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Properties

Label 775.2.ck.a
Level $775$
Weight $2$
Character orbit 775.ck
Analytic conductor $6.188$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [775,2,Mod(49,775)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(775, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([15, 26])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("775.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.ck (of order \(30\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{30})\)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 28 q^{4} + 22 q^{6} + 20 q^{9} - 14 q^{11} + 12 q^{14} - 4 q^{16} - 32 q^{19} + 18 q^{21} + 40 q^{24} + 18 q^{26} + 28 q^{29} + 30 q^{31} + 64 q^{34} + 2 q^{36} + 6 q^{39} - 16 q^{41} - 78 q^{44}+ \cdots - 12 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.35633 − 1.86683i 1.03422 + 2.32289i −1.02738 + 3.16196i 0 2.93370 − 5.08132i −1.19043 + 1.07187i 2.90713 − 0.944583i −2.31884 + 2.57533i 0
49.2 −0.405274 − 0.557811i −0.367040 − 0.824384i 0.471127 − 1.44998i 0 −0.311099 + 0.538840i 0.567312 − 0.510810i −2.31124 + 0.750969i 1.46250 − 1.62427i 0
49.3 0.405274 + 0.557811i 0.367040 + 0.824384i 0.471127 − 1.44998i 0 −0.311099 + 0.538840i −0.567312 + 0.510810i 2.31124 − 0.750969i 1.46250 − 1.62427i 0
49.4 1.35633 + 1.86683i −1.03422 − 2.32289i −1.02738 + 3.16196i 0 2.93370 − 5.08132i 1.19043 − 1.07187i −2.90713 + 0.944583i −2.31884 + 2.57533i 0
174.1 −1.35633 + 1.86683i 1.03422 − 2.32289i −1.02738 − 3.16196i 0 2.93370 + 5.08132i −1.19043 − 1.07187i 2.90713 + 0.944583i −2.31884 − 2.57533i 0
174.2 −0.405274 + 0.557811i −0.367040 + 0.824384i 0.471127 + 1.44998i 0 −0.311099 − 0.538840i 0.567312 + 0.510810i −2.31124 − 0.750969i 1.46250 + 1.62427i 0
174.3 0.405274 − 0.557811i 0.367040 − 0.824384i 0.471127 + 1.44998i 0 −0.311099 − 0.538840i −0.567312 − 0.510810i 2.31124 + 0.750969i 1.46250 + 1.62427i 0
174.4 1.35633 − 1.86683i −1.03422 + 2.32289i −1.02738 − 3.16196i 0 2.93370 + 5.08132i 1.19043 + 1.07187i −2.90713 − 0.944583i −2.31884 − 2.57533i 0
224.1 −1.75965 + 0.571745i −0.103822 − 0.488442i 1.15144 − 0.836573i 0 0.461954 + 0.800128i 1.51837 − 3.41030i 0.627215 − 0.863288i 2.51284 − 1.11879i 0
224.2 −1.17187 + 0.380762i 0.431412 + 2.02963i −0.389745 + 0.283166i 0 −1.27836 − 2.21419i 1.54713 − 3.47491i 1.79742 − 2.47393i −1.19265 + 0.531003i 0
224.3 1.17187 − 0.380762i −0.431412 − 2.02963i −0.389745 + 0.283166i 0 −1.27836 − 2.21419i −1.54713 + 3.47491i −1.79742 + 2.47393i −1.19265 + 0.531003i 0
224.4 1.75965 − 0.571745i 0.103822 + 0.488442i 1.15144 − 0.836573i 0 0.461954 + 0.800128i −1.51837 + 3.41030i −0.627215 + 0.863288i 2.51284 − 1.11879i 0
299.1 −2.55849 + 0.831304i −1.05464 + 0.949606i 4.23677 − 3.07819i 0 1.90889 − 3.30629i 1.71742 − 0.180508i −5.11835 + 7.04481i −0.103062 + 0.980572i 0
299.2 −1.97070 + 0.640321i −1.58988 + 1.43153i 1.85563 − 1.34820i 0 2.21654 − 3.83916i −3.65441 + 0.384094i −0.357701 + 0.492333i 0.164841 − 1.56836i 0
299.3 1.97070 − 0.640321i 1.58988 − 1.43153i 1.85563 − 1.34820i 0 2.21654 − 3.83916i 3.65441 − 0.384094i 0.357701 − 0.492333i 0.164841 − 1.56836i 0
299.4 2.55849 − 0.831304i 1.05464 − 0.949606i 4.23677 − 3.07819i 0 1.90889 − 3.30629i −1.71742 + 0.180508i 5.11835 − 7.04481i −0.103062 + 0.980572i 0
324.1 −2.55849 − 0.831304i −1.05464 − 0.949606i 4.23677 + 3.07819i 0 1.90889 + 3.30629i 1.71742 + 0.180508i −5.11835 − 7.04481i −0.103062 − 0.980572i 0
324.2 −1.97070 − 0.640321i −1.58988 − 1.43153i 1.85563 + 1.34820i 0 2.21654 + 3.83916i −3.65441 − 0.384094i −0.357701 − 0.492333i 0.164841 + 1.56836i 0
324.3 1.97070 + 0.640321i 1.58988 + 1.43153i 1.85563 + 1.34820i 0 2.21654 + 3.83916i 3.65441 + 0.384094i 0.357701 + 0.492333i 0.164841 + 1.56836i 0
324.4 2.55849 + 0.831304i 1.05464 + 0.949606i 4.23677 + 3.07819i 0 1.90889 + 3.30629i −1.71742 − 0.180508i 5.11835 + 7.04481i −0.103062 − 0.980572i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.g even 15 1 inner
155.u even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.ck.a 32
5.b even 2 1 inner 775.2.ck.a 32
5.c odd 4 1 31.2.g.a ✓ 16
5.c odd 4 1 775.2.bl.a 16
15.e even 4 1 279.2.y.c 16
20.e even 4 1 496.2.bg.c 16
31.g even 15 1 inner 775.2.ck.a 32
155.f even 4 1 961.2.g.l 16
155.o odd 12 1 961.2.d.o 16
155.o odd 12 1 961.2.g.k 16
155.p even 12 1 961.2.d.n 16
155.p even 12 1 961.2.g.j 16
155.r even 20 1 961.2.c.i 16
155.r even 20 1 961.2.g.j 16
155.r even 20 1 961.2.g.m 16
155.r even 20 1 961.2.g.n 16
155.s odd 20 1 961.2.c.j 16
155.s odd 20 1 961.2.g.k 16
155.s odd 20 1 961.2.g.s 16
155.s odd 20 1 961.2.g.t 16
155.u even 30 1 inner 775.2.ck.a 32
155.w odd 60 1 31.2.g.a ✓ 16
155.w odd 60 1 775.2.bl.a 16
155.w odd 60 1 961.2.a.i 8
155.w odd 60 1 961.2.c.j 16
155.w odd 60 1 961.2.d.o 16
155.w odd 60 2 961.2.d.p 16
155.w odd 60 1 961.2.g.s 16
155.w odd 60 1 961.2.g.t 16
155.x even 60 1 961.2.a.j 8
155.x even 60 1 961.2.c.i 16
155.x even 60 1 961.2.d.n 16
155.x even 60 2 961.2.d.q 16
155.x even 60 1 961.2.g.l 16
155.x even 60 1 961.2.g.m 16
155.x even 60 1 961.2.g.n 16
465.bt even 60 1 279.2.y.c 16
465.bt even 60 1 8649.2.a.bf 8
465.bv odd 60 1 8649.2.a.be 8
620.bt even 60 1 496.2.bg.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.g.a ✓ 16 5.c odd 4 1
31.2.g.a ✓ 16 155.w odd 60 1
279.2.y.c 16 15.e even 4 1
279.2.y.c 16 465.bt even 60 1
496.2.bg.c 16 20.e even 4 1
496.2.bg.c 16 620.bt even 60 1
775.2.bl.a 16 5.c odd 4 1
775.2.bl.a 16 155.w odd 60 1
775.2.ck.a 32 1.a even 1 1 trivial
775.2.ck.a 32 5.b even 2 1 inner
775.2.ck.a 32 31.g even 15 1 inner
775.2.ck.a 32 155.u even 30 1 inner
961.2.a.i 8 155.w odd 60 1
961.2.a.j 8 155.x even 60 1
961.2.c.i 16 155.r even 20 1
961.2.c.i 16 155.x even 60 1
961.2.c.j 16 155.s odd 20 1
961.2.c.j 16 155.w odd 60 1
961.2.d.n 16 155.p even 12 1
961.2.d.n 16 155.x even 60 1
961.2.d.o 16 155.o odd 12 1
961.2.d.o 16 155.w odd 60 1
961.2.d.p 16 155.w odd 60 2
961.2.d.q 16 155.x even 60 2
961.2.g.j 16 155.p even 12 1
961.2.g.j 16 155.r even 20 1
961.2.g.k 16 155.o odd 12 1
961.2.g.k 16 155.s odd 20 1
961.2.g.l 16 155.f even 4 1
961.2.g.l 16 155.x even 60 1
961.2.g.m 16 155.r even 20 1
961.2.g.m 16 155.x even 60 1
961.2.g.n 16 155.r even 20 1
961.2.g.n 16 155.x even 60 1
961.2.g.s 16 155.s odd 20 1
961.2.g.s 16 155.w odd 60 1
961.2.g.t 16 155.s odd 20 1
961.2.g.t 16 155.w odd 60 1
8649.2.a.be 8 465.bv odd 60 1
8649.2.a.bf 8 465.bt even 60 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 22 T_{2}^{30} + 241 T_{2}^{28} - 1733 T_{2}^{26} + 10266 T_{2}^{24} - 51961 T_{2}^{22} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display