Properties

Label 1805.2.b.b
Level $1805$
Weight $2$
Character orbit 1805.b
Analytic conductor $14.413$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 95)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} - 2 q^{4} + ( - \beta - 1) q^{5} - 2 \beta q^{7} + 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 2 q^{4} + ( - \beta - 1) q^{5} - 2 \beta q^{7} + 3 q^{9} + ( - \beta + 4) q^{10} - q^{11} + \beta q^{13} + 8 q^{14} - 4 q^{16} - \beta q^{17} + 3 \beta q^{18} + (2 \beta + 2) q^{20} - \beta q^{22} - 3 \beta q^{23} + (2 \beta - 3) q^{25} - 4 q^{26} + 4 \beta q^{28} - 9 q^{29} - 7 q^{31} - 4 \beta q^{32} + 4 q^{34} + (2 \beta - 8) q^{35} - 6 q^{36} + \beta q^{37} + 2 q^{41} - \beta q^{43} + 2 q^{44} + ( - 3 \beta - 3) q^{45} + 12 q^{46} + 3 \beta q^{47} - 9 q^{49} + ( - 3 \beta - 8) q^{50} - 2 \beta q^{52} - 2 \beta q^{53} + (\beta + 1) q^{55} - 9 \beta q^{58} - 9 q^{59} - 7 q^{61} - 7 \beta q^{62} - 6 \beta q^{63} + 8 q^{64} + ( - \beta + 4) q^{65} - 5 \beta q^{67} + 2 \beta q^{68} + ( - 8 \beta - 8) q^{70} + q^{71} - 5 \beta q^{73} - 4 q^{74} + 2 \beta q^{77} - q^{79} + (4 \beta + 4) q^{80} + 9 q^{81} + 2 \beta q^{82} + 3 \beta q^{83} + (\beta - 4) q^{85} + 4 q^{86} + 11 q^{89} + ( - 3 \beta + 12) q^{90} + 8 q^{91} + 6 \beta q^{92} - 12 q^{94} - 3 \beta q^{97} - 9 \beta q^{98} - 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 2 q^{5} + 6 q^{9} + 8 q^{10} - 2 q^{11} + 16 q^{14} - 8 q^{16} + 4 q^{20} - 6 q^{25} - 8 q^{26} - 18 q^{29} - 14 q^{31} + 8 q^{34} - 16 q^{35} - 12 q^{36} + 4 q^{41} + 4 q^{44} - 6 q^{45} + 24 q^{46} - 18 q^{49} - 16 q^{50} + 2 q^{55} - 18 q^{59} - 14 q^{61} + 16 q^{64} + 8 q^{65} - 16 q^{70} + 2 q^{71} - 8 q^{74} - 2 q^{79} + 8 q^{80} + 18 q^{81} - 8 q^{85} + 8 q^{86} + 22 q^{89} + 24 q^{90} + 16 q^{91} - 24 q^{94} - 6 q^{99}+O(q^{100}) \) 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
1.00000i
1.00000i
2.00000i 0 −2.00000 −1.00000 + 2.00000i 0 4.00000i 0 3.00000 4.00000 + 2.00000i
1084.2 2.00000i 0 −2.00000 −1.00000 2.00000i 0 4.00000i 0 3.00000 4.00000 2.00000i
\(n\): 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 1805.2.b.b 2
5.b even 2 1 inner 1805.2.b.b 2
5.c odd 4 1 9025.2.a.b 1
5.c odd 4 1 9025.2.a.i 1
19.b odd 2 1 1805.2.b.a 2
19.c even 3 2 95.2.i.a 4
57.h odd 6 2 855.2.be.a 4
95.d odd 2 1 1805.2.b.a 2
95.g even 4 1 9025.2.a.a 1
95.g even 4 1 9025.2.a.j 1
95.i even 6 2 95.2.i.a 4
95.m odd 12 2 475.2.e.a 2
95.m odd 12 2 475.2.e.c 2
285.n odd 6 2 855.2.be.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.a 4 19.c even 3 2
95.2.i.a 4 95.i even 6 2
475.2.e.a 2 95.m odd 12 2
475.2.e.c 2 95.m odd 12 2
855.2.be.a 4 57.h odd 6 2
855.2.be.a 4 285.n odd 6 2
1805.2.b.a 2 19.b odd 2 1
1805.2.b.a 2 95.d odd 2 1
1805.2.b.b 2 1.a even 1 1 trivial
1805.2.b.b 2 5.b even 2 1 inner
9025.2.a.a 1 95.g even 4 1
9025.2.a.b 1 5.c odd 4 1
9025.2.a.i 1 5.c odd 4 1
9025.2.a.j 1 95.g even 4 1

Hecke kernels

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

\( T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{29} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T + 9)^{2} \) Copy content Toggle raw display
$61$ \( (T + 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( (T - 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 11)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 36 \) Copy content Toggle raw display
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