Properties

Label 170.2.b.b
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: \(\Z[a_1, \ldots, a_{5}]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

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 3.00000i 1.00000 1.00000i 3.00000i 4.00000i 1.00000 −6.00000 1.00000i
101.2 1.00000 3.00000i 1.00000 1.00000i 3.00000i 4.00000i 1.00000 −6.00000 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.b 2
3.b odd 2 1 1530.2.c.b 2
4.b odd 2 1 1360.2.c.a 2
5.b even 2 1 850.2.b.a 2
5.c odd 4 1 850.2.d.a 2
5.c odd 4 1 850.2.d.h 2
17.b even 2 1 inner 170.2.b.b 2
17.c even 4 1 2890.2.a.a 1
17.c even 4 1 2890.2.a.l 1
51.c odd 2 1 1530.2.c.b 2
68.d odd 2 1 1360.2.c.a 2
85.c even 2 1 850.2.b.a 2
85.g odd 4 1 850.2.d.a 2
85.g odd 4 1 850.2.d.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.b.b 2 1.a even 1 1 trivial
170.2.b.b 2 17.b even 2 1 inner
850.2.b.a 2 5.b even 2 1
850.2.b.a 2 85.c even 2 1
850.2.d.a 2 5.c odd 4 1
850.2.d.a 2 85.g odd 4 1
850.2.d.h 2 5.c odd 4 1
850.2.d.h 2 85.g odd 4 1
1360.2.c.a 2 4.b odd 2 1
1360.2.c.a 2 68.d odd 2 1
1530.2.c.b 2 3.b odd 2 1
1530.2.c.b 2 51.c odd 2 1
2890.2.a.a 1 17.c even 4 1
2890.2.a.l 1 17.c even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\).