Properties

Label 170.2.a.f
Level $170$
Weight $2$
Character orbit 170.a
Self dual yes
Analytic conductor $1.357$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56155
−1.56155
1.00000 −2.56155 1.00000 1.00000 −2.56155 5.12311 1.00000 3.56155 1.00000
1.2 1.00000 1.56155 1.00000 1.00000 1.56155 −3.12311 1.00000 −0.561553 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(17\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.a.f 2
3.b odd 2 1 1530.2.a.r 2
4.b odd 2 1 1360.2.a.m 2
5.b even 2 1 850.2.a.n 2
5.c odd 4 2 850.2.c.i 4
7.b odd 2 1 8330.2.a.bq 2
8.b even 2 1 5440.2.a.bj 2
8.d odd 2 1 5440.2.a.bd 2
15.d odd 2 1 7650.2.a.de 2
17.b even 2 1 2890.2.a.u 2
17.c even 4 2 2890.2.b.i 4
20.d odd 2 1 6800.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.a.f 2 1.a even 1 1 trivial
850.2.a.n 2 5.b even 2 1
850.2.c.i 4 5.c odd 4 2
1360.2.a.m 2 4.b odd 2 1
1530.2.a.r 2 3.b odd 2 1
2890.2.a.u 2 17.b even 2 1
2890.2.b.i 4 17.c even 4 2
5440.2.a.bd 2 8.d odd 2 1
5440.2.a.bj 2 8.b even 2 1
6800.2.a.be 2 20.d odd 2 1
7650.2.a.de 2 15.d odd 2 1
8330.2.a.bq 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(170))\):

\( T_{3}^{2} + T_{3} - 4 \)
\( T_{7}^{2} - 2 T_{7} - 16 \)
\( T_{13}^{2} - 5 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( ( 1 - T )^{2} \)
$7$ \( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} \)
$11$ \( ( 1 + 4 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - T )^{2} \)
$19$ \( 1 + T + 34 T^{2} + 19 T^{3} + 361 T^{4} \)
$23$ \( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( 1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4} \)
$31$ \( 1 + 9 T + 78 T^{2} + 279 T^{3} + 961 T^{4} \)
$37$ \( 1 + 6 T + 66 T^{2} + 222 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 8 T + 30 T^{2} + 328 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 6 T + 78 T^{2} - 258 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 9 T + 110 T^{2} - 423 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 3 T + 104 T^{2} - 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 13 T + 122 T^{2} + 767 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 19 T + 208 T^{2} - 1159 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 10 T + 142 T^{2} + 670 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 21 T + 214 T^{2} + 1491 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 7 T + 52 T^{2} - 511 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( 1 - 4 T + 102 T^{2} - 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 7 T + 152 T^{2} + 623 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 9 T + 176 T^{2} - 873 T^{3} + 9409 T^{4} \)
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