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: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} -2 q^{4} + ( -1 - 2 i ) q^{5} -4 i q^{7} + 3 q^{9} +O(q^{10})\) \( q + 2 i q^{2} -2 q^{4} + ( -1 - 2 i ) q^{5} -4 i q^{7} + 3 q^{9} + ( 4 - 2 i ) q^{10} - q^{11} + 2 i q^{13} + 8 q^{14} -4 q^{16} -2 i q^{17} + 6 i q^{18} + ( 2 + 4 i ) q^{20} -2 i q^{22} -6 i q^{23} + ( -3 + 4 i ) q^{25} -4 q^{26} + 8 i q^{28} -9 q^{29} -7 q^{31} -8 i q^{32} + 4 q^{34} + ( -8 + 4 i ) q^{35} -6 q^{36} + 2 i q^{37} + 2 q^{41} -2 i q^{43} + 2 q^{44} + ( -3 - 6 i ) q^{45} + 12 q^{46} + 6 i q^{47} -9 q^{49} + ( -8 - 6 i ) q^{50} -4 i q^{52} -4 i q^{53} + ( 1 + 2 i ) q^{55} -18 i q^{58} -9 q^{59} -7 q^{61} -14 i q^{62} -12 i q^{63} + 8 q^{64} + ( 4 - 2 i ) q^{65} -10 i q^{67} + 4 i q^{68} + ( -8 - 16 i ) q^{70} + q^{71} -10 i q^{73} -4 q^{74} + 4 i q^{77} - q^{79} + ( 4 + 8 i ) q^{80} + 9 q^{81} + 4 i q^{82} + 6 i q^{83} + ( -4 + 2 i ) q^{85} + 4 q^{86} + 11 q^{89} + ( 12 - 6 i ) q^{90} + 8 q^{91} + 12 i q^{92} -12 q^{94} -6 i q^{97} -18 i q^{98} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 2 q^{5} + 6 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{4} - 2 q^{5} + 6 q^{9} + 8 q^{10} - 2 q^{11} + 16 q^{14} - 8 q^{16} + 4 q^{20} - 6 q^{25} - 8 q^{26} - 18 q^{29} - 14 q^{31} + 8 q^{34} - 16 q^{35} - 12 q^{36} + 4 q^{41} + 4 q^{44} - 6 q^{45} + 24 q^{46} - 18 q^{49} - 16 q^{50} + 2 q^{55} - 18 q^{59} - 14 q^{61} + 16 q^{64} + 8 q^{65} - 16 q^{70} + 2 q^{71} - 8 q^{74} - 2 q^{79} + 8 q^{80} + 18 q^{81} - 8 q^{85} + 8 q^{86} + 22 q^{89} + 24 q^{90} + 16 q^{91} - 24 q^{94} - 6 q^{99} + O(q^{100}) \)

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 \(S_{2}^{\mathrm{new}}(1805, [\chi])\):

\( T_{2}^{2} + 4 \)
\( T_{29} + 9 \)

Hecke characteristic polynomials

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