Properties

Label 700.2.p.c
Level $700$
Weight $2$
Character orbit 700.p
Analytic conductor $5.590$
Analytic rank $0$
Dimension $32$
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] = [700,2,Mod(451,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("700.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.p (of order \(6\), 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} - 2 q^{4} + 4 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} - 2 q^{4} + 4 q^{8} - 16 q^{9} + 30 q^{12} + 2 q^{14} - 14 q^{16} - 12 q^{21} + 8 q^{22} + 36 q^{24} + 30 q^{26} - 2 q^{28} - 40 q^{29} - 2 q^{32} + 60 q^{36} - 8 q^{37} + 60 q^{38} + 62 q^{42} - 18 q^{44} + 2 q^{46} - 16 q^{49} + 36 q^{52} + 8 q^{53} + 12 q^{54} - 4 q^{56} - 48 q^{57} - 2 q^{58} + 24 q^{61} + 4 q^{64} + 24 q^{66} - 60 q^{68} - 4 q^{72} + 72 q^{73} + 38 q^{74} + 40 q^{77} - 120 q^{78} - 36 q^{81} - 42 q^{82} - 20 q^{84} + 28 q^{86} - 4 q^{88} - 60 q^{89} + 4 q^{92} + 8 q^{93} + 18 q^{94} - 60 q^{96} - 78 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1 −1.38207 + 0.299797i 1.36859 2.37047i 1.82024 0.828682i 0 −1.18083 + 3.68646i −1.02440 2.43939i −2.26727 + 1.69100i −2.24609 3.89033i 0
451.2 −1.37405 0.334637i 0.556469 0.963833i 1.77604 + 0.919616i 0 −1.08715 + 1.13814i 2.32410 + 1.26433i −2.13263 1.85793i 0.880685 + 1.52539i 0
451.3 −1.17265 + 0.790498i −0.331177 + 0.573616i 0.750225 1.85396i 0 −0.0650866 0.934447i 1.68748 + 2.03775i 0.585797 + 2.76710i 1.28064 + 2.21814i 0
451.4 −1.05472 0.942109i −0.450639 + 0.780530i 0.224860 + 1.98732i 0 1.21064 0.398687i −2.29962 + 1.30833i 1.63511 2.30790i 1.09385 + 1.89460i 0
451.5 −0.906567 + 1.08542i −0.406021 + 0.703249i −0.356272 1.96801i 0 −0.395235 1.07825i 0.336411 2.62428i 2.45910 + 1.39743i 1.17029 + 2.02701i 0
451.6 −0.836177 1.14053i −1.51353 + 2.62152i −0.601615 + 1.90737i 0 4.25550 0.465823i 2.57616 0.602834i 2.67847 0.908739i −3.08156 5.33743i 0
451.7 −0.569639 1.29442i 1.51353 2.62152i −1.35102 + 1.47470i 0 −4.25550 0.465823i −2.57616 + 0.602834i 2.67847 + 0.908739i −3.08156 5.33743i 0
451.8 −0.288532 1.38447i 0.450639 0.780530i −1.83350 + 0.798926i 0 −1.21064 0.398687i 2.29962 1.30833i 1.63511 + 2.30790i 1.09385 + 1.89460i 0
451.9 −0.0915727 + 1.41125i −1.49907 + 2.59647i −1.98323 0.258463i 0 −3.52698 2.35332i −2.06101 1.65899i 0.546365 2.77516i −2.99443 5.18651i 0
451.10 0.397222 1.35728i −0.556469 + 0.963833i −1.68443 1.07828i 0 1.08715 + 1.13814i −2.32410 1.26433i −2.13263 + 1.85793i 0.880685 + 1.52539i 0
451.11 0.501653 + 1.32225i 0.895374 1.55083i −1.49669 + 1.32662i 0 2.49976 + 0.405928i 0.644798 2.56598i −2.50494 1.31349i −0.103389 0.179074i 0
451.12 0.894275 + 1.09557i −0.895374 + 1.55083i −0.400544 + 1.95948i 0 −2.49976 + 0.405928i −0.644798 + 2.56598i −2.50494 + 1.31349i −0.103389 0.179074i 0
451.13 0.950668 1.04701i −1.36859 + 2.37047i −0.192463 1.99072i 0 1.18083 + 3.68646i 1.02440 + 2.43939i −2.26727 1.69100i −2.24609 3.89033i 0
451.14 1.26796 + 0.626319i 1.49907 2.59647i 1.21545 + 1.58830i 0 3.52698 2.35332i 2.06101 + 1.65899i 0.546365 + 2.77516i −2.99443 5.18651i 0
451.15 1.27092 0.620297i 0.331177 0.573616i 1.23046 1.57669i 0 0.0650866 0.934447i −1.68748 2.03775i 0.585797 2.76710i 1.28064 + 2.21814i 0
451.16 1.39328 0.242400i 0.406021 0.703249i 1.88248 0.675465i 0 0.395235 1.07825i −0.336411 + 2.62428i 2.45910 1.39743i 1.17029 + 2.02701i 0
551.1 −1.38207 0.299797i 1.36859 + 2.37047i 1.82024 + 0.828682i 0 −1.18083 3.68646i −1.02440 + 2.43939i −2.26727 1.69100i −2.24609 + 3.89033i 0
551.2 −1.37405 + 0.334637i 0.556469 + 0.963833i 1.77604 0.919616i 0 −1.08715 1.13814i 2.32410 1.26433i −2.13263 + 1.85793i 0.880685 1.52539i 0
551.3 −1.17265 0.790498i −0.331177 0.573616i 0.750225 + 1.85396i 0 −0.0650866 + 0.934447i 1.68748 2.03775i 0.585797 2.76710i 1.28064 2.21814i 0
551.4 −1.05472 + 0.942109i −0.450639 0.780530i 0.224860 1.98732i 0 1.21064 + 0.398687i −2.29962 1.30833i 1.63511 + 2.30790i 1.09385 1.89460i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.p.c 32
4.b odd 2 1 inner 700.2.p.c 32
5.b even 2 1 140.2.o.a 32
5.c odd 4 1 700.2.t.c 32
5.c odd 4 1 700.2.t.d 32
7.d odd 6 1 inner 700.2.p.c 32
20.d odd 2 1 140.2.o.a 32
20.e even 4 1 700.2.t.c 32
20.e even 4 1 700.2.t.d 32
28.f even 6 1 inner 700.2.p.c 32
35.c odd 2 1 980.2.o.f 32
35.i odd 6 1 140.2.o.a 32
35.i odd 6 1 980.2.g.a 32
35.j even 6 1 980.2.g.a 32
35.j even 6 1 980.2.o.f 32
35.k even 12 1 700.2.t.c 32
35.k even 12 1 700.2.t.d 32
140.c even 2 1 980.2.o.f 32
140.p odd 6 1 980.2.g.a 32
140.p odd 6 1 980.2.o.f 32
140.s even 6 1 140.2.o.a 32
140.s even 6 1 980.2.g.a 32
140.x odd 12 1 700.2.t.c 32
140.x odd 12 1 700.2.t.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.o.a 32 5.b even 2 1
140.2.o.a 32 20.d odd 2 1
140.2.o.a 32 35.i odd 6 1
140.2.o.a 32 140.s even 6 1
700.2.p.c 32 1.a even 1 1 trivial
700.2.p.c 32 4.b odd 2 1 inner
700.2.p.c 32 7.d odd 6 1 inner
700.2.p.c 32 28.f even 6 1 inner
700.2.t.c 32 5.c odd 4 1
700.2.t.c 32 20.e even 4 1
700.2.t.c 32 35.k even 12 1
700.2.t.c 32 140.x odd 12 1
700.2.t.d 32 5.c odd 4 1
700.2.t.d 32 20.e even 4 1
700.2.t.d 32 35.k even 12 1
700.2.t.d 32 140.x odd 12 1
980.2.g.a 32 35.i odd 6 1
980.2.g.a 32 35.j even 6 1
980.2.g.a 32 140.p odd 6 1
980.2.g.a 32 140.s even 6 1
980.2.o.f 32 35.c odd 2 1
980.2.o.f 32 35.j even 6 1
980.2.o.f 32 140.c even 2 1
980.2.o.f 32 140.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{32} + 32 T_{3}^{30} + 629 T_{3}^{28} + 7904 T_{3}^{26} + 73006 T_{3}^{24} + 483232 T_{3}^{22} + 2399561 T_{3}^{20} + 8429248 T_{3}^{18} + 21956782 T_{3}^{16} + 39983424 T_{3}^{14} + 54411189 T_{3}^{12} + \cdots + 331776 \) Copy content Toggle raw display
\( T_{17}^{16} - 64 T_{17}^{14} + 2988 T_{17}^{12} - 672 T_{17}^{11} - 61696 T_{17}^{10} + 39360 T_{17}^{9} + 922000 T_{17}^{8} - 1774464 T_{17}^{7} - 3374592 T_{17}^{6} + 10285056 T_{17}^{5} + 10186752 T_{17}^{4} + \cdots + 9437184 \) Copy content Toggle raw display