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Properties

Label 222.3.l.c
Level $222$
Weight $3$
Character orbit 222.l
Analytic conductor $6.049$
Analytic rank $0$
Dimension $16$
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] = [222,3,Mod(97,222)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(222, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("222.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 222 = 2 \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 222.l (of order \(12\), 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: \(6.04906186880\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 174 x^{14} + 12083 x^{12} + 425514 x^{10} + 7984353 x^{8} + 77125728 x^{6} + 345835640 x^{4} + \cdots + 179238544 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{2} + (\beta_1 - 1) q^{3} + 2 \beta_{4} q^{4} + (\beta_{11} + \beta_{6} - \beta_{4} + \cdots - 1) q^{5}+ \cdots - 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{2} + (\beta_1 - 1) q^{3} + 2 \beta_{4} q^{4} + (\beta_{11} + \beta_{6} - \beta_{4} + \cdots - 1) q^{5}+ \cdots + ( - 3 \beta_{15} - 6 \beta_{14} + \cdots - 3 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 24 q^{3} - 6 q^{5} - 2 q^{7} + 32 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} - 24 q^{3} - 6 q^{5} - 2 q^{7} + 32 q^{8} + 24 q^{9} - 24 q^{10} + 32 q^{13} - 4 q^{14} + 18 q^{15} + 32 q^{16} + 20 q^{17} - 24 q^{18} + 66 q^{19} - 12 q^{20} + 6 q^{21} - 14 q^{22} - 10 q^{23} - 48 q^{24} - 84 q^{25} + 56 q^{26} + 38 q^{29} + 36 q^{30} - 82 q^{31} - 32 q^{32} + 6 q^{33} + 2 q^{34} + 246 q^{35} - 86 q^{37} - 36 q^{38} - 60 q^{39} + 54 q^{41} + 6 q^{42} - 56 q^{43} + 16 q^{44} - 36 q^{45} - 10 q^{46} - 284 q^{47} - 62 q^{49} - 68 q^{50} - 42 q^{51} + 64 q^{52} - 230 q^{53} + 72 q^{54} + 12 q^{55} - 4 q^{56} - 48 q^{57} + 114 q^{58} + 252 q^{59} - 274 q^{61} - 6 q^{62} - 12 q^{63} + 126 q^{65} + 12 q^{66} + 660 q^{67} + 4 q^{68} + 96 q^{69} - 222 q^{70} - 38 q^{71} + 48 q^{72} - 18 q^{74} + 168 q^{75} + 132 q^{76} - 486 q^{77} - 84 q^{78} - 58 q^{79} - 48 q^{80} - 72 q^{81} - 244 q^{82} + 44 q^{83} - 56 q^{86} - 114 q^{87} + 32 q^{88} + 272 q^{89} - 36 q^{90} - 72 q^{91} + 172 q^{92} + 126 q^{93} - 172 q^{94} + 552 q^{95} + 96 q^{96} + 170 q^{97} - 82 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 174 x^{14} + 12083 x^{12} + 425514 x^{10} + 7984353 x^{8} + 77125728 x^{6} + 345835640 x^{4} + \cdots + 179238544 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6341229 \nu^{15} + 1018360046 \nu^{13} + 62460488691 \nu^{11} + 1787482849362 \nu^{9} + \cdots - 893964106822656 ) / 17\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 33\!\cdots\!21 \nu^{15} + \cdots + 27\!\cdots\!48 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27870815259471 \nu^{15} + 49638015950816 \nu^{14} + \cdots + 56\!\cdots\!20 ) / 37\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27870815259471 \nu^{15} - 49638015950816 \nu^{14} + \cdots - 56\!\cdots\!20 ) / 37\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 85\!\cdots\!85 \nu^{15} + \cdots + 96\!\cdots\!00 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 85\!\cdots\!85 \nu^{15} + \cdots + 96\!\cdots\!00 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20\!\cdots\!29 \nu^{15} + \cdots + 55\!\cdots\!76 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!29 \nu^{15} + \cdots + 55\!\cdots\!76 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21\!\cdots\!01 \nu^{15} + \cdots - 90\!\cdots\!60 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!01 \nu^{15} + \cdots - 90\!\cdots\!60 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!59 \nu^{15} + \cdots - 50\!\cdots\!68 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24\!\cdots\!59 \nu^{15} + \cdots + 50\!\cdots\!68 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 31\!\cdots\!38 \nu^{15} + \cdots - 81\!\cdots\!16 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 34\!\cdots\!96 \nu^{15} + \cdots - 59\!\cdots\!12 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 36\!\cdots\!65 \nu^{15} + \cdots - 55\!\cdots\!92 ) / 92\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \cdots - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + 5 \beta_{14} - 5 \beta_{13} + 5 \beta_{12} + \beta_{11} + 6 \beta_{10} + \beta_{9} + \cdots - 63 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 39 \beta_{15} - 34 \beta_{14} + 44 \beta_{13} - 50 \beta_{12} - 55 \beta_{11} - 17 \beta_{10} + \cdots + 22 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 13 \beta_{15} - 185 \beta_{14} + 185 \beta_{13} - 197 \beta_{12} - \beta_{11} - 264 \beta_{10} + \cdots + 2184 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1571 \beta_{15} + 1314 \beta_{14} - 1828 \beta_{13} + 2203 \beta_{12} + 2460 \beta_{11} + 812 \beta_{10} + \cdots - 780 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 60 \beta_{15} + 2074 \beta_{14} - 2074 \beta_{13} + 2598 \beta_{12} - 584 \beta_{11} + 3420 \beta_{10} + \cdots - 28021 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 64382 \beta_{15} - 54890 \beta_{14} + 73874 \beta_{13} - 96497 \beta_{12} - 105989 \beta_{11} + \cdots + 25370 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 26149 \beta_{15} - 207745 \beta_{14} + 207745 \beta_{13} - 312751 \beta_{12} + 131155 \beta_{11} + \cdots + 3362235 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2663373 \beta_{15} + 2369174 \beta_{14} - 2957572 \beta_{13} + 4253548 \beta_{12} + 4547747 \beta_{11} + \cdots - 641636 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1608175 \beta_{15} + 6939475 \beta_{14} - 6939475 \beta_{13} + 12613045 \beta_{12} - 7281745 \beta_{11} + \cdots - 137201040 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 110942341 \beta_{15} - 103759794 \beta_{14} + 118124888 \beta_{13} - 188474105 \beta_{12} + \cdots + 4987110 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 27420120 \beta_{15} - 76784214 \beta_{14} + 76784214 \beta_{13} - 170032152 \beta_{12} + \cdots + 1890991807 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 4647754564 \beta_{15} + 4578032056 \beta_{14} - 4717477072 \beta_{13} + 8380294561 \beta_{12} + \cdots + 846693716 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 3904300529 \beta_{15} + 7505700143 \beta_{14} - 7505700143 \beta_{13} + 20690765357 \beta_{12} + \cdots - 236887827381 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 195685611405 \beta_{15} - 202856855908 \beta_{14} + 188514366902 \beta_{13} - 373475618042 \beta_{12} + \cdots - 84107165156 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/222\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(187\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
6.70611i
− 3.45790i
− 6.32750i
− 0.652766i
− 6.70611i
3.45790i
6.32750i
0.652766i
1.53333i
− 5.25672i
− 2.78080i
6.23624i
− 1.53333i
5.25672i
2.78080i
− 6.23624i
−0.366025 − 1.36603i −1.50000 + 0.866025i −1.73205 + 1.00000i −1.88911 + 7.05025i 1.73205 + 1.73205i −3.56475 − 6.17433i 2.00000 + 2.00000i 1.50000 − 2.59808i 10.3223
97.2 −0.366025 − 1.36603i −1.50000 + 0.866025i −1.73205 + 1.00000i 0.221548 − 0.826827i 1.73205 + 1.73205i −3.07794 − 5.33114i 2.00000 + 2.00000i 1.50000 − 2.59808i −1.21056
97.3 −0.366025 − 1.36603i −1.50000 + 0.866025i −1.73205 + 1.00000i 0.541156 − 2.01962i 1.73205 + 1.73205i 1.44189 + 2.49743i 2.00000 + 2.00000i 1.50000 − 2.59808i −2.95693
97.4 −0.366025 − 1.36603i −1.50000 + 0.866025i −1.73205 + 1.00000i 2.22448 − 8.30188i 1.73205 + 1.73205i 4.70080 + 8.14202i 2.00000 + 2.00000i 1.50000 − 2.59808i −12.1548
103.1 −0.366025 + 1.36603i −1.50000 − 0.866025i −1.73205 − 1.00000i −1.88911 − 7.05025i 1.73205 − 1.73205i −3.56475 + 6.17433i 2.00000 − 2.00000i 1.50000 + 2.59808i 10.3223
103.2 −0.366025 + 1.36603i −1.50000 − 0.866025i −1.73205 − 1.00000i 0.221548 + 0.826827i 1.73205 − 1.73205i −3.07794 + 5.33114i 2.00000 − 2.00000i 1.50000 + 2.59808i −1.21056
103.3 −0.366025 + 1.36603i −1.50000 − 0.866025i −1.73205 − 1.00000i 0.541156 + 2.01962i 1.73205 − 1.73205i 1.44189 − 2.49743i 2.00000 − 2.00000i 1.50000 + 2.59808i −2.95693
103.4 −0.366025 + 1.36603i −1.50000 − 0.866025i −1.73205 − 1.00000i 2.22448 + 8.30188i 1.73205 − 1.73205i 4.70080 − 8.14202i 2.00000 − 2.00000i 1.50000 + 2.59808i −12.1548
193.1 1.36603 + 0.366025i −1.50000 − 0.866025i 1.73205 + 1.00000i −5.47176 + 1.46615i −1.73205 − 1.73205i −1.43309 + 2.48218i 2.00000 + 2.00000i 1.50000 + 2.59808i −8.01121
193.2 1.36603 + 0.366025i −1.50000 − 0.866025i 1.73205 + 1.00000i −4.86572 + 1.30377i −1.73205 − 1.73205i 4.50599 − 7.80460i 2.00000 + 2.00000i 1.50000 + 2.59808i −7.12391
193.3 1.36603 + 0.366025i −1.50000 − 0.866025i 1.73205 + 1.00000i 1.52902 − 0.409700i −1.73205 − 1.73205i −6.16834 + 10.6839i 2.00000 + 2.00000i 1.50000 + 2.59808i 2.23864
193.4 1.36603 + 0.366025i −1.50000 − 0.866025i 1.73205 + 1.00000i 4.71038 − 1.26214i −1.73205 − 1.73205i 2.59543 − 4.49542i 2.00000 + 2.00000i 1.50000 + 2.59808i 6.89648
199.1 1.36603 − 0.366025i −1.50000 + 0.866025i 1.73205 − 1.00000i −5.47176 − 1.46615i −1.73205 + 1.73205i −1.43309 − 2.48218i 2.00000 − 2.00000i 1.50000 − 2.59808i −8.01121
199.2 1.36603 − 0.366025i −1.50000 + 0.866025i 1.73205 − 1.00000i −4.86572 − 1.30377i −1.73205 + 1.73205i 4.50599 + 7.80460i 2.00000 − 2.00000i 1.50000 − 2.59808i −7.12391
199.3 1.36603 − 0.366025i −1.50000 + 0.866025i 1.73205 − 1.00000i 1.52902 + 0.409700i −1.73205 + 1.73205i −6.16834 − 10.6839i 2.00000 − 2.00000i 1.50000 − 2.59808i 2.23864
199.4 1.36603 − 0.366025i −1.50000 + 0.866025i 1.73205 − 1.00000i 4.71038 + 1.26214i −1.73205 + 1.73205i 2.59543 + 4.49542i 2.00000 − 2.00000i 1.50000 − 2.59808i 6.89648
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.g odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 222.3.l.c ✓ 16
37.g odd 12 1 inner 222.3.l.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.3.l.c ✓ 16 1.a even 1 1 trivial
222.3.l.c ✓ 16 37.g odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 6 T_{5}^{15} + 60 T_{5}^{14} + 498 T_{5}^{13} + 312 T_{5}^{12} - 5850 T_{5}^{11} + \cdots + 611671824 \) acting on \(S_{3}^{\mathrm{new}}(222, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 611671824 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 3873874222656 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 2840654059776 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 48\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 71\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 75\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 24\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 41\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 50\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 3944670586128)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 46\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 63\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 33\!\cdots\!49 \) Copy content Toggle raw display
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