Properties

Label 2624.1.h.b
Level $2624$
Weight $1$
Character orbit 2624.h
Self dual yes
Analytic conductor $1.310$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -164
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2624 = 2^{6} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2624.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.30954659315\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.6724.1
Artin image: $D_8$
Artin field: Galois closure of 8.0.4516806656.3

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} -\beta q^{7} + q^{9} +O(q^{10})\) \( q -\beta q^{3} -\beta q^{7} + q^{9} + \beta q^{11} -\beta q^{19} + 2 q^{21} - q^{25} -2 q^{33} + q^{41} + \beta q^{47} + q^{49} + 2 q^{57} + 2 q^{61} -\beta q^{63} + \beta q^{67} + \beta q^{71} + \beta q^{75} -2 q^{77} + \beta q^{79} - q^{81} + \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{9} + 4 q^{21} - 2 q^{25} - 4 q^{33} + 2 q^{41} + 2 q^{49} + 4 q^{57} + 4 q^{61} - 4 q^{77} - 2 q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2624\mathbb{Z}\right)^\times\).

\(n\) \(129\) \(575\) \(1477\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2623.1
1.41421
−1.41421
0 −1.41421 0 0 0 −1.41421 0 1.00000 0
2623.2 0 1.41421 0 0 0 1.41421 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
164.d odd 2 1 CM by \(\Q(\sqrt{-41}) \)
4.b odd 2 1 inner
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2624.1.h.b 2
4.b odd 2 1 inner 2624.1.h.b 2
8.b even 2 1 164.1.d.b 2
8.d odd 2 1 164.1.d.b 2
24.f even 2 1 1476.1.h.b 2
24.h odd 2 1 1476.1.h.b 2
41.b even 2 1 inner 2624.1.h.b 2
164.d odd 2 1 CM 2624.1.h.b 2
328.c odd 2 1 164.1.d.b 2
328.g even 2 1 164.1.d.b 2
984.m odd 2 1 1476.1.h.b 2
984.p even 2 1 1476.1.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.1.d.b 2 8.b even 2 1
164.1.d.b 2 8.d odd 2 1
164.1.d.b 2 328.c odd 2 1
164.1.d.b 2 328.g even 2 1
1476.1.h.b 2 24.f even 2 1
1476.1.h.b 2 24.h odd 2 1
1476.1.h.b 2 984.m odd 2 1
1476.1.h.b 2 984.p even 2 1
2624.1.h.b 2 1.a even 1 1 trivial
2624.1.h.b 2 4.b odd 2 1 inner
2624.1.h.b 2 41.b even 2 1 inner
2624.1.h.b 2 164.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 \) acting on \(S_{1}^{\mathrm{new}}(2624, [\chi])\).

Hecke characteristic polynomials

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