Properties

Label 1856.2.e.g
Level $1856$
Weight $2$
Character orbit 1856.e
Analytic conductor $14.820$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.8202346151\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \(x^{2} + 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 3 q^{5} + 2 q^{7} -2 q^{9} +O(q^{10})\) \( q + \beta q^{3} + 3 q^{5} + 2 q^{7} -2 q^{9} -\beta q^{11} + q^{13} + 3 \beta q^{15} -2 \beta q^{17} + 2 \beta q^{21} + 6 q^{23} + 4 q^{25} + \beta q^{27} + ( 3 - 2 \beta ) q^{29} + 3 \beta q^{31} + 5 q^{33} + 6 q^{35} + \beta q^{39} -2 \beta q^{41} + 3 \beta q^{43} -6 q^{45} + \beta q^{47} -3 q^{49} + 10 q^{51} + 9 q^{53} -3 \beta q^{55} -6 q^{59} -6 \beta q^{61} -4 q^{63} + 3 q^{65} -8 q^{67} + 6 \beta q^{69} + 4 \beta q^{75} -2 \beta q^{77} -3 \beta q^{79} -11 q^{81} + 6 q^{83} -6 \beta q^{85} + ( 10 + 3 \beta ) q^{87} -2 \beta q^{89} + 2 q^{91} -15 q^{93} + 6 \beta q^{97} + 2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} + 4 q^{7} - 4 q^{9} + O(q^{10}) \) \( 2 q + 6 q^{5} + 4 q^{7} - 4 q^{9} + 2 q^{13} + 12 q^{23} + 8 q^{25} + 6 q^{29} + 10 q^{33} + 12 q^{35} - 12 q^{45} - 6 q^{49} + 20 q^{51} + 18 q^{53} - 12 q^{59} - 8 q^{63} + 6 q^{65} - 16 q^{67} - 22 q^{81} + 12 q^{83} + 20 q^{87} + 4 q^{91} - 30 q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(639\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1217.1
2.23607i
2.23607i
0 2.23607i 0 3.00000 0 2.00000 0 −2.00000 0
1217.2 0 2.23607i 0 3.00000 0 2.00000 0 −2.00000 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
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.2.e.g 2
4.b odd 2 1 1856.2.e.f 2
8.b even 2 1 29.2.b.a 2
8.d odd 2 1 464.2.e.a 2
24.f even 2 1 4176.2.o.k 2
24.h odd 2 1 261.2.c.a 2
29.b even 2 1 inner 1856.2.e.g 2
40.f even 2 1 725.2.c.c 2
40.i odd 4 2 725.2.d.a 4
56.h odd 2 1 1421.2.b.b 2
116.d odd 2 1 1856.2.e.f 2
232.b odd 2 1 464.2.e.a 2
232.g even 2 1 29.2.b.a 2
232.l odd 4 2 841.2.a.b 2
232.o even 14 6 841.2.e.g 12
232.s even 14 6 841.2.e.g 12
232.u odd 28 12 841.2.d.h 12
696.l even 2 1 4176.2.o.k 2
696.n odd 2 1 261.2.c.a 2
696.t even 4 2 7569.2.a.i 2
1160.e even 2 1 725.2.c.c 2
1160.be odd 4 2 725.2.d.a 4
1624.i odd 2 1 1421.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.b.a 2 8.b even 2 1
29.2.b.a 2 232.g even 2 1
261.2.c.a 2 24.h odd 2 1
261.2.c.a 2 696.n odd 2 1
464.2.e.a 2 8.d odd 2 1
464.2.e.a 2 232.b odd 2 1
725.2.c.c 2 40.f even 2 1
725.2.c.c 2 1160.e even 2 1
725.2.d.a 4 40.i odd 4 2
725.2.d.a 4 1160.be odd 4 2
841.2.a.b 2 232.l odd 4 2
841.2.d.h 12 232.u odd 28 12
841.2.e.g 12 232.o even 14 6
841.2.e.g 12 232.s even 14 6
1421.2.b.b 2 56.h odd 2 1
1421.2.b.b 2 1624.i odd 2 1
1856.2.e.f 2 4.b odd 2 1
1856.2.e.f 2 116.d odd 2 1
1856.2.e.g 2 1.a even 1 1 trivial
1856.2.e.g 2 29.b even 2 1 inner
4176.2.o.k 2 24.f even 2 1
4176.2.o.k 2 696.l even 2 1
7569.2.a.i 2 696.t even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1856, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 5 + T^{2} \)
$5$ \( ( -3 + T )^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( 5 + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( 20 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( 29 - 6 T + T^{2} \)
$31$ \( 45 + T^{2} \)
$37$ \( T^{2} \)
$41$ \( 20 + T^{2} \)
$43$ \( 45 + T^{2} \)
$47$ \( 5 + T^{2} \)
$53$ \( ( -9 + T )^{2} \)
$59$ \( ( 6 + T )^{2} \)
$61$ \( 180 + T^{2} \)
$67$ \( ( 8 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( 45 + T^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 20 + T^{2} \)
$97$ \( 180 + T^{2} \)
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