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Properties

Label 198.2.i.a
Level $198$
Weight $2$
Character orbit 198.i
Analytic conductor $1.581$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [198,2,Mod(65,198)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(198, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("198.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 198.i (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: \(1.58103796002\)
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: \( x^{12} - 2 x^{11} + 7 x^{10} - 12 x^{9} + 24 x^{8} - 36 x^{7} + 75 x^{6} - 108 x^{5} + 216 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + \beta_{3} q^{3} + ( - \beta_{4} - 1) q^{4} + ( - \beta_{10} + \beta_{2}) q^{5} - \beta_{6} q^{6} - \beta_{11} q^{7} + q^{8} + (\beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + \beta_{3} q^{3} + ( - \beta_{4} - 1) q^{4} + ( - \beta_{10} + \beta_{2}) q^{5} - \beta_{6} q^{6} - \beta_{11} q^{7} + q^{8} + (\beta_{5} - 1) q^{9} - \beta_{2} q^{10} + (\beta_{9} + \beta_{6} - \beta_{5} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{10} - 2 \beta_{9} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 2 q^{3} - 6 q^{4} + 3 q^{5} - q^{6} + 12 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{2} + 2 q^{3} - 6 q^{4} + 3 q^{5} - q^{6} + 12 q^{8} - 10 q^{9} + 6 q^{11} - q^{12} - 12 q^{13} + 7 q^{15} - 6 q^{16} + 6 q^{17} + 5 q^{18} - 3 q^{20} - 9 q^{22} - 18 q^{23} + 2 q^{24} + 9 q^{25} + 2 q^{27} + 6 q^{29} - 11 q^{30} - 3 q^{31} - 6 q^{32} - 4 q^{33} - 3 q^{34} + 12 q^{35} + 5 q^{36} + 6 q^{37} - 15 q^{38} + 36 q^{39} + 3 q^{40} + 3 q^{41} + 12 q^{42} - 33 q^{43} + 3 q^{44} + 10 q^{45} - 3 q^{47} - q^{48} + 6 q^{49} + 9 q^{50} - 39 q^{51} + 12 q^{52} - 10 q^{54} - 6 q^{55} - 27 q^{57} + 6 q^{58} + 6 q^{59} + 4 q^{60} + 6 q^{62} + 36 q^{63} + 12 q^{64} - 4 q^{66} - 3 q^{68} - 38 q^{69} - 6 q^{70} - 10 q^{72} - 3 q^{74} - 9 q^{75} + 15 q^{76} - 42 q^{77} + 42 q^{79} + 2 q^{81} - 6 q^{82} - 18 q^{83} - 12 q^{84} + 6 q^{85} + 33 q^{86} - 18 q^{87} + 6 q^{88} + 19 q^{90} - 24 q^{91} + 18 q^{92} + 25 q^{93} + 3 q^{94} - 12 q^{95} - q^{96} - 12 q^{98} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 7 x^{10} - 12 x^{9} + 24 x^{8} - 36 x^{7} + 75 x^{6} - 108 x^{5} + 216 x^{4} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27 \nu^{11} + 239 \nu^{10} - 391 \nu^{9} + 842 \nu^{8} - 1053 \nu^{7} + 1713 \nu^{6} + \cdots + 28998 ) / 13446 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} - 7 \nu^{9} + 12 \nu^{8} - 24 \nu^{7} + 36 \nu^{6} - 75 \nu^{5} + 108 \nu^{4} + \cdots + 486 ) / 243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 247 \nu^{11} + 445 \nu^{10} - 653 \nu^{9} + 1458 \nu^{8} - 1821 \nu^{7} + 6813 \nu^{6} + \cdots + 195372 ) / 40338 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{11} + \nu^{10} - 8\nu^{9} + 3\nu^{8} - 12\nu^{7} - 42\nu^{5} - 9\nu^{4} - 108\nu^{3} - 162\nu - 486 ) / 243 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 268 \nu^{11} - 783 \nu^{10} + 1431 \nu^{9} - 2563 \nu^{8} + 4974 \nu^{7} - 7827 \nu^{6} + \cdots - 74439 ) / 13446 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 313 \nu^{11} - 794 \nu^{10} + 1474 \nu^{9} - 2583 \nu^{8} + 5235 \nu^{7} - 7362 \nu^{6} + \cdots - 60021 ) / 13446 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 313 \nu^{11} + 794 \nu^{10} - 1474 \nu^{9} + 2583 \nu^{8} - 5235 \nu^{7} + 7362 \nu^{6} + \cdots + 60021 ) / 13446 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1004 \nu^{11} + 37 \nu^{10} - 1367 \nu^{9} + 105 \nu^{8} - 2802 \nu^{7} - 4869 \nu^{6} + \cdots - 368145 ) / 40338 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1163 \nu^{11} - 2920 \nu^{10} + 5846 \nu^{9} - 10689 \nu^{8} + 19461 \nu^{7} - 29178 \nu^{6} + \cdots - 279207 ) / 40338 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 227 \nu^{11} - 85 \nu^{10} + 287 \nu^{9} - 147 \nu^{8} + 636 \nu^{7} + 243 \nu^{6} + 3408 \nu^{5} + \cdots + 71685 ) / 6723 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{7} - 3\beta _1 - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{10} - 2\beta_{8} + \beta_{7} - 6\beta_{6} - 3\beta_{4} + 3\beta_{3} + 3\beta_{2} - 3\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{10} + 9 \beta_{9} + \beta_{8} - 2 \beta_{7} + 3 \beta_{6} - 9 \beta_{5} + 15 \beta_{4} + \cdots + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{11} - 6 \beta_{10} + 9 \beta_{9} - 5 \beta_{8} + 4 \beta_{7} - 6 \beta_{6} - 3 \beta_{5} + \cdots + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 15 \beta_{11} + 3 \beta_{10} - 18 \beta_{9} - 14 \beta_{8} + 16 \beta_{7} - 33 \beta_{6} + 3 \beta_{5} + \cdots - 30 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 45 \beta_{11} - 15 \beta_{10} - 9 \beta_{9} - 26 \beta_{8} - 2 \beta_{7} + 30 \beta_{6} - 18 \beta_{5} + \cdots + 36 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 6 \beta_{11} - 60 \beta_{10} - 50 \beta_{8} + 22 \beta_{7} + 12 \beta_{6} - 39 \beta_{5} - 69 \beta_{4} + \cdots - 24 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3 \beta_{11} - 33 \beta_{10} - 45 \beta_{9} + 31 \beta_{8} - 20 \beta_{7} + 39 \beta_{6} - 87 \beta_{5} + \cdots - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 15 \beta_{10} - 117 \beta_{9} + 73 \beta_{8} - 92 \beta_{7} + 57 \beta_{6} + 135 \beta_{5} + \cdots - 63 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 141 \beta_{11} + 75 \beta_{10} - 54 \beta_{9} + 121 \beta_{8} - 131 \beta_{7} + 147 \beta_{6} + \cdots - 1104 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/198\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(155\)
\(\chi(n)\) \(-1\) \(1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.43086 − 0.976035i
−0.673399 + 1.59579i
0.130431 − 1.72713i
0.198381 + 1.72065i
1.17411 − 1.27337i
1.60133 + 0.660096i
−1.43086 + 0.976035i
−0.673399 − 1.59579i
0.130431 + 1.72713i
0.198381 − 1.72065i
1.17411 + 1.27337i
1.60133 − 0.660096i
−0.500000 + 0.866025i −1.43086 + 0.976035i −0.500000 − 0.866025i 2.73591 − 1.57958i −0.129841 − 1.72718i 2.02416 + 1.16865i 1.00000 1.09471 − 2.79314i 3.15916i
65.2 −0.500000 + 0.866025i −0.673399 − 1.59579i −0.500000 − 0.866025i −3.04011 + 1.75521i 1.71869 + 0.214712i 2.61375 + 1.50905i 1.00000 −2.09307 + 2.14920i − 3.51042i
65.3 −0.500000 + 0.866025i 0.130431 + 1.72713i −0.500000 − 0.866025i −1.99236 + 1.15029i −1.56096 − 0.750610i −4.25017 − 2.45384i 1.00000 −2.96598 + 0.450543i − 2.30058i
65.4 −0.500000 + 0.866025i 0.198381 − 1.72065i −0.500000 − 0.866025i 1.52547 − 0.880729i 1.39094 + 1.03213i −2.28060 − 1.31671i 1.00000 −2.92129 − 0.682689i 1.76146i
65.5 −0.500000 + 0.866025i 1.17411 + 1.27337i −0.500000 − 0.866025i 2.48592 − 1.43525i −1.68982 + 0.380126i 0.861219 + 0.497225i 1.00000 −0.242927 + 2.99015i 2.87049i
65.6 −0.500000 + 0.866025i 1.60133 − 0.660096i −0.500000 − 0.866025i −0.214825 + 0.124029i −0.229008 + 1.71684i 1.03164 + 0.595618i 1.00000 2.12855 − 2.11407i − 0.248058i
131.1 −0.500000 − 0.866025i −1.43086 − 0.976035i −0.500000 + 0.866025i 2.73591 + 1.57958i −0.129841 + 1.72718i 2.02416 − 1.16865i 1.00000 1.09471 + 2.79314i − 3.15916i
131.2 −0.500000 − 0.866025i −0.673399 + 1.59579i −0.500000 + 0.866025i −3.04011 − 1.75521i 1.71869 − 0.214712i 2.61375 − 1.50905i 1.00000 −2.09307 − 2.14920i 3.51042i
131.3 −0.500000 − 0.866025i 0.130431 − 1.72713i −0.500000 + 0.866025i −1.99236 − 1.15029i −1.56096 + 0.750610i −4.25017 + 2.45384i 1.00000 −2.96598 − 0.450543i 2.30058i
131.4 −0.500000 − 0.866025i 0.198381 + 1.72065i −0.500000 + 0.866025i 1.52547 + 0.880729i 1.39094 − 1.03213i −2.28060 + 1.31671i 1.00000 −2.92129 + 0.682689i − 1.76146i
131.5 −0.500000 − 0.866025i 1.17411 − 1.27337i −0.500000 + 0.866025i 2.48592 + 1.43525i −1.68982 − 0.380126i 0.861219 − 0.497225i 1.00000 −0.242927 − 2.99015i − 2.87049i
131.6 −0.500000 − 0.866025i 1.60133 + 0.660096i −0.500000 + 0.866025i −0.214825 − 0.124029i −0.229008 − 1.71684i 1.03164 − 0.595618i 1.00000 2.12855 + 2.11407i 0.248058i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
99.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 198.2.i.a ✓ 12
3.b odd 2 1 594.2.i.b 12
9.c even 3 1 594.2.i.a 12
9.c even 3 1 1782.2.b.f 12
9.d odd 6 1 198.2.i.b yes 12
9.d odd 6 1 1782.2.b.e 12
11.b odd 2 1 198.2.i.b yes 12
33.d even 2 1 594.2.i.a 12
99.g even 6 1 inner 198.2.i.a ✓ 12
99.g even 6 1 1782.2.b.f 12
99.h odd 6 1 594.2.i.b 12
99.h odd 6 1 1782.2.b.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.2.i.a ✓ 12 1.a even 1 1 trivial
198.2.i.a ✓ 12 99.g even 6 1 inner
198.2.i.b yes 12 9.d odd 6 1
198.2.i.b yes 12 11.b odd 2 1
594.2.i.a 12 9.c even 3 1
594.2.i.a 12 33.d even 2 1
594.2.i.b 12 3.b odd 2 1
594.2.i.b 12 99.h odd 6 1
1782.2.b.e 12 9.d odd 6 1
1782.2.b.e 12 99.h odd 6 1
1782.2.b.f 12 9.c even 3 1
1782.2.b.f 12 99.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} - 24 T_{7}^{10} + 474 T_{7}^{8} - 918 T_{7}^{7} - 1692 T_{7}^{6} + 5508 T_{7}^{5} + \cdots + 11664 \) acting on \(S_{2}^{\mathrm{new}}(198, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{12} - 24 T^{10} + \cdots + 11664 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 12 T^{11} + \cdots + 104976 \) Copy content Toggle raw display
$17$ \( (T^{6} - 3 T^{5} + \cdots - 162)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 135 T^{10} + \cdots + 419904 \) Copy content Toggle raw display
$23$ \( T^{12} + 18 T^{11} + \cdots + 5798464 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} + \cdots + 19607184 \) Copy content Toggle raw display
$31$ \( T^{12} + 3 T^{11} + \cdots + 91853056 \) Copy content Toggle raw display
$37$ \( (T^{6} - 3 T^{5} - 60 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 22422666564 \) Copy content Toggle raw display
$43$ \( T^{12} + 33 T^{11} + \cdots + 492804 \) Copy content Toggle raw display
$47$ \( T^{12} + 3 T^{11} + \cdots + 27556 \) Copy content Toggle raw display
$53$ \( T^{12} + 135 T^{10} + \cdots + 891136 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 125462401 \) Copy content Toggle raw display
$61$ \( T^{12} - 96 T^{10} + \cdots + 47775744 \) Copy content Toggle raw display
$67$ \( T^{12} + 207 T^{10} + \cdots + 426409 \) Copy content Toggle raw display
$71$ \( T^{12} + 315 T^{10} + \cdots + 5400976 \) Copy content Toggle raw display
$73$ \( T^{12} + 387 T^{10} + \cdots + 25220484 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 649638144 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 8546632704 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 9799416064 \) Copy content Toggle raw display
$97$ \( T^{12} + 207 T^{10} + \cdots + 4489 \) Copy content Toggle raw display
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