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Properties

Label 1098.2.o.b
Level $1098$
Weight $2$
Character orbit 1098.o
Analytic conductor $8.768$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1098,2,Mod(109,1098)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1098, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1098.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1098 = 2 \cdot 3^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1098.o (of order \(6\), degree \(2\), 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: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} + 22 x^{9} + 139 x^{8} - 68 x^{7} + 100 x^{6} + 1016 x^{5} + 3551 x^{4} + \cdots + 3721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 366)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_{7} q^{4} + \beta_{8} q^{5} + ( - \beta_{7} + \beta_{3} + \beta_1 - 1) q^{7} + \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + \beta_{7} q^{4} + \beta_{8} q^{5} + ( - \beta_{7} + \beta_{3} + \beta_1 - 1) q^{7} + \beta_{2} q^{8} + \beta_{9} q^{10} + (\beta_{10} - 2 \beta_{7} + \beta_{5} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{11} - \beta_{9} + \beta_{8} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} - 2 q^{5} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} - 2 q^{5} - 12 q^{7} - 4 q^{14} - 6 q^{16} - 6 q^{17} + 2 q^{19} - 4 q^{20} - 8 q^{22} - 8 q^{25} - 30 q^{29} - 18 q^{31} - 4 q^{34} - 6 q^{35} + 28 q^{41} + 6 q^{43} + 12 q^{44} + 6 q^{46} + 24 q^{47} + 6 q^{49} - 6 q^{55} + 4 q^{56} + 8 q^{58} + 28 q^{61} + 4 q^{62} - 12 q^{64} + 12 q^{65} + 12 q^{67} - 6 q^{68} + 20 q^{70} + 6 q^{71} - 10 q^{73} + 14 q^{74} - 2 q^{76} - 40 q^{77} + 6 q^{79} - 2 q^{80} + 12 q^{82} - 16 q^{83} - 2 q^{86} - 16 q^{88} - 36 q^{91} + 6 q^{92} - 4 q^{95} + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} + 22 x^{9} + 139 x^{8} - 68 x^{7} + 100 x^{6} + 1016 x^{5} + 3551 x^{4} + \cdots + 3721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2297796143 \nu^{11} - 128526274987 \nu^{10} + 279836268504 \nu^{9} + \cdots + 12910994060252 ) / 571276603542854 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 20802575051 \nu^{11} - 1034371874151 \nu^{10} + 4368142181789 \nu^{9} + \cdots - 12\!\cdots\!30 ) / 880327880869316 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2572883423835 \nu^{11} - 16505546927925 \nu^{10} + 124424416835651 \nu^{9} + \cdots + 48\!\cdots\!39 ) / 53\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 184336040091 \nu^{11} - 1472676708811 \nu^{10} + 4848667251339 \nu^{9} + \cdots - 12\!\cdots\!92 ) / 880327880869316 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20296656769272 \nu^{11} + 29348815092993 \nu^{10} + 49239965698927 \nu^{9} + \cdots - 11\!\cdots\!41 ) / 53\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1121867 \nu^{11} + 5150750 \nu^{10} - 10675520 \nu^{9} - 14199566 \nu^{8} - 88309118 \nu^{7} + \cdots + 876160080 ) / 2480952716 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35543957363250 \nu^{11} + 16283400199308 \nu^{10} - 294037733323597 \nu^{9} + \cdots + 86\!\cdots\!94 ) / 53\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 48933551170275 \nu^{11} + 194829016848345 \nu^{10} - 480600510464232 \nu^{9} + \cdots + 13\!\cdots\!27 ) / 53\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47656 \nu^{11} - 138226 \nu^{10} + 171828 \nu^{9} + 1108695 \nu^{8} + 5189640 \nu^{7} + \cdots + 68433887 ) / 40671356 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 55470129011831 \nu^{11} + 200098938577296 \nu^{10} - 397883547460587 \nu^{9} + \cdots - 11\!\cdots\!11 ) / 26\!\cdots\!38 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - 4\beta_{7} + \beta_{5} - \beta_{3} - 6\beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{10} + \beta_{9} - \beta_{8} - 5\beta_{7} - 5\beta_{6} + 9\beta_{5} - 9\beta_{3} - 3\beta_{2} - 2\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{11} - 8 \beta_{10} - 14 \beta_{9} - 10 \beta_{8} - 62 \beta_{6} + 19 \beta_{5} + 10 \beta_{4} + \cdots - 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 5 \beta_{9} - 5 \beta_{8} + 116 \beta_{7} - 116 \beta_{6} + 41 \beta_{5} + 9 \beta_{4} + 41 \beta_{3} + \cdots - 170 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 139 \beta_{11} + 194 \beta_{10} + 36 \beta_{9} - 36 \beta_{8} + 926 \beta_{7} - 126 \beta_{5} + \cdots - 463 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 290 \beta_{11} + 1386 \beta_{10} + 200 \beta_{9} + 90 \beta_{8} + 2123 \beta_{7} + 2123 \beta_{6} + \cdots + 861 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1754 \beta_{11} + 3564 \beta_{10} + 2302 \beta_{9} + 1206 \beta_{8} + 14120 \beta_{6} - 5228 \beta_{5} + \cdots + 12559 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2926 \beta_{9} + 2926 \beta_{8} - 35826 \beta_{7} + 35826 \beta_{6} - 11172 \beta_{5} - 1116 \beta_{4} + \cdots + 49500 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 24167 \beta_{11} - 59980 \beta_{10} - 8246 \beta_{9} + 8246 \beta_{8} - 218924 \beta_{7} + 24275 \beta_{5} + \cdots + 109462 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 103684 \beta_{11} - 355376 \beta_{10} - 87655 \beta_{9} - 16029 \beta_{8} - 584615 \beta_{7} + \cdots - 217631 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(245\) \(307\)
\(\chi(n)\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.615449 + 0.615449i
1.79116 + 1.79116i
−1.04059 − 1.04059i
−1.17841 + 1.17841i
2.81570 − 2.81570i
−2.00332 + 2.00332i
0.615449 − 0.615449i
1.79116 − 1.79116i
−1.04059 + 1.04059i
−1.17841 − 1.17841i
2.81570 + 2.81570i
−2.00332 − 2.00332i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.50462 + 2.60608i 0 −0.0438317 − 0.0253062i − 1.00000i 0 2.60608 − 1.50462i
109.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.149157 − 0.258348i 0 2.73794 + 1.58075i − 1.00000i 0 −0.258348 + 0.149157i
109.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.855464 − 1.48171i 0 −3.96206 − 2.28750i − 1.00000i 0 −1.48171 + 0.855464i
109.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.16500 + 3.74989i 0 −2.24708 − 1.29735i 1.00000i 0 −3.74989 + 2.16500i
109.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.380432 − 0.658928i 0 0.285085 + 0.164594i 1.00000i 0 0.658928 − 0.380432i
109.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.28457 − 2.22493i 0 −2.77005 − 1.59929i 1.00000i 0 2.22493 − 1.28457i
685.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.50462 − 2.60608i 0 −0.0438317 + 0.0253062i 1.00000i 0 2.60608 + 1.50462i
685.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.149157 + 0.258348i 0 2.73794 − 1.58075i 1.00000i 0 −0.258348 − 0.149157i
685.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.855464 + 1.48171i 0 −3.96206 + 2.28750i 1.00000i 0 −1.48171 − 0.855464i
685.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i −2.16500 − 3.74989i 0 −2.24708 + 1.29735i − 1.00000i 0 −3.74989 − 2.16500i
685.5 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.380432 + 0.658928i 0 0.285085 − 0.164594i − 1.00000i 0 0.658928 + 0.380432i
685.6 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.28457 + 2.22493i 0 −2.77005 + 1.59929i − 1.00000i 0 2.22493 + 1.28457i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1098.2.o.b 12
3.b odd 2 1 366.2.i.a ✓ 12
61.f even 6 1 inner 1098.2.o.b 12
183.i odd 6 1 366.2.i.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
366.2.i.a ✓ 12 3.b odd 2 1
366.2.i.a ✓ 12 183.i odd 6 1
1098.2.o.b 12 1.a even 1 1 trivial
1098.2.o.b 12 61.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 2 T_{5}^{11} + 21 T_{5}^{10} - 18 T_{5}^{9} + 226 T_{5}^{8} - 246 T_{5}^{7} + 1565 T_{5}^{6} + \cdots + 169 \) acting on \(S_{2}^{\mathrm{new}}(1098, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{11} + \cdots + 169 \) Copy content Toggle raw display
$7$ \( T^{12} + 12 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{12} + 56 T^{10} + \cdots + 676 \) Copy content Toggle raw display
$13$ \( T^{12} + 68 T^{10} + \cdots + 3748096 \) Copy content Toggle raw display
$17$ \( T^{12} + 6 T^{11} + \cdots + 1024144 \) Copy content Toggle raw display
$19$ \( T^{12} - 2 T^{11} + \cdots + 238144 \) Copy content Toggle raw display
$23$ \( T^{12} + 100 T^{10} + \cdots + 4778596 \) Copy content Toggle raw display
$29$ \( T^{12} + 30 T^{11} + \cdots + 2819041 \) Copy content Toggle raw display
$31$ \( T^{12} + 18 T^{11} + \cdots + 4840000 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 28507283281 \) Copy content Toggle raw display
$41$ \( (T^{6} - 14 T^{5} + \cdots - 108131)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 5839099396 \) Copy content Toggle raw display
$47$ \( T^{12} - 24 T^{11} + \cdots + 676 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 63800192569 \) Copy content Toggle raw display
$59$ \( T^{12} - 184 T^{10} + \cdots + 64513024 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 51520374361 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 807582724 \) Copy content Toggle raw display
$71$ \( T^{12} - 6 T^{11} + \cdots + 10816 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1983188089 \) Copy content Toggle raw display
$79$ \( T^{12} - 6 T^{11} + \cdots + 10816 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 52432756324 \) Copy content Toggle raw display
$89$ \( T^{12} + 170 T^{10} + \cdots + 15389929 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 823906582249 \) Copy content Toggle raw display
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