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Properties

Label 63.3.t.a
Level $63$
Weight $3$
Character orbit 63.t
Analytic conductor $1.717$
Analytic rank $0$
Dimension $28$
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] = [63,3,Mod(40,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.40");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 63.t (of order \(6\), degree \(2\), 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: \(1.71662566547\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} - 3 q^{3} + 46 q^{4} - 3 q^{5} - 12 q^{6} - 16 q^{8} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} - 3 q^{3} + 46 q^{4} - 3 q^{5} - 12 q^{6} - 16 q^{8} - 15 q^{9} - 6 q^{10} + 7 q^{11} - 30 q^{12} - 15 q^{13} + 10 q^{14} - 18 q^{15} + 54 q^{16} - 33 q^{17} - 42 q^{18} - 6 q^{19} - 108 q^{20} + 21 q^{21} - 10 q^{22} + 34 q^{23} - 78 q^{24} + 31 q^{25} + 54 q^{26} + 81 q^{27} - 16 q^{28} + 70 q^{29} - 27 q^{30} - 306 q^{32} - 3 q^{33} - 12 q^{34} + 18 q^{35} - 174 q^{36} + 9 q^{37} + 87 q^{38} + 129 q^{39} - 102 q^{40} + 234 q^{41} + 306 q^{42} + 30 q^{43} + 51 q^{44} + 273 q^{45} - 22 q^{46} - 147 q^{48} - 38 q^{49} + 241 q^{50} + 12 q^{51} - 219 q^{52} + 148 q^{53} + 171 q^{54} + 110 q^{56} + 189 q^{57} + 17 q^{58} + 33 q^{60} - 471 q^{63} - 48 q^{64} - 228 q^{65} + 258 q^{66} + 68 q^{67} - 18 q^{68} - 78 q^{69} - 225 q^{70} - 350 q^{71} + 162 q^{72} - 6 q^{73} + 359 q^{74} - 510 q^{75} - 72 q^{76} - 224 q^{77} - 375 q^{78} + 164 q^{79} - 609 q^{80} - 435 q^{81} - 18 q^{82} - 738 q^{83} - 21 q^{84} + 3 q^{85} + 17 q^{86} - 561 q^{87} + 25 q^{88} + 21 q^{89} + 543 q^{90} + 39 q^{91} + 288 q^{92} - 222 q^{93} - 1014 q^{95} + 231 q^{96} + 57 q^{97} + 811 q^{98} - 162 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −3.83603 −0.610361 + 2.93725i 10.7151 0.187534 − 0.108273i 2.34136 − 11.2674i −5.01978 + 4.87871i −25.7594 −8.25492 − 3.58557i −0.719386 + 0.415338i
40.2 −3.25435 2.46512 − 1.70973i 6.59082 −2.94001 + 1.69741i −8.02237 + 5.56407i 2.50562 − 6.53620i −8.43145 3.15364 − 8.42939i 9.56782 − 5.52398i
40.3 −2.83394 −2.47969 − 1.68854i 4.03122 −2.07720 + 1.19927i 7.02729 + 4.78521i 6.05681 + 3.50928i −0.0884848 3.29769 + 8.37408i 5.88667 − 3.39867i
40.4 −1.80456 2.82582 + 1.00733i −0.743548 4.98393 − 2.87747i −5.09938 − 1.81779i −0.851442 + 6.94802i 8.56004 6.97057 + 5.69308i −8.99383 + 5.19259i
40.5 −1.65335 −2.65959 + 1.38801i −1.26644 6.81496 − 3.93462i 4.39723 − 2.29486i −0.460386 − 6.98484i 8.70726 5.14685 − 7.38308i −11.2675 + 6.50529i
40.6 −1.32480 1.42175 + 2.64171i −2.24491 −6.26581 + 3.61757i −1.88353 − 3.49973i −3.07531 − 6.28828i 8.27324 −4.95727 + 7.51169i 8.30093 − 4.79254i
40.7 −0.396136 −1.19743 − 2.75066i −3.84308 −2.57417 + 1.48620i 0.474346 + 1.08964i −6.97569 − 0.582933i 3.10693 −6.13231 + 6.58747i 1.01972 − 0.588737i
40.8 0.357823 1.44862 − 2.62707i −3.87196 3.97509 − 2.29502i 0.518348 − 0.940026i 6.10412 − 3.42632i −2.81677 −4.80301 − 7.61125i 1.42238 − 0.821210i
40.9 0.455152 −2.48323 + 1.68332i −3.79284 −3.78523 + 2.18540i −1.13025 + 0.766164i 1.42833 + 6.85273i −3.54692 3.33290 − 8.36013i −1.72285 + 0.994690i
40.10 1.68199 1.37442 + 2.66664i −1.17091 2.03050 − 1.17231i 2.31176 + 4.48526i 6.98121 − 0.512524i −8.69742 −5.22193 + 7.33017i 3.41529 − 1.97182i
40.11 2.24050 2.95463 − 0.519775i 1.01982 −1.67528 + 0.967222i 6.61983 − 1.16455i −6.98642 + 0.435817i −6.67708 8.45967 − 3.07148i −3.75345 + 2.16706i
40.12 2.65681 −2.49911 − 1.65965i 3.05866 7.97090 − 4.60200i −6.63967 − 4.40939i −2.88812 + 6.37642i −2.50096 3.49110 + 8.29531i 21.1772 − 12.2267i
40.13 3.35512 −1.69559 + 2.47487i 7.25684 −0.769575 + 0.444314i −5.68892 + 8.30348i −3.70266 − 5.94056i 10.9271 −3.24993 − 8.39273i −2.58202 + 1.49073i
40.14 3.35577 −0.365354 − 2.97767i 7.26121 −7.37564 + 4.25833i −1.22605 − 9.99238i 6.88370 + 1.27067i 10.9439 −8.73303 + 2.17581i −24.7510 + 14.2900i
52.1 −3.83603 −0.610361 − 2.93725i 10.7151 0.187534 + 0.108273i 2.34136 + 11.2674i −5.01978 − 4.87871i −25.7594 −8.25492 + 3.58557i −0.719386 − 0.415338i
52.2 −3.25435 2.46512 + 1.70973i 6.59082 −2.94001 − 1.69741i −8.02237 − 5.56407i 2.50562 + 6.53620i −8.43145 3.15364 + 8.42939i 9.56782 + 5.52398i
52.3 −2.83394 −2.47969 + 1.68854i 4.03122 −2.07720 − 1.19927i 7.02729 − 4.78521i 6.05681 − 3.50928i −0.0884848 3.29769 − 8.37408i 5.88667 + 3.39867i
52.4 −1.80456 2.82582 − 1.00733i −0.743548 4.98393 + 2.87747i −5.09938 + 1.81779i −0.851442 − 6.94802i 8.56004 6.97057 − 5.69308i −8.99383 − 5.19259i
52.5 −1.65335 −2.65959 − 1.38801i −1.26644 6.81496 + 3.93462i 4.39723 + 2.29486i −0.460386 + 6.98484i 8.70726 5.14685 + 7.38308i −11.2675 − 6.50529i
52.6 −1.32480 1.42175 − 2.64171i −2.24491 −6.26581 − 3.61757i −1.88353 + 3.49973i −3.07531 + 6.28828i 8.27324 −4.95727 − 7.51169i 8.30093 + 4.79254i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.3.t.a yes 28
3.b odd 2 1 189.3.t.a 28
7.b odd 2 1 441.3.t.a 28
7.c even 3 1 441.3.k.b 28
7.c even 3 1 441.3.l.b 28
7.d odd 6 1 63.3.k.a ✓ 28
7.d odd 6 1 441.3.l.a 28
9.c even 3 1 63.3.k.a ✓ 28
9.d odd 6 1 189.3.k.a 28
21.g even 6 1 189.3.k.a 28
63.g even 3 1 441.3.l.a 28
63.h even 3 1 441.3.t.a 28
63.i even 6 1 189.3.t.a 28
63.k odd 6 1 441.3.l.b 28
63.l odd 6 1 441.3.k.b 28
63.t odd 6 1 inner 63.3.t.a yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.k.a ✓ 28 7.d odd 6 1
63.3.k.a ✓ 28 9.c even 3 1
63.3.t.a yes 28 1.a even 1 1 trivial
63.3.t.a yes 28 63.t odd 6 1 inner
189.3.k.a 28 9.d odd 6 1
189.3.k.a 28 21.g even 6 1
189.3.t.a 28 3.b odd 2 1
189.3.t.a 28 63.i even 6 1
441.3.k.b 28 7.c even 3 1
441.3.k.b 28 63.l odd 6 1
441.3.l.a 28 7.d odd 6 1
441.3.l.a 28 63.g even 3 1
441.3.l.b 28 7.c even 3 1
441.3.l.b 28 63.k odd 6 1
441.3.t.a 28 7.b odd 2 1
441.3.t.a 28 63.h even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(63, [\chi])\).