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Properties

Label 171.4.f.e
Level $171$
Weight $4$
Character orbit 171.f
Analytic conductor $10.089$
Analytic rank $0$
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] = [171,4,Mod(64,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.64");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 171.f (of order \(3\), 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: \(10.0893266110\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{55})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 55x^{2} + 3025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (7 \beta_{2} + 7) q^{4} + (\beta_{3} + 7 \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - 7) q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (7 \beta_{2} + 7) q^{4} + (\beta_{3} + 7 \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - 7) q^{7} + 15 q^{8} + (7 \beta_{2} + \beta_1 + 7) q^{10} + (7 \beta_{3} - 14) q^{11} + (14 \beta_{2} - 8 \beta_1 + 14) q^{13} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{14} + 41 \beta_{2} q^{16} + ( - 8 \beta_{3} + 56 \beta_{2} - 8 \beta_1) q^{17} + (8 \beta_{3} - 42 \beta_{2} + \cdots - 70) q^{19}+ \cdots + (14 \beta_{3} + 239 \beta_{2} + 14 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 14 q^{4} - 14 q^{5} - 28 q^{7} + 60 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 14 q^{4} - 14 q^{5} - 28 q^{7} + 60 q^{8} + 14 q^{10} - 56 q^{11} + 28 q^{13} - 14 q^{14} - 82 q^{16} - 112 q^{17} - 196 q^{19} - 196 q^{20} - 28 q^{22} + 114 q^{23} + 42 q^{25} + 56 q^{26} - 98 q^{28} + 222 q^{29} + 532 q^{31} + 322 q^{32} + 112 q^{34} - 12 q^{35} + 364 q^{37} - 224 q^{38} - 210 q^{40} + 154 q^{41} - 268 q^{43} - 196 q^{44} + 228 q^{46} + 126 q^{47} - 956 q^{49} + 84 q^{50} - 196 q^{52} - 884 q^{53} + 966 q^{55} - 420 q^{56} + 444 q^{58} + 112 q^{59} - 546 q^{61} + 266 q^{62} - 668 q^{64} + 1368 q^{65} + 740 q^{67} - 1568 q^{68} + 12 q^{70} + 432 q^{71} - 350 q^{73} + 182 q^{74} + 196 q^{76} - 1148 q^{77} + 152 q^{79} - 574 q^{80} - 154 q^{82} + 3808 q^{83} + 96 q^{85} + 268 q^{86} - 840 q^{88} + 112 q^{89} - 1076 q^{91} - 798 q^{92} + 252 q^{94} + 908 q^{95} + 546 q^{97} - 478 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 55x^{2} + 3025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 55 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 55\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 55\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
3.70810 − 6.42262i
−3.70810 + 6.42262i
3.70810 + 6.42262i
−3.70810 − 6.42262i
0.500000 + 0.866025i 0 3.50000 − 6.06218i −7.20810 − 12.4848i 0 0.416198 15.0000 0 7.20810 − 12.4848i
64.2 0.500000 + 0.866025i 0 3.50000 − 6.06218i 0.208099 + 0.360438i 0 −14.4162 15.0000 0 −0.208099 + 0.360438i
163.1 0.500000 − 0.866025i 0 3.50000 + 6.06218i −7.20810 + 12.4848i 0 0.416198 15.0000 0 7.20810 + 12.4848i
163.2 0.500000 − 0.866025i 0 3.50000 + 6.06218i 0.208099 − 0.360438i 0 −14.4162 15.0000 0 −0.208099 − 0.360438i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.4.f.e 4
3.b odd 2 1 19.4.c.a ✓ 4
12.b even 2 1 304.4.i.c 4
19.c even 3 1 inner 171.4.f.e 4
57.f even 6 1 361.4.a.d 2
57.h odd 6 1 19.4.c.a ✓ 4
57.h odd 6 1 361.4.a.g 2
228.m even 6 1 304.4.i.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.c.a ✓ 4 3.b odd 2 1
19.4.c.a ✓ 4 57.h odd 6 1
171.4.f.e 4 1.a even 1 1 trivial
171.4.f.e 4 19.c even 3 1 inner
304.4.i.c 4 12.b even 2 1
304.4.i.c 4 228.m even 6 1
361.4.a.d 2 57.f even 6 1
361.4.a.g 2 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(171, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 14T_{7} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 14 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( (T^{2} + 14 T - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 28 T - 2499)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 28 T^{3} + \cdots + 11048976 \) Copy content Toggle raw display
$17$ \( T^{4} + 112 T^{3} + \cdots + 147456 \) Copy content Toggle raw display
$19$ \( T^{4} + 196 T^{3} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{4} - 114 T^{3} + \cdots + 306916 \) Copy content Toggle raw display
$29$ \( T^{4} - 222 T^{3} + \cdots + 92659876 \) Copy content Toggle raw display
$31$ \( (T^{2} - 266 T + 14994)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 182 T + 5586)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 3323637801 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 6251116096 \) Copy content Toggle raw display
$47$ \( T^{4} - 126 T^{3} + \cdots + 70660836 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 23178235536 \) Copy content Toggle raw display
$59$ \( T^{4} - 112 T^{3} + \cdots + 37933281 \) Copy content Toggle raw display
$61$ \( T^{4} + 546 T^{3} + \cdots + 799419076 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 6625146025 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 2536532496 \) Copy content Toggle raw display
$73$ \( T^{4} + 350 T^{3} + \cdots + 820536025 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 27790223616 \) Copy content Toggle raw display
$83$ \( (T^{2} - 1904 T + 838929)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 350160961536 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 18739145881 \) Copy content Toggle raw display
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