Properties

Label 68.2.b.a
Level $68$
Weight $2$
Character orbit 68.b
Analytic conductor $0.543$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 68.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.542982733745\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 \beta q^{5} - 3 \beta q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 2 \beta q^{5} - 3 \beta q^{7} + q^{9} - \beta q^{11} - 4 q^{13} - 4 q^{15} + ( - 2 \beta + 3) q^{17} - 4 q^{19} + 6 q^{21} - \beta q^{23} - 3 q^{25} + 4 \beta q^{27} + 2 \beta q^{29} + 3 \beta q^{31} + 2 q^{33} + 12 q^{35} - 6 \beta q^{37} - 4 \beta q^{39} + 8 \beta q^{41} + 8 q^{43} + 2 \beta q^{45} - 12 q^{47} - 11 q^{49} + (3 \beta + 4) q^{51} - 6 q^{53} + 4 q^{55} - 4 \beta q^{57} - 6 \beta q^{61} - 3 \beta q^{63} - 8 \beta q^{65} - 4 q^{67} + 2 q^{69} + 5 \beta q^{71} - 3 \beta q^{75} - 6 q^{77} + 3 \beta q^{79} - 5 q^{81} + (6 \beta + 8) q^{85} - 4 q^{87} + 12 q^{89} + 12 \beta q^{91} - 6 q^{93} - 8 \beta q^{95} - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{9} - 8 q^{13} - 8 q^{15} + 6 q^{17} - 8 q^{19} + 12 q^{21} - 6 q^{25} + 4 q^{33} + 24 q^{35} + 16 q^{43} - 24 q^{47} - 22 q^{49} + 8 q^{51} - 12 q^{53} + 8 q^{55} - 8 q^{67} + 4 q^{69} - 12 q^{77} - 10 q^{81} + 16 q^{85} - 8 q^{87} + 24 q^{89} - 12 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
1.41421i
1.41421i
0 1.41421i 0 2.82843i 0 4.24264i 0 1.00000 0
33.2 0 1.41421i 0 2.82843i 0 4.24264i 0 1.00000 0
\(n\): 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 68.2.b.a 2
3.b odd 2 1 612.2.b.a 2
4.b odd 2 1 272.2.b.c 2
5.b even 2 1 1700.2.c.a 2
5.c odd 4 2 1700.2.g.a 4
7.b odd 2 1 3332.2.b.a 2
8.b even 2 1 1088.2.b.e 2
8.d odd 2 1 1088.2.b.f 2
12.b even 2 1 2448.2.c.d 2
17.b even 2 1 inner 68.2.b.a 2
17.c even 4 2 1156.2.a.c 2
17.d even 8 2 1156.2.e.a 2
17.d even 8 2 1156.2.e.b 2
17.e odd 16 8 1156.2.h.d 8
51.c odd 2 1 612.2.b.a 2
68.d odd 2 1 272.2.b.c 2
68.f odd 4 2 4624.2.a.n 2
85.c even 2 1 1700.2.c.a 2
85.g odd 4 2 1700.2.g.a 4
119.d odd 2 1 3332.2.b.a 2
136.e odd 2 1 1088.2.b.f 2
136.h even 2 1 1088.2.b.e 2
204.h even 2 1 2448.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.b.a 2 1.a even 1 1 trivial
68.2.b.a 2 17.b even 2 1 inner
272.2.b.c 2 4.b odd 2 1
272.2.b.c 2 68.d odd 2 1
612.2.b.a 2 3.b odd 2 1
612.2.b.a 2 51.c odd 2 1
1088.2.b.e 2 8.b even 2 1
1088.2.b.e 2 136.h even 2 1
1088.2.b.f 2 8.d odd 2 1
1088.2.b.f 2 136.e odd 2 1
1156.2.a.c 2 17.c even 4 2
1156.2.e.a 2 17.d even 8 2
1156.2.e.b 2 17.d even 8 2
1156.2.h.d 8 17.e odd 16 8
1700.2.c.a 2 5.b even 2 1
1700.2.c.a 2 85.c even 2 1
1700.2.g.a 4 5.c odd 4 2
1700.2.g.a 4 85.g odd 4 2
2448.2.c.d 2 12.b even 2 1
2448.2.c.d 2 204.h even 2 1
3332.2.b.a 2 7.b odd 2 1
3332.2.b.a 2 119.d odd 2 1
4624.2.a.n 2 68.f odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(68, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 8 \) Copy content Toggle raw display
$7$ \( T^{2} + 18 \) Copy content Toggle raw display
$11$ \( T^{2} + 2 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 17 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2 \) Copy content Toggle raw display
$29$ \( T^{2} + 8 \) Copy content Toggle raw display
$31$ \( T^{2} + 18 \) Copy content Toggle raw display
$37$ \( T^{2} + 72 \) Copy content Toggle raw display
$41$ \( T^{2} + 128 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( (T + 12)^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 72 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 50 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 18 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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