Properties

Label 170.2.b.a
Level $170$
Weight $2$
Character orbit 170.b
Analytic conductor $1.357$
Analytic rank $0$
Dimension $2$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
1.00000i
1.00000i
−1.00000 1.00000i 1.00000 1.00000i 1.00000i 0 −1.00000 2.00000 1.00000i
101.2 −1.00000 1.00000i 1.00000 1.00000i 1.00000i 0 −1.00000 2.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.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.b.a 2
3.b odd 2 1 1530.2.c.d 2
4.b odd 2 1 1360.2.c.b 2
5.b even 2 1 850.2.b.j 2
5.c odd 4 1 850.2.d.b 2
5.c odd 4 1 850.2.d.g 2
17.b even 2 1 inner 170.2.b.a 2
17.c even 4 1 2890.2.a.m 1
17.c even 4 1 2890.2.a.q 1
51.c odd 2 1 1530.2.c.d 2
68.d odd 2 1 1360.2.c.b 2
85.c even 2 1 850.2.b.j 2
85.g odd 4 1 850.2.d.b 2
85.g odd 4 1 850.2.d.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.b.a 2 1.a even 1 1 trivial
170.2.b.a 2 17.b even 2 1 inner
850.2.b.j 2 5.b even 2 1
850.2.b.j 2 85.c even 2 1
850.2.d.b 2 5.c odd 4 1
850.2.d.b 2 85.g odd 4 1
850.2.d.g 2 5.c odd 4 1
850.2.d.g 2 85.g odd 4 1
1360.2.c.b 2 4.b odd 2 1
1360.2.c.b 2 68.d odd 2 1
1530.2.c.d 2 3.b odd 2 1
1530.2.c.d 2 51.c odd 2 1
2890.2.a.m 1 17.c even 4 1
2890.2.a.q 1 17.c even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\).