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Properties

Label 1323.4.a.bo
Level $1323$
Weight $4$
Character orbit 1323.a
Self dual yes
Analytic conductor $78.060$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 54x^{6} - 6x^{5} + 555x^{4} + 642x^{3} - 218x^{2} - 54x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_1 + 6) q^{4} + \beta_{3} q^{5} + ( - \beta_{4} - 5 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_1 + 6) q^{4} + \beta_{3} q^{5} + ( - \beta_{4} - 5 \beta_{2}) q^{8} + (\beta_{6} + 5) q^{10} + ( - \beta_{5} - \beta_{4} + \cdots - 3 \beta_{2}) q^{11}+ \cdots + (2 \beta_{7} + 8 \beta_{6} + 46 \beta_1 + 69) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 48 q^{4} + 44 q^{10} - 84 q^{13} + 156 q^{16} + 12 q^{19} + 224 q^{22} + 408 q^{25} + 800 q^{31} - 948 q^{34} + 692 q^{37} + 96 q^{40} + 1456 q^{43} + 1524 q^{46} + 1972 q^{52} - 1280 q^{55} + 2372 q^{58} + 216 q^{61} + 4964 q^{64} + 684 q^{67} - 4564 q^{73} - 380 q^{76} + 556 q^{79} + 3340 q^{82} + 1296 q^{85} + 6696 q^{88} - 492 q^{94} + 584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 54x^{6} - 6x^{5} + 555x^{4} + 642x^{3} - 218x^{2} - 54x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 43\nu^{7} - 264\nu^{6} - 1786\nu^{5} + 12816\nu^{4} - 1106\nu^{3} - 63870\nu^{2} + 6267\nu - 2010 ) / 714 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -482\nu^{7} + 618\nu^{6} + 24611\nu^{5} - 27567\nu^{4} - 201047\nu^{3} - 103161\nu^{2} + 10425\nu + 12543 ) / 3570 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 739\nu^{7} + 519\nu^{6} - 41122\nu^{5} - 29901\nu^{4} + 467824\nu^{3} + 644097\nu^{2} - 262695\nu - 55416 ) / 3570 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1159 \nu^{7} + 1884 \nu^{6} - 65167 \nu^{5} - 102981 \nu^{4} + 757939 \nu^{3} + 1525047 \nu^{2} + \cdots - 164091 ) / 3570 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -12\nu^{7} + 29\nu^{6} + 602\nu^{5} - 1397\nu^{4} - 4532\nu^{3} + 3815\nu^{2} + 4496\nu - 516 ) / 34 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -359\nu^{7} - 270\nu^{6} + 19826\nu^{5} + 15801\nu^{4} - 219296\nu^{3} - 336171\nu^{2} + 55359\nu + 22626 ) / 357 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 128\nu^{7} - 162\nu^{6} - 6563\nu^{5} + 7257\nu^{4} + 54671\nu^{3} + 25941\nu^{2} - 8217\nu + 642 ) / 51 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{4} + 5\beta_{3} + 14\beta_{2} + \beta _1 - 1 ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{7} + 4\beta_{6} - 6\beta_{5} + 9\beta_{4} + 12\beta_{3} - 63\beta_{2} + 9\beta _1 + 565 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 65\beta_{7} + 144\beta_{6} + 115\beta_{5} + 83\beta_{4} + 449\beta_{3} + 749\beta_{2} + 58\beta _1 + 117 ) / 84 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -218\beta_{7} + 166\beta_{6} - 285\beta_{5} + 312\beta_{4} + 339\beta_{3} - 3318\beta_{2} - 99\beta _1 + 18880 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2577 \beta_{7} + 5672 \beta_{6} + 5063 \beta_{5} + 2377 \beta_{4} + 17713 \beta_{3} + 25375 \beta_{2} + \cdots - 19531 ) / 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1577 \beta_{7} + 983 \beta_{6} - 2046 \beta_{5} + 1823 \beta_{4} + 1614 \beta_{3} - 25221 \beta_{2} + \cdots + 118967 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 53663 \beta_{7} + 112030 \beta_{6} + 107141 \beta_{5} + 37672 \beta_{4} + 349141 \beta_{3} + \cdots - 807479 ) / 42 \) 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
4.35333
−2.07765
0.128152
−0.256959
0.356871
6.13087
−6.51461
−2.12000
−5.46178 0 21.8311 −0.199136 0 0 −75.5424 0 1.08764
1.2 −3.49236 0 4.19658 7.87531 0 0 13.2829 0 −27.5034
1.3 −3.29273 0 2.84210 −21.7165 0 0 16.9836 0 71.5066
1.4 −1.76924 0 −4.86977 13.0512 0 0 22.7698 0 −23.0908
1.5 1.76924 0 −4.86977 −13.0512 0 0 −22.7698 0 −23.0908
1.6 3.29273 0 2.84210 21.7165 0 0 −16.9836 0 71.5066
1.7 3.49236 0 4.19658 −7.87531 0 0 −13.2829 0 −27.5034
1.8 5.46178 0 21.8311 0.199136 0 0 75.5424 0 1.08764
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.4.a.bo 8
3.b odd 2 1 inner 1323.4.a.bo 8
7.b odd 2 1 1323.4.a.bn 8
7.c even 3 2 189.4.e.h ✓ 16
21.c even 2 1 1323.4.a.bn 8
21.h odd 6 2 189.4.e.h ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.h ✓ 16 7.c even 3 2
189.4.e.h ✓ 16 21.h odd 6 2
1323.4.a.bn 8 7.b odd 2 1
1323.4.a.bn 8 21.c even 2 1
1323.4.a.bo 8 1.a even 1 1 trivial
1323.4.a.bo 8 3.b odd 2 1 inner

Hecke kernels

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

\( T_{2}^{8} - 56T_{2}^{6} + 985T_{2}^{4} - 6510T_{2}^{2} + 12348 \) Copy content Toggle raw display
\( T_{5}^{8} - 704T_{5}^{6} + 120172T_{5}^{4} - 4986912T_{5}^{2} + 197568 \) Copy content Toggle raw display
\( T_{13}^{4} + 42T_{13}^{3} - 4475T_{13}^{2} - 94812T_{13} + 5259952 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 56 T^{6} + \cdots + 12348 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 704 T^{6} + \cdots + 197568 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 11628770291712 \) Copy content Toggle raw display
$13$ \( (T^{4} + 42 T^{3} + \cdots + 5259952)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 5799554111232 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + \cdots - 11890928)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 19258163854272 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 18\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( (T^{4} - 400 T^{3} + \cdots - 343595259)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 346 T^{3} + \cdots - 405399024)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( (T^{4} - 728 T^{3} + \cdots - 12025459739)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 44\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 19\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( (T^{4} - 108 T^{3} + \cdots - 1314437615)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 342 T^{3} + \cdots - 16835689556)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + 2282 T^{3} + \cdots - 330036608412)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 278 T^{3} + \cdots - 2954434736)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 16\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 69\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( (T^{4} - 292 T^{3} + \cdots + 100316149821)^{2} \) Copy content Toggle raw display
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