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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Newspace parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
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

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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} + 10 \nu^{2} - 121 \nu + 100 \)\()/121\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{5} + 27 \nu^{4} + 35 \nu^{3} - 14 \nu^{2} + 223 \)\()/121\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258 \)\()/242\)
\(\beta_{5}\)\(=\)\((\)\( -65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1331 \nu + 574 \)\()/242\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} - 5 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(5 \beta_{3} + 7 \beta_{2} + 7 \beta_{1} - 15\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} - 16 \beta_{4} - 2 \beta_{3} - 29 \beta_{1} + 16\)

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.

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 \( T_{3}^{6} + 17 T_{3}^{4} + 72 T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( 1 - T^{2} + 3 T^{4} - 38 T^{6} + 27 T^{8} - 81 T^{10} + 729 T^{12} \)
$5$ \( 1 - 2 T - 3 T^{2} + 24 T^{3} - 15 T^{4} - 50 T^{5} + 125 T^{6} \)
$7$ \( 1 - 22 T^{2} + 275 T^{4} - 2316 T^{6} + 13475 T^{8} - 52822 T^{10} + 117649 T^{12} \)
$11$ \( ( 1 - 2 T + 11 T^{2} )^{6} \)
$13$ \( 1 - 61 T^{2} + 1723 T^{4} - 28558 T^{6} + 291187 T^{8} - 1742221 T^{10} + 4826809 T^{12} \)
$17$ \( ( 1 + T^{2} )^{3} \)
$19$ \( ( 1 + T + 33 T^{2} + 58 T^{3} + 627 T^{4} + 361 T^{5} + 6859 T^{6} )^{2} \)
$23$ \( 1 - 70 T^{2} + 2355 T^{4} - 57580 T^{6} + 1245795 T^{8} - 19588870 T^{10} + 148035889 T^{12} \)
$29$ \( ( 1 + 17 T + 153 T^{2} + 936 T^{3} + 4437 T^{4} + 14297 T^{5} + 24389 T^{6} )^{2} \)
$31$ \( ( 1 - 3 T + 47 T^{2} - 260 T^{3} + 1457 T^{4} - 2883 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( ( 1 - 10 T + 93 T^{2} - 472 T^{3} + 3441 T^{4} - 13690 T^{5} + 50653 T^{6} )( 1 + 10 T + 93 T^{2} + 472 T^{3} + 3441 T^{4} + 13690 T^{5} + 50653 T^{6} ) \)
$41$ \( ( 1 - 16 T + 183 T^{2} - 1296 T^{3} + 7503 T^{4} - 26896 T^{5} + 68921 T^{6} )^{2} \)
$43$ \( 1 - 186 T^{2} + 16391 T^{4} - 879788 T^{6} + 30306959 T^{8} - 635896986 T^{10} + 6321363049 T^{12} \)
$47$ \( 1 - 25 T^{2} + 4995 T^{4} - 125830 T^{6} + 11033955 T^{8} - 121992025 T^{10} + 10779215329 T^{12} \)
$53$ \( 1 - 53 T^{2} + 7995 T^{4} - 263774 T^{6} + 22457955 T^{8} - 418195493 T^{10} + 22164361129 T^{12} \)
$59$ \( ( 1 + 9 T + 161 T^{2} + 902 T^{3} + 9499 T^{4} + 31329 T^{5} + 205379 T^{6} )^{2} \)
$61$ \( ( 1 + 9 T + 153 T^{2} + 1064 T^{3} + 9333 T^{4} + 33489 T^{5} + 226981 T^{6} )^{2} \)
$67$ \( 1 - 270 T^{2} + 34775 T^{4} - 2833220 T^{6} + 156104975 T^{8} - 5440802670 T^{10} + 90458382169 T^{12} \)
$71$ \( ( 1 + T + 95 T^{2} + 476 T^{3} + 6745 T^{4} + 5041 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 - 421 T^{2} + 75043 T^{4} - 7247278 T^{6} + 399904147 T^{8} - 11955659461 T^{10} + 151334226289 T^{12} \)
$79$ \( ( 1 + 4 T + 163 T^{2} + 472 T^{3} + 12877 T^{4} + 24964 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 - 170 T^{2} + 855 T^{4} + 1073140 T^{6} + 5890095 T^{8} - 8067914570 T^{10} + 326940373369 T^{12} \)
$89$ \( ( 1 - 9 T + 251 T^{2} - 1442 T^{3} + 22339 T^{4} - 71289 T^{5} + 704969 T^{6} )^{2} \)
$97$ \( 1 - 493 T^{2} + 108803 T^{4} - 13645854 T^{6} + 1023727427 T^{8} - 43644935533 T^{10} + 832972004929 T^{12} \)
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