Properties

Label 1840.2.m.f
Level $1840$
Weight $2$
Character orbit 1840.m
Analytic conductor $14.692$
Analytic rank $0$
Dimension $8$
CM no
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: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + \beta_{6} q^{5} + 2 \beta_{5} q^{7} + 2 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + \beta_{6} q^{5} + 2 \beta_{5} q^{7} + 2 q^{9} + \beta_{4} q^{13} + ( -3 \beta_{5} - \beta_{7} ) q^{15} + ( \beta_{1} - \beta_{6} ) q^{17} + ( -\beta_{5} - 2 \beta_{7} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{6} ) q^{21} + ( -\beta_{2} + 3 \beta_{5} ) q^{23} + \beta_{4} q^{25} + \beta_{2} q^{27} + 9 q^{29} -3 \beta_{3} q^{31} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{35} -5 \beta_{3} q^{39} -3 q^{41} -\beta_{5} q^{43} + 2 \beta_{6} q^{45} -3 \beta_{2} q^{47} - q^{49} + ( \beta_{5} + 2 \beta_{7} ) q^{51} + ( -2 \beta_{1} + 2 \beta_{6} ) q^{53} + ( -5 \beta_{1} + 5 \beta_{6} ) q^{57} + 2 \beta_{3} q^{59} + ( -3 \beta_{1} - 3 \beta_{6} ) q^{61} + 4 \beta_{5} q^{63} + 5 \beta_{1} q^{65} + 2 \beta_{5} q^{67} + ( 5 - 3 \beta_{1} - 3 \beta_{6} ) q^{69} -3 \beta_{3} q^{71} -3 \beta_{4} q^{73} -5 \beta_{3} q^{75} + ( \beta_{5} + 2 \beta_{7} ) q^{79} -11 q^{81} + 9 \beta_{5} q^{83} + ( -5 - \beta_{4} ) q^{85} -9 \beta_{2} q^{87} + ( \beta_{1} + \beta_{6} ) q^{89} + ( -2 \beta_{5} - 4 \beta_{7} ) q^{91} + 3 \beta_{4} q^{93} + ( -5 \beta_{2} - 5 \beta_{3} ) q^{95} + ( 6 \beta_{1} - 6 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{9} + O(q^{10}) \) \( 8q + 16q^{9} + 72q^{29} - 24q^{41} - 8q^{49} + 40q^{69} - 88q^{81} - 40q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 11 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{4} + 7 \)\()/3\)
\(\beta_{3}\)\(=\)\( -\nu^{6} - 6 \nu^{2} \)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{6} - 40 \nu^{2} \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{7} + \nu^{5} + 13 \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{7} + 29 \nu^{3} \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{7} - 3 \nu^{5} + 26 \nu^{3} - 15 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 2 \beta_{5} + 5 \beta_{1}\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{4} + 5 \beta_{3}\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{7} + 5 \beta_{6} - 6 \beta_{5}\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{2} - 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{7} + 22 \beta_{5} - 25 \beta_{1}\)\()/10\)
\(\nu^{6}\)\(=\)\((\)\(9 \beta_{4} - 20 \beta_{3}\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(29 \beta_{7} - 65 \beta_{6} + 87 \beta_{5}\)\()/10\)

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
0.437016 0.437016i
0.437016 + 0.437016i
−0.437016 0.437016i
−0.437016 + 0.437016i
1.14412 1.14412i
1.14412 + 1.14412i
−1.14412 1.14412i
−1.14412 + 1.14412i
0 −2.23607 0 −1.58114 1.58114i 0 2.82843i 0 2.00000 0
1839.2 0 −2.23607 0 −1.58114 + 1.58114i 0 2.82843i 0 2.00000 0
1839.3 0 −2.23607 0 1.58114 1.58114i 0 2.82843i 0 2.00000 0
1839.4 0 −2.23607 0 1.58114 + 1.58114i 0 2.82843i 0 2.00000 0
1839.5 0 2.23607 0 −1.58114 1.58114i 0 2.82843i 0 2.00000 0
1839.6 0 2.23607 0 −1.58114 + 1.58114i 0 2.82843i 0 2.00000 0
1839.7 0 2.23607 0 1.58114 1.58114i 0 2.82843i 0 2.00000 0
1839.8 0 2.23607 0 1.58114 + 1.58114i 0 2.82843i 0 2.00000 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
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
115.c 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.f 8
4.b odd 2 1 inner 1840.2.m.f 8
5.b even 2 1 inner 1840.2.m.f 8
20.d odd 2 1 inner 1840.2.m.f 8
23.b odd 2 1 inner 1840.2.m.f 8
92.b even 2 1 inner 1840.2.m.f 8
115.c odd 2 1 inner 1840.2.m.f 8
460.g even 2 1 inner 1840.2.m.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.f 8 1.a even 1 1 trivial
1840.2.m.f 8 4.b odd 2 1 inner
1840.2.m.f 8 5.b even 2 1 inner
1840.2.m.f 8 20.d odd 2 1 inner
1840.2.m.f 8 23.b odd 2 1 inner
1840.2.m.f 8 92.b even 2 1 inner
1840.2.m.f 8 115.c odd 2 1 inner
1840.2.m.f 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 \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3}^{2} - 5 \)
\( T_{7}^{2} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -5 + T^{2} )^{4} \)
$5$ \( ( 25 + T^{4} )^{2} \)
$7$ \( ( 8 + T^{2} )^{4} \)
$11$ \( T^{8} \)
$13$ \( ( 25 + T^{2} )^{4} \)
$17$ \( ( -10 + T^{2} )^{4} \)
$19$ \( ( -50 + T^{2} )^{4} \)
$23$ \( ( 529 + 26 T^{2} + T^{4} )^{2} \)
$29$ \( ( -9 + T )^{8} \)
$31$ \( ( 45 + T^{2} )^{4} \)
$37$ \( T^{8} \)
$41$ \( ( 3 + T )^{8} \)
$43$ \( ( 2 + T^{2} )^{4} \)
$47$ \( ( -45 + T^{2} )^{4} \)
$53$ \( ( -40 + T^{2} )^{4} \)
$59$ \( ( 20 + T^{2} )^{4} \)
$61$ \( ( 90 + T^{2} )^{4} \)
$67$ \( ( 8 + T^{2} )^{4} \)
$71$ \( ( 45 + T^{2} )^{4} \)
$73$ \( ( 225 + T^{2} )^{4} \)
$79$ \( ( -50 + T^{2} )^{4} \)
$83$ \( ( 162 + T^{2} )^{4} \)
$89$ \( ( 10 + T^{2} )^{4} \)
$97$ \( ( -360 + T^{2} )^{4} \)
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