Properties

Label 441.3.v.c
Level $441$
Weight $3$
Character orbit 441.v
Analytic conductor $12.016$
Analytic rank $0$
Dimension $108$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,3,Mod(55,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.55");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.v (of order \(14\), degree \(6\), 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: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{14})\)
Twist minimal: no (minimal twist has level 147)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 32 q^{4} - 2 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - 32 q^{4} - 2 q^{7} - 12 q^{8} - 66 q^{11} + 2 q^{14} - 96 q^{16} + 98 q^{17} + 112 q^{20} - 116 q^{22} - 64 q^{23} + 130 q^{25} - 224 q^{26} - 204 q^{28} - 72 q^{29} + 220 q^{32} + 784 q^{34} + 376 q^{35} + 156 q^{37} + 280 q^{38} - 728 q^{40} + 196 q^{41} - 56 q^{43} + 840 q^{44} - 16 q^{46} - 266 q^{47} + 122 q^{49} + 244 q^{50} + 168 q^{52} - 148 q^{53} - 252 q^{55} - 686 q^{56} + 252 q^{58} - 700 q^{59} - 112 q^{61} - 392 q^{62} + 496 q^{64} + 12 q^{65} - 196 q^{67} + 898 q^{70} + 732 q^{71} + 126 q^{73} + 508 q^{74} - 210 q^{76} + 230 q^{77} + 136 q^{79} - 1960 q^{82} + 574 q^{83} - 480 q^{85} + 392 q^{86} - 108 q^{88} + 742 q^{89} + 152 q^{91} + 42 q^{92} + 98 q^{94} - 68 q^{95} + 508 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −2.34463 + 2.94007i 0 −2.25666 9.88706i −1.32428 + 2.74990i 0 −6.90317 + 1.16027i 20.8074 + 10.0203i 0 −4.97996 10.3410i
55.2 −2.15774 + 2.70572i 0 −1.77499 7.77675i −1.41126 + 2.93051i 0 1.51875 + 6.83326i 12.3996 + 5.97132i 0 −4.88401 10.1418i
55.3 −1.90714 + 2.39148i 0 −1.19191 5.22209i 0.609499 1.26564i 0 −1.84331 6.75294i 3.73807 + 1.80016i 0 1.86435 + 3.87136i
55.4 −1.68215 + 2.10935i 0 −0.729644 3.19678i −1.89988 + 3.94513i 0 5.36283 4.49889i −1.75261 0.844013i 0 −5.12579 10.6438i
55.5 −1.45374 + 1.82294i 0 −0.319642 1.40044i 2.35105 4.88201i 0 1.00834 + 6.92699i −5.38528 2.59341i 0 5.48176 + 11.3830i
55.6 −0.988314 + 1.23931i 0 0.330967 + 1.45006i −1.75892 + 3.65243i 0 5.32542 + 4.54311i −7.83679 3.77400i 0 −2.78812 5.78959i
55.7 −0.622826 + 0.780999i 0 0.668036 + 2.92686i 3.23603 6.71968i 0 6.96696 0.679280i −6.30199 3.03488i 0 3.23258 + 6.71253i
55.8 −0.310648 + 0.389540i 0 0.834844 + 3.65769i −3.33592 + 6.92711i 0 −5.25652 4.62266i −3.47976 1.67576i 0 −1.66209 3.45137i
55.9 −0.192098 + 0.240883i 0 0.868961 + 3.80717i −3.14124 + 6.52285i 0 2.01189 + 6.70465i −2.19437 1.05675i 0 −0.967820 2.00970i
55.10 −0.141675 + 0.177654i 0 0.878594 + 3.84937i 1.28346 2.66513i 0 −6.17180 3.30286i −1.62724 0.783636i 0 0.291639 + 0.605594i
55.11 0.573681 0.719373i 0 0.701696 + 3.07433i 1.16880 2.42703i 0 −4.29685 + 5.52604i 5.93011 + 2.85579i 0 −1.07542 2.23314i
55.12 0.903917 1.13348i 0 0.422382 + 1.85058i 3.47572 7.21742i 0 6.98724 + 0.422439i 7.70417 + 3.71013i 0 −5.03900 10.4636i
55.13 1.18389 1.48455i 0 0.0877927 + 0.384645i −3.80801 + 7.90742i 0 1.93330 6.72773i 7.51802 + 3.62049i 0 7.23068 + 15.0147i
55.14 1.33639 1.67578i 0 −0.132215 0.579271i −0.288815 + 0.599731i 0 −6.52547 + 2.53342i 6.57713 + 3.16738i 0 0.619047 + 1.28546i
55.15 1.50073 1.88185i 0 −0.399101 1.74858i −0.465335 + 0.966277i 0 6.93265 0.968726i 4.78495 + 2.30431i 0 1.12005 + 2.32581i
55.16 1.61994 2.03134i 0 −0.612049 2.68156i −0.534134 + 1.10914i 0 −4.39676 5.44688i 2.92487 + 1.40855i 0 1.38777 + 2.88174i
55.17 2.29233 2.87449i 0 −2.11783 9.27881i 3.51947 7.30825i 0 2.61430 + 6.49349i −18.2765 8.80151i 0 −12.9397 26.8695i
55.18 2.39011 2.99710i 0 −2.37991 10.4270i 0.631746 1.31183i 0 −0.971914 6.93220i −23.1239 11.1359i 0 −2.42176 5.02883i
118.1 −3.30424 1.59124i 0 5.89202 + 7.38836i 7.61062 1.73707i 0 4.61863 + 5.26006i −4.44770 19.4866i 0 −27.9114 6.37060i
118.2 −3.08824 1.48722i 0 4.83146 + 6.05846i 0.636277 0.145226i 0 −6.79347 + 1.68784i −2.85954 12.5284i 0 −2.18096 0.497790i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.f odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.v.c 108
3.b odd 2 1 147.3.j.a 108
49.f odd 14 1 inner 441.3.v.c 108
147.k even 14 1 147.3.j.a 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.3.j.a 108 3.b odd 2 1
147.3.j.a 108 147.k even 14 1
441.3.v.c 108 1.a even 1 1 trivial
441.3.v.c 108 49.f odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{108} + 52 T_{2}^{106} + 4 T_{2}^{105} + 1616 T_{2}^{104} + 68 T_{2}^{103} + 38576 T_{2}^{102} - 776 T_{2}^{101} + 779621 T_{2}^{100} - 85794 T_{2}^{99} + 14214595 T_{2}^{98} - 2738404 T_{2}^{97} + \cdots + 32\!\cdots\!69 \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display