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 $2$

Related objects

Downloads

Learn more about

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: \(x^{2} + 2\)
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.

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

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

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