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Properties

Label 441.3.m.j
Level $441$
Weight $3$
Character orbit 441.m
Analytic conductor $12.016$
Analytic rank $0$
Dimension $8$
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(19,441)]
 
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([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.m (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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: no (minimal twist has level 147)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{5} - \beta_{3} - 1) q^{2} + ( - \beta_{7} + 5 \beta_{3} + \cdots + 2 \beta_1) q^{4}+ \cdots + (\beta_{6} - 3 \beta_{5} + \beta_{4} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{5} - \beta_{3} - 1) q^{2} + ( - \beta_{7} + 5 \beta_{3} + \cdots + 2 \beta_1) q^{4}+ \cdots + ( - 28 \beta_{7} - 20 \beta_{6} + \cdots - 20) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 20 q^{4} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 20 q^{4} - 8 q^{8} - 48 q^{10} - 32 q^{11} - 60 q^{16} + 48 q^{17} + 96 q^{19} - 64 q^{22} - 72 q^{23} + 28 q^{25} + 120 q^{26} - 16 q^{29} - 144 q^{31} - 28 q^{32} - 32 q^{37} - 264 q^{38} + 16 q^{43} - 104 q^{44} - 88 q^{46} - 120 q^{47} - 264 q^{50} + 480 q^{52} + 56 q^{53} + 72 q^{58} + 24 q^{59} - 144 q^{61} + 152 q^{64} + 56 q^{65} - 96 q^{67} - 216 q^{68} + 160 q^{71} + 144 q^{73} - 264 q^{74} + 88 q^{79} + 816 q^{80} - 336 q^{82} - 544 q^{85} + 232 q^{86} + 344 q^{88} - 192 q^{89} + 448 q^{92} - 480 q^{94} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 6\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 7\nu^{5} - 14\nu^{3} + 8\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 2 ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} + 2\nu^{6} + 7\nu^{5} - 14\nu^{4} - 28\nu^{3} + 42\nu^{2} + 16\nu - 44 ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{7} + \nu^{6} + 21\nu^{5} - 70\nu^{3} + 74\nu + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{6} + 14\nu^{4} - 42\nu^{2} + 14\nu - 16 ) / 14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} + 4\nu^{6} - 21\nu^{5} - 14\nu^{4} + 70\nu^{3} + 42\nu^{2} + 28\nu - 4 ) / 14 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} - 3\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 4\beta_{6} - 2\beta_{4} - 14\beta_{3} - \beta_{2} + 2\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 10\beta_{2} - 10\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} + 8\beta_{6} + 4\beta_{5} - 16\beta_{4} - 42\beta_{3} - 8\beta_{2} + 4\beta _1 - 42 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{7} - 4\beta_{5} - 2\beta_{4} + 34\beta_{2} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{6} + 2\beta_{5} - 4\beta_{4} - 2\beta_{2} - 2\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -6\beta_{7} - 6\beta_{6} - 6\beta_{5} + 116\beta_1 ) / 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(1 + \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
1.60021 + 0.923880i
−0.662827 − 0.382683i
−1.60021 − 0.923880i
0.662827 + 0.382683i
1.60021 − 0.923880i
−0.662827 + 0.382683i
−1.60021 + 0.923880i
0.662827 − 0.382683i
−1.86993 + 3.23882i 0 −4.99331 − 8.64866i −3.33325 − 1.92445i 0 0 22.3891 0 12.4659 − 7.19720i
19.2 −1.39310 + 2.41292i 0 −1.88145 − 3.25877i 8.24759 + 4.76175i 0 0 −0.660594 0 −22.9794 + 13.2672i
19.3 −0.544280 + 0.942720i 0 1.40752 + 2.43790i −0.909389 − 0.525036i 0 0 −7.41857 0 0.989924 − 0.571533i
19.4 1.80731 − 3.13036i 0 −4.53276 − 7.85097i −4.00495 − 2.31226i 0 0 −18.3100 0 −14.4764 + 8.35796i
325.1 −1.86993 − 3.23882i 0 −4.99331 + 8.64866i −3.33325 + 1.92445i 0 0 22.3891 0 12.4659 + 7.19720i
325.2 −1.39310 − 2.41292i 0 −1.88145 + 3.25877i 8.24759 − 4.76175i 0 0 −0.660594 0 −22.9794 − 13.2672i
325.3 −0.544280 − 0.942720i 0 1.40752 − 2.43790i −0.909389 + 0.525036i 0 0 −7.41857 0 0.989924 + 0.571533i
325.4 1.80731 + 3.13036i 0 −4.53276 + 7.85097i −4.00495 + 2.31226i 0 0 −18.3100 0 −14.4764 − 8.35796i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d 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.m.j 8
3.b odd 2 1 147.3.f.g 8
7.b odd 2 1 441.3.m.k 8
7.c even 3 1 441.3.d.h 8
7.c even 3 1 441.3.m.k 8
7.d odd 6 1 441.3.d.h 8
7.d odd 6 1 inner 441.3.m.j 8
21.c even 2 1 147.3.f.f 8
21.g even 6 1 147.3.d.d ✓ 8
21.g even 6 1 147.3.f.g 8
21.h odd 6 1 147.3.d.d ✓ 8
21.h odd 6 1 147.3.f.f 8
84.j odd 6 1 2352.3.f.l 8
84.n even 6 1 2352.3.f.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.3.d.d ✓ 8 21.g even 6 1
147.3.d.d ✓ 8 21.h odd 6 1
147.3.f.f 8 21.c even 2 1
147.3.f.f 8 21.h odd 6 1
147.3.f.g 8 3.b odd 2 1
147.3.f.g 8 21.g even 6 1
441.3.d.h 8 7.c even 3 1
441.3.d.h 8 7.d odd 6 1
441.3.m.j 8 1.a even 1 1 trivial
441.3.m.j 8 7.d odd 6 1 inner
441.3.m.k 8 7.b odd 2 1
441.3.m.k 8 7.c even 3 1
2352.3.f.l 8 84.j odd 6 1
2352.3.f.l 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{8} + 4T_{2}^{7} + 26T_{2}^{6} + 64T_{2}^{5} + 349T_{2}^{4} + 848T_{2}^{3} + 2294T_{2}^{2} + 2132T_{2} + 1681 \) Copy content Toggle raw display
\( T_{5}^{8} - 64T_{5}^{6} + 4274T_{5}^{4} + 26112T_{5}^{3} + 66880T_{5}^{2} + 72624T_{5} + 31684 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} + \cdots + 1681 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 64 T^{6} + \cdots + 31684 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 32 T^{7} + \cdots + 868624 \) Copy content Toggle raw display
$13$ \( T^{8} + 904 T^{6} + \cdots + 10640644 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 1039546564 \) Copy content Toggle raw display
$19$ \( T^{8} - 96 T^{7} + \cdots + 85082176 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 4519603984 \) Copy content Toggle raw display
$29$ \( (T^{4} + 8 T^{3} + \cdots + 301468)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 144 T^{7} + \cdots + 851238976 \) Copy content Toggle raw display
$37$ \( T^{8} + 32 T^{7} + \cdots + 146216464 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 1143056063044 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + \cdots + 1217296)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 36741733758016 \) Copy content Toggle raw display
$53$ \( T^{8} - 56 T^{7} + \cdots + 29073664 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 68212902976 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 10776263729284 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 567936769867776 \) Copy content Toggle raw display
$71$ \( (T^{4} - 80 T^{3} + \cdots + 1658524)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 25117918721284 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 7398574081024 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 7946907588676 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
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