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Properties

Label 847.2.p.b
Level $847$
Weight $2$
Character orbit 847.p
Analytic conductor $6.763$
Analytic rank $0$
Dimension $840$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,2,Mod(76,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.p (of order \(22\), degree \(10\), 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: \(840\)
Relative dimension: \(84\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 840 q - 22 q^{2} + 58 q^{4} - 11 q^{7} - 22 q^{8} - 924 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 840 q - 22 q^{2} + 58 q^{4} - 11 q^{7} - 22 q^{8} - 924 q^{9} - 32 q^{11} - 41 q^{14} + 46 q^{15} - 78 q^{16} + 22 q^{21} + 14 q^{22} + 100 q^{23} + 58 q^{25} - 11 q^{28} - 22 q^{29} + 44 q^{30} - 22 q^{32} - 11 q^{35} - 144 q^{36} + 40 q^{37} + 44 q^{39} - 4 q^{42} - 22 q^{43} + 110 q^{44} - 22 q^{46} - 53 q^{49} - 22 q^{50} + 132 q^{51} - 62 q^{53} - 106 q^{56} + 44 q^{57} - 98 q^{58} + 32 q^{60} + 194 q^{64} - 22 q^{65} - 10 q^{67} + 21 q^{70} - 42 q^{71} - 132 q^{72} - 22 q^{74} - 131 q^{77} + 108 q^{78} - 110 q^{79} + 632 q^{81} + 22 q^{84} + 88 q^{85} - 46 q^{86} + 182 q^{88} + 70 q^{91} + 442 q^{92} - 74 q^{93} + 66 q^{95} + 154 q^{98} - 68 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −2.04803 + 1.77463i − 2.53561i 0.760496 − 5.28936i 0.00682295 − 0.0106167i 4.49977 + 5.19301i 0.0499523 − 2.64528i 4.89895 + 7.62291i −3.42930 0.00486713 + 0.0338516i
76.2 −2.04803 + 1.77463i 2.53561i 0.760496 − 5.28936i −0.00682295 + 0.0106167i −4.49977 − 5.19301i −1.47217 − 2.19834i 4.89895 + 7.62291i −3.42930 −0.00486713 − 0.0338516i
76.3 −2.02099 + 1.75120i − 2.60581i 0.733081 − 5.09869i 0.552411 − 0.859568i 4.56331 + 5.26633i −0.791762 + 2.52450i 4.55576 + 7.08891i −3.79027 0.388858 + 2.70456i
76.4 −2.02099 + 1.75120i 2.60581i 0.733081 − 5.09869i −0.552411 + 0.859568i −4.56331 − 5.26633i 2.03092 + 1.69569i 4.55576 + 7.08891i −3.79027 −0.388858 − 2.70456i
76.5 −2.01326 + 1.74450i − 0.672588i 0.725304 − 5.04460i −2.27709 + 3.54322i 1.17333 + 1.35409i −2.49690 − 0.874926i 4.45962 + 6.93931i 2.54762 −1.59677 − 11.1058i
76.6 −2.01326 + 1.74450i 0.672588i 0.725304 − 5.04460i 2.27709 − 3.54322i −1.17333 − 1.35409i 1.62750 − 2.08596i 4.45962 + 6.93931i 2.54762 1.59677 + 11.1058i
76.7 −1.89347 + 1.64070i − 0.176213i 0.608703 − 4.23362i 1.55515 − 2.41985i 0.289114 + 0.333655i −1.06911 + 2.42013i 3.08448 + 4.79955i 2.96895 1.02564 + 7.13347i
76.8 −1.89347 + 1.64070i 0.176213i 0.608703 − 4.23362i −1.55515 + 2.41985i −0.289114 − 0.333655i 2.20781 + 1.45794i 3.08448 + 4.79955i 2.96895 −1.02564 − 7.13347i
76.9 −1.71232 + 1.48373i − 3.10793i 0.445941 − 3.10159i −1.79697 + 2.79614i 4.61133 + 5.32176i 1.12659 + 2.39391i 1.38845 + 2.16047i −6.65922 −1.07174 − 7.45409i
76.10 −1.71232 + 1.48373i 3.10793i 0.445941 − 3.10159i 1.79697 − 2.79614i −4.61133 − 5.32176i 0.346502 + 2.62296i 1.38845 + 2.16047i −6.65922 1.07174 + 7.45409i
76.11 −1.63207 + 1.41419i − 1.61058i 0.379068 − 2.63647i 1.07408 − 1.67131i 2.27767 + 2.62858i −2.62444 − 0.335147i 0.774755 + 1.20554i 0.406026 0.610575 + 4.24664i
76.12 −1.63207 + 1.41419i 1.61058i 0.379068 − 2.63647i −1.07408 + 1.67131i −2.27767 − 2.62858i 2.02662 − 1.70082i 0.774755 + 1.20554i 0.406026 −0.610575 − 4.24664i
76.13 −1.62088 + 1.40450i − 1.65558i 0.370000 − 2.57341i −0.766471 + 1.19265i 2.32526 + 2.68350i 2.42376 − 1.06084i 0.695569 + 1.08233i 0.259056 −0.432723 − 3.00965i
76.14 −1.62088 + 1.40450i 1.65558i 0.370000 − 2.57341i 0.766471 − 1.19265i −2.32526 − 2.68350i −2.61253 + 0.417949i 0.695569 + 1.08233i 0.259056 0.432723 + 3.00965i
76.15 −1.51315 + 1.31115i − 2.96764i 0.285875 − 1.98830i 1.43587 − 2.23426i 3.89103 + 4.49048i 1.81027 − 1.92949i 0.00946863 + 0.0147335i −5.80687 0.756765 + 5.26341i
76.16 −1.51315 + 1.31115i 2.96764i 0.285875 − 1.98830i −1.43587 + 2.23426i −3.89103 − 4.49048i −2.56606 − 0.644483i 0.00946863 + 0.0147335i −5.80687 −0.756765 − 5.26341i
76.17 −1.37112 + 1.18808i − 1.66383i 0.183802 − 1.27837i 0.808470 − 1.25800i 1.97677 + 2.28132i 1.99569 + 1.73702i −0.694925 − 1.08132i 0.231658 0.386103 + 2.68541i
76.18 −1.37112 + 1.18808i 1.66383i 0.183802 − 1.27837i −0.808470 + 1.25800i −1.97677 − 2.28132i −0.739778 + 2.54022i −0.694925 − 1.08132i 0.231658 −0.386103 − 2.68541i
76.19 −1.33834 + 1.15968i − 0.108070i 0.161673 − 1.12446i 1.43945 − 2.23982i 0.125327 + 0.144635i 2.59681 + 0.506516i −0.827180 − 1.28712i 2.98832 0.671006 + 4.66695i
76.20 −1.33834 + 1.15968i 0.108070i 0.161673 − 1.12446i −1.43945 + 2.23982i −0.125327 − 0.144635i −1.91074 + 1.83005i −0.827180 − 1.28712i 2.98832 −0.671006 − 4.66695i
See next 80 embeddings (of 840 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.84
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
121.f odd 22 1 inner
847.p even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.p.b ✓ 840
7.b odd 2 1 inner 847.2.p.b ✓ 840
121.f odd 22 1 inner 847.2.p.b ✓ 840
847.p even 22 1 inner 847.2.p.b ✓ 840
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.p.b ✓ 840 1.a even 1 1 trivial
847.2.p.b ✓ 840 7.b odd 2 1 inner
847.2.p.b ✓ 840 121.f odd 22 1 inner
847.2.p.b ✓ 840 847.p even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{420} + 11 T_{2}^{419} + 4 T_{2}^{418} - 396 T_{2}^{417} - 1018 T_{2}^{416} + \cdots + 10\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display