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Properties

Label 700.2.p.d
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: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{2} + q^{4} - 4 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{2} + q^{4} - 4 q^{8} - 16 q^{9} + 15 q^{12} - 13 q^{14} + q^{16} - 15 q^{18} + 12 q^{21} + 34 q^{22} - 18 q^{24} - 15 q^{26} + 17 q^{28} + 8 q^{29} + 14 q^{32} - 30 q^{36} - 16 q^{37} + 30 q^{38} + q^{42} + 12 q^{44} + 2 q^{46} + 20 q^{49} + 18 q^{52} - 20 q^{53} + 57 q^{54} - 31 q^{56} - 24 q^{57} - 4 q^{58} - 12 q^{61} + 40 q^{64} - 66 q^{66} + 15 q^{68} + 13 q^{72} - 72 q^{73} - q^{74} + 8 q^{77} + 60 q^{78} - 36 q^{81} - 66 q^{82} + 67 q^{84} + 4 q^{86} + 34 q^{88} - 60 q^{89} - 148 q^{92} - 20 q^{93} + 45 q^{94} + 93 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1 −1.41372 − 0.0373956i 0.266293 − 0.461233i 1.99720 + 0.105734i 0 −0.393712 + 0.642096i −2.27248 − 1.35494i −2.81953 − 0.224164i 1.35818 + 2.35243i 0
451.2 −1.40227 + 0.183411i −1.34536 + 2.33023i 1.93272 − 0.514384i 0 1.45917 − 3.51437i 2.64444 − 0.0832596i −2.61585 + 1.07579i −2.11999 − 3.67193i 0
451.3 −1.22420 − 0.708043i 1.64638 − 2.85162i 0.997350 + 1.73358i 0 −4.03458 + 2.32525i 2.24133 − 1.40586i 0.00648826 − 2.82842i −3.92116 − 6.79165i 0
451.4 −1.07322 + 0.920983i 1.28534 − 2.22627i 0.303582 − 1.97683i 0 0.670911 + 3.57303i −0.304602 + 2.62816i 1.49481 + 2.40115i −1.80417 − 3.12492i 0
451.5 −1.05045 − 0.946868i −0.217629 + 0.376945i 0.206882 + 1.98927i 0 0.585526 − 0.189895i 0.322869 + 2.62598i 1.66626 − 2.28552i 1.40527 + 2.43401i 0
451.6 −0.819270 + 1.15273i −0.878165 + 1.52103i −0.657592 − 1.88880i 0 −1.03388 − 2.25842i −2.00509 + 1.72616i 2.71603 + 0.789410i −0.0423472 − 0.0733475i 0
451.7 −0.294788 − 1.38315i 0.217629 − 0.376945i −1.82620 + 0.815471i 0 −0.585526 − 0.189895i −0.322869 − 2.62598i 1.66626 + 2.28552i 1.40527 + 2.43401i 0
451.8 −0.237248 + 1.39417i −0.359818 + 0.623223i −1.88743 − 0.661529i 0 −0.783513 − 0.649506i 2.46698 − 0.956035i 1.37007 − 2.47445i 1.24106 + 2.14958i 0
451.9 −0.00108138 − 1.41421i −1.64638 + 2.85162i −2.00000 + 0.00305860i 0 4.03458 + 2.32525i −2.24133 + 1.40586i 0.00648826 + 2.82842i −3.92116 − 6.79165i 0
451.10 0.672242 + 1.24422i 0.899123 − 1.55733i −1.09618 + 1.67284i 0 2.54209 + 0.0718084i 1.73661 + 1.99604i −2.81828 − 0.239340i −0.116845 − 0.202382i 0
451.11 0.674474 − 1.24301i −0.266293 + 0.461233i −1.09017 − 1.67676i 0 0.393712 + 0.642096i 2.27248 + 1.35494i −2.81953 + 0.224164i 1.35818 + 2.35243i 0
451.12 0.741407 + 1.20429i −0.899123 + 1.55733i −0.900630 + 1.78574i 0 −2.54209 + 0.0718084i −1.73661 − 1.99604i −2.81828 + 0.239340i −0.116845 − 0.202382i 0
451.13 0.859974 − 1.12270i 1.34536 − 2.33023i −0.520890 − 1.93098i 0 −1.45917 − 3.51437i −2.64444 + 0.0832596i −2.61585 − 1.07579i −2.11999 − 3.67193i 0
451.14 1.32601 + 0.491623i 0.359818 − 0.623223i 1.51661 + 1.30380i 0 0.783513 − 0.649506i −2.46698 + 0.956035i 1.37007 + 2.47445i 1.24106 + 2.14958i 0
451.15 1.33420 − 0.468940i −1.28534 + 2.22627i 1.56019 − 1.25132i 0 −0.670911 + 3.57303i 0.304602 − 2.62816i 1.49481 − 2.40115i −1.80417 − 3.12492i 0
451.16 1.40793 − 0.133142i 0.878165 − 1.52103i 1.96455 − 0.374909i 0 1.03388 − 2.25842i 2.00509 − 1.72616i 2.71603 − 0.789410i −0.0423472 − 0.0733475i 0
551.1 −1.41372 + 0.0373956i 0.266293 + 0.461233i 1.99720 − 0.105734i 0 −0.393712 − 0.642096i −2.27248 + 1.35494i −2.81953 + 0.224164i 1.35818 − 2.35243i 0
551.2 −1.40227 − 0.183411i −1.34536 − 2.33023i 1.93272 + 0.514384i 0 1.45917 + 3.51437i 2.64444 + 0.0832596i −2.61585 − 1.07579i −2.11999 + 3.67193i 0
551.3 −1.22420 + 0.708043i 1.64638 + 2.85162i 0.997350 − 1.73358i 0 −4.03458 − 2.32525i 2.24133 + 1.40586i 0.00648826 + 2.82842i −3.92116 + 6.79165i 0
551.4 −1.07322 − 0.920983i 1.28534 + 2.22627i 0.303582 + 1.97683i 0 0.670911 − 3.57303i −0.304602 − 2.62816i 1.49481 − 2.40115i −1.80417 + 3.12492i 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.d ✓ 32
4.b odd 2 1 inner 700.2.p.d ✓ 32
5.b even 2 1 700.2.p.f yes 32
5.c odd 4 2 700.2.t.e 64
7.d odd 6 1 inner 700.2.p.d ✓ 32
20.d odd 2 1 700.2.p.f yes 32
20.e even 4 2 700.2.t.e 64
28.f even 6 1 inner 700.2.p.d ✓ 32
35.i odd 6 1 700.2.p.f yes 32
35.k even 12 2 700.2.t.e 64
140.s even 6 1 700.2.p.f yes 32
140.x odd 12 2 700.2.t.e 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.p.d ✓ 32 1.a even 1 1 trivial
700.2.p.d ✓ 32 4.b odd 2 1 inner
700.2.p.d ✓ 32 7.d odd 6 1 inner
700.2.p.d ✓ 32 28.f even 6 1 inner
700.2.p.f yes 32 5.b even 2 1
700.2.p.f yes 32 20.d odd 2 1
700.2.p.f yes 32 35.i odd 6 1
700.2.p.f yes 32 140.s even 6 1
700.2.t.e 64 5.c odd 4 2
700.2.t.e 64 20.e even 4 2
700.2.t.e 64 35.k even 12 2
700.2.t.e 64 140.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 \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{32} + 32 T_{3}^{30} + 629 T_{3}^{28} + 7868 T_{3}^{26} + 72313 T_{3}^{24} + 482422 T_{3}^{22} + \cdots + 20736 \) Copy content Toggle raw display
\( T_{17}^{16} - 73 T_{17}^{14} + 3924 T_{17}^{12} + 7782 T_{17}^{11} - 78301 T_{17}^{10} - 199524 T_{17}^{9} + \cdots + 4601025 \) Copy content Toggle raw display