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Properties

Label 198.3.k.a
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,-24] 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: \( x^{16} + 14 x^{14} + 255 x^{12} + 3946 x^{10} + 33929 x^{8} + 477466 x^{6} + 3733455 x^{4} + \cdots + 214358881 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + 2 \beta_{2} q^{4} + ( - 2 \beta_{15} - \beta_{14} + \cdots - \beta_1) q^{5} + ( - \beta_{13} + \beta_{10} - 2 \beta_{6} - 2) q^{7} - 2 \beta_{11} q^{8} + ( - \beta_{12} + \beta_{10} - 3 \beta_{6} + \cdots - 3) q^{10}+ \cdots + ( - 8 \beta_{14} + 8 \beta_{11} + \cdots + 10 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 24 q^{7} - 24 q^{10} + 48 q^{13} - 16 q^{16} - 36 q^{19} + 48 q^{22} + 192 q^{25} - 32 q^{28} + 4 q^{31} - 112 q^{34} + 76 q^{37} - 72 q^{40} - 440 q^{43} - 36 q^{46} - 168 q^{49} + 24 q^{52}+ \cdots + 156 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 14 x^{14} + 255 x^{12} + 3946 x^{10} + 33929 x^{8} + 477466 x^{6} + 3733455 x^{4} + \cdots + 214358881 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 692398 \nu^{14} - 19071919 \nu^{12} + 350781678 \nu^{10} - 3605467857 \nu^{8} + \cdots + 90959465559567 ) / 48221803613511 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 961320 \nu^{14} + 47328316 \nu^{12} + 268312940 \nu^{10} + 8307578761 \nu^{8} + \cdots + 36902579359184 ) / 48221803613511 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2362906 \nu^{15} - 276804813 \nu^{13} - 751331705 \nu^{11} - 24768172562 \nu^{9} + \cdots + 23950549848621 \nu ) / 530439839748621 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3431265 \nu^{15} + 105048949 \nu^{13} - 545984223 \nu^{11} - 5955245818 \nu^{9} + \cdots - 28939704971749 \nu ) / 530439839748621 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3871994 \nu^{14} - 29093035 \nu^{12} - 663011678 \nu^{10} - 7064834300 \nu^{8} + \cdots - 42877459367688 ) / 48221803613511 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28957 \nu^{15} + 37800 \nu^{13} - 5126760 \nu^{11} - 2854845 \nu^{9} + \cdots - 769380084495 \nu ) / 2032336550761 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 692398 \nu^{15} - 19071919 \nu^{13} + 350781678 \nu^{11} - 3605467857 \nu^{9} + \cdots + 90959465559567 \nu ) / 48221803613511 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5542502 \nu^{14} + 286825929 \nu^{12} + 1765125061 \nu^{10} + 28227539005 \nu^{8} + \cdots + 61664571465123 ) / 48221803613511 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7572181 \nu^{14} - 47699517 \nu^{12} + 1828090519 \nu^{10} + 17722191022 \nu^{8} + \cdots + 199679209258188 ) / 48221803613511 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 20771943 \nu^{15} - 24617413 \nu^{13} + 2105576582 \nu^{11} + 9326199912 \nu^{9} + \cdots + 39428970613186 \nu ) / 530439839748621 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21464341 \nu^{14} - 5545494 \nu^{12} + 1754794904 \nu^{10} + 12931667769 \nu^{8} + \cdots - 3308691332870 ) / 48221803613511 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 23841651 \nu^{14} - 213694002 \nu^{12} - 5672301488 \nu^{10} - 56167145898 \nu^{8} + \cdots - 325006366516381 ) / 48221803613511 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3871994 \nu^{15} - 29093035 \nu^{13} - 663011678 \nu^{11} - 7064834300 \nu^{9} + \cdots - 42877459367688 \nu ) / 48221803613511 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4140916 \nu^{15} + 57349432 \nu^{13} + 1282106296 \nu^{11} + 11766945204 \nu^{9} + \cdots + 122517700672928 \nu ) / 48221803613511 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{10} + \beta_{9} + 9\beta_{6} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} + 9\beta_{14} + 11\beta_{11} - 11\beta_{5} - 11\beta_{4} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{13} + 2\beta_{12} - 16\beta_{10} - 16\beta_{9} - 43\beta_{6} + 81\beta_{3} + 43\beta_{2} - 81 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 63\beta_{15} + 54\beta_{14} - 54\beta_{8} + 22\beta_{7} + 198\beta_{5} + 198\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 207\beta_{13} + 207\beta_{10} + 131\beta_{9} - 725\beta_{6} + 239\beta_{2} - 725 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -856\beta_{14} - 2277\beta_{11} + 577\beta_{8} + 2277\beta_{7} + 836\beta_{4} - 856\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -1700\beta_{13} - 1692\beta_{12} - 1700\beta_{9} - 1369\beta_{3} - 9424\beta_{2} + 9424 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 331\beta_{15} + 331\beta_{14} + 88\beta_{11} - 9763\beta_{8} - 18700\beta_{7} - 18700\beta_{5} + 9763\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -28044\beta_{13} - 8937\beta_{12} - 8937\beta_{10} + 84645\beta_{6} - 84645\beta_{3} - 12578 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 65538 \beta_{15} + 210177 \beta_{11} + 65538 \beta_{8} - 308484 \beta_{7} - 210177 \beta_{5} + \cdots - 87053 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 11254\beta_{12} - 362768\beta_{10} + 11254\beta_{9} - 606069\beta_{6} + 358150\beta_{3} + 358150\beta_{2} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -4618\beta_{15} - 259173\beta_{14} + 123794\beta_{11} + 3990448\beta_{5} - 123794\beta_{4} - 4618\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4109624 \beta_{13} + 4109624 \beta_{12} + 4240385 \beta_{10} + 4240385 \beta_{9} - 1571041 \beta_{6} + \cdots + 2720142 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2589381 \beta_{15} - 8531568 \beta_{14} + 8531568 \beta_{8} + 45205864 \beta_{7} + \cdots - 1438371 \beta_{4} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(145\) \(155\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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
−2.40294 − 2.28602i
2.91668 + 1.57891i
−2.91668 − 1.57891i
2.40294 + 2.28602i
−2.40294 + 2.28602i
2.91668 − 1.57891i
−2.91668 + 1.57891i
2.40294 − 2.28602i
2.05011 + 2.60712i
0.126145 − 3.31423i
−0.126145 + 3.31423i
−2.05011 − 2.60712i
2.05011 − 2.60712i
0.126145 + 3.31423i
−0.126145 − 3.31423i
−2.05011 + 2.60712i
−1.34500 − 0.437016i 0 1.61803 + 1.17557i −2.68314 + 0.871805i 0 1.80392 + 1.31063i −1.66251 − 2.28825i 0 3.98981
53.2 −1.34500 − 0.437016i 0 1.61803 + 1.17557i 9.21189 − 2.99313i 0 −7.03999 − 5.11485i −1.66251 − 2.28825i 0 −13.6980
53.3 1.34500 + 0.437016i 0 1.61803 + 1.17557i −9.21189 + 2.99313i 0 −7.03999 − 5.11485i 1.66251 + 2.28825i 0 −13.6980
53.4 1.34500 + 0.437016i 0 1.61803 + 1.17557i 2.68314 − 0.871805i 0 1.80392 + 1.31063i 1.66251 + 2.28825i 0 3.98981
71.1 −1.34500 + 0.437016i 0 1.61803 − 1.17557i −2.68314 − 0.871805i 0 1.80392 − 1.31063i −1.66251 + 2.28825i 0 3.98981
71.2 −1.34500 + 0.437016i 0 1.61803 − 1.17557i 9.21189 + 2.99313i 0 −7.03999 + 5.11485i −1.66251 + 2.28825i 0 −13.6980
71.3 1.34500 − 0.437016i 0 1.61803 − 1.17557i −9.21189 − 2.99313i 0 −7.03999 + 5.11485i 1.66251 − 2.28825i 0 −13.6980
71.4 1.34500 − 0.437016i 0 1.61803 − 1.17557i 2.68314 + 0.871805i 0 1.80392 − 1.31063i 1.66251 − 2.28825i 0 3.98981
125.1 −0.831254 + 1.14412i 0 −0.618034 − 1.90211i −2.92167 − 4.02133i 0 2.20576 + 6.78862i 2.68999 + 0.874032i 0 7.02955
125.2 −0.831254 + 1.14412i 0 −0.618034 − 1.90211i 1.38044 + 1.90001i 0 −2.96969 − 9.13976i 2.68999 + 0.874032i 0 −3.32134
125.3 0.831254 − 1.14412i 0 −0.618034 − 1.90211i −1.38044 − 1.90001i 0 −2.96969 − 9.13976i −2.68999 − 0.874032i 0 −3.32134
125.4 0.831254 − 1.14412i 0 −0.618034 − 1.90211i 2.92167 + 4.02133i 0 2.20576 + 6.78862i −2.68999 − 0.874032i 0 7.02955
179.1 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i −2.92167 + 4.02133i 0 2.20576 − 6.78862i 2.68999 − 0.874032i 0 7.02955
179.2 −0.831254 − 1.14412i 0 −0.618034 + 1.90211i 1.38044 − 1.90001i 0 −2.96969 + 9.13976i 2.68999 − 0.874032i 0 −3.32134
179.3 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −1.38044 + 1.90001i 0 −2.96969 + 9.13976i −2.68999 + 0.874032i 0 −3.32134
179.4 0.831254 + 1.14412i 0 −0.618034 + 1.90211i 2.92167 − 4.02133i 0 2.20576 − 6.78862i −2.68999 + 0.874032i 0 7.02955
\(n\): 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.a ✓ 16
3.b odd 2 1 inner 198.3.k.a ✓ 16
11.c even 5 1 inner 198.3.k.a ✓ 16
11.c even 5 1 2178.3.c.p 8
11.d odd 10 1 2178.3.c.m 8
33.f even 10 1 2178.3.c.m 8
33.h odd 10 1 inner 198.3.k.a ✓ 16
33.h odd 10 1 2178.3.c.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.3.k.a ✓ 16 1.a even 1 1 trivial
198.3.k.a ✓ 16 3.b odd 2 1 inner
198.3.k.a ✓ 16 11.c even 5 1 inner
198.3.k.a ✓ 16 33.h odd 10 1 inner
2178.3.c.m 8 11.d odd 10 1
2178.3.c.m 8 33.f even 10 1
2178.3.c.p 8 11.c even 5 1
2178.3.c.p 8 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 146 T_{5}^{14} + 8437 T_{5}^{12} - 32428 T_{5}^{10} + 5357550 T_{5}^{8} + \cdots + 10355301121 \) 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
$5$ \( T^{16} + \cdots + 10355301121 \) Copy content Toggle raw display
$7$ \( (T^{8} + 12 T^{7} + \cdots + 1771561)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( (T^{8} - 24 T^{7} + \cdots + 1515361)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 27\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( (T^{8} + 18 T^{7} + \cdots + 938135641)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1342 T^{6} + \cdots + 92140801)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{8} - 2 T^{7} + \cdots + 25471840801)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 38 T^{7} + \cdots + 9034312401)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 79\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( (T^{4} + 110 T^{3} + \cdots - 2538225)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( (T^{8} - 20 T^{7} + \cdots + 14882780025)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 138 T^{3} + \cdots - 15268559)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 45\!\cdots\!81 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 304825201051536)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 18649260873441)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 34\!\cdots\!61 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 301784681546401)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 18\!\cdots\!01)^{2} \) Copy content Toggle raw display
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