Properties

Label 170.2.d.a
Level 170
Weight 2
Character orbit 170.d
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.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1
1.00000i
1.00000i
1.00000i −1.00000 −1.00000 −2.00000 1.00000i 1.00000i −2.00000 1.00000i −2.00000 −1.00000 + 2.00000i
169.2 1.00000i −1.00000 −1.00000 −2.00000 + 1.00000i 1.00000i −2.00000 1.00000i −2.00000 −1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
85.c Even 1 yes

Hecke kernels

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