Properties

Label 68.1.d.a
Level 68
Weight 1
Character orbit 68.d
Self dual yes
Analytic conductor 0.034
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -4, -68, 17
Inner twists 4

Related objects

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.0339364208590\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(i, \sqrt{17})\)
Artin image $D_4$
Artin field Galois closure of 4.0.272.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{8} - q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{8} - q^{9} - 2q^{13} + q^{16} + q^{17} + q^{18} + q^{25} + 2q^{26} - q^{32} - q^{34} - q^{36} - q^{49} - q^{50} - 2q^{52} + 2q^{53} + q^{64} + q^{68} + q^{72} + q^{81} - 2q^{89} + q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0
−1.00000 0 1.00000 0 0 0 −1.00000 −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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
17.b even 2 1 RM by \(\Q(\sqrt{17}) \)
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.1.d.a 1
3.b odd 2 1 612.1.e.a 1
4.b odd 2 1 CM 68.1.d.a 1
5.b even 2 1 1700.1.h.d 1
5.c odd 4 2 1700.1.d.b 2
7.b odd 2 1 3332.1.g.a 1
7.c even 3 2 3332.1.o.c 2
7.d odd 6 2 3332.1.o.d 2
8.b even 2 1 1088.1.g.a 1
8.d odd 2 1 1088.1.g.a 1
12.b even 2 1 612.1.e.a 1
17.b even 2 1 RM 68.1.d.a 1
17.c even 4 2 1156.1.c.a 1
17.d even 8 4 1156.1.f.a 2
17.e odd 16 8 1156.1.g.a 4
20.d odd 2 1 1700.1.h.d 1
20.e even 4 2 1700.1.d.b 2
28.d even 2 1 3332.1.g.a 1
28.f even 6 2 3332.1.o.d 2
28.g odd 6 2 3332.1.o.c 2
51.c odd 2 1 612.1.e.a 1
68.d odd 2 1 CM 68.1.d.a 1
68.f odd 4 2 1156.1.c.a 1
68.g odd 8 4 1156.1.f.a 2
68.i even 16 8 1156.1.g.a 4
85.c even 2 1 1700.1.h.d 1
85.g odd 4 2 1700.1.d.b 2
119.d odd 2 1 3332.1.g.a 1
119.h odd 6 2 3332.1.o.d 2
119.j even 6 2 3332.1.o.c 2
136.e odd 2 1 1088.1.g.a 1
136.h even 2 1 1088.1.g.a 1
204.h even 2 1 612.1.e.a 1
340.d odd 2 1 1700.1.h.d 1
340.r even 4 2 1700.1.d.b 2
476.e even 2 1 3332.1.g.a 1
476.o odd 6 2 3332.1.o.c 2
476.q even 6 2 3332.1.o.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a 1 1.a even 1 1 trivial
68.1.d.a 1 4.b odd 2 1 CM
68.1.d.a 1 17.b even 2 1 RM
68.1.d.a 1 68.d odd 2 1 CM
612.1.e.a 1 3.b odd 2 1
612.1.e.a 1 12.b even 2 1
612.1.e.a 1 51.c odd 2 1
612.1.e.a 1 204.h even 2 1
1088.1.g.a 1 8.b even 2 1
1088.1.g.a 1 8.d odd 2 1
1088.1.g.a 1 136.e odd 2 1
1088.1.g.a 1 136.h even 2 1
1156.1.c.a 1 17.c even 4 2
1156.1.c.a 1 68.f odd 4 2
1156.1.f.a 2 17.d even 8 4
1156.1.f.a 2 68.g odd 8 4
1156.1.g.a 4 17.e odd 16 8
1156.1.g.a 4 68.i even 16 8
1700.1.d.b 2 5.c odd 4 2
1700.1.d.b 2 20.e even 4 2
1700.1.d.b 2 85.g odd 4 2
1700.1.d.b 2 340.r even 4 2
1700.1.h.d 1 5.b even 2 1
1700.1.h.d 1 20.d odd 2 1
1700.1.h.d 1 85.c even 2 1
1700.1.h.d 1 340.d odd 2 1
3332.1.g.a 1 7.b odd 2 1
3332.1.g.a 1 28.d even 2 1
3332.1.g.a 1 119.d odd 2 1
3332.1.g.a 1 476.e even 2 1
3332.1.o.c 2 7.c even 3 2
3332.1.o.c 2 28.g odd 6 2
3332.1.o.c 2 119.j even 6 2
3332.1.o.c 2 476.o odd 6 2
3332.1.o.d 2 7.d odd 6 2
3332.1.o.d 2 28.f even 6 2
3332.1.o.d 2 119.h odd 6 2
3332.1.o.d 2 476.q even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(68, [\chi])\).

Hecke Characteristic Polynomials

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