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Properties

Label 539.2.s.b
Level $539$
Weight $2$
Character orbit 539.s
Analytic conductor $4.304$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,2,Mod(19,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([25, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.s (of order \(30\), degree \(8\), 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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{30})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 20x^{14} + 260x^{12} + 2030x^{10} + 11605x^{8} + 42100x^{6} + 106925x^{4} + 113575x^{2} + 87025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{14} - \beta_{11} + \cdots + \beta_{4}) q^{2}+ \cdots + (\beta_{14} - 2 \beta_{12} - \beta_{9} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{14} - \beta_{11} + \cdots + \beta_{4}) q^{2}+ \cdots + ( - 5 \beta_{14} + 8 \beta_{12} + \cdots - 12 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} - 10 q^{4} + 10 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} - 10 q^{4} + 10 q^{8} + 26 q^{9} - 4 q^{11} + 22 q^{16} + 4 q^{18} + 42 q^{22} + 4 q^{23} + 10 q^{29} + 50 q^{30} + 20 q^{36} + 58 q^{37} - 40 q^{44} + 26 q^{46} - 60 q^{50} + 50 q^{51} + 30 q^{53} - 90 q^{57} + 16 q^{58} - 90 q^{60} - 38 q^{64} - 60 q^{65} + 2 q^{67} - 56 q^{71} - 48 q^{72} - 18 q^{74} - 20 q^{78} - 58 q^{79} + 8 q^{81} + 80 q^{85} - 18 q^{86} - 20 q^{88} + 20 q^{92} - 40 q^{93} + 90 q^{95} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 20x^{14} + 260x^{12} + 2030x^{10} + 11605x^{8} + 42100x^{6} + 106925x^{4} + 113575x^{2} + 87025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 473414 \nu^{15} - 121962046 \nu^{13} - 1914724533 \nu^{11} - 19540502651 \nu^{9} + \cdots - 700136714820 \nu ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1553143 \nu^{15} - 83115610 \nu^{13} - 738702950 \nu^{11} - 5169909042 \nu^{9} + \cdots + 389963137595 \nu ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2026557 \nu^{14} + 205077656 \nu^{12} + 2653427483 \nu^{10} + 24710411693 \nu^{8} + \cdots + 310173577225 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2307412 \nu^{14} - 42954924 \nu^{12} - 546435360 \nu^{10} - 4022687807 \nu^{8} + \cdots - 70674046800 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2307412 \nu^{15} + 42954924 \nu^{13} + 546435360 \nu^{11} + 4022687807 \nu^{9} + \cdots + 70674046800 \nu ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8126617 \nu^{15} + 107242387 \nu^{13} + 1727473773 \nu^{11} + 21511248812 \nu^{9} + \cdots + 534354646485 \nu ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9624825 \nu^{14} + 326813127 \nu^{12} + 4112826637 \nu^{10} + 33537013348 \nu^{8} + \cdots + 564950170875 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 85328 \nu^{14} - 3129227 \nu^{12} - 37034577 \nu^{10} - 295472690 \nu^{8} + \cdots - 2416417865 ) / 1080638985 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 85328 \nu^{15} - 3129227 \nu^{13} - 37034577 \nu^{11} - 295472690 \nu^{9} + \cdots - 2416417865 \nu ) / 1080638985 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19576647 \nu^{14} + 220271693 \nu^{12} + 1820750374 \nu^{10} + 4608222307 \nu^{8} + \cdots - 312330974765 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23957031 \nu^{14} - 304074935 \nu^{12} - 2842215611 \nu^{10} - 12038362652 \nu^{8} + \cdots + 147206723390 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 198348 \nu^{15} - 3886661 \nu^{13} - 41951474 \nu^{11} - 280524995 \nu^{9} + \cdots - 1244021785 \nu ) / 1080638985 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 26536505 \nu^{14} - 264615202 \nu^{12} - 2085345531 \nu^{10} - 3473599188 \nu^{8} + \cdots + 29496038150 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 38187887 \nu^{15} - 796505985 \nu^{13} - 8851599651 \nu^{11} - 61721024229 \nu^{9} + \cdots - 987191416985 \nu ) / 141563707035 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} - 2\beta_{11} - 2\beta_{9} - 2\beta_{8} - 5\beta_{5} - \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - 2\beta_{13} + \beta_{10} + 2\beta_{7} + 7\beta_{6} + \beta_{3} + 2\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{14} + 8\beta_{12} + \beta_{11} + 7\beta_{9} + 16\beta_{8} + 37\beta_{5} - 2\beta_{4} - 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{15} + 9\beta_{13} - 2\beta_{10} - \beta_{7} - 38\beta_{6} + 9\beta_{3} + \beta_{2} - 47\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 84\beta_{14} - 29\beta_{12} + 98\beta_{11} + 42\beta_{9} - 13\beta_{8} - 42\beta_{5} + 56\beta_{4} + 231 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -14\beta_{15} + 69\beta_{13} - 29\beta_{10} - 98\beta_{7} - 56\beta_{6} - 138\beta_{3} - 111\beta_{2} + 146\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -245\beta_{14} - 260\beta_{12} - 750\beta_{11} - 490\beta_{9} - 630\beta_{8} - 1210\beta_{5} - 125\beta_{4} - 245 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 505 \beta_{15} - 1010 \beta_{13} + 385 \beta_{10} + 750 \beta_{7} + 1960 \beta_{6} + 505 \beta_{3} + \cdots + 1135 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1420 \beta_{14} + 2645 \beta_{12} + 965 \beta_{11} + 1420 \beta_{9} + 5030 \beta_{8} + 9235 \beta_{5} + \cdots - 7815 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2385 \beta_{15} + 3610 \beta_{13} - 2190 \beta_{10} - 965 \beta_{7} - 10200 \beta_{6} + \cdots - 13810 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 16410 \beta_{14} - 995 \beta_{12} + 26405 \beta_{11} + 8205 \beta_{9} - 7210 \beta_{8} - 8205 \beta_{5} + \cdots + 59020 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9995 \beta_{15} + 25410 \beta_{13} - 995 \beta_{10} - 26405 \beta_{7} - 18200 \beta_{6} + \cdots + 39825 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 47235 \beta_{14} - 129635 \beta_{12} - 224105 \beta_{11} - 94470 \beta_{9} - 172640 \beta_{8} + \cdots - 47235 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 176870 \beta_{15} - 353740 \beta_{13} + 125405 \beta_{10} + 224105 \beta_{7} + 556400 \beta_{6} + \cdots + 349510 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(442\)
\(\chi(n)\) \(1 - \beta_{5}\) \(-\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.551501 + 0.955228i
0.551501 − 0.955228i
1.29877 − 2.24954i
−1.29877 + 2.24954i
−1.27939 + 2.21596i
1.27939 − 2.21596i
−1.27939 − 2.21596i
1.27939 + 2.21596i
1.17141 + 2.02895i
−1.17141 − 2.02895i
1.17141 − 2.02895i
−1.17141 + 2.02895i
−0.551501 − 0.955228i
0.551501 + 0.955228i
1.29877 + 2.24954i
−1.29877 − 2.24954i
0.669131 − 0.0703285i −1.97125 − 1.77492i −1.51351 + 0.321706i −1.44385 − 3.24294i −1.44385 − 1.04902i 0 −2.26988 + 0.737529i 0.421892 + 4.01404i −1.19419 − 2.06840i
19.2 0.669131 − 0.0703285i 1.97125 + 1.77492i −1.51351 + 0.321706i 1.44385 + 3.24294i 1.44385 + 1.04902i 0 −2.26988 + 0.737529i 0.421892 + 4.01404i 1.19419 + 2.06840i
68.1 0.913545 − 0.822560i −0.531168 + 1.19302i −0.0510966 + 0.486152i 0.496087 − 2.33391i 0.496087 + 1.52680i 0 1.79833 + 2.47520i 0.866227 + 0.962043i −1.46658 − 2.54019i
68.2 0.913545 − 0.822560i 0.531168 − 1.19302i −0.0510966 + 0.486152i −0.496087 + 2.33391i −0.496087 − 1.52680i 0 1.79833 + 2.47520i 0.866227 + 0.962043i 1.46658 + 2.54019i
117.1 −0.978148 − 2.19696i −0.357937 − 1.68396i −2.53158 + 2.81160i −3.34948 + 0.352044i −3.34948 + 2.43354i 0 4.07890 + 1.32531i 0.0330227 − 0.0147027i 4.04971 + 7.01430i
117.2 −0.978148 − 2.19696i 0.357937 + 1.68396i −2.53158 + 2.81160i 3.34948 − 0.352044i 3.34948 − 2.43354i 0 4.07890 + 1.32531i 0.0330227 − 0.0147027i −4.04971 − 7.01430i
129.1 −0.978148 + 2.19696i −0.357937 + 1.68396i −2.53158 − 2.81160i −3.34948 − 0.352044i −3.34948 − 2.43354i 0 4.07890 − 1.32531i 0.0330227 + 0.0147027i 4.04971 − 7.01430i
129.2 −0.978148 + 2.19696i 0.357937 − 1.68396i −2.53158 − 2.81160i 3.34948 + 0.352044i 3.34948 + 2.43354i 0 4.07890 − 1.32531i 0.0330227 + 0.0147027i −4.04971 + 7.01430i
178.1 −0.104528 − 0.491768i −2.86425 + 0.301045i 1.59618 − 0.710666i 0.447440 + 0.402877i 0.447440 + 1.37708i 0 −1.10735 − 1.52414i 5.17886 − 1.10080i 0.151352 − 0.262149i
178.2 −0.104528 − 0.491768i 2.86425 − 0.301045i 1.59618 − 0.710666i −0.447440 − 0.402877i −0.447440 − 1.37708i 0 −1.10735 − 1.52414i 5.17886 − 1.10080i −0.151352 + 0.262149i
215.1 −0.104528 + 0.491768i −2.86425 − 0.301045i 1.59618 + 0.710666i 0.447440 − 0.402877i 0.447440 − 1.37708i 0 −1.10735 + 1.52414i 5.17886 + 1.10080i 0.151352 + 0.262149i
215.2 −0.104528 + 0.491768i 2.86425 + 0.301045i 1.59618 + 0.710666i −0.447440 + 0.402877i −0.447440 + 1.37708i 0 −1.10735 + 1.52414i 5.17886 + 1.10080i −0.151352 − 0.262149i
227.1 0.669131 + 0.0703285i −1.97125 + 1.77492i −1.51351 − 0.321706i −1.44385 + 3.24294i −1.44385 + 1.04902i 0 −2.26988 − 0.737529i 0.421892 − 4.01404i −1.19419 + 2.06840i
227.2 0.669131 + 0.0703285i 1.97125 − 1.77492i −1.51351 − 0.321706i 1.44385 − 3.24294i 1.44385 − 1.04902i 0 −2.26988 − 0.737529i 0.421892 − 4.01404i 1.19419 − 2.06840i
325.1 0.913545 + 0.822560i −0.531168 − 1.19302i −0.0510966 − 0.486152i 0.496087 + 2.33391i 0.496087 − 1.52680i 0 1.79833 − 2.47520i 0.866227 − 0.962043i −1.46658 + 2.54019i
325.2 0.913545 + 0.822560i 0.531168 + 1.19302i −0.0510966 − 0.486152i −0.496087 − 2.33391i −0.496087 + 1.52680i 0 1.79833 − 2.47520i 0.866227 − 0.962043i 1.46658 − 2.54019i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
77.n even 30 1 inner
77.o odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.s.b 16
7.b odd 2 1 inner 539.2.s.b 16
7.c even 3 1 77.2.l.b ✓ 16
7.c even 3 1 539.2.s.c 16
7.d odd 6 1 77.2.l.b ✓ 16
7.d odd 6 1 539.2.s.c 16
11.d odd 10 1 539.2.s.c 16
21.g even 6 1 693.2.bu.d 16
21.h odd 6 1 693.2.bu.d 16
77.h odd 6 1 847.2.l.i 16
77.i even 6 1 847.2.l.i 16
77.l even 10 1 539.2.s.c 16
77.m even 15 1 847.2.b.f 16
77.m even 15 1 847.2.l.e 16
77.m even 15 1 847.2.l.i 16
77.m even 15 1 847.2.l.j 16
77.n even 30 1 77.2.l.b ✓ 16
77.n even 30 1 inner 539.2.s.b 16
77.n even 30 1 847.2.b.f 16
77.n even 30 1 847.2.l.e 16
77.n even 30 1 847.2.l.j 16
77.o odd 30 1 77.2.l.b ✓ 16
77.o odd 30 1 inner 539.2.s.b 16
77.o odd 30 1 847.2.b.f 16
77.o odd 30 1 847.2.l.e 16
77.o odd 30 1 847.2.l.j 16
77.p odd 30 1 847.2.b.f 16
77.p odd 30 1 847.2.l.e 16
77.p odd 30 1 847.2.l.i 16
77.p odd 30 1 847.2.l.j 16
231.be even 30 1 693.2.bu.d 16
231.bf odd 30 1 693.2.bu.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.l.b ✓ 16 7.c even 3 1
77.2.l.b ✓ 16 7.d odd 6 1
77.2.l.b ✓ 16 77.n even 30 1
77.2.l.b ✓ 16 77.o odd 30 1
539.2.s.b 16 1.a even 1 1 trivial
539.2.s.b 16 7.b odd 2 1 inner
539.2.s.b 16 77.n even 30 1 inner
539.2.s.b 16 77.o odd 30 1 inner
539.2.s.c 16 7.c even 3 1
539.2.s.c 16 7.d odd 6 1
539.2.s.c 16 11.d odd 10 1
539.2.s.c 16 77.l even 10 1
693.2.bu.d 16 21.g even 6 1
693.2.bu.d 16 21.h odd 6 1
693.2.bu.d 16 231.be even 30 1
693.2.bu.d 16 231.bf odd 30 1
847.2.b.f 16 77.m even 15 1
847.2.b.f 16 77.n even 30 1
847.2.b.f 16 77.o odd 30 1
847.2.b.f 16 77.p odd 30 1
847.2.l.e 16 77.m even 15 1
847.2.l.e 16 77.n even 30 1
847.2.l.e 16 77.o odd 30 1
847.2.l.e 16 77.p odd 30 1
847.2.l.i 16 77.h odd 6 1
847.2.l.i 16 77.i even 6 1
847.2.l.i 16 77.m even 15 1
847.2.l.i 16 77.p odd 30 1
847.2.l.j 16 77.m even 15 1
847.2.l.j 16 77.n even 30 1
847.2.l.j 16 77.o odd 30 1
847.2.l.j 16 77.p odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} + 4T_{2}^{6} - 12T_{2}^{5} + 19T_{2}^{4} - 14T_{2}^{3} + 6T_{2}^{2} - 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{7} + 4 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 10 T^{14} + \cdots + 87025 \) Copy content Toggle raw display
$5$ \( T^{16} + 5 T^{14} + \cdots + 87025 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} + 21 T^{2} + \cdots + 121)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + 15 T^{14} + \cdots + 87025 \) Copy content Toggle raw display
$17$ \( T^{16} + 15 T^{14} + \cdots + 54390625 \) Copy content Toggle raw display
$19$ \( T^{16} + 95 T^{14} + \cdots + 87025 \) Copy content Toggle raw display
$23$ \( (T^{4} - T^{3} + 12 T^{2} + \cdots + 121)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} - 5 T^{7} + \cdots + 961)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + 35 T^{14} + \cdots + 87025 \) Copy content Toggle raw display
$37$ \( (T^{8} - 29 T^{7} + \cdots + 477481)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 61551129025 \) Copy content Toggle raw display
$43$ \( (T^{2} + 3)^{8} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 570971025 \) Copy content Toggle raw display
$53$ \( (T^{8} - 15 T^{7} + \cdots + 25)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1204934313025 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 283945921983025 \) Copy content Toggle raw display
$67$ \( (T^{8} - T^{7} + \cdots + 58081)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 28 T^{7} + \cdots + 8288641)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1204934313025 \) Copy content Toggle raw display
$79$ \( (T^{8} + 29 T^{7} + \cdots + 44521)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1204934313025 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 570971025 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
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