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Properties

Label 735.3.e.a
Level $735$
Weight $3$
Character orbit 735.e
Analytic conductor $20.027$
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] = [735,3,Mod(244,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.244");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 735.e (of order \(2\), degree \(1\), 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: \(20.0272994305\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 64 q^{4} + 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 64 q^{4} + 96 q^{9} + 56 q^{11} - 24 q^{15} + 80 q^{16} + 68 q^{25} - 88 q^{29} - 192 q^{36} - 72 q^{39} - 640 q^{44} + 120 q^{46} - 24 q^{51} + 396 q^{60} - 400 q^{64} + 92 q^{65} + 344 q^{71} - 1800 q^{74} + 40 q^{79} + 288 q^{81} + 304 q^{85} - 288 q^{86} + 684 q^{95} + 168 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1 − 3.77385i −1.73205 −10.2419 4.71188 − 1.67279i 6.53650i 0 23.5561i 3.00000 −6.31286 − 17.7819i
244.2 3.77385i −1.73205 −10.2419 4.71188 + 1.67279i − 6.53650i 0 − 23.5561i 3.00000 −6.31286 + 17.7819i
244.3 − 1.98186i −1.73205 0.0722490 4.19079 − 2.72714i 3.43267i 0 − 8.07061i 3.00000 −5.40480 − 8.30553i
244.4 1.98186i −1.73205 0.0722490 4.19079 + 2.72714i − 3.43267i 0 8.07061i 3.00000 −5.40480 + 8.30553i
244.5 − 3.40813i 1.73205 −7.61537 −1.24390 − 4.84280i − 5.90306i 0 12.3217i 3.00000 −16.5049 + 4.23938i
244.6 3.40813i 1.73205 −7.61537 −1.24390 + 4.84280i 5.90306i 0 − 12.3217i 3.00000 −16.5049 − 4.23938i
244.7 − 2.41267i 1.73205 −1.82098 0.329338 − 4.98914i − 4.17887i 0 − 5.25727i 3.00000 −12.0372 − 0.794583i
244.8 2.41267i 1.73205 −1.82098 0.329338 + 4.98914i 4.17887i 0 5.25727i 3.00000 −12.0372 + 0.794583i
244.9 − 1.53035i −1.73205 1.65803 4.75041 − 1.56000i 2.65064i 0 − 8.65876i 3.00000 −2.38734 − 7.26979i
244.10 1.53035i −1.73205 1.65803 4.75041 + 1.56000i − 2.65064i 0 8.65876i 3.00000 −2.38734 + 7.26979i
244.11 − 0.494258i 1.73205 3.75571 4.99109 − 0.298311i − 0.856079i 0 − 3.83332i 3.00000 −0.147443 − 2.46689i
244.12 0.494258i 1.73205 3.75571 4.99109 + 0.298311i 0.856079i 0 3.83332i 3.00000 −0.147443 + 2.46689i
244.13 − 3.40813i −1.73205 −7.61537 1.24390 + 4.84280i 5.90306i 0 12.3217i 3.00000 16.5049 − 4.23938i
244.14 3.40813i −1.73205 −7.61537 1.24390 − 4.84280i − 5.90306i 0 − 12.3217i 3.00000 16.5049 + 4.23938i
244.15 − 2.93230i 1.73205 −4.59841 3.73497 + 3.32416i − 5.07890i 0 1.75471i 3.00000 9.74744 − 10.9521i
244.16 2.93230i 1.73205 −4.59841 3.73497 − 3.32416i 5.07890i 0 − 1.75471i 3.00000 9.74744 + 10.9521i
244.17 − 0.494258i −1.73205 3.75571 −4.99109 + 0.298311i 0.856079i 0 − 3.83332i 3.00000 0.147443 + 2.46689i
244.18 0.494258i −1.73205 3.75571 −4.99109 − 0.298311i − 0.856079i 0 3.83332i 3.00000 0.147443 − 2.46689i
244.19 − 3.77385i 1.73205 −10.2419 −4.71188 + 1.67279i − 6.53650i 0 23.5561i 3.00000 6.31286 + 17.7819i
244.20 3.77385i 1.73205 −10.2419 −4.71188 − 1.67279i 6.53650i 0 − 23.5561i 3.00000 6.31286 − 17.7819i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 244.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.3.e.a 32
5.b even 2 1 inner 735.3.e.a 32
7.b odd 2 1 inner 735.3.e.a 32
7.c even 3 1 105.3.r.a ✓ 32
7.d odd 6 1 105.3.r.a ✓ 32
21.g even 6 1 315.3.bi.e 32
21.h odd 6 1 315.3.bi.e 32
35.c odd 2 1 inner 735.3.e.a 32
35.i odd 6 1 105.3.r.a ✓ 32
35.j even 6 1 105.3.r.a ✓ 32
35.k even 12 1 525.3.o.p 16
35.k even 12 1 525.3.o.q 16
35.l odd 12 1 525.3.o.p 16
35.l odd 12 1 525.3.o.q 16
105.o odd 6 1 315.3.bi.e 32
105.p even 6 1 315.3.bi.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.r.a ✓ 32 7.c even 3 1
105.3.r.a ✓ 32 7.d odd 6 1
105.3.r.a ✓ 32 35.i odd 6 1
105.3.r.a ✓ 32 35.j even 6 1
315.3.bi.e 32 21.g even 6 1
315.3.bi.e 32 21.h odd 6 1
315.3.bi.e 32 105.o odd 6 1
315.3.bi.e 32 105.p even 6 1
525.3.o.p 16 35.k even 12 1
525.3.o.p 16 35.l odd 12 1
525.3.o.q 16 35.k even 12 1
525.3.o.q 16 35.l odd 12 1
735.3.e.a 32 1.a even 1 1 trivial
735.3.e.a 32 5.b even 2 1 inner
735.3.e.a 32 7.b odd 2 1 inner
735.3.e.a 32 35.c odd 2 1 inner