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Properties

Label 980.2.k.k
Level $980$
Weight $2$
Character orbit 980.k
Analytic conductor $7.825$
Analytic rank $0$
Dimension $36$
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] = [980,2,Mod(687,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.687");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.k (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: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 140)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} + 8 q^{5} - 8 q^{6} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} + 8 q^{5} - 8 q^{6} - 2 q^{8} - 2 q^{10} - 10 q^{12} + 28 q^{16} - 4 q^{17} + 20 q^{18} - 28 q^{20} - 8 q^{22} + 16 q^{25} + 4 q^{26} + 32 q^{30} + 38 q^{32} + 64 q^{33} + 8 q^{36} + 4 q^{37} - 12 q^{38} - 2 q^{40} - 20 q^{41} + 12 q^{45} + 28 q^{46} + 6 q^{48} - 14 q^{50} - 48 q^{52} + 24 q^{53} - 8 q^{57} - 30 q^{58} + 10 q^{60} + 20 q^{61} + 28 q^{62} - 4 q^{65} - 44 q^{66} + 12 q^{68} - 44 q^{72} + 12 q^{73} + 56 q^{76} + 32 q^{78} - 52 q^{80} + 52 q^{81} + 34 q^{82} + 8 q^{85} - 64 q^{86} - 16 q^{88} - 16 q^{90} + 22 q^{92} - 12 q^{93} + 48 q^{96} - 12 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
687.1 −1.38434 − 0.289136i −0.881483 − 0.881483i 1.83280 + 0.800525i −1.71735 + 1.43203i 0.965404 + 1.47514i 0 −2.30576 − 1.63813i − 1.44598i 2.79145 − 1.48587i
687.2 −1.38156 + 0.302141i −1.98735 − 1.98735i 1.81742 − 0.834853i 1.84087 − 1.26933i 3.34610 + 2.14518i 0 −2.25864 + 1.70252i 4.89911i −2.15976 + 2.30986i
687.3 −1.29583 + 0.566415i 0.792756 + 0.792756i 1.35835 − 1.46795i 0.427372 + 2.19485i −1.47630 − 0.578247i 0 −0.928715 + 2.67161i − 1.74308i −1.79700 − 2.60208i
687.4 −1.25695 + 0.648130i 1.74573 + 1.74573i 1.15986 − 1.62934i −1.61944 − 1.54189i −3.32575 − 1.06284i 0 −0.401862 + 2.79973i 3.09512i 3.03490 + 0.888479i
687.5 −1.20158 − 0.745796i 1.48476 + 1.48476i 0.887576 + 1.79226i 2.14305 + 0.638231i −0.676724 − 2.89137i 0 0.270171 − 2.81549i 1.40900i −2.09905 − 2.36516i
687.6 −0.745796 − 1.20158i −1.48476 − 1.48476i −0.887576 + 1.79226i 2.14305 + 0.638231i −0.676724 + 2.89137i 0 2.81549 − 0.270171i 1.40900i −0.831394 − 3.05103i
687.7 −0.738625 + 1.20600i −0.294434 − 0.294434i −0.908866 − 1.78156i 0.950617 − 2.02394i 0.572564 − 0.137611i 0 2.81987 + 0.219816i − 2.82662i 1.73872 + 2.64137i
687.8 −0.289136 − 1.38434i 0.881483 + 0.881483i −1.83280 + 0.800525i −1.71735 + 1.43203i 0.965404 − 1.47514i 0 1.63813 + 2.30576i − 1.44598i 2.47896 + 1.96335i
687.9 −0.121020 + 1.40903i −0.404049 − 0.404049i −1.97071 − 0.341041i −2.23575 − 0.0378402i 0.618214 − 0.520418i 0 0.719030 − 2.73551i − 2.67349i 0.323888 − 3.14565i
687.10 0.0374590 + 1.41372i 1.81487 + 1.81487i −1.99719 + 0.105913i 0.00568855 + 2.23606i −2.49773 + 2.63369i 0 −0.224544 − 2.81950i 3.58748i −3.16094 + 0.0918026i
687.11 0.302141 − 1.38156i 1.98735 + 1.98735i −1.81742 − 0.834853i 1.84087 − 1.26933i 3.34610 − 2.14518i 0 −1.70252 + 2.25864i 4.89911i −1.19745 − 2.92679i
687.12 0.514741 + 1.31721i −0.590951 − 0.590951i −1.47008 + 1.35604i 2.20494 + 0.371830i 0.474220 − 1.08259i 0 −2.54291 − 1.23840i − 2.30155i 0.645193 + 3.09576i
687.13 0.566415 − 1.29583i −0.792756 − 0.792756i −1.35835 − 1.46795i 0.427372 + 2.19485i −1.47630 + 0.578247i 0 −2.67161 + 0.928715i − 1.74308i 3.08622 + 0.689394i
687.14 0.648130 − 1.25695i −1.74573 − 1.74573i −1.15986 − 1.62934i −1.61944 − 1.54189i −3.32575 + 1.06284i 0 −2.79973 + 0.401862i 3.09512i −2.98769 + 1.03621i
687.15 1.20600 − 0.738625i 0.294434 + 0.294434i 0.908866 − 1.78156i 0.950617 − 2.02394i 0.572564 + 0.137611i 0 −0.219816 − 2.81987i − 2.82662i −0.348490 − 3.14302i
687.16 1.31721 + 0.514741i 0.590951 + 0.590951i 1.47008 + 1.35604i 2.20494 + 0.371830i 0.474220 + 1.08259i 0 1.23840 + 2.54291i − 2.30155i 2.71297 + 1.62475i
687.17 1.40903 − 0.121020i 0.404049 + 0.404049i 1.97071 − 0.341041i −2.23575 − 0.0378402i 0.618214 + 0.520418i 0 2.73551 − 0.719030i − 2.67349i −3.15481 + 0.217252i
687.18 1.41372 + 0.0374590i −1.81487 − 1.81487i 1.99719 + 0.105913i 0.00568855 + 2.23606i −2.49773 − 2.63369i 0 2.81950 + 0.224544i 3.58748i −0.0757186 + 3.16137i
883.1 −1.38434 + 0.289136i −0.881483 + 0.881483i 1.83280 − 0.800525i −1.71735 − 1.43203i 0.965404 − 1.47514i 0 −2.30576 + 1.63813i 1.44598i 2.79145 + 1.48587i
883.2 −1.38156 − 0.302141i −1.98735 + 1.98735i 1.81742 + 0.834853i 1.84087 + 1.26933i 3.34610 − 2.14518i 0 −2.25864 − 1.70252i − 4.89911i −2.15976 − 2.30986i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 687.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.k.k 36
4.b odd 2 1 inner 980.2.k.k 36
5.c odd 4 1 inner 980.2.k.k 36
7.b odd 2 1 980.2.k.j 36
7.c even 3 2 140.2.w.b ✓ 72
7.d odd 6 2 980.2.x.m 72
20.e even 4 1 inner 980.2.k.k 36
28.d even 2 1 980.2.k.j 36
28.f even 6 2 980.2.x.m 72
28.g odd 6 2 140.2.w.b ✓ 72
35.f even 4 1 980.2.k.j 36
35.j even 6 2 700.2.be.e 72
35.k even 12 2 980.2.x.m 72
35.l odd 12 2 140.2.w.b ✓ 72
35.l odd 12 2 700.2.be.e 72
140.j odd 4 1 980.2.k.j 36
140.p odd 6 2 700.2.be.e 72
140.w even 12 2 140.2.w.b ✓ 72
140.w even 12 2 700.2.be.e 72
140.x odd 12 2 980.2.x.m 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.w.b ✓ 72 7.c even 3 2
140.2.w.b ✓ 72 28.g odd 6 2
140.2.w.b ✓ 72 35.l odd 12 2
140.2.w.b ✓ 72 140.w even 12 2
700.2.be.e 72 35.j even 6 2
700.2.be.e 72 35.l odd 12 2
700.2.be.e 72 140.p odd 6 2
700.2.be.e 72 140.w even 12 2
980.2.k.j 36 7.b odd 2 1
980.2.k.j 36 28.d even 2 1
980.2.k.j 36 35.f even 4 1
980.2.k.j 36 140.j odd 4 1
980.2.k.k 36 1.a even 1 1 trivial
980.2.k.k 36 4.b odd 2 1 inner
980.2.k.k 36 5.c odd 4 1 inner
980.2.k.k 36 20.e even 4 1 inner
980.2.x.m 72 7.d odd 6 2
980.2.x.m 72 28.f even 6 2
980.2.x.m 72 35.k even 12 2
980.2.x.m 72 140.x odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{36} + 167 T_{3}^{32} + 10173 T_{3}^{28} + 274163 T_{3}^{24} + 3076651 T_{3}^{20} + 10522885 T_{3}^{16} + \cdots + 11664 \) Copy content Toggle raw display
\( T_{13}^{18} + 36 T_{13}^{15} + 1009 T_{13}^{14} + 1068 T_{13}^{13} + 648 T_{13}^{12} + 5320 T_{13}^{11} + \cdots + 12800 \) Copy content Toggle raw display