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Properties

Label 950.2.bd.a
Level $950$
Weight $2$
Character orbit 950.bd
Analytic conductor $7.586$
Analytic rank $0$
Dimension $800$
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] = [950,2,Mod(27,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(60))
 
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("950.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.bd (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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(800\)
Relative dimension: \(50\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 800 q - 4 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 800 q - 4 q^{5} - 8 q^{7} - 36 q^{15} - 100 q^{16} - 4 q^{17} + 40 q^{19} - 96 q^{22} - 44 q^{23} - 4 q^{25} + 4 q^{28} + 120 q^{29} - 64 q^{30} + 132 q^{33} + 28 q^{35} + 100 q^{36} - 4 q^{38} - 80 q^{39} - 32 q^{42} - 36 q^{43} + 216 q^{45} + 76 q^{47} + 36 q^{53} + 88 q^{55} - 124 q^{57} + 120 q^{59} + 48 q^{60} - 8 q^{62} - 72 q^{63} - 24 q^{67} + 88 q^{68} + 24 q^{70} - 24 q^{73} - 256 q^{77} + 24 q^{78} + 4 q^{80} - 100 q^{81} - 8 q^{82} - 24 q^{83} + 28 q^{85} - 112 q^{87} - 12 q^{90} + 4 q^{92} - 108 q^{93} - 28 q^{95} + 12 q^{97} - 120 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −0.777146 − 0.629320i −0.174328 + 3.32637i 0.207912 + 0.978148i −2.18227 + 0.487544i 2.22883 − 2.47537i 2.37586 + 2.37586i 0.453990 − 0.891007i −8.05081 − 0.846175i 2.00276 + 0.994454i
27.2 −0.777146 − 0.629320i −0.165111 + 3.15050i 0.207912 + 0.978148i 1.29881 − 1.82019i 2.11099 − 2.34449i −1.30050 − 1.30050i 0.453990 − 0.891007i −6.91481 − 0.726775i −2.15485 + 0.597183i
27.3 −0.777146 − 0.629320i −0.146876 + 2.80257i 0.207912 + 0.978148i 0.359551 + 2.20697i 1.87786 − 2.08557i −2.28843 − 2.28843i 0.453990 − 0.891007i −4.84924 − 0.509675i 1.10947 − 1.94141i
27.4 −0.777146 − 0.629320i −0.125909 + 2.40249i 0.207912 + 0.978148i 2.21427 − 0.311457i 1.60978 − 1.78784i 1.52376 + 1.52376i 0.453990 − 0.891007i −2.77252 − 0.291403i −1.91682 − 1.15144i
27.5 −0.777146 − 0.629320i −0.102836 + 1.96222i 0.207912 + 0.978148i −1.42891 − 1.71995i 1.31478 − 1.46022i −2.76852 − 2.76852i 0.453990 − 0.891007i −0.856176 − 0.0899877i 0.0280713 + 2.23589i
27.6 −0.777146 − 0.629320i −0.0968046 + 1.84714i 0.207912 + 0.978148i 1.09978 + 1.94692i 1.23768 − 1.37458i −0.632973 − 0.632973i 0.453990 − 0.891007i −0.418995 − 0.0440382i 0.370549 − 2.20515i
27.7 −0.777146 − 0.629320i −0.0959375 + 1.83060i 0.207912 + 0.978148i −1.73058 + 1.41601i 1.22659 − 1.36227i 1.20076 + 1.20076i 0.453990 − 0.891007i −0.358316 − 0.0376605i 2.23604 − 0.0113576i
27.8 −0.777146 − 0.629320i −0.0775294 + 1.47935i 0.207912 + 0.978148i −1.34338 − 1.78755i 0.991237 − 1.10088i 1.75517 + 1.75517i 0.453990 − 0.891007i 0.801101 + 0.0841991i −0.0809353 + 2.23460i
27.9 −0.777146 − 0.629320i −0.0710372 + 1.35547i 0.207912 + 0.978148i −1.70399 + 1.44791i 0.908231 − 1.00869i −0.514487 − 0.514487i 0.453990 − 0.891007i 1.15131 + 0.121008i 2.23544 − 0.0528825i
27.10 −0.777146 − 0.629320i −0.0494370 + 0.943315i 0.207912 + 0.978148i 1.73843 − 1.40637i 0.632067 − 0.701982i 3.25050 + 3.25050i 0.453990 − 0.891007i 2.09617 + 0.220316i −2.23607 − 0.00107286i
27.11 −0.777146 − 0.629320i −0.0107868 + 0.205824i 0.207912 + 0.978148i 0.365512 − 2.20599i 0.137912 − 0.153167i −1.79103 − 1.79103i 0.453990 − 0.891007i 2.94132 + 0.309145i −1.67233 + 1.48435i
27.12 −0.777146 − 0.629320i 0.00690609 − 0.131776i 0.207912 + 0.978148i −2.11597 − 0.722972i −0.0882964 + 0.0980631i 0.298792 + 0.298792i 0.453990 − 0.891007i 2.96625 + 0.311765i 1.18943 + 1.89348i
27.13 −0.777146 − 0.629320i 0.00693568 − 0.132341i 0.207912 + 0.978148i 1.23454 − 1.86438i −0.0886748 + 0.0984833i −2.47768 − 2.47768i 0.453990 − 0.891007i 2.96610 + 0.311750i −2.13271 + 0.671975i
27.14 −0.777146 − 0.629320i 0.0138662 − 0.264584i 0.207912 + 0.978148i 0.889244 + 2.05164i −0.177284 + 0.196894i 2.41122 + 2.41122i 0.453990 − 0.891007i 2.91375 + 0.306248i 0.600069 − 2.15405i
27.15 −0.777146 − 0.629320i 0.0178542 − 0.340678i 0.207912 + 0.978148i 1.74069 + 1.40356i −0.228271 + 0.253521i −0.826843 − 0.826843i 0.453990 − 0.891007i 2.86782 + 0.301420i −0.469482 − 2.18623i
27.16 −0.777146 − 0.629320i 0.0239777 − 0.457521i 0.207912 + 0.978148i 0.165650 − 2.22992i −0.306561 + 0.340471i 1.81231 + 1.81231i 0.453990 − 0.891007i 2.77482 + 0.291645i −1.53207 + 1.62873i
27.17 −0.777146 − 0.629320i 0.0402384 − 0.767794i 0.207912 + 0.978148i −1.26479 + 1.84399i −0.514459 + 0.571365i −3.24113 − 3.24113i 0.453990 − 0.891007i 2.39568 + 0.251796i 2.14339 − 0.637094i
27.18 −0.777146 − 0.629320i 0.0828442 − 1.58076i 0.207912 + 0.978148i −2.09573 − 0.779702i −1.05919 + 1.17635i 0.596283 + 0.596283i 0.453990 − 0.891007i 0.491621 + 0.0516714i 1.13800 + 1.92483i
27.19 −0.777146 − 0.629320i 0.0945439 − 1.80400i 0.207912 + 0.978148i 2.05987 − 0.870013i −1.20877 + 1.34248i 0.220705 + 0.220705i 0.453990 − 0.891007i −0.261928 − 0.0275297i −2.14834 − 0.620193i
27.20 −0.777146 − 0.629320i 0.108889 − 2.07773i 0.207912 + 0.978148i −1.28740 + 1.82828i −1.39218 + 1.54617i 0.660854 + 0.660854i 0.453990 − 0.891007i −1.32154 − 0.138899i 2.15107 − 0.610647i
See next 80 embeddings (of 800 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner
25.f odd 20 1 inner
475.be even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.bd.a ✓ 800
19.d odd 6 1 inner 950.2.bd.a ✓ 800
25.f odd 20 1 inner 950.2.bd.a ✓ 800
475.be even 60 1 inner 950.2.bd.a ✓ 800
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bd.a ✓ 800 1.a even 1 1 trivial
950.2.bd.a ✓ 800 19.d odd 6 1 inner
950.2.bd.a ✓ 800 25.f odd 20 1 inner
950.2.bd.a ✓ 800 475.be even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(950, [\chi])\).