Properties

Label 1840.2.e.a
Level $1840$
Weight $2$
Character orbit 1840.e
Analytic conductor $14.692$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{3} + ( 2 + i ) q^{5} -3 i q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} + ( 2 + i ) q^{5} -3 i q^{7} - q^{9} -6 i q^{13} + ( -2 + 4 i ) q^{15} + 7 i q^{17} -4 q^{19} + 6 q^{21} + i q^{23} + ( 3 + 4 i ) q^{25} + 4 i q^{27} + 9 q^{29} + 3 q^{31} + ( 3 - 6 i ) q^{35} + 7 i q^{37} + 12 q^{39} + 9 q^{41} + 4 i q^{43} + ( -2 - i ) q^{45} -2 i q^{47} -2 q^{49} -14 q^{51} -7 i q^{53} -8 i q^{57} + 9 q^{59} -2 q^{61} + 3 i q^{63} + ( 6 - 12 i ) q^{65} + 13 i q^{67} -2 q^{69} + 13 q^{71} -4 i q^{73} + ( -8 + 6 i ) q^{75} -2 q^{79} -11 q^{81} + 11 i q^{83} + ( -7 + 14 i ) q^{85} + 18 i q^{87} + 10 q^{89} -18 q^{91} + 6 i q^{93} + ( -8 - 4 i ) q^{95} -2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{5} - 2q^{9} - 4q^{15} - 8q^{19} + 12q^{21} + 6q^{25} + 18q^{29} + 6q^{31} + 6q^{35} + 24q^{39} + 18q^{41} - 4q^{45} - 4q^{49} - 28q^{51} + 18q^{59} - 4q^{61} + 12q^{65} - 4q^{69} + 26q^{71} - 16q^{75} - 4q^{79} - 22q^{81} - 14q^{85} + 20q^{89} - 36q^{91} - 16q^{95} + O(q^{100}) \)

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
1.00000i
1.00000i
0 2.00000i 0 2.00000 1.00000i 0 3.00000i 0 −1.00000 0
369.2 0 2.00000i 0 2.00000 + 1.00000i 0 3.00000i 0 −1.00000 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
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.a 2
4.b odd 2 1 920.2.e.a 2
5.b even 2 1 inner 1840.2.e.a 2
5.c odd 4 1 9200.2.a.d 1
5.c odd 4 1 9200.2.a.bi 1
20.d odd 2 1 920.2.e.a 2
20.e even 4 1 4600.2.a.b 1
20.e even 4 1 4600.2.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.a 2 4.b odd 2 1
920.2.e.a 2 20.d odd 2 1
1840.2.e.a 2 1.a even 1 1 trivial
1840.2.e.a 2 5.b even 2 1 inner
4600.2.a.b 1 20.e even 4 1
4600.2.a.o 1 20.e even 4 1
9200.2.a.d 1 5.c odd 4 1
9200.2.a.bi 1 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 \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{7}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 49 + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( -9 + T )^{2} \)
$31$ \( ( -3 + T )^{2} \)
$37$ \( 49 + T^{2} \)
$41$ \( ( -9 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 4 + T^{2} \)
$53$ \( 49 + T^{2} \)
$59$ \( ( -9 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 169 + T^{2} \)
$71$ \( ( -13 + T )^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( ( 2 + T )^{2} \)
$83$ \( 121 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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