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Properties

Label 171.2.u.d
Level $171$
Weight $2$
Character orbit 171.u
Analytic conductor $1.365$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [171,2,Mod(28,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.28");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.u (of order \(9\), degree \(6\), 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: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 9x^{10} + 57x^{8} - 182x^{6} + 423x^{4} - 408x^{2} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{9} - \beta_{7} + 2 \beta_{6} - 1) q^{4} + ( - \beta_{11} + \beta_{10}) q^{5} + (\beta_{9} + \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + (\beta_{8} - \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{9} - \beta_{7} + 2 \beta_{6} - 1) q^{4} + ( - \beta_{11} + \beta_{10}) q^{5} + (\beta_{9} + \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{10} + \cdots + 7 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{4} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{4} + 6 q^{7} + 12 q^{10} - 24 q^{13} + 18 q^{16} + 12 q^{19} + 12 q^{25} - 12 q^{28} + 24 q^{31} - 78 q^{34} - 48 q^{37} - 30 q^{40} - 24 q^{43} + 6 q^{46} + 54 q^{52} + 18 q^{55} - 48 q^{58} + 30 q^{61} + 30 q^{64} + 48 q^{67} + 60 q^{70} - 42 q^{73} + 12 q^{76} + 48 q^{79} + 66 q^{82} - 78 q^{85} - 12 q^{91} - 12 q^{94} + 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 9x^{10} + 57x^{8} - 182x^{6} + 423x^{4} - 408x^{2} + 289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -49\nu^{11} + 815\nu^{9} - 13741\nu^{7} + 69676\nu^{5} - 257078\nu^{3} + 315775\nu ) / 115821 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -193\nu^{10} + 3998\nu^{8} - 21031\nu^{6} + 90070\nu^{4} - 136430\nu^{2} + 215560 ) / 115821 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -193\nu^{11} + 3998\nu^{9} - 21031\nu^{7} + 90070\nu^{5} - 136430\nu^{3} + 331381\nu ) / 115821 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -245\nu^{10} + 4075\nu^{8} - 30098\nu^{6} + 116738\nu^{4} - 243001\nu^{2} + 150416 ) / 115821 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -214\nu^{11} - 1168\nu^{9} + 11687\nu^{7} - 95165\nu^{5} + 222193\nu^{3} - 366095\nu ) / 115821 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 407\nu^{10} - 2830\nu^{8} + 9344\nu^{6} + 5095\nu^{4} - 85763\nu^{2} + 150535 ) / 115821 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 152\nu^{10} - 1215\nu^{8} + 7695\nu^{6} - 21527\nu^{4} + 57105\nu^{2} - 55080 ) / 38607 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 652\nu^{11} - 6905\nu^{9} + 39442\nu^{7} - 111643\nu^{5} + 157238\nu^{3} + 115940\nu ) / 115821 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -1123\nu^{10} + 6860\nu^{8} - 39157\nu^{6} + 77005\nu^{4} - 155123\nu^{2} - 2159 ) / 115821 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -1561\nu^{11} + 14933\nu^{9} - 90286\nu^{7} + 283813\nu^{5} - 534554\nu^{3} + 363817\nu ) / 115821 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -1772\nu^{11} + 14503\nu^{9} - 83273\nu^{7} + 231656\nu^{5} - 462868\nu^{3} + 378641\nu ) / 115821 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} + \beta_{8} + 2\beta_{5} + 2\beta_{3} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 3\beta_{7} + \beta_{4} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{11} + 10\beta_{10} + 7\beta_{8} + 5\beta_{5} + 2\beta_{3} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{9} + 11\beta_{7} + 6\beta_{6} - \beta_{4} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{11} + 22\beta_{10} + 25\beta_{8} - 22\beta_{5} - 25\beta_{3} - 29\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{9} + 21\beta_{6} - 30\beta_{4} + 30\beta_{2} - 44 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 122\beta_{11} - 95\beta_{10} - 5\beta_{8} - 190\beta_{5} - 127\beta_{3} - 122\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -86\beta_{9} - 183\beta_{7} - 57\beta_{6} - 86\beta_{4} + 57\beta_{2} - 183 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 566\beta_{11} - 824\beta_{10} - 566\beta_{8} - 412\beta_{5} + 17\beta_{3} - 17\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -669\beta_{9} - 778\beta_{7} - 669\beta_{6} + 314\beta_{4} - 355\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -205\beta_{11} - 1802\beta_{10} - 2744\beta_{8} + 1802\beta_{5} + 2744\beta_{3} + 2539\beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
1.58445 − 0.914781i
−1.58445 + 0.914781i
1.58445 + 0.914781i
−1.58445 − 0.914781i
1.84366 + 1.06444i
−1.84366 − 1.06444i
1.84366 − 1.06444i
−1.84366 + 1.06444i
0.916767 + 0.529295i
−0.916767 − 0.529295i
0.916767 − 0.529295i
−0.916767 + 0.529295i
−0.408427 − 2.31631i 0 −3.31908 + 1.20805i −3.38621 − 1.23248i 0 −0.0923963 + 0.160035i 1.80177 + 3.12075i 0 −1.47178 + 8.34689i
28.2 0.408427 + 2.31631i 0 −3.31908 + 1.20805i 3.38621 + 1.23248i 0 −0.0923963 + 0.160035i −1.80177 − 3.12075i 0 −1.47178 + 8.34689i
55.1 −0.408427 + 2.31631i 0 −3.31908 − 1.20805i −3.38621 + 1.23248i 0 −0.0923963 − 0.160035i 1.80177 − 3.12075i 0 −1.47178 − 8.34689i
55.2 0.408427 − 2.31631i 0 −3.31908 − 1.20805i 3.38621 − 1.23248i 0 −0.0923963 − 0.160035i −1.80177 + 3.12075i 0 −1.47178 − 8.34689i
73.1 −1.11554 − 0.936049i 0 0.0209445 + 0.118782i −0.475244 + 2.69524i 0 2.20574 + 3.82045i −1.36841 + 2.37016i 0 3.05303 − 2.56180i
73.2 1.11554 + 0.936049i 0 0.0209445 + 0.118782i 0.475244 − 2.69524i 0 2.20574 + 3.82045i 1.36841 − 2.37016i 0 3.05303 − 2.56180i
82.1 −1.11554 + 0.936049i 0 0.0209445 − 0.118782i −0.475244 − 2.69524i 0 2.20574 − 3.82045i −1.36841 − 2.37016i 0 3.05303 + 2.56180i
82.2 1.11554 − 0.936049i 0 0.0209445 − 0.118782i 0.475244 + 2.69524i 0 2.20574 − 3.82045i 1.36841 + 2.37016i 0 3.05303 + 2.56180i
100.1 −1.95928 + 0.713118i 0 1.79813 − 1.50881i −0.554707 − 0.465455i 0 −0.613341 − 1.06234i −0.362059 + 0.627105i 0 1.41875 + 0.516382i
100.2 1.95928 − 0.713118i 0 1.79813 − 1.50881i 0.554707 + 0.465455i 0 −0.613341 − 1.06234i 0.362059 − 0.627105i 0 1.41875 + 0.516382i
118.1 −1.95928 − 0.713118i 0 1.79813 + 1.50881i −0.554707 + 0.465455i 0 −0.613341 + 1.06234i −0.362059 − 0.627105i 0 1.41875 − 0.516382i
118.2 1.95928 + 0.713118i 0 1.79813 + 1.50881i 0.554707 − 0.465455i 0 −0.613341 + 1.06234i 0.362059 + 0.627105i 0 1.41875 − 0.516382i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.e even 9 1 inner
57.l odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.u.d ✓ 12
3.b odd 2 1 inner 171.2.u.d ✓ 12
19.e even 9 1 inner 171.2.u.d ✓ 12
19.e even 9 1 3249.2.a.bi 6
19.f odd 18 1 3249.2.a.bj 6
57.j even 18 1 3249.2.a.bj 6
57.l odd 18 1 inner 171.2.u.d ✓ 12
57.l odd 18 1 3249.2.a.bi 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.u.d ✓ 12 1.a even 1 1 trivial
171.2.u.d ✓ 12 3.b odd 2 1 inner
171.2.u.d ✓ 12 19.e even 9 1 inner
171.2.u.d ✓ 12 57.l odd 18 1 inner
3249.2.a.bi 6 19.e even 9 1
3249.2.a.bi 6 57.l odd 18 1
3249.2.a.bj 6 19.f odd 18 1
3249.2.a.bj 6 57.j even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 3T_{2}^{10} - 18T_{2}^{8} + 24T_{2}^{6} + 495T_{2}^{4} - 459T_{2}^{2} + 2601 \) acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{10} + \cdots + 2601 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{10} + \cdots + 2601 \) Copy content Toggle raw display
$7$ \( (T^{6} - 3 T^{5} + 15 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 27 T^{10} + \cdots + 210681 \) Copy content Toggle raw display
$13$ \( (T^{6} + 12 T^{5} + \cdots + 289)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 338964921 \) Copy content Toggle raw display
$19$ \( (T^{6} - 6 T^{5} + \cdots + 6859)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 33 T^{10} + \cdots + 2601 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 217238121 \) Copy content Toggle raw display
$31$ \( (T^{6} - 12 T^{5} + \cdots + 1369)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 12 T^{2} + \cdots + 37)^{4} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 4874692761 \) Copy content Toggle raw display
$43$ \( (T^{6} + 12 T^{5} + 42 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 20523141081 \) Copy content Toggle raw display
$53$ \( T^{12} + 174 T^{10} + \cdots + 2601 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 73863824841 \) Copy content Toggle raw display
$61$ \( (T^{6} - 15 T^{5} + \cdots + 32041)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 24 T^{5} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 20523141081 \) Copy content Toggle raw display
$73$ \( (T^{6} + 21 T^{5} + \cdots + 26569)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 24 T^{5} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 207 T^{10} + \cdots + 210681 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 367152376761 \) Copy content Toggle raw display
$97$ \( (T^{6} - 15 T^{5} + \cdots + 289)^{2} \) Copy content Toggle raw display
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