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Properties

Label 143.2.j.b
Level $143$
Weight $2$
Character orbit 143.j
Analytic conductor $1.142$
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] = [143,2,Mod(23,143)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(143, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("143.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 143 = 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 143.j (of order \(6\), 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: \(1.14186074890\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 20x^{14} + 150x^{12} + 530x^{10} + 915x^{8} + 758x^{6} + 287x^{4} + 42x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{13} - \beta_{11} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{15} + \beta_{14} - \beta_{13} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{13} - \beta_{11} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{15} + \beta_{13} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{2} + 4 q^{4} - 24 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{2} + 4 q^{4} - 24 q^{6} - 2 q^{9} + 10 q^{10} + 16 q^{12} + 4 q^{13} - 12 q^{14} - 6 q^{15} - 12 q^{16} + 6 q^{17} - 12 q^{19} + 6 q^{20} + 2 q^{22} - 14 q^{23} - 30 q^{24} - 24 q^{25} + 32 q^{26} + 48 q^{27} + 24 q^{28} + 10 q^{29} + 14 q^{30} + 18 q^{32} - 6 q^{33} - 10 q^{35} - 30 q^{37} - 24 q^{38} - 30 q^{39} - 28 q^{40} + 30 q^{41} + 2 q^{42} + 14 q^{43} - 6 q^{45} + 6 q^{46} + 6 q^{48} + 2 q^{49} + 42 q^{50} - 24 q^{51} + 10 q^{52} - 16 q^{53} - 36 q^{54} + 8 q^{55} - 28 q^{56} + 6 q^{58} - 42 q^{59} + 4 q^{61} + 48 q^{63} - 12 q^{64} + 14 q^{65} - 8 q^{66} + 34 q^{68} - 10 q^{69} + 18 q^{71} + 24 q^{72} - 2 q^{74} - 18 q^{75} - 30 q^{76} + 28 q^{77} + 12 q^{78} + 52 q^{79} + 36 q^{80} + 16 q^{81} - 64 q^{82} + 12 q^{84} - 30 q^{85} + 18 q^{87} + 6 q^{88} + 42 q^{89} + 40 q^{90} - 6 q^{91} - 60 q^{92} - 6 q^{93} + 4 q^{94} + 22 q^{95} + 6 q^{97} - 96 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 20x^{14} + 150x^{12} + 530x^{10} + 915x^{8} + 758x^{6} + 287x^{4} + 42x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8\nu^{14} + 153\nu^{12} + 1069\nu^{10} + 3342\nu^{8} + 4528\nu^{6} + 2171\nu^{4} + 244\nu^{2} + 46\nu + 19 ) / 92 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{14} - 153\nu^{12} - 1069\nu^{10} - 3342\nu^{8} - 4528\nu^{6} - 2171\nu^{4} - 244\nu^{2} + 46\nu - 19 ) / 92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31 \nu^{15} - 155 \nu^{14} + 567 \nu^{13} - 3019 \nu^{12} + 3645 \nu^{11} - 21721 \nu^{10} + \cdots - 1153 ) / 736 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\nu^{15} + 372\nu^{13} + 2697\nu^{11} + 9001\nu^{9} + 14043\nu^{7} + 9874\nu^{5} + 3282\nu^{3} + 554\nu - 46 ) / 92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -119\nu^{14} - 2319\nu^{12} - 16669\nu^{10} - 54663\nu^{8} - 81752\nu^{6} - 50754\nu^{4} - 10587\nu^{2} - 205 ) / 368 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 205 \nu^{15} - 207 \nu^{14} + 3981 \nu^{13} - 4071 \nu^{12} + 28431 \nu^{11} - 29693 \nu^{10} + \cdots + 115 ) / 736 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 207 \nu^{15} - 119 \nu^{14} - 4071 \nu^{13} - 2319 \nu^{12} - 29693 \nu^{11} - 16669 \nu^{10} + \cdots - 205 ) / 736 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 205 \nu^{15} - 207 \nu^{14} - 3981 \nu^{13} - 4071 \nu^{12} - 28431 \nu^{11} - 29693 \nu^{10} + \cdots + 115 ) / 736 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 383 \nu^{15} - 143 \nu^{14} + 7575 \nu^{13} - 2847 \nu^{12} + 55741 \nu^{11} - 21141 \nu^{10} + \cdots - 469 ) / 736 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 118 \nu^{15} - 149 \nu^{14} - 2274 \nu^{13} - 2933 \nu^{12} - 16038 \nu^{11} - 21431 \nu^{10} + \cdots - 443 ) / 368 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 108 \nu^{15} - 249 \nu^{14} + 2192 \nu^{13} - 4857 \nu^{12} + 16812 \nu^{11} - 34943 \nu^{10} + \cdots + 21 ) / 368 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 383 \nu^{15} - 143 \nu^{14} - 7575 \nu^{13} - 2847 \nu^{12} - 55741 \nu^{11} - 21141 \nu^{10} + \cdots - 469 ) / 736 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 7 \nu^{15} - 619 \nu^{14} - 223 \nu^{13} - 12123 \nu^{12} - 2669 \nu^{11} - 87817 \nu^{10} + \cdots - 2025 ) / 736 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 52 \nu^{15} - 22 \nu^{14} - 1029 \nu^{13} - 438 \nu^{12} - 7581 \nu^{11} - 3256 \nu^{10} + \cdots - 173 ) / 92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 413 \nu^{15} + 88 \nu^{14} - 8097 \nu^{13} + 1752 \nu^{12} - 58755 \nu^{11} + 13024 \nu^{10} + \cdots + 692 ) / 368 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - \beta_{14} + \beta_{12} + \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + \cdots - 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} - \beta_{13} + 7 \beta_{12} - \beta_{11} + 7 \beta_{9} - 6 \beta_{8} + \beta_{7} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7 \beta_{15} + 7 \beta_{14} - 7 \beta_{12} - 10 \beta_{10} + 7 \beta_{9} + 12 \beta_{8} - 20 \beta_{7} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{15} + 7 \beta_{14} + 10 \beta_{13} - 47 \beta_{12} + 10 \beta_{11} + 3 \beta_{10} - 47 \beta_{9} + \cdots - 70 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 44 \beta_{15} - 47 \beta_{14} + 3 \beta_{13} + 43 \beta_{12} - 3 \beta_{11} + 83 \beta_{10} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 40 \beta_{15} - 43 \beta_{14} - 83 \beta_{13} + 322 \beta_{12} - 83 \beta_{11} - 42 \beta_{10} + \cdots + 468 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 280 \beta_{15} + 322 \beta_{14} - 42 \beta_{13} - 268 \beta_{12} + 42 \beta_{11} - 651 \beta_{10} + \cdots - 330 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 383 \beta_{15} + 268 \beta_{14} + 651 \beta_{13} - 2245 \beta_{12} + 651 \beta_{11} + 405 \beta_{10} + \cdots - 3215 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1840 \beta_{15} - 2245 \beta_{14} + 405 \beta_{13} + 1743 \beta_{12} - 405 \beta_{11} + 4970 \beta_{10} + \cdots + 2884 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3227 \beta_{15} - 1743 \beta_{14} - 4970 \beta_{13} + 15852 \beta_{12} - 4970 \beta_{11} + \cdots + 22487 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12463 \beta_{15} + 15852 \beta_{14} - 3389 \beta_{13} - 11796 \beta_{12} + 3389 \beta_{11} + \cdots - 23302 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 25550 \beta_{15} + 11796 \beta_{14} + 37346 \beta_{13} - 112964 \beta_{12} + 37346 \beta_{11} + \cdots - 159273 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 86420 \beta_{15} - 112964 \beta_{14} + 26544 \beta_{13} + 82184 \beta_{12} - 26544 \beta_{11} + \cdots + 180576 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(67\) \(79\)
\(\chi(n)\) \(-\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
− 2.69931i
− 2.07742i
− 1.37460i
− 0.920076i
0.170890i
0.549080i
0.681121i
2.20621i
2.69931i
2.07742i
1.37460i
0.920076i
− 0.170890i
− 0.549080i
− 0.681121i
− 2.20621i
−2.33767 + 1.34965i 0.798396 + 1.38286i 2.64313 − 4.57804i 1.39555i −3.73277 − 2.15512i 3.53213 + 2.03927i 8.87064i 0.225128 − 0.389933i −1.88351 − 3.26233i
23.2 −1.79910 + 1.03871i 0.567997 + 0.983799i 1.15784 − 2.00544i − 4.26888i −2.04377 − 1.17997i −2.99836 − 1.73110i 0.655817i 0.854760 − 1.48049i 4.43414 + 7.68015i
23.3 −1.19044 + 0.687299i 1.04247 + 1.80561i −0.0552408 + 0.0956798i 2.34290i −2.48199 − 1.43298i −1.12231 − 0.647967i − 2.90106i −0.673485 + 1.16651i −1.61027 − 2.78907i
23.4 −0.796809 + 0.460038i −0.449421 − 0.778420i −0.576730 + 0.998926i − 2.76396i 0.716205 + 0.413501i 3.02456 + 1.74623i − 2.90142i 1.09604 − 1.89840i 1.27153 + 2.20235i
23.5 0.147995 − 0.0854449i −1.47439 − 2.55372i −0.985398 + 1.70676i − 1.29847i −0.436405 − 0.251959i −3.16341 − 1.82640i 0.678569i −2.84767 + 4.93230i −0.110947 − 0.192166i
23.6 0.475517 − 0.274540i 0.729952 + 1.26431i −0.849255 + 1.47095i − 0.775548i 0.694210 + 0.400802i 1.22191 + 0.705469i 2.03078i 0.434341 − 0.752300i −0.212919 − 0.368787i
23.7 0.589868 − 0.340560i 0.0274983 + 0.0476285i −0.768037 + 1.33028i 3.68496i 0.0324408 + 0.0187297i −0.203284 − 0.117366i 2.40849i 1.49849 − 2.59546i 1.25495 + 2.17364i
23.8 1.91064 − 1.10311i −1.24250 − 2.15207i 1.43369 − 2.48322i 1.68345i −4.74793 − 2.74122i −0.291222 − 0.168137i − 1.91361i −1.58761 + 2.74981i 1.85703 + 3.21646i
56.1 −2.33767 − 1.34965i 0.798396 − 1.38286i 2.64313 + 4.57804i − 1.39555i −3.73277 + 2.15512i 3.53213 − 2.03927i − 8.87064i 0.225128 + 0.389933i −1.88351 + 3.26233i
56.2 −1.79910 − 1.03871i 0.567997 − 0.983799i 1.15784 + 2.00544i 4.26888i −2.04377 + 1.17997i −2.99836 + 1.73110i − 0.655817i 0.854760 + 1.48049i 4.43414 − 7.68015i
56.3 −1.19044 − 0.687299i 1.04247 − 1.80561i −0.0552408 − 0.0956798i − 2.34290i −2.48199 + 1.43298i −1.12231 + 0.647967i 2.90106i −0.673485 − 1.16651i −1.61027 + 2.78907i
56.4 −0.796809 − 0.460038i −0.449421 + 0.778420i −0.576730 − 0.998926i 2.76396i 0.716205 − 0.413501i 3.02456 − 1.74623i 2.90142i 1.09604 + 1.89840i 1.27153 − 2.20235i
56.5 0.147995 + 0.0854449i −1.47439 + 2.55372i −0.985398 − 1.70676i 1.29847i −0.436405 + 0.251959i −3.16341 + 1.82640i − 0.678569i −2.84767 − 4.93230i −0.110947 + 0.192166i
56.6 0.475517 + 0.274540i 0.729952 − 1.26431i −0.849255 − 1.47095i 0.775548i 0.694210 − 0.400802i 1.22191 − 0.705469i − 2.03078i 0.434341 + 0.752300i −0.212919 + 0.368787i
56.7 0.589868 + 0.340560i 0.0274983 − 0.0476285i −0.768037 − 1.33028i − 3.68496i 0.0324408 − 0.0187297i −0.203284 + 0.117366i − 2.40849i 1.49849 + 2.59546i 1.25495 − 2.17364i
56.8 1.91064 + 1.10311i −1.24250 + 2.15207i 1.43369 + 2.48322i − 1.68345i −4.74793 + 2.74122i −0.291222 + 0.168137i 1.91361i −1.58761 − 2.74981i 1.85703 − 3.21646i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.2.j.b ✓ 16
13.e even 6 1 inner 143.2.j.b ✓ 16
13.f odd 12 1 1859.2.a.o 8
13.f odd 12 1 1859.2.a.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.j.b ✓ 16 1.a even 1 1 trivial
143.2.j.b ✓ 16 13.e even 6 1 inner
1859.2.a.o 8 13.f odd 12 1
1859.2.a.p 8 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 6 T_{2}^{15} + 8 T_{2}^{14} - 24 T_{2}^{13} - 51 T_{2}^{12} + 108 T_{2}^{11} + 374 T_{2}^{10} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(143, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 6 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} + 13 T^{14} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{16} + 52 T^{14} + \cdots + 58081 \) Copy content Toggle raw display
$7$ \( T^{16} - 29 T^{14} + \cdots + 676 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 6 T^{15} + \cdots + 9728161 \) Copy content Toggle raw display
$19$ \( T^{16} + 12 T^{15} + \cdots + 4072324 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1145416336 \) Copy content Toggle raw display
$29$ \( T^{16} - 10 T^{15} + \cdots + 4255969 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 2559145744 \) Copy content Toggle raw display
$37$ \( T^{16} + 30 T^{15} + \cdots + 3721 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 2930714596489 \) Copy content Toggle raw display
$43$ \( T^{16} - 14 T^{15} + \cdots + 13366336 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 2946969796 \) Copy content Toggle raw display
$53$ \( (T^{8} + 8 T^{7} + \cdots + 1610773)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 598262028676 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 264773022721 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 764141309327236 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 152655612630544 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 18869223952384 \) Copy content Toggle raw display
$79$ \( (T^{8} - 26 T^{7} + \cdots + 2764672)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 213821191135876 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 180410863504 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 9967280410000 \) Copy content Toggle raw display
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