Properties

Label 441.3.j.d
Level $441$
Weight $3$
Character orbit 441.j
Analytic conductor $12.016$
Analytic rank $0$
Dimension $8$
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(263,441)]
 
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([1, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.263");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.j (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: 8.0.3317760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_1) q^{2} + (\beta_{7} - \beta_{6} - \beta_{3}) q^{3} - q^{4} + (\beta_{6} - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{3} - 3 \beta_{2}) q^{6} + (3 \beta_{5} - 3 \beta_1) q^{8} + ( - 3 \beta_{5} + 6 \beta_{4} - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_1) q^{2} + (\beta_{7} - \beta_{6} - \beta_{3}) q^{3} - q^{4} + (\beta_{6} - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{3} - 3 \beta_{2}) q^{6} + (3 \beta_{5} - 3 \beta_1) q^{8} + ( - 3 \beta_{5} + 6 \beta_{4} - 6) q^{9} + (4 \beta_{7} + \beta_{6} + \beta_{3} + 2 \beta_{2}) q^{10} + ( - 4 \beta_{5} + 6 \beta_{4} + 6) q^{11} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{12} + (\beta_{7} + 2 \beta_{6} - 3 \beta_{2}) q^{13} + (6 \beta_{4} - 3 \beta_1 + 9) q^{15} - 19 q^{16} + ( - 2 \beta_{6} + 2 \beta_{3} + 8 \beta_{2}) q^{17} + ( - 6 \beta_{5} - 15 \beta_{4} + 15) q^{18} + (5 \beta_{7} - 5 \beta_{6}) q^{19} + ( - \beta_{6} + \beta_{3} + \beta_{2}) q^{20} + (6 \beta_{5} - 20 \beta_{4} - 12 \beta_1 + 20) q^{22} + (15 \beta_{4} - 4 \beta_1 - 30) q^{23} + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{3} - 9 \beta_{2}) q^{24} + (3 \beta_{5} + \beta_{4} - 6 \beta_1 - 1) q^{25} + ( - \beta_{7} + 4 \beta_{6} + 8 \beta_{3} + 3 \beta_{2}) q^{26} + ( - 12 \beta_{7} + 3 \beta_{3}) q^{27} - 4 \beta_1 q^{29} + (9 \beta_{5} - 15 \beta_{4} - 15 \beta_1) q^{30} + (10 \beta_{7} - 10 \beta_{3} + 10 \beta_{2}) q^{31} + ( - 7 \beta_{5} + 7 \beta_1) q^{32} + ( - 2 \beta_{7} - 12 \beta_{6} - 10 \beta_{3}) q^{33} + (4 \beta_{7} - 14 \beta_{6} - 14 \beta_{3} + 2 \beta_{2}) q^{34} + (3 \beta_{5} - 6 \beta_{4} + 6) q^{36} + (12 \beta_{5} - 24 \beta_1) q^{37} + ( - 20 \beta_{7} + 5 \beta_{6} + 10 \beta_{3} - 15 \beta_{2}) q^{38} + ( - 3 \beta_{5} - 12 \beta_{4} - 9 \beta_1 + 21) q^{39} + (12 \beta_{7} + 3 \beta_{6} + 3 \beta_{3} + 6 \beta_{2}) q^{40} + (30 \beta_{7} + 30 \beta_{2}) q^{41} + ( - 12 \beta_{5} + 6 \beta_1) q^{43} + (4 \beta_{5} - 6 \beta_{4} - 6) q^{44} + ( - 12 \beta_{6} - 9 \beta_{3} - 9 \beta_{2}) q^{45} + ( - 30 \beta_{5} - 20 \beta_{4} + 15 \beta_1) q^{46} + ( - 28 \beta_{7} + 20 \beta_{6} + 10 \beta_{3} - 10 \beta_{2}) q^{47} + ( - 19 \beta_{7} + 19 \beta_{6} + 19 \beta_{3}) q^{48} + ( - \beta_{5} - 15 \beta_{4} - 15) q^{50} + (18 \beta_{5} + 6 \beta_{4} + 6 \beta_1 - 36) q^{51} + ( - \beta_{7} - 2 \beta_{6} + 3 \beta_{2}) q^{52} + ( - 30 \beta_{4} - 4 \beta_1 + 60) q^{53} + (3 \beta_{7} - 21 \beta_{3}) q^{54} + ( - 2 \beta_{7} - 22 \beta_{3} - 26 \beta_{2}) q^{55} + ( - 15 \beta_{5} - 15 \beta_{4} - 30) q^{57} - 20 \beta_{4} q^{58} + ( - 31 \beta_{7} + 8 \beta_{6} + 4 \beta_{3} - 4 \beta_{2}) q^{59} + ( - 6 \beta_{4} + 3 \beta_1 - 9) q^{60} + (5 \beta_{7} - 5 \beta_{3} + 5 \beta_{2}) q^{61} + (40 \beta_{7} - 20 \beta_{6} - 10 \beta_{3} + 10 \beta_{2}) q^{62} - 41 q^{64} + (21 \beta_{5} - 42 \beta_{4} - 21 \beta_1 + 21) q^{65} + ( - 4 \beta_{7} + 12 \beta_{6} - 14 \beta_{3} - 36 \beta_{2}) q^{66} + (12 \beta_{5} + 12 \beta_1 + 40) q^{67} + (2 \beta_{6} - 2 \beta_{3} - 8 \beta_{2}) q^{68} + ( - 42 \beta_{7} + 19 \beta_{6} + 30 \beta_{3} - 12 \beta_{2}) q^{69} + ( - 17 \beta_{5} - 12 \beta_{4} + 17 \beta_1 + 6) q^{71} + ( - 18 \beta_{5} - 45 \beta_{4} + 45) q^{72} + (40 \beta_{7} + 4 \beta_{6} + 4 \beta_{3} + 20 \beta_{2}) q^{73} + ( - 60 \beta_{4} - 60) q^{74} + ( - 13 \beta_{7} + 6 \beta_{6} + 4 \beta_{3} - 18 \beta_{2}) q^{75} + ( - 5 \beta_{7} + 5 \beta_{6}) q^{76} + (21 \beta_{5} - 60 \beta_{4} - 9 \beta_1 + 15) q^{78} + (21 \beta_{5} + 21 \beta_1 + 26) q^{79} + ( - 19 \beta_{6} + 19 \beta_{3} + 19 \beta_{2}) q^{80} + (9 \beta_{4} + 36 \beta_1) q^{81} + (30 \beta_{7} - 60 \beta_{6} + 30 \beta_{2}) q^{82} + ( - 16 \beta_{6} + 16 \beta_{3} + 25 \beta_{2}) q^{83} + ( - 24 \beta_{5} + 66 \beta_{4} + 48 \beta_1 - 66) q^{85} + ( - 30 \beta_{4} + 60) q^{86} + ( - 12 \beta_{7} + 4 \beta_{6} - 12 \beta_{2}) q^{87} + (18 \beta_{5} - 60 \beta_{4} - 36 \beta_1 + 60) q^{88} + ( - 34 \beta_{7} + 16 \beta_{6} + 32 \beta_{3} - 18 \beta_{2}) q^{89} + ( - 27 \beta_{7} + 30 \beta_{6} + 9 \beta_{3} - 45 \beta_{2}) q^{90} + ( - 15 \beta_{4} + 4 \beta_1 + 30) q^{92} + (30 \beta_{5} + 90 \beta_{4} - 30 \beta_1 - 30) q^{93} + (38 \beta_{7} - 26 \beta_{3} + 50 \beta_{2}) q^{94} + (15 \beta_{5} + 60 \beta_{4} - 15 \beta_1 - 30) q^{95} + (7 \beta_{7} - 7 \beta_{6} - 7 \beta_{3} + 21 \beta_{2}) q^{96} + (32 \beta_{7} - 46 \beta_{6} - 46 \beta_{3} + 16 \beta_{2}) q^{97} + ( - 36 \beta_{5} + 96 \beta_{4} + 42 \beta_1 - 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 24 q^{9} + 72 q^{11} + 96 q^{15} - 152 q^{16} + 60 q^{18} + 80 q^{22} - 180 q^{23} - 4 q^{25} - 60 q^{30} + 24 q^{36} + 120 q^{39} - 72 q^{44} - 80 q^{46} - 180 q^{50} - 264 q^{51} + 360 q^{53} - 300 q^{57} - 80 q^{58} - 96 q^{60} - 328 q^{64} + 320 q^{67} + 180 q^{72} - 720 q^{74} - 120 q^{78} + 208 q^{79} + 36 q^{81} - 264 q^{85} + 360 q^{86} + 240 q^{88} + 180 q^{92} + 120 q^{93} - 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 14\nu^{5} + 70\nu^{3} - 99\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} - 28\nu^{5} + 49\nu^{3} - 180\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{6} + 14\nu^{4} + 7\nu^{2} + 162 ) / 63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{7} - 7\nu^{5} + 28\nu^{3} - 18\nu ) / 63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} - \nu^{5} - 5\nu^{3} + 63\nu ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + \beta_{3} + 6\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{7} - 5\beta_{6} - 10\beta_{3} + 12\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{5} + 7\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} + 29\beta_{6} - 29\beta_{3} - 21\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1
−1.72286 0.178197i
1.72286 + 0.178197i
−1.01575 1.40294i
1.01575 + 1.40294i
−1.01575 + 1.40294i
1.01575 1.40294i
−1.72286 + 0.178197i
1.72286 0.178197i
2.23607i −0.308646 + 2.98408i −1.00000 −3.04726 1.75934i 6.67261 + 0.690154i 0 6.70820i −8.80948 1.84205i −3.93399 + 6.81388i
263.2 2.23607i 0.308646 2.98408i −1.00000 3.04726 + 1.75934i −6.67261 0.690154i 0 6.70820i −8.80948 1.84205i 3.93399 6.81388i
263.3 2.23607i −2.42997 + 1.75934i −1.00000 −5.16858 2.98408i −3.93399 5.43357i 0 6.70820i 2.80948 8.55025i 6.67261 11.5573i
263.4 2.23607i 2.42997 1.75934i −1.00000 5.16858 + 2.98408i 3.93399 + 5.43357i 0 6.70820i 2.80948 8.55025i −6.67261 + 11.5573i
275.1 2.23607i −2.42997 1.75934i −1.00000 −5.16858 + 2.98408i −3.93399 + 5.43357i 0 6.70820i 2.80948 + 8.55025i 6.67261 + 11.5573i
275.2 2.23607i 2.42997 + 1.75934i −1.00000 5.16858 2.98408i 3.93399 5.43357i 0 6.70820i 2.80948 + 8.55025i −6.67261 11.5573i
275.3 2.23607i −0.308646 2.98408i −1.00000 −3.04726 + 1.75934i 6.67261 0.690154i 0 6.70820i −8.80948 + 1.84205i −3.93399 6.81388i
275.4 2.23607i 0.308646 + 2.98408i −1.00000 3.04726 1.75934i −6.67261 + 0.690154i 0 6.70820i −8.80948 + 1.84205i 3.93399 + 6.81388i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 263.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.i even 6 1 inner
63.j 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.j.d 8
7.b odd 2 1 inner 441.3.j.d 8
7.c even 3 1 441.3.n.d 8
7.c even 3 1 441.3.r.d 8
7.d odd 6 1 441.3.n.d 8
7.d odd 6 1 441.3.r.d 8
9.d odd 6 1 441.3.n.d 8
63.i even 6 1 inner 441.3.j.d 8
63.j odd 6 1 inner 441.3.j.d 8
63.n odd 6 1 441.3.r.d 8
63.o even 6 1 441.3.n.d 8
63.s even 6 1 441.3.r.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.3.j.d 8 1.a even 1 1 trivial
441.3.j.d 8 7.b odd 2 1 inner
441.3.j.d 8 63.i even 6 1 inner
441.3.j.d 8 63.j odd 6 1 inner
441.3.n.d 8 7.c even 3 1
441.3.n.d 8 7.d odd 6 1
441.3.n.d 8 9.d odd 6 1
441.3.n.d 8 63.o even 6 1
441.3.r.d 8 7.c even 3 1
441.3.r.d 8 7.d odd 6 1
441.3.r.d 8 63.n odd 6 1
441.3.r.d 8 63.s even 6 1

Hecke kernels

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

\( T_{2}^{2} + 5 \) Copy content Toggle raw display
\( T_{5}^{8} - 48T_{5}^{6} + 1863T_{5}^{4} - 21168T_{5}^{2} + 194481 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 12 T^{6} + 63 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} - 48 T^{6} + 1863 T^{4} + \cdots + 194481 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 36 T^{3} + 460 T^{2} - 1008 T + 784)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 204 T^{6} + 39852 T^{4} + \cdots + 3111696 \) Copy content Toggle raw display
$17$ \( T^{8} - 768 T^{6} + \cdots + 1731891456 \) Copy content Toggle raw display
$19$ \( T^{8} + 600 T^{6} + \cdots + 31640625 \) Copy content Toggle raw display
$23$ \( (T^{4} + 90 T^{3} + 3295 T^{2} + \cdots + 354025)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 80 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2400 T^{2} + 90000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2160 T^{2} + 4665600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 5400 T^{2} + 29160000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 540 T^{2} + 291600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 6528 T^{2} + 1527696)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 180 T^{3} + 13420 T^{2} + \cdots + 6864400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 8220 T^{2} + 11492100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 600 T^{2} + 5625)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 80 T - 560)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 3106 T^{2} + 1787569)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 17664 T^{6} + \cdots + 54\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{2} - 52 T - 5939)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 262721177672976 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 708922575360000 \) Copy content Toggle raw display
$97$ \( T^{8} + 33504 T^{6} + \cdots + 50\!\cdots\!36 \) Copy content Toggle raw display
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