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Properties

Label 111.4.e.a
Level $111$
Weight $4$
Character orbit 111.e
Analytic conductor $6.549$
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] = [111,4,Mod(10,111)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(111, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("111.10");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 111 = 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 111.e (of order \(3\), degree \(2\), 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: \(6.54921201064\)
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: \( x^{18} - 3 x^{17} + 56 x^{16} - 83 x^{15} + 1848 x^{14} - 2187 x^{13} + 34326 x^{12} - 11181 x^{11} + \cdots + 14379264 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{4} q^{3} + ( - \beta_{7} + 3 \beta_{4} - \beta_1) q^{4} + (\beta_{13} + \beta_1) q^{5} + 3 \beta_{2} q^{6} + ( - \beta_{12} - 4 \beta_{4} + \beta_1) q^{7} + ( - \beta_{11} - \beta_{9} + \beta_{8} + \cdots + 5) q^{8}+ \cdots + ( - 9 \beta_{4} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{4} q^{3} + ( - \beta_{7} + 3 \beta_{4} - \beta_1) q^{4} + (\beta_{13} + \beta_1) q^{5} + 3 \beta_{2} q^{6} + ( - \beta_{12} - 4 \beta_{4} + \beta_1) q^{7} + ( - \beta_{11} - \beta_{9} + \beta_{8} + \cdots + 5) q^{8}+ \cdots + (9 \beta_{17} - 9 \beta_{16} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{2} - 27 q^{3} - 31 q^{4} + 4 q^{5} + 18 q^{6} + 43 q^{7} + 132 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 3 q^{2} - 27 q^{3} - 31 q^{4} + 4 q^{5} + 18 q^{6} + 43 q^{7} + 132 q^{8} - 81 q^{9} - 164 q^{10} - 20 q^{11} - 93 q^{12} - 53 q^{13} - 266 q^{14} + 12 q^{15} - 163 q^{16} - 8 q^{17} - 27 q^{18} + 36 q^{19} + 291 q^{20} + 129 q^{21} - 105 q^{22} - 284 q^{23} - 198 q^{24} - 131 q^{25} - 110 q^{26} + 486 q^{27} + 359 q^{28} + 16 q^{29} + 246 q^{30} - 394 q^{31} - 361 q^{32} + 30 q^{33} - 239 q^{34} - 234 q^{35} + 558 q^{36} + 568 q^{37} - 1582 q^{38} - 159 q^{39} + 1877 q^{40} + 1032 q^{41} + 399 q^{42} - 366 q^{43} - 944 q^{44} - 72 q^{45} + 754 q^{46} + 296 q^{47} + 978 q^{48} + 256 q^{49} - 2092 q^{50} + 48 q^{51} + 30 q^{52} + 900 q^{53} - 81 q^{54} + 458 q^{55} + 2378 q^{56} + 108 q^{57} + 3 q^{58} - 438 q^{59} - 1746 q^{60} - 652 q^{61} + 1668 q^{62} - 774 q^{63} + 2796 q^{64} + 168 q^{65} + 630 q^{66} + 1393 q^{67} + 490 q^{68} + 426 q^{69} - 1350 q^{70} - 492 q^{71} - 594 q^{72} - 2442 q^{73} + 321 q^{74} + 786 q^{75} - 300 q^{76} + 2388 q^{77} + 165 q^{78} + 329 q^{79} - 6540 q^{80} - 729 q^{81} - 5742 q^{82} + 936 q^{83} - 2154 q^{84} - 1712 q^{85} + 1013 q^{86} - 24 q^{87} + 1718 q^{88} + 752 q^{89} + 738 q^{90} + 665 q^{91} + 523 q^{92} + 591 q^{93} - 517 q^{94} - 676 q^{95} - 1083 q^{96} + 286 q^{97} - 996 q^{98} + 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 56 x^{16} - 83 x^{15} + 1848 x^{14} - 2187 x^{13} + 34326 x^{12} - 11181 x^{11} + \cdots + 14379264 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 63\!\cdots\!21 \nu^{17} + \cdots + 23\!\cdots\!84 ) / 31\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 93\!\cdots\!76 \nu^{17} + \cdots + 36\!\cdots\!20 ) / 31\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25\!\cdots\!73 \nu^{17} + \cdots - 14\!\cdots\!28 ) / 49\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 52\!\cdots\!59 \nu^{17} + \cdots - 23\!\cdots\!48 ) / 39\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 89\!\cdots\!43 \nu^{17} + \cdots + 51\!\cdots\!92 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 26\!\cdots\!77 \nu^{17} + \cdots - 16\!\cdots\!60 ) / 49\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16\!\cdots\!25 \nu^{17} + \cdots + 91\!\cdots\!00 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33\!\cdots\!13 \nu^{17} + \cdots - 12\!\cdots\!12 ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 45\!\cdots\!25 \nu^{17} + \cdots + 28\!\cdots\!88 ) / 39\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!43 \nu^{17} + \cdots + 52\!\cdots\!32 ) / 83\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 28\!\cdots\!53 \nu^{17} + \cdots - 31\!\cdots\!92 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 86\!\cdots\!91 \nu^{17} + \cdots - 10\!\cdots\!76 ) / 39\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 39\!\cdots\!59 \nu^{17} + \cdots - 98\!\cdots\!96 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 71\!\cdots\!79 \nu^{17} + \cdots + 11\!\cdots\!64 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 79\!\cdots\!33 \nu^{17} + \cdots - 91\!\cdots\!60 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 50\!\cdots\!41 \nu^{17} + \cdots + 16\!\cdots\!44 ) / 12\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 11\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{8} - 20\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{16} + 3\beta_{12} - 2\beta_{10} - 27\beta_{7} + 213\beta_{4} - 33\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{17} - 5 \beta_{16} - 2 \beta_{15} + 5 \beta_{14} - 34 \beta_{13} + 37 \beta_{12} - 34 \beta_{11} + \cdots + 184 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6 \beta_{17} + 41 \beta_{14} - 35 \beta_{11} - 134 \beta_{9} + 15 \beta_{8} + 90 \beta_{6} - 695 \beta_{3} + \cdots + 4918 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 212 \beta_{16} + 98 \beta_{15} + 1048 \beta_{13} - 1152 \beta_{12} + 398 \beta_{10} + 236 \beta_{7} + \cdots + 12319 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 372 \beta_{17} + 1354 \beta_{16} + 372 \beta_{15} - 1354 \beta_{14} + 1998 \beta_{13} - 4616 \beta_{12} + \cdots - 122991 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3656 \beta_{17} - 7054 \beta_{14} + 31463 \beta_{11} + 34341 \beta_{9} - 29593 \beta_{8} + \cdots - 190505 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 41835 \beta_{16} - 15504 \beta_{15} - 80962 \beta_{13} + 146799 \beta_{12} - 103168 \beta_{10} + \cdots - 953335 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 122666 \beta_{17} - 218959 \beta_{16} - 122666 \beta_{15} + 218959 \beta_{14} - 931460 \beta_{13} + \cdots + 6208464 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 552942 \beta_{17} + 1260123 \beta_{14} - 2872561 \beta_{11} - 4525654 \beta_{9} + 1511655 \beta_{8} + \cdots + 86405846 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6628586 \beta_{16} + 3897386 \beta_{15} + 27345850 \beta_{13} - 29595944 \beta_{12} + \cdots + 256428053 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 18271908 \beta_{17} + 37565712 \beta_{16} + 18271908 \beta_{15} - 37565712 \beta_{14} + \cdots - 2368438907 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 120082232 \beta_{17} - 198623364 \beta_{14} + 799226277 \beta_{11} + 867027465 \beta_{9} + \cdots - 6441470333 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1115068613 \beta_{16} - 579164496 \beta_{15} - 3051223940 \beta_{13} + 4150024795 \beta_{12} + \cdots - 27924240957 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 3634357202 \beta_{17} - 5923943017 \beta_{16} - 3634357202 \beta_{15} + 5923943017 \beta_{14} + \cdots + 203676262456 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\) \(38\) \(76\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
2.72151 + 4.71379i
2.14681 + 3.71838i
1.67767 + 2.90580i
0.519373 + 0.899581i
0.414193 + 0.717403i
−0.755419 − 1.30842i
−1.23412 − 2.13757i
−1.53406 − 2.65706i
−2.45595 − 4.25383i
2.72151 − 4.71379i
2.14681 − 3.71838i
1.67767 − 2.90580i
0.519373 − 0.899581i
0.414193 − 0.717403i
−0.755419 + 1.30842i
−1.23412 + 2.13757i
−1.53406 + 2.65706i
−2.45595 + 4.25383i
−2.72151 + 4.71379i −1.50000 − 2.59808i −10.8132 − 18.7290i 10.0400 + 17.3898i 16.3291 13.4362 + 23.2722i 74.1689 −4.50000 + 7.79423i −109.296
10.2 −2.14681 + 3.71838i −1.50000 − 2.59808i −5.21757 − 9.03710i −5.76551 − 9.98616i 12.8809 4.16197 + 7.20875i 10.4556 −4.50000 + 7.79423i 49.5098
10.3 −1.67767 + 2.90580i −1.50000 − 2.59808i −1.62913 − 2.82173i 2.54174 + 4.40242i 10.0660 −8.87729 − 15.3759i −15.9101 −4.50000 + 7.79423i −17.0568
10.4 −0.519373 + 0.899581i −1.50000 − 2.59808i 3.46050 + 5.99377i 4.05427 + 7.02220i 3.11624 −1.62039 − 2.80660i −15.4991 −4.50000 + 7.79423i −8.42271
10.5 −0.414193 + 0.717403i −1.50000 − 2.59808i 3.65689 + 6.33392i −9.54990 − 16.5409i 2.48516 7.92289 + 13.7229i −12.6857 −4.50000 + 7.79423i 15.8220
10.6 0.755419 − 1.30842i −1.50000 − 2.59808i 2.85868 + 4.95138i −1.15801 − 2.00574i −4.53252 12.2136 + 21.1545i 20.7247 −4.50000 + 7.79423i −3.49915
10.7 1.23412 − 2.13757i −1.50000 − 2.59808i 0.953873 + 1.65216i 7.07746 + 12.2585i −7.40475 6.20191 + 10.7420i 24.4548 −4.50000 + 7.79423i 34.9379
10.8 1.53406 − 2.65706i −1.50000 − 2.59808i −0.706658 − 1.22397i −2.02895 − 3.51425i −9.20434 −13.2766 − 22.9957i 20.2087 −4.50000 + 7.79423i −12.4501
10.9 2.45595 − 4.25383i −1.50000 − 2.59808i −8.06337 − 13.9662i −3.21110 − 5.56178i −14.7357 1.33771 + 2.31698i −39.9177 −4.50000 + 7.79423i −31.5452
100.1 −2.72151 − 4.71379i −1.50000 + 2.59808i −10.8132 + 18.7290i 10.0400 − 17.3898i 16.3291 13.4362 − 23.2722i 74.1689 −4.50000 − 7.79423i −109.296
100.2 −2.14681 − 3.71838i −1.50000 + 2.59808i −5.21757 + 9.03710i −5.76551 + 9.98616i 12.8809 4.16197 − 7.20875i 10.4556 −4.50000 − 7.79423i 49.5098
100.3 −1.67767 − 2.90580i −1.50000 + 2.59808i −1.62913 + 2.82173i 2.54174 − 4.40242i 10.0660 −8.87729 + 15.3759i −15.9101 −4.50000 − 7.79423i −17.0568
100.4 −0.519373 − 0.899581i −1.50000 + 2.59808i 3.46050 − 5.99377i 4.05427 − 7.02220i 3.11624 −1.62039 + 2.80660i −15.4991 −4.50000 − 7.79423i −8.42271
100.5 −0.414193 − 0.717403i −1.50000 + 2.59808i 3.65689 − 6.33392i −9.54990 + 16.5409i 2.48516 7.92289 − 13.7229i −12.6857 −4.50000 − 7.79423i 15.8220
100.6 0.755419 + 1.30842i −1.50000 + 2.59808i 2.85868 − 4.95138i −1.15801 + 2.00574i −4.53252 12.2136 − 21.1545i 20.7247 −4.50000 − 7.79423i −3.49915
100.7 1.23412 + 2.13757i −1.50000 + 2.59808i 0.953873 − 1.65216i 7.07746 − 12.2585i −7.40475 6.20191 − 10.7420i 24.4548 −4.50000 − 7.79423i 34.9379
100.8 1.53406 + 2.65706i −1.50000 + 2.59808i −0.706658 + 1.22397i −2.02895 + 3.51425i −9.20434 −13.2766 + 22.9957i 20.2087 −4.50000 − 7.79423i −12.4501
100.9 2.45595 + 4.25383i −1.50000 + 2.59808i −8.06337 + 13.9662i −3.21110 + 5.56178i −14.7357 1.33771 − 2.31698i −39.9177 −4.50000 − 7.79423i −31.5452
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.9
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 111.4.e.a ✓ 18
3.b odd 2 1 333.4.f.d 18
37.c even 3 1 inner 111.4.e.a ✓ 18
111.i odd 6 1 333.4.f.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.4.e.a ✓ 18 1.a even 1 1 trivial
111.4.e.a ✓ 18 37.c even 3 1 inner
333.4.f.d 18 3.b odd 2 1
333.4.f.d 18 111.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 3 T_{2}^{17} + 56 T_{2}^{16} + 83 T_{2}^{15} + 1848 T_{2}^{14} + 2187 T_{2}^{13} + \cdots + 14379264 \) acting on \(S_{4}^{\mathrm{new}}(111, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 3 T^{17} + \cdots + 14379264 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{9} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{9} + 10 T^{8} + \cdots + 661281265440)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{9} + \cdots + 48\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots + 26\!\cdots\!88)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 63\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 21\!\cdots\!33 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 91\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots - 82\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots + 83\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 51\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 62\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 71\!\cdots\!68)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 44\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots - 10\!\cdots\!09)^{2} \) Copy content Toggle raw display
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