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Properties

Label 143.2.b.a
Level $143$
Weight $2$
Character orbit 143.b
Analytic conductor $1.142$
Analytic rank $0$
Dimension $12$
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] = [143,2,Mod(12,143)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(143, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("143.12");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 143 = 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 143.b (of order \(2\), degree \(1\), 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: \(1.14186074890\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 19x^{10} + 133x^{8} + 423x^{6} + 601x^{4} + 312x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + ( - \beta_{5} + \beta_{4} - 1) q^{4} + ( - \beta_{8} - \beta_{6}) q^{5} + (\beta_{11} - \beta_{9} - \beta_{8} + \cdots - \beta_1) q^{6}+ \cdots + ( - \beta_{10} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + ( - \beta_{5} + \beta_{4} - 1) q^{4} + ( - \beta_{8} - \beta_{6}) q^{5} + (\beta_{11} - \beta_{9} - \beta_{8} + \cdots - \beta_1) q^{6}+ \cdots + ( - \beta_{8} - \beta_{6} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} - 14 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} - 14 q^{4} + 12 q^{9} + 8 q^{10} - 8 q^{13} + 10 q^{16} - 12 q^{17} - 2 q^{22} - 4 q^{23} - 16 q^{25} + 10 q^{26} - 28 q^{27} + 8 q^{29} + 4 q^{30} + 16 q^{35} - 16 q^{36} + 18 q^{38} + 4 q^{39} + 16 q^{40} - 2 q^{42} + 12 q^{43} - 58 q^{48} + 32 q^{49} + 36 q^{52} - 20 q^{53} - 8 q^{55} + 22 q^{56} - 12 q^{61} - 72 q^{62} - 10 q^{64} - 20 q^{65} + 2 q^{66} + 68 q^{68} + 52 q^{69} + 20 q^{74} - 8 q^{75} + 8 q^{77} + 6 q^{78} - 48 q^{79} - 36 q^{81} - 44 q^{82} + 12 q^{87} + 30 q^{88} + 68 q^{90} - 6 q^{92} - 64 q^{94} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 19x^{10} + 133x^{8} + 423x^{6} + 601x^{4} + 312x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 205\nu^{9} + 2767\nu^{7} + 11877\nu^{5} + 15235\nu^{3} - 1482\nu ) / 2088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 31\nu^{8} + 331\nu^{6} + 1437\nu^{4} + 2185\nu^{2} + 432 ) / 174 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{10} + 167\nu^{8} + 857\nu^{6} + 1887\nu^{4} + 2285\nu^{2} + 1446 ) / 348 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{10} + 167\nu^{8} + 857\nu^{6} + 1887\nu^{4} + 1937\nu^{2} + 402 ) / 348 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -67\nu^{11} - 1207\nu^{9} - 7909\nu^{7} - 23199\nu^{5} - 28945\nu^{3} - 9282\nu ) / 2088 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -15\nu^{11} - 291\nu^{9} - 2065\nu^{7} - 6475\nu^{5} - 8357\nu^{3} - 3058\nu ) / 232 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -55\nu^{11} - 835\nu^{9} - 3937\nu^{7} - 4911\nu^{5} + 5627\nu^{3} + 8430\nu ) / 1044 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 50 \nu^{11} - 81 \nu^{10} + 854 \nu^{9} - 1293 \nu^{8} + 5066 \nu^{7} - 6975 \nu^{6} + 12342 \nu^{5} + \cdots - 1062 ) / 696 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25\nu^{10} + 427\nu^{8} + 2533\nu^{6} + 6171\nu^{4} + 5557\nu^{2} + 1230 ) / 174 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 50 \nu^{11} - 81 \nu^{10} - 854 \nu^{9} - 1293 \nu^{8} - 5066 \nu^{7} - 6975 \nu^{6} - 12342 \nu^{5} + \cdots - 1062 ) / 696 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + 2\beta_{6} - \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} + 9\beta_{5} - 6\beta_{4} - \beta_{3} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} + 2\beta_{9} + 3\beta_{8} + 10\beta_{7} - 16\beta_{6} + 8\beta_{2} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{11} - 12\beta_{10} - 11\beta_{9} - 67\beta_{5} + 38\beta_{4} + 14\beta_{3} - 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 26\beta_{11} - 26\beta_{9} - 36\beta_{8} - 81\beta_{7} + 105\beta_{6} - 60\beta_{2} - 195\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96\beta_{11} + 112\beta_{10} + 96\beta_{9} + 477\beta_{5} - 255\beta_{4} - 133\beta_{3} + 340 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -245\beta_{11} + 245\beta_{9} + 325\beta_{8} + 610\beta_{7} - 659\beta_{6} + 447\beta_{2} + 1317\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -772\beta_{11} - 937\beta_{10} - 772\beta_{9} - 3358\beta_{5} + 1764\beta_{4} + 1100\beta_{3} - 1914 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2037\beta_{11} - 2037\beta_{9} - 2644\beta_{8} - 4458\beta_{7} + 4122\beta_{6} - 3308\beta_{2} - 9073\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/143\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1
− 2.66546i
− 2.23753i
− 1.92437i
− 1.50594i
− 0.871160i
− 0.398488i
0.398488i
0.871160i
1.50594i
1.92437i
2.23753i
2.66546i
− 2.66546i −2.17072 −5.10468 0.458686i 5.78598i − 1.17072i 8.27540i 1.71204 1.22261
12.2 − 2.23753i 1.75439 −3.00653 − 1.83227i − 3.92549i 2.75439i 2.25214i 0.0778811 −4.09976
12.3 − 1.92437i 2.13208 −1.70321 3.67785i − 4.10292i − 3.13208i − 0.571131i 1.54577 7.07756
12.4 − 1.50594i −1.34063 −0.267864 − 2.54333i 2.01892i 0.340634i − 2.60850i −1.20270 −3.83012
12.5 − 0.871160i −3.06458 1.24108 3.32707i 2.66974i 2.06458i − 2.82350i 6.39165 2.89841
12.6 − 0.398488i 0.689466 1.84121 1.83517i − 0.274744i 1.68947i − 1.53067i −2.52464 0.731293
12.7 0.398488i 0.689466 1.84121 − 1.83517i 0.274744i − 1.68947i 1.53067i −2.52464 0.731293
12.8 0.871160i −3.06458 1.24108 − 3.32707i − 2.66974i − 2.06458i 2.82350i 6.39165 2.89841
12.9 1.50594i −1.34063 −0.267864 2.54333i − 2.01892i − 0.340634i 2.60850i −1.20270 −3.83012
12.10 1.92437i 2.13208 −1.70321 − 3.67785i 4.10292i 3.13208i 0.571131i 1.54577 7.07756
12.11 2.23753i 1.75439 −3.00653 1.83227i 3.92549i − 2.75439i − 2.25214i 0.0778811 −4.09976
12.12 2.66546i −2.17072 −5.10468 − 0.458686i − 5.78598i 1.17072i − 8.27540i 1.71204 1.22261
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.2.b.a ✓ 12
3.b odd 2 1 1287.2.b.b 12
4.b odd 2 1 2288.2.j.k 12
13.b even 2 1 inner 143.2.b.a ✓ 12
13.d odd 4 1 1859.2.a.j 6
13.d odd 4 1 1859.2.a.n 6
39.d odd 2 1 1287.2.b.b 12
52.b odd 2 1 2288.2.j.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.b.a ✓ 12 1.a even 1 1 trivial
143.2.b.a ✓ 12 13.b even 2 1 inner
1287.2.b.b 12 3.b odd 2 1
1287.2.b.b 12 39.d odd 2 1
1859.2.a.j 6 13.d odd 4 1
1859.2.a.n 6 13.d odd 4 1
2288.2.j.k 12 4.b odd 2 1
2288.2.j.k 12 52.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(143, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 19 T^{10} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( (T^{6} + 2 T^{5} - 10 T^{4} + \cdots - 23)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 38 T^{10} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{12} + 26 T^{10} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} + 8 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} + 6 T^{5} - 34 T^{4} + \cdots - 96)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 118 T^{10} + \cdots + 16112196 \) Copy content Toggle raw display
$23$ \( (T^{6} + 2 T^{5} + \cdots + 17625)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 4 T^{5} - 42 T^{4} + \cdots - 96)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 162 T^{10} + \cdots + 1937664 \) Copy content Toggle raw display
$37$ \( T^{12} + 226 T^{10} + \cdots + 53934336 \) Copy content Toggle raw display
$41$ \( T^{12} + 142 T^{10} + \cdots + 5062500 \) Copy content Toggle raw display
$43$ \( (T^{6} - 6 T^{5} - 86 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 220 T^{10} + \cdots + 36864 \) Copy content Toggle raw display
$53$ \( (T^{6} + 10 T^{5} + \cdots + 6024)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 2477650176 \) Copy content Toggle raw display
$61$ \( (T^{6} + 6 T^{5} + \cdots + 27296)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 296941824 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 93401139456 \) Copy content Toggle raw display
$73$ \( T^{12} + 234 T^{10} + \cdots + 9216 \) Copy content Toggle raw display
$79$ \( (T^{6} + 24 T^{5} + \cdots - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 478 T^{10} + \cdots + 2762244 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 38598103296 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 152176896 \) Copy content Toggle raw display
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