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Properties

Label 171.3.ba.b
Level $171$
Weight $3$
Character orbit 171.ba
Analytic conductor $4.659$
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] = [171,3,Mod(10,171)]
 
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, 17]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 171.ba (of order \(18\), 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: \(4.65941252056\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 24x^{10} + 216x^{8} + 905x^{6} + 1770x^{4} + 1395x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{9} + \beta_{6} - \beta_{4} + 1) q^{2} + ( - \beta_{11} - \beta_{9} + \cdots - \beta_{5}) q^{4}+ \cdots + ( - \beta_{10} - \beta_{9} + 3 \beta_{8} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} + \beta_{6} - \beta_{4} + 1) q^{2} + ( - \beta_{11} - \beta_{9} + \cdots - \beta_{5}) q^{4}+ \cdots + (20 \beta_{11} - \beta_{10} + \cdots - 51) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 6 q^{5} + 6 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} + 6 q^{5} + 6 q^{7} + 9 q^{8} + 51 q^{10} + 18 q^{11} + 21 q^{13} - 9 q^{14} - 12 q^{16} + 3 q^{17} - 24 q^{19} + 90 q^{20} - 78 q^{22} + 102 q^{23} - 156 q^{25} - 21 q^{26} + 12 q^{28} - 147 q^{29} + 99 q^{31} - 165 q^{32} + 132 q^{34} - 96 q^{35} - 72 q^{38} - 138 q^{40} + 144 q^{41} - 27 q^{43} + 123 q^{44} - 54 q^{46} + 99 q^{47} - 24 q^{49} - 72 q^{50} + 93 q^{52} - 111 q^{53} + 162 q^{55} - 132 q^{58} - 3 q^{59} + 150 q^{61} - 108 q^{62} + 27 q^{64} - 126 q^{65} + 135 q^{67} + 30 q^{68} + 225 q^{70} + 168 q^{71} - 90 q^{73} + 231 q^{74} + 42 q^{76} - 246 q^{77} - 75 q^{79} - 21 q^{80} - 117 q^{82} + 156 q^{83} - 300 q^{85} + 144 q^{86} - 405 q^{88} + 558 q^{89} - 453 q^{91} - 48 q^{92} + 69 q^{95} + 465 q^{97} - 777 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 216x^{8} + 905x^{6} + 1770x^{4} + 1395x^{2} + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} - 27 \nu^{9} + 28 \nu^{8} - 277 \nu^{7} + 38 \nu^{6} - 1304 \nu^{5} + \cdots - 1520 ) / 304 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} - 5 \nu^{10} - \nu^{9} - 89 \nu^{8} + 239 \nu^{7} - 437 \nu^{6} + 2078 \nu^{5} + \cdots + 1881 ) / 304 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - \nu^{9} + 89 \nu^{8} + 239 \nu^{7} + 437 \nu^{6} + 2078 \nu^{5} + \cdots - 1881 ) / 304 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} - 2 \nu^{10} - 27 \nu^{9} - 28 \nu^{8} - 277 \nu^{7} - 38 \nu^{6} - 1304 \nu^{5} + \cdots + 1520 ) / 304 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{11} + 17 \nu^{10} + 55 \nu^{9} + 333 \nu^{8} + 315 \nu^{7} + 2185 \nu^{6} + 530 \nu^{5} + \cdots + 171 ) / 608 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{11} - 17 \nu^{10} + 55 \nu^{9} - 333 \nu^{8} + 315 \nu^{7} - 2185 \nu^{6} + 530 \nu^{5} + \cdots - 171 ) / 608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} + 21\nu^{8} + 153\nu^{6} + 454\nu^{4} + 504\nu^{2} + 8\nu + 171 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - 21\nu^{8} - 153\nu^{6} - 454\nu^{4} - 504\nu^{2} + 8\nu - 171 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{11} + \nu^{10} - 56 \nu^{9} + 14 \nu^{8} - 329 \nu^{7} + 19 \nu^{6} - 549 \nu^{5} + \cdots - 380 ) / 152 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17 \nu^{11} + 17 \nu^{10} + 333 \nu^{9} + 333 \nu^{8} + 2185 \nu^{7} + 2185 \nu^{6} + 5334 \nu^{5} + \cdots + 1083 ) / 608 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9\nu^{11} + 197\nu^{9} + 1545\nu^{7} + 5238\nu^{5} + 7304\nu^{3} + 2979\nu - 152 ) / 304 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{8} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{5} + \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{8} - 5\beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{10} - 7 \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} + 5 \beta_{5} - 11 \beta_{4} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + 32 \beta_{8} + 32 \beta_{7} - 18 \beta_{6} - 15 \beta_{5} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 50 \beta_{10} + 50 \beta_{9} + 14 \beta_{8} - 14 \beta_{7} - 30 \beta_{6} - 20 \beta_{5} + 102 \beta_{4} + \cdots - 153 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 38 \beta_{11} + 44 \beta_{10} - 44 \beta_{9} - 226 \beta_{8} - 226 \beta_{7} + 158 \beta_{6} + \cdots + 25 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 384 \beta_{10} - 384 \beta_{9} - 149 \beta_{8} + 149 \beta_{7} + 334 \beta_{6} + 50 \beta_{5} + \cdots + 1104 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 494 \beta_{11} - 483 \beta_{10} + 483 \beta_{9} + 1688 \beta_{8} + 1688 \beta_{7} - 1400 \beta_{6} + \cdots - 236 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3088 \beta_{10} + 3088 \beta_{9} + 1433 \beta_{8} - 1433 \beta_{7} - 3332 \beta_{6} + 244 \beta_{5} + \cdots - 8372 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5420 \beta_{11} + 4765 \beta_{10} - 4765 \beta_{9} - 13049 \beta_{8} - 13049 \beta_{7} + 12374 \beta_{6} + \cdots + 2055 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(-\beta_{4} + \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
− 0.918492i
2.88811i
1.89323i
− 2.57727i
− 1.89323i
2.57727i
0.728740i
− 2.01431i
− 0.728740i
2.01431i
0.918492i
− 2.88811i
−0.904538 + 0.159494i 0 −2.96602 + 1.07954i −0.295437 − 0.107530i 0 −0.328846 + 0.569578i 5.69245 − 3.28654i 0 0.284385 + 0.0501447i
10.2 2.84423 − 0.501515i 0 4.07936 − 1.48477i 3.00117 + 1.09234i 0 3.87208 − 6.70664i 0.853313 − 0.492661i 0 9.08386 + 1.60173i
91.1 −0.647524 + 1.77906i 0 0.318417 + 0.267183i 2.13959 − 1.79533i 0 −1.20796 + 2.09224i −7.23987 + 4.17994i 0 1.80856 + 4.96897i
91.2 0.881480 − 2.42185i 0 −2.02415 − 1.69847i −1.73199 + 1.45331i 0 5.72163 − 9.91015i 3.03027 − 1.74952i 0 1.99298 + 5.47566i
109.1 −0.647524 − 1.77906i 0 0.318417 − 0.267183i 2.13959 + 1.79533i 0 −1.20796 − 2.09224i −7.23987 − 4.17994i 0 1.80856 − 4.96897i
109.2 0.881480 + 2.42185i 0 −2.02415 + 1.69847i −1.73199 − 1.45331i 0 5.72163 + 9.91015i 3.03027 + 1.74952i 0 1.99298 − 5.47566i
127.1 −0.468425 − 0.558247i 0 0.602375 − 3.41624i 1.13111 + 6.41483i 0 −5.47943 + 9.49065i −4.71370 + 2.72146i 0 3.05122 − 3.63630i
127.2 1.29478 + 1.54305i 0 −0.00997859 + 0.0565914i −1.24445 − 7.05761i 0 0.422527 − 0.731838i 6.87755 − 3.97075i 0 9.27900 − 11.0583i
136.1 −0.468425 + 0.558247i 0 0.602375 + 3.41624i 1.13111 − 6.41483i 0 −5.47943 − 9.49065i −4.71370 − 2.72146i 0 3.05122 + 3.63630i
136.2 1.29478 − 1.54305i 0 −0.00997859 − 0.0565914i −1.24445 + 7.05761i 0 0.422527 + 0.731838i 6.87755 + 3.97075i 0 9.27900 + 11.0583i
154.1 −0.904538 − 0.159494i 0 −2.96602 − 1.07954i −0.295437 + 0.107530i 0 −0.328846 − 0.569578i 5.69245 + 3.28654i 0 0.284385 − 0.0501447i
154.2 2.84423 + 0.501515i 0 4.07936 + 1.48477i 3.00117 − 1.09234i 0 3.87208 + 6.70664i 0.853313 + 0.492661i 0 9.08386 − 1.60173i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.3.ba.b 12
3.b odd 2 1 19.3.f.a ✓ 12
12.b even 2 1 304.3.z.a 12
19.f odd 18 1 inner 171.3.ba.b 12
57.d even 2 1 361.3.f.g 12
57.f even 6 1 361.3.f.b 12
57.f even 6 1 361.3.f.c 12
57.h odd 6 1 361.3.f.e 12
57.h odd 6 1 361.3.f.f 12
57.j even 18 1 19.3.f.a ✓ 12
57.j even 18 1 361.3.b.c 12
57.j even 18 1 361.3.d.d 12
57.j even 18 1 361.3.d.f 12
57.j even 18 1 361.3.f.e 12
57.j even 18 1 361.3.f.f 12
57.l odd 18 1 361.3.b.c 12
57.l odd 18 1 361.3.d.d 12
57.l odd 18 1 361.3.d.f 12
57.l odd 18 1 361.3.f.b 12
57.l odd 18 1 361.3.f.c 12
57.l odd 18 1 361.3.f.g 12
228.u odd 18 1 304.3.z.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.f.a ✓ 12 3.b odd 2 1
19.3.f.a ✓ 12 57.j even 18 1
171.3.ba.b 12 1.a even 1 1 trivial
171.3.ba.b 12 19.f odd 18 1 inner
304.3.z.a 12 12.b even 2 1
304.3.z.a 12 228.u odd 18 1
361.3.b.c 12 57.j even 18 1
361.3.b.c 12 57.l odd 18 1
361.3.d.d 12 57.j even 18 1
361.3.d.d 12 57.l odd 18 1
361.3.d.f 12 57.j even 18 1
361.3.d.f 12 57.l odd 18 1
361.3.f.b 12 57.f even 6 1
361.3.f.b 12 57.l odd 18 1
361.3.f.c 12 57.f even 6 1
361.3.f.c 12 57.l odd 18 1
361.3.f.e 12 57.h odd 6 1
361.3.f.e 12 57.j even 18 1
361.3.f.f 12 57.h odd 6 1
361.3.f.f 12 57.j even 18 1
361.3.f.g 12 57.d even 2 1
361.3.f.g 12 57.l odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 6 T_{2}^{11} + 18 T_{2}^{10} - 39 T_{2}^{9} + 48 T_{2}^{8} - 57 T_{2}^{7} + 74 T_{2}^{6} + \cdots + 361 \) acting on \(S_{3}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 6 T^{11} + \cdots + 361 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + \cdots + 87616 \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + \cdots + 1700416 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 131774082049 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 3568865052736 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 21556993329 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 114585206984704 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 484594358358016 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 125789503910464 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 58527273697344 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 17\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 126559282520449 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 48\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 42\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 15\!\cdots\!89 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!89 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
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