Properties

Label 170.2.g.f
Level $170$
Weight $2$
Character orbit 170.g
Analytic conductor $1.357$
Analytic rank $0$
Dimension $4$
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.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(71\) \(137\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.00000 0.292893 + 0.292893i 1.00000 0.707107 + 2.12132i 0.292893 + 0.292893i −1.00000 + 1.00000i 1.00000 2.82843i 0.707107 + 2.12132i
89.2 1.00000 1.70711 + 1.70711i 1.00000 −0.707107 2.12132i 1.70711 + 1.70711i −1.00000 + 1.00000i 1.00000 2.82843i −0.707107 2.12132i
149.1 1.00000 0.292893 0.292893i 1.00000 0.707107 2.12132i 0.292893 0.292893i −1.00000 1.00000i 1.00000 2.82843i 0.707107 2.12132i
149.2 1.00000 1.70711 1.70711i 1.00000 −0.707107 + 2.12132i 1.70711 1.70711i −1.00000 1.00000i 1.00000 2.82843i −0.707107 + 2.12132i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.g.f yes 4
3.b odd 2 1 1530.2.n.j 4
5.b even 2 1 170.2.g.e 4
5.c odd 4 1 850.2.h.h 4
5.c odd 4 1 850.2.h.k 4
15.d odd 2 1 1530.2.n.o 4
17.c even 4 1 170.2.g.e 4
51.f odd 4 1 1530.2.n.o 4
85.f odd 4 1 850.2.h.h 4
85.i odd 4 1 850.2.h.k 4
85.j even 4 1 inner 170.2.g.f yes 4
255.i odd 4 1 1530.2.n.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.g.e 4 5.b even 2 1
170.2.g.e 4 17.c even 4 1
170.2.g.f yes 4 1.a even 1 1 trivial
170.2.g.f yes 4 85.j even 4 1 inner
850.2.h.h 4 5.c odd 4 1
850.2.h.h 4 85.f odd 4 1
850.2.h.k 4 5.c odd 4 1
850.2.h.k 4 85.i odd 4 1
1530.2.n.j 4 3.b odd 2 1
1530.2.n.j 4 255.i odd 4 1
1530.2.n.o 4 15.d odd 2 1
1530.2.n.o 4 51.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\):

\( T_{3}^{4} - 4 T_{3}^{3} + 8 T_{3}^{2} - 4 T_{3} + 1 \)
\( T_{7}^{2} + 2 T_{7} + 2 \)