Properties

Label 1840.2.e.c
Level $1840$
Weight $2$
Character orbit 1840.e
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (2 \beta_{2} + 1) q^{5} + (3 \beta_{3} + 3 \beta_1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (2 \beta_{2} + 1) q^{5} + (3 \beta_{3} + 3 \beta_1) q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_{2} + 4) q^{11} + (\beta_{3} - \beta_1) q^{13} + (2 \beta_{3} - \beta_1) q^{15} + (4 \beta_{3} + 3 \beta_1) q^{17} + ( - 3 \beta_{2} - 5) q^{19} - 3 q^{21} - \beta_{3} q^{23} + 5 q^{25} + (\beta_{3} + 4 \beta_1) q^{27} - 6 \beta_{2} q^{29} + (3 \beta_{2} + 7) q^{31} + ( - \beta_{3} + 5 \beta_1) q^{33} + (9 \beta_{3} + 3 \beta_1) q^{35} - 6 \beta_1 q^{37} + ( - 2 \beta_{2} + 1) q^{39} + (\beta_{2} - 4) q^{41} + ( - 8 \beta_{3} + 2 \beta_1) q^{43} + (3 \beta_{2} + 4) q^{45} + ( - 8 \beta_{3} - 6 \beta_1) q^{47} + ( - 9 \beta_{2} - 11) q^{49} + ( - \beta_{2} - 3) q^{51} - 2 \beta_{3} q^{53} + (9 \beta_{2} + 2) q^{55} + ( - 3 \beta_{3} - 2 \beta_1) q^{57} - 6 q^{59} + ( - 3 \beta_{2} - 2) q^{61} + (9 \beta_{3} + 6 \beta_1) q^{63} + ( - \beta_{3} + 3 \beta_1) q^{65} + (2 \beta_{3} - 2 \beta_1) q^{67} + \beta_{2} q^{69} + ( - \beta_{2} - 2) q^{71} + ( - 10 \beta_{3} + 4 \beta_1) q^{73} + 5 \beta_1 q^{75} + (9 \beta_{3} + 12 \beta_1) q^{77} + (6 \beta_{2} + 2) q^{79} + (6 \beta_{2} + 2) q^{81} + (2 \beta_{3} + 6 \beta_1) q^{83} + (10 \beta_{3} + 5 \beta_1) q^{85} + ( - 6 \beta_{3} + 6 \beta_1) q^{87} + (6 \beta_{2} + 6) q^{89} - 3 \beta_{2} q^{91} + (3 \beta_{3} + 4 \beta_1) q^{93} + ( - 7 \beta_{2} - 11) q^{95} + (8 \beta_{3} + 13 \beta_1) q^{97} + (3 \beta_{2} + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{9} + 18 q^{11} - 14 q^{19} - 12 q^{21} + 20 q^{25} + 12 q^{29} + 22 q^{31} + 8 q^{39} - 18 q^{41} + 10 q^{45} - 26 q^{49} - 10 q^{51} - 10 q^{55} - 24 q^{59} - 2 q^{61} - 2 q^{69} - 6 q^{71} - 4 q^{79} - 4 q^{81} + 12 q^{89} + 6 q^{91} - 30 q^{95} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

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
1.61803i
0.618034i
0.618034i
1.61803i
0 1.61803i 0 −2.23607 0 1.85410i 0 0.381966 0
369.2 0 0.618034i 0 2.23607 0 4.85410i 0 2.61803 0
369.3 0 0.618034i 0 2.23607 0 4.85410i 0 2.61803 0
369.4 0 1.61803i 0 −2.23607 0 1.85410i 0 0.381966 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.c 4
4.b odd 2 1 230.2.b.a 4
5.b even 2 1 inner 1840.2.e.c 4
5.c odd 4 1 9200.2.a.bo 2
5.c odd 4 1 9200.2.a.by 2
12.b even 2 1 2070.2.d.c 4
20.d odd 2 1 230.2.b.a 4
20.e even 4 1 1150.2.a.l 2
20.e even 4 1 1150.2.a.n 2
60.h even 2 1 2070.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.b.a 4 4.b odd 2 1
230.2.b.a 4 20.d odd 2 1
1150.2.a.l 2 20.e even 4 1
1150.2.a.n 2 20.e even 4 1
1840.2.e.c 4 1.a even 1 1 trivial
1840.2.e.c 4 5.b even 2 1 inner
2070.2.d.c 4 12.b even 2 1
2070.2.d.c 4 60.h even 2 1
9200.2.a.bo 2 5.c odd 4 1
9200.2.a.by 2 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 \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3}^{4} + 3T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 27T_{7}^{2} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 27T^{2} + 81 \) Copy content Toggle raw display
$11$ \( (T^{2} - 9 T + 19)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + 35T^{2} + 25 \) Copy content Toggle raw display
$19$ \( (T^{2} + 7 T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T - 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 11 T + 19)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
$41$ \( (T^{2} + 9 T + 19)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 172T^{2} + 5776 \) Copy content Toggle raw display
$47$ \( T^{4} + 140T^{2} + 400 \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + T - 11)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 328 T^{2} + 15376 \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T - 44)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 92T^{2} + 1936 \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T - 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 427 T^{2} + 43681 \) Copy content Toggle raw display
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