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Properties

Label 350.2.x.a
Level $350$
Weight $2$
Character orbit 350.x
Analytic conductor $2.795$
Analytic rank $0$
Dimension $320$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,2,Mod(3,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([21, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.x (of order \(60\), degree \(16\), 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: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(20\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 320 q + 12 q^{5} - 8 q^{7} + 12 q^{10} - 16 q^{15} - 40 q^{16} + 36 q^{17} + 8 q^{18} - 72 q^{22} + 44 q^{23} - 12 q^{25} - 24 q^{28} - 80 q^{29} + 20 q^{30} - 48 q^{33} - 28 q^{35} + 80 q^{36} - 4 q^{37}+ \cdots - 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.544639 + 0.838671i −1.15652 + 3.01283i −0.406737 − 0.913545i −1.77963 − 1.35386i −1.89689 − 2.61084i −0.00275235 + 2.64575i 0.987688 + 0.156434i −5.51019 − 4.96140i 2.10469 − 0.755156i
3.2 −0.544639 + 0.838671i −0.817783 + 2.13040i −0.406737 − 0.913545i 2.21313 − 0.319493i −1.34131 − 1.84615i 2.63815 − 0.200418i 0.987688 + 0.156434i −1.64039 − 1.47701i −0.937405 + 2.03009i
3.3 −0.544639 + 0.838671i −0.476017 + 1.24007i −0.406737 − 0.913545i 1.14772 + 1.91905i −0.780750 − 1.07461i −2.35038 + 1.21478i 0.987688 + 0.156434i 0.918261 + 0.826806i −2.23454 − 0.0826285i
3.4 −0.544639 + 0.838671i −0.316438 + 0.824348i −0.406737 − 0.913545i 0.514287 − 2.17612i −0.519012 − 0.714359i −2.61096 − 0.427657i 0.987688 + 0.156434i 1.65002 + 1.48568i 1.54495 + 1.61652i
3.5 −0.544639 + 0.838671i −0.255867 + 0.666556i −0.406737 − 0.913545i −2.11039 − 0.739089i −0.419666 − 0.577620i 2.27088 − 1.35761i 0.987688 + 0.156434i 1.85061 + 1.66629i 1.76925 − 1.36739i
3.6 −0.544639 + 0.838671i 0.172615 − 0.449679i −0.406737 − 0.913545i −2.02394 + 0.950624i 0.283119 + 0.389680i −1.46618 − 2.20234i 0.987688 + 0.156434i 2.05702 + 1.85215i 0.305054 − 2.21516i
3.7 −0.544639 + 0.838671i 0.375167 − 0.977343i −0.406737 − 0.913545i 1.91741 + 1.15045i 0.615338 + 0.846940i 2.64280 + 0.124862i 0.987688 + 0.156434i 1.41499 + 1.27406i −2.00915 + 0.981495i
3.8 −0.544639 + 0.838671i 0.637721 − 1.66132i −0.406737 − 0.913545i −1.80729 + 1.31671i 1.04597 + 1.43966i −0.155100 + 2.64120i 0.987688 + 0.156434i −0.123863 − 0.111527i −0.119964 − 2.23285i
3.9 −0.544639 + 0.838671i 0.828696 − 2.15883i −0.406737 − 0.913545i 1.91840 − 1.14880i 1.35920 + 1.87078i −1.79641 − 1.94240i 0.987688 + 0.156434i −1.74436 − 1.57063i −0.0813718 + 2.23459i
3.10 −0.544639 + 0.838671i 1.00842 − 2.62703i −0.406737 − 0.913545i −0.333527 − 2.21105i 1.65399 + 2.27652i 2.63222 + 0.267259i 0.987688 + 0.156434i −3.65494 − 3.29092i 2.03600 + 0.924507i
3.11 0.544639 − 0.838671i −1.20499 + 3.13910i −0.406737 − 0.913545i 1.05534 + 1.97136i 1.97639 + 2.72026i 2.18246 + 1.49561i −0.987688 − 0.156434i −6.17252 − 5.55776i 2.22810 + 0.188596i
3.12 0.544639 − 0.838671i −1.02776 + 2.67740i −0.406737 − 0.913545i −1.46986 − 1.68509i 1.68570 + 2.32016i 0.0482142 − 2.64531i −0.987688 − 0.156434i −3.88274 − 3.49603i −2.21377 + 0.314962i
3.13 0.544639 − 0.838671i −0.439528 + 1.14501i −0.406737 − 0.913545i −2.19147 + 0.444347i 0.720902 + 0.992236i −2.02541 + 1.70227i −0.987688 − 0.156434i 1.11157 + 1.00086i −0.820901 + 2.07993i
3.14 0.544639 − 0.838671i −0.327039 + 0.851967i −0.406737 − 0.913545i −1.80326 + 1.32222i 0.536401 + 0.738293i 2.28229 − 1.33834i −0.987688 − 0.156434i 1.61054 + 1.45014i 0.126777 + 2.23247i
3.15 0.544639 − 0.838671i −0.110635 + 0.288213i −0.406737 − 0.913545i 1.72772 − 1.41950i 0.181460 + 0.249758i 2.11003 + 1.59618i −0.987688 − 0.156434i 2.15861 + 1.94362i −0.249509 − 2.22210i
3.16 0.544639 − 0.838671i −0.0130481 + 0.0339916i −0.406737 − 0.913545i 0.859199 + 2.06441i 0.0214012 + 0.0294562i −1.30429 + 2.30192i −0.987688 − 0.156434i 2.22845 + 2.00650i 2.19931 + 0.403772i
3.17 0.544639 − 0.838671i 0.309740 − 0.806901i −0.406737 − 0.913545i −0.415279 − 2.19717i −0.508027 − 0.699240i −1.18335 − 2.36636i −0.987688 − 0.156434i 1.67428 + 1.50753i −2.06888 − 0.848381i
3.18 0.544639 − 0.838671i 0.822850 − 2.14360i −0.406737 − 0.913545i −2.04873 − 0.895939i −1.34962 − 1.85759i 2.63801 + 0.202309i −0.987688 − 0.156434i −1.68849 − 1.52033i −1.86722 + 1.23025i
3.19 0.544639 − 0.838671i 0.894901 − 2.33130i −0.406737 − 0.913545i 0.406439 + 2.19882i −1.46779 − 2.02024i −1.33531 − 2.28406i −0.987688 − 0.156434i −2.40466 − 2.16517i 2.06545 + 0.856694i
3.20 0.544639 − 0.838671i 1.09550 − 2.85388i −0.406737 − 0.913545i 1.64939 − 1.50980i −1.79681 − 2.47310i −1.61036 + 2.09922i −0.987688 − 0.156434i −4.71508 − 4.24548i −0.367903 − 2.20559i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
25.f odd 20 1 inner
175.x even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.x.a ✓ 320
7.d odd 6 1 inner 350.2.x.a ✓ 320
25.f odd 20 1 inner 350.2.x.a ✓ 320
175.x even 60 1 inner 350.2.x.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.x.a ✓ 320 1.a even 1 1 trivial
350.2.x.a ✓ 320 7.d odd 6 1 inner
350.2.x.a ✓ 320 25.f odd 20 1 inner
350.2.x.a ✓ 320 175.x even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(350, [\chi])\).