Properties

Label 68.2.h.a
Level $68$
Weight $2$
Character orbit 68.h
Analytic conductor $0.543$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 68.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.542982733745\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{3} + ( - \zeta_{8} + 1) q^{5} + (\zeta_{8}^{2} + \zeta_{8}) q^{7} + ( - \zeta_{8}^{2} - \zeta_{8} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{3} + ( - \zeta_{8} + 1) q^{5} + (\zeta_{8}^{2} + \zeta_{8}) q^{7} + ( - \zeta_{8}^{2} - \zeta_{8} - 1) q^{9} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} - 2) q^{11} + (2 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 2 \zeta_{8}) q^{13} + ( - 2 \zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{15} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 3) q^{17} + (4 \zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{19} + (\zeta_{8}^{3} + \zeta_{8}) q^{21} + ( - 2 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 3 \zeta_{8} + 2) q^{23} + (\zeta_{8}^{2} + 3 \zeta_{8} + 1) q^{25} + ( - 3 \zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8} - 3) q^{27} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 3 \zeta_{8} + 3) q^{29} + (\zeta_{8}^{3} - \zeta_{8}^{2} + 6 \zeta_{8} + 6) q^{31} + (3 \zeta_{8}^{3} - 3 \zeta_{8} + 2) q^{33} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{35} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + \zeta_{8} + 1) q^{37} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 6 \zeta_{8} + 6) q^{39} + (\zeta_{8}^{3} + 4 \zeta_{8}^{2} + 4 \zeta_{8} + 1) q^{41} + ( - 3 \zeta_{8}^{2} - 3) q^{43} + (\zeta_{8}^{3} - 1) q^{45} + ( - 4 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 4 \zeta_{8}) q^{47} + ( - 5 \zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{49} + (\zeta_{8}^{3} - 5 \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{51} + (4 \zeta_{8}^{3} + 7 \zeta_{8}^{2} - 7) q^{53} + (3 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 3 \zeta_{8}) q^{55} + (\zeta_{8}^{3} + 3 \zeta_{8}^{2} - 3 \zeta_{8} - 1) q^{57} + ( - 7 \zeta_{8}^{2} + 4 \zeta_{8} - 7) q^{59} + (\zeta_{8}^{3} + 1) q^{61} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - \zeta_{8} + 1) q^{63} + (6 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 2 \zeta_{8} + 2) q^{65} + (8 \zeta_{8}^{3} - 8 \zeta_{8}) q^{67} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8} - 6) q^{69} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + 4 \zeta_{8} + 4) q^{71} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 3 \zeta_{8} - 3) q^{73} + (2 \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} + 2) q^{75} + ( - \zeta_{8}^{2} - 4 \zeta_{8} - 1) q^{77} + (2 \zeta_{8}^{3} + 7 \zeta_{8}^{2} - 7 \zeta_{8} - 2) q^{79} + (5 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 5 \zeta_{8}) q^{81} + ( - 4 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3) q^{83} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + \zeta_{8} - 5) q^{85} + ( - 6 \zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{87} + ( - 2 \zeta_{8}^{3} + 8 \zeta_{8}^{2} - 2 \zeta_{8}) q^{89} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2 \zeta_{8} + 2) q^{91} + (7 \zeta_{8}^{2} - 2 \zeta_{8} + 7) q^{93} + (3 \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} + 3) q^{95} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8}^{2} - 3 \zeta_{8} + 3) q^{97} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 3 \zeta_{8} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} - 4 q^{9} - 8 q^{11} - 4 q^{15} - 12 q^{17} - 4 q^{19} + 8 q^{23} + 4 q^{25} - 12 q^{27} + 12 q^{29} + 24 q^{31} + 8 q^{33} + 4 q^{37} + 24 q^{39} + 4 q^{41} - 12 q^{43} - 4 q^{45} - 4 q^{49} - 8 q^{51} - 28 q^{53} - 4 q^{57} - 28 q^{59} + 4 q^{61} + 4 q^{63} + 8 q^{65} - 24 q^{69} + 16 q^{71} - 12 q^{73} + 8 q^{75} - 4 q^{77} - 8 q^{79} + 12 q^{83} - 20 q^{85} - 4 q^{87} + 8 q^{91} + 28 q^{93} + 12 q^{95} + 12 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(35\) \(37\)
\(\chi(n)\) \(1\) \(\zeta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 0.707107 + 0.292893i 0 0.292893 0.707107i 0 0.707107 + 1.70711i 0 −1.70711 1.70711i 0
25.1 0 −0.707107 + 1.70711i 0 1.70711 + 0.707107i 0 −0.707107 + 0.292893i 0 −0.292893 0.292893i 0
49.1 0 −0.707107 1.70711i 0 1.70711 0.707107i 0 −0.707107 0.292893i 0 −0.292893 + 0.292893i 0
53.1 0 0.707107 0.292893i 0 0.292893 + 0.707107i 0 0.707107 1.70711i 0 −1.70711 + 1.70711i 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.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.2.h.a 4
3.b odd 2 1 612.2.w.a 4
4.b odd 2 1 272.2.v.c 4
17.b even 2 1 1156.2.h.b 4
17.c even 4 1 1156.2.h.a 4
17.c even 4 1 1156.2.h.c 4
17.d even 8 1 inner 68.2.h.a 4
17.d even 8 1 1156.2.h.a 4
17.d even 8 1 1156.2.h.b 4
17.d even 8 1 1156.2.h.c 4
17.e odd 16 2 1156.2.a.g 4
17.e odd 16 2 1156.2.b.d 4
17.e odd 16 4 1156.2.e.f 8
51.g odd 8 1 612.2.w.a 4
68.g odd 8 1 272.2.v.c 4
68.i even 16 2 4624.2.a.bl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.h.a 4 1.a even 1 1 trivial
68.2.h.a 4 17.d even 8 1 inner
272.2.v.c 4 4.b odd 2 1
272.2.v.c 4 68.g odd 8 1
612.2.w.a 4 3.b odd 2 1
612.2.w.a 4 51.g odd 8 1
1156.2.a.g 4 17.e odd 16 2
1156.2.b.d 4 17.e odd 16 2
1156.2.e.f 8 17.e odd 16 4
1156.2.h.a 4 17.c even 4 1
1156.2.h.a 4 17.d even 8 1
1156.2.h.b 4 17.b even 2 1
1156.2.h.b 4 17.d even 8 1
1156.2.h.c 4 17.c even 4 1
1156.2.h.c 4 17.d even 8 1
4624.2.a.bl 4 68.i even 16 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$11$ \( T^{4} + 8 T^{3} + 34 T^{2} + 84 T + 98 \) Copy content Toggle raw display
$13$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T + 17)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + 8 T^{2} - 56 T + 196 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + 66 T^{2} - 140 T + 98 \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{3} + 86 T^{2} + \cdots + 578 \) Copy content Toggle raw display
$31$ \( T^{4} - 24 T^{3} + 242 T^{2} + \cdots + 4418 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + 22 T^{2} + \cdots + 1058 \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} + 54 T^{2} + 140 T + 98 \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$53$ \( T^{4} + 28 T^{3} + 392 T^{2} + \cdots + 6724 \) Copy content Toggle raw display
$59$ \( T^{4} + 28 T^{3} + 392 T^{2} + \cdots + 6724 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$67$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 16 T^{3} + 82 T^{2} - 84 T + 98 \) Copy content Toggle raw display
$73$ \( T^{4} + 12 T^{3} + 38 T^{2} + \cdots + 1922 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + 66 T^{2} + \cdots + 10658 \) Copy content Toggle raw display
$83$ \( T^{4} - 12 T^{3} + 72 T^{2} - 24 T + 4 \) Copy content Toggle raw display
$89$ \( T^{4} + 144T^{2} + 3136 \) Copy content Toggle raw display
$97$ \( T^{4} - 12 T^{3} + 198 T^{2} + \cdots + 162 \) Copy content Toggle raw display
show more
show less