Properties

Label 819.2.c.a
Level $819$
Weight $2$
Character orbit 819.c
Analytic conductor $6.540$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} -2 q^{4} + 3 i q^{5} + i q^{7} +O(q^{10})\) \( q + 2 i q^{2} -2 q^{4} + 3 i q^{5} + i q^{7} -6 q^{10} + ( 2 + 3 i ) q^{13} -2 q^{14} -4 q^{16} + 2 q^{17} -i q^{19} -6 i q^{20} + q^{23} -4 q^{25} + ( -6 + 4 i ) q^{26} -2 i q^{28} -5 q^{29} -5 i q^{31} -8 i q^{32} + 4 i q^{34} -3 q^{35} + 8 i q^{37} + 2 q^{38} -10 i q^{41} + 9 q^{43} + 2 i q^{46} + 7 i q^{47} - q^{49} -8 i q^{50} + ( -4 - 6 i ) q^{52} -9 q^{53} -10 i q^{58} -4 i q^{59} -8 q^{61} + 10 q^{62} + 8 q^{64} + ( -9 + 6 i ) q^{65} -2 i q^{67} -4 q^{68} -6 i q^{70} -9 i q^{73} -16 q^{74} + 2 i q^{76} + 15 q^{79} -12 i q^{80} + 20 q^{82} + 9 i q^{83} + 6 i q^{85} + 18 i q^{86} -9 i q^{89} + ( -3 + 2 i ) q^{91} -2 q^{92} -14 q^{94} + 3 q^{95} + 13 i q^{97} -2 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + O(q^{10}) \) \( 2q - 4q^{4} - 12q^{10} + 4q^{13} - 4q^{14} - 8q^{16} + 4q^{17} + 2q^{23} - 8q^{25} - 12q^{26} - 10q^{29} - 6q^{35} + 4q^{38} + 18q^{43} - 2q^{49} - 8q^{52} - 18q^{53} - 16q^{61} + 20q^{62} + 16q^{64} - 18q^{65} - 8q^{68} - 32q^{74} + 30q^{79} + 40q^{82} - 6q^{91} - 4q^{92} - 28q^{94} + 6q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
1.00000i
1.00000i
2.00000i 0 −2.00000 3.00000i 0 1.00000i 0 0 −6.00000
64.2 2.00000i 0 −2.00000 3.00000i 0 1.00000i 0 0 −6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.c.a 2
3.b odd 2 1 273.2.c.a 2
12.b even 2 1 4368.2.h.e 2
13.b even 2 1 inner 819.2.c.a 2
21.c even 2 1 1911.2.c.a 2
39.d odd 2 1 273.2.c.a 2
39.f even 4 1 3549.2.a.a 1
39.f even 4 1 3549.2.a.e 1
156.h even 2 1 4368.2.h.e 2
273.g even 2 1 1911.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.a 2 3.b odd 2 1
273.2.c.a 2 39.d odd 2 1
819.2.c.a 2 1.a even 1 1 trivial
819.2.c.a 2 13.b even 2 1 inner
1911.2.c.a 2 21.c even 2 1
1911.2.c.a 2 273.g even 2 1
3549.2.a.a 1 39.f even 4 1
3549.2.a.e 1 39.f even 4 1
4368.2.h.e 2 12.b even 2 1
4368.2.h.e 2 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 9 + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 - 4 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 1 + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( 25 + T^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( 100 + T^{2} \)
$43$ \( ( -9 + T )^{2} \)
$47$ \( 49 + T^{2} \)
$53$ \( ( 9 + T )^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 81 + T^{2} \)
$79$ \( ( -15 + T )^{2} \)
$83$ \( 81 + T^{2} \)
$89$ \( 81 + T^{2} \)
$97$ \( 169 + T^{2} \)
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