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Properties

Label 198.6.f.h
Level $198$
Weight $6$
Character orbit 198.f
Analytic conductor $31.756$
Analytic rank $0$
Dimension $20$
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] = [198,6,Mod(37,198)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(198, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("198.37");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 198.f (of order \(5\), degree \(4\), 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: \(31.7559963230\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} - 32717 x^{18} - 175765 x^{17} + 429989344 x^{16} + 5846276963 x^{15} + \cdots + 29\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8}\cdot 11^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_{2} q^{2} + 16 \beta_{3} q^{4} + (\beta_{5} - 6 \beta_{3} + \cdots + 10 \beta_1) q^{5}+ \cdots + (64 \beta_{3} + 64 \beta_{2} + \cdots + 64) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_{2} q^{2} + 16 \beta_{3} q^{4} + (\beta_{5} - 6 \beta_{3} + \cdots + 10 \beta_1) q^{5}+ \cdots + ( - 44 \beta_{19} - 68 \beta_{18} + \cdots - 19740) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} - 80 q^{4} + 112 q^{5} - 392 q^{7} + 320 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{2} - 80 q^{4} + 112 q^{5} - 392 q^{7} + 320 q^{8} + 552 q^{10} - 60 q^{11} - 420 q^{13} + 1568 q^{14} - 1280 q^{16} - 712 q^{17} - 898 q^{19} + 1792 q^{20} + 2020 q^{22} - 1180 q^{23} - 1079 q^{25} + 2880 q^{26} + 4688 q^{28} - 517 q^{29} - 5551 q^{31} - 20480 q^{32} - 6992 q^{34} - 14325 q^{35} - 7584 q^{37} + 1832 q^{38} + 2752 q^{40} + 16868 q^{41} - 704 q^{43} - 8080 q^{44} - 11400 q^{46} + 38866 q^{47} - 22573 q^{49} + 5416 q^{50} - 11520 q^{52} + 97517 q^{53} + 14404 q^{55} - 12672 q^{56} + 2068 q^{58} + 52682 q^{59} + 73874 q^{61} - 21136 q^{62} - 20480 q^{64} - 236352 q^{65} - 267432 q^{67} - 11392 q^{68} - 67660 q^{70} - 20588 q^{71} + 97257 q^{73} + 30336 q^{74} + 43392 q^{76} - 100582 q^{77} + 37498 q^{79} - 11008 q^{80} - 3672 q^{82} + 140952 q^{83} - 158376 q^{85} + 75136 q^{86} - 63040 q^{88} - 168796 q^{89} + 173196 q^{91} - 36160 q^{92} + 45376 q^{94} + 518002 q^{95} - 225802 q^{97} - 396808 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} - 32717 x^{18} - 175765 x^{17} + 429989344 x^{16} + 5846276963 x^{15} + \cdots + 29\!\cdots\!01 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 16\!\cdots\!83 \nu^{19} + \cdots + 40\!\cdots\!61 ) / 20\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 61\!\cdots\!39 \nu^{19} + \cdots + 65\!\cdots\!20 ) / 20\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61\!\cdots\!55 \nu^{19} + \cdots + 60\!\cdots\!82 ) / 20\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!11 \nu^{19} + \cdots - 10\!\cdots\!66 ) / 20\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 69\!\cdots\!08 \nu^{19} + \cdots - 54\!\cdots\!65 ) / 20\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!92 \nu^{19} + \cdots - 10\!\cdots\!64 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!17 \nu^{19} + \cdots + 17\!\cdots\!30 ) / 20\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!86 \nu^{19} + \cdots - 65\!\cdots\!11 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21\!\cdots\!24 \nu^{19} + \cdots - 28\!\cdots\!21 ) / 18\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 36\!\cdots\!11 \nu^{19} + \cdots - 11\!\cdots\!98 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 38\!\cdots\!08 \nu^{19} + \cdots - 24\!\cdots\!96 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 44\!\cdots\!08 \nu^{19} + \cdots - 13\!\cdots\!62 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 60\!\cdots\!05 \nu^{19} + \cdots + 12\!\cdots\!17 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 78\!\cdots\!73 \nu^{19} + \cdots + 21\!\cdots\!62 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 91\!\cdots\!72 \nu^{19} + \cdots - 10\!\cdots\!09 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 99\!\cdots\!66 \nu^{19} + \cdots - 63\!\cdots\!05 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 13\!\cdots\!93 \nu^{19} + \cdots - 19\!\cdots\!81 ) / 17\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 17\!\cdots\!76 \nu^{19} + \cdots + 27\!\cdots\!73 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 19\!\cdots\!52 \nu^{19} + \cdots - 30\!\cdots\!62 ) / 19\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 10 \beta_{19} - 10 \beta_{17} - 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + 3 \beta_{12} - 7 \beta_{11} + \cdots + 3322 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 231 \beta_{19} - 386 \beta_{18} - 246 \beta_{17} + 30 \beta_{16} + 104 \beta_{15} + 344 \beta_{14} + \cdots + 82318 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 93510 \beta_{19} + 16 \beta_{18} - 93524 \beta_{17} - 1788 \beta_{16} - 21914 \beta_{15} + \cdots + 20970073 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2204169 \beta_{19} - 4045498 \beta_{18} - 2394974 \beta_{17} + 333390 \beta_{16} + 1600226 \beta_{15} + \cdots + 944273100 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 792570901 \beta_{19} - 17803629 \beta_{18} - 786364685 \beta_{17} - 22508408 \beta_{16} + \cdots + 163794635056 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 19335253315 \beta_{19} - 37036978579 \beta_{18} - 21152744641 \beta_{17} + 3452167978 \beta_{16} + \cdots + 9204288459593 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6685432675256 \beta_{19} - 332846671518 \beta_{18} - 6565037951510 \beta_{17} - 242498132228 \beta_{16} + \cdots + 13\!\cdots\!62 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 174954133546377 \beta_{19} - 329329248607384 \beta_{18} - 189885832161432 \beta_{17} + \cdots + 85\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 56\!\cdots\!91 \beta_{19} + \cdots + 11\!\cdots\!90 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16\!\cdots\!87 \beta_{19} + \cdots + 79\!\cdots\!27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 48\!\cdots\!58 \beta_{19} + \cdots + 10\!\cdots\!97 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 15\!\cdots\!71 \beta_{19} + \cdots + 73\!\cdots\!22 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 42\!\cdots\!77 \beta_{19} + \cdots + 91\!\cdots\!59 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 14\!\cdots\!97 \beta_{19} + \cdots + 67\!\cdots\!61 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 36\!\cdots\!16 \beta_{19} + \cdots + 81\!\cdots\!32 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 14\!\cdots\!53 \beta_{19} + \cdots + 62\!\cdots\!91 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 32\!\cdots\!86 \beta_{19} + \cdots + 72\!\cdots\!83 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 13\!\cdots\!30 \beta_{19} + \cdots + 57\!\cdots\!46 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−90.3506 + 0.951057i
−20.8963 + 0.951057i
1.30358 + 0.951057i
28.0249 + 0.951057i
80.9913 + 0.951057i
−90.3506 − 0.951057i
−20.8963 − 0.951057i
1.30358 − 0.951057i
28.0249 − 0.951057i
80.9913 − 0.951057i
95.4367 − 0.587785i
37.6203 − 0.587785i
−24.7410 − 0.587785i
−48.5080 − 0.587785i
−57.3810 − 0.587785i
95.4367 + 0.587785i
37.6203 + 0.587785i
−24.7410 + 0.587785i
−48.5080 + 0.587785i
−57.3810 + 0.587785i
3.23607 + 2.35114i 0 4.94427 + 15.2169i −61.7550 + 44.8676i 0 21.9027 + 67.4096i −19.7771 + 60.8676i 0 −305.333
37.2 3.23607 + 2.35114i 0 4.94427 + 15.2169i −5.56531 + 4.04343i 0 −20.6422 − 63.5301i −19.7771 + 60.8676i 0 −27.5164
37.3 3.23607 + 2.35114i 0 4.94427 + 15.2169i 12.3948 − 9.00534i 0 −75.6508 − 232.829i −19.7771 + 60.8676i 0 61.2832
37.4 3.23607 + 2.35114i 0 4.94427 + 15.2169i 34.0128 − 24.7118i 0 58.4057 + 179.754i −19.7771 + 60.8676i 0 168.169
37.5 3.23607 + 2.35114i 0 4.94427 + 15.2169i 76.8635 − 55.8446i 0 −16.0513 − 49.4010i −19.7771 + 60.8676i 0 380.034
91.1 3.23607 − 2.35114i 0 4.94427 − 15.2169i −61.7550 − 44.8676i 0 21.9027 − 67.4096i −19.7771 − 60.8676i 0 −305.333
91.2 3.23607 − 2.35114i 0 4.94427 − 15.2169i −5.56531 − 4.04343i 0 −20.6422 + 63.5301i −19.7771 − 60.8676i 0 −27.5164
91.3 3.23607 − 2.35114i 0 4.94427 − 15.2169i 12.3948 + 9.00534i 0 −75.6508 + 232.829i −19.7771 − 60.8676i 0 61.2832
91.4 3.23607 − 2.35114i 0 4.94427 − 15.2169i 34.0128 + 24.7118i 0 58.4057 − 179.754i −19.7771 − 60.8676i 0 168.169
91.5 3.23607 − 2.35114i 0 4.94427 − 15.2169i 76.8635 + 55.8446i 0 −16.0513 + 49.4010i −19.7771 − 60.8676i 0 380.034
163.1 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i −29.3317 − 90.2738i 0 −134.482 − 97.7069i 51.7771 − 37.6183i 0 379.678
163.2 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i −11.4655 − 35.2872i 0 9.62554 + 6.99336i 51.7771 − 37.6183i 0 148.412
163.3 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i 7.80522 + 24.0220i 0 84.3699 + 61.2983i 51.7771 − 37.6183i 0 −101.033
163.4 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i 15.1496 + 46.6258i 0 77.8429 + 56.5562i 51.7771 − 37.6183i 0 −196.101
163.5 −1.23607 + 3.80423i 0 −12.9443 − 9.40456i 17.8915 + 55.0645i 0 −201.320 − 146.268i 51.7771 − 37.6183i 0 −231.593
181.1 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i −29.3317 + 90.2738i 0 −134.482 + 97.7069i 51.7771 + 37.6183i 0 379.678
181.2 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i −11.4655 + 35.2872i 0 9.62554 − 6.99336i 51.7771 + 37.6183i 0 148.412
181.3 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i 7.80522 − 24.0220i 0 84.3699 − 61.2983i 51.7771 + 37.6183i 0 −101.033
181.4 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i 15.1496 − 46.6258i 0 77.8429 − 56.5562i 51.7771 + 37.6183i 0 −196.101
181.5 −1.23607 − 3.80423i 0 −12.9443 + 9.40456i 17.8915 − 55.0645i 0 −201.320 + 146.268i 51.7771 + 37.6183i 0 −231.593
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
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 198.6.f.h yes 20
3.b odd 2 1 198.6.f.g ✓ 20
11.c even 5 1 inner 198.6.f.h yes 20
33.h odd 10 1 198.6.f.g ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.6.f.g ✓ 20 3.b odd 2 1
198.6.f.g ✓ 20 33.h odd 10 1
198.6.f.h yes 20 1.a even 1 1 trivial
198.6.f.h yes 20 11.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - 112 T_{5}^{19} + 14624 T_{5}^{18} - 1380421 T_{5}^{17} + 149885534 T_{5}^{16} + \cdots + 65\!\cdots\!25 \) acting on \(S_{6}^{\mathrm{new}}(198, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{3} + \cdots + 256)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 77\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 46\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 67\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 40\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 63\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 39\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 20\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
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