Properties

Label 170.2.c.b
Level $170$
Weight $2$
Character orbit 170.c
Analytic conductor $1.357$
Analytic rank $0$
Dimension $6$
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] = [170,2,Mod(69,170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(170, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("170.69");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.c (of order \(2\), degree \(1\), 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: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{5} + \beta_{2} - \beta_1) q^{3} - q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{6} + (\beta_{2} - \beta_1) q^{7} - \beta_{4} q^{8} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{5} + \beta_{2} - \beta_1) q^{3} - q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{6} + (\beta_{2} - \beta_1) q^{7} - \beta_{4} q^{8} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{9} + ( - \beta_{5} + \beta_1) q^{10} + 2 q^{11} + ( - \beta_{5} - \beta_{2} + \beta_1) q^{12} + (\beta_{5} + \beta_{2} - \beta_1) q^{13} + ( - \beta_{2} - \beta_1) q^{14} + ( - \beta_{5} - 4 \beta_{4} - \beta_{2} - \beta_1 + 2) q^{15} + q^{16} + \beta_{4} q^{17} + ( - \beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_1) q^{18} + (\beta_{3} - \beta_{2} - \beta_1) q^{19} + (\beta_{3} + \beta_{2}) q^{20} + (2 \beta_{2} + 2 \beta_1 - 4) q^{21} + 2 \beta_{4} q^{22} + (4 \beta_{4} - \beta_{2} + \beta_1) q^{23} + (\beta_{3} + \beta_{2} + \beta_1) q^{24} + (\beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 + 2) q^{25} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{26} + ( - \beta_{5} + 2 \beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{27} + ( - \beta_{2} + \beta_1) q^{28} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 6) q^{29} + (2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 4) q^{30} + (3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{31} + \beta_{4} q^{32} + (2 \beta_{5} + 2 \beta_{2} - 2 \beta_1) q^{33} - q^{34} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1 + 3) q^{35} + (\beta_{3} - \beta_{2} - \beta_1 + 3) q^{36} + (4 \beta_{5} + 4 \beta_{4} - \beta_{2} + \beta_1) q^{37} + (\beta_{5} - \beta_{2} + \beta_1) q^{38} + ( - \beta_{3} + \beta_{2} + \beta_1 - 6) q^{39} + (\beta_{5} - \beta_1) q^{40} + (2 \beta_{3} + 6) q^{41} + ( - 4 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{42} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{43} - 2 q^{44} + (2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} + \beta_1 + 2) q^{45} + (\beta_{2} + \beta_1 - 4) q^{46} + ( - 3 \beta_{5} - 6 \beta_{4} + \beta_{2} - \beta_1) q^{47} + (\beta_{5} + \beta_{2} - \beta_1) q^{48} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{49} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 1) q^{50} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{51} + ( - \beta_{5} - \beta_{2} + \beta_1) q^{52} + ( - 5 \beta_{5} - 3 \beta_{2} + 3 \beta_1) q^{53} + (\beta_{3} + 3 \beta_{2} + 3 \beta_1 - 2) q^{54} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{55} + (\beta_{2} + \beta_1) q^{56} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{2} - 3 \beta_1) q^{57} + ( - \beta_{5} - 6 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{58} + (3 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{59} + (\beta_{5} + 4 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{60} + (3 \beta_{3} - 2) q^{61} + (3 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{62} + ( - 4 \beta_{5} + 8 \beta_{4} - 5 \beta_{2} + 5 \beta_1) q^{63} - q^{64} + ( - \beta_{5} - 4 \beta_{4} - \beta_{2} - \beta_1 + 2) q^{65} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{66} + (2 \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{67} - \beta_{4} q^{68} + ( - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_1 + 4) q^{69} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + 1) q^{70} + ( - 5 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{71} + (\beta_{5} + 3 \beta_{4} - \beta_{2} + \beta_1) q^{72} + (\beta_{5} + \beta_{2} - \beta_1) q^{73} + ( - 4 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{74} + (3 \beta_{5} + 6 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 2) q^{75} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{76} + (2 \beta_{2} - 2 \beta_1) q^{77} + ( - \beta_{5} - 6 \beta_{4} + \beta_{2} - \beta_1) q^{78} + (4 \beta_{3} + \beta_{2} + \beta_1) q^{79} + ( - \beta_{3} - \beta_{2}) q^{80} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 5) q^{81} + (2 \beta_{5} + 6 \beta_{4}) q^{82} + ( - 2 \beta_{5} + 6 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{83} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{84} + ( - \beta_{5} + \beta_1) q^{85} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{86} + ( - 7 \beta_{5} - 10 \beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{87} - 2 \beta_{4} q^{88} + ( - 3 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{89} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 6 \beta_1 + 4) q^{90} + (2 \beta_{2} + 2 \beta_1 - 4) q^{91} + ( - 4 \beta_{4} + \beta_{2} - \beta_1) q^{92} + (5 \beta_{5} + 14 \beta_{4} + \beta_{2} - \beta_1) q^{93} + (3 \beta_{3} - \beta_{2} - \beta_1 + 6) q^{94} + ( - 2 \beta_{5} + 4 \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{95} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{96} + ( - 3 \beta_{5} - \beta_{2} + \beta_1) q^{97} + (2 \beta_{5} + \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{98} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 2 q^{5} + 2 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 2 q^{5} + 2 q^{6} - 16 q^{9} + 12 q^{11} + 12 q^{15} + 6 q^{16} - 2 q^{19} - 2 q^{20} - 24 q^{21} - 2 q^{24} + 10 q^{25} + 2 q^{26} - 34 q^{29} + 22 q^{30} + 6 q^{31} - 6 q^{34} + 20 q^{35} + 16 q^{36} - 34 q^{39} + 32 q^{41} - 12 q^{44} + 8 q^{45} - 24 q^{46} + 2 q^{49} - 4 q^{50} + 2 q^{51} - 14 q^{54} + 4 q^{55} - 18 q^{59} - 12 q^{60} - 18 q^{61} - 6 q^{64} + 12 q^{65} + 4 q^{66} + 32 q^{69} + 4 q^{70} - 2 q^{71} - 16 q^{74} + 16 q^{75} + 2 q^{76} - 8 q^{79} + 2 q^{80} + 38 q^{81} + 24 q^{84} + 8 q^{86} + 18 q^{89} + 28 q^{90} - 24 q^{91} + 30 q^{94} - 14 q^{95} + 2 q^{96} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} - 5\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).

\(n\) \(71\) \(137\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
0.432320 + 0.432320i
1.32001 + 1.32001i
−1.75233 1.75233i
−1.75233 + 1.75233i
1.32001 1.32001i
0.432320 0.432320i
1.00000i 2.62620i −1.00000 −2.19388 + 0.432320i −2.62620 0.864641i 1.00000i −3.89692 0.432320 + 2.19388i
69.2 1.00000i 0.484862i −1.00000 1.80487 + 1.32001i 0.484862 2.64002i 1.00000i 2.76491 1.32001 1.80487i
69.3 1.00000i 3.14134i −1.00000 1.38900 1.75233i 3.14134 3.50466i 1.00000i −6.86799 −1.75233 1.38900i
69.4 1.00000i 3.14134i −1.00000 1.38900 + 1.75233i 3.14134 3.50466i 1.00000i −6.86799 −1.75233 + 1.38900i
69.5 1.00000i 0.484862i −1.00000 1.80487 1.32001i 0.484862 2.64002i 1.00000i 2.76491 1.32001 + 1.80487i
69.6 1.00000i 2.62620i −1.00000 −2.19388 0.432320i −2.62620 0.864641i 1.00000i −3.89692 0.432320 2.19388i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.c.b 6
3.b odd 2 1 1530.2.d.g 6
4.b odd 2 1 1360.2.e.c 6
5.b even 2 1 inner 170.2.c.b 6
5.c odd 4 1 850.2.a.p 3
5.c odd 4 1 850.2.a.q 3
15.d odd 2 1 1530.2.d.g 6
15.e even 4 1 7650.2.a.dj 3
15.e even 4 1 7650.2.a.do 3
20.d odd 2 1 1360.2.e.c 6
20.e even 4 1 6800.2.a.bk 3
20.e even 4 1 6800.2.a.bp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.c.b 6 1.a even 1 1 trivial
170.2.c.b 6 5.b even 2 1 inner
850.2.a.p 3 5.c odd 4 1
850.2.a.q 3 5.c odd 4 1
1360.2.e.c 6 4.b odd 2 1
1360.2.e.c 6 20.d odd 2 1
1530.2.d.g 6 3.b odd 2 1
1530.2.d.g 6 15.d odd 2 1
6800.2.a.bk 3 20.e even 4 1
6800.2.a.bp 3 20.e even 4 1
7650.2.a.dj 3 15.e even 4 1
7650.2.a.do 3 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 17T_{3}^{4} + 72T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 17 T^{4} + 72 T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} - 3 T^{4} + 24 T^{3} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 20 T^{4} + 100 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T - 2)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 17 T^{4} + 72 T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$19$ \( (T^{3} + T^{2} - 24 T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 68 T^{4} + 676 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( (T^{3} + 17 T^{2} + 66 T - 50)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 3 T^{2} - 46 T - 74)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 308 T^{4} + 31140 T^{2} + \cdots + 1032256 \) Copy content Toggle raw display
$41$ \( (T^{3} - 16 T^{2} + 60 T + 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 72 T^{4} + 1040 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( T^{6} + 257 T^{4} + 20176 T^{2} + \cdots + 440896 \) Copy content Toggle raw display
$53$ \( T^{6} + 265 T^{4} + 22040 T^{2} + \cdots + 583696 \) Copy content Toggle raw display
$59$ \( (T^{3} + 9 T^{2} - 16 T - 160)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 9 T^{2} - 30 T - 34)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 132 T^{4} + 2816 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T^{3} + T^{2} - 118 T + 334)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 17 T^{4} + 72 T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{3} + 4 T^{2} - 74 T - 160)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 328 T^{4} + 6416 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$89$ \( (T^{3} - 9 T^{2} - 16 T + 160)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 89 T^{4} + 2200 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
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