Properties

Label 805.2.c.a
Level 805
Weight 2
Character orbit 805.c
Analytic conductor 6.428
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 805.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.42795736271\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + i q^{3} + 2 q^{4} + ( 2 - i ) q^{5} + i q^{7} + 2 q^{9} +O(q^{10})\) \( q + i q^{3} + 2 q^{4} + ( 2 - i ) q^{5} + i q^{7} + 2 q^{9} - q^{11} + 2 i q^{12} + i q^{13} + ( 1 + 2 i ) q^{15} + 4 q^{16} -i q^{17} -2 q^{19} + ( 4 - 2 i ) q^{20} - q^{21} + i q^{23} + ( 3 - 4 i ) q^{25} + 5 i q^{27} + 2 i q^{28} -7 q^{29} + 4 q^{31} -i q^{33} + ( 1 + 2 i ) q^{35} + 4 q^{36} -8 i q^{37} - q^{39} -6 q^{41} -8 i q^{43} -2 q^{44} + ( 4 - 2 i ) q^{45} + 7 i q^{47} + 4 i q^{48} - q^{49} + q^{51} + 2 i q^{52} + 4 i q^{53} + ( -2 + i ) q^{55} -2 i q^{57} + 4 q^{59} + ( 2 + 4 i ) q^{60} -10 q^{61} + 2 i q^{63} + 8 q^{64} + ( 1 + 2 i ) q^{65} + 14 i q^{67} -2 i q^{68} - q^{69} + 2 i q^{73} + ( 4 + 3 i ) q^{75} -4 q^{76} -i q^{77} + 15 q^{79} + ( 8 - 4 i ) q^{80} + q^{81} -8 i q^{83} -2 q^{84} + ( -1 - 2 i ) q^{85} -7 i q^{87} -6 q^{89} - q^{91} + 2 i q^{92} + 4 i q^{93} + ( -4 + 2 i ) q^{95} + 7 i q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{4} + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{4} + 4q^{5} + 4q^{9} - 2q^{11} + 2q^{15} + 8q^{16} - 4q^{19} + 8q^{20} - 2q^{21} + 6q^{25} - 14q^{29} + 8q^{31} + 2q^{35} + 8q^{36} - 2q^{39} - 12q^{41} - 4q^{44} + 8q^{45} - 2q^{49} + 2q^{51} - 4q^{55} + 8q^{59} + 4q^{60} - 20q^{61} + 16q^{64} + 2q^{65} - 2q^{69} + 8q^{75} - 8q^{76} + 30q^{79} + 16q^{80} + 2q^{81} - 4q^{84} - 2q^{85} - 12q^{89} - 2q^{91} - 8q^{95} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/805\mathbb{Z}\right)^\times\).

\(n\) \(162\) \(281\) \(346\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
484.1
1.00000i
1.00000i
0 1.00000i 2.00000 2.00000 + 1.00000i 0 1.00000i 0 2.00000 0
484.2 0 1.00000i 2.00000 2.00000 1.00000i 0 1.00000i 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
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 805.2.c.a 2
5.b even 2 1 inner 805.2.c.a 2
5.c odd 4 1 4025.2.a.c 1
5.c odd 4 1 4025.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.c.a 2 1.a even 1 1 trivial
805.2.c.a 2 5.b even 2 1 inner
4025.2.a.c 1 5.c odd 4 1
4025.2.a.d 1 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{2} \)
$3$ \( 1 - 5 T^{2} + 9 T^{4} \)
$5$ \( 1 - 4 T + 5 T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + T + 11 T^{2} )^{2} \)
$13$ \( 1 - 25 T^{2} + 169 T^{4} \)
$17$ \( 1 - 33 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 2 T + 19 T^{2} )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( 1 + 7 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 10 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 22 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 45 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} ) \)
$59$ \( ( 1 - 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 10 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 62 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 15 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 102 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 145 T^{2} + 9409 T^{4} \)
show more
show less