Properties

Label 170.2.d.b
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 \(2\), 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: \(\Z[a_1, a_2]\)
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.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
85.c 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.d.b yes 2
3.b odd 2 1 1530.2.f.b 2
4.b odd 2 1 1360.2.o.a 2
5.b even 2 1 170.2.d.a 2
5.c odd 4 1 850.2.b.c 2
5.c odd 4 1 850.2.b.i 2
15.d odd 2 1 1530.2.f.e 2
17.b even 2 1 170.2.d.a 2
20.d odd 2 1 1360.2.o.b 2
51.c odd 2 1 1530.2.f.e 2
68.d odd 2 1 1360.2.o.b 2
85.c even 2 1 inner 170.2.d.b yes 2
85.g odd 4 1 850.2.b.c 2
85.g odd 4 1 850.2.b.i 2
255.h odd 2 1 1530.2.f.b 2
340.d odd 2 1 1360.2.o.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.d.a 2 5.b even 2 1
170.2.d.a 2 17.b even 2 1
170.2.d.b yes 2 1.a even 1 1 trivial
170.2.d.b yes 2 85.c even 2 1 inner
850.2.b.c 2 5.c odd 4 1
850.2.b.c 2 85.g odd 4 1
850.2.b.i 2 5.c odd 4 1
850.2.b.i 2 85.g odd 4 1
1360.2.o.a 2 4.b odd 2 1
1360.2.o.a 2 340.d odd 2 1
1360.2.o.b 2 20.d odd 2 1
1360.2.o.b 2 68.d odd 2 1
1530.2.f.b 2 3.b odd 2 1
1530.2.f.b 2 255.h odd 2 1
1530.2.f.e 2 15.d odd 2 1
1530.2.f.e 2 51.c odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( 1 - T + 3 T^{2} )^{2} \)
$5$ \( 1 - 4 T + 5 T^{2} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( 1 - 25 T^{2} + 169 T^{4} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( ( 1 + 5 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 4 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 23 T^{2} + 841 T^{4} \)
$31$ \( 1 - 37 T^{2} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 8 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} ) \)
$43$ \( 1 - 50 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 45 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 105 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 5 T + 59 T^{2} )^{2} \)
$61$ \( 1 - 97 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 130 T^{2} + 4489 T^{4} \)
$71$ \( 1 - 117 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 - 11 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 98 T^{2} + 6241 T^{4} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 5 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 7 T + 97 T^{2} )^{2} \)
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