Properties

Label 441.2.f.c
Level $441$
Weight $2$
Character orbit 441.f
Analytic conductor $3.521$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,2,Mod(148,441)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.148");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.f (of order \(3\), 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_1 - 1) q^{2} + ( - \beta_{5} + \beta_{3}) q^{3} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + ( - 2 \beta_{5} + \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_1 - 1) q^{2} + ( - \beta_{5} + \beta_{3}) q^{3} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + (3 \beta_{5} - 6 \beta_{4} - 3 \beta_{3} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} + 3 q^{5} + 9 q^{6} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{4} + 3 q^{5} + 9 q^{6} + 12 q^{8} - 6 q^{11} - 18 q^{12} - 3 q^{13} - 9 q^{15} - 3 q^{16} - 12 q^{17} + 9 q^{18} + 6 q^{19} - 6 q^{20} - 9 q^{22} - 12 q^{23} + 27 q^{24} + 6 q^{25} + 6 q^{26} - 27 q^{27} - 9 q^{29} + 18 q^{30} - 3 q^{31} + 9 q^{34} - 27 q^{36} - 6 q^{37} + 6 q^{38} - 18 q^{39} - 9 q^{40} + 3 q^{43} + 30 q^{44} - 9 q^{45} + 3 q^{47} + 6 q^{50} + 9 q^{51} - 21 q^{52} + 12 q^{53} + 27 q^{54} - 9 q^{57} + 9 q^{58} - 3 q^{59} + 6 q^{61} + 60 q^{62} + 24 q^{64} - 15 q^{65} - 36 q^{66} + 12 q^{67} + 6 q^{68} + 9 q^{69} + 18 q^{71} - 36 q^{72} + 42 q^{73} + 30 q^{74} + 9 q^{75} + 15 q^{76} + 54 q^{78} + 21 q^{79} + 30 q^{80} + 18 q^{82} - 18 q^{83} - 9 q^{85} - 6 q^{86} - 9 q^{87} - 27 q^{88} - 24 q^{89} - 27 q^{90} - 3 q^{92} - 27 q^{93} - 18 q^{94} + 12 q^{95} - 27 q^{96} - 3 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{18}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{18}^{5} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{18}^{5} + \zeta_{18}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \) Copy content Toggle raw display
\(\zeta_{18}\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{2}\)\(=\) \( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{18}^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{5}\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1
−0.766044 0.642788i
−0.173648 + 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.173648 0.984808i
0.939693 + 0.342020i
−1.26604 2.19285i 1.11334 + 1.32683i −2.20574 + 3.82045i −0.439693 + 0.761570i 1.50000 4.12122i 0 6.10607 −0.520945 + 2.95442i 2.22668
148.2 −0.673648 1.16679i −1.70574 + 0.300767i 0.0923963 0.160035i 1.26604 2.19285i 1.50000 + 1.78763i 0 −2.94356 2.81908 1.02606i −3.41147
148.3 0.439693 + 0.761570i 0.592396 1.62760i 0.613341 1.06234i 0.673648 1.16679i 1.50000 0.264490i 0 2.83750 −2.29813 1.92836i 1.18479
295.1 −1.26604 + 2.19285i 1.11334 1.32683i −2.20574 3.82045i −0.439693 0.761570i 1.50000 + 4.12122i 0 6.10607 −0.520945 2.95442i 2.22668
295.2 −0.673648 + 1.16679i −1.70574 0.300767i 0.0923963 + 0.160035i 1.26604 + 2.19285i 1.50000 1.78763i 0 −2.94356 2.81908 + 1.02606i −3.41147
295.3 0.439693 0.761570i 0.592396 + 1.62760i 0.613341 + 1.06234i 0.673648 + 1.16679i 1.50000 + 0.264490i 0 2.83750 −2.29813 + 1.92836i 1.18479
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 148.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.f.c 6
3.b odd 2 1 1323.2.f.d 6
7.b odd 2 1 63.2.f.a 6
7.c even 3 1 441.2.g.b 6
7.c even 3 1 441.2.h.e 6
7.d odd 6 1 441.2.g.c 6
7.d odd 6 1 441.2.h.d 6
9.c even 3 1 inner 441.2.f.c 6
9.c even 3 1 3969.2.a.q 3
9.d odd 6 1 1323.2.f.d 6
9.d odd 6 1 3969.2.a.l 3
21.c even 2 1 189.2.f.b 6
21.g even 6 1 1323.2.g.d 6
21.g even 6 1 1323.2.h.c 6
21.h odd 6 1 1323.2.g.e 6
21.h odd 6 1 1323.2.h.b 6
28.d even 2 1 1008.2.r.h 6
63.g even 3 1 441.2.h.e 6
63.h even 3 1 441.2.g.b 6
63.i even 6 1 1323.2.g.d 6
63.j odd 6 1 1323.2.g.e 6
63.k odd 6 1 441.2.h.d 6
63.l odd 6 1 63.2.f.a 6
63.l odd 6 1 567.2.a.h 3
63.n odd 6 1 1323.2.h.b 6
63.o even 6 1 189.2.f.b 6
63.o even 6 1 567.2.a.c 3
63.s even 6 1 1323.2.h.c 6
63.t odd 6 1 441.2.g.c 6
84.h odd 2 1 3024.2.r.k 6
252.s odd 6 1 3024.2.r.k 6
252.s odd 6 1 9072.2.a.bs 3
252.bi even 6 1 1008.2.r.h 6
252.bi even 6 1 9072.2.a.ca 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 7.b odd 2 1
63.2.f.a 6 63.l odd 6 1
189.2.f.b 6 21.c even 2 1
189.2.f.b 6 63.o even 6 1
441.2.f.c 6 1.a even 1 1 trivial
441.2.f.c 6 9.c even 3 1 inner
441.2.g.b 6 7.c even 3 1
441.2.g.b 6 63.h even 3 1
441.2.g.c 6 7.d odd 6 1
441.2.g.c 6 63.t odd 6 1
441.2.h.d 6 7.d odd 6 1
441.2.h.d 6 63.k odd 6 1
441.2.h.e 6 7.c even 3 1
441.2.h.e 6 63.g even 3 1
567.2.a.c 3 63.o even 6 1
567.2.a.h 3 63.l odd 6 1
1008.2.r.h 6 28.d even 2 1
1008.2.r.h 6 252.bi even 6 1
1323.2.f.d 6 3.b odd 2 1
1323.2.f.d 6 9.d odd 6 1
1323.2.g.d 6 21.g even 6 1
1323.2.g.d 6 63.i even 6 1
1323.2.g.e 6 21.h odd 6 1
1323.2.g.e 6 63.j odd 6 1
1323.2.h.b 6 21.h odd 6 1
1323.2.h.b 6 63.n odd 6 1
1323.2.h.c 6 21.g even 6 1
1323.2.h.c 6 63.s even 6 1
3024.2.r.k 6 84.h odd 2 1
3024.2.r.k 6 252.s odd 6 1
3969.2.a.l 3 9.d odd 6 1
3969.2.a.q 3 9.c even 3 1
9072.2.a.bs 3 252.s odd 6 1
9072.2.a.ca 3 252.bi even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{6} + 3T_{2}^{5} + 9T_{2}^{4} + 6T_{2}^{3} + 9T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{6} - 3T_{5}^{5} + 9T_{5}^{4} - 6T_{5}^{3} + 9T_{5}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} + 9T^{3} + 27 \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + \cdots + 11449 \) Copy content Toggle raw display
$17$ \( (T^{3} + 6 T^{2} + 9 T + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 3 T^{2} - 6 T + 17)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{6} + 9 T^{5} + \cdots + 110889 \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} + \cdots + 104329 \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} + \cdots - 323)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 9 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$53$ \( (T^{3} - 6 T^{2} - 9 T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 3 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$67$ \( T^{6} - 12 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$71$ \( (T^{3} - 9 T^{2} - 54 T - 27)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 21 T^{2} + \cdots + 269)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 21 T^{5} + \cdots + 32761 \) Copy content Toggle raw display
$83$ \( T^{6} + 18 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$89$ \( (T^{3} + 12 T^{2} + \cdots - 813)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} + \cdots + 104329 \) Copy content Toggle raw display
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