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Properties

Label 1323.2.o.c
Level $1323$
Weight $2$
Character orbit 1323.o
Analytic conductor $10.564$
Analytic rank $0$
Dimension $10$
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] = [1323,2,Mod(440,1323)]
 
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([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.440");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.o (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: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.288778218147.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} + \beta_{7} + \beta_{4}) q^{2} + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots + 1) q^{4}+ \cdots + ( - 2 \beta_{9} + 2 \beta_{8} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} + \beta_{7} + \beta_{4}) q^{2} + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{9} - 5 \beta_{8} + \cdots - 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{4} - 12 q^{11} - 6 q^{13} - 6 q^{16} + 24 q^{17} - 3 q^{20} + 5 q^{22} - 15 q^{23} + 7 q^{25} - 6 q^{26} + 15 q^{29} - 9 q^{31} + 48 q^{32} - 3 q^{34} - 12 q^{37} - 18 q^{38} - 15 q^{40} - 9 q^{41} + 3 q^{43} + 26 q^{46} + 15 q^{47} - 3 q^{50} + 12 q^{52} + 8 q^{58} - 18 q^{59} + 12 q^{61} + 12 q^{62} + 6 q^{64} - 3 q^{65} - 10 q^{67} + 27 q^{68} - 30 q^{74} - 9 q^{76} + 20 q^{79} + 60 q^{80} - 15 q^{83} + 18 q^{85} + 54 q^{86} - 8 q^{88} - 48 q^{89} - 39 q^{92} - 3 q^{94} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 339 \nu^{9} + 1348 \nu^{8} - 4381 \nu^{7} + 7882 \nu^{6} - 19883 \nu^{5} + 36059 \nu^{4} + \cdots + 29709 ) / 72795 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 658 \nu^{9} + 2394 \nu^{8} + 4352 \nu^{7} + 10326 \nu^{6} + 25351 \nu^{5} + 51907 \nu^{4} + \cdots + 98232 ) / 72795 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4192 \nu^{9} - 796 \nu^{8} - 21678 \nu^{7} - 20279 \nu^{6} - 85319 \nu^{5} - 118353 \nu^{4} + \cdots - 398583 ) / 218385 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8236 \nu^{9} - 9272 \nu^{8} + 54399 \nu^{7} - 28438 \nu^{6} + 233822 \nu^{5} - 150966 \nu^{4} + \cdots - 336546 ) / 218385 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + \cdots - 54156 ) / 72795 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} + \cdots + 11514 ) / 14559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3085 \nu^{9} - 1373 \nu^{8} + 17808 \nu^{7} + 1181 \nu^{6} + 84554 \nu^{5} + 5736 \nu^{4} + \cdots + 17856 ) / 43677 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18476 \nu^{9} + 18997 \nu^{8} - 128469 \nu^{7} + 65033 \nu^{6} - 601717 \nu^{5} + 295851 \nu^{4} + \cdots + 178101 ) / 218385 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} + 3\beta_{6} - \beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{9} + 5\beta_{8} + \beta_{7} - 12\beta_{6} - 5\beta_{5} - \beta_{3} - 5\beta_{2} + 5\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{9} - \beta_{8} - \beta_{7} - 7\beta_{5} + 4\beta_{4} - 2\beta_{3} + 11\beta_{2} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{9} - 8\beta_{8} - 14\beta_{7} + 16\beta_{5} + 9\beta_{4} - 7\beta_{3} + 22\beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{9} - 31\beta_{8} - 8\beta_{7} + 31\beta_{5} - 30\beta_{4} + 8\beta_{3} + \beta_{2} + 43\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 75 \beta_{9} - 66 \beta_{8} + 38 \beta_{7} + 222 \beta_{6} + 47 \beta_{5} - 37 \beta_{4} + \cdots - 112 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 95\beta_{9} + 189\beta_{8} + 94\beta_{7} + 37\beta_{5} + 84\beta_{4} + 47\beta_{3} - 194\beta_{2} - 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
440.1
1.07065 + 1.85442i
0.187540 + 0.324828i
−0.539982 − 0.935277i
−1.04536 − 1.81062i
0.827154 + 1.43267i
1.07065 − 1.85442i
0.187540 − 0.324828i
−0.539982 + 0.935277i
−1.04536 + 1.81062i
0.827154 − 1.43267i
−2.24607 − 1.29677i 0 2.36322 + 4.09323i −0.626493 − 1.08512i 0 0 − 7.07116i 0 3.24967i
440.2 −0.621951 − 0.359083i 0 −0.742118 − 1.28539i 0.723774 + 1.25361i 0 0 2.50226i 0 − 1.03958i
440.3 −0.254498 − 0.146935i 0 −0.956820 − 1.65726i −1.53014 − 2.65027i 0 0 1.15010i 0 0.899320i
440.4 1.30778 + 0.755047i 0 0.140193 + 0.242822i 0.387938 + 0.671929i 0 0 − 2.59678i 0 1.17165i
440.5 1.81474 + 1.04774i 0 1.19552 + 2.07070i 1.04492 + 1.80985i 0 0 0.819421i 0 4.37920i
881.1 −2.24607 + 1.29677i 0 2.36322 − 4.09323i −0.626493 + 1.08512i 0 0 7.07116i 0 − 3.24967i
881.2 −0.621951 + 0.359083i 0 −0.742118 + 1.28539i 0.723774 − 1.25361i 0 0 − 2.50226i 0 1.03958i
881.3 −0.254498 + 0.146935i 0 −0.956820 + 1.65726i −1.53014 + 2.65027i 0 0 − 1.15010i 0 − 0.899320i
881.4 1.30778 − 0.755047i 0 0.140193 − 0.242822i 0.387938 − 0.671929i 0 0 2.59678i 0 − 1.17165i
881.5 1.81474 − 1.04774i 0 1.19552 − 2.07070i 1.04492 − 1.80985i 0 0 − 0.819421i 0 − 4.37920i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 440.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.o.c 10
3.b odd 2 1 441.2.o.d 10
7.b odd 2 1 1323.2.o.d 10
7.c even 3 1 189.2.i.b 10
7.c even 3 1 1323.2.s.b 10
7.d odd 6 1 189.2.s.b 10
7.d odd 6 1 1323.2.i.b 10
9.c even 3 1 441.2.o.c 10
9.d odd 6 1 1323.2.o.d 10
21.c even 2 1 441.2.o.c 10
21.g even 6 1 63.2.s.b yes 10
21.g even 6 1 441.2.i.b 10
21.h odd 6 1 63.2.i.b ✓ 10
21.h odd 6 1 441.2.s.b 10
28.f even 6 1 3024.2.df.b 10
28.g odd 6 1 3024.2.ca.b 10
63.g even 3 1 441.2.i.b 10
63.g even 3 1 567.2.p.c 10
63.h even 3 1 63.2.s.b yes 10
63.i even 6 1 567.2.p.c 10
63.i even 6 1 1323.2.s.b 10
63.j odd 6 1 189.2.s.b 10
63.k odd 6 1 63.2.i.b ✓ 10
63.l odd 6 1 441.2.o.d 10
63.n odd 6 1 567.2.p.d 10
63.n odd 6 1 1323.2.i.b 10
63.o even 6 1 inner 1323.2.o.c 10
63.s even 6 1 189.2.i.b 10
63.t odd 6 1 441.2.s.b 10
63.t odd 6 1 567.2.p.d 10
84.j odd 6 1 1008.2.df.b 10
84.n even 6 1 1008.2.ca.b 10
252.n even 6 1 1008.2.ca.b 10
252.u odd 6 1 1008.2.df.b 10
252.bb even 6 1 3024.2.df.b 10
252.bn odd 6 1 3024.2.ca.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.i.b ✓ 10 21.h odd 6 1
63.2.i.b ✓ 10 63.k odd 6 1
63.2.s.b yes 10 21.g even 6 1
63.2.s.b yes 10 63.h even 3 1
189.2.i.b 10 7.c even 3 1
189.2.i.b 10 63.s even 6 1
189.2.s.b 10 7.d odd 6 1
189.2.s.b 10 63.j odd 6 1
441.2.i.b 10 21.g even 6 1
441.2.i.b 10 63.g even 3 1
441.2.o.c 10 9.c even 3 1
441.2.o.c 10 21.c even 2 1
441.2.o.d 10 3.b odd 2 1
441.2.o.d 10 63.l odd 6 1
441.2.s.b 10 21.h odd 6 1
441.2.s.b 10 63.t odd 6 1
567.2.p.c 10 63.g even 3 1
567.2.p.c 10 63.i even 6 1
567.2.p.d 10 63.n odd 6 1
567.2.p.d 10 63.t odd 6 1
1008.2.ca.b 10 84.n even 6 1
1008.2.ca.b 10 252.n even 6 1
1008.2.df.b 10 84.j odd 6 1
1008.2.df.b 10 252.u odd 6 1
1323.2.i.b 10 7.d odd 6 1
1323.2.i.b 10 63.n odd 6 1
1323.2.o.c 10 1.a even 1 1 trivial
1323.2.o.c 10 63.o even 6 1 inner
1323.2.o.d 10 7.b odd 2 1
1323.2.o.d 10 9.d odd 6 1
1323.2.s.b 10 7.c even 3 1
1323.2.s.b 10 63.i even 6 1
3024.2.ca.b 10 28.g odd 6 1
3024.2.ca.b 10 252.bn odd 6 1
3024.2.df.b 10 28.f even 6 1
3024.2.df.b 10 252.bb even 6 1

Hecke kernels

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

\( T_{2}^{10} - 7T_{2}^{8} + 42T_{2}^{6} - 24T_{2}^{5} - 46T_{2}^{4} + 21T_{2}^{3} + 52T_{2}^{2} + 21T_{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{10} + 9T_{5}^{8} - 12T_{5}^{7} + 69T_{5}^{6} - 63T_{5}^{5} + 144T_{5}^{4} - 90T_{5}^{3} + 198T_{5}^{2} - 108T_{5} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 7 T^{8} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 9 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + 12 T^{9} + \cdots + 2883 \) Copy content Toggle raw display
$13$ \( T^{10} + 6 T^{9} + \cdots + 3267 \) Copy content Toggle raw display
$17$ \( (T^{5} - 12 T^{4} + \cdots + 513)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 57 T^{8} + \cdots + 2187 \) Copy content Toggle raw display
$23$ \( T^{10} + 15 T^{9} + \cdots + 27 \) Copy content Toggle raw display
$29$ \( T^{10} - 15 T^{9} + \cdots + 186003 \) Copy content Toggle raw display
$31$ \( T^{10} + 9 T^{9} + \cdots + 16875 \) Copy content Toggle raw display
$37$ \( (T^{5} + 6 T^{4} + \cdots + 608)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 9 T^{9} + \cdots + 40487769 \) Copy content Toggle raw display
$43$ \( T^{10} - 3 T^{9} + \cdots + 12243001 \) Copy content Toggle raw display
$47$ \( T^{10} - 15 T^{9} + \cdots + 321489 \) Copy content Toggle raw display
$53$ \( T^{10} + 137 T^{8} + \cdots + 871563 \) Copy content Toggle raw display
$59$ \( T^{10} + 18 T^{9} + \cdots + 4100625 \) Copy content Toggle raw display
$61$ \( T^{10} - 12 T^{9} + \cdots + 826875 \) Copy content Toggle raw display
$67$ \( T^{10} + 10 T^{9} + \cdots + 361 \) Copy content Toggle raw display
$71$ \( T^{10} + 359 T^{8} + \cdots + 46216875 \) Copy content Toggle raw display
$73$ \( T^{10} + 219 T^{8} + \cdots + 789507 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 1067655625 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 340734681 \) Copy content Toggle raw display
$89$ \( (T^{5} + 24 T^{4} + \cdots + 5697)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 6 T^{9} + \cdots + 9687627 \) Copy content Toggle raw display
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