↑

Properties

Label 215.2.l.c
Level $215$
Weight $2$
Character orbit 215.l
Analytic conductor $1.717$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [215,2,Mod(7,215)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(215, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("215.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 215 = 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 215.l (of order \(12\), 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: \(1.71678364346\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{3} - 6 q^{5} - 16 q^{6} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{3} - 6 q^{5} - 16 q^{6} - 12 q^{7} - 4 q^{10} - 40 q^{11} + 54 q^{12} + 2 q^{13} - 96 q^{16} + 4 q^{17} + 24 q^{18} + 6 q^{20} - 40 q^{21} - 18 q^{23} - 10 q^{25} + 36 q^{26} - 30 q^{28} + 90 q^{30} + 4 q^{31} - 60 q^{33} + 60 q^{35} - 8 q^{36} + 42 q^{37} + 40 q^{38} + 12 q^{40} + 48 q^{41} - 22 q^{43} - 120 q^{46} - 56 q^{47} + 96 q^{48} - 72 q^{50} + 40 q^{52} - 16 q^{53} - 84 q^{55} + 84 q^{56} - 20 q^{57} + 2 q^{58} - 4 q^{60} + 24 q^{61} - 72 q^{62} - 36 q^{63} + 28 q^{66} + 24 q^{67} - 86 q^{68} - 30 q^{72} - 72 q^{73} + 132 q^{76} + 60 q^{77} + 72 q^{78} - 24 q^{80} + 44 q^{81} - 22 q^{83} - 56 q^{86} - 92 q^{87} - 4 q^{90} + 36 q^{91} + 6 q^{92} + 90 q^{93} + 42 q^{95} + 56 q^{96} + 8 q^{97} + 108 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.81575 + 1.81575i −0.830791 + 3.10055i − 4.59392i −0.0448054 + 2.23562i −4.12133 − 7.13835i −0.904220 − 3.37460i 4.70991 + 4.70991i −6.32514 − 3.65182i −3.97798 − 4.14069i
7.2 −1.76536 + 1.76536i −0.212315 + 0.792369i − 4.23299i 1.26617 − 1.84304i −1.02400 − 1.77363i 0.530036 + 1.97812i 3.94202 + 3.94202i 2.01530 + 1.16354i 1.01838 + 5.48888i
7.3 −1.58894 + 1.58894i 0.00254100 − 0.00948315i − 3.04945i −2.16557 − 0.557048i 0.0110306 + 0.0191056i −0.480636 − 1.79376i 1.66752 + 1.66752i 2.59799 + 1.49995i 4.32608 − 2.55584i
7.4 −1.42052 + 1.42052i 0.558117 − 2.08292i − 2.03573i 2.18449 + 0.477515i 2.16601 + 3.75164i −1.08174 − 4.03712i 0.0507618 + 0.0507618i −1.42899 − 0.825025i −3.78142 + 2.42478i
7.5 −0.888257 + 0.888257i −0.301799 + 1.12633i 0.422000i 1.61771 + 1.54370i −0.732395 − 1.26855i 0.358131 + 1.33656i −2.15136 − 2.15136i 1.42054 + 0.820148i −2.80815 + 0.0657357i
7.6 −0.866315 + 0.866315i −0.790305 + 2.94946i 0.498996i −1.88347 − 1.20522i −1.87051 − 3.23981i 0.880897 + 3.28755i −2.16492 − 2.16492i −5.47664 − 3.16194i 2.67578 − 0.587575i
7.7 −0.199891 + 0.199891i 0.784503 − 2.92781i 1.92009i −2.02054 − 0.957810i 0.428428 + 0.742059i −0.859254 − 3.20678i −0.783591 − 0.783591i −5.35853 − 3.09375i 0.595347 − 0.212431i
7.8 −0.192675 + 0.192675i −0.342310 + 1.27752i 1.92575i −2.04945 + 0.894293i −0.180192 − 0.312101i −0.391968 − 1.46284i −0.756395 − 0.756395i 1.08320 + 0.625383i 0.222570 − 0.567186i
7.9 0.108782 − 0.108782i 0.463714 − 1.73061i 1.97633i −0.269521 + 2.21977i −0.137815 − 0.238702i 0.825451 + 3.08062i 0.432553 + 0.432553i −0.181887 − 0.105012i 0.212151 + 0.270789i
7.10 0.326485 − 0.326485i −0.735211 + 2.74384i 1.78682i 2.19401 − 0.431633i 0.655788 + 1.13586i −0.341869 − 1.27587i 1.23634 + 1.23634i −4.39007 − 2.53461i 0.575391 − 0.857234i
7.11 0.647075 − 0.647075i −0.223794 + 0.835211i 1.16259i −0.180018 − 2.22881i 0.395633 + 0.685256i 1.02365 + 3.82030i 2.04643 + 2.04643i 1.95058 + 1.12617i −1.55869 − 1.32572i
7.12 0.723681 − 0.723681i 0.164500 − 0.613921i 0.952571i 1.04593 + 1.97637i −0.325238 − 0.563328i −1.24630 − 4.65125i 2.13672 + 2.13672i 2.24824 + 1.29802i 2.18718 + 0.673346i
7.13 0.777897 − 0.777897i 0.696612 − 2.59979i 0.789752i 1.99938 − 1.00124i −1.48048 − 2.56426i 0.312346 + 1.16569i 2.17014 + 2.17014i −3.67558 − 2.12210i 0.776456 − 2.33417i
7.14 1.29059 − 1.29059i −0.594483 + 2.21864i − 1.33123i −1.36711 + 1.76946i 2.09612 + 3.63058i 0.0485393 + 0.181151i 0.863108 + 0.863108i −1.97088 − 1.13789i 0.519267 + 4.04802i
7.15 1.37861 − 1.37861i −0.00453645 + 0.0169302i − 1.80111i −1.03619 − 1.98149i 0.0170862 + 0.0295941i −0.898652 − 3.35381i 0.274190 + 0.274190i 2.59781 + 1.49985i −4.16019 − 1.30321i
7.16 1.45267 − 1.45267i 0.564012 − 2.10492i − 2.22049i −2.22139 − 0.255819i −2.23843 − 3.87707i 0.515817 + 1.92506i −0.320292 − 0.320292i −1.51451 − 0.874402i −3.59856 + 2.85531i
7.17 1.85718 − 1.85718i −0.659639 + 2.46181i − 4.89823i 1.02456 − 1.98753i 3.34695 + 5.79708i 0.150722 + 0.562504i −5.38253 − 5.38253i −3.02729 − 1.74780i −1.78840 − 5.59399i
7.18 1.90680 − 1.90680i 0.193235 − 0.721162i − 5.27175i 0.405810 + 2.19894i −1.00665 − 1.74357i 0.291106 + 1.08642i −6.23856 − 6.23856i 2.11534 + 1.22129i 4.96672 + 3.41913i
37.1 −1.90680 + 1.90680i 0.721162 − 0.193235i − 5.27175i −1.70143 + 1.45091i −1.00665 + 1.74357i 1.08642 + 0.291106i 6.23856 + 6.23856i −2.11534 + 1.22129i 0.477689 − 6.01087i
37.2 −1.85718 + 1.85718i −2.46181 + 0.659639i − 4.89823i 2.23353 − 0.106469i 3.34695 − 5.79708i 0.562504 + 0.150722i 5.38253 + 5.38253i 3.02729 − 1.74780i −3.95034 + 4.34580i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
43.d odd 6 1 inner
215.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 215.2.l.c ✓ 72
5.c odd 4 1 inner 215.2.l.c ✓ 72
43.d odd 6 1 inner 215.2.l.c ✓ 72
215.l even 12 1 inner 215.2.l.c ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
215.2.l.c ✓ 72 1.a even 1 1 trivial
215.2.l.c ✓ 72 5.c odd 4 1 inner
215.2.l.c ✓ 72 43.d odd 6 1 inner
215.2.l.c ✓ 72 215.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} + 276 T_{2}^{68} + 33068 T_{2}^{64} + 2261044 T_{2}^{60} + 97639674 T_{2}^{56} + 2785724158 T_{2}^{52} + \cdots + 28561 \) acting on \(S_{2}^{\mathrm{new}}(215, [\chi])\). Copy content Toggle raw display