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Properties

Label 420.2.br.a
Level $420$
Weight $2$
Character orbit 420.br
Analytic conductor $3.354$
Analytic rank $0$
Dimension $352$
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] = [420,2,Mod(47,420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(420, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 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("420.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.br (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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(352\)
Relative dimension: \(88\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 352 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 352 q - 12 q^{10} - 6 q^{12} - 8 q^{16} + 6 q^{18} - 8 q^{21} - 8 q^{25} - 24 q^{28} - 32 q^{30} - 12 q^{33} - 48 q^{36} - 8 q^{37} - 72 q^{40} - 26 q^{42} - 12 q^{45} - 8 q^{46} - 60 q^{52} - 72 q^{57} - 52 q^{58} + 10 q^{60} - 48 q^{61} + 60 q^{66} - 104 q^{70} - 38 q^{72} - 24 q^{73} - 56 q^{78} - 48 q^{81} - 96 q^{82} - 32 q^{85} - 12 q^{88} + 20 q^{93} + 72 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −1.41399 − 0.0249681i −1.69730 + 0.345215i 1.99875 + 0.0706094i −2.18051 + 0.495354i 2.40859 − 0.445753i 0.721133 + 2.54558i −2.82446 − 0.149746i 2.76165 − 1.17187i 3.09559 − 0.645984i
47.2 −1.41282 − 0.0626810i 1.44560 − 0.954069i 1.99214 + 0.177114i 1.69994 + 1.45266i −2.10218 + 1.25732i 1.41372 − 2.23638i −2.80344 − 0.375101i 1.17951 − 2.75840i −2.31066 − 2.15890i
47.3 −1.40811 + 0.131214i −0.600368 + 1.62467i 1.96557 − 0.369528i 0.132022 + 2.23217i 0.632207 − 2.36650i −2.64287 − 0.123399i −2.71925 + 0.778246i −2.27912 − 1.95080i −0.478793 − 3.12582i
47.4 −1.40266 − 0.180377i −1.66665 − 0.471450i 1.93493 + 0.506016i 0.242551 − 2.22287i 2.25272 + 0.961911i 0.722305 − 2.54525i −2.62278 − 1.05879i 2.55547 + 1.57149i −0.741173 + 3.07419i
47.5 −1.39828 + 0.211707i −0.525221 − 1.65050i 1.91036 − 0.592049i −1.22857 + 1.86832i 1.08383 + 2.19666i −1.21490 − 2.35032i −2.54587 + 1.23228i −2.44829 + 1.73375i 1.32235 − 2.87252i
47.6 −1.39382 + 0.239277i 1.70593 − 0.299686i 1.88549 − 0.667020i −1.97012 − 1.05766i −2.30606 + 0.825899i −2.50417 − 0.853900i −2.46844 + 1.38086i 2.82038 − 1.02249i 2.99907 + 1.00279i
47.7 −1.37971 − 0.310470i 0.650798 + 1.60514i 1.80722 + 0.856719i −0.0650945 − 2.23512i −0.399568 − 2.41668i −2.59900 + 0.495153i −2.22746 − 1.74311i −2.15292 + 2.08924i −0.604126 + 3.10403i
47.8 −1.36891 − 0.355066i 0.909705 + 1.47392i 1.74786 + 0.972111i 1.53861 + 1.62255i −0.721970 − 2.34068i 1.81998 + 1.92033i −2.04750 − 1.95134i −1.34487 + 2.68166i −1.53012 − 2.76744i
47.9 −1.35775 − 0.395629i −0.121522 − 1.72778i 1.68696 + 1.07433i 2.07531 − 0.832520i −0.518564 + 2.39397i −2.37213 + 1.17174i −1.86542 − 2.12607i −2.97046 + 0.419928i −3.14712 + 0.309300i
47.10 −1.32673 + 0.489692i −1.32753 − 1.11250i 1.52040 − 1.29937i 1.97012 + 1.05766i 2.30606 + 0.825899i 2.50417 + 0.853900i −1.38086 + 2.46844i 0.524689 + 2.95376i −3.13173 − 0.438475i
47.11 −1.31680 + 0.515796i 1.28010 − 1.16676i 1.46791 − 1.35840i 1.22857 − 1.86832i −1.08383 + 2.19666i 1.21490 + 2.35032i −1.23228 + 2.54587i 0.277329 − 2.98715i −0.654111 + 3.09389i
47.12 −1.28507 + 0.590422i −0.292402 + 1.70719i 1.30280 − 1.51747i −0.132022 − 2.23217i −0.632207 − 2.36650i 2.64287 + 0.123399i −0.778246 + 2.71925i −2.82900 − 0.998371i 1.48758 + 2.79054i
47.13 −1.26441 − 0.633457i 1.25380 − 1.19498i 1.19746 + 1.60190i −1.21060 + 1.88001i −2.34229 + 0.716715i −0.583432 + 2.58062i −0.499352 − 2.78400i 0.144042 − 2.99654i 2.72160 − 1.61025i
47.14 −1.21207 + 0.728620i 1.29730 + 1.14761i 0.938227 − 1.76628i 2.18051 − 0.495354i −2.40859 − 0.445753i −0.721133 − 2.54558i 0.149746 + 2.82446i 0.365961 + 2.97760i −2.28201 + 2.18917i
47.15 −1.21161 − 0.729377i −0.696637 + 1.58578i 0.936019 + 1.76745i −2.23596 − 0.0223025i 2.00069 − 1.41324i 1.09488 − 2.40857i 0.155041 − 2.82417i −2.02939 − 2.20943i 2.69285 + 1.65788i
47.16 −1.19220 + 0.760695i −0.774890 − 1.54905i 0.842686 − 1.81380i −1.69994 − 1.45266i 2.10218 + 1.25732i −1.41372 + 2.23638i 0.375101 + 2.80344i −1.79909 + 2.40068i 3.13170 + 0.438726i
47.17 −1.16222 − 0.805756i 1.72906 + 0.101774i 0.701514 + 1.87293i 0.944430 − 2.02683i −1.92754 − 1.51148i 2.06344 − 1.65596i 0.693813 − 2.74201i 2.97928 + 0.351948i −2.73077 + 1.59465i
47.18 −1.14166 − 0.834632i −1.73150 + 0.0438386i 0.606780 + 1.90573i 1.29039 + 1.82617i 2.01337 + 1.39511i −2.09581 − 1.61480i 0.897848 − 2.68214i 2.99616 − 0.151813i 0.0509867 − 3.16187i
47.19 −1.12678 − 0.854615i −1.28409 + 1.16237i 0.539267 + 1.92593i 2.15438 − 0.598853i 2.44027 − 0.212332i 1.67737 + 2.04607i 1.03829 − 2.63096i 0.297787 − 2.98518i −2.93931 − 1.16639i
47.20 −1.12455 + 0.857543i 1.67909 + 0.425039i 0.529241 − 1.92871i −0.242551 + 2.22287i −2.25272 + 0.961911i −0.722305 + 2.54525i 1.05879 + 2.62278i 2.63868 + 1.42736i −1.63345 − 2.70774i
See next 80 embeddings (of 352 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.c odd 4 1 inner
7.d odd 6 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
21.g even 6 1 inner
28.f even 6 1 inner
35.k even 12 1 inner
60.l odd 4 1 inner
84.j odd 6 1 inner
105.w odd 12 1 inner
140.x odd 12 1 inner
420.br even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.br.a ✓ 352
3.b odd 2 1 inner 420.2.br.a ✓ 352
4.b odd 2 1 inner 420.2.br.a ✓ 352
5.c odd 4 1 inner 420.2.br.a ✓ 352
7.d odd 6 1 inner 420.2.br.a ✓ 352
12.b even 2 1 inner 420.2.br.a ✓ 352
15.e even 4 1 inner 420.2.br.a ✓ 352
20.e even 4 1 inner 420.2.br.a ✓ 352
21.g even 6 1 inner 420.2.br.a ✓ 352
28.f even 6 1 inner 420.2.br.a ✓ 352
35.k even 12 1 inner 420.2.br.a ✓ 352
60.l odd 4 1 inner 420.2.br.a ✓ 352
84.j odd 6 1 inner 420.2.br.a ✓ 352
105.w odd 12 1 inner 420.2.br.a ✓ 352
140.x odd 12 1 inner 420.2.br.a ✓ 352
420.br even 12 1 inner 420.2.br.a ✓ 352
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.br.a ✓ 352 1.a even 1 1 trivial
420.2.br.a ✓ 352 3.b odd 2 1 inner
420.2.br.a ✓ 352 4.b odd 2 1 inner
420.2.br.a ✓ 352 5.c odd 4 1 inner
420.2.br.a ✓ 352 7.d odd 6 1 inner
420.2.br.a ✓ 352 12.b even 2 1 inner
420.2.br.a ✓ 352 15.e even 4 1 inner
420.2.br.a ✓ 352 20.e even 4 1 inner
420.2.br.a ✓ 352 21.g even 6 1 inner
420.2.br.a ✓ 352 28.f even 6 1 inner
420.2.br.a ✓ 352 35.k even 12 1 inner
420.2.br.a ✓ 352 60.l odd 4 1 inner
420.2.br.a ✓ 352 84.j odd 6 1 inner
420.2.br.a ✓ 352 105.w odd 12 1 inner
420.2.br.a ✓ 352 140.x odd 12 1 inner
420.2.br.a ✓ 352 420.br even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(420, [\chi])\).