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Properties

Label 869.2.l.c
Level $869$
Weight $2$
Character orbit 869.l
Analytic conductor $6.939$
Analytic rank $0$
Dimension $264$
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] = [869,2,Mod(315,869)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("869.315");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 869.l (of order \(10\), degree \(4\), 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: \(6.93899993565\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(66\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 264 q - 20 q^{2} + 48 q^{4} - 20 q^{5} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 264 q - 20 q^{2} + 48 q^{4} - 20 q^{5} + 72 q^{9} - 6 q^{11} + 8 q^{16} - 70 q^{18} - 10 q^{19} + 54 q^{20} - 26 q^{22} + 8 q^{23} - 22 q^{25} - 80 q^{26} - 14 q^{31} - 138 q^{36} - 8 q^{38} + 50 q^{40} + 26 q^{42} + 76 q^{44} - 76 q^{45} - 70 q^{46} - 118 q^{49} + 120 q^{50} - 60 q^{51} + 20 q^{52} + 26 q^{55} - 20 q^{62} - 112 q^{64} - 52 q^{67} - 70 q^{72} - 120 q^{73} + 40 q^{79} + 130 q^{80} - 50 q^{81} + 180 q^{83} + 110 q^{84} + 224 q^{88} + 28 q^{89} + 90 q^{90} + 54 q^{92} - 140 q^{95} + 36 q^{97} + 34 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
315.1 −1.58224 + 2.17776i −2.36867 − 0.769626i −1.62114 − 4.98937i −0.842391 + 0.612033i 5.42386 − 3.94066i −0.638950 − 1.96649i 8.31047 + 2.70024i 2.59120 + 1.88262i − 2.80291i
315.2 −1.58224 + 2.17776i 2.36867 + 0.769626i −1.62114 − 4.98937i −0.842391 + 0.612033i −5.42386 + 3.94066i 0.638950 + 1.96649i 8.31047 + 2.70024i 2.59120 + 1.88262i − 2.80291i
315.3 −1.44017 + 1.98222i −2.62751 − 0.853730i −1.23708 − 3.80733i 3.01046 − 2.18722i 5.47633 − 3.97879i −0.337328 − 1.03819i 4.66808 + 1.51675i 3.74791 + 2.72302i 9.11735i
315.4 −1.44017 + 1.98222i 2.62751 + 0.853730i −1.23708 − 3.80733i 3.01046 − 2.18722i −5.47633 + 3.97879i 0.337328 + 1.03819i 4.66808 + 1.51675i 3.74791 + 2.72302i 9.11735i
315.5 −1.40592 + 1.93508i −1.90194 − 0.617979i −1.14990 − 3.53903i −2.29137 + 1.66478i 3.86982 − 2.81159i 1.27309 + 3.91818i 3.91532 + 1.27217i 0.808441 + 0.587367i − 6.77454i
315.6 −1.40592 + 1.93508i 1.90194 + 0.617979i −1.14990 − 3.53903i −2.29137 + 1.66478i −3.86982 + 2.81159i −1.27309 − 3.91818i 3.91532 + 1.27217i 0.808441 + 0.587367i − 6.77454i
315.7 −1.34764 + 1.85486i −0.649294 − 0.210968i −1.00636 − 3.09725i −0.0378096 + 0.0274703i 1.26633 − 0.920043i −1.01174 − 3.11382i 2.74013 + 0.890322i −2.04998 − 1.48939i − 0.107152i
315.8 −1.34764 + 1.85486i 0.649294 + 0.210968i −1.00636 − 3.09725i −0.0378096 + 0.0274703i −1.26633 + 0.920043i 1.01174 + 3.11382i 2.74013 + 0.890322i −2.04998 − 1.48939i − 0.107152i
315.9 −1.32033 + 1.81727i −1.61562 − 0.524946i −0.941184 − 2.89667i 0.527998 − 0.383613i 3.08711 − 2.24292i 0.183715 + 0.565417i 2.23403 + 0.725882i −0.0923951 − 0.0671290i 1.46601i
315.10 −1.32033 + 1.81727i 1.61562 + 0.524946i −0.941184 − 2.89667i 0.527998 − 0.383613i −3.08711 + 2.24292i −0.183715 − 0.565417i 2.23403 + 0.725882i −0.0923951 − 0.0671290i 1.46601i
315.11 −1.31252 + 1.80653i −0.555715 − 0.180563i −0.922803 − 2.84010i 2.37097 − 1.72261i 1.05558 − 0.766923i 1.53852 + 4.73508i 2.09451 + 0.680548i −2.15083 − 1.56267i 6.54418i
315.12 −1.31252 + 1.80653i 0.555715 + 0.180563i −0.922803 − 2.84010i 2.37097 − 1.72261i −1.05558 + 0.766923i −1.53852 − 4.73508i 2.09451 + 0.680548i −2.15083 − 1.56267i 6.54418i
315.13 −1.08927 + 1.49926i −3.26624 − 1.06127i −0.443219 − 1.36409i −2.95764 + 2.14885i 5.14894 − 3.74092i −0.919352 − 2.82947i −0.997057 − 0.323963i 7.11499 + 5.16935i − 6.77496i
315.14 −1.08927 + 1.49926i 3.26624 + 1.06127i −0.443219 − 1.36409i −2.95764 + 2.14885i −5.14894 + 3.74092i 0.919352 + 2.82947i −0.997057 − 0.323963i 7.11499 + 5.16935i − 6.77496i
315.15 −0.994033 + 1.36817i −1.02386 − 0.332672i −0.265751 − 0.817896i −2.02378 + 1.47036i 1.47290 − 1.07013i −0.693795 − 2.13528i −1.83357 − 0.595764i −1.48943 − 1.08214i − 4.23046i
315.16 −0.994033 + 1.36817i 1.02386 + 0.332672i −0.265751 − 0.817896i −2.02378 + 1.47036i −1.47290 + 1.07013i 0.693795 + 2.13528i −1.83357 − 0.595764i −1.48943 − 1.08214i − 4.23046i
315.17 −0.959385 + 1.32048i −1.91908 − 0.623546i −0.205214 − 0.631585i 0.00922593 − 0.00670303i 2.66451 − 1.93588i 0.0213735 + 0.0657810i −2.07376 − 0.673806i 0.866998 + 0.629911i 0.0186135i
315.18 −0.959385 + 1.32048i 1.91908 + 0.623546i −0.205214 − 0.631585i 0.00922593 − 0.00670303i −2.66451 + 1.93588i −0.0213735 − 0.0657810i −2.07376 − 0.673806i 0.866998 + 0.629911i 0.0186135i
315.19 −0.930178 + 1.28028i −3.07115 − 0.997878i −0.155852 − 0.479663i 1.81430 − 1.31816i 4.13428 − 3.00373i 0.805599 + 2.47938i −2.25105 − 0.731409i 6.00917 + 4.36592i 3.54893i
315.20 −0.930178 + 1.28028i 3.07115 + 0.997878i −0.155852 − 0.479663i 1.81430 − 1.31816i −4.13428 + 3.00373i −0.805599 − 2.47938i −2.25105 − 0.731409i 6.00917 + 4.36592i 3.54893i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 315.66
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
79.b odd 2 1 inner
869.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 869.2.l.c ✓ 264
11.d odd 10 1 inner 869.2.l.c ✓ 264
79.b odd 2 1 inner 869.2.l.c ✓ 264
869.l even 10 1 inner 869.2.l.c ✓ 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.2.l.c ✓ 264 1.a even 1 1 trivial
869.2.l.c ✓ 264 11.d odd 10 1 inner
869.2.l.c ✓ 264 79.b odd 2 1 inner
869.2.l.c ✓ 264 869.l even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{132} + 10 T_{2}^{131} + 5 T_{2}^{130} - 290 T_{2}^{129} - 779 T_{2}^{128} + 4510 T_{2}^{127} + \cdots + 3125 \) acting on \(S_{2}^{\mathrm{new}}(869, [\chi])\). Copy content Toggle raw display