Properties

Label 1638.2.c.h
Level $1638$
Weight $2$
Character orbit 1638.c
Analytic conductor $13.079$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} - q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} -\beta_{2} q^{7} + \beta_{2} q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} - q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} -\beta_{2} q^{7} + \beta_{2} q^{8} + ( -2 + \beta_{3} ) q^{10} + ( \beta_{1} - 3 \beta_{2} ) q^{11} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{13} - q^{14} + q^{16} + ( 2 - 3 \beta_{3} ) q^{17} + ( \beta_{1} + \beta_{2} ) q^{19} + ( -\beta_{1} + \beta_{2} ) q^{20} + ( -4 + \beta_{3} ) q^{22} + ( 2 - 3 \beta_{3} ) q^{23} + ( -3 + 3 \beta_{3} ) q^{25} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{26} + \beta_{2} q^{28} + \beta_{3} q^{29} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{31} -\beta_{2} q^{32} + ( 3 \beta_{1} + \beta_{2} ) q^{34} + ( -2 + \beta_{3} ) q^{35} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{37} + \beta_{3} q^{38} + ( 2 - \beta_{3} ) q^{40} -4 \beta_{2} q^{41} + ( -8 - \beta_{3} ) q^{43} + ( -\beta_{1} + 3 \beta_{2} ) q^{44} + ( 3 \beta_{1} + \beta_{2} ) q^{46} -4 \beta_{1} q^{47} - q^{49} -3 \beta_{1} q^{50} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{52} + ( -8 - 2 \beta_{3} ) q^{53} + ( -12 + 5 \beta_{3} ) q^{55} + q^{56} + ( -\beta_{1} - \beta_{2} ) q^{58} + ( 4 \beta_{1} + 6 \beta_{2} ) q^{59} + \beta_{3} q^{61} + 4 \beta_{3} q^{62} - q^{64} + ( -2 + 3 \beta_{1} - 5 \beta_{2} ) q^{65} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{67} + ( -2 + 3 \beta_{3} ) q^{68} + ( -\beta_{1} + \beta_{2} ) q^{70} + ( 6 \beta_{1} + 6 \beta_{2} ) q^{71} + ( -\beta_{1} + 9 \beta_{2} ) q^{73} + ( 6 - 3 \beta_{3} ) q^{74} + ( -\beta_{1} - \beta_{2} ) q^{76} + ( -4 + \beta_{3} ) q^{77} + 16 q^{79} + ( \beta_{1} - \beta_{2} ) q^{80} -4 q^{82} + 2 \beta_{2} q^{83} + ( 5 \beta_{1} - 11 \beta_{2} ) q^{85} + ( \beta_{1} + 9 \beta_{2} ) q^{86} + ( 4 - \beta_{3} ) q^{88} + 8 \beta_{2} q^{89} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{91} + ( -2 + 3 \beta_{3} ) q^{92} + ( 4 - 4 \beta_{3} ) q^{94} + ( -4 + \beta_{3} ) q^{95} + 10 \beta_{2} q^{97} + \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} - 6q^{10} + 6q^{13} - 4q^{14} + 4q^{16} + 2q^{17} - 14q^{22} + 2q^{23} - 6q^{25} + 6q^{26} + 2q^{29} - 6q^{35} + 2q^{38} + 6q^{40} - 34q^{43} - 4q^{49} - 6q^{52} - 36q^{53} - 38q^{55} + 4q^{56} + 2q^{61} + 8q^{62} - 4q^{64} - 8q^{65} - 2q^{68} + 18q^{74} - 14q^{77} + 64q^{79} - 16q^{82} + 14q^{88} + 6q^{91} - 2q^{92} + 8q^{94} - 14q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 5\)
\(\nu^{3}\)\(=\)\(4 \beta_{2} - 5 \beta_{1}\)

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
2.56155i
1.56155i
1.56155i
2.56155i
1.00000i 0 −1.00000 3.56155i 0 1.00000i 1.00000i 0 −3.56155
883.2 1.00000i 0 −1.00000 0.561553i 0 1.00000i 1.00000i 0 0.561553
883.3 1.00000i 0 −1.00000 0.561553i 0 1.00000i 1.00000i 0 0.561553
883.4 1.00000i 0 −1.00000 3.56155i 0 1.00000i 1.00000i 0 −3.56155
\(n\): 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.h 4
3.b odd 2 1 546.2.c.e 4
12.b even 2 1 4368.2.h.n 4
13.b even 2 1 inner 1638.2.c.h 4
21.c even 2 1 3822.2.c.h 4
39.d odd 2 1 546.2.c.e 4
39.f even 4 1 7098.2.a.bg 2
39.f even 4 1 7098.2.a.bv 2
156.h even 2 1 4368.2.h.n 4
273.g even 2 1 3822.2.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.e 4 3.b odd 2 1
546.2.c.e 4 39.d odd 2 1
1638.2.c.h 4 1.a even 1 1 trivial
1638.2.c.h 4 13.b even 2 1 inner
3822.2.c.h 4 21.c even 2 1
3822.2.c.h 4 273.g even 2 1
4368.2.h.n 4 12.b even 2 1
4368.2.h.n 4 156.h even 2 1
7098.2.a.bg 2 39.f even 4 1
7098.2.a.bv 2 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 \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{4} + 13 T_{5}^{2} + 4 \)
\( T_{11}^{4} + 33 T_{11}^{2} + 64 \)
\( T_{17}^{2} - T_{17} - 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 4 + 13 T^{2} + T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( 64 + 33 T^{2} + T^{4} \)
$13$ \( 169 - 78 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$17$ \( ( -38 - T + T^{2} )^{2} \)
$19$ \( 16 + 9 T^{2} + T^{4} \)
$23$ \( ( -38 - T + T^{2} )^{2} \)
$29$ \( ( -4 - T + T^{2} )^{2} \)
$31$ \( 4096 + 144 T^{2} + T^{4} \)
$37$ \( 324 + 117 T^{2} + T^{4} \)
$41$ \( ( 16 + T^{2} )^{2} \)
$43$ \( ( 68 + 17 T + T^{2} )^{2} \)
$47$ \( 4096 + 144 T^{2} + T^{4} \)
$53$ \( ( 64 + 18 T + T^{2} )^{2} \)
$59$ \( 2704 + 168 T^{2} + T^{4} \)
$61$ \( ( -4 - T + T^{2} )^{2} \)
$67$ \( 64 + 52 T^{2} + T^{4} \)
$71$ \( 20736 + 324 T^{2} + T^{4} \)
$73$ \( 7396 + 189 T^{2} + T^{4} \)
$79$ \( ( -16 + T )^{4} \)
$83$ \( ( 4 + T^{2} )^{2} \)
$89$ \( ( 64 + T^{2} )^{2} \)
$97$ \( ( 100 + T^{2} )^{2} \)
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