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Properties

Label 1098.2.k.l
Level $1098$
Weight $2$
Character orbit 1098.k
Analytic conductor $8.768$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1098,2,Mod(217,1098)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1098, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1098.217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1098 = 2 \cdot 3^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1098.k (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: \(8.76757414194\)
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: \( x^{16} - 3 x^{15} + 18 x^{14} - 39 x^{13} + 151 x^{12} - 259 x^{11} + 842 x^{10} - 662 x^{9} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 122)
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.

\(f(q)\) \(=\) \( q + (\beta_{8} + \beta_{6} - \beta_{5} + 1) q^{2} - \beta_{5} q^{4} + \beta_{13} q^{5} + (\beta_{15} + \beta_{12} + \cdots + \beta_{7}) q^{7}+ \cdots - \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} + \beta_{6} - \beta_{5} + 1) q^{2} - \beta_{5} q^{4} + \beta_{13} q^{5} + (\beta_{15} + \beta_{12} + \cdots + \beta_{7}) q^{7}+ \cdots + (3 \beta_{14} + \beta_{13} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 4 q^{4} - 3 q^{5} - 2 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 4 q^{4} - 3 q^{5} - 2 q^{7} + 4 q^{8} + 3 q^{10} + 2 q^{13} - 8 q^{14} - 4 q^{16} + 7 q^{17} - 12 q^{19} + 2 q^{20} + 5 q^{22} + 16 q^{23} + 9 q^{25} + 3 q^{26} - 2 q^{28} - 4 q^{29} - 19 q^{31} - 16 q^{32} - 7 q^{34} + 21 q^{35} - 30 q^{37} - 8 q^{38} - 2 q^{40} - 4 q^{41} - 36 q^{43} + 5 q^{44} - 16 q^{46} + 30 q^{47} - 32 q^{49} - 14 q^{50} + 2 q^{52} - 55 q^{53} - 27 q^{55} + 2 q^{56} + 14 q^{58} - 8 q^{59} + 57 q^{61} - 46 q^{62} - 4 q^{64} + 41 q^{65} - 19 q^{67} - 8 q^{68} + 4 q^{70} - 31 q^{71} + 4 q^{73} - 30 q^{74} + 8 q^{76} + 29 q^{77} + 2 q^{79} + 2 q^{80} - 16 q^{82} - 12 q^{83} + 52 q^{85} - 24 q^{86} - 5 q^{88} + 15 q^{89} - 3 q^{91} - 14 q^{92} + 3 q^{95} - 18 q^{97} + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 18 x^{14} - 39 x^{13} + 151 x^{12} - 259 x^{11} + 842 x^{10} - 662 x^{9} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 34\!\cdots\!89 \nu^{15} + \cdots + 13\!\cdots\!00 ) / 33\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 70\!\cdots\!11 \nu^{15} + \cdots + 29\!\cdots\!28 ) / 67\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 82\!\cdots\!81 \nu^{15} + \cdots - 56\!\cdots\!44 ) / 67\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 42\!\cdots\!75 \nu^{15} + \cdots + 60\!\cdots\!12 ) / 26\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 45\!\cdots\!77 \nu^{15} + \cdots + 36\!\cdots\!92 ) / 26\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\!\cdots\!09 \nu^{15} + \cdots + 59\!\cdots\!22 ) / 84\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 85\!\cdots\!69 \nu^{15} + \cdots + 61\!\cdots\!76 ) / 26\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 51\!\cdots\!93 \nu^{15} + \cdots - 10\!\cdots\!76 ) / 13\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 26\!\cdots\!35 \nu^{15} + \cdots + 23\!\cdots\!60 ) / 26\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 28\!\cdots\!87 \nu^{15} + \cdots + 39\!\cdots\!88 ) / 26\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35\!\cdots\!29 \nu^{15} + \cdots + 30\!\cdots\!68 ) / 26\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 63\!\cdots\!21 \nu^{15} + \cdots - 58\!\cdots\!88 ) / 44\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 53\!\cdots\!75 \nu^{15} + \cdots - 43\!\cdots\!08 ) / 26\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 71\!\cdots\!23 \nu^{15} + \cdots - 24\!\cdots\!52 ) / 26\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} + \beta_{14} - \beta_{13} - \beta_{10} + \beta_{9} - 5\beta_{8} - 5\beta_{6} + 6\beta_{5} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{9} - \beta_{7} + 8\beta_{4} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + 11 \beta_{13} + \beta_{12} + 9 \beta_{11} + 12 \beta_{10} + 16 \beta_{8} - 58 \beta_{5} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{14} + 14 \beta_{13} - \beta_{12} + \beta_{11} - 16 \beta_{9} + 10 \beta_{7} + 11 \beta_{6} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 19 \beta_{15} - 114 \beta_{14} - 100 \beta_{12} - 135 \beta_{10} - 135 \beta_{9} - 213 \beta_{8} + \cdots + 9 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 109 \beta_{15} + 44 \beta_{14} - 176 \beta_{13} - 18 \beta_{11} - 44 \beta_{10} + 176 \beta_{9} + \cdots - 121 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 748 \beta_{15} + 1504 \beta_{14} - 1192 \beta_{13} + 1016 \beta_{12} - 748 \beta_{11} + 1192 \beta_{10} + \cdots - 2604 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 245 \beta_{15} + 720 \beta_{13} + 245 \beta_{12} - 1029 \beta_{11} + 2157 \beta_{10} + 3017 \beta_{8} + \cdots + 3017 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 16708 \beta_{14} + 16708 \beta_{13} - 3383 \beta_{12} + 3383 \beta_{11} - 29313 \beta_{9} + \cdots + 65812 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3065 \beta_{15} - 10409 \beta_{14} + 6299 \beta_{12} - 26079 \beta_{10} - 26079 \beta_{9} + \cdots + 102190 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 68028 \beta_{15} + 134568 \beta_{14} - 185438 \beta_{13} - 40461 \beta_{11} - 134568 \beta_{10} + \cdots - 702687 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 82302 \beta_{15} + 312330 \beta_{14} - 140449 \beta_{13} - 44989 \beta_{12} + 82302 \beta_{11} + \cdots - 488858 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 469691 \beta_{15} + 1448040 \beta_{13} + 469691 \beta_{12} + 666019 \beta_{11} + 2058180 \beta_{10} + \cdots + 3945747 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3713440 \beta_{14} + 3713440 \beta_{13} - 451112 \beta_{12} + 451112 \beta_{11} - 5527728 \beta_{9} + \cdots + 7036820 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1098\mathbb{Z}\right)^\times\).

\(n\) \(245\) \(307\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
0.315412 + 0.970740i
−0.182902 − 0.562916i
−0.982624 − 3.02421i
1.04110 + 3.20417i
0.315412 − 0.970740i
−0.182902 + 0.562916i
−0.982624 + 3.02421i
1.04110 − 3.20417i
0.376781 + 0.273748i
−2.05627 − 1.49397i
2.42480 + 1.76172i
0.563712 + 0.409561i
0.376781 − 0.273748i
−2.05627 + 1.49397i
2.42480 − 1.76172i
0.563712 − 0.409561i
0.809017 − 0.587785i 0 0.309017 − 0.951057i −1.04535 + 3.21727i 0 −3.46273 − 2.51582i −0.309017 − 0.951057i 0 1.04535 + 3.21727i
217.2 0.809017 − 0.587785i 0 0.309017 − 0.951057i −0.0570675 + 0.175636i 0 3.87052 + 2.81210i −0.309017 − 0.951057i 0 0.0570675 + 0.175636i
217.3 0.809017 − 0.587785i 0 0.309017 − 0.951057i 0.140397 − 0.432098i 0 −2.67289 − 1.94197i −0.309017 − 0.951057i 0 −0.140397 − 0.432098i
217.4 0.809017 − 0.587785i 0 0.309017 − 0.951057i 0.771041 − 2.37302i 0 0.647069 + 0.470123i −0.309017 − 0.951057i 0 −0.771041 − 2.37302i
253.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i −1.04535 − 3.21727i 0 −3.46273 + 2.51582i −0.309017 + 0.951057i 0 1.04535 − 3.21727i
253.2 0.809017 + 0.587785i 0 0.309017 + 0.951057i −0.0570675 − 0.175636i 0 3.87052 − 2.81210i −0.309017 + 0.951057i 0 0.0570675 − 0.175636i
253.3 0.809017 + 0.587785i 0 0.309017 + 0.951057i 0.140397 + 0.432098i 0 −2.67289 + 1.94197i −0.309017 + 0.951057i 0 −0.140397 + 0.432098i
253.4 0.809017 + 0.587785i 0 0.309017 + 0.951057i 0.771041 + 2.37302i 0 0.647069 − 0.470123i −0.309017 + 0.951057i 0 −0.771041 + 2.37302i
325.1 −0.309017 − 0.951057i 0 −0.809017 + 0.587785i −2.83793 + 2.06188i 0 −1.39211 + 4.28447i 0.809017 + 0.587785i 0 2.83793 + 2.06188i
325.2 −0.309017 − 0.951057i 0 −0.809017 + 0.587785i −0.906816 + 0.658840i 0 0.818925 − 2.52039i 0.809017 + 0.587785i 0 0.906816 + 0.658840i
325.3 −0.309017 − 0.951057i 0 −0.809017 + 0.587785i −0.679133 + 0.493419i 0 1.63182 − 5.02222i 0.809017 + 0.587785i 0 0.679133 + 0.493419i
325.4 −0.309017 − 0.951057i 0 −0.809017 + 0.587785i 3.11486 − 2.26308i 0 −0.440601 + 1.35603i 0.809017 + 0.587785i 0 −3.11486 − 2.26308i
973.1 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −2.83793 − 2.06188i 0 −1.39211 − 4.28447i 0.809017 − 0.587785i 0 2.83793 − 2.06188i
973.2 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −0.906816 − 0.658840i 0 0.818925 + 2.52039i 0.809017 − 0.587785i 0 0.906816 − 0.658840i
973.3 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −0.679133 − 0.493419i 0 1.63182 + 5.02222i 0.809017 − 0.587785i 0 0.679133 − 0.493419i
973.4 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i 3.11486 + 2.26308i 0 −0.440601 − 1.35603i 0.809017 − 0.587785i 0 −3.11486 + 2.26308i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.e even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1098.2.k.l 16
3.b odd 2 1 122.2.e.b ✓ 16
12.b even 2 1 976.2.v.c 16
61.e even 5 1 inner 1098.2.k.l 16
183.l odd 10 1 7442.2.a.q 8
183.n odd 10 1 122.2.e.b ✓ 16
183.n odd 10 1 7442.2.a.s 8
732.bb even 10 1 976.2.v.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
122.2.e.b ✓ 16 3.b odd 2 1
122.2.e.b ✓ 16 183.n odd 10 1
976.2.v.c 16 12.b even 2 1
976.2.v.c 16 732.bb even 10 1
1098.2.k.l 16 1.a even 1 1 trivial
1098.2.k.l 16 61.e even 5 1 inner
7442.2.a.q 8 183.l odd 10 1
7442.2.a.s 8 183.n odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1098, [\chi])\):

\( T_{5}^{16} + 3 T_{5}^{15} + 10 T_{5}^{14} + 11 T_{5}^{13} + 138 T_{5}^{12} + 610 T_{5}^{11} + 3196 T_{5}^{10} + \cdots + 81 \) Copy content Toggle raw display
\( T_{7}^{16} + 2 T_{7}^{15} + 32 T_{7}^{14} + 99 T_{7}^{13} + 790 T_{7}^{12} + 1874 T_{7}^{11} + \cdots + 23658496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 3 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{15} + \cdots + 23658496 \) Copy content Toggle raw display
$11$ \( (T^{8} - 48 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - T^{7} - 51 T^{6} + \cdots + 2421)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 7 T^{15} + \cdots + 531441 \) Copy content Toggle raw display
$19$ \( T^{16} + 12 T^{15} + \cdots + 86118400 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1916338176 \) Copy content Toggle raw display
$29$ \( (T^{8} + 2 T^{7} + \cdots + 152595)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1955762176 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 208742989456 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 28631685681 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 128686336 \) Copy content Toggle raw display
$47$ \( (T^{8} - 15 T^{7} + \cdots - 79344)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 203156631441 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23079686400 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 191707312997281 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1261254779136 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 20212433438976 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 130051216 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 263870135046400 \) Copy content Toggle raw display
$83$ \( T^{16} + 12 T^{15} + \cdots + 7485696 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 2705054537025 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 36349710336 \) Copy content Toggle raw display
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