Properties

Label 975.2.n.q
Level $975$
Weight $2$
Character orbit 975.n
Analytic conductor $7.785$
Analytic rank $0$
Dimension $40$
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] = [975,2,Mod(749,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.n (of order \(4\), degree \(2\), not 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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{6} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{6} - 16 q^{7} - 24 q^{12} + 24 q^{13} - 64 q^{16} + 4 q^{18} + 16 q^{19} - 12 q^{21} - 8 q^{24} + 32 q^{28} + 32 q^{31} + 4 q^{33} + 16 q^{34} + 32 q^{37} + 8 q^{39} - 32 q^{43} - 40 q^{46} - 8 q^{52} + 32 q^{54} + 36 q^{57} + 24 q^{58} + 8 q^{61} - 8 q^{63} - 48 q^{66} + 32 q^{67} + 132 q^{72} + 64 q^{73} + 16 q^{76} - 108 q^{78} - 40 q^{79} + 72 q^{81} - 128 q^{82} - 124 q^{84} - 80 q^{88} + 8 q^{91} - 108 q^{93} + 32 q^{94} - 76 q^{96} - 24 q^{97} + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1 −1.90946 1.90946i −1.66974 0.460384i 5.29209i 0 2.30923 + 4.06740i −1.58484 1.58484i 6.28612 6.28612i 2.57609 + 1.53745i 0
749.2 −1.89167 1.89167i 0.588468 + 1.62902i 5.15683i 0 1.96838 4.19475i 0.157581 + 0.157581i 5.97167 5.97167i −2.30741 + 1.91725i 0
749.3 −1.57927 1.57927i 0.0822389 1.73010i 2.98819i 0 −2.86217 + 2.60241i −2.29263 2.29263i 1.56062 1.56062i −2.98647 0.284563i 0
749.4 −1.43871 1.43871i 1.72478 + 0.158548i 2.13978i 0 −2.25335 2.70956i −1.21328 1.21328i 0.201099 0.201099i 2.94973 + 0.546921i 0
749.5 −1.35176 1.35176i −0.875587 + 1.49444i 1.65452i 0 3.20371 0.836541i 3.04341 + 3.04341i −0.467005 + 0.467005i −1.46670 2.61702i 0
749.6 −1.11881 1.11881i −1.67111 0.455401i 0.503463i 0 1.36015 + 2.37916i 1.46633 + 1.46633i −1.67434 + 1.67434i 2.58522 + 1.52205i 0
749.7 −0.789301 0.789301i −0.548503 + 1.64291i 0.754007i 0 1.72968 0.863815i −1.97746 1.97746i −2.17374 + 2.17374i −2.39829 1.80228i 0
749.8 −0.483812 0.483812i 0.793397 1.53965i 1.53185i 0 −1.12876 + 0.361045i −1.24531 1.24531i −1.70875 + 1.70875i −1.74104 2.44311i 0
749.9 −0.455718 0.455718i −1.67352 + 0.446465i 1.58464i 0 0.966114 + 0.559191i −2.89711 2.89711i −1.63358 + 1.63358i 2.60134 1.49434i 0
749.10 −0.260415 0.260415i 1.26244 1.18585i 1.86437i 0 −0.637571 0.0199472i 2.54331 + 2.54331i −1.00634 + 1.00634i 0.187534 2.99413i 0
749.11 0.260415 + 0.260415i −1.26244 1.18585i 1.86437i 0 −0.0199472 0.637571i 2.54331 + 2.54331i 1.00634 1.00634i 0.187534 + 2.99413i 0
749.12 0.455718 + 0.455718i 1.67352 + 0.446465i 1.58464i 0 0.559191 + 0.966114i −2.89711 2.89711i 1.63358 1.63358i 2.60134 + 1.49434i 0
749.13 0.483812 + 0.483812i −0.793397 1.53965i 1.53185i 0 0.361045 1.12876i −1.24531 1.24531i 1.70875 1.70875i −1.74104 + 2.44311i 0
749.14 0.789301 + 0.789301i 0.548503 + 1.64291i 0.754007i 0 −0.863815 + 1.72968i −1.97746 1.97746i 2.17374 2.17374i −2.39829 + 1.80228i 0
749.15 1.11881 + 1.11881i 1.67111 0.455401i 0.503463i 0 2.37916 + 1.36015i 1.46633 + 1.46633i 1.67434 1.67434i 2.58522 1.52205i 0
749.16 1.35176 + 1.35176i 0.875587 + 1.49444i 1.65452i 0 −0.836541 + 3.20371i 3.04341 + 3.04341i 0.467005 0.467005i −1.46670 + 2.61702i 0
749.17 1.43871 + 1.43871i −1.72478 + 0.158548i 2.13978i 0 −2.70956 2.25335i −1.21328 1.21328i −0.201099 + 0.201099i 2.94973 0.546921i 0
749.18 1.57927 + 1.57927i −0.0822389 1.73010i 2.98819i 0 2.60241 2.86217i −2.29263 2.29263i −1.56062 + 1.56062i −2.98647 + 0.284563i 0
749.19 1.89167 + 1.89167i −0.588468 + 1.62902i 5.15683i 0 −4.19475 + 1.96838i 0.157581 + 0.157581i −5.97167 + 5.97167i −2.30741 1.91725i 0
749.20 1.90946 + 1.90946i 1.66974 0.460384i 5.29209i 0 4.06740 + 2.30923i −1.58484 1.58484i −6.28612 + 6.28612i 2.57609 1.53745i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
65.g odd 4 1 inner
195.n even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.n.q 40
3.b odd 2 1 inner 975.2.n.q 40
5.b even 2 1 975.2.n.r 40
5.c odd 4 1 195.2.o.a 40
5.c odd 4 1 975.2.o.p 40
13.d odd 4 1 975.2.n.r 40
15.d odd 2 1 975.2.n.r 40
15.e even 4 1 195.2.o.a 40
15.e even 4 1 975.2.o.p 40
39.f even 4 1 975.2.n.r 40
65.f even 4 1 195.2.o.a 40
65.g odd 4 1 inner 975.2.n.q 40
65.k even 4 1 975.2.o.p 40
195.j odd 4 1 975.2.o.p 40
195.n even 4 1 inner 975.2.n.q 40
195.u odd 4 1 195.2.o.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.o.a 40 5.c odd 4 1
195.2.o.a 40 15.e even 4 1
195.2.o.a 40 65.f even 4 1
195.2.o.a 40 195.u odd 4 1
975.2.n.q 40 1.a even 1 1 trivial
975.2.n.q 40 3.b odd 2 1 inner
975.2.n.q 40 65.g odd 4 1 inner
975.2.n.q 40 195.n even 4 1 inner
975.2.n.r 40 5.b even 2 1
975.2.n.r 40 13.d odd 4 1
975.2.n.r 40 15.d odd 2 1
975.2.n.r 40 39.f even 4 1
975.2.o.p 40 5.c odd 4 1
975.2.o.p 40 15.e even 4 1
975.2.o.p 40 65.k even 4 1
975.2.o.p 40 195.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{40} + 168 T_{2}^{36} + 10820 T_{2}^{32} + 339816 T_{2}^{28} + 5544982 T_{2}^{24} + 45953376 T_{2}^{20} + \cdots + 104976 \) Copy content Toggle raw display
\( T_{7}^{20} + 8 T_{7}^{19} + 32 T_{7}^{18} + 80 T_{7}^{17} + 641 T_{7}^{16} + 4304 T_{7}^{15} + \cdots + 3240000 \) Copy content Toggle raw display
\( T_{11}^{40} + 2442 T_{11}^{36} + 2101553 T_{11}^{32} + 730668016 T_{11}^{28} + 82164365552 T_{11}^{24} + \cdots + 84934656 \) Copy content Toggle raw display
\( T_{37}^{20} - 16 T_{37}^{19} + 128 T_{37}^{18} - 208 T_{37}^{17} + 15825 T_{37}^{16} + \cdots + 101455659150400 \) Copy content Toggle raw display