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Properties

Label 1323.4.a.bj
Level $1323$
Weight $4$
Character orbit 1323.a
Self dual yes
Analytic conductor $78.060$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,4,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

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: yes
Analytic conductor: \(78.0595269376\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 43x^{5} + 10x^{4} + 513x^{3} + 258x^{2} - 936x - 504 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 189)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 4) q^{4} - \beta_{4} q^{5} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 11) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 4) q^{4} - \beta_{4} q^{5} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 11) q^{8} + ( - 2 \beta_{6} - 2 \beta_{4} + \cdots + 3) q^{10}+ \cdots + (18 \beta_{6} - 18 \beta_{5} + \cdots - 526) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8} + 12 q^{10} + 98 q^{11} - 124 q^{13} + 139 q^{16} + 30 q^{17} + 182 q^{19} - 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} - 245 q^{26} + 323 q^{29} + 26 q^{31} + 398 q^{32} + 114 q^{34} - 112 q^{37} + 1015 q^{38} - 147 q^{40} - 524 q^{41} + 8 q^{43} + 937 q^{44} - 339 q^{46} + 288 q^{47} + 2576 q^{50} - 1075 q^{52} + 1353 q^{53} + 156 q^{55} - 81 q^{58} + 165 q^{59} + 56 q^{61} - 1215 q^{62} - 1706 q^{64} + 1694 q^{65} - 988 q^{67} + 2625 q^{68} + 792 q^{71} + 1487 q^{73} + 2736 q^{74} + 1952 q^{76} - 1273 q^{79} - 2501 q^{80} - 2049 q^{82} - 1170 q^{83} + 216 q^{85} - 160 q^{86} + 9 q^{88} + 1058 q^{89} + 3834 q^{92} + 1653 q^{94} + 3260 q^{95} - 3730 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 43x^{5} + 10x^{4} + 513x^{3} + 258x^{2} - 936x - 504 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 19\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} + 52\nu^{4} - 103\nu^{3} - 729\nu^{2} + 207\nu + 1008 ) / 42 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} + 31\nu^{4} - 103\nu^{3} - 246\nu^{2} + 480\nu + 294 ) / 21 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{6} + \nu^{5} - 197\nu^{4} - 52\nu^{3} + 1965\nu^{2} + 372\nu - 2436 ) / 84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 20\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} + 2\beta_{4} + 23\beta_{2} + 36\beta _1 + 242 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{6} + 3\beta_{5} + 4\beta_{4} + 27\beta_{3} + 38\beta_{2} + 445\beta _1 + 407 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16\beta_{6} - 40\beta_{5} + 78\beta_{4} + 5\beta_{3} + 516\beta_{2} + 1070\beta _1 + 5339 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.49279
−3.62964
−1.63185
−0.541672
1.44197
4.76215
5.09184
−4.49279 0 12.1851 −8.19814 0 0 −18.8030 0 36.8325
1.2 −3.62964 0 5.17430 4.84163 0 0 10.2563 0 −17.5734
1.3 −1.63185 0 −5.33705 12.3790 0 0 21.7641 0 −20.2007
1.4 −0.541672 0 −7.70659 −16.7289 0 0 8.50782 0 9.06158
1.5 1.44197 0 −5.92074 6.60364 0 0 −20.0732 0 9.52221
1.6 4.76215 0 14.6781 18.7013 0 0 31.8020 0 89.0586
1.7 5.09184 0 17.9269 −18.5985 0 0 50.5460 0 −94.7009
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.4.a.bj 7
3.b odd 2 1 1323.4.a.bi 7
7.b odd 2 1 1323.4.a.bk 7
7.c even 3 2 189.4.e.f ✓ 14
21.c even 2 1 1323.4.a.bh 7
21.h odd 6 2 189.4.e.g yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.f ✓ 14 7.c even 3 2
189.4.e.g yes 14 21.h odd 6 2
1323.4.a.bh 7 21.c even 2 1
1323.4.a.bi 7 3.b odd 2 1
1323.4.a.bj 7 1.a even 1 1 trivial
1323.4.a.bk 7 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1323))\):

\( T_{2}^{7} - T_{2}^{6} - 43T_{2}^{5} + 10T_{2}^{4} + 513T_{2}^{3} + 258T_{2}^{2} - 936T_{2} - 504 \) Copy content Toggle raw display
\( T_{5}^{7} + T_{5}^{6} - 631T_{5}^{5} + 311T_{5}^{4} + 112338T_{5}^{3} - 287178T_{5}^{2} - 4846491T_{5} + 18879633 \) Copy content Toggle raw display
\( T_{13}^{7} + 124 T_{13}^{6} - 354 T_{13}^{5} - 620994 T_{13}^{4} - 26760399 T_{13}^{3} + \cdots + 86896171316 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} + \cdots - 504 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + T^{6} + \cdots + 18879633 \) Copy content Toggle raw display
$7$ \( T^{7} \) Copy content Toggle raw display
$11$ \( T^{7} - 98 T^{6} + \cdots + 934983936 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots + 86896171316 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 37518015618 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots - 2945814380716 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots - 20028475364274 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots - 495493458583509 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 379691567407224 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 36979101568584 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 6874920774882 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 765727707363728 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 22\!\cdots\!06 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots - 33\!\cdots\!73 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 18\!\cdots\!05 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 46\!\cdots\!86 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 95\!\cdots\!50 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots + 23\!\cdots\!58 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots - 39\!\cdots\!45 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 25\!\cdots\!31 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots + 86\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 12\!\cdots\!14 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
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