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Properties

Label 1521.4.a.bk
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
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] = [1521,4,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, 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("1521.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} + 6) q^{4} + ( - \beta_{8} + \beta_{2} + \beta_1) q^{5} + (\beta_{9} - 2 \beta_{2}) q^{7} + (\beta_{9} - \beta_{8} + \cdots + 7 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} + 6) q^{4} + ( - \beta_{8} + \beta_{2} + \beta_1) q^{5} + (\beta_{9} - 2 \beta_{2}) q^{7} + (\beta_{9} - \beta_{8} + \cdots + 7 \beta_1) q^{8}+ \cdots + ( - 33 \beta_{9} + 21 \beta_{8} + \cdots + 11 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 60 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 35\nu^{3} - 210\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 35\nu^{4} - 210\nu^{2} ) / 96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 51\nu^{4} - 706\nu^{2} + 1792 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 49\nu^{5} + 700\nu^{3} - 2940\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} - 61\nu^{6} + 1168\nu^{4} - 7140\nu^{2} + 8064 ) / 96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - 64\nu^{7} + 1309\nu^{5} - 9030\nu^{3} + 15912\nu ) / 576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} - 70\nu^{7} + 1603\nu^{5} - 12654\nu^{3} + 20304\nu ) / 576 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{6} + 23\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 31\beta_{4} - 6\beta_{3} + 322 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 35\beta_{9} - 35\beta_{8} + 35\beta_{6} - 96\beta_{2} + 595\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 70\beta_{5} + 875\beta_{4} - 306\beta_{3} + 8330 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1015\beta_{9} - 1015\beta_{8} + 1111\beta_{6} - 4704\beta_{2} + 15995\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96\beta_{7} + 1934\beta_{5} + 24307\beta_{4} - 11658\beta_{3} + 223930 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 28175\beta_{9} - 27599\beta_{8} + 34319\beta_{6} - 175392\beta_{2} + 436603\beta_1 \) 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
−5.36472
−5.04537
−3.27897
−2.04224
−0.917374
0.917374
2.04224
3.27897
5.04537
5.36472
−5.36472 0 20.7803 2.69631 0 −15.2025 −68.5626 0 −14.4650
1.2 −5.04537 0 17.4557 −20.1174 0 −15.4279 −47.7076 0 101.500
1.3 −3.27897 0 2.75167 17.5414 0 26.6999 17.2091 0 −57.5178
1.4 −2.04224 0 −3.82924 −12.0825 0 29.7373 24.1582 0 24.6753
1.5 −0.917374 0 −7.15843 15.4704 0 −20.5833 13.9059 0 −14.1922
1.6 0.917374 0 −7.15843 −15.4704 0 20.5833 −13.9059 0 −14.1922
1.7 2.04224 0 −3.82924 12.0825 0 −29.7373 −24.1582 0 24.6753
1.8 3.27897 0 2.75167 −17.5414 0 −26.6999 −17.2091 0 −57.5178
1.9 5.04537 0 17.4557 20.1174 0 15.4279 47.7076 0 101.500
1.10 5.36472 0 20.7803 −2.69631 0 15.2025 68.5626 0 −14.4650
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.4.a.bk 10
3.b odd 2 1 507.4.a.r 10
13.b even 2 1 inner 1521.4.a.bk 10
13.f odd 12 2 117.4.q.e 10
39.d odd 2 1 507.4.a.r 10
39.f even 4 2 507.4.b.i 10
39.k even 12 2 39.4.j.c ✓ 10
156.v odd 12 2 624.4.bv.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.c ✓ 10 39.k even 12 2
117.4.q.e 10 13.f odd 12 2
507.4.a.r 10 3.b odd 2 1
507.4.a.r 10 39.d odd 2 1
507.4.b.i 10 39.f even 4 2
624.4.bv.h 10 156.v odd 12 2
1521.4.a.bk 10 1.a even 1 1 trivial
1521.4.a.bk 10 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{10} - 70T_{2}^{8} + 1645T_{2}^{6} - 14700T_{2}^{4} + 44100T_{2}^{2} - 27648 \) Copy content Toggle raw display
\( T_{5}^{10} - 1105T_{5}^{8} + 441955T_{5}^{6} - 76029795T_{5}^{4} + 4880780280T_{5}^{2} - 31632011568 \) Copy content Toggle raw display
\( T_{7}^{10} - 2490T_{7}^{8} + 2310165T_{7}^{6} - 991459980T_{7}^{4} + 197203450500T_{7}^{2} - 14692478786352 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 70 T^{8} + \cdots - 27648 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 31632011568 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 14692478786352 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots - 50\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( (T^{5} + 105 T^{4} + \cdots - 18224352)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( (T^{5} - 60 T^{4} + \cdots + 8153671248)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 495 T^{4} + \cdots - 427627836)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 48\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 36\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( (T^{5} + 370 T^{4} + \cdots - 227329236796)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 21\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( (T^{5} + 165 T^{4} + \cdots + 46733997168)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 41\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( (T^{5} - 1375 T^{4} + \cdots - 933851008945)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 13\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 20\!\cdots\!75 \) Copy content Toggle raw display
$79$ \( (T^{5} - 550 T^{4} + \cdots - 920208867136)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 16\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 28\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 15\!\cdots\!68 \) Copy content Toggle raw display
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