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Properties

Label 539.2.f.e
Level $539$
Weight $2$
Character orbit 539.f
Analytic conductor $4.304$
Analytic rank $0$
Dimension $16$
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] = [539,2,Mod(148,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.148");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.f (of order \(5\), degree \(4\), 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: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{5}) q^{2} + ( - \beta_{13} + \beta_{11} - \beta_{9}) q^{3} + (\beta_{11} + \beta_{7} + \beta_{2}) q^{4} + ( - \beta_{13} + \beta_{7} + \cdots - \beta_1) q^{5}+ \cdots + (\beta_{14} - \beta_{12} + \cdots - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{5}) q^{2} + ( - \beta_{13} + \beta_{11} - \beta_{9}) q^{3} + (\beta_{11} + \beta_{7} + \beta_{2}) q^{4} + ( - \beta_{13} + \beta_{7} + \cdots - \beta_1) q^{5}+ \cdots + ( - 3 \beta_{15} + 2 \beta_{12} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{2} + 2 q^{3} - 11 q^{4} + 5 q^{5} - 3 q^{6} - 5 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 3 q^{2} + 2 q^{3} - 11 q^{4} + 5 q^{5} - 3 q^{6} - 5 q^{8} - 12 q^{9} - 12 q^{10} - 3 q^{11} - 18 q^{12} + 7 q^{13} - 18 q^{15} + 17 q^{16} + 5 q^{17} + 11 q^{18} - 19 q^{19} - q^{20} - 33 q^{22} + 32 q^{23} + 35 q^{24} + 7 q^{25} + 27 q^{26} - 10 q^{27} + 3 q^{29} - 2 q^{30} + 7 q^{31} + 32 q^{32} + 26 q^{33} + 24 q^{34} + 52 q^{36} + 4 q^{37} + 5 q^{38} + 11 q^{39} + 10 q^{40} + 10 q^{41} - 8 q^{43} - 38 q^{44} - 70 q^{45} - 42 q^{46} + 23 q^{47} + 36 q^{48} + 52 q^{50} - 29 q^{51} - 33 q^{52} + 4 q^{53} - 60 q^{54} + 12 q^{55} - 11 q^{57} + 20 q^{58} - 17 q^{59} - 30 q^{60} + 7 q^{61} - 79 q^{62} + 7 q^{64} - 8 q^{65} - 8 q^{66} - 38 q^{67} + 2 q^{68} - 10 q^{69} - 14 q^{71} + 35 q^{73} - 29 q^{74} - 9 q^{75} - 52 q^{76} - 58 q^{78} + 15 q^{79} + 87 q^{80} - 14 q^{81} - 19 q^{82} - 5 q^{83} + 6 q^{85} - 52 q^{86} + 72 q^{87} + 55 q^{88} - 74 q^{89} + 14 q^{90} - 55 q^{92} + 32 q^{93} + 24 q^{94} + 32 q^{95} + 42 q^{96} - 20 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 33722118880975 \nu^{15} - 215936836813390 \nu^{14} + 154302737507927 \nu^{13} + \cdots + 88\!\cdots\!50 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 686277383720070 \nu^{15} + \cdots - 15\!\cdots\!35 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 874936745197272 \nu^{15} - 514706296938285 \nu^{14} + \cdots - 92\!\cdots\!35 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!39 \nu^{15} + \cdots + 95\!\cdots\!50 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!90 \nu^{15} + \cdots - 18\!\cdots\!25 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!90 \nu^{15} + \cdots + 18\!\cdots\!25 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35\!\cdots\!14 \nu^{15} + \cdots + 82\!\cdots\!05 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38\!\cdots\!02 \nu^{15} + \cdots + 22\!\cdots\!25 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41\!\cdots\!60 \nu^{15} + \cdots - 29\!\cdots\!75 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 73\!\cdots\!73 \nu^{15} + \cdots - 84\!\cdots\!55 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 54\!\cdots\!19 \nu^{15} + \cdots + 86\!\cdots\!50 ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{15} + \cdots + 59\!\cdots\!75 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16\!\cdots\!01 \nu^{15} + \cdots + 16\!\cdots\!55 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 16\!\cdots\!14 \nu^{15} + \cdots - 10\!\cdots\!35 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 31\!\cdots\!64 \nu^{15} + \cdots + 34\!\cdots\!95 ) / 45\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 2\beta_{10} - 2\beta_{8} + 3\beta_{7} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{8} - 5\beta_{6} - \beta_{5} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - \beta_{14} - \beta_{11} - 8 \beta_{10} - 6 \beta_{9} - 15 \beta_{7} + \beta_{5} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{14} - 7 \beta_{13} - 7 \beta_{12} + 7 \beta_{11} - 8 \beta_{8} + 8 \beta_{7} + 10 \beta_{6} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{15} + \beta_{14} - 7 \beta_{13} + 11 \beta_{12} + 54 \beta_{10} + 35 \beta_{9} + 40 \beta_{8} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 42 \beta_{15} + 54 \beta_{14} + 13 \beta_{12} - 45 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} + 30 \beta_{8} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 9 \beta_{15} + 38 \beta_{13} - 210 \beta_{12} + 89 \beta_{11} - 234 \beta_{10} - 89 \beta_{9} + \cdots - 98 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 346 \beta_{15} - 346 \beta_{14} + 248 \beta_{13} + 117 \beta_{11} + 38 \beta_{10} + 290 \beta_{9} + \cdots - 94 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 56 \beta_{14} - 56 \beta_{13} + 1290 \beta_{12} - 1290 \beta_{11} + 1992 \beta_{8} - 1992 \beta_{7} + \cdots + 1406 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2174 \beta_{15} + 1477 \beta_{14} - 2174 \beta_{13} - 913 \beta_{12} - 447 \beta_{10} + \cdots - 3461 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 284 \beta_{15} + 848 \beta_{14} - 4374 \beta_{12} + 8050 \beta_{11} + 8579 \beta_{10} + 4374 \beta_{9} + \cdots - 8669 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 8898 \beta_{15} + 13556 \beta_{13} + 12425 \beta_{12} - 6647 \beta_{11} + 2237 \beta_{10} + \cdots + 5394 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 3382 \beta_{15} - 3382 \beta_{14} + 1131 \beta_{13} - 29041 \beta_{11} - 85364 \beta_{10} + \cdots + 31232 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 54132 \beta_{14} - 54132 \beta_{13} - 82100 \beta_{12} + 82100 \beta_{11} - 120596 \beta_{8} + \cdots - 67213 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(442\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7} - \beta_{8} + \beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1
0.751051 + 2.31150i
0.435488 + 1.34029i
−0.206962 − 0.636964i
−0.788594 − 2.42704i
1.60551 + 1.16647i
0.901622 + 0.655067i
0.183009 + 0.132964i
−1.38112 − 1.00344i
0.751051 − 2.31150i
0.435488 − 1.34029i
−0.206962 + 0.636964i
−0.788594 + 2.42704i
1.60551 − 1.16647i
0.901622 − 0.655067i
0.183009 − 0.132964i
−1.38112 + 1.00344i
−0.751051 − 2.31150i 1.16030 + 0.843005i −3.16091 + 2.29654i −0.388938 + 1.19703i 1.07716 − 3.31516i 0 3.74989 + 2.72445i −0.291419 − 0.896896i 3.05904
148.2 −0.435488 − 1.34029i −1.75021 − 1.27160i 0.0112975 − 0.00820814i 0.565930 − 1.74175i −0.942126 + 2.89957i 0 −2.29616 − 1.66826i 0.519216 + 1.59798i −2.58091
148.3 0.206962 + 0.636964i 2.54013 + 1.84551i 1.25514 − 0.911915i −0.662464 + 2.03885i −0.649815 + 1.99992i 0 1.92429 + 1.39808i 2.11929 + 6.52251i −1.43578
148.4 0.788594 + 2.42704i −0.332181 − 0.241344i −3.65062 + 2.65233i −1.05961 + 3.26115i 0.323795 − 0.996539i 0 −5.18703 − 3.76860i −0.874954 − 2.69283i −8.75055
246.1 −1.60551 − 1.16647i −0.861043 + 2.65002i 0.598967 + 1.84343i −0.0217822 + 0.0158257i 4.47357 − 3.25024i 0 −0.0378378 + 0.116453i −3.85415 − 2.80020i 0.0534317
246.2 −0.901622 − 0.655067i 0.883423 − 2.71890i −0.234224 − 0.720867i 2.79603 − 2.03143i −2.57757 + 1.87272i 0 −0.949813 + 2.92322i −4.18492 − 3.04052i −3.85168
246.3 −0.183009 − 0.132964i 0.0677147 − 0.208405i −0.602221 − 1.85345i −2.01892 + 1.46683i −0.0401026 + 0.0291363i 0 −0.276036 + 0.849550i 2.38820 + 1.73513i 0.564516
246.4 1.38112 + 1.00344i −0.708129 + 2.17940i 0.282562 + 0.869638i 3.28976 − 2.39015i −3.16491 + 2.29944i 0 0.572703 − 1.76260i −1.82128 − 1.32323i 6.94194
295.1 −0.751051 + 2.31150i 1.16030 − 0.843005i −3.16091 − 2.29654i −0.388938 − 1.19703i 1.07716 + 3.31516i 0 3.74989 − 2.72445i −0.291419 + 0.896896i 3.05904
295.2 −0.435488 + 1.34029i −1.75021 + 1.27160i 0.0112975 + 0.00820814i 0.565930 + 1.74175i −0.942126 − 2.89957i 0 −2.29616 + 1.66826i 0.519216 − 1.59798i −2.58091
295.3 0.206962 − 0.636964i 2.54013 − 1.84551i 1.25514 + 0.911915i −0.662464 − 2.03885i −0.649815 − 1.99992i 0 1.92429 − 1.39808i 2.11929 − 6.52251i −1.43578
295.4 0.788594 − 2.42704i −0.332181 + 0.241344i −3.65062 − 2.65233i −1.05961 − 3.26115i 0.323795 + 0.996539i 0 −5.18703 + 3.76860i −0.874954 + 2.69283i −8.75055
344.1 −1.60551 + 1.16647i −0.861043 − 2.65002i 0.598967 − 1.84343i −0.0217822 − 0.0158257i 4.47357 + 3.25024i 0 −0.0378378 − 0.116453i −3.85415 + 2.80020i 0.0534317
344.2 −0.901622 + 0.655067i 0.883423 + 2.71890i −0.234224 + 0.720867i 2.79603 + 2.03143i −2.57757 − 1.87272i 0 −0.949813 − 2.92322i −4.18492 + 3.04052i −3.85168
344.3 −0.183009 + 0.132964i 0.0677147 + 0.208405i −0.602221 + 1.85345i −2.01892 − 1.46683i −0.0401026 − 0.0291363i 0 −0.276036 − 0.849550i 2.38820 − 1.73513i 0.564516
344.4 1.38112 − 1.00344i −0.708129 − 2.17940i 0.282562 − 0.869638i 3.28976 + 2.39015i −3.16491 − 2.29944i 0 0.572703 + 1.76260i −1.82128 + 1.32323i 6.94194
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 148.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.f.e 16
7.b odd 2 1 77.2.f.b ✓ 16
7.c even 3 2 539.2.q.f 32
7.d odd 6 2 539.2.q.g 32
11.c even 5 1 inner 539.2.f.e 16
11.c even 5 1 5929.2.a.bt 8
11.d odd 10 1 5929.2.a.bs 8
21.c even 2 1 693.2.m.i 16
77.b even 2 1 847.2.f.x 16
77.j odd 10 1 77.2.f.b ✓ 16
77.j odd 10 1 847.2.a.p 8
77.j odd 10 2 847.2.f.w 16
77.l even 10 1 847.2.a.o 8
77.l even 10 2 847.2.f.v 16
77.l even 10 1 847.2.f.x 16
77.m even 15 2 539.2.q.f 32
77.p odd 30 2 539.2.q.g 32
231.r odd 10 1 7623.2.a.cw 8
231.u even 10 1 693.2.m.i 16
231.u even 10 1 7623.2.a.ct 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b ✓ 16 7.b odd 2 1
77.2.f.b ✓ 16 77.j odd 10 1
539.2.f.e 16 1.a even 1 1 trivial
539.2.f.e 16 11.c even 5 1 inner
539.2.q.f 32 7.c even 3 2
539.2.q.f 32 77.m even 15 2
539.2.q.g 32 7.d odd 6 2
539.2.q.g 32 77.p odd 30 2
693.2.m.i 16 21.c even 2 1
693.2.m.i 16 231.u even 10 1
847.2.a.o 8 77.l even 10 1
847.2.a.p 8 77.j odd 10 1
847.2.f.v 16 77.l even 10 2
847.2.f.w 16 77.j odd 10 2
847.2.f.x 16 77.b even 2 1
847.2.f.x 16 77.l even 10 1
5929.2.a.bs 8 11.d odd 10 1
5929.2.a.bt 8 11.c even 5 1
7623.2.a.ct 8 231.u even 10 1
7623.2.a.cw 8 231.r odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\( T_{2}^{16} + 3 T_{2}^{15} + 14 T_{2}^{14} + 32 T_{2}^{13} + 86 T_{2}^{12} + 145 T_{2}^{11} + 245 T_{2}^{10} + \cdots + 25 \) Copy content Toggle raw display
\( T_{3}^{16} - 2 T_{3}^{15} + 14 T_{3}^{14} - 26 T_{3}^{13} + 124 T_{3}^{12} - 100 T_{3}^{11} + 747 T_{3}^{10} + \cdots + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 3 T^{15} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{16} - 2 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{16} - 5 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 188897536 \) Copy content Toggle raw display
$17$ \( T^{16} - 5 T^{15} + \cdots + 2611456 \) Copy content Toggle raw display
$19$ \( T^{16} + 19 T^{15} + \cdots + 62726400 \) Copy content Toggle raw display
$23$ \( (T^{8} - 16 T^{7} + \cdots + 859)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} - 3 T^{15} + \cdots + 245025 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 2629638400 \) Copy content Toggle raw display
$37$ \( T^{16} - 4 T^{15} + \cdots + 212521 \) Copy content Toggle raw display
$41$ \( T^{16} - 10 T^{15} + \cdots + 13424896 \) Copy content Toggle raw display
$43$ \( (T^{8} + 4 T^{7} + \cdots - 971)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 5345713926400 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 310840815961 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 187142400 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 253446400 \) Copy content Toggle raw display
$67$ \( (T^{8} + 19 T^{7} + \cdots - 27395)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 379119841 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 105069332736 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 15858514175625 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 756470016 \) Copy content Toggle raw display
$89$ \( (T^{8} + 37 T^{7} + \cdots + 952400)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 4647025244416 \) Copy content Toggle raw display
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