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Properties

Label 99.2.j.a
Level $99$
Weight $2$
Character orbit 99.j
Analytic conductor $0.791$
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] = [99,2,Mod(8,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.j (of order \(10\), 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: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + (2 \beta_{15} - \beta_{14} + \cdots + \beta_{8}) q^{2}+ \cdots + ( - \beta_{15} + 2 \beta_{14} + \cdots + 2 \beta_{10}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{15} - \beta_{14} + \cdots + \beta_{8}) q^{2}+ \cdots + ( - 12 \beta_{15} + 12 \beta_{14} + \cdots + 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} - 20 q^{16} - 48 q^{22} - 32 q^{25} + 40 q^{28} + 16 q^{31} + 40 q^{34} - 12 q^{37} + 60 q^{40} - 40 q^{46} - 24 q^{49} - 40 q^{52} + 16 q^{55} + 12 q^{58} + 36 q^{64} + 96 q^{67} + 76 q^{70} - 20 q^{73} - 12 q^{82} - 100 q^{85} - 12 q^{88} - 72 q^{91} - 80 q^{94} + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 155 \nu^{14} - 1408 \nu^{12} - 4611 \nu^{10} + 9188 \nu^{8} + 65146 \nu^{6} - 26953 \nu^{4} + \cdots + 108500 ) / 176275 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5996 \nu^{14} + 4807 \nu^{12} - 95306 \nu^{10} - 277252 \nu^{8} + 500966 \nu^{6} + 1108500 \nu^{4} + \cdots + 2787000 ) / 881375 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8474 \nu^{14} + 2743 \nu^{12} - 149594 \nu^{10} - 389798 \nu^{8} + 955934 \nu^{6} + 2009635 \nu^{4} + \cdots + 3175500 ) / 881375 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8672 \nu^{14} - 3649 \nu^{12} + 153342 \nu^{10} + 402539 \nu^{8} - 938462 \nu^{6} + \cdots - 3730750 ) / 881375 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12916 \nu^{14} - 10637 \nu^{12} + 225196 \nu^{10} + 690882 \nu^{8} - 1135131 \nu^{6} + \cdots - 6749625 ) / 881375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14403 \nu^{14} - 14916 \nu^{12} + 224078 \nu^{10} + 748351 \nu^{8} - 1023533 \nu^{6} + \cdots - 6732875 ) / 881375 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} + \cdots - 598500 \nu ) / 400625 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22098 \nu^{14} - 13706 \nu^{12} + 377023 \nu^{10} + 1116041 \nu^{8} - 1983428 \nu^{6} + \cdots - 11238000 ) / 881375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22194 \nu^{15} + 11523 \nu^{13} - 339409 \nu^{11} - 1024903 \nu^{9} + 1669899 \nu^{7} + \cdots + 9814875 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25903 \nu^{15} - 3944 \nu^{13} - 422373 \nu^{11} - 826866 \nu^{9} + 3248903 \nu^{7} + \cdots + 4286500 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 8474 \nu^{15} + 2743 \nu^{13} - 149594 \nu^{11} - 389798 \nu^{9} + 955934 \nu^{7} + \cdots + 2294125 \nu ) / 881375 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8672 \nu^{15} - 3649 \nu^{13} + 153342 \nu^{11} + 402539 \nu^{9} - 938462 \nu^{7} + \cdots - 3730750 \nu ) / 881375 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 50508 \nu^{15} - 41726 \nu^{13} + 817683 \nu^{11} + 2587261 \nu^{9} - 4065438 \nu^{7} + \cdots - 22711875 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11661 \nu^{15} - 8721 \nu^{13} + 202398 \nu^{11} + 611766 \nu^{9} - 1064278 \nu^{7} + \cdots - 5486525 \nu ) / 881375 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - 2\beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 3\beta_{10} + 2\beta_{8} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 2\beta_{6} + 8\beta_{5} + 5\beta_{4} + 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{15} + 13\beta_{14} + 12\beta_{13} + 7\beta_{11} + 7\beta_{10} - 5\beta_{8} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{9} - 11\beta_{7} - 6\beta_{6} + 11\beta_{4} - 15\beta_{3} + 5\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 20\beta_{15} - 43\beta_{14} + 9\beta_{12} - 4\beta_{11} - 43\beta_{10} + 20\beta_{8} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 32\beta_{9} - 52\beta_{6} + 53\beta_{5} + 18\beta_{4} + 53\beta_{3} + 32\beta_{2} - 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -209\beta_{15} + 137\beta_{14} + 137\beta_{13} - 87\beta_{12} + 137\beta_{11} + 21\beta_{10} - 21\beta_{8} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 21\beta_{9} - 36\beta_{7} + 275\beta_{5} + 275\beta_{4} + 166 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 78\beta_{15} - 129\beta_{14} + 239\beta_{13} + 239\beta_{12} - 78\beta_{10} + 384\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 752\beta_{9} - 462\beta_{7} - 752\beta_{6} - 110\beta_{5} - 177\beta_{3} + 462\beta_{2} + 110 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1037 \beta_{15} - 639 \beta_{14} + 642 \beta_{13} - 1037 \beta_{12} + 1037 \beta_{11} + \cdots - 642 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 639\beta_{7} - 639\beta_{6} + 5431\beta_{5} + 3360\beta_{4} + 3360\beta_{3} + 400\beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -6716\beta_{15} + 6716\beta_{14} + 7109\beta_{13} + 4399\beta_{11} + 4399\beta_{10} - 2960\beta_{8} + 4399\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(\beta_{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
−0.0783900 − 1.17295i
−0.752864 + 0.902863i
0.752864 − 0.902863i
0.0783900 + 1.17295i
0.556839 − 1.81878i
1.90184 + 0.0324487i
−1.90184 − 0.0324487i
−0.556839 + 1.81878i
0.556839 + 1.81878i
1.90184 − 0.0324487i
−1.90184 + 0.0324487i
−0.556839 − 1.81878i
−0.0783900 + 1.17295i
−0.752864 − 0.902863i
0.752864 + 0.902863i
0.0783900 − 1.17295i
−1.97102 + 1.43203i 0 1.21618 − 3.74302i 2.23109 − 3.07083i 0 −0.349790 − 0.113654i 1.45728 + 4.48505i 0 9.24768i
8.2 −0.205228 + 0.149107i 0 −0.598148 + 1.84091i −1.71735 + 2.36373i 0 2.58586 + 0.840196i −0.308515 − 0.949513i 0 − 0.741170i
8.3 0.205228 − 0.149107i 0 −0.598148 + 1.84091i 1.71735 − 2.36373i 0 2.58586 + 0.840196i 0.308515 + 0.949513i 0 − 0.741170i
8.4 1.97102 − 1.43203i 0 1.21618 − 3.74302i −2.23109 + 3.07083i 0 −0.349790 − 0.113654i −1.45728 − 4.48505i 0 9.24768i
17.1 −0.726437 − 2.23574i 0 −2.85280 + 2.07268i −2.13811 − 0.694712i 0 −2.38116 − 3.27739i 2.90269 + 2.10893i 0 5.28492i
17.2 −0.212694 − 0.654604i 0 1.23477 − 0.897110i 0.0381457 + 0.0123943i 0 0.145094 + 0.199704i −1.96356 − 1.42661i 0 − 0.0276065i
17.3 0.212694 + 0.654604i 0 1.23477 − 0.897110i −0.0381457 − 0.0123943i 0 0.145094 + 0.199704i 1.96356 + 1.42661i 0 − 0.0276065i
17.4 0.726437 + 2.23574i 0 −2.85280 + 2.07268i 2.13811 + 0.694712i 0 −2.38116 − 3.27739i −2.90269 − 2.10893i 0 5.28492i
35.1 −0.726437 + 2.23574i 0 −2.85280 − 2.07268i −2.13811 + 0.694712i 0 −2.38116 + 3.27739i 2.90269 − 2.10893i 0 − 5.28492i
35.2 −0.212694 + 0.654604i 0 1.23477 + 0.897110i 0.0381457 − 0.0123943i 0 0.145094 − 0.199704i −1.96356 + 1.42661i 0 0.0276065i
35.3 0.212694 − 0.654604i 0 1.23477 + 0.897110i −0.0381457 + 0.0123943i 0 0.145094 − 0.199704i 1.96356 − 1.42661i 0 0.0276065i
35.4 0.726437 − 2.23574i 0 −2.85280 − 2.07268i 2.13811 − 0.694712i 0 −2.38116 + 3.27739i −2.90269 + 2.10893i 0 − 5.28492i
62.1 −1.97102 − 1.43203i 0 1.21618 + 3.74302i 2.23109 + 3.07083i 0 −0.349790 + 0.113654i 1.45728 − 4.48505i 0 − 9.24768i
62.2 −0.205228 − 0.149107i 0 −0.598148 − 1.84091i −1.71735 − 2.36373i 0 2.58586 − 0.840196i −0.308515 + 0.949513i 0 0.741170i
62.3 0.205228 + 0.149107i 0 −0.598148 − 1.84091i 1.71735 + 2.36373i 0 2.58586 − 0.840196i 0.308515 − 0.949513i 0 0.741170i
62.4 1.97102 + 1.43203i 0 1.21618 + 3.74302i −2.23109 − 3.07083i 0 −0.349790 + 0.113654i −1.45728 + 4.48505i 0 − 9.24768i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.j.a ✓ 16
3.b odd 2 1 inner 99.2.j.a ✓ 16
4.b odd 2 1 1584.2.cd.c 16
9.c even 3 2 891.2.u.c 32
9.d odd 6 2 891.2.u.c 32
11.c even 5 1 1089.2.d.g 16
11.d odd 10 1 inner 99.2.j.a ✓ 16
11.d odd 10 1 1089.2.d.g 16
12.b even 2 1 1584.2.cd.c 16
33.f even 10 1 inner 99.2.j.a ✓ 16
33.f even 10 1 1089.2.d.g 16
33.h odd 10 1 1089.2.d.g 16
44.g even 10 1 1584.2.cd.c 16
99.o odd 30 2 891.2.u.c 32
99.p even 30 2 891.2.u.c 32
132.n odd 10 1 1584.2.cd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.j.a ✓ 16 1.a even 1 1 trivial
99.2.j.a ✓ 16 3.b odd 2 1 inner
99.2.j.a ✓ 16 11.d odd 10 1 inner
99.2.j.a ✓ 16 33.f even 10 1 inner
891.2.u.c 32 9.c even 3 2
891.2.u.c 32 9.d odd 6 2
891.2.u.c 32 99.o odd 30 2
891.2.u.c 32 99.p even 30 2
1089.2.d.g 16 11.c even 5 1
1089.2.d.g 16 11.d odd 10 1
1089.2.d.g 16 33.f even 10 1
1089.2.d.g 16 33.h odd 10 1
1584.2.cd.c 16 4.b odd 2 1
1584.2.cd.c 16 12.b even 2 1
1584.2.cd.c 16 44.g even 10 1
1584.2.cd.c 16 132.n odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(99, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 6 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 6 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{8} - T^{6} - 50 T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} - 19 T^{6} + \cdots + 841)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 10 T^{14} + \cdots + 625 \) Copy content Toggle raw display
$19$ \( (T^{8} - 4 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 82 T^{6} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 3544535296 \) Copy content Toggle raw display
$31$ \( (T^{8} - 8 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 6 T^{7} + \cdots + 1565001)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 11943113486161 \) Copy content Toggle raw display
$43$ \( (T^{8} + 164 T^{6} + \cdots + 9801)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 4 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 166726039041 \) Copy content Toggle raw display
$59$ \( T^{16} - 66 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( (T^{8} - 39 T^{6} + \cdots + 450241)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 24 T^{3} + \cdots - 649)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 373301041 \) Copy content Toggle raw display
$73$ \( (T^{8} + 10 T^{7} + \cdots + 1488400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 45 T^{6} + \cdots + 24025)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 228745085711041 \) Copy content Toggle raw display
$89$ \( (T^{8} + 130 T^{6} + \cdots + 126025)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 30 T^{7} + \cdots + 1575025)^{2} \) Copy content Toggle raw display
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