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Properties

Label 735.2.y.i
Level $735$
Weight $2$
Character orbit 735.y
Analytic conductor $5.869$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(128,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.128");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.y (of order \(12\), degree \(4\), not 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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{3} + 24 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{3} + 24 q^{6} + 8 q^{10} + 10 q^{12} + 16 q^{13} + 4 q^{15} - 8 q^{16} + 14 q^{18} - 8 q^{22} + 4 q^{25} - 40 q^{27} + 40 q^{30} + 24 q^{31} + 4 q^{33} + 8 q^{36} + 4 q^{37} + 16 q^{40} + 16 q^{43} - 40 q^{45} - 32 q^{46} - 44 q^{48} + 8 q^{51} - 36 q^{52} + 40 q^{55} - 88 q^{57} + 56 q^{58} - 50 q^{60} + 8 q^{61} - 76 q^{66} + 12 q^{67} - 34 q^{72} - 52 q^{73} - 6 q^{75} - 64 q^{76} - 120 q^{78} + 20 q^{81} - 104 q^{82} - 24 q^{85} + 46 q^{87} + 84 q^{90} - 44 q^{93} - 12 q^{96} + 120 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
128.1 −2.35640 + 0.631395i −1.72899 − 0.102851i 3.42191 − 1.97564i −0.0540016 + 2.23542i 4.13914 − 0.849321i 0 −3.36596 + 3.36596i 2.97884 + 0.355658i −1.28418 − 5.30163i
128.2 −2.17249 + 0.582118i 0.644934 − 1.60750i 2.64881 − 1.52929i −1.39781 − 1.74531i −0.465359 + 3.86771i 0 −1.68355 + 1.68355i −2.16812 − 2.07346i 4.05271 + 2.97799i
128.3 −1.46015 + 0.391246i −1.49243 + 0.879005i 0.246919 − 0.142558i 0.207883 − 2.22638i 1.83527 − 1.86739i 0 1.83305 − 1.83305i 1.45470 − 2.62371i 0.567525 + 3.33219i
128.4 −1.26950 + 0.340162i 1.59946 + 0.664627i −0.236127 + 0.136328i −2.23063 − 0.155895i −2.25660 − 0.299670i 0 2.11207 − 2.11207i 2.11654 + 2.12609i 2.88481 − 0.560865i
128.5 −0.907300 + 0.243110i 0.315275 + 1.70312i −0.967960 + 0.558852i 2.12501 + 0.695932i −0.700094 − 1.46859i 0 2.07075 − 2.07075i −2.80120 + 1.07390i −2.09721 − 0.114806i
128.6 −0.298314 + 0.0799329i −1.15464 − 1.29105i −1.64945 + 0.952310i −1.56830 + 1.59387i 0.447643 + 0.292843i 0 0.852694 − 0.852694i −0.333606 + 2.98139i 0.340444 − 0.600832i
128.7 0.298314 − 0.0799329i 1.64547 − 0.540759i −1.64945 + 0.952310i 1.56830 − 1.59387i 0.447643 − 0.292843i 0 −0.852694 + 0.852694i 2.41516 − 1.77961i 0.340444 − 0.600832i
128.8 0.907300 − 0.243110i −1.12459 + 1.31730i −0.967960 + 0.558852i −2.12501 − 0.695932i −0.700094 + 1.46859i 0 −2.07075 + 2.07075i −0.470578 − 2.96286i −2.09721 − 0.114806i
128.9 1.26950 − 0.340162i −1.71749 − 0.224146i −0.236127 + 0.136328i 2.23063 + 0.155895i −2.25660 + 0.299670i 0 −2.11207 + 2.11207i 2.89952 + 0.769934i 2.88481 − 0.560865i
128.10 1.46015 − 0.391246i 0.852980 + 1.50746i 0.246919 − 0.142558i −0.207883 + 2.22638i 1.83527 + 1.86739i 0 −1.83305 + 1.83305i −1.54485 + 2.57166i 0.567525 + 3.33219i
128.11 2.17249 − 0.582118i 0.245221 − 1.71460i 2.64881 − 1.52929i 1.39781 + 1.74531i −0.465359 − 3.86771i 0 1.68355 − 1.68355i −2.87973 − 0.840915i 4.05271 + 2.97799i
128.12 2.35640 − 0.631395i 1.54878 + 0.775426i 3.42191 − 1.97564i 0.0540016 − 2.23542i 4.13914 + 0.849321i 0 3.36596 − 3.36596i 1.79743 + 2.40193i −1.28418 − 5.30163i
263.1 −0.631395 + 2.35640i −0.102851 − 1.72899i −3.42191 − 1.97564i 1.90893 + 1.16447i 4.13914 + 0.849321i 0 3.36596 − 3.36596i −2.97884 + 0.355658i −3.94925 + 3.76295i
263.2 −0.582118 + 2.17249i −1.60750 + 0.644934i −2.64881 − 1.52929i −2.21039 + 0.337883i −0.465359 − 3.86771i 0 1.68355 − 1.68355i 2.16812 − 2.07346i 0.552660 − 4.99875i
263.3 −0.391246 + 1.46015i 0.879005 − 1.49243i −0.246919 − 0.142558i −1.82416 − 1.29322i 1.83527 + 1.86739i 0 −1.83305 + 1.83305i −1.45470 − 2.62371i 2.60200 − 2.15759i
263.4 −0.340162 + 1.26950i 0.664627 + 1.59946i 0.236127 + 0.136328i −1.25032 + 1.85383i −2.25660 + 0.299670i 0 −2.11207 + 2.11207i −2.11654 + 2.12609i −1.92813 − 2.21789i
263.5 −0.243110 + 0.907300i 1.70312 + 0.315275i 0.967960 + 0.558852i 1.66520 − 1.49235i −0.700094 + 1.46859i 0 −2.07075 + 2.07075i 2.80120 + 1.07390i 0.949181 + 1.87364i
263.6 −0.0799329 + 0.298314i −1.29105 − 1.15464i 1.64945 + 0.952310i 0.596180 + 2.15513i 0.447643 − 0.292843i 0 −0.852694 + 0.852694i 0.333606 + 2.98139i −0.690558 + 0.00558322i
263.7 0.0799329 − 0.298314i −0.540759 + 1.64547i 1.64945 + 0.952310i −0.596180 − 2.15513i 0.447643 + 0.292843i 0 0.852694 − 0.852694i −2.41516 − 1.77961i −0.690558 + 0.00558322i
263.8 0.243110 − 0.907300i 1.31730 − 1.12459i 0.967960 + 0.558852i −1.66520 + 1.49235i −0.700094 − 1.46859i 0 2.07075 − 2.07075i 0.470578 − 2.96286i 0.949181 + 1.87364i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 128.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.y.i 48
3.b odd 2 1 inner 735.2.y.i 48
5.c odd 4 1 inner 735.2.y.i 48
7.b odd 2 1 105.2.x.a ✓ 48
7.c even 3 1 735.2.j.e 24
7.c even 3 1 inner 735.2.y.i 48
7.d odd 6 1 105.2.x.a ✓ 48
7.d odd 6 1 735.2.j.g 24
15.e even 4 1 inner 735.2.y.i 48
21.c even 2 1 105.2.x.a ✓ 48
21.g even 6 1 105.2.x.a ✓ 48
21.g even 6 1 735.2.j.g 24
21.h odd 6 1 735.2.j.e 24
21.h odd 6 1 inner 735.2.y.i 48
35.c odd 2 1 525.2.bf.f 48
35.f even 4 1 105.2.x.a ✓ 48
35.f even 4 1 525.2.bf.f 48
35.i odd 6 1 525.2.bf.f 48
35.k even 12 1 105.2.x.a ✓ 48
35.k even 12 1 525.2.bf.f 48
35.k even 12 1 735.2.j.g 24
35.l odd 12 1 735.2.j.e 24
35.l odd 12 1 inner 735.2.y.i 48
105.g even 2 1 525.2.bf.f 48
105.k odd 4 1 105.2.x.a ✓ 48
105.k odd 4 1 525.2.bf.f 48
105.p even 6 1 525.2.bf.f 48
105.w odd 12 1 105.2.x.a ✓ 48
105.w odd 12 1 525.2.bf.f 48
105.w odd 12 1 735.2.j.g 24
105.x even 12 1 735.2.j.e 24
105.x even 12 1 inner 735.2.y.i 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.x.a ✓ 48 7.b odd 2 1
105.2.x.a ✓ 48 7.d odd 6 1
105.2.x.a ✓ 48 21.c even 2 1
105.2.x.a ✓ 48 21.g even 6 1
105.2.x.a ✓ 48 35.f even 4 1
105.2.x.a ✓ 48 35.k even 12 1
105.2.x.a ✓ 48 105.k odd 4 1
105.2.x.a ✓ 48 105.w odd 12 1
525.2.bf.f 48 35.c odd 2 1
525.2.bf.f 48 35.f even 4 1
525.2.bf.f 48 35.i odd 6 1
525.2.bf.f 48 35.k even 12 1
525.2.bf.f 48 105.g even 2 1
525.2.bf.f 48 105.k odd 4 1
525.2.bf.f 48 105.p even 6 1
525.2.bf.f 48 105.w odd 12 1
735.2.j.e 24 7.c even 3 1
735.2.j.e 24 21.h odd 6 1
735.2.j.e 24 35.l odd 12 1
735.2.j.e 24 105.x even 12 1
735.2.j.g 24 7.d odd 6 1
735.2.j.g 24 21.g even 6 1
735.2.j.g 24 35.k even 12 1
735.2.j.g 24 105.w odd 12 1
735.2.y.i 48 1.a even 1 1 trivial
735.2.y.i 48 3.b odd 2 1 inner
735.2.y.i 48 5.c odd 4 1 inner
735.2.y.i 48 7.c even 3 1 inner
735.2.y.i 48 15.e even 4 1 inner
735.2.y.i 48 21.h odd 6 1 inner
735.2.y.i 48 35.l odd 12 1 inner
735.2.y.i 48 105.x even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{48} - 70 T_{2}^{44} + 3423 T_{2}^{40} - 84374 T_{2}^{36} + 1495233 T_{2}^{32} - 11151456 T_{2}^{28} + \cdots + 10000 \) Copy content Toggle raw display
\( T_{11}^{24} - 50 T_{11}^{22} + 1650 T_{11}^{20} - 31664 T_{11}^{18} + 443086 T_{11}^{16} + \cdots + 1000000 \) Copy content Toggle raw display
\( T_{13}^{12} - 4 T_{13}^{11} + 8 T_{13}^{10} - 26 T_{13}^{9} + 334 T_{13}^{8} - 1504 T_{13}^{7} + \cdots + 30976 \) Copy content Toggle raw display
\( T_{17}^{48} - 3624 T_{17}^{44} + 10945740 T_{17}^{40} - 7351219384 T_{17}^{36} + 3727850045900 T_{17}^{32} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display