Properties

Label 950.2.bb.e
Level $950$
Weight $2$
Character orbit 950.bb
Analytic conductor $7.586$
Analytic rank $0$
Dimension $120$
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(143,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([27, 34]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.143");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.bb (of order \(36\), degree \(12\), 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: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 12 q^{7} - 36 q^{17} - 96 q^{21} - 24 q^{22} - 12 q^{26} + 96 q^{33} - 12 q^{41} + 72 q^{43} + 24 q^{47} + 24 q^{51} - 36 q^{53} - 84 q^{57} + 48 q^{61} + 24 q^{62} - 36 q^{63} - 24 q^{66} + 96 q^{67} + 12 q^{68} + 36 q^{73} + 12 q^{76} - 96 q^{78} + 144 q^{81} - 48 q^{82} - 24 q^{83} + 48 q^{86} - 72 q^{87} + 72 q^{91} - 72 q^{92} - 156 q^{93} - 120 q^{97} - 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1 −0.573576 + 0.819152i −0.243772 + 2.78632i −0.342020 0.939693i 0 −2.14260 1.79785i 0.966713 + 3.60782i 0.965926 + 0.258819i −4.74974 0.837508i 0
143.2 −0.573576 + 0.819152i −0.197676 + 2.25945i −0.342020 0.939693i 0 −1.73745 1.45789i −0.925273 3.45316i 0.965926 + 0.258819i −2.11161 0.372333i 0
143.3 −0.573576 + 0.819152i 0.0801625 0.916262i −0.342020 0.939693i 0 0.704578 + 0.591211i −0.322803 1.20472i 0.965926 + 0.258819i 2.12131 + 0.374045i 0
143.4 −0.573576 + 0.819152i 0.0833140 0.952283i −0.342020 0.939693i 0 0.732278 + 0.614454i 0.790865 + 2.95155i 0.965926 + 0.258819i 2.05452 + 0.362268i 0
143.5 −0.573576 + 0.819152i 0.277971 3.17723i −0.342020 0.939693i 0 2.44319 + 2.05008i −1.05257 3.92825i 0.965926 + 0.258819i −7.06307 1.24541i 0
143.6 0.573576 0.819152i −0.255151 + 2.91639i −0.342020 0.939693i 0 2.24262 + 1.88178i −0.0607618 0.226766i −0.965926 0.258819i −5.48583 0.967300i 0
143.7 0.573576 0.819152i −0.0418747 + 0.478630i −0.342020 0.939693i 0 0.368052 + 0.308832i 0.971585 + 3.62600i −0.965926 0.258819i 2.72709 + 0.480860i 0
143.8 0.573576 0.819152i 0.0201525 0.230344i −0.342020 0.939693i 0 −0.177128 0.148628i −1.23887 4.62353i −0.965926 0.258819i 2.90177 + 0.511661i 0
143.9 0.573576 0.819152i 0.0709170 0.810585i −0.342020 0.939693i 0 −0.623316 0.523024i 0.211848 + 0.790626i −0.965926 0.258819i 2.30240 + 0.405976i 0
143.10 0.573576 0.819152i 0.205957 2.35409i −0.342020 0.939693i 0 −1.81023 1.51896i −0.0727825 0.271628i −0.965926 0.258819i −2.54492 0.448738i 0
193.1 −0.0871557 0.996195i −1.35298 + 2.90149i −0.984808 + 0.173648i 0 3.00836 + 1.09496i 1.43361 + 0.384135i 0.258819 + 0.965926i −4.65969 5.55320i 0
193.2 −0.0871557 0.996195i −0.507694 + 1.08875i −0.984808 + 0.173648i 0 1.12886 + 0.410871i −2.08332 0.558224i 0.258819 + 0.965926i 1.00073 + 1.19263i 0
193.3 −0.0871557 0.996195i −0.186197 + 0.399301i −0.984808 + 0.173648i 0 0.414010 + 0.150687i −1.97694 0.529719i 0.258819 + 0.965926i 1.80359 + 2.14944i 0
193.4 −0.0871557 0.996195i 0.684861 1.46869i −0.984808 + 0.173648i 0 −1.52279 0.554250i 3.13506 + 0.840038i 0.258819 + 0.965926i 0.240351 + 0.286439i 0
193.5 −0.0871557 0.996195i 1.36202 2.92085i −0.984808 + 0.173648i 0 −3.02844 1.10226i −1.04490 0.279979i 0.258819 + 0.965926i −4.74793 5.65836i 0
193.6 0.0871557 + 0.996195i −1.08832 + 2.33390i −0.984808 + 0.173648i 0 −2.41987 0.880761i 3.72976 + 0.999386i −0.258819 0.965926i −2.33430 2.78191i 0
193.7 0.0871557 + 0.996195i −0.602383 + 1.29181i −0.984808 + 0.173648i 0 −1.33940 0.487501i 1.39748 + 0.374454i −0.258819 0.965926i 0.622444 + 0.741800i 0
193.8 0.0871557 + 0.996195i 0.202610 0.434499i −0.984808 + 0.173648i 0 0.450504 + 0.163970i −3.80940 1.02072i −0.258819 0.965926i 1.78062 + 2.12207i 0
193.9 0.0871557 + 0.996195i 0.715886 1.53522i −0.984808 + 0.173648i 0 1.59177 + 0.579359i 4.29599 + 1.15111i −0.258819 0.965926i 0.0839459 + 0.100043i 0
193.10 0.0871557 + 0.996195i 0.772202 1.65599i −0.984808 + 0.173648i 0 1.71699 + 0.624934i −2.34531 0.628423i −0.258819 0.965926i −0.217652 0.259388i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.f odd 18 1 inner
95.r even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.bb.e 120
5.b even 2 1 190.2.r.a 120
5.c odd 4 1 190.2.r.a 120
5.c odd 4 1 inner 950.2.bb.e 120
19.f odd 18 1 inner 950.2.bb.e 120
95.o odd 18 1 190.2.r.a 120
95.r even 36 1 190.2.r.a 120
95.r even 36 1 inner 950.2.bb.e 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.r.a 120 5.b even 2 1
190.2.r.a 120 5.c odd 4 1
190.2.r.a 120 95.o odd 18 1
190.2.r.a 120 95.r even 36 1
950.2.bb.e 120 1.a even 1 1 trivial
950.2.bb.e 120 5.c odd 4 1 inner
950.2.bb.e 120 19.f odd 18 1 inner
950.2.bb.e 120 95.r even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{120} - 36 T_{3}^{116} + 120 T_{3}^{115} - 192 T_{3}^{113} + 2682 T_{3}^{112} - 4320 T_{3}^{111} + 7200 T_{3}^{110} + 8988 T_{3}^{109} - 1056716 T_{3}^{108} - 337368 T_{3}^{107} - 240768 T_{3}^{106} + \cdots + 110075314176 \) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\). Copy content Toggle raw display