Properties

Label 68.1.f.a
Level 68
Weight 1
Character orbit 68.f
Analytic conductor 0.034
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM disc. -4
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 68.f (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.033936420859\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.19652.1
Artin image size \(32\)
Artin image $C_4\wr C_2$
Artin field Galois closure of 8.0.1257728.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q\) \( + i q^{2} \) \(- q^{4}\) \( + ( -1 - i ) q^{5} \) \( -i q^{8} \) \( + i q^{9} \) \(+O(q^{10})\) \( q\) \( + i q^{2} \) \(- q^{4}\) \( + ( -1 - i ) q^{5} \) \( -i q^{8} \) \( + i q^{9} \) \( + ( 1 - i ) q^{10} \) \(+ q^{16}\) \(- q^{17}\) \(- q^{18}\) \( + ( 1 + i ) q^{20} \) \( + i q^{25} \) \( + ( 1 + i ) q^{29} \) \( + i q^{32} \) \( -i q^{34} \) \( -i q^{36} \) \( + ( -1 - i ) q^{37} \) \( + ( -1 + i ) q^{40} \) \( + ( 1 - i ) q^{41} \) \( + ( 1 - i ) q^{45} \) \( -i q^{49} \) \(- q^{50}\) \( + ( -1 + i ) q^{58} \) \( + ( -1 + i ) q^{61} \) \(- q^{64}\) \(+ q^{68}\) \(+ q^{72}\) \( + ( 1 + i ) q^{73} \) \( + ( 1 - i ) q^{74} \) \( + ( -1 - i ) q^{80} \) \(- q^{81}\) \( + ( 1 + i ) q^{82} \) \( + ( 1 + i ) q^{85} \) \( + ( 1 + i ) q^{90} \) \( + ( -1 - i ) q^{97} \) \(+ q^{98}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 2q^{20} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 2q^{40} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 2q^{58} \) \(\mathstrut -\mathstrut 2q^{61} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut 2q^{68} \) \(\mathstrut +\mathstrut 2q^{72} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 2q^{74} \) \(\mathstrut -\mathstrut 2q^{80} \) \(\mathstrut -\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 2q^{82} \) \(\mathstrut +\mathstrut 2q^{85} \) \(\mathstrut +\mathstrut 2q^{90} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.00000i
1.00000i
1.00000i 0 −1.00000 −1.00000 1.00000i 0 0 1.00000i 1.00000i 1.00000 1.00000i
55.1 1.00000i 0 −1.00000 −1.00000 + 1.00000i 0 0 1.00000i 1.00000i 1.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
17.c Even 1 yes
68.f Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(68, [\chi])\).