Properties

Label 143.2.h.a
Level $143$
Weight $2$
Character orbit 143.h
Analytic conductor $1.142$
Analytic rank $1$
Dimension $4$
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(14,143)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(143, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("143.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 143 = 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 143.h (of order \(5\), 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: \(1.14186074890\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{2} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{3} - 3 \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{5} + (2 \zeta_{10}^{2} + 4 \zeta_{10} + 2) q^{6} + (\zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{7} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{8} + (5 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 5) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{2} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{3} - 3 \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{5} + (2 \zeta_{10}^{2} + 4 \zeta_{10} + 2) q^{6} + (\zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{7} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{8} + (5 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 5) q^{9} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 4) q^{10} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2) q^{11} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 6) q^{12} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{13} + (\zeta_{10}^{3} - 8 \zeta_{10}^{2} + \zeta_{10}) q^{14} + 4 \zeta_{10}^{3} q^{15} + \zeta_{10} q^{16} + (\zeta_{10}^{2} - 4 \zeta_{10} + 1) q^{17} + ( - 13 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{18} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{19} + (6 \zeta_{10}^{2} - 6 \zeta_{10}) q^{20} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 8) q^{21} + ( - 4 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{22} - 8 q^{23} + ( - 6 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 6) q^{24} + ( - 4 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 4 \zeta_{10}) q^{25} + (\zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{26} + (4 \zeta_{10}^{2} + 8 \zeta_{10} + 4) q^{27} + (9 \zeta_{10}^{2} - 6 \zeta_{10} + 9) q^{28} + (4 \zeta_{10}^{3} - \zeta_{10} + 1) q^{29} + ( - 8 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 8 \zeta_{10}) q^{30} + (4 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 4) q^{31} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 3) q^{32} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 4 \zeta_{10} + 6) q^{33} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} + 1) q^{34} + ( - 6 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 10 \zeta_{10} + 6) q^{35} + (12 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 12 \zeta_{10}) q^{36} - 4 \zeta_{10}^{3} q^{37} + (\zeta_{10}^{2} - 3 \zeta_{10} + 1) q^{38} + ( - 2 \zeta_{10}^{2} - 2) q^{39} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{40} + (12 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 12) q^{42} - 4 q^{43} + (6 \zeta_{10}^{3} + 3 \zeta_{10} + 6) q^{44} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 8) q^{45} + ( - 8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} + 8) q^{46} + ( - 7 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 7 \zeta_{10}) q^{47} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{48} + (3 \zeta_{10}^{2} - 6 \zeta_{10} + 3) q^{49} + ( - 2 \zeta_{10}^{2} + 11 \zeta_{10} - 2) q^{50} + (4 \zeta_{10}^{3} + 6 \zeta_{10} - 6) q^{51} - 3 \zeta_{10}^{2} q^{52} + ( - 8 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 8) q^{53} + (20 \zeta_{10}^{3} - 20 \zeta_{10}^{2} - 20) q^{54} + (2 \zeta_{10}^{2} - 6 \zeta_{10} + 8) q^{55} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 7) q^{56} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{57} + ( - 9 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 9 \zeta_{10}) q^{58} + (8 \zeta_{10}^{3} - \zeta_{10} + 1) q^{59} + 12 \zeta_{10} q^{60} + ( - 7 \zeta_{10}^{2} + 3 \zeta_{10} - 7) q^{61} + ( - 2 \zeta_{10}^{3} - 9 \zeta_{10} + 9) q^{62} + ( - 7 \zeta_{10}^{3} - 10 \zeta_{10}^{2} - 7 \zeta_{10}) q^{63} + ( - 13 \zeta_{10}^{3} + 13 \zeta_{10}^{2} - 13 \zeta_{10} + 13) q^{64} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2}) q^{65} + (18 \zeta_{10}^{3} - 8 \zeta_{10}^{2} - 14) q^{66} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 8) q^{67} + (9 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 12 \zeta_{10} - 9) q^{68} + (16 \zeta_{10}^{3} + 16 \zeta_{10}) q^{69} + ( - 2 \zeta_{10}^{3} + 16 \zeta_{10} - 16) q^{70} + ( - 9 \zeta_{10}^{2} + 6 \zeta_{10} - 9) q^{71} + ( - 6 \zeta_{10}^{2} - 7 \zeta_{10} - 6) q^{72} + (4 \zeta_{10}^{3} + 6 \zeta_{10} - 6) q^{73} + (8 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 8 \zeta_{10}) q^{74} + (10 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 10) q^{75} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2}) q^{76} + (4 \zeta_{10}^{3} - 9 \zeta_{10}^{2} - 4 \zeta_{10} + 1) q^{77} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 6) q^{78} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{79} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{80} + ( - 17 \zeta_{10}^{3} - 12 \zeta_{10} + 12) q^{81} + (5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{83} + ( - 24 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{84} + (8 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 8 \zeta_{10}) q^{85} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 4) q^{86} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 6) q^{87} + (2 \zeta_{10}^{2} - 6 \zeta_{10} - 3) q^{88} + ( - 12 \zeta_{10}^{3} + 12 \zeta_{10}^{2} + 4) q^{89} + (12 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 12) q^{90} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 3 \zeta_{10}) q^{91} + 24 \zeta_{10}^{3} q^{92} + (8 \zeta_{10}^{2} - 2 \zeta_{10} + 8) q^{93} + (3 \zeta_{10}^{2} + 16 \zeta_{10} + 3) q^{94} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{95} + (6 \zeta_{10}^{3} + 12 \zeta_{10}^{2} + 6 \zeta_{10}) q^{96} + (6 \zeta_{10}^{2} - 6 \zeta_{10}) q^{97} + ( - 9 \zeta_{10}^{3} + 9 \zeta_{10}^{2} - 3) q^{98} + ( - 14 \zeta_{10}^{3} - 7 \zeta_{10}^{2} - 8 \zeta_{10} + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{2} - 4 q^{3} - 3 q^{4} - 4 q^{5} + 10 q^{6} - 8 q^{7} + 5 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 5 q^{2} - 4 q^{3} - 3 q^{4} - 4 q^{5} + 10 q^{6} - 8 q^{7} + 5 q^{8} - 13 q^{9} + 20 q^{10} - 9 q^{11} - 12 q^{12} + q^{13} + 10 q^{14} + 4 q^{15} + q^{16} - q^{17} + 5 q^{18} + 3 q^{19} - 12 q^{20} + 28 q^{21} - 5 q^{22} - 32 q^{23} + 10 q^{24} - 11 q^{25} - 5 q^{26} + 20 q^{27} + 21 q^{28} + 7 q^{29} - 20 q^{30} - 2 q^{31} + 24 q^{33} - 10 q^{34} - 2 q^{35} + 21 q^{36} - 4 q^{37} - 6 q^{39} - 10 q^{40} - 40 q^{42} - 16 q^{43} + 33 q^{44} + 28 q^{45} + 40 q^{46} - 16 q^{47} + 4 q^{48} + 3 q^{49} + 5 q^{50} - 14 q^{51} + 3 q^{52} + 18 q^{53} - 40 q^{54} + 24 q^{55} - 30 q^{56} + 2 q^{57} - 25 q^{58} + 11 q^{59} + 12 q^{60} - 18 q^{61} + 25 q^{62} - 4 q^{63} + 13 q^{64} + 4 q^{65} - 30 q^{66} - 24 q^{67} - 3 q^{68} + 32 q^{69} - 50 q^{70} - 21 q^{71} - 25 q^{72} - 14 q^{73} + 20 q^{74} - 14 q^{75} - 6 q^{76} + 13 q^{77} + 20 q^{78} + 2 q^{79} - 6 q^{80} + 19 q^{81} + 10 q^{83} - 6 q^{84} + 26 q^{85} + 20 q^{86} + 8 q^{87} - 20 q^{88} - 8 q^{89} - 40 q^{90} + 8 q^{91} + 24 q^{92} + 22 q^{93} + 25 q^{94} - 8 q^{95} - 12 q^{97} - 30 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(67\) \(79\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
0.809017 + 0.587785i
−0.309017 + 0.951057i
−0.309017 0.951057i
0.809017 0.587785i
−1.80902 + 1.31433i −1.00000 3.07768i 0.927051 2.85317i −1.00000 0.726543i 5.85410 + 4.25325i −0.881966 + 2.71441i 0.690983 + 2.12663i −6.04508 + 4.39201i 2.76393
27.1 −0.690983 2.12663i −1.00000 0.726543i −2.42705 + 1.76336i −1.00000 + 3.07768i −0.854102 + 2.62866i −3.11803 + 2.26538i 1.80902 + 1.31433i −0.454915 1.40008i 7.23607
53.1 −0.690983 + 2.12663i −1.00000 + 0.726543i −2.42705 1.76336i −1.00000 3.07768i −0.854102 2.62866i −3.11803 2.26538i 1.80902 1.31433i −0.454915 + 1.40008i 7.23607
92.1 −1.80902 1.31433i −1.00000 + 3.07768i 0.927051 + 2.85317i −1.00000 + 0.726543i 5.85410 4.25325i −0.881966 2.71441i 0.690983 2.12663i −6.04508 4.39201i 2.76393
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.2.h.a 4
11.c even 5 1 inner 143.2.h.a 4
11.c even 5 1 1573.2.a.d 2
11.d odd 10 1 1573.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.h.a 4 1.a even 1 1 trivial
143.2.h.a 4 11.c even 5 1 inner
1573.2.a.d 2 11.c even 5 1
1573.2.a.e 2 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 5T_{2}^{3} + 15T_{2}^{2} + 25T_{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(143, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$7$ \( T^{4} + 8 T^{3} + 34 T^{2} + 77 T + 121 \) Copy content Toggle raw display
$11$ \( T^{4} + 9 T^{3} + 41 T^{2} + 99 T + 121 \) Copy content Toggle raw display
$13$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} + 16 T^{2} + 66 T + 121 \) Copy content Toggle raw display
$19$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$23$ \( (T + 8)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 7 T^{3} + 24 T^{2} - 38 T + 361 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + 24 T^{2} + 133 T + 361 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T + 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 16 T^{3} + 186 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$53$ \( T^{4} - 18 T^{3} + 124 T^{2} - 7 T + 1 \) Copy content Toggle raw display
$59$ \( T^{4} - 11 T^{3} + 76 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$61$ \( T^{4} + 18 T^{3} + 184 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$67$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 21 T^{3} + 306 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + 136 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$79$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$83$ \( T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$89$ \( (T^{2} + 4 T - 176)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} + 144 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
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