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Properties

Label 539.2.q.b
Level $539$
Weight $2$
Character orbit 539.q
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(214,539)]
 
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([20, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.214");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.q (of order \(15\), 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 \(\Q(\zeta_{15})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 5 x^{14} + 16 x^{13} + 4 x^{12} - 29 x^{11} + 10 x^{10} - 156 x^{9} + 251 x^{8} + \cdots + 625 \) 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: $\mathrm{SU}(2)[C_{15}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{12} + \beta_{11} + \cdots + \beta_{2}) q^{2}+ \cdots + (\beta_{14} + 2 \beta_{12} + \beta_{6} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{12} + \beta_{11} + \cdots + \beta_{2}) q^{2}+ \cdots + ( - 2 \beta_{10} + 3 \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{2} - 4 q^{3} - 3 q^{4} + 3 q^{5} - 6 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{2} - 4 q^{3} - 3 q^{4} + 3 q^{5} - 6 q^{6} + 6 q^{8} + 2 q^{9} - 28 q^{10} - 5 q^{11} - 14 q^{12} - 10 q^{13} + 12 q^{15} + 3 q^{16} - 11 q^{17} - 4 q^{18} - 9 q^{19} - 42 q^{20} - 2 q^{22} + 16 q^{23} + 21 q^{24} - 5 q^{25} + 21 q^{26} + 44 q^{27} - 18 q^{29} - 14 q^{30} - 11 q^{31} + 20 q^{32} + 10 q^{33} + 48 q^{34} - 4 q^{36} - 6 q^{37} + 35 q^{38} + 5 q^{39} - 16 q^{40} + 44 q^{41} + 32 q^{43} - 29 q^{44} + 18 q^{45} - 29 q^{46} + 7 q^{47} - 8 q^{48} - 68 q^{50} - 3 q^{51} + 21 q^{52} - 2 q^{53} + 4 q^{54} - 52 q^{55} - 6 q^{57} + 39 q^{58} + 25 q^{59} + 38 q^{60} + 7 q^{61} + 10 q^{62} + 2 q^{64} - 24 q^{65} + 18 q^{66} + 30 q^{67} + 8 q^{68} - 16 q^{69} - 28 q^{71} - 3 q^{72} + 3 q^{73} + 9 q^{74} + 5 q^{75} + 104 q^{76} - 36 q^{78} + 9 q^{79} - 33 q^{80} + 28 q^{81} + 31 q^{82} - 46 q^{83} - 20 q^{85} + 17 q^{86} + 12 q^{87} + 7 q^{88} - 34 q^{89} - 4 q^{90} - 68 q^{92} - 8 q^{93} - 30 q^{94} - 24 q^{95} + 10 q^{96} - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - 5 x^{14} + 16 x^{13} + 4 x^{12} - 29 x^{11} + 10 x^{10} - 156 x^{9} + 251 x^{8} + \cdots + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10747578565 \nu^{15} - 390646149564 \nu^{14} + 595438889614 \nu^{13} + \cdots - 81680866488250 ) / 451236007807750 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11294463 \nu^{15} - 216782207 \nu^{14} + 202974090 \nu^{13} + 547473112 \nu^{12} + \cdots + 153447866875 ) / 262194077750 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4442741005 \nu^{15} - 50549404333 \nu^{14} + 57359873198 \nu^{13} + 277539827085 \nu^{12} + \cdots - 11984339635250 ) / 90247201561550 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45506971144 \nu^{15} + 350661785078 \nu^{14} - 149323399494 \nu^{13} + \cdots - 322941345212125 ) / 451236007807750 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 83632252305 \nu^{15} + 367332418783 \nu^{14} - 21593146433 \nu^{13} + \cdots - 334386818347500 ) / 451236007807750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 95874717082 \nu^{15} - 73661012057 \nu^{14} - 732120607075 \nu^{13} + \cdots - 20998151463750 ) / 451236007807750 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 30198383055 \nu^{15} + 119405641263 \nu^{14} + 4417433767 \nu^{13} + \cdots + 38634849837750 ) / 90247201561550 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 154162898659 \nu^{15} - 220169951101 \nu^{14} + 919001199770 \nu^{13} + \cdots - 221616106153750 ) / 451236007807750 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 169638367088 \nu^{15} + 295656442235 \nu^{14} - 1142785999873 \nu^{13} + \cdots + 288196456635125 ) / 451236007807750 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 176090223129 \nu^{15} - 541590004299 \nu^{14} - 1156604341685 \nu^{13} + \cdots + 224933071900000 ) / 451236007807750 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 354585769846 \nu^{15} + 508748668505 \nu^{14} + 1993098800331 \nu^{13} + \cdots - 236118590095250 ) / 451236007807750 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 379898570999 \nu^{15} + 649176782439 \nu^{14} + 1851783118665 \nu^{13} + \cdots - 6717236603125 ) / 451236007807750 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 245516587 \nu^{15} + 234222124 \nu^{14} + 1010800728 \nu^{13} - 3725291302 \nu^{12} + \cdots - 228316588875 ) / 262194077750 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 493169051379 \nu^{15} - 864985162329 \nu^{14} - 2475405530945 \nu^{13} + \cdots + 516245422374375 ) / 451236007807750 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} - \beta_{13} + \beta_{12} - \beta_{9} + 3\beta_{7} - \beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{6} + 4\beta_{4} - 2\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} + 7 \beta_{14} - 7 \beta_{13} + 7 \beta_{12} + 13 \beta_{10} + 2 \beta_{8} + 13 \beta_{7} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20\beta_{15} + 12\beta_{14} + 8\beta_{13} - 11\beta_{12} - 20\beta_{11} + 8\beta_{9} - 11\beta_{7} - 8\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36\beta_{10} + 19\beta_{9} + 32\beta_{6} - 19\beta_{4} + 24\beta_{3} + 36\beta_{2} - 24\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 111 \beta_{15} + 111 \beta_{13} - 81 \beta_{12} - 55 \beta_{11} - 148 \beta_{10} - 111 \beta_{8} + \cdots + 148 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -136\beta_{15} - 221\beta_{14} + 167\beta_{12} + 259\beta_{11} + 221\beta_{5} + 136\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -913\beta_{10} - 647\beta_{9} - 357\beta_{8} - 357\beta_{3} - 531\beta_{2} + 357\beta _1 + 531 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1560 \beta_{15} - 937 \beta_{14} - 888 \beta_{13} + 1361 \beta_{12} + 1560 \beta_{11} + 1361 \beta_{10} + \cdots - 2298 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3336\beta_{14} - 2249\beta_{13} - 2249\beta_{11} - 2249\beta_{9} + 3336\beta_{7} - 5568\beta_{5} - 1609\beta_{3} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 13823 \beta_{10} + 9426 \beta_{9} + 9426 \beta_{8} - 5467 \beta_{6} + 5585 \beta_{4} + 5585 \beta_{3} + \cdots - 13823 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13941 \beta_{15} + 20596 \beta_{14} - 13267 \beta_{12} - 23249 \beta_{11} - 13941 \beta_{8} + \cdots + 20596 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 34537 \beta_{15} - 32557 \beta_{14} + 57112 \beta_{13} - 32557 \beta_{12} + 57112 \beta_{9} + \cdots + 83688 \beta_{5} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 126186 \beta_{10} - 85668 \beta_{9} - 140800 \beta_{8} + 126186 \beta_{6} - 140800 \beta_{4} + \cdots + 205873 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(442\)
\(\chi(n)\) \(-1 - \beta_{14}\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1
1.62381 + 0.722968i
−0.710267 − 0.316231i
−0.981435 + 1.08999i
1.65057 − 1.83314i
0.0812692 + 0.773225i
−0.185798 − 1.76775i
0.0812692 − 0.773225i
−0.185798 + 1.76775i
−2.41283 + 0.512862i
1.43468 − 0.304951i
−2.41283 − 0.512862i
1.43468 + 0.304951i
1.62381 − 0.722968i
−0.710267 + 0.316231i
−0.981435 − 1.08999i
1.65057 + 1.83314i
−0.520239 + 0.577783i −0.564602 + 0.251377i 0.145871 + 1.38787i −0.217653 + 0.0462636i 0.148486 − 0.456994i 0 −2.13577 − 1.55173i −1.75181 + 1.94558i 0.0865012 − 0.149825i
214.2 1.18937 − 1.32093i −0.564602 + 0.251377i −0.121196 − 1.15310i −2.71679 + 0.577471i −0.339469 + 1.04478i 0 1.20872 + 0.878189i −1.75181 + 1.94558i −2.46847 + 4.27551i
312.1 −0.257844 + 2.45322i 1.08268 + 1.20243i −3.99550 − 0.849271i 3.16702 + 1.41005i −3.22899 + 2.34600i 0 1.58914 − 4.89086i 0.0399263 − 0.379874i −4.27575 + 7.40581i
312.2 0.153315 − 1.45870i 1.08268 + 1.20243i −0.147996 − 0.0314575i −0.426381 − 0.189837i 1.91998 − 1.39494i 0 0.837913 − 2.57883i 0.0399263 − 0.379874i −0.342285 + 0.592855i
324.1 −1.73864 − 0.369560i 0.0646021 − 0.614648i 1.05921 + 0.471591i 1.85850 + 2.06407i −0.339469 + 1.04478i 0 1.20872 + 0.878189i 2.56082 + 0.544320i −2.46847 − 4.27551i
324.2 0.760494 + 0.161648i 0.0646021 − 0.614648i −1.27487 − 0.567608i 0.148892 + 0.165361i 0.148486 − 0.456994i 0 −2.13577 − 1.55173i 2.56082 + 0.544320i 0.0865012 + 0.149825i
361.1 −1.73864 + 0.369560i 0.0646021 + 0.614648i 1.05921 − 0.471591i 1.85850 − 2.06407i −0.339469 − 1.04478i 0 1.20872 − 0.878189i 2.56082 − 0.544320i −2.46847 + 4.27551i
361.2 0.760494 − 0.161648i 0.0646021 + 0.614648i −1.27487 + 0.567608i 0.148892 − 0.165361i 0.148486 + 0.456994i 0 −2.13577 + 1.55173i 2.56082 − 0.544320i 0.0865012 − 0.149825i
410.1 −1.33993 − 0.596574i −1.58268 − 0.336408i 0.101241 + 0.112439i 0.0487868 − 0.464175i 1.91998 + 1.39494i 0 0.837913 + 2.57883i −0.348943 − 0.155360i −0.342285 + 0.592855i
410.2 2.25347 + 1.00331i −1.58268 − 0.336408i 2.73324 + 3.03557i −0.362372 + 3.44774i −3.22899 − 2.34600i 0 1.58914 + 4.89086i −0.348943 − 0.155360i −4.27575 + 7.40581i
422.1 −1.33993 + 0.596574i −1.58268 + 0.336408i 0.101241 − 0.112439i 0.0487868 + 0.464175i 1.91998 − 1.39494i 0 0.837913 − 2.57883i −0.348943 + 0.155360i −0.342285 − 0.592855i
422.2 2.25347 − 1.00331i −1.58268 + 0.336408i 2.73324 − 3.03557i −0.362372 − 3.44774i −3.22899 + 2.34600i 0 1.58914 − 4.89086i −0.348943 + 0.155360i −4.27575 − 7.40581i
471.1 −0.520239 − 0.577783i −0.564602 − 0.251377i 0.145871 − 1.38787i −0.217653 − 0.0462636i 0.148486 + 0.456994i 0 −2.13577 + 1.55173i −1.75181 − 1.94558i 0.0865012 + 0.149825i
471.2 1.18937 + 1.32093i −0.564602 − 0.251377i −0.121196 + 1.15310i −2.71679 − 0.577471i −0.339469 − 1.04478i 0 1.20872 − 0.878189i −1.75181 − 1.94558i −2.46847 − 4.27551i
520.1 −0.257844 − 2.45322i 1.08268 − 1.20243i −3.99550 + 0.849271i 3.16702 − 1.41005i −3.22899 − 2.34600i 0 1.58914 + 4.89086i 0.0399263 + 0.379874i −4.27575 − 7.40581i
520.2 0.153315 + 1.45870i 1.08268 − 1.20243i −0.147996 + 0.0314575i −0.426381 + 0.189837i 1.91998 + 1.39494i 0 0.837913 + 2.57883i 0.0399263 + 0.379874i −0.342285 − 0.592855i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 214.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.q.b 16
7.b odd 2 1 539.2.q.c 16
7.c even 3 1 539.2.f.d 8
7.c even 3 1 inner 539.2.q.b 16
7.d odd 6 1 77.2.f.a ✓ 8
7.d odd 6 1 539.2.q.c 16
11.c even 5 1 inner 539.2.q.b 16
21.g even 6 1 693.2.m.g 8
77.i even 6 1 847.2.f.q 8
77.j odd 10 1 539.2.q.c 16
77.m even 15 1 539.2.f.d 8
77.m even 15 1 inner 539.2.q.b 16
77.m even 15 1 5929.2.a.bi 4
77.n even 30 1 847.2.a.k 4
77.n even 30 1 847.2.f.q 8
77.n even 30 2 847.2.f.s 8
77.o odd 30 1 5929.2.a.bb 4
77.p odd 30 1 77.2.f.a ✓ 8
77.p odd 30 1 539.2.q.c 16
77.p odd 30 1 847.2.a.l 4
77.p odd 30 2 847.2.f.p 8
231.bc even 30 1 693.2.m.g 8
231.bc even 30 1 7623.2.a.ch 4
231.bf odd 30 1 7623.2.a.co 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a ✓ 8 7.d odd 6 1
77.2.f.a ✓ 8 77.p odd 30 1
539.2.f.d 8 7.c even 3 1
539.2.f.d 8 77.m even 15 1
539.2.q.b 16 1.a even 1 1 trivial
539.2.q.b 16 7.c even 3 1 inner
539.2.q.b 16 11.c even 5 1 inner
539.2.q.b 16 77.m even 15 1 inner
539.2.q.c 16 7.b odd 2 1
539.2.q.c 16 7.d odd 6 1
539.2.q.c 16 77.j odd 10 1
539.2.q.c 16 77.p odd 30 1
693.2.m.g 8 21.g even 6 1
693.2.m.g 8 231.bc even 30 1
847.2.a.k 4 77.n even 30 1
847.2.a.l 4 77.p odd 30 1
847.2.f.p 8 77.p odd 30 2
847.2.f.q 8 77.i even 6 1
847.2.f.q 8 77.n even 30 1
847.2.f.s 8 77.n even 30 2
5929.2.a.bb 4 77.o odd 30 1
5929.2.a.bi 4 77.m even 15 1
7623.2.a.ch 4 231.bc even 30 1
7623.2.a.co 4 231.bf odd 30 1

Hecke kernels

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

\( T_{2}^{16} - T_{2}^{15} + T_{2}^{13} - 16 T_{2}^{12} + 56 T_{2}^{11} + 80 T_{2}^{10} - 101 T_{2}^{9} + \cdots + 625 \) Copy content Toggle raw display
\( T_{3}^{8} + 2T_{3}^{7} + 2T_{3}^{5} + 9T_{3}^{4} + 8T_{3}^{3} + 5T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{15} + \cdots + 625 \) Copy content Toggle raw display
$3$ \( (T^{8} + 2 T^{7} + 2 T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} - 3 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} + 5 T^{7} + \cdots + 841)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 11 T^{15} + \cdots + 25411681 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 442050625 \) Copy content Toggle raw display
$23$ \( (T^{8} - 8 T^{7} + \cdots + 42025)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 9 T^{7} + \cdots + 164025)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + 11 T^{15} + \cdots + 625 \) Copy content Toggle raw display
$37$ \( T^{16} + 6 T^{15} + \cdots + 625 \) Copy content Toggle raw display
$41$ \( (T^{8} - 22 T^{7} + \cdots + 6241)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 41)^{8} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 8653650625 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 570268135921 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1998607065841 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 980149500625 \) Copy content Toggle raw display
$67$ \( (T^{8} - 15 T^{7} + \cdots + 39601)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 14 T^{7} + \cdots + 982081)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 612593437890625 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 5393580481 \) Copy content Toggle raw display
$83$ \( (T^{8} + 23 T^{7} + \cdots + 5040025)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 17 T^{7} + \cdots + 570025)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 30 T^{7} + \cdots + 17850625)^{2} \) Copy content Toggle raw display
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