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Properties

Label 441.3.t.a
Level $441$
Weight $3$
Character orbit 441.t
Analytic conductor $12.016$
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] = [441,3,Mod(166,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, 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("441.166");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.t (of order \(6\), degree \(2\), 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: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 63)
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
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} + 3 q^{3} + 46 q^{4} + 3 q^{5} + 12 q^{6} - 16 q^{8} - 15 q^{9} + 6 q^{10} + 7 q^{11} + 30 q^{12} + 15 q^{13} - 18 q^{15} + 54 q^{16} + 33 q^{17} - 42 q^{18} + 6 q^{19} + 108 q^{20} - 10 q^{22} + 34 q^{23} + 78 q^{24} + 31 q^{25} - 54 q^{26} - 81 q^{27} + 70 q^{29} - 27 q^{30} - 306 q^{32} + 3 q^{33} + 12 q^{34} - 174 q^{36} + 9 q^{37} - 87 q^{38} + 129 q^{39} + 102 q^{40} - 234 q^{41} + 30 q^{43} + 51 q^{44} - 273 q^{45} - 22 q^{46} + 147 q^{48} + 241 q^{50} + 12 q^{51} + 219 q^{52} + 148 q^{53} - 171 q^{54} + 189 q^{57} + 17 q^{58} + 33 q^{60} - 48 q^{64} - 228 q^{65} - 258 q^{66} + 68 q^{67} + 18 q^{68} + 78 q^{69} - 350 q^{71} + 162 q^{72} + 6 q^{73} + 359 q^{74} + 510 q^{75} + 72 q^{76} - 375 q^{78} + 164 q^{79} + 609 q^{80} - 435 q^{81} + 18 q^{82} + 738 q^{83} + 3 q^{85} + 17 q^{86} + 561 q^{87} + 25 q^{88} - 21 q^{89} - 543 q^{90} + 288 q^{92} - 222 q^{93} - 1014 q^{95} - 231 q^{96} - 57 q^{97} - 162 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
166.1 −3.83603 0.610361 − 2.93725i 10.7151 −0.187534 + 0.108273i −2.34136 + 11.2674i 0 −25.7594 −8.25492 − 3.58557i 0.719386 − 0.415338i
166.2 −3.25435 −2.46512 + 1.70973i 6.59082 2.94001 − 1.69741i 8.02237 − 5.56407i 0 −8.43145 3.15364 − 8.42939i −9.56782 + 5.52398i
166.3 −2.83394 2.47969 + 1.68854i 4.03122 2.07720 − 1.19927i −7.02729 − 4.78521i 0 −0.0884848 3.29769 + 8.37408i −5.88667 + 3.39867i
166.4 −1.80456 −2.82582 − 1.00733i −0.743548 −4.98393 + 2.87747i 5.09938 + 1.81779i 0 8.56004 6.97057 + 5.69308i 8.99383 − 5.19259i
166.5 −1.65335 2.65959 − 1.38801i −1.26644 −6.81496 + 3.93462i −4.39723 + 2.29486i 0 8.70726 5.14685 − 7.38308i 11.2675 − 6.50529i
166.6 −1.32480 −1.42175 − 2.64171i −2.24491 6.26581 − 3.61757i 1.88353 + 3.49973i 0 8.27324 −4.95727 + 7.51169i −8.30093 + 4.79254i
166.7 −0.396136 1.19743 + 2.75066i −3.84308 2.57417 − 1.48620i −0.474346 − 1.08964i 0 3.10693 −6.13231 + 6.58747i −1.01972 + 0.588737i
166.8 0.357823 −1.44862 + 2.62707i −3.87196 −3.97509 + 2.29502i −0.518348 + 0.940026i 0 −2.81677 −4.80301 − 7.61125i −1.42238 + 0.821210i
166.9 0.455152 2.48323 − 1.68332i −3.79284 3.78523 − 2.18540i 1.13025 − 0.766164i 0 −3.54692 3.33290 − 8.36013i 1.72285 − 0.994690i
166.10 1.68199 −1.37442 − 2.66664i −1.17091 −2.03050 + 1.17231i −2.31176 − 4.48526i 0 −8.69742 −5.22193 + 7.33017i −3.41529 + 1.97182i
166.11 2.24050 −2.95463 + 0.519775i 1.01982 1.67528 − 0.967222i −6.61983 + 1.16455i 0 −6.67708 8.45967 − 3.07148i 3.75345 − 2.16706i
166.12 2.65681 2.49911 + 1.65965i 3.05866 −7.97090 + 4.60200i 6.63967 + 4.40939i 0 −2.50096 3.49110 + 8.29531i −21.1772 + 12.2267i
166.13 3.35512 1.69559 − 2.47487i 7.25684 0.769575 − 0.444314i 5.68892 − 8.30348i 0 10.9271 −3.24993 − 8.39273i 2.58202 − 1.49073i
166.14 3.35577 0.365354 + 2.97767i 7.26121 7.37564 − 4.25833i 1.22605 + 9.99238i 0 10.9439 −8.73303 + 2.17581i 24.7510 − 14.2900i
178.1 −3.83603 0.610361 + 2.93725i 10.7151 −0.187534 − 0.108273i −2.34136 − 11.2674i 0 −25.7594 −8.25492 + 3.58557i 0.719386 + 0.415338i
178.2 −3.25435 −2.46512 − 1.70973i 6.59082 2.94001 + 1.69741i 8.02237 + 5.56407i 0 −8.43145 3.15364 + 8.42939i −9.56782 − 5.52398i
178.3 −2.83394 2.47969 − 1.68854i 4.03122 2.07720 + 1.19927i −7.02729 + 4.78521i 0 −0.0884848 3.29769 − 8.37408i −5.88667 − 3.39867i
178.4 −1.80456 −2.82582 + 1.00733i −0.743548 −4.98393 − 2.87747i 5.09938 − 1.81779i 0 8.56004 6.97057 − 5.69308i 8.99383 + 5.19259i
178.5 −1.65335 2.65959 + 1.38801i −1.26644 −6.81496 − 3.93462i −4.39723 − 2.29486i 0 8.70726 5.14685 + 7.38308i 11.2675 + 6.50529i
178.6 −1.32480 −1.42175 + 2.64171i −2.24491 6.26581 + 3.61757i 1.88353 − 3.49973i 0 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 166.14
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 441.3.t.a 28
7.b odd 2 1 63.3.t.a yes 28
7.c even 3 1 63.3.k.a ✓ 28
7.c even 3 1 441.3.l.a 28
7.d odd 6 1 441.3.k.b 28
7.d odd 6 1 441.3.l.b 28
9.c even 3 1 441.3.k.b 28
21.c even 2 1 189.3.t.a 28
21.h odd 6 1 189.3.k.a 28
63.g even 3 1 441.3.l.b 28
63.h even 3 1 63.3.t.a yes 28
63.j odd 6 1 189.3.t.a 28
63.k odd 6 1 441.3.l.a 28
63.l odd 6 1 63.3.k.a ✓ 28
63.o even 6 1 189.3.k.a 28
63.t odd 6 1 inner 441.3.t.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.k.a ✓ 28 7.c even 3 1
63.3.k.a ✓ 28 63.l odd 6 1
63.3.t.a yes 28 7.b odd 2 1
63.3.t.a yes 28 63.h even 3 1
189.3.k.a 28 21.h odd 6 1
189.3.k.a 28 63.o even 6 1
189.3.t.a 28 21.c even 2 1
189.3.t.a 28 63.j odd 6 1
441.3.k.b 28 7.d odd 6 1
441.3.k.b 28 9.c even 3 1
441.3.l.a 28 7.c even 3 1
441.3.l.a 28 63.k odd 6 1
441.3.l.b 28 7.d odd 6 1
441.3.l.b 28 63.g even 3 1
441.3.t.a 28 1.a even 1 1 trivial
441.3.t.a 28 63.t odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + T_{2}^{13} - 39 T_{2}^{12} - 34 T_{2}^{11} + 578 T_{2}^{10} + 450 T_{2}^{9} - 4069 T_{2}^{8} + \cdots - 1017 \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display