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Properties

Label 43.3.f.a
Level $43$
Weight $3$
Character orbit 43.f
Analytic conductor $1.172$
Analytic rank $0$
Dimension $42$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [43,3,Mod(2,43)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(43, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 43.f (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: \(1.17166513675\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9} - 5 q^{10} - 24 q^{11} - 35 q^{12} - 34 q^{13} + 69 q^{14} + 7 q^{15} - 39 q^{16} + 22 q^{17} - 70 q^{18} - 49 q^{19} + 133 q^{20} + 77 q^{22} + 42 q^{23} - 349 q^{24} + 10 q^{25} + 49 q^{26} - 7 q^{27} + 105 q^{28} + 63 q^{29} - 252 q^{30} - 152 q^{31} + 343 q^{32} + 329 q^{33} + 161 q^{34} + 58 q^{35} + 576 q^{36} - 289 q^{38} + 77 q^{39} - 101 q^{40} + 133 q^{41} - 79 q^{43} + 148 q^{44} + 84 q^{45} - 504 q^{46} + 6 q^{47} - 595 q^{48} - 302 q^{49} + 161 q^{51} - 267 q^{52} - 394 q^{53} - 227 q^{54} - 637 q^{55} + 355 q^{56} - 7 q^{57} + 165 q^{58} - 46 q^{59} - 657 q^{60} - 175 q^{61} - 91 q^{62} + 511 q^{63} + 725 q^{64} + 161 q^{65} - 227 q^{66} - 756 q^{67} - 586 q^{68} + 441 q^{69} + 1526 q^{70} + 266 q^{71} + 1078 q^{72} - 252 q^{73} + 204 q^{74} + 112 q^{75} + 994 q^{76} + 791 q^{77} + 94 q^{78} - 178 q^{79} - 428 q^{81} + 245 q^{82} + 238 q^{83} + 66 q^{84} + 365 q^{86} + 426 q^{87} - 119 q^{88} + 252 q^{89} - 926 q^{90} - 224 q^{91} - 764 q^{92} + 133 q^{94} + 11 q^{95} - 2602 q^{96} - 491 q^{97} - 553 q^{98} + 431 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.57834 − 3.27745i −1.75180 + 3.63765i −5.75657 + 7.21851i −7.35771 − 1.67935i 14.6871 − 2.47266i 18.5581 + 4.23577i −4.55228 − 5.70838i 6.10896 + 26.7651i
2.2 −1.52706 − 3.17098i 2.15990 − 4.48508i −5.22924 + 6.55726i 5.44765 + 1.24339i −17.5204 9.10137i 15.0532 + 3.43580i −9.83935 − 12.3382i −4.37614 − 19.1731i
2.3 −0.686008 − 1.42451i 0.358272 − 0.743960i 0.935338 − 1.17288i −1.02237 − 0.233349i −1.30556 − 7.62808i −8.47820 − 1.93509i 5.18629 + 6.50340i 0.368944 + 1.61645i
2.4 −0.468493 − 0.972836i −1.93369 + 4.01535i 1.76703 − 2.21579i 8.30688 + 1.89599i 4.81220 6.84462i −7.19423 − 1.64204i −6.77248 − 8.49242i −2.04723 − 8.96949i
2.5 0.311899 + 0.647664i 1.16241 − 2.41377i 2.17177 − 2.72331i −3.49769 − 0.798324i 1.92587 9.27538i 5.24449 + 1.19702i 1.13631 + 1.42489i −0.573878 − 2.51432i
2.6 1.03788 + 2.15517i −1.08703 + 2.25725i −1.07362 + 1.34627i −0.904592 − 0.206467i −5.99296 − 5.06547i 5.31261 + 1.21257i 1.69788 + 2.12908i −0.493882 − 2.16384i
2.7 1.68761 + 3.50435i 2.07233 − 4.30323i −6.93850 + 8.70061i −2.02712 − 0.462677i 18.5773 − 2.90918i −27.0314 − 6.16973i −8.61185 − 10.7989i −1.79960 − 7.88456i
8.1 −3.53859 − 0.807660i 1.86243 − 0.425087i 8.26544 + 3.98042i −5.65332 − 4.50837i −6.93370 − 11.0869i −14.6823 − 11.7087i −4.82078 + 2.32157i 16.3636 + 20.5192i
8.2 −2.63550 − 0.601536i −1.75981 + 0.401664i 2.98015 + 1.43516i 7.00305 + 5.58475i 4.87959 3.35098i 1.46315 + 1.16682i −5.17313 + 2.49125i −15.0971 − 18.9312i
8.3 −0.844666 − 0.192789i 4.86248 − 1.10983i −2.92758 − 1.40985i 0.831553 + 0.663142i −4.32114 − 0.302383i 4.91050 + 3.91600i 14.3033 − 6.88811i −0.574538 − 0.720448i
8.4 −0.811544 − 0.185230i −3.07466 + 0.701771i −2.97958 − 1.43489i −3.71827 − 2.96522i 2.62521 − 0.589128i 4.75551 + 3.79239i 0.852334 − 0.410462i 2.46829 + 3.09514i
8.5 1.71853 + 0.392244i 0.160301 − 0.0365876i −0.804371 − 0.387365i 5.40310 + 4.30882i 0.289833 − 6.65286i −6.74303 − 5.37738i −8.08436 + 3.89322i 7.59529 + 9.52420i
8.6 2.11328 + 0.482341i 1.61544 − 0.368714i 0.629407 + 0.303106i −3.78878 − 3.02145i 3.59172 10.9446i −5.59495 − 4.46183i −5.63502 + 2.71368i −6.54936 − 8.21264i
8.7 3.62198 + 0.826694i −3.87513 + 0.884474i 8.83146 + 4.25301i −2.82431 − 2.25231i −14.7669 − 9.33217i 16.8530 + 13.4399i 6.12565 − 2.94996i −8.36764 − 10.4927i
22.1 −1.57834 + 3.27745i −1.75180 − 3.63765i −5.75657 − 7.21851i −7.35771 + 1.67935i 14.6871 2.47266i 18.5581 − 4.23577i −4.55228 + 5.70838i 6.10896 − 26.7651i
22.2 −1.52706 + 3.17098i 2.15990 + 4.48508i −5.22924 − 6.55726i 5.44765 − 1.24339i −17.5204 − 9.10137i 15.0532 − 3.43580i −9.83935 + 12.3382i −4.37614 + 19.1731i
22.3 −0.686008 + 1.42451i 0.358272 + 0.743960i 0.935338 + 1.17288i −1.02237 + 0.233349i −1.30556 7.62808i −8.47820 + 1.93509i 5.18629 − 6.50340i 0.368944 − 1.61645i
22.4 −0.468493 + 0.972836i −1.93369 − 4.01535i 1.76703 + 2.21579i 8.30688 − 1.89599i 4.81220 − 6.84462i −7.19423 + 1.64204i −6.77248 + 8.49242i −2.04723 + 8.96949i
22.5 0.311899 − 0.647664i 1.16241 + 2.41377i 2.17177 + 2.72331i −3.49769 + 0.798324i 1.92587 − 9.27538i 5.24449 − 1.19702i 1.13631 − 1.42489i −0.573878 + 2.51432i
22.6 1.03788 − 2.15517i −1.08703 − 2.25725i −1.07362 − 1.34627i −0.904592 + 0.206467i −5.99296 5.06547i 5.31261 − 1.21257i 1.69788 − 2.12908i −0.493882 + 2.16384i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.3.f.a ✓ 42
3.b odd 2 1 387.3.w.b 42
43.f odd 14 1 inner 43.3.f.a ✓ 42
129.j even 14 1 387.3.w.b 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.f.a ✓ 42 1.a even 1 1 trivial
43.3.f.a ✓ 42 43.f odd 14 1 inner
387.3.w.b 42 3.b odd 2 1
387.3.w.b 42 129.j even 14 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(43, [\chi])\).