Properties

Label 34.2.a.a
Level $34$
Weight $2$
Character orbit 34.a
Self dual yes
Analytic conductor $0.271$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 34 = 2 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 34.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.271491366872\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - 4q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - 4q^{7} + q^{8} + q^{9} + 6q^{11} - 2q^{12} + 2q^{13} - 4q^{14} + q^{16} - q^{17} + q^{18} - 4q^{19} + 8q^{21} + 6q^{22} - 2q^{24} - 5q^{25} + 2q^{26} + 4q^{27} - 4q^{28} - 4q^{31} + q^{32} - 12q^{33} - q^{34} + q^{36} - 4q^{37} - 4q^{38} - 4q^{39} + 6q^{41} + 8q^{42} + 8q^{43} + 6q^{44} - 2q^{48} + 9q^{49} - 5q^{50} + 2q^{51} + 2q^{52} - 6q^{53} + 4q^{54} - 4q^{56} + 8q^{57} - 4q^{61} - 4q^{62} - 4q^{63} + q^{64} - 12q^{66} + 8q^{67} - q^{68} + q^{72} + 2q^{73} - 4q^{74} + 10q^{75} - 4q^{76} - 24q^{77} - 4q^{78} + 8q^{79} - 11q^{81} + 6q^{82} + 8q^{84} + 8q^{86} + 6q^{88} - 6q^{89} - 8q^{91} + 8q^{93} - 2q^{96} + 14q^{97} + 9q^{98} + 6q^{99} + O(q^{100}) \)

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
0
1.00000 −2.00000 1.00000 0 −2.00000 −4.00000 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 34.2.a.a 1
3.b odd 2 1 306.2.a.a 1
4.b odd 2 1 272.2.a.d 1
5.b even 2 1 850.2.a.e 1
5.c odd 4 2 850.2.c.b 2
7.b odd 2 1 1666.2.a.m 1
8.b even 2 1 1088.2.a.l 1
8.d odd 2 1 1088.2.a.d 1
11.b odd 2 1 4114.2.a.a 1
12.b even 2 1 2448.2.a.k 1
13.b even 2 1 5746.2.a.b 1
15.d odd 2 1 7650.2.a.ci 1
17.b even 2 1 578.2.a.a 1
17.c even 4 2 578.2.b.a 2
17.d even 8 4 578.2.c.e 4
17.e odd 16 8 578.2.d.e 8
20.d odd 2 1 6800.2.a.b 1
24.f even 2 1 9792.2.a.bj 1
24.h odd 2 1 9792.2.a.y 1
51.c odd 2 1 5202.2.a.d 1
68.d odd 2 1 4624.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a 1 1.a even 1 1 trivial
272.2.a.d 1 4.b odd 2 1
306.2.a.a 1 3.b odd 2 1
578.2.a.a 1 17.b even 2 1
578.2.b.a 2 17.c even 4 2
578.2.c.e 4 17.d even 8 4
578.2.d.e 8 17.e odd 16 8
850.2.a.e 1 5.b even 2 1
850.2.c.b 2 5.c odd 4 2
1088.2.a.d 1 8.d odd 2 1
1088.2.a.l 1 8.b even 2 1
1666.2.a.m 1 7.b odd 2 1
2448.2.a.k 1 12.b even 2 1
4114.2.a.a 1 11.b odd 2 1
4624.2.a.a 1 68.d odd 2 1
5202.2.a.d 1 51.c odd 2 1
5746.2.a.b 1 13.b even 2 1
6800.2.a.b 1 20.d odd 2 1
7650.2.a.ci 1 15.d odd 2 1
9792.2.a.y 1 24.h odd 2 1
9792.2.a.bj 1 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(34))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 + 2 T + 3 T^{2} \)
$5$ \( 1 + 5 T^{2} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 6 T + 11 T^{2} \)
$13$ \( 1 - 2 T + 13 T^{2} \)
$17$ \( 1 + T \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 + 4 T + 31 T^{2} \)
$37$ \( 1 + 4 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 - 8 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 + 4 T + 61 T^{2} \)
$67$ \( 1 - 8 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 - 8 T + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 - 14 T + 97 T^{2} \)
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