Properties

Label 1638.2.c.f
Level $1638$
Weight $2$
Character orbit 1638.c
Analytic conductor $13.079$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.0794958511\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 546)
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.

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} + i q^{5} + i q^{7} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} + i q^{5} + i q^{7} + i q^{8} + q^{10} + i q^{11} + ( 3 - 2 i ) q^{13} + q^{14} + q^{16} - q^{17} + i q^{19} -i q^{20} + q^{22} -3 q^{23} + 4 q^{25} + ( -2 - 3 i ) q^{26} -i q^{28} -9 q^{29} + 4 i q^{31} -i q^{32} + i q^{34} - q^{35} + 9 i q^{37} + q^{38} - q^{40} + 8 i q^{41} + 7 q^{43} -i q^{44} + 3 i q^{46} + 8 i q^{47} - q^{49} -4 i q^{50} + ( -3 + 2 i ) q^{52} + 10 q^{53} - q^{55} - q^{56} + 9 i q^{58} -6 i q^{59} + 11 q^{61} + 4 q^{62} - q^{64} + ( 2 + 3 i ) q^{65} + 12 i q^{67} + q^{68} + i q^{70} + 6 i q^{71} -11 i q^{73} + 9 q^{74} -i q^{76} - q^{77} -12 q^{79} + i q^{80} + 8 q^{82} + 6 i q^{83} -i q^{85} -7 i q^{86} - q^{88} + 12 i q^{89} + ( 2 + 3 i ) q^{91} + 3 q^{92} + 8 q^{94} - q^{95} + 2 i q^{97} + i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{10} + 6q^{13} + 2q^{14} + 2q^{16} - 2q^{17} + 2q^{22} - 6q^{23} + 8q^{25} - 4q^{26} - 18q^{29} - 2q^{35} + 2q^{38} - 2q^{40} + 14q^{43} - 2q^{49} - 6q^{52} + 20q^{53} - 2q^{55} - 2q^{56} + 22q^{61} + 8q^{62} - 2q^{64} + 4q^{65} + 2q^{68} + 18q^{74} - 2q^{77} - 24q^{79} + 16q^{82} - 2q^{88} + 4q^{91} + 6q^{92} + 16q^{94} - 2q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
1.00000i
1.00000i
1.00000i 0 −1.00000 1.00000i 0 1.00000i 1.00000i 0 1.00000
883.2 1.00000i 0 −1.00000 1.00000i 0 1.00000i 1.00000i 0 1.00000
\(n\): 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 1638.2.c.f 2
3.b odd 2 1 546.2.c.a 2
12.b even 2 1 4368.2.h.k 2
13.b even 2 1 inner 1638.2.c.f 2
21.c even 2 1 3822.2.c.e 2
39.d odd 2 1 546.2.c.a 2
39.f even 4 1 7098.2.a.d 1
39.f even 4 1 7098.2.a.s 1
156.h even 2 1 4368.2.h.k 2
273.g even 2 1 3822.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.a 2 3.b odd 2 1
546.2.c.a 2 39.d odd 2 1
1638.2.c.f 2 1.a even 1 1 trivial
1638.2.c.f 2 13.b even 2 1 inner
3822.2.c.e 2 21.c even 2 1
3822.2.c.e 2 273.g even 2 1
4368.2.h.k 2 12.b even 2 1
4368.2.h.k 2 156.h even 2 1
7098.2.a.d 1 39.f even 4 1
7098.2.a.s 1 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{2} + 1 \)
\( T_{11}^{2} + 1 \)
\( T_{17} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( 1 + T^{2} \)
$13$ \( 13 - 6 T + T^{2} \)
$17$ \( ( 1 + T )^{2} \)
$19$ \( 1 + T^{2} \)
$23$ \( ( 3 + T )^{2} \)
$29$ \( ( 9 + T )^{2} \)
$31$ \( 16 + T^{2} \)
$37$ \( 81 + T^{2} \)
$41$ \( 64 + T^{2} \)
$43$ \( ( -7 + T )^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( ( -10 + T )^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( ( -11 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( 36 + T^{2} \)
$73$ \( 121 + T^{2} \)
$79$ \( ( 12 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( 144 + T^{2} \)
$97$ \( 4 + T^{2} \)
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