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Properties

Label 441.3.k.a
Level $441$
Weight $3$
Character orbit 441.k
Analytic conductor $12.016$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,3,Mod(31,441)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.k (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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} - 26 q^{4} + 8 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} - 26 q^{4} + 8 q^{8} - 12 q^{9} - 8 q^{11} - 54 q^{15} - 42 q^{16} - 138 q^{18} + 14 q^{22} - 8 q^{23} - 56 q^{25} - 38 q^{29} - 294 q^{30} - 168 q^{32} + 234 q^{36} - 18 q^{37} + 84 q^{39} - 66 q^{43} - 54 q^{44} + 20 q^{46} + 196 q^{50} + 318 q^{51} - 260 q^{53} - 198 q^{57} + 68 q^{58} + 366 q^{60} + 72 q^{64} - 102 q^{65} + 68 q^{67} - 332 q^{71} + 714 q^{72} + 1232 q^{74} - 168 q^{78} + 146 q^{79} - 516 q^{81} + 78 q^{85} + 680 q^{86} + 148 q^{88} + 606 q^{92} - 1146 q^{93} - 360 q^{95} + 900 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.85100 + 3.20603i −2.30657 − 1.91826i −4.85242 − 8.40464i − 5.48291i 10.4195 − 3.84424i 0 21.1194 1.64056 + 8.84921i 17.5784 + 10.1489i
31.2 −1.85100 + 3.20603i 2.30657 + 1.91826i −4.85242 − 8.40464i 5.48291i −10.4195 + 3.84424i 0 21.1194 1.64056 + 8.84921i −17.5784 − 10.1489i
31.3 −1.06935 + 1.85216i −2.45962 + 1.71764i −0.287009 − 0.497114i − 1.60844i −0.551161 − 6.39237i 0 −7.32713 3.09944 − 8.44947i 2.97910 + 1.71998i
31.4 −1.06935 + 1.85216i 2.45962 − 1.71764i −0.287009 − 0.497114i 1.60844i 0.551161 + 6.39237i 0 −7.32713 3.09944 − 8.44947i −2.97910 − 1.71998i
31.5 −0.952438 + 1.64967i −1.69506 − 2.47523i 0.185723 + 0.321682i 8.23162i 5.69776 − 0.438798i 0 −8.32706 −3.25351 + 8.39135i −13.5795 − 7.84010i
31.6 −0.952438 + 1.64967i 1.69506 + 2.47523i 0.185723 + 0.321682i − 8.23162i −5.69776 + 0.438798i 0 −8.32706 −3.25351 + 8.39135i 13.5795 + 7.84010i
31.7 −0.0644125 + 0.111566i −0.0616869 + 2.99937i 1.99170 + 3.44973i 6.63133i −0.330653 − 0.200079i 0 −1.02846 −8.99239 − 0.370043i −0.739829 − 0.427141i
31.8 −0.0644125 + 0.111566i 0.0616869 − 2.99937i 1.99170 + 3.44973i − 6.63133i 0.330653 + 0.200079i 0 −1.02846 −8.99239 − 0.370043i 0.739829 + 0.427141i
31.9 0.435704 − 0.754662i −2.98422 + 0.307314i 1.62032 + 2.80648i − 2.19279i −1.06832 + 2.38597i 0 6.30956 8.81112 − 1.83419i −1.65481 − 0.955407i
31.10 0.435704 − 0.754662i 2.98422 − 0.307314i 1.62032 + 2.80648i 2.19279i 1.06832 − 2.38597i 0 6.30956 8.81112 − 1.83419i 1.65481 + 0.955407i
31.11 1.30753 − 2.26471i −2.30352 − 1.92193i −1.41927 − 2.45825i 3.68512i −7.36452 + 2.70382i 0 3.03728 1.61239 + 8.85439i 8.34571 + 4.81840i
31.12 1.30753 − 2.26471i 2.30352 + 1.92193i −1.41927 − 2.45825i − 3.68512i 7.36452 − 2.70382i 0 3.03728 1.61239 + 8.85439i −8.34571 − 4.81840i
31.13 1.69397 − 2.93404i −1.24145 + 2.73108i −3.73905 − 6.47622i 5.12135i 5.91012 + 8.26882i 0 −11.7836 −5.91761 − 6.78100i 15.0262 + 8.67539i
31.14 1.69397 − 2.93404i 1.24145 − 2.73108i −3.73905 − 6.47622i − 5.12135i −5.91012 − 8.26882i 0 −11.7836 −5.91761 − 6.78100i −15.0262 − 8.67539i
313.1 −1.85100 − 3.20603i −2.30657 + 1.91826i −4.85242 + 8.40464i 5.48291i 10.4195 + 3.84424i 0 21.1194 1.64056 − 8.84921i 17.5784 − 10.1489i
313.2 −1.85100 − 3.20603i 2.30657 − 1.91826i −4.85242 + 8.40464i − 5.48291i −10.4195 − 3.84424i 0 21.1194 1.64056 − 8.84921i −17.5784 + 10.1489i
313.3 −1.06935 − 1.85216i −2.45962 − 1.71764i −0.287009 + 0.497114i 1.60844i −0.551161 + 6.39237i 0 −7.32713 3.09944 + 8.44947i 2.97910 − 1.71998i
313.4 −1.06935 − 1.85216i 2.45962 + 1.71764i −0.287009 + 0.497114i − 1.60844i 0.551161 − 6.39237i 0 −7.32713 3.09944 + 8.44947i −2.97910 + 1.71998i
313.5 −0.952438 − 1.64967i −1.69506 + 2.47523i 0.185723 − 0.321682i − 8.23162i 5.69776 + 0.438798i 0 −8.32706 −3.25351 − 8.39135i −13.5795 + 7.84010i
313.6 −0.952438 − 1.64967i 1.69506 − 2.47523i 0.185723 − 0.321682i 8.23162i −5.69776 − 0.438798i 0 −8.32706 −3.25351 − 8.39135i 13.5795 − 7.84010i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.g even 3 1 inner
63.k 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.k.a 28
7.b odd 2 1 inner 441.3.k.a 28
7.c even 3 1 63.3.l.a ✓ 28
7.c even 3 1 441.3.t.b 28
7.d odd 6 1 63.3.l.a ✓ 28
7.d odd 6 1 441.3.t.b 28
9.c even 3 1 441.3.t.b 28
21.g even 6 1 189.3.l.a 28
21.h odd 6 1 189.3.l.a 28
63.g even 3 1 inner 441.3.k.a 28
63.g even 3 1 567.3.d.h 14
63.h even 3 1 63.3.l.a ✓ 28
63.i even 6 1 189.3.l.a 28
63.j odd 6 1 189.3.l.a 28
63.k odd 6 1 inner 441.3.k.a 28
63.k odd 6 1 567.3.d.h 14
63.l odd 6 1 441.3.t.b 28
63.n odd 6 1 567.3.d.g 14
63.s even 6 1 567.3.d.g 14
63.t odd 6 1 63.3.l.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.l.a ✓ 28 7.c even 3 1
63.3.l.a ✓ 28 7.d odd 6 1
63.3.l.a ✓ 28 63.h even 3 1
63.3.l.a ✓ 28 63.t odd 6 1
189.3.l.a 28 21.g even 6 1
189.3.l.a 28 21.h odd 6 1
189.3.l.a 28 63.i even 6 1
189.3.l.a 28 63.j odd 6 1
441.3.k.a 28 1.a even 1 1 trivial
441.3.k.a 28 7.b odd 2 1 inner
441.3.k.a 28 63.g even 3 1 inner
441.3.k.a 28 63.k odd 6 1 inner
441.3.t.b 28 7.c even 3 1
441.3.t.b 28 7.d odd 6 1
441.3.t.b 28 9.c even 3 1
441.3.t.b 28 63.l odd 6 1
567.3.d.g 14 63.n odd 6 1
567.3.d.g 14 63.s even 6 1
567.3.d.h 14 63.g even 3 1
567.3.d.h 14 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + T_{2}^{13} + 21 T_{2}^{12} + 14 T_{2}^{11} + 314 T_{2}^{10} + 213 T_{2}^{9} + 2054 T_{2}^{8} + \cdots + 225 \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display