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Properties

Label 429.2.bq.a
Level $429$
Weight $2$
Character orbit 429.bq
Analytic conductor $3.426$
Analytic rank $0$
Dimension $416$
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] = [429,2,Mod(29,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 21, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bq (of order \(30\), degree \(8\), 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: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(416\)
Relative dimension: \(52\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 416 q - 3 q^{3} + 42 q^{4} - 5 q^{6} - 10 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 416 q - 3 q^{3} + 42 q^{4} - 5 q^{6} - 10 q^{7} + 3 q^{9} - 52 q^{12} - 20 q^{13} - 3 q^{15} + 30 q^{16} - 20 q^{18} - 10 q^{19} + 4 q^{22} + 5 q^{24} + 60 q^{25} - 24 q^{27} - 10 q^{28} - 65 q^{30} - 8 q^{31} + 49 q^{33} - 24 q^{34} + 3 q^{36} - 6 q^{37} - 35 q^{39} - 120 q^{40} - 13 q^{42} - 32 q^{45} - 30 q^{46} - 33 q^{48} - 26 q^{49} - 100 q^{51} - 60 q^{52} - 42 q^{55} - 20 q^{57} - 34 q^{58} + 32 q^{60} - 30 q^{61} - 45 q^{63} + 40 q^{64} + 42 q^{66} + 32 q^{67} - 33 q^{69} - 268 q^{70} + 55 q^{72} - 40 q^{73} + 9 q^{75} - 32 q^{78} - 37 q^{81} + 28 q^{82} - 180 q^{84} - 10 q^{85} - 142 q^{88} - 30 q^{90} + 8 q^{91} - 26 q^{93} - 90 q^{94} + 330 q^{96} - 14 q^{97} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.276807 + 2.63364i −1.73182 − 0.0281544i −4.90316 − 1.04220i −1.71868 − 2.36556i 0.553529 − 4.55321i −0.0575585 + 0.270792i 2.46537 − 7.58762i 2.99841 + 0.0975169i 6.70579 − 3.87159i
29.2 −0.274469 + 2.61140i 1.03166 − 1.39129i −4.78776 − 1.01767i 1.02930 + 1.41670i 3.35004 + 3.07594i 0.916640 − 4.31245i 2.34881 − 7.22888i −0.871350 − 2.87067i −3.98208 + 2.29906i
29.3 −0.270097 + 2.56980i 0.0599513 − 1.73101i −4.57463 − 0.972368i −0.821874 − 1.13121i 4.43217 + 0.621604i −0.850348 + 4.00057i 2.13742 − 6.57829i −2.99281 − 0.207553i 3.12898 − 1.80652i
29.4 −0.260719 + 2.48057i 1.72227 + 0.183844i −4.12897 − 0.877640i −2.22258 − 3.05912i −0.905066 + 4.22428i 0.668489 − 3.14500i 1.71203 − 5.26908i 2.93240 + 0.633258i 8.16784 − 4.71570i
29.5 −0.256618 + 2.44156i 1.38266 + 1.04320i −3.93907 − 0.837274i 1.76595 + 2.43063i −2.90184 + 3.10814i −0.156026 + 0.734044i 1.53781 − 4.73290i 0.823485 + 2.88477i −6.38769 + 3.68794i
29.6 −0.244335 + 2.32469i −1.09963 + 1.33821i −3.38819 − 0.720181i 0.672844 + 0.926090i −2.84225 − 2.88327i −0.747086 + 3.51476i 1.05740 − 3.25434i −0.581631 − 2.94308i −2.31727 + 1.33788i
29.7 −0.237841 + 2.26290i −0.837076 − 1.51635i −3.10787 − 0.660597i 0.619671 + 0.852904i 3.63043 − 1.53357i 0.272540 − 1.28220i 0.827791 − 2.54768i −1.59861 + 2.53859i −2.07742 + 1.19940i
29.8 −0.230857 + 2.19646i −1.65359 − 0.515394i −2.81485 − 0.598316i 1.87210 + 2.57672i 1.51379 − 3.51307i −0.269880 + 1.26968i 0.599043 − 1.84366i 2.46874 + 1.70450i −6.09186 + 3.51714i
29.9 −0.219545 + 2.08883i 1.04164 + 1.38383i −2.35871 − 0.501359i −0.372847 − 0.513179i −3.11927 + 1.87199i −0.107574 + 0.506098i 0.267016 − 0.821790i −0.829974 + 2.88291i 1.15380 − 0.666147i
29.10 −0.198720 + 1.89069i −1.52365 + 0.823714i −1.57894 − 0.335614i −0.924919 − 1.27304i −1.25461 − 3.04444i 0.668009 − 3.14273i −0.226639 + 0.697522i 1.64299 − 2.51010i 2.59073 − 1.49576i
29.11 −0.188305 + 1.79160i 1.68236 − 0.411909i −1.21807 − 0.258909i −1.00123 − 1.37808i 0.421180 + 3.09168i −0.457516 + 2.15245i −0.420139 + 1.29305i 2.66066 − 1.38596i 2.65749 − 1.53430i
29.12 −0.179632 + 1.70909i 1.33049 − 1.10896i −0.932422 − 0.198192i 1.96134 + 2.69955i 1.65631 + 2.47313i −0.393158 + 1.84966i −0.555871 + 1.71080i 0.540418 − 2.95092i −4.96609 + 2.86717i
29.13 −0.170529 + 1.62248i −0.280130 + 1.70925i −0.647056 − 0.137536i −2.20891 − 3.04030i −2.72544 − 0.745981i −0.757508 + 3.56379i −0.674778 + 2.07675i −2.84305 − 0.957623i 5.30950 − 3.06544i
29.14 −0.159737 + 1.51979i −1.11400 − 1.32627i −0.327964 − 0.0697108i −0.0308880 − 0.0425136i 2.19361 − 1.48120i 0.593356 − 2.79152i −0.786124 + 2.41944i −0.517989 + 2.95494i 0.0695459 − 0.0401523i
29.15 −0.144098 + 1.37101i 0.143318 + 1.72611i 0.0974043 + 0.0207039i 0.756574 + 1.04133i −2.38716 − 0.0522407i 0.619507 − 2.91455i −0.894416 + 2.75273i −2.95892 + 0.494764i −1.53670 + 0.887212i
29.16 −0.127461 + 1.21271i 0.478976 − 1.66451i 0.501877 + 0.106677i −1.76681 − 2.43180i 1.95751 + 0.793018i 0.706704 − 3.32478i −0.946962 + 2.91445i −2.54116 − 1.59452i 3.17427 − 1.83266i
29.17 −0.123261 + 1.17275i −1.50937 − 0.849596i 0.596149 + 0.126715i −2.37562 − 3.26976i 1.18241 − 1.66539i −0.489968 + 2.30512i −0.950878 + 2.92650i 1.55637 + 2.56470i 4.12743 − 2.38297i
29.18 −0.115554 + 1.09942i 1.72615 − 0.142897i 0.760914 + 0.161737i 1.07486 + 1.47941i −0.0423587 + 1.91428i 0.576184 − 2.71073i −0.948969 + 2.92063i 2.95916 − 0.493323i −1.75071 + 1.01077i
29.19 −0.0983746 + 0.935971i −0.0424277 + 1.73153i 1.08993 + 0.231672i 1.83190 + 2.52139i −1.61649 − 0.210050i −0.116038 + 0.545914i −0.905708 + 2.78748i −2.99640 − 0.146930i −2.54016 + 1.46656i
29.20 −0.0790544 + 0.752152i −1.44283 + 0.958255i 1.39681 + 0.296902i 0.221314 + 0.304612i −0.606692 − 1.16098i 0.0924831 − 0.435099i −0.801155 + 2.46570i 1.16349 − 2.76519i −0.246611 + 0.142381i
See next 80 embeddings (of 416 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
13.c even 3 1 inner
33.f even 10 1 inner
39.i odd 6 1 inner
143.t odd 30 1 inner
429.bq even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bq.a ✓ 416
3.b odd 2 1 inner 429.2.bq.a ✓ 416
11.d odd 10 1 inner 429.2.bq.a ✓ 416
13.c even 3 1 inner 429.2.bq.a ✓ 416
33.f even 10 1 inner 429.2.bq.a ✓ 416
39.i odd 6 1 inner 429.2.bq.a ✓ 416
143.t odd 30 1 inner 429.2.bq.a ✓ 416
429.bq even 30 1 inner 429.2.bq.a ✓ 416
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bq.a ✓ 416 1.a even 1 1 trivial
429.2.bq.a ✓ 416 3.b odd 2 1 inner
429.2.bq.a ✓ 416 11.d odd 10 1 inner
429.2.bq.a ✓ 416 13.c even 3 1 inner
429.2.bq.a ✓ 416 33.f even 10 1 inner
429.2.bq.a ✓ 416 39.i odd 6 1 inner
429.2.bq.a ✓ 416 143.t odd 30 1 inner
429.2.bq.a ✓ 416 429.bq even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(429, [\chi])\).