Properties

Label 34.2.b.a
Level 34
Weight 2
Character orbit 34.b
Analytic conductor 0.271
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 34 = 2 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 34.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.271491366872\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/34\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
1.41421i
1.41421i
−1.00000 2.82843i 1.00000 2.82843i 2.82843i 0 −1.00000 −5.00000 2.82843i
33.2 −1.00000 2.82843i 1.00000 2.82843i 2.82843i 0 −1.00000 −5.00000 2.82843i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 34.2.b.a 2
3.b odd 2 1 306.2.b.d 2
4.b odd 2 1 272.2.b.a 2
5.b even 2 1 850.2.b.f 2
5.c odd 4 2 850.2.d.i 4
7.b odd 2 1 1666.2.b.c 2
8.b even 2 1 1088.2.b.b 2
8.d odd 2 1 1088.2.b.a 2
12.b even 2 1 2448.2.c.n 2
17.b even 2 1 inner 34.2.b.a 2
17.c even 4 2 578.2.a.d 2
17.d even 8 2 578.2.c.a 2
17.d even 8 2 578.2.c.d 2
17.e odd 16 8 578.2.d.g 8
51.c odd 2 1 306.2.b.d 2
51.f odd 4 2 5202.2.a.u 2
68.d odd 2 1 272.2.b.a 2
68.f odd 4 2 4624.2.a.s 2
85.c even 2 1 850.2.b.f 2
85.g odd 4 2 850.2.d.i 4
119.d odd 2 1 1666.2.b.c 2
136.e odd 2 1 1088.2.b.a 2
136.h even 2 1 1088.2.b.b 2
204.h even 2 1 2448.2.c.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.b.a 2 1.a even 1 1 trivial
34.2.b.a 2 17.b even 2 1 inner
272.2.b.a 2 4.b odd 2 1
272.2.b.a 2 68.d odd 2 1
306.2.b.d 2 3.b odd 2 1
306.2.b.d 2 51.c odd 2 1
578.2.a.d 2 17.c even 4 2
578.2.c.a 2 17.d even 8 2
578.2.c.d 2 17.d even 8 2
578.2.d.g 8 17.e odd 16 8
850.2.b.f 2 5.b even 2 1
850.2.b.f 2 85.c even 2 1
850.2.d.i 4 5.c odd 4 2
850.2.d.i 4 85.g odd 4 2
1088.2.b.a 2 8.d odd 2 1
1088.2.b.a 2 136.e odd 2 1
1088.2.b.b 2 8.b even 2 1
1088.2.b.b 2 136.h even 2 1
1666.2.b.c 2 7.b odd 2 1
1666.2.b.c 2 119.d odd 2 1
2448.2.c.n 2 12.b even 2 1
2448.2.c.n 2 204.h even 2 1
4624.2.a.s 2 68.f odd 4 2
5202.2.a.u 2 51.f odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(34, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} ) \)
$5$ \( 1 - 2 T^{2} + 25 T^{4} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} ) \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 6 T + 17 T^{2} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 14 T^{2} + 529 T^{4} \)
$29$ \( 1 - 50 T^{2} + 841 T^{4} \)
$31$ \( ( 1 - 31 T^{2} )^{2} \)
$37$ \( 1 - 2 T^{2} + 1369 T^{4} \)
$41$ \( 1 - 50 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 6 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 12 T + 59 T^{2} )^{2} \)
$61$ \( 1 - 50 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{2} \)
$71$ \( 1 - 110 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 - 73 T^{2} )^{2} \)
$79$ \( 1 + 130 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 + 12 T + 83 T^{2} )^{2} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )( 1 + 10 T + 97 T^{2} ) \)
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