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Properties

Label 975.2.o.p
Level $975$
Weight $2$
Character orbit 975.o
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(476,975)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.476");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.o (of order \(4\), degree \(2\), 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
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{6} + 16 q^{7} - 64 q^{16} - 4 q^{18} - 16 q^{19} - 12 q^{21} + 8 q^{24} + 24 q^{27} - 32 q^{28} + 32 q^{31} + 4 q^{33} - 16 q^{34} - 32 q^{37} - 8 q^{39} - 8 q^{42} - 40 q^{46} - 8 q^{48} - 32 q^{54} + 36 q^{57} - 24 q^{58} + 8 q^{61} - 8 q^{63} - 48 q^{66} + 32 q^{67} - 132 q^{72} + 64 q^{73} + 16 q^{76} + 12 q^{78} + 40 q^{79} + 72 q^{81} + 124 q^{84} - 16 q^{87} + 8 q^{91} + 108 q^{93} - 32 q^{94} - 76 q^{96} - 24 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
476.1 −1.90946 + 1.90946i −0.460384 − 1.66974i − 5.29209i 0 4.06740 + 2.30923i 1.58484 − 1.58484i 6.28612 + 6.28612i −2.57609 + 1.53745i 0
476.2 −1.89167 + 1.89167i 1.62902 + 0.588468i − 5.15683i 0 −4.19475 + 1.96838i −0.157581 + 0.157581i 5.97167 + 5.97167i 2.30741 + 1.91725i 0
476.3 −1.57927 + 1.57927i −1.73010 + 0.0822389i − 2.98819i 0 2.60241 − 2.86217i 2.29263 − 2.29263i 1.56062 + 1.56062i 2.98647 − 0.284563i 0
476.4 −1.43871 + 1.43871i 0.158548 + 1.72478i − 2.13978i 0 −2.70956 − 2.25335i 1.21328 − 1.21328i 0.201099 + 0.201099i −2.94973 + 0.546921i 0
476.5 −1.35176 + 1.35176i 1.49444 − 0.875587i − 1.65452i 0 −0.836541 + 3.20371i −3.04341 + 3.04341i −0.467005 − 0.467005i 1.46670 − 2.61702i 0
476.6 −1.11881 + 1.11881i −0.455401 − 1.67111i − 0.503463i 0 2.37916 + 1.36015i −1.46633 + 1.46633i −1.67434 − 1.67434i −2.58522 + 1.52205i 0
476.7 −0.789301 + 0.789301i 1.64291 − 0.548503i 0.754007i 0 −0.863815 + 1.72968i 1.97746 − 1.97746i −2.17374 − 2.17374i 2.39829 − 1.80228i 0
476.8 −0.483812 + 0.483812i −1.53965 + 0.793397i 1.53185i 0 0.361045 − 1.12876i 1.24531 − 1.24531i −1.70875 − 1.70875i 1.74104 − 2.44311i 0
476.9 −0.455718 + 0.455718i 0.446465 − 1.67352i 1.58464i 0 0.559191 + 0.966114i 2.89711 − 2.89711i −1.63358 − 1.63358i −2.60134 − 1.49434i 0
476.10 −0.260415 + 0.260415i −1.18585 + 1.26244i 1.86437i 0 −0.0199472 − 0.637571i −2.54331 + 2.54331i −1.00634 − 1.00634i −0.187534 − 2.99413i 0
476.11 0.260415 − 0.260415i −1.18585 − 1.26244i 1.86437i 0 −0.637571 − 0.0199472i −2.54331 + 2.54331i 1.00634 + 1.00634i −0.187534 + 2.99413i 0
476.12 0.455718 − 0.455718i 0.446465 + 1.67352i 1.58464i 0 0.966114 + 0.559191i 2.89711 − 2.89711i 1.63358 + 1.63358i −2.60134 + 1.49434i 0
476.13 0.483812 − 0.483812i −1.53965 − 0.793397i 1.53185i 0 −1.12876 + 0.361045i 1.24531 − 1.24531i 1.70875 + 1.70875i 1.74104 + 2.44311i 0
476.14 0.789301 − 0.789301i 1.64291 + 0.548503i 0.754007i 0 1.72968 − 0.863815i 1.97746 − 1.97746i 2.17374 + 2.17374i 2.39829 + 1.80228i 0
476.15 1.11881 − 1.11881i −0.455401 + 1.67111i − 0.503463i 0 1.36015 + 2.37916i −1.46633 + 1.46633i 1.67434 + 1.67434i −2.58522 − 1.52205i 0
476.16 1.35176 − 1.35176i 1.49444 + 0.875587i − 1.65452i 0 3.20371 − 0.836541i −3.04341 + 3.04341i 0.467005 + 0.467005i 1.46670 + 2.61702i 0
476.17 1.43871 − 1.43871i 0.158548 − 1.72478i − 2.13978i 0 −2.25335 − 2.70956i 1.21328 − 1.21328i −0.201099 − 0.201099i −2.94973 − 0.546921i 0
476.18 1.57927 − 1.57927i −1.73010 − 0.0822389i − 2.98819i 0 −2.86217 + 2.60241i 2.29263 − 2.29263i −1.56062 − 1.56062i 2.98647 + 0.284563i 0
476.19 1.89167 − 1.89167i 1.62902 − 0.588468i − 5.15683i 0 1.96838 − 4.19475i −0.157581 + 0.157581i −5.97167 − 5.97167i 2.30741 − 1.91725i 0
476.20 1.90946 − 1.90946i −0.460384 + 1.66974i − 5.29209i 0 2.30923 + 4.06740i 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 476.20
Significant digits:
Format:

Inner twists

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

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