Properties

Label 1323.2.i.d
Level $1323$
Weight $2$
Character orbit 1323.i
Analytic conductor $10.564$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 441)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48q - 48q^{4} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - 48q^{4} - 24q^{11} + 48q^{16} - 48q^{23} - 24q^{25} + 96q^{44} - 48q^{50} + 48q^{53} - 48q^{64} - 168q^{74} + 48q^{79} - 24q^{85} + 24q^{86} + 144q^{92} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1 1.17820i 0 0.611843 2.16601 3.75164i 0 0 3.07728i 0 −4.42019 2.55200i
521.2 2.57819i 0 −4.64709 −1.16595 + 2.01948i 0 0 6.82470i 0 −5.20660 3.00603i
521.3 0.424201i 0 1.82005 1.80381 3.12430i 0 0 1.62047i 0 −1.32533 0.765180i
521.4 2.37274i 0 −3.62990 −1.71774 + 2.97522i 0 0 3.86732i 0 7.05942 + 4.07576i
521.5 1.48451i 0 −0.203760 0.154215 0.267109i 0 0 2.66653i 0 −0.396525 0.228934i
521.6 2.70883i 0 −5.33776 −0.601464 + 1.04177i 0 0 9.04141i 0 2.82197 + 1.62926i
521.7 1.86894i 0 −1.49292 1.25287 2.17003i 0 0 0.947692i 0 4.05565 + 2.34153i
521.8 0.981621i 0 1.03642 −0.940599 + 1.62916i 0 0 2.98061i 0 1.59922 + 0.923312i
521.9 0.664297i 0 1.55871 −0.0141520 + 0.0245119i 0 0 2.36404i 0 −0.0162832 0.00940110i
521.10 0.664297i 0 1.55871 0.0141520 0.0245119i 0 0 2.36404i 0 0.0162832 + 0.00940110i
521.11 0.122344i 0 1.98503 0.264715 0.458500i 0 0 0.487547i 0 0.0560949 + 0.0323864i
521.12 0.122344i 0 1.98503 −0.264715 + 0.458500i 0 0 0.487547i 0 −0.0560949 0.0323864i
521.13 1.48451i 0 −0.203760 −0.154215 + 0.267109i 0 0 2.66653i 0 0.396525 + 0.228934i
521.14 2.57819i 0 −4.64709 1.16595 2.01948i 0 0 6.82470i 0 5.20660 + 3.00603i
521.15 1.83202i 0 −1.35631 0.322784 0.559079i 0 0 1.17925i 0 1.02425 + 0.591348i
521.16 2.08430i 0 −2.34432 1.65233 2.86191i 0 0 0.717672i 0 5.96509 + 3.44395i
521.17 1.83202i 0 −1.35631 −0.322784 + 0.559079i 0 0 1.17925i 0 −1.02425 0.591348i
521.18 0.981621i 0 1.03642 0.940599 1.62916i 0 0 2.98061i 0 −1.59922 0.923312i
521.19 0.424201i 0 1.82005 −1.80381 + 3.12430i 0 0 1.62047i 0 1.32533 + 0.765180i
521.20 1.17820i 0 0.611843 −2.16601 + 3.75164i 0 0 3.07728i 0 4.42019 + 2.55200i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1097.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.i.d 48
3.b odd 2 1 441.2.i.d 48
7.b odd 2 1 inner 1323.2.i.d 48
7.c even 3 1 1323.2.o.e 48
7.c even 3 1 1323.2.s.d 48
7.d odd 6 1 1323.2.o.e 48
7.d odd 6 1 1323.2.s.d 48
9.c even 3 1 441.2.s.d 48
9.d odd 6 1 1323.2.s.d 48
21.c even 2 1 441.2.i.d 48
21.g even 6 1 441.2.o.e 48
21.g even 6 1 441.2.s.d 48
21.h odd 6 1 441.2.o.e 48
21.h odd 6 1 441.2.s.d 48
63.g even 3 1 441.2.o.e 48
63.h even 3 1 441.2.i.d 48
63.i even 6 1 inner 1323.2.i.d 48
63.j odd 6 1 inner 1323.2.i.d 48
63.k odd 6 1 441.2.o.e 48
63.l odd 6 1 441.2.s.d 48
63.n odd 6 1 1323.2.o.e 48
63.o even 6 1 1323.2.s.d 48
63.s even 6 1 1323.2.o.e 48
63.t odd 6 1 441.2.i.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.i.d 48 3.b odd 2 1
441.2.i.d 48 21.c even 2 1
441.2.i.d 48 63.h even 3 1
441.2.i.d 48 63.t odd 6 1
441.2.o.e 48 21.g even 6 1
441.2.o.e 48 21.h odd 6 1
441.2.o.e 48 63.g even 3 1
441.2.o.e 48 63.k odd 6 1
441.2.s.d 48 9.c even 3 1
441.2.s.d 48 21.g even 6 1
441.2.s.d 48 21.h odd 6 1
441.2.s.d 48 63.l odd 6 1
1323.2.i.d 48 1.a even 1 1 trivial
1323.2.i.d 48 7.b odd 2 1 inner
1323.2.i.d 48 63.i even 6 1 inner
1323.2.i.d 48 63.j odd 6 1 inner
1323.2.o.e 48 7.c even 3 1
1323.2.o.e 48 7.d odd 6 1
1323.2.o.e 48 63.n odd 6 1
1323.2.o.e 48 63.s even 6 1
1323.2.s.d 48 7.c even 3 1
1323.2.s.d 48 7.d odd 6 1
1323.2.s.d 48 9.d odd 6 1
1323.2.s.d 48 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{24} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\).