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Properties

Label 1323.2.h.f
Level $1323$
Weight $2$
Character orbit 1323.h
Analytic conductor $10.564$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,2,Mod(226,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.h (of order \(3\), 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: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.991381711347.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{3} + 1) q^{4} + (\beta_{9} - \beta_{6}) q^{5} + ( - \beta_{8} - \beta_{4} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{3} + 1) q^{4} + (\beta_{9} - \beta_{6}) q^{5} + ( - \beta_{8} - \beta_{4} + 1) q^{8} + (\beta_{9} - \beta_{8} - 2 \beta_{6}) q^{10} + (\beta_{7} - \beta_{6} + \beta_{4} - 1) q^{11} + (\beta_{6} + \beta_{2} + \beta_1 + 1) q^{13} + (\beta_{9} - \beta_{8} + \cdots - \beta_{2}) q^{16}+ \cdots + ( - 2 \beta_{9} + \beta_{8} + \cdots + 4 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} + 8 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{2} + 8 q^{4} + 4 q^{5} + 6 q^{8} + 7 q^{10} - 4 q^{11} + 8 q^{13} - 4 q^{16} + 12 q^{17} - q^{19} + 5 q^{20} - q^{22} - 3 q^{23} - q^{25} + 11 q^{26} - 7 q^{29} - 6 q^{31} - 4 q^{32} - 3 q^{34} + 20 q^{38} + 3 q^{40} + 5 q^{41} - 7 q^{43} + 10 q^{44} + 3 q^{46} - 54 q^{47} - 19 q^{50} + 10 q^{52} + 21 q^{53} - 4 q^{55} - 10 q^{58} - 60 q^{59} - 28 q^{61} - 12 q^{62} - 50 q^{64} - 22 q^{65} + 4 q^{67} + 27 q^{68} + 6 q^{71} - 15 q^{73} + 36 q^{74} - 5 q^{76} + 8 q^{79} + 20 q^{80} + 5 q^{82} + 9 q^{83} - 6 q^{85} + 8 q^{86} - 18 q^{88} + 28 q^{89} - 27 q^{92} - 6 q^{94} - 28 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} - 9\nu^{8} - 3\nu^{7} - 61\nu^{6} - 72\nu^{5} - 282\nu^{4} - 204\nu^{3} - 387\nu^{2} - 873\nu - 117 ) / 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{9} - 12\nu^{8} + 48\nu^{7} - 23\nu^{6} + 204\nu^{5} - 240\nu^{4} + 303\nu^{3} - 108\nu^{2} + 36\nu - 1557 ) / 567 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{9} - \nu^{8} + 12\nu^{7} + 8\nu^{6} + 68\nu^{5} + 30\nu^{4} + 123\nu^{3} + 204\nu^{2} + 270\nu + 63 ) / 63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} + \cdots - 180 ) / 567 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + \cdots - 63 ) / 567 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 53 \nu^{9} + 60 \nu^{8} - 375 \nu^{7} - 11 \nu^{6} - 1668 \nu^{5} - 69 \nu^{4} - 2757 \nu^{3} + \cdots - 1368 ) / 567 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} + \cdots + 720 ) / 567 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 91 \nu^{9} + 174 \nu^{8} - 813 \nu^{7} + 704 \nu^{6} - 3633 \nu^{5} + 3174 \nu^{4} - 8070 \nu^{3} + \cdots + 801 ) / 567 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 3\beta_{6} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 4\beta_{5} + \beta_{4} - 4\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{7} - 14\beta_{6} + \beta_{4} + \beta_{2} - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} - 7\beta_{8} - \beta_{7} - 9\beta_{6} - 17\beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{9} - 10\beta_{8} - \beta_{5} - 10\beta_{4} + 24\beta_{3} - 9\beta_{2} + \beta _1 + 70 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{7} + 65\beta_{6} - 43\beta_{4} - 19\beta_{2} + 75\beta _1 + 65 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -62\beta_{9} + 73\beta_{8} + 118\beta_{7} + 360\beta_{6} + 14\beta_{5} - 118\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -135\beta_{9} + 253\beta_{8} + 343\beta_{5} + 253\beta_{4} - 87\beta_{3} + 135\beta_{2} - 343\beta _1 - 430 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1 - \beta_{6}\) \(-1 - \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
−1.02682 + 1.77851i
−0.335166 + 0.580525i
0.247934 − 0.429435i
0.920620 − 1.59456i
1.19343 − 2.06709i
−1.02682 − 1.77851i
−0.335166 − 0.580525i
0.247934 + 0.429435i
0.920620 + 1.59456i
1.19343 + 2.06709i
−2.05365 0 2.21746 0.0731228 + 0.126652i 0 0 −0.446582 0 −0.150168 − 0.260099i
226.2 −0.670333 0 −1.55065 −0.712469 − 1.23403i 0 0 2.38012 0 0.477591 + 0.827212i
226.3 0.495868 0 −1.75411 1.84629 + 3.19787i 0 0 −1.86155 0 0.915516 + 1.58572i
226.4 1.84124 0 1.39017 −0.667377 − 1.15593i 0 0 −1.12285 0 −1.22880 − 2.12835i
226.5 2.38687 0 3.69714 1.46043 + 2.52954i 0 0 4.05086 0 3.48586 + 6.03769i
802.1 −2.05365 0 2.21746 0.0731228 − 0.126652i 0 0 −0.446582 0 −0.150168 + 0.260099i
802.2 −0.670333 0 −1.55065 −0.712469 + 1.23403i 0 0 2.38012 0 0.477591 − 0.827212i
802.3 0.495868 0 −1.75411 1.84629 − 3.19787i 0 0 −1.86155 0 0.915516 − 1.58572i
802.4 1.84124 0 1.39017 −0.667377 + 1.15593i 0 0 −1.12285 0 −1.22880 + 2.12835i
802.5 2.38687 0 3.69714 1.46043 − 2.52954i 0 0 4.05086 0 3.48586 − 6.03769i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.h.f 10
3.b odd 2 1 441.2.h.f 10
7.b odd 2 1 189.2.h.b 10
7.c even 3 1 1323.2.f.f 10
7.c even 3 1 1323.2.g.f 10
7.d odd 6 1 189.2.g.b 10
7.d odd 6 1 1323.2.f.e 10
9.c even 3 1 1323.2.g.f 10
9.d odd 6 1 441.2.g.f 10
21.c even 2 1 63.2.h.b yes 10
21.g even 6 1 63.2.g.b ✓ 10
21.g even 6 1 441.2.f.e 10
21.h odd 6 1 441.2.f.f 10
21.h odd 6 1 441.2.g.f 10
28.d even 2 1 3024.2.q.i 10
28.f even 6 1 3024.2.t.i 10
63.g even 3 1 1323.2.f.f 10
63.h even 3 1 inner 1323.2.h.f 10
63.h even 3 1 3969.2.a.bb 5
63.i even 6 1 63.2.h.b yes 10
63.i even 6 1 3969.2.a.z 5
63.j odd 6 1 441.2.h.f 10
63.j odd 6 1 3969.2.a.ba 5
63.k odd 6 1 567.2.e.e 10
63.k odd 6 1 1323.2.f.e 10
63.l odd 6 1 189.2.g.b 10
63.l odd 6 1 567.2.e.e 10
63.n odd 6 1 441.2.f.f 10
63.o even 6 1 63.2.g.b ✓ 10
63.o even 6 1 567.2.e.f 10
63.s even 6 1 441.2.f.e 10
63.s even 6 1 567.2.e.f 10
63.t odd 6 1 189.2.h.b 10
63.t odd 6 1 3969.2.a.bc 5
84.h odd 2 1 1008.2.q.i 10
84.j odd 6 1 1008.2.t.i 10
252.r odd 6 1 1008.2.q.i 10
252.s odd 6 1 1008.2.t.i 10
252.bi even 6 1 3024.2.t.i 10
252.bj even 6 1 3024.2.q.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b ✓ 10 21.g even 6 1
63.2.g.b ✓ 10 63.o even 6 1
63.2.h.b yes 10 21.c even 2 1
63.2.h.b yes 10 63.i even 6 1
189.2.g.b 10 7.d odd 6 1
189.2.g.b 10 63.l odd 6 1
189.2.h.b 10 7.b odd 2 1
189.2.h.b 10 63.t odd 6 1
441.2.f.e 10 21.g even 6 1
441.2.f.e 10 63.s even 6 1
441.2.f.f 10 21.h odd 6 1
441.2.f.f 10 63.n odd 6 1
441.2.g.f 10 9.d odd 6 1
441.2.g.f 10 21.h odd 6 1
441.2.h.f 10 3.b odd 2 1
441.2.h.f 10 63.j odd 6 1
567.2.e.e 10 63.k odd 6 1
567.2.e.e 10 63.l odd 6 1
567.2.e.f 10 63.o even 6 1
567.2.e.f 10 63.s even 6 1
1008.2.q.i 10 84.h odd 2 1
1008.2.q.i 10 252.r odd 6 1
1008.2.t.i 10 84.j odd 6 1
1008.2.t.i 10 252.s odd 6 1
1323.2.f.e 10 7.d odd 6 1
1323.2.f.e 10 63.k odd 6 1
1323.2.f.f 10 7.c even 3 1
1323.2.f.f 10 63.g even 3 1
1323.2.g.f 10 7.c even 3 1
1323.2.g.f 10 9.c even 3 1
1323.2.h.f 10 1.a even 1 1 trivial
1323.2.h.f 10 63.h even 3 1 inner
3024.2.q.i 10 28.d even 2 1
3024.2.q.i 10 252.bj even 6 1
3024.2.t.i 10 28.f even 6 1
3024.2.t.i 10 252.bi even 6 1
3969.2.a.z 5 63.i even 6 1
3969.2.a.ba 5 63.j odd 6 1
3969.2.a.bb 5 63.h even 3 1
3969.2.a.bc 5 63.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\):

\( T_{2}^{5} - 2T_{2}^{4} - 5T_{2}^{3} + 9T_{2}^{2} + 3T_{2} - 3 \) Copy content Toggle raw display
\( T_{5}^{10} - 4T_{5}^{9} + 21T_{5}^{8} - 16T_{5}^{7} + 79T_{5}^{6} + 51T_{5}^{5} + 402T_{5}^{4} + 294T_{5}^{3} + 378T_{5}^{2} - 54T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} - 2 T^{4} - 5 T^{3} + \cdots - 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 4 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + 4 T^{9} + \cdots + 225 \) Copy content Toggle raw display
$13$ \( T^{10} - 8 T^{9} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( T^{10} - 12 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{10} + T^{9} + \cdots + 185761 \) Copy content Toggle raw display
$23$ \( T^{10} + 3 T^{9} + \cdots + 2595321 \) Copy content Toggle raw display
$29$ \( T^{10} + 7 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$31$ \( (T^{5} + 3 T^{4} + \cdots + 285)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 96 T^{8} + \cdots + 82944 \) Copy content Toggle raw display
$41$ \( T^{10} - 5 T^{9} + \cdots + 2025 \) Copy content Toggle raw display
$43$ \( T^{10} + 7 T^{9} + \cdots + 687241 \) Copy content Toggle raw display
$47$ \( (T^{5} + 27 T^{4} + \cdots - 6615)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 21 T^{9} + \cdots + 178929 \) Copy content Toggle raw display
$59$ \( (T^{5} + 30 T^{4} + \cdots - 5625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 14 T^{4} + 34 T^{3} + \cdots - 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 2 T^{4} + \cdots - 7121)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 3 T^{4} - 168 T^{3} + \cdots + 81)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 15 T^{9} + \cdots + 772641 \) Copy content Toggle raw display
$79$ \( (T^{5} - 4 T^{4} + \cdots + 193)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 218123361 \) Copy content Toggle raw display
$89$ \( T^{10} - 28 T^{9} + \cdots + 7080921 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 2307745521 \) Copy content Toggle raw display
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