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Properties

Label 189.4.f.c
Level $189$
Weight $4$
Character orbit 189.f
Analytic conductor $11.151$
Analytic rank $0$
Dimension $18$
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] = [189,4,Mod(64,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.64");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 189.f (of order \(3\), degree \(2\), not 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: \(11.1513609911\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 6 x^{16} - 23 x^{15} - 6 x^{14} + 255 x^{13} - 56 x^{12} - 81 x^{11} + \cdots + 387420489 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{16} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{3}) q^{2} + ( - \beta_{10} - 4 \beta_{4} + \cdots - 4) q^{4}+ \cdots + (2 \beta_{5} - \beta_{2} + 4 \beta_1 + 9) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{3}) q^{2} + ( - \beta_{10} - 4 \beta_{4} + \cdots - 4) q^{4}+ \cdots + (49 \beta_1 + 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 6 q^{2} - 36 q^{4} - 24 q^{5} - 63 q^{7} + 150 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 6 q^{2} - 36 q^{4} - 24 q^{5} - 63 q^{7} + 150 q^{8} - 111 q^{11} - 18 q^{13} - 42 q^{14} - 144 q^{16} + 546 q^{17} + 90 q^{19} - 402 q^{20} + 162 q^{22} - 312 q^{23} - 279 q^{25} - 102 q^{26} + 504 q^{28} - 378 q^{29} - 18 q^{31} - 891 q^{32} + 324 q^{34} + 336 q^{35} - 72 q^{37} - 147 q^{38} - 405 q^{40} - 477 q^{41} + 171 q^{43} + 1896 q^{44} - 756 q^{46} - 654 q^{47} - 441 q^{49} - 429 q^{50} - 747 q^{52} + 1896 q^{53} - 432 q^{55} - 525 q^{56} - 297 q^{58} - 957 q^{59} + 198 q^{61} + 600 q^{62} + 4770 q^{64} - 2478 q^{65} + 333 q^{67} - 1443 q^{68} + 5652 q^{71} + 306 q^{73} - 2100 q^{74} + 144 q^{76} - 777 q^{77} - 1152 q^{79} + 8418 q^{80} - 6048 q^{82} - 1890 q^{83} + 648 q^{85} - 3837 q^{86} + 2268 q^{88} + 2604 q^{89} + 252 q^{91} - 987 q^{92} - 324 q^{94} - 3144 q^{95} + 1737 q^{97} + 588 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 6 x^{16} - 23 x^{15} - 6 x^{14} + 255 x^{13} - 56 x^{12} - 81 x^{11} + \cdots + 387420489 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 33291362 \nu^{17} - 49543827 \nu^{16} + 1047096165 \nu^{15} - 321711301 \nu^{14} + \cdots - 11\!\cdots\!96 ) / 32\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 37129573 \nu^{17} + 979472019 \nu^{16} + 10255968003 \nu^{15} + 5505932219 \nu^{14} + \cdots - 45\!\cdots\!52 ) / 32\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 100884080039 \nu^{17} + 189431315142 \nu^{16} + 536203636539 \nu^{15} + \cdots + 34\!\cdots\!28 ) / 47\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 934928795 \nu^{17} - 1529236356 \nu^{16} + 4688064201 \nu^{15} - 16147747558 \nu^{14} + \cdots - 13\!\cdots\!90 ) / 36\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 125740550 \nu^{17} + 239756565 \nu^{16} + 3019695132 \nu^{15} - 6332440258 \nu^{14} + \cdots - 39\!\cdots\!90 ) / 32\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29406205898 \nu^{17} + 195345729363 \nu^{16} - 595559286294 \nu^{15} + \cdots - 19\!\cdots\!20 ) / 67\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7881112439 \nu^{17} - 150593598264 \nu^{16} - 115305213819 \nu^{15} + \cdots - 89\!\cdots\!21 ) / 10\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1789705618 \nu^{17} - 4808034645 \nu^{16} + 7842191781 \nu^{15} - 75966958000 \nu^{14} + \cdots - 17\!\cdots\!67 ) / 18\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 471729716684 \nu^{17} + 1671338975199 \nu^{16} - 124452258888 \nu^{15} + \cdots + 93\!\cdots\!97 ) / 47\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 705247659434 \nu^{17} + 1068128624931 \nu^{16} - 2710590961395 \nu^{15} + \cdots + 96\!\cdots\!63 ) / 47\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 193556779 \nu^{17} + 276829698 \nu^{16} + 2133154329 \nu^{15} + 3107835478 \nu^{14} + \cdots - 29\!\cdots\!04 ) / 10\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 774309029 \nu^{17} + 815271315 \nu^{16} - 2460306774 \nu^{15} - 1874093159 \nu^{14} + \cdots + 29\!\cdots\!06 ) / 32\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 806254553 \nu^{17} - 655552110 \nu^{16} - 2598929994 \nu^{15} - 8913639005 \nu^{14} + \cdots + 13\!\cdots\!46 ) / 32\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3136900 \nu^{17} - 1886397 \nu^{16} + 4689417 \nu^{15} - 76992469 \nu^{14} + \cdots - 155998648007298 ) / 12471223378887 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2875873 \nu^{17} + 10890546 \nu^{16} + 33736470 \nu^{15} + 54188278 \nu^{14} + \cdots - 32918171922468 ) / 10689620039046 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 8369887993 \nu^{17} + 5990233296 \nu^{16} + 8097101286 \nu^{15} - 246444400754 \nu^{14} + \cdots - 17\!\cdots\!53 ) / 27\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 1463891803664 \nu^{17} - 741429596631 \nu^{16} - 8190740135685 \nu^{15} + \cdots + 13\!\cdots\!69 ) / 47\!\cdots\!98 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} + \beta_{12} + \beta_{8} + 3\beta_{4} + \beta _1 + 3 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{16} - \beta_{14} + 3 \beta_{13} + 3 \beta_{11} + 6 \beta_{10} - 3 \beta_{9} - 3 \beta_{5} + \cdots + 6 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{17} + 2 \beta_{16} - 5 \beta_{15} + 3 \beta_{14} - \beta_{13} - \beta_{12} + 6 \beta_{10} + \cdots + 60 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18 \beta_{17} + 4 \beta_{16} - 3 \beta_{15} - 2 \beta_{14} - 6 \beta_{13} - \beta_{12} + 27 \beta_{11} + \cdots + 51 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 27 \beta_{17} - 9 \beta_{16} + 15 \beta_{15} - 12 \beta_{14} + 27 \beta_{13} + 11 \beta_{12} + \cdots - 597 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 111 \beta_{17} - 34 \beta_{16} - 143 \beta_{15} + 84 \beta_{14} - 16 \beta_{13} - 52 \beta_{12} + \cdots - 1707 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 63 \beta_{17} + 350 \beta_{16} - 213 \beta_{15} + 189 \beta_{14} + 114 \beta_{13} - 780 \beta_{12} + \cdots - 540 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 63 \beta_{17} - 1597 \beta_{16} + 768 \beta_{15} + 304 \beta_{14} - 552 \beta_{13} + 1255 \beta_{12} + \cdots - 22953 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2628 \beta_{17} + 140 \beta_{16} + 1165 \beta_{15} + 2154 \beta_{14} + 1223 \beta_{13} - 2665 \beta_{12} + \cdots - 62460 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5031 \beta_{17} - 5229 \beta_{16} - 8109 \beta_{15} - 3169 \beta_{14} - 14319 \beta_{13} - 8720 \beta_{12} + \cdots + 67377 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 27405 \beta_{17} + 22510 \beta_{16} + 3906 \beta_{15} - 23005 \beta_{14} + 18777 \beta_{13} + \cdots - 99813 ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 33801 \beta_{17} - 35926 \beta_{16} + 153850 \beta_{15} + 91929 \beta_{14} - 78256 \beta_{13} + \cdots - 339681 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 18045 \beta_{17} - 64571 \beta_{16} - 99426 \beta_{15} + 192526 \beta_{14} - 192939 \beta_{13} + \cdots - 2556039 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 867564 \beta_{17} + 398538 \beta_{16} + 169863 \beta_{15} - 751026 \beta_{14} - 470115 \beta_{13} + \cdots - 11832240 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 35751 \beta_{17} - 2662414 \beta_{16} + 1402246 \beta_{15} + 1602174 \beta_{14} + 1476011 \beta_{13} + \cdots + 22364742 ) / 9 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 9813114 \beta_{17} + 6223553 \beta_{16} + 4940670 \beta_{15} + 7118199 \beta_{14} - 7080519 \beta_{13} + \cdots + 14398830 ) / 9 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 15641064 \beta_{17} - 11815708 \beta_{16} + 2218944 \beta_{15} - 21059822 \beta_{14} + \cdots + 407436882 ) / 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
2.37763 − 1.82944i
−1.07135 + 2.80218i
−2.61694 − 1.46684i
2.81021 − 1.05012i
2.85089 + 0.934028i
−0.831471 − 2.88247i
−0.0718325 + 2.99914i
−2.90795 + 0.737429i
0.960810 − 2.84198i
2.37763 + 1.82944i
−1.07135 − 2.80218i
−2.61694 + 1.46684i
2.81021 + 1.05012i
2.85089 − 0.934028i
−0.831471 + 2.88247i
−0.0718325 − 2.99914i
−2.90795 − 0.737429i
0.960810 + 2.84198i
−2.80688 − 4.86166i 0 −11.7571 + 20.3640i −5.31469 + 9.20530i 0 −3.50000 − 6.06218i 87.0935 0 59.6707
64.2 −2.21232 − 3.83185i 0 −5.78871 + 10.0263i 1.92707 − 3.33779i 0 −3.50000 − 6.06218i 15.8288 0 −17.0532
64.3 −1.66614 − 2.88585i 0 −1.55207 + 2.68826i −8.37356 + 14.5034i 0 −3.50000 − 6.06218i −16.3144 0 55.8062
64.4 −1.32666 − 2.29785i 0 0.479932 − 0.831266i 10.0300 − 17.3725i 0 −3.50000 − 6.06218i −23.7734 0 −53.2258
64.5 −0.377604 − 0.654029i 0 3.71483 − 6.43428i −6.62462 + 11.4742i 0 −3.50000 − 6.06218i −11.6526 0 10.0059
64.6 0.231183 + 0.400421i 0 3.89311 − 6.74306i 2.26496 − 3.92303i 0 −3.50000 − 6.06218i 7.29902 0 2.09449
64.7 1.21836 + 2.11026i 0 1.03120 − 1.78609i −2.61762 + 4.53384i 0 −3.50000 − 6.06218i 24.5192 0 −12.7568
64.8 1.61753 + 2.80164i 0 −1.23279 + 2.13526i 4.95111 − 8.57557i 0 −3.50000 − 6.06218i 17.9041 0 32.0342
64.9 2.32254 + 4.02275i 0 −6.78835 + 11.7578i −8.24268 + 14.2767i 0 −3.50000 − 6.06218i −25.9042 0 −76.5757
127.1 −2.80688 + 4.86166i 0 −11.7571 − 20.3640i −5.31469 − 9.20530i 0 −3.50000 + 6.06218i 87.0935 0 59.6707
127.2 −2.21232 + 3.83185i 0 −5.78871 − 10.0263i 1.92707 + 3.33779i 0 −3.50000 + 6.06218i 15.8288 0 −17.0532
127.3 −1.66614 + 2.88585i 0 −1.55207 − 2.68826i −8.37356 − 14.5034i 0 −3.50000 + 6.06218i −16.3144 0 55.8062
127.4 −1.32666 + 2.29785i 0 0.479932 + 0.831266i 10.0300 + 17.3725i 0 −3.50000 + 6.06218i −23.7734 0 −53.2258
127.5 −0.377604 + 0.654029i 0 3.71483 + 6.43428i −6.62462 − 11.4742i 0 −3.50000 + 6.06218i −11.6526 0 10.0059
127.6 0.231183 − 0.400421i 0 3.89311 + 6.74306i 2.26496 + 3.92303i 0 −3.50000 + 6.06218i 7.29902 0 2.09449
127.7 1.21836 − 2.11026i 0 1.03120 + 1.78609i −2.61762 − 4.53384i 0 −3.50000 + 6.06218i 24.5192 0 −12.7568
127.8 1.61753 − 2.80164i 0 −1.23279 − 2.13526i 4.95111 + 8.57557i 0 −3.50000 + 6.06218i 17.9041 0 32.0342
127.9 2.32254 − 4.02275i 0 −6.78835 − 11.7578i −8.24268 − 14.2767i 0 −3.50000 + 6.06218i −25.9042 0 −76.5757
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.4.f.c 18
3.b odd 2 1 63.4.f.c ✓ 18
9.c even 3 1 inner 189.4.f.c 18
9.c even 3 1 567.4.a.k 9
9.d odd 6 1 63.4.f.c ✓ 18
9.d odd 6 1 567.4.a.j 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.f.c ✓ 18 3.b odd 2 1
63.4.f.c ✓ 18 9.d odd 6 1
189.4.f.c 18 1.a even 1 1 trivial
189.4.f.c 18 9.c even 3 1 inner
567.4.a.j 9 9.d odd 6 1
567.4.a.k 9 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 6 T_{2}^{17} + 72 T_{2}^{16} + 246 T_{2}^{15} + 2340 T_{2}^{14} + 6831 T_{2}^{13} + \cdots + 7884864 \) acting on \(S_{4}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 6 T^{17} + \cdots + 7884864 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 49\!\cdots\!24 \) Copy content Toggle raw display
$7$ \( (T^{2} + 7 T + 49)^{9} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots - 50\!\cdots\!16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{9} - 45 T^{8} + \cdots - 278256660857)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 57\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 93\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots + 84\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 33\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots - 32\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 34\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 15\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
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