Properties

Label 2646.2.h.m
Level $2646$
Weight $2$
Character orbit 2646.h
Analytic conductor $21.128$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_1 - 1) q^{4} + ( - \beta_{3} - 1) q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_1 - 1) q^{4} + ( - \beta_{3} - 1) q^{5} - q^{8} + ( - \beta_{2} - \beta_1) q^{10} - 2 q^{11} + 2 \beta_{2} q^{13} - \beta_1 q^{16} + 2 \beta_1 q^{17} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 - 5) q^{19} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{20} - 2 \beta_1 q^{22} + q^{23} + (2 \beta_{3} + 2) q^{25} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{26} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{29} + (6 \beta_1 - 6) q^{31} + ( - \beta_1 + 1) q^{32} + (2 \beta_1 - 2) q^{34} + (4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 2) q^{37} + ( - \beta_{3} - 5) q^{38} + (\beta_{3} + 1) q^{40} - 4 \beta_{2} q^{41} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{43} + ( - 2 \beta_1 + 2) q^{44} + \beta_1 q^{46} - 4 \beta_{2} q^{47} + (2 \beta_{2} + 2 \beta_1) q^{50} - 2 \beta_{3} q^{52} + (2 \beta_{2} - 6 \beta_1) q^{53} + (2 \beta_{3} + 2) q^{55} + (2 \beta_{3} - 2) q^{58} + (2 \beta_1 - 2) q^{59} + ( - \beta_{2} - 9 \beta_1) q^{61} - 6 q^{62} + q^{64} + ( - 2 \beta_{2} - 12 \beta_1) q^{65} + ( - 2 \beta_{3} + 2 \beta_{2} - 8 \beta_1 + 8) q^{67} - 2 q^{68} + ( - 2 \beta_{3} - 5) q^{71} + ( - 2 \beta_{2} + 2 \beta_1) q^{73} + (4 \beta_{3} - 2) q^{74} + ( - \beta_{2} - 5 \beta_1) q^{76} + ( - 2 \beta_{2} - 3 \beta_1) q^{79} + (\beta_{2} + \beta_1) q^{80} + (4 \beta_{3} - 4 \beta_{2}) q^{82} + ( - 2 \beta_1 + 2) q^{83} + ( - 2 \beta_{2} - 2 \beta_1) q^{85} + (2 \beta_{3} - 2) q^{86} + 2 q^{88} + (2 \beta_{3} - 2 \beta_{2} + 12 \beta_1 - 12) q^{89} + (\beta_1 - 1) q^{92} + (4 \beta_{3} - 4 \beta_{2}) q^{94} + (6 \beta_{3} - 6 \beta_{2} - 11 \beta_1 + 11) q^{95} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} - 4 q^{8} - 2 q^{10} - 8 q^{11} - 2 q^{16} + 4 q^{17} - 10 q^{19} + 2 q^{20} - 4 q^{22} + 4 q^{23} + 8 q^{25} - 4 q^{29} - 12 q^{31} + 2 q^{32} - 4 q^{34} - 4 q^{37} - 20 q^{38} + 4 q^{40} - 4 q^{43} + 4 q^{44} + 2 q^{46} + 4 q^{50} - 12 q^{53} + 8 q^{55} - 8 q^{58} - 4 q^{59} - 18 q^{61} - 24 q^{62} + 4 q^{64} - 24 q^{65} + 16 q^{67} - 8 q^{68} - 20 q^{71} + 4 q^{73} - 8 q^{74} - 10 q^{76} - 6 q^{79} + 2 q^{80} + 4 q^{83} - 4 q^{85} - 8 q^{86} + 8 q^{88} - 24 q^{89} - 2 q^{92} + 22 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-\beta_{1}\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
0.500000 0.866025i 0 −0.500000 0.866025i −3.44949 0 0 −1.00000 0 −1.72474 + 2.98735i
361.2 0.500000 0.866025i 0 −0.500000 0.866025i 1.44949 0 0 −1.00000 0 0.724745 1.25529i
667.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.44949 0 0 −1.00000 0 −1.72474 2.98735i
667.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.44949 0 0 −1.00000 0 0.724745 + 1.25529i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.h.m 4
3.b odd 2 1 882.2.h.k 4
7.b odd 2 1 2646.2.h.n 4
7.c even 3 1 378.2.f.d 4
7.c even 3 1 2646.2.e.l 4
7.d odd 6 1 2646.2.e.k 4
7.d odd 6 1 2646.2.f.k 4
9.c even 3 1 2646.2.e.l 4
9.d odd 6 1 882.2.e.m 4
21.c even 2 1 882.2.h.l 4
21.g even 6 1 882.2.e.n 4
21.g even 6 1 882.2.f.j 4
21.h odd 6 1 126.2.f.c 4
21.h odd 6 1 882.2.e.m 4
28.g odd 6 1 3024.2.r.e 4
63.g even 3 1 1134.2.a.i 2
63.g even 3 1 inner 2646.2.h.m 4
63.h even 3 1 378.2.f.d 4
63.i even 6 1 882.2.f.j 4
63.j odd 6 1 126.2.f.c 4
63.k odd 6 1 2646.2.h.n 4
63.k odd 6 1 7938.2.a.bm 2
63.l odd 6 1 2646.2.e.k 4
63.n odd 6 1 882.2.h.k 4
63.n odd 6 1 1134.2.a.p 2
63.o even 6 1 882.2.e.n 4
63.s even 6 1 882.2.h.l 4
63.s even 6 1 7938.2.a.bn 2
63.t odd 6 1 2646.2.f.k 4
84.n even 6 1 1008.2.r.e 4
252.o even 6 1 9072.2.a.bk 2
252.u odd 6 1 3024.2.r.e 4
252.bb even 6 1 1008.2.r.e 4
252.bl odd 6 1 9072.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 21.h odd 6 1
126.2.f.c 4 63.j odd 6 1
378.2.f.d 4 7.c even 3 1
378.2.f.d 4 63.h even 3 1
882.2.e.m 4 9.d odd 6 1
882.2.e.m 4 21.h odd 6 1
882.2.e.n 4 21.g even 6 1
882.2.e.n 4 63.o even 6 1
882.2.f.j 4 21.g even 6 1
882.2.f.j 4 63.i even 6 1
882.2.h.k 4 3.b odd 2 1
882.2.h.k 4 63.n odd 6 1
882.2.h.l 4 21.c even 2 1
882.2.h.l 4 63.s even 6 1
1008.2.r.e 4 84.n even 6 1
1008.2.r.e 4 252.bb even 6 1
1134.2.a.i 2 63.g even 3 1
1134.2.a.p 2 63.n odd 6 1
2646.2.e.k 4 7.d odd 6 1
2646.2.e.k 4 63.l odd 6 1
2646.2.e.l 4 7.c even 3 1
2646.2.e.l 4 9.c even 3 1
2646.2.f.k 4 7.d odd 6 1
2646.2.f.k 4 63.t odd 6 1
2646.2.h.m 4 1.a even 1 1 trivial
2646.2.h.m 4 63.g even 3 1 inner
2646.2.h.n 4 7.b odd 2 1
2646.2.h.n 4 63.k odd 6 1
3024.2.r.e 4 28.g odd 6 1
3024.2.r.e 4 252.u odd 6 1
7938.2.a.bm 2 63.k odd 6 1
7938.2.a.bn 2 63.s even 6 1
9072.2.a.bd 2 252.bl odd 6 1
9072.2.a.bk 2 252.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2646, [\chi])\):

\( T_{5}^{2} + 2T_{5} - 5 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{13}^{4} + 24T_{13}^{2} + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 2)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 10 T^{3} + 81 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( (T - 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400 \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + 108 T^{2} + \cdots + 8464 \) Copy content Toggle raw display
$41$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400 \) Copy content Toggle raw display
$47$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + 132 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 18 T^{3} + 249 T^{2} + \cdots + 5625 \) Copy content Toggle raw display
$67$ \( T^{4} - 16 T^{3} + 216 T^{2} + \cdots + 1600 \) Copy content Toggle raw display
$71$ \( (T^{2} + 10 T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + 51 T^{2} - 90 T + 225 \) Copy content Toggle raw display
$83$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + 456 T^{2} + \cdots + 14400 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400 \) Copy content Toggle raw display
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