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Properties

Label 1098.2.k.i
Level $1098$
Weight $2$
Character orbit 1098.k
Analytic conductor $8.768$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 19 x^{10} - 37 x^{9} + 166 x^{8} + 23 x^{7} + 984 x^{6} + 2859 x^{5} + 4794 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_{8} q^{4} + ( - \beta_{7} + \beta_{4} + 1) q^{5} + (\beta_{4} + \beta_1) q^{7} + ( - \beta_{8} - \beta_{5} - \beta_{4} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + \beta_{8} q^{4} + ( - \beta_{7} + \beta_{4} + 1) q^{5} + (\beta_{4} + \beta_1) q^{7} + ( - \beta_{8} - \beta_{5} - \beta_{4} - 1) q^{8} + (\beta_{6} + \beta_{5} + 1) q^{10} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{11} + (\beta_{11} - \beta_{10} + \cdots + \beta_{3}) q^{13}+ \cdots + (\beta_{9} + 2 \beta_{7} + \cdots + 2 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 3 q^{4} + 5 q^{5} + q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} - 3 q^{4} + 5 q^{5} + q^{7} - 3 q^{8} + 5 q^{10} + 16 q^{13} + 6 q^{14} - 3 q^{16} - 16 q^{17} - 5 q^{19} - 5 q^{20} - 5 q^{22} + 19 q^{23} - 9 q^{26} + q^{28} + 16 q^{29} + 7 q^{31} + 12 q^{32} - 16 q^{34} - 15 q^{35} - 18 q^{37} - 10 q^{38} - 21 q^{41} + 25 q^{43} + 5 q^{44} + 19 q^{46} - 30 q^{47} - 6 q^{49} + 20 q^{50} + q^{52} + 29 q^{53} + 7 q^{55} - 4 q^{56} + q^{58} + 7 q^{59} - 9 q^{61} - 48 q^{62} - 3 q^{64} - 29 q^{65} + 14 q^{67} + 4 q^{68} - 20 q^{70} - 36 q^{71} - 25 q^{73} + 62 q^{74} - 10 q^{76} + 7 q^{77} - 30 q^{79} - 5 q^{80} + 4 q^{82} + 12 q^{83} - 20 q^{85} - 10 q^{86} + 5 q^{88} - 7 q^{89} + 19 q^{91} - 21 q^{92} + 15 q^{94} - 10 q^{95} - 16 q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 19 x^{10} - 37 x^{9} + 166 x^{8} + 23 x^{7} + 984 x^{6} + 2859 x^{5} + 4794 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\!\cdots\!58 \nu^{11} + \cdots - 14\!\cdots\!76 ) / 34\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30\!\cdots\!57 \nu^{11} + \cdots + 16\!\cdots\!54 ) / 34\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 79\!\cdots\!52 \nu^{11} + \cdots - 13\!\cdots\!18 ) / 34\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 87\!\cdots\!48 \nu^{11} + \cdots - 55\!\cdots\!42 ) / 34\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!10 \nu^{11} + \cdots - 31\!\cdots\!13 ) / 34\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!71 \nu^{11} + \cdots - 52\!\cdots\!92 ) / 34\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!76 \nu^{11} + \cdots + 54\!\cdots\!72 ) / 34\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21\!\cdots\!54 \nu^{11} + \cdots + 37\!\cdots\!01 ) / 34\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 38\!\cdots\!73 \nu^{11} + \cdots + 68\!\cdots\!50 ) / 34\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 40\!\cdots\!91 \nu^{11} + \cdots - 25\!\cdots\!96 ) / 34\!\cdots\!53 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - 6\beta_{8} - 2\beta_{7} - 6\beta_{5} - 4\beta_{4} - \beta_{3} + 2\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{11} - \beta_{10} - \beta_{9} - 17 \beta_{8} - 11 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{11} - 11 \beta_{9} - 15 \beta_{8} - 32 \beta_{6} + 30 \beta_{5} - 32 \beta_{4} + 11 \beta_{3} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{11} + 14 \beta_{10} + 137 \beta_{7} - 137 \beta_{6} - 50 \beta_{5} - 50 \beta_{4} + \cdots - 127 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 42\beta_{10} + 130\beta_{9} + 402\beta_{8} + 494\beta_{7} - 92\beta_{5} + 597\beta_{4} + 494\beta_{2} - 610\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 204 \beta_{11} + 364 \beta_{10} + 364 \beta_{9} + 4303 \beta_{8} + 1170 \beta_{7} + 1807 \beta_{6} + \cdots + 1779 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 966 \beta_{11} + 1647 \beta_{10} + 966 \beta_{9} + 18664 \beta_{8} + 2516 \beta_{7} + 7672 \beta_{6} + \cdots + 8126 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6025 \beta_{11} + 3197 \beta_{9} + 42511 \beta_{8} + 21796 \beta_{6} + 20001 \beta_{5} + 21796 \beta_{4} + \cdots + 42511 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 22185 \beta_{11} - 22185 \beta_{10} - 48584 \beta_{7} + 48584 \beta_{6} + 21017 \beta_{5} + \cdots + 146586 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 98173 \beta_{10} - 52170 \beta_{9} - 701759 \beta_{8} - 384050 \beta_{7} - 317709 \beta_{5} + \cdots + 744953 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(245\) \(307\)
\(\chi(n)\) \(1\) \(\beta_{8}\)

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
3.26592 + 2.37283i
0.0202500 + 0.0147125i
−1.16813 − 0.848698i
3.26592 − 2.37283i
0.0202500 − 0.0147125i
−1.16813 + 0.848698i
1.02634 − 3.15876i
−0.211167 + 0.649907i
−0.933208 + 2.87212i
1.02634 + 3.15876i
−0.211167 − 0.649907i
−0.933208 − 2.87212i
−0.809017 + 0.587785i 0 0.309017 − 0.951057i −0.747469 + 2.30047i 0 2.45690 + 1.78504i 0.309017 + 0.951057i 0 −0.747469 − 2.30047i
217.2 −0.809017 + 0.587785i 0 0.309017 − 0.951057i 0.492265 − 1.51504i 0 −0.788767 − 0.573073i 0.309017 + 0.951057i 0 0.492265 + 1.51504i
217.3 −0.809017 + 0.587785i 0 0.309017 − 0.951057i 0.946187 − 2.91206i 0 −1.97715 − 1.43648i 0.309017 + 0.951057i 0 0.946187 + 2.91206i
253.1 −0.809017 − 0.587785i 0 0.309017 + 0.951057i −0.747469 − 2.30047i 0 2.45690 − 1.78504i 0.309017 − 0.951057i 0 −0.747469 + 2.30047i
253.2 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 0.492265 + 1.51504i 0 −0.788767 + 0.573073i 0.309017 − 0.951057i 0 0.492265 − 1.51504i
253.3 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 0.946187 + 2.91206i 0 −1.97715 + 1.43648i 0.309017 − 0.951057i 0 0.946187 − 2.91206i
325.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −2.18700 + 1.58895i 0 1.33536 − 4.10981i −0.809017 − 0.587785i 0 −2.18700 − 1.58895i
325.2 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 1.05284 − 0.764936i 0 0.0978495 − 0.301150i −0.809017 − 0.587785i 0 1.05284 + 0.764936i
325.3 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 2.94317 − 2.13834i 0 −0.624191 + 1.92106i −0.809017 − 0.587785i 0 2.94317 + 2.13834i
973.1 0.309017 − 0.951057i 0 −0.809017 − 0.587785i −2.18700 − 1.58895i 0 1.33536 + 4.10981i −0.809017 + 0.587785i 0 −2.18700 + 1.58895i
973.2 0.309017 − 0.951057i 0 −0.809017 − 0.587785i 1.05284 + 0.764936i 0 0.0978495 + 0.301150i −0.809017 + 0.587785i 0 1.05284 − 0.764936i
973.3 0.309017 − 0.951057i 0 −0.809017 − 0.587785i 2.94317 + 2.13834i 0 −0.624191 − 1.92106i −0.809017 + 0.587785i 0 2.94317 − 2.13834i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.3
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.i ✓ 12
3.b odd 2 1 1098.2.k.k yes 12
61.e even 5 1 inner 1098.2.k.i ✓ 12
183.n odd 10 1 1098.2.k.k yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1098.2.k.i ✓ 12 1.a even 1 1 trivial
1098.2.k.i ✓ 12 61.e even 5 1 inner
1098.2.k.k yes 12 3.b odd 2 1
1098.2.k.k yes 12 183.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 \(S_{2}^{\mathrm{new}}(1098, [\chi])\):

\( T_{5}^{12} - 5 T_{5}^{11} + 20 T_{5}^{10} - 42 T_{5}^{9} + 140 T_{5}^{8} - 245 T_{5}^{7} + 1319 T_{5}^{6} + \cdots + 22801 \) Copy content Toggle raw display
\( T_{7}^{12} - T_{7}^{11} + 14 T_{7}^{10} + 28 T_{7}^{9} + 49 T_{7}^{8} - 24 T_{7}^{7} + 766 T_{7}^{6} + \cdots + 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 5 T^{11} + \cdots + 22801 \) Copy content Toggle raw display
$7$ \( T^{12} - T^{11} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( (T^{6} - 45 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 8 T^{5} + \cdots + 1881)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 16 T^{11} + \cdots + 15376 \) Copy content Toggle raw display
$19$ \( T^{12} + 5 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{12} - 19 T^{11} + \cdots + 524176 \) Copy content Toggle raw display
$29$ \( (T^{6} - 8 T^{5} + \cdots + 6849)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 7 T^{11} + \cdots + 7929856 \) Copy content Toggle raw display
$37$ \( T^{12} + 18 T^{11} + \cdots + 23814400 \) Copy content Toggle raw display
$41$ \( T^{12} + 21 T^{11} + \cdots + 245025 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 474368400 \) Copy content Toggle raw display
$47$ \( (T^{6} + 15 T^{5} + \cdots - 34124)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 14917935321 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 17086672656 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 51520374361 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 152670736 \) Copy content Toggle raw display
$71$ \( T^{12} + 36 T^{11} + \cdots + 1210000 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1497999616 \) Copy content Toggle raw display
$79$ \( T^{12} + 30 T^{11} + \cdots + 2509056 \) Copy content Toggle raw display
$83$ \( T^{12} - 12 T^{11} + \cdots + 1048576 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 372837481 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 4326482176 \) Copy content Toggle raw display
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