Properties

Label 17.16.b.a
Level $17$
Weight $16$
Character orbit 17.b
Analytic conductor $24.258$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [17,16,Mod(16,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.16");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 17.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.2578958670\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 258 q^{2} + 414386 q^{4} - 12648450 q^{8} - 78109330 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 258 q^{2} + 414386 q^{4} - 12648450 q^{8} - 78109330 q^{9} + 702506672 q^{13} - 1787378376 q^{15} + 3524081474 q^{16} - 2245058454 q^{17} + 6803778314 q^{18} + 9958891784 q^{19} - 4168893668 q^{21} - 238696683970 q^{25} - 33467295588 q^{26} - 62541989808 q^{30} - 43445086338 q^{32} + 213283309748 q^{33} + 521524562854 q^{34} - 467785613304 q^{35} - 2300588654186 q^{36} + 3162083165688 q^{38} - 3011205093968 q^{42} - 2215728209008 q^{43} - 7793870107128 q^{47} - 1555224751482 q^{49} + 30118817411766 q^{50} - 21451923375880 q^{51} + 51163160044372 q^{52} - 6062965973460 q^{53} - 11679154373592 q^{55} + 22772194849344 q^{59} - 86295684546192 q^{60} - 28567749560318 q^{64} + 251781147903680 q^{66} + 153875904272808 q^{67} - 48849686100870 q^{68} + 60664072036996 q^{69} - 150925771647648 q^{70} - 293782759569702 q^{72} - 388479948338264 q^{76} - 622427249887884 q^{77} + 983865215787034 q^{81} - 15\!\cdots\!44 q^{83}+ \cdots + 26\!\cdots\!90 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −343.151 2662.06i 84984.7 187278.i 913491.i 2.90764e6i −1.79182e7 7.26232e6 6.42646e7i
16.2 −343.151 2662.06i 84984.7 187278.i 913491.i 2.90764e6i −1.79182e7 7.26232e6 6.42646e7i
16.3 −271.837 6935.71i 41127.3 305155.i 1.88538e6i 956533.i −2.27238e6 −3.37551e7 8.29523e7i
16.4 −271.837 6935.71i 41127.3 305155.i 1.88538e6i 956533.i −2.27238e6 −3.37551e7 8.29523e7i
16.5 −252.822 1625.59i 31151.0 14176.6i 410984.i 2.61491e6i 408809. 1.17064e7 3.58415e6i
16.6 −252.822 1625.59i 31151.0 14176.6i 410984.i 2.61491e6i 408809. 1.17064e7 3.58415e6i
16.7 −202.905 5602.01i 8402.47 284728.i 1.13668e6i 2.21629e6i 4.94389e6 −1.70336e7 5.77728e7i
16.8 −202.905 5602.01i 8402.47 284728.i 1.13668e6i 2.21629e6i 4.94389e6 −1.70336e7 5.77728e7i
16.9 −103.869 2983.33i −21979.3 46006.2i 309875.i 1.62872e6i 5.68653e6 5.44863e6 4.77860e6i
16.10 −103.869 2983.33i −21979.3 46006.2i 309875.i 1.62872e6i 5.68653e6 5.44863e6 4.77860e6i
16.11 12.5353 1940.16i −32610.9 330469.i 24320.6i 2.72193e6i −819546. 1.05847e7 4.14255e6i
16.12 12.5353 1940.16i −32610.9 330469.i 24320.6i 2.72193e6i −819546. 1.05847e7 4.14255e6i
16.13 52.4616 5953.93i −30015.8 78941.6i 312353.i 404894.i −3.29374e6 −2.11004e7 4.14140e6i
16.14 52.4616 5953.93i −30015.8 78941.6i 312353.i 404894.i −3.29374e6 −2.11004e7 4.14140e6i
16.15 158.526 159.452i −7637.56 199360.i 25277.2i 4.18240e6i −6.40533e6 1.43235e7 3.16037e7i
16.16 158.526 159.452i −7637.56 199360.i 25277.2i 4.18240e6i −6.40533e6 1.43235e7 3.16037e7i
16.17 210.994 3417.77i 11750.3 46569.8i 721128.i 204211.i −4.43460e6 2.66774e6 9.82593e6i
16.18 210.994 3417.77i 11750.3 46569.8i 721128.i 204211.i −4.43460e6 2.66774e6 9.82593e6i
16.19 284.859 6544.95i 48376.7 188557.i 1.86439e6i 714240.i 4.44627e6 −2.84875e7 5.37121e7i
16.20 284.859 6544.95i 48376.7 188557.i 1.86439e6i 714240.i 4.44627e6 −2.84875e7 5.37121e7i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.16.b.a 22
17.b even 2 1 inner 17.16.b.a 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.16.b.a 22 1.a even 1 1 trivial
17.16.b.a 22 17.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(17, [\chi])\).