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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [198,4,Mod(37,198)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("198.37"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(198, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 198.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,6,0,-12,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.6823781811\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 651x^{10} + 154866x^{8} + 16636791x^{6} + 828488506x^{4} + 17109953235x^{2} + 84670385805 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

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

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} - 4 \beta_{4} q^{4} + ( - \beta_{10} - 3 \beta_{5} + \cdots - 3) q^{5} + ( - \beta_{8} + \beta_{7} + \cdots + \beta_1) q^{7} - 8 \beta_{5} q^{8} + ( - 2 \beta_{9} + 2 \beta_{8} + \cdots - 6) q^{10}+ \cdots + (20 \beta_{11} - 2 \beta_{9} + \cdots + 60) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} - 12 q^{4} - 16 q^{5} + 6 q^{7} + 24 q^{8} - 68 q^{10} + 116 q^{11} - 46 q^{13} - 12 q^{14} - 48 q^{16} - 24 q^{17} - 6 q^{19} - 64 q^{20} - 22 q^{22} + 420 q^{23} - 431 q^{25} - 228 q^{26}+ \cdots + 1756 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 651x^{10} + 154866x^{8} + 16636791x^{6} + 828488506x^{4} + 17109953235x^{2} + 84670385805 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 32346861182 \nu^{10} - 19879731324123 \nu^{8} + \cdots - 41\!\cdots\!43 ) / 13\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 38834054737 \nu^{10} + 23147079084069 \nu^{8} + \cdots + 58\!\cdots\!51 ) / 44\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 415508698554 \nu^{11} + 2923381535767 \nu^{10} - 452982742786617 \nu^{9} + \cdots - 46\!\cdots\!69 ) / 38\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1503212638064 \nu^{11} + 8770144607301 \nu^{10} + \cdots - 81\!\cdots\!43 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1503212638064 \nu^{11} - 8770144607301 \nu^{10} + \cdots + 81\!\cdots\!43 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 203789071538872 \nu^{11} + \cdots - 61\!\cdots\!97 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 566133449216986 \nu^{11} + \cdots - 76\!\cdots\!81 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 573400381186223 \nu^{11} + \cdots + 42\!\cdots\!12 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 573400381186223 \nu^{11} + \cdots - 42\!\cdots\!12 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 762398902244219 \nu^{11} + \cdots - 17\!\cdots\!11 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 778692665363159 \nu^{11} + \cdots + 13\!\cdots\!88 ) / 11\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{8} - 4 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots - 1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{11} + 4\beta_{9} + \beta_{8} - 5\beta_{6} + 5\beta_{5} - 4\beta_{2} + 4\beta _1 - 106 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 174 \beta_{11} - 388 \beta_{10} - 318 \beta_{9} - 144 \beta_{8} + 696 \beta_{7} - 174 \beta_{6} + \cdots + 1169 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -1301\beta_{11} - 1352\beta_{9} + 51\beta_{8} + 1301\beta_{6} - 1301\beta_{5} + 928\beta_{2} - 1072\beta _1 + 18549 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 38741 \beta_{11} + 99362 \beta_{10} + 56372 \beta_{9} + 17631 \beta_{8} - 143564 \beta_{7} + \cdots - 320796 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 300976 \beta_{11} + 344808 \beta_{9} - 43832 \beta_{8} - 300976 \beta_{6} + 492060 \beta_{5} + \cdots - 3708435 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 9265399 \beta_{11} - 24607878 \beta_{10} - 11359698 \beta_{9} - 2094299 \beta_{8} + 31155036 \beta_{7} + \cdots + 78216999 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 67615211 \beta_{11} - 82383456 \beta_{9} + 14768245 \beta_{8} + 67615211 \beta_{6} - 153971299 \beta_{5} + \cdots + 781978026 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2235530906 \beta_{11} + 5910194572 \beta_{10} + 2453278402 \beta_{9} + 217747496 \beta_{8} + \cdots - 18885207371 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 15055964539 \beta_{11} + 19293550100 \beta_{9} - 4237585561 \beta_{8} - 15055964539 \beta_{6} + \cdots - 168880585499 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 537580425579 \beta_{11} - 1397216863238 \beta_{10} - 547774713528 \beta_{9} - 10194287949 \beta_{8} + \cdots + 4563366131224 ) / 5 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
10.0232i
− 14.5748i
2.64944i
− 10.0232i
14.5748i
− 2.64944i
15.2352i
− 7.29793i
− 6.76166i
− 15.2352i
7.29793i
6.76166i
1.61803 + 1.17557i 0 1.23607 + 3.80423i −11.0898 + 8.05721i 0 −6.50955 − 20.0343i −2.47214 + 7.60845i 0 −27.4155
37.2 1.61803 + 1.17557i 0 1.23607 + 3.80423i −2.20285 + 1.60046i 0 7.94882 + 24.4639i −2.47214 + 7.60845i 0 −5.44574
37.3 1.61803 + 1.17557i 0 1.23607 + 3.80423i 3.70247 − 2.69000i 0 −2.17534 − 6.69500i −2.47214 + 7.60845i 0 9.15301
91.1 1.61803 − 1.17557i 0 1.23607 − 3.80423i −11.0898 − 8.05721i 0 −6.50955 + 20.0343i −2.47214 − 7.60845i 0 −27.4155
91.2 1.61803 − 1.17557i 0 1.23607 − 3.80423i −2.20285 − 1.60046i 0 7.94882 − 24.4639i −2.47214 − 7.60845i 0 −5.44574
91.3 1.61803 − 1.17557i 0 1.23607 − 3.80423i 3.70247 + 2.69000i 0 −2.17534 + 6.69500i −2.47214 − 7.60845i 0 9.15301
163.1 −0.618034 + 1.90211i 0 −3.23607 − 2.35114i −3.57139 − 10.9916i 0 −12.8715 − 9.35167i 6.47214 − 4.70228i 0 23.1145
163.2 −0.618034 + 1.90211i 0 −3.23607 − 2.35114i −1.25198 − 3.85319i 0 8.55878 + 6.21832i 6.47214 − 4.70228i 0 8.10296
163.3 −0.618034 + 1.90211i 0 −3.23607 − 2.35114i 6.41354 + 19.7388i 0 8.04876 + 5.84777i 6.47214 − 4.70228i 0 −41.5093
181.1 −0.618034 − 1.90211i 0 −3.23607 + 2.35114i −3.57139 + 10.9916i 0 −12.8715 + 9.35167i 6.47214 + 4.70228i 0 23.1145
181.2 −0.618034 − 1.90211i 0 −3.23607 + 2.35114i −1.25198 + 3.85319i 0 8.55878 − 6.21832i 6.47214 + 4.70228i 0 8.10296
181.3 −0.618034 − 1.90211i 0 −3.23607 + 2.35114i 6.41354 − 19.7388i 0 8.04876 − 5.84777i 6.47214 + 4.70228i 0 −41.5093
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 37.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 198.4.f.h yes 12
3.b odd 2 1 198.4.f.g ✓ 12
11.c even 5 1 inner 198.4.f.h yes 12
11.c even 5 1 2178.4.a.ce 6
11.d odd 10 1 2178.4.a.cg 6
33.f even 10 1 2178.4.a.cd 6
33.h odd 10 1 198.4.f.g ✓ 12
33.h odd 10 1 2178.4.a.cf 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.4.f.g ✓ 12 3.b odd 2 1
198.4.f.g ✓ 12 33.h odd 10 1
198.4.f.h yes 12 1.a even 1 1 trivial
198.4.f.h yes 12 11.c even 5 1 inner
2178.4.a.cd 6 33.f even 10 1
2178.4.a.ce 6 11.c even 5 1
2178.4.a.cf 6 33.h odd 10 1
2178.4.a.cg 6 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 16 T_{5}^{11} + 531 T_{5}^{10} + 10569 T_{5}^{9} + 173547 T_{5}^{8} + 1486769 T_{5}^{7} + \cdots + 27556332001 \) acting on \(S_{4}^{\mathrm{new}}(198, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 27556332001 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 40799760225961 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 55\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 861685496070400 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{6} - 210 T^{5} + \cdots - 383709146084)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 31\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 46\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 1672466963904)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 69\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 16\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 36\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 19\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 30\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 69\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 68\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
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