Properties

Label 1617.2.c.b
Level $1617$
Weight $2$
Character orbit 1617.c
Analytic conductor $12.912$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9118100068\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 64 q^{4} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 64 q^{4} - 48 q^{9} - 16 q^{11} + 64 q^{16} + 16 q^{22} + 32 q^{23} - 80 q^{25} + 64 q^{36} - 96 q^{37} - 32 q^{44} + 64 q^{53} + 48 q^{58} - 48 q^{60} - 240 q^{64} + 96 q^{67} - 32 q^{71} + 48 q^{78} + 48 q^{81} - 96 q^{86} - 48 q^{88} - 32 q^{92} + 96 q^{93} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
538.1 2.80191i 1.00000i −5.85072 1.80767i −2.80191 0 10.7894i −1.00000 5.06493
538.2 2.80191i 1.00000i −5.85072 1.80767i 2.80191 0 10.7894i −1.00000 −5.06493
538.3 2.69641i 1.00000i −5.27065 3.60829i −2.69641 0 8.81904i −1.00000 −9.72943
538.4 2.69641i 1.00000i −5.27065 3.60829i 2.69641 0 8.81904i −1.00000 9.72943
538.5 2.34878i 1.00000i −3.51679 3.84365i −2.34878 0 3.56261i −1.00000 9.02791
538.6 2.34878i 1.00000i −3.51679 3.84365i 2.34878 0 3.56261i −1.00000 −9.02791
538.7 2.14374i 1.00000i −2.59563 1.55945i −2.14374 0 1.27689i −1.00000 3.34306
538.8 2.14374i 1.00000i −2.59563 1.55945i 2.14374 0 1.27689i −1.00000 −3.34306
538.9 2.06268i 1.00000i −2.25465 3.23777i −2.06268 0 0.525252i −1.00000 6.67848
538.10 2.06268i 1.00000i −2.25465 3.23777i 2.06268 0 0.525252i −1.00000 −6.67848
538.11 1.89228i 1.00000i −1.58073 1.86532i −1.89228 0 0.793378i −1.00000 −3.52970
538.12 1.89228i 1.00000i −1.58073 1.86532i 1.89228 0 0.793378i −1.00000 3.52970
538.13 1.47317i 1.00000i −0.170220 1.04370i −1.47317 0 2.69557i −1.00000 −1.53754
538.14 1.47317i 1.00000i −0.170220 1.04370i 1.47317 0 2.69557i −1.00000 1.53754
538.15 1.38648i 1.00000i 0.0776704 3.79946i −1.38648 0 2.88065i −1.00000 −5.26789
538.16 1.38648i 1.00000i 0.0776704 3.79946i 1.38648 0 2.88065i −1.00000 5.26789
538.17 1.03215i 1.00000i 0.934667 3.55499i −1.03215 0 3.02902i −1.00000 −3.66928
538.18 1.03215i 1.00000i 0.934667 3.55499i 1.03215 0 3.02902i −1.00000 3.66928
538.19 1.01496i 1.00000i 0.969853 1.37003i −1.01496 0 3.01429i −1.00000 1.39053
538.20 1.01496i 1.00000i 0.969853 1.37003i 1.01496 0 3.01429i −1.00000 −1.39053
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 538.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1617.2.c.b 48
7.b odd 2 1 inner 1617.2.c.b 48
11.b odd 2 1 inner 1617.2.c.b 48
77.b even 2 1 inner 1617.2.c.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.2.c.b 48 1.a even 1 1 trivial
1617.2.c.b 48 7.b odd 2 1 inner
1617.2.c.b 48 11.b odd 2 1 inner
1617.2.c.b 48 77.b even 2 1 inner