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Properties

Label 198.3.k.b
Level $198$
Weight $3$
Character orbit 198.k
Analytic conductor $5.395$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [198,3,Mod(53,198)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(198, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 6])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("198.53"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 198.k (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,8,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.39510923433\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6x^{14} + 240x^{12} - 3964x^{10} + 32714x^{8} - 117184x^{6} + 384005x^{4} - 1203596x^{2} + 2825761 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + 2 \beta_{3} q^{4} + (\beta_{15} + \beta_{14} + \cdots - \beta_1) q^{5} + (\beta_{6} + \beta_{5} + \cdots + 4 \beta_{2}) q^{7} + 2 \beta_{10} q^{8} + (\beta_{8} - \beta_{7} + \cdots - \beta_{3}) q^{10}+ \cdots + ( - 8 \beta_{14} - 24 \beta_{13} + \cdots - 8 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 40 q^{7} - 8 q^{10} - 64 q^{13} - 16 q^{16} + 116 q^{19} + 128 q^{25} + 80 q^{28} + 172 q^{31} - 16 q^{34} - 100 q^{37} - 24 q^{40} - 392 q^{43} - 316 q^{46} - 56 q^{49} - 72 q^{52} - 500 q^{55}+ \cdots - 500 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6x^{14} + 240x^{12} - 3964x^{10} + 32714x^{8} - 117184x^{6} + 384005x^{4} - 1203596x^{2} + 2825761 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 158531431901020 \nu^{14} + \cdots - 48\!\cdots\!56 ) / 10\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 791772175552960 \nu^{14} + \cdots + 17\!\cdots\!66 ) / 10\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19\!\cdots\!78 \nu^{14} + \cdots + 67\!\cdots\!04 ) / 10\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 34\!\cdots\!62 \nu^{14} + \cdots - 31\!\cdots\!35 ) / 10\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44\!\cdots\!32 \nu^{14} + \cdots + 52\!\cdots\!17 ) / 10\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 46\!\cdots\!72 \nu^{14} + \cdots + 29\!\cdots\!07 ) / 10\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 71\!\cdots\!95 \nu^{14} + \cdots + 24\!\cdots\!28 ) / 10\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27\!\cdots\!54 \nu^{15} + \cdots - 25\!\cdots\!33 \nu ) / 44\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18793390963876 \nu^{15} - 181492756469898 \nu^{13} + \cdots - 18\!\cdots\!97 \nu ) / 26\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 41\!\cdots\!15 \nu^{15} + \cdots - 28\!\cdots\!35 \nu ) / 44\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 41\!\cdots\!57 \nu^{15} + \cdots + 44\!\cdots\!91 \nu ) / 44\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 42\!\cdots\!19 \nu^{15} + \cdots - 55\!\cdots\!84 \nu ) / 44\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!38 \nu^{15} + \cdots - 16\!\cdots\!63 \nu ) / 10\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 19\!\cdots\!78 \nu^{15} + \cdots + 67\!\cdots\!04 \nu ) / 10\!\cdots\!69 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} + 2\beta_{3} + 9\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 11\beta_{15} + 11\beta_{14} + 8\beta_{13} + 7\beta_{12} - \beta_{11} + 8\beta_{10} - \beta_{9} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -22\beta_{8} + 18\beta_{6} + 18\beta_{5} + 51\beta_{4} - 120\beta_{3} - 51\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -73\beta_{14} + 87\beta_{13} - 42\beta_{12} - 259\beta_{11} + 259\beta_{10} - 42\beta_{9} - 73\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 290\beta_{8} + 290\beta_{7} - 405\beta_{6} - 885\beta_{4} - 1003\beta_{2} + 1003 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2583 \beta_{15} - 1408 \beta_{14} - 1408 \beta_{13} - 980 \beta_{12} + 1408 \beta_{11} + \cdots + 2145 \beta_{9} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4728\beta_{8} - 2388\beta_{7} - 4728\beta_{5} - 13186\beta_{4} + 31333\beta_{3} + 31333\beta_{2} - 13186 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 25263\beta_{15} + 43177\beta_{14} + 35484\beta_{12} + 54540\beta_{11} - 35484\beta_{10} + 25263\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 122980 \beta_{8} - 44319 \beta_{7} + 122980 \beta_{6} + 44319 \beta_{5} + 317735 \beta_{4} + \cdots - 211408 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 440715\beta_{15} + 440715\beta_{13} - 1035661\beta_{11} + 1380236\beta_{10} - 594946\beta_{9} - 290069\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1325730\beta_{7} - 1325730\beta_{6} + 785290\beta_{5} - 5486094\beta_{3} - 8992953\beta_{2} + 5486094 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 12429703 \beta_{15} - 12429703 \beta_{14} - 4832589 \beta_{13} - 10065400 \beta_{12} + \cdots - 4832589 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 36129687 \beta_{8} - 22495103 \beta_{6} - 22495103 \beta_{5} - 94142074 \beta_{4} + \cdots + 94142074 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 130271761 \beta_{14} - 81582854 \beta_{13} + 104302607 \beta_{12} + 356985766 \beta_{11} + \cdots + 130271761 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(145\) \(155\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
2.42869 + 3.34281i
−1.08370 − 1.49158i
1.08370 + 1.49158i
−2.42869 − 3.34281i
2.42869 − 3.34281i
−1.08370 + 1.49158i
1.08370 − 1.49158i
−2.42869 + 3.34281i
2.66080 + 0.864545i
−1.82954 − 0.594454i
1.82954 + 0.594454i
−2.66080 − 0.864545i
2.66080 − 0.864545i
−1.82954 + 0.594454i
1.82954 − 0.594454i
−2.66080 + 0.864545i
−1.34500 − 0.437016i 0 1.61803 + 1.17557i −5.26585 + 1.71098i 0 1.50533 + 1.09369i −1.66251 − 2.28825i 0 7.83027
53.2 −1.34500 − 0.437016i 0 1.61803 + 1.17557i 7.44210 − 2.41808i 0 5.73074 + 4.16363i −1.66251 − 2.28825i 0 −11.0663
53.3 1.34500 + 0.437016i 0 1.61803 + 1.17557i −7.44210 + 2.41808i 0 5.73074 + 4.16363i 1.66251 + 2.28825i 0 −11.0663
53.4 1.34500 + 0.437016i 0 1.61803 + 1.17557i 5.26585 − 1.71098i 0 1.50533 + 1.09369i 1.66251 + 2.28825i 0 7.83027
71.1 −1.34500 + 0.437016i 0 1.61803 − 1.17557i −5.26585 − 1.71098i 0 1.50533 − 1.09369i −1.66251 + 2.28825i 0 7.83027
71.2 −1.34500 + 0.437016i 0 1.61803 − 1.17557i 7.44210 + 2.41808i 0 5.73074 − 4.16363i −1.66251 + 2.28825i 0 −11.0663
71.3 1.34500 − 0.437016i 0 1.61803 − 1.17557i −7.44210 − 2.41808i 0 5.73074 − 4.16363i 1.66251 − 2.28825i 0 −11.0663
71.4 1.34500 − 0.437016i 0 1.61803 − 1.17557i 5.26585 + 1.71098i 0 1.50533 − 1.09369i 1.66251 − 2.28825i 0 7.83027
125.1 −0.831254 + 1.14412i 0 −0.618034 − 1.90211i −3.35962 − 4.62412i 0 3.05124 + 9.39075i 2.68999 + 0.874032i 0 8.08325
125.2 −0.831254 + 1.14412i 0 −0.618034 − 1.90211i 2.84587 + 3.91701i 0 −0.287308 − 0.884242i 2.68999 + 0.874032i 0 −6.84718
125.3 0.831254 − 1.14412i 0 −0.618034 − 1.90211i −2.84587 − 3.91701i 0 −0.287308 − 0.884242i −2.68999 − 0.874032i 0 −6.84718
125.4 0.831254 − 1.14412i 0 −0.618034 − 1.90211i 3.35962 + 4.62412i 0 3.05124 + 9.39075i −2.68999 − 0.874032i 0 8.08325
179.1 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i −3.35962 + 4.62412i 0 3.05124 − 9.39075i 2.68999 − 0.874032i 0 8.08325
179.2 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i 2.84587 − 3.91701i 0 −0.287308 + 0.884242i 2.68999 − 0.874032i 0 −6.84718
179.3 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −2.84587 + 3.91701i 0 −0.287308 + 0.884242i −2.68999 + 0.874032i 0 −6.84718
179.4 0.831254 + 1.14412i 0 −0.618034 + 1.90211i 3.35962 − 4.62412i 0 3.05124 − 9.39075i −2.68999 + 0.874032i 0 8.08325
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 198.3.k.b ✓ 16
3.b odd 2 1 inner 198.3.k.b ✓ 16
11.c even 5 1 inner 198.3.k.b ✓ 16
11.c even 5 1 2178.3.c.n 8
11.d odd 10 1 2178.3.c.o 8
33.f even 10 1 2178.3.c.o 8
33.h odd 10 1 inner 198.3.k.b ✓ 16
33.h odd 10 1 2178.3.c.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.3.k.b ✓ 16 1.a even 1 1 trivial
198.3.k.b ✓ 16 3.b odd 2 1 inner
198.3.k.b ✓ 16 11.c even 5 1 inner
198.3.k.b ✓ 16 33.h odd 10 1 inner
2178.3.c.n 8 11.c even 5 1
2178.3.c.n 8 33.h odd 10 1
2178.3.c.o 8 11.d odd 10 1
2178.3.c.o 8 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 114 T_{5}^{14} + 6357 T_{5}^{12} - 203372 T_{5}^{10} + 8819630 T_{5}^{8} + \cdots + 2066696635201 \) acting on \(S_{3}^{\mathrm{new}}(198, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 2066696635201 \) Copy content Toggle raw display
$7$ \( (T^{8} - 20 T^{7} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( (T^{8} + 32 T^{7} + \cdots + 340365601)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( (T^{8} - 58 T^{7} + \cdots + 11977332481)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2646 T^{6} + \cdots + 8607942841)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{8} - 86 T^{7} + \cdots + 1771561)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 50 T^{7} + \cdots + 432598401)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{4} + 98 T^{3} + \cdots + 30699)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 20\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 95392941158761)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 82 T^{3} + \cdots + 1520741)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 37\!\cdots\!01 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 272178856648336)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 18\!\cdots\!01)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 570189878065201)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 351507039687441)^{2} \) Copy content Toggle raw display
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