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Properties

Label 847.2.r.f
Level $847$
Weight $2$
Character orbit 847.r
Analytic conductor $6.763$
Analytic rank $0$
Dimension $224$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,2,Mod(40,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([25, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.40");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.r (of order \(30\), degree \(8\), not 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(28\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 32 q^{4} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q - 32 q^{4} - 32 q^{9} - 240 q^{12} + 4 q^{14} + 32 q^{15} + 56 q^{16} - 32 q^{23} - 36 q^{25} + 24 q^{31} + 64 q^{36} - 16 q^{37} - 168 q^{38} + 80 q^{42} + 288 q^{45} + 48 q^{47} - 8 q^{53} + 128 q^{56} - 32 q^{58} + 96 q^{59} + 96 q^{60} + 80 q^{64} - 64 q^{67} - 60 q^{70} - 88 q^{71} - 240 q^{75} + 32 q^{78} + 60 q^{80} - 44 q^{81} + 156 q^{82} - 136 q^{86} - 96 q^{89} - 20 q^{91} - 40 q^{92} + 88 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −1.09991 − 2.47044i 0.368985 + 1.73594i −3.55501 + 3.94824i −1.46209 + 0.153672i 3.88268 − 2.82093i 1.84824 + 1.89315i 8.52033 + 2.76842i −0.136687 + 0.0608572i 1.98781 + 3.44299i
40.2 −1.09666 − 2.46313i −0.215315 − 1.01298i −3.52610 + 3.91613i 2.33193 − 0.245096i −2.25897 + 1.64124i −2.09464 + 1.61632i 8.38431 + 2.72423i 1.76088 − 0.783992i −3.16104 − 5.47507i
40.3 −1.00417 − 2.25539i 0.596160 + 2.80471i −2.74019 + 3.04329i −2.63216 + 0.276651i 5.72709 − 4.16097i −2.64453 − 0.0802694i 4.91942 + 1.59842i −4.77037 + 2.12391i 3.26708 + 5.65875i
40.4 −0.997947 − 2.24143i −0.0324834 − 0.152822i −2.68983 + 2.98736i −3.40076 + 0.357434i −0.310123 + 0.225318i 2.34817 − 1.21906i 4.71332 + 1.53145i 2.71834 − 1.21028i 4.19494 + 7.26584i
40.5 −0.879644 − 1.97571i −0.356823 − 1.67872i −1.79141 + 1.98956i −0.239376 + 0.0251595i −3.00279 + 2.18166i 2.53490 − 0.757825i 1.39292 + 0.452588i 0.0498604 − 0.0221993i 0.260274 + 0.450807i
40.6 −0.796055 − 1.78797i −0.308143 − 1.44970i −1.22487 + 1.36035i 4.21807 − 0.443337i −2.34671 + 1.70499i 0.479415 − 2.60195i −0.315444 − 0.102494i 0.733966 − 0.326783i −4.15048 − 7.18885i
40.7 −0.772849 − 1.73585i 0.470659 + 2.21427i −1.07761 + 1.19681i 2.57475 − 0.270617i 3.47989 − 2.52829i −1.99648 + 1.73611i −0.703944 − 0.228725i −1.94086 + 0.864124i −2.45965 − 4.26023i
40.8 −0.557342 − 1.25181i −0.430195 − 2.02391i 0.0818623 − 0.0909173i −3.94430 + 0.414562i −2.29379 + 1.66653i −1.50566 − 2.17555i −2.76586 − 0.898682i −1.17051 + 0.521145i 2.71728 + 4.70646i
40.9 −0.499976 − 1.12297i 0.559624 + 2.63282i 0.327186 − 0.363377i 0.0207770 − 0.00218375i 2.67677 − 1.94479i −1.54998 + 2.14419i −2.90980 − 0.945450i −3.87795 + 1.72657i −0.0128403 − 0.0222400i
40.10 −0.320579 − 0.720033i 0.453325 + 2.13272i 0.922585 − 1.02463i −1.25761 + 0.132180i 1.39031 − 1.01012i −1.74749 − 1.98653i −2.53273 − 0.822933i −1.60238 + 0.713424i 0.498336 + 0.863143i
40.11 −0.304911 − 0.684841i 0.213467 + 1.00428i 0.962224 − 1.06866i 1.19428 − 0.125524i 0.622686 − 0.452408i 0.164380 + 2.64064i −2.45118 − 0.796436i 1.77762 − 0.791448i −0.450112 − 0.779616i
40.12 −0.242181 − 0.543947i −0.661680 − 3.11296i 1.10103 − 1.22282i −2.20883 + 0.232157i −1.53304 + 1.11382i −0.985858 + 2.45522i −2.06436 − 0.670752i −6.51206 + 2.89935i 0.661216 + 1.14526i
40.13 −0.203569 − 0.457223i −0.0827501 − 0.389309i 1.17065 − 1.30014i 1.53646 − 0.161489i −0.161155 + 0.117086i 0.372716 − 2.61937i −1.78475 − 0.579901i 2.59592 − 1.15578i −0.386612 − 0.669631i
40.14 −0.0347467 − 0.0780423i −0.574830 − 2.70436i 1.33338 − 1.48087i 3.26884 − 0.343569i −0.191081 + 0.138829i 2.26147 + 1.37322i −0.324394 − 0.105402i −4.24250 + 1.88888i −0.140394 − 0.243170i
40.15 0.0347467 + 0.0780423i −0.574830 − 2.70436i 1.33338 − 1.48087i 3.26884 − 0.343569i 0.191081 − 0.138829i −2.26147 − 1.37322i 0.324394 + 0.105402i −4.24250 + 1.88888i 0.140394 + 0.243170i
40.16 0.203569 + 0.457223i −0.0827501 − 0.389309i 1.17065 − 1.30014i 1.53646 − 0.161489i 0.161155 − 0.117086i −0.372716 + 2.61937i 1.78475 + 0.579901i 2.59592 − 1.15578i 0.386612 + 0.669631i
40.17 0.242181 + 0.543947i −0.661680 − 3.11296i 1.10103 − 1.22282i −2.20883 + 0.232157i 1.53304 − 1.11382i 0.985858 − 2.45522i 2.06436 + 0.670752i −6.51206 + 2.89935i −0.661216 − 1.14526i
40.18 0.304911 + 0.684841i 0.213467 + 1.00428i 0.962224 − 1.06866i 1.19428 − 0.125524i −0.622686 + 0.452408i −0.164380 − 2.64064i 2.45118 + 0.796436i 1.77762 − 0.791448i 0.450112 + 0.779616i
40.19 0.320579 + 0.720033i 0.453325 + 2.13272i 0.922585 − 1.02463i −1.25761 + 0.132180i −1.39031 + 1.01012i 1.74749 + 1.98653i 2.53273 + 0.822933i −1.60238 + 0.713424i −0.498336 − 0.863143i
40.20 0.499976 + 1.12297i 0.559624 + 2.63282i 0.327186 − 0.363377i 0.0207770 − 0.00218375i −2.67677 + 1.94479i 1.54998 − 2.14419i 2.90980 + 0.945450i −3.87795 + 1.72657i 0.0128403 + 0.0222400i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
77.i even 6 1 inner
77.n even 30 3 inner
77.p odd 30 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.r.f 224
7.d odd 6 1 inner 847.2.r.f 224
11.b odd 2 1 inner 847.2.r.f 224
11.c even 5 1 847.2.i.c ✓ 56
11.c even 5 3 inner 847.2.r.f 224
11.d odd 10 1 847.2.i.c ✓ 56
11.d odd 10 3 inner 847.2.r.f 224
77.i even 6 1 inner 847.2.r.f 224
77.n even 30 1 847.2.i.c ✓ 56
77.n even 30 3 inner 847.2.r.f 224
77.p odd 30 1 847.2.i.c ✓ 56
77.p odd 30 3 inner 847.2.r.f 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.i.c ✓ 56 11.c even 5 1
847.2.i.c ✓ 56 11.d odd 10 1
847.2.i.c ✓ 56 77.n even 30 1
847.2.i.c ✓ 56 77.p odd 30 1
847.2.r.f 224 1.a even 1 1 trivial
847.2.r.f 224 7.d odd 6 1 inner
847.2.r.f 224 11.b odd 2 1 inner
847.2.r.f 224 11.c even 5 3 inner
847.2.r.f 224 11.d odd 10 3 inner
847.2.r.f 224 77.i even 6 1 inner
847.2.r.f 224 77.n even 30 3 inner
847.2.r.f 224 77.p odd 30 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{224} + 44 T_{2}^{222} + 850 T_{2}^{220} + 8048 T_{2}^{218} + 4479 T_{2}^{216} + \cdots + 18\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display