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Properties

Label 847.2.r.e
Level $847$
Weight $2$
Character orbit 847.r
Analytic conductor $6.763$
Analytic rank $0$
Dimension $96$
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: \(96\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 8 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 8 q^{4} - 8 q^{9} + 240 q^{12} + 44 q^{14} - 32 q^{15} + 8 q^{16} + 32 q^{23} - 4 q^{25} - 24 q^{31} - 24 q^{36} - 8 q^{37} - 12 q^{38} + 28 q^{42} - 288 q^{45} - 84 q^{47} - 36 q^{49} + 20 q^{53} - 272 q^{56} - 64 q^{58} + 48 q^{59} - 24 q^{60} - 48 q^{64} - 80 q^{67} + 60 q^{70} - 8 q^{71} + 96 q^{75} - 704 q^{78} + 84 q^{80} + 36 q^{81} - 12 q^{82} + 64 q^{86} + 240 q^{89} + 28 q^{91} + 160 q^{92} - 36 q^{93}+O(q^{100}) \) 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 −1.00361 − 2.25413i −0.601360 − 2.82918i −2.73564 + 3.03823i −1.52819 + 0.160620i −5.77382 + 4.19492i −1.57269 − 2.12759i 4.90069 + 1.59233i −4.90198 + 2.18250i 1.89576 + 3.28355i
40.2 −0.954546 − 2.14395i −0.0527516 − 0.248177i −2.34708 + 2.60670i −0.519673 + 0.0546199i −0.481723 + 0.349993i −2.11176 − 1.59389i 3.36507 + 1.09338i 2.68183 − 1.19403i 0.613154 + 1.06201i
40.3 −0.649326 − 1.45841i 0.210703 + 0.991278i −0.367072 + 0.407675i 0.324736 − 0.0341312i 1.30887 − 0.950952i 1.86039 − 1.88121i −2.20368 − 0.716019i 1.80240 − 0.802480i −0.260637 − 0.451436i
40.4 −0.476756 − 1.07081i −0.349081 − 1.64230i 0.418919 − 0.465257i 2.20062 − 0.231294i −1.59216 + 1.15678i 1.70095 + 2.02652i −2.92749 − 0.951198i 0.165356 − 0.0736214i −1.29683 − 2.24618i
40.5 −0.276787 − 0.621674i 0.626798 + 2.94885i 1.02839 − 1.14215i 3.22411 − 0.338868i 1.65974 − 1.20587i −0.720115 − 2.54587i −2.28909 − 0.743770i −5.56223 + 2.47647i −1.10306 − 1.91055i
40.6 −0.0551589 − 0.123889i 0.165692 + 0.779518i 1.32596 − 1.47262i −3.70160 + 0.389054i 0.0874343 − 0.0635247i −2.46959 + 0.949266i −0.513531 − 0.166856i 2.16044 − 0.961890i 0.252376 + 0.437127i
40.7 0.0551589 + 0.123889i 0.165692 + 0.779518i 1.32596 − 1.47262i −3.70160 + 0.389054i −0.0874343 + 0.0635247i 2.46959 − 0.949266i 0.513531 + 0.166856i 2.16044 − 0.961890i −0.252376 − 0.437127i
40.8 0.276787 + 0.621674i 0.626798 + 2.94885i 1.02839 − 1.14215i 3.22411 − 0.338868i −1.65974 + 1.20587i 0.720115 + 2.54587i 2.28909 + 0.743770i −5.56223 + 2.47647i 1.10306 + 1.91055i
40.9 0.476756 + 1.07081i −0.349081 − 1.64230i 0.418919 − 0.465257i 2.20062 − 0.231294i 1.59216 − 1.15678i −1.70095 − 2.02652i 2.92749 + 0.951198i 0.165356 − 0.0736214i 1.29683 + 2.24618i
40.10 0.649326 + 1.45841i 0.210703 + 0.991278i −0.367072 + 0.407675i 0.324736 − 0.0341312i −1.30887 + 0.950952i −1.86039 + 1.88121i 2.20368 + 0.716019i 1.80240 − 0.802480i 0.260637 + 0.451436i
40.11 0.954546 + 2.14395i −0.0527516 − 0.248177i −2.34708 + 2.60670i −0.519673 + 0.0546199i 0.481723 − 0.349993i 2.11176 + 1.59389i −3.36507 − 1.09338i 2.68183 − 1.19403i −0.613154 − 1.06201i
40.12 1.00361 + 2.25413i −0.601360 − 2.82918i −2.73564 + 3.03823i −1.52819 + 0.160620i 5.77382 − 4.19492i 1.57269 + 2.12759i −4.90069 − 1.59233i −4.90198 + 2.18250i −1.89576 − 3.28355i
94.1 −1.83368 − 1.65105i −1.17644 − 2.64232i 0.427348 + 4.06595i −0.319479 − 1.50303i −2.20540 + 6.78753i −2.52290 + 0.796859i 3.02880 − 4.16878i −3.59048 + 3.98763i −1.89576 + 3.28355i
94.2 −1.74404 − 1.57034i −0.103198 − 0.231786i 0.366650 + 3.48845i −0.108641 − 0.511117i −0.184002 + 0.566300i −2.64531 + 0.0482259i 2.07973 − 2.86250i 1.96432 − 2.18160i −0.613154 + 1.06201i
94.3 −1.18638 − 1.06822i 0.412196 + 0.925808i 0.0573423 + 0.545575i 0.0678884 + 0.319390i 0.499945 − 1.53867i 0.399345 + 2.61544i −1.36195 + 1.87456i 1.32018 − 1.46620i 0.260637 − 0.451436i
94.4 −0.871077 − 0.784321i −0.682905 − 1.53383i −0.0654416 − 0.622635i 0.460054 + 2.16439i −0.608153 + 1.87170i 2.56725 − 0.639692i −1.80929 + 2.49027i 0.121116 − 0.134513i 1.29683 − 2.24618i
94.5 −0.505715 − 0.455348i 1.22620 + 2.75410i −0.160651 − 1.52849i 0.674023 + 3.17103i 0.633963 − 1.95114i −2.07901 + 1.63638i −1.41474 + 1.94722i −4.07408 + 4.52473i 1.10306 − 1.91055i
94.6 −0.100780 − 0.0907430i 0.324142 + 0.728035i −0.207135 − 1.97075i −0.773845 − 3.64066i 0.0333969 − 0.102785i −1.43998 − 2.21956i −0.317380 + 0.436836i 1.58243 − 1.75746i −0.252376 + 0.437127i
94.7 0.100780 + 0.0907430i 0.324142 + 0.728035i −0.207135 − 1.97075i −0.773845 − 3.64066i −0.0333969 + 0.102785i 1.43998 + 2.21956i 0.317380 − 0.436836i 1.58243 − 1.75746i 0.252376 − 0.437127i
94.8 0.505715 + 0.455348i 1.22620 + 2.75410i −0.160651 − 1.52849i 0.674023 + 3.17103i −0.633963 + 1.95114i 2.07901 − 1.63638i 1.41474 − 1.94722i −4.07408 + 4.52473i −1.10306 + 1.91055i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.12
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.e 96
7.d odd 6 1 inner 847.2.r.e 96
11.b odd 2 1 inner 847.2.r.e 96
11.c even 5 1 847.2.i.a ✓ 24
11.c even 5 3 inner 847.2.r.e 96
11.d odd 10 1 847.2.i.a ✓ 24
11.d odd 10 3 inner 847.2.r.e 96
77.i even 6 1 inner 847.2.r.e 96
77.n even 30 1 847.2.i.a ✓ 24
77.n even 30 3 inner 847.2.r.e 96
77.p odd 30 1 847.2.i.a ✓ 24
77.p odd 30 3 inner 847.2.r.e 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.i.a ✓ 24 11.c even 5 1
847.2.i.a ✓ 24 11.d odd 10 1
847.2.i.a ✓ 24 77.n even 30 1
847.2.i.a ✓ 24 77.p odd 30 1
847.2.r.e 96 1.a even 1 1 trivial
847.2.r.e 96 7.d odd 6 1 inner
847.2.r.e 96 11.b odd 2 1 inner
847.2.r.e 96 11.c even 5 3 inner
847.2.r.e 96 11.d odd 10 3 inner
847.2.r.e 96 77.i even 6 1 inner
847.2.r.e 96 77.n even 30 3 inner
847.2.r.e 96 77.p odd 30 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} + 16 T_{2}^{94} + 90 T_{2}^{92} - 200 T_{2}^{90} - 6391 T_{2}^{88} - 50568 T_{2}^{86} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display