Properties

Label 1805.2.b.d
Level $1805$
Weight $2$
Character orbit 1805.b
Analytic conductor $14.413$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
Defining polynomial: \( x^{2} - x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-19})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{4} + (\beta - 1) q^{5} + (2 \beta - 1) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{4} + (\beta - 1) q^{5} + (2 \beta - 1) q^{7} + 3 q^{9} + 5 q^{11} + 4 q^{16} + (2 \beta - 1) q^{17} + (2 \beta - 2) q^{20} + ( - 4 \beta + 2) q^{23} + ( - \beta - 4) q^{25} + (4 \beta - 2) q^{28} + ( - \beta - 9) q^{35} + 6 q^{36} + ( - 6 \beta + 3) q^{43} + 10 q^{44} + (3 \beta - 3) q^{45} + (2 \beta - 1) q^{47} - 12 q^{49} + (5 \beta - 5) q^{55} - 15 q^{61} + (6 \beta - 3) q^{63} + 8 q^{64} + (4 \beta - 2) q^{68} + ( - 6 \beta + 3) q^{73} + (10 \beta - 5) q^{77} + (4 \beta - 4) q^{80} + 9 q^{81} + (4 \beta - 2) q^{83} + ( - \beta - 9) q^{85} + ( - 8 \beta + 4) q^{92} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} - q^{5} + 6 q^{9} + 10 q^{11} + 8 q^{16} - 2 q^{20} - 9 q^{25} - 19 q^{35} + 12 q^{36} + 20 q^{44} - 3 q^{45} - 24 q^{49} - 5 q^{55} - 30 q^{61} + 16 q^{64} - 4 q^{80} + 18 q^{81} - 19 q^{85} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(1446\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1084.1
0.500000 2.17945i
0.500000 + 2.17945i
0 0 2.00000 −0.500000 2.17945i 0 4.35890i 0 3.00000 0
1084.2 0 0 2.00000 −0.500000 + 2.17945i 0 4.35890i 0 3.00000 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
5.b even 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.b.d 2
5.b even 2 1 inner 1805.2.b.d 2
5.c odd 4 2 9025.2.a.p 2
19.b odd 2 1 CM 1805.2.b.d 2
95.d odd 2 1 inner 1805.2.b.d 2
95.g even 4 2 9025.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.b.d 2 1.a even 1 1 trivial
1805.2.b.d 2 5.b even 2 1 inner
1805.2.b.d 2 19.b odd 2 1 CM
1805.2.b.d 2 95.d odd 2 1 inner
9025.2.a.p 2 5.c odd 4 2
9025.2.a.p 2 95.g even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1805, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 19 \) Copy content Toggle raw display
$11$ \( (T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 19 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 76 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 171 \) Copy content Toggle raw display
$47$ \( T^{2} + 19 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 15)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 171 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 76 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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