{3} - 315 \nu \)\()/98\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + \beta_{5} - 5 \beta_{2} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{3} - 8 \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{6} + 9 \beta_{5} + 13 \beta_{2} + 13\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(32 \beta_{7} + 31 \beta_{4} - 63 \beta_{3} - 32 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(-32 \beta_{6} + 16 \beta_{2} + 297\)
\(\nu^{7}\)\(=\)\((\)\(224 \beta_{7} + 377 \beta_{4} + 153 \beta_{3} + 224 \beta_{1}\)\()/2\)

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
2.63567 0.230712i
−1.51764 2.16720i
−1.51764 + 2.16720i
2.63567 + 0.230712i
−2.63567 0.230712i
1.51764 2.16720i
1.51764 + 2.16720i
−2.63567 + 0.230712i
0 0 0 −2.23607 0 4.33441i 0 −3.00000 0
1839.2 0 0 0 −2.23607 0 0.461424i 0 −3.00000 0
1839.3 0 0 0 −2.23607 0 0.461424i 0 −3.00000 0
1839.4 0 0 0 −2.23607 0 4.33441i 0 −3.00000 0
1839.5 0 0 0 2.23607 0 4.33441i 0 −3.00000 0
1839.6 0 0 0 2.23607 0 0.461424i 0 −3.00000 0
1839.7 0 0 0 2.23607 0 0.461424i 0 −3.00000 0
1839.8 0 0 0 2.23607 0 4.33441i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1839.8
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.e 8
4.b odd 2 1 inner 1840.2.m.e 8
5.b even 2 1 inner 1840.2.m.e 8
20.d odd 2 1 inner 1840.2.m.e 8
23.b odd 2 1 inner 1840.2.m.e 8
92.b even 2 1 inner 1840.2.m.e 8
115.c odd 2 1 CM 1840.2.m.e 8
460.g even 2 1 inner 1840.2.m.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.e 8 1.a even 1 1 trivial
1840.2.m.e 8 4.b odd 2 1 inner
1840.2.m.e 8 5.b even 2 1 inner
1840.2.m.e 8 20.d odd 2 1 inner
1840.2.m.e 8 23.b odd 2 1 inner
1840.2.m.e 8 92.b even 2 1 inner
1840.2.m.e 8 115.c odd 2 1 CM
1840.2.m.e 8 460.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 \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3} \)
\( T_{7}^{4} + 19 T_{7}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( -5 + T^{2} )^{4} \)
$7$ \( ( 4 + 19 T^{2} + T^{4} )^{2} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( ( 36 - 57 T^{2} + T^{4} )^{2} \)
$19$ \( T^{8} \)
$23$ \( ( 23 + T^{2} )^{4} \)
$29$ \( ( -86 - T + T^{2} )^{4} \)
$31$ \( ( 484 + 71 T^{2} + T^{4} )^{2} \)
$37$ \( ( 196 - 97 T^{2} + T^{4} )^{2} \)
$41$ \( ( -74 + 7 T + T^{2} )^{4} \)
$43$ \( ( 92 + T^{2} )^{4} \)
$47$ \( T^{8} \)
$53$ \( ( 23716 - 313 T^{2} + T^{4} )^{2} \)
$59$ \( ( 3844 + 239 T^{2} + T^{4} )^{2} \)
$61$ \( T^{8} \)
$67$ \( ( 31684 + 379 T^{2} + T^{4} )^{2} \)
$71$ \( ( 9604 + 311 T^{2} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( ( 1764 + 291 T^{2} + T^{4} )^{2} \)
$89$ \( T^{8} \)
$97$ \( ( -20 + T^{2} )^{4} \)
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