Properties

Label 1638.2.bj.c
Level $1638$
Weight $2$
Character orbit 1638.bj
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.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + \zeta_{12}^{3} q^{5} - \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + \zeta_{12}^{3} q^{5} - \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{2} q^{10} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 2) q^{11} + (\zeta_{12}^{3} + 3 \zeta_{12}) q^{13} + q^{14} - \zeta_{12}^{2} q^{16} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + \zeta_{12} + 4) q^{17} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{19} + \zeta_{12} q^{20} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 1) q^{22} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12}) q^{23} + 4 q^{25} + ( - \zeta_{12}^{2} - 3) q^{26} + (\zeta_{12}^{3} - \zeta_{12}) q^{28} - 3 \zeta_{12}^{2} q^{29} + (7 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{31} + \zeta_{12} q^{32} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{34} + ( - \zeta_{12}^{2} + 1) q^{35} + ( - 3 \zeta_{12}^{2} + 6) q^{37} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 2) q^{38} - q^{40} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 4) q^{41} + (6 \zeta_{12}^{3} + 7 \zeta_{12}^{2} - 3 \zeta_{12} - 7) q^{43} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{44} + ( - \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{46} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{47} + \zeta_{12}^{2} q^{49} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{50} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}) q^{52} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 5) q^{53} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{55} + ( - \zeta_{12}^{2} + 1) q^{56} + 3 \zeta_{12} q^{58} + (\zeta_{12}^{2} + 9 \zeta_{12} + 1) q^{59} + ( - 2 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + \zeta_{12} + 10) q^{61} + (\zeta_{12}^{3} - 7 \zeta_{12}^{2} + \zeta_{12}) q^{62} - q^{64} + (3 \zeta_{12}^{2} - 4) q^{65} + ( - 3 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 3 \zeta_{12} + 10) q^{67} + ( - \zeta_{12}^{3} - 4 \zeta_{12}^{2} - \zeta_{12}) q^{68} + \zeta_{12}^{3} q^{70} + (8 \zeta_{12}^{2} + 8) q^{71} + (\zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{73} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{74} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{76} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 1) q^{77} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} - 9) q^{79} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{80} + (4 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{82} + (9 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{83} + (\zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{85} + ( - 7 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{86} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{88} + (6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 6 \zeta_{12} + 4) q^{89} + ( - 4 \zeta_{12}^{2} + 1) q^{91} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 3) q^{92} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{94} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{95} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{97} - \zeta_{12} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{10} - 6 q^{11} + 4 q^{14} - 2 q^{16} + 8 q^{17} - 12 q^{19} - 2 q^{22} - 6 q^{23} + 16 q^{25} - 14 q^{26} - 6 q^{29} + 2 q^{35} + 18 q^{37} - 8 q^{38} - 4 q^{40} + 12 q^{41} - 14 q^{43} - 6 q^{46} + 2 q^{49} - 20 q^{53} + 2 q^{55} + 2 q^{56} + 6 q^{59} + 20 q^{61} - 14 q^{62} - 4 q^{64} - 10 q^{65} + 30 q^{67} - 8 q^{68} + 48 q^{71} - 12 q^{76} - 4 q^{77} - 36 q^{79} + 2 q^{82} + 6 q^{85} + 2 q^{88} + 12 q^{89} - 4 q^{91} - 12 q^{92} + 8 q^{94} - 4 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(1 - \zeta_{12}^{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 1.00000i 0 −0.866025 0.500000i 1.00000i 0 −0.500000 0.866025i
127.2 0.866025 0.500000i 0 0.500000 0.866025i 1.00000i 0 0.866025 + 0.500000i 1.00000i 0 −0.500000 0.866025i
1135.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 −0.866025 + 0.500000i 1.00000i 0 −0.500000 + 0.866025i
1135.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 0.866025 0.500000i 1.00000i 0 −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.bj.c 4
3.b odd 2 1 182.2.m.a 4
12.b even 2 1 1456.2.cc.b 4
13.e even 6 1 inner 1638.2.bj.c 4
21.c even 2 1 1274.2.m.a 4
21.g even 6 1 1274.2.o.a 4
21.g even 6 1 1274.2.v.b 4
21.h odd 6 1 1274.2.o.b 4
21.h odd 6 1 1274.2.v.a 4
39.h odd 6 1 182.2.m.a 4
39.h odd 6 1 2366.2.d.k 4
39.i odd 6 1 2366.2.d.k 4
39.k even 12 1 2366.2.a.q 2
39.k even 12 1 2366.2.a.s 2
156.r even 6 1 1456.2.cc.b 4
273.u even 6 1 1274.2.m.a 4
273.x odd 6 1 1274.2.o.b 4
273.y even 6 1 1274.2.o.a 4
273.bp odd 6 1 1274.2.v.a 4
273.br even 6 1 1274.2.v.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.m.a 4 3.b odd 2 1
182.2.m.a 4 39.h odd 6 1
1274.2.m.a 4 21.c even 2 1
1274.2.m.a 4 273.u even 6 1
1274.2.o.a 4 21.g even 6 1
1274.2.o.a 4 273.y even 6 1
1274.2.o.b 4 21.h odd 6 1
1274.2.o.b 4 273.x odd 6 1
1274.2.v.a 4 21.h odd 6 1
1274.2.v.a 4 273.bp odd 6 1
1274.2.v.b 4 21.g even 6 1
1274.2.v.b 4 273.br even 6 1
1456.2.cc.b 4 12.b even 2 1
1456.2.cc.b 4 156.r even 6 1
1638.2.bj.c 4 1.a even 1 1 trivial
1638.2.bj.c 4 13.e even 6 1 inner
2366.2.a.q 2 39.k even 12 1
2366.2.a.s 2 39.k even 12 1
2366.2.d.k 4 39.h odd 6 1
2366.2.d.k 4 39.i odd 6 1

Hecke kernels

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

\( T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 6T_{11}^{3} + 14T_{11}^{2} + 12T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + 14 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + 51 T^{2} - 104 T + 169 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + 56 T^{2} + 96 T + 64 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 104T^{2} + 2116 \) Copy content Toggle raw display
$37$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + 59 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$43$ \( T^{4} + 14 T^{3} + 174 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$47$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$53$ \( (T^{2} + 10 T + 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} - 66 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$61$ \( T^{4} - 20 T^{3} + 303 T^{2} + \cdots + 9409 \) Copy content Toggle raw display
$67$ \( T^{4} - 30 T^{3} + 366 T^{2} + \cdots + 4356 \) Copy content Toggle raw display
$71$ \( (T^{2} - 24 T + 192)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 218 T^{2} + 11449 \) Copy content Toggle raw display
$79$ \( (T^{2} + 18 T + 54)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 216T^{2} + 2916 \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + 24 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} + 56 T^{2} + 96 T + 64 \) Copy content Toggle raw display
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