Properties

Label 28.2.d.a
Level 28
Weight 2
Character orbit 28.d
Analytic conductor 0.224
Analytic rank 0
Dimension 2
CM disc. -7
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 28.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.22358112566\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-7})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta q^{2} \) \( + ( -2 + \beta ) q^{4} \) \( + ( -1 + 2 \beta ) q^{7} \) \( + ( 2 + \beta ) q^{8} \) \( -3 q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta q^{2} \) \( + ( -2 + \beta ) q^{4} \) \( + ( -1 + 2 \beta ) q^{7} \) \( + ( 2 + \beta ) q^{8} \) \( -3 q^{9} \) \( + ( 2 - 4 \beta ) q^{11} \) \( + ( 4 - \beta ) q^{14} \) \( + ( 2 - 3 \beta ) q^{16} \) \( + 3 \beta q^{18} \) \( + ( -8 + 2 \beta ) q^{22} \) \( + ( -2 + 4 \beta ) q^{23} \) \( + 5 q^{25} \) \( + ( -2 - 3 \beta ) q^{28} \) \( -2 q^{29} \) \( + ( -6 + \beta ) q^{32} \) \( + ( 6 - 3 \beta ) q^{36} \) \( + 6 q^{37} \) \( + ( 2 - 4 \beta ) q^{43} \) \( + ( 4 + 6 \beta ) q^{44} \) \( + ( 8 - 2 \beta ) q^{46} \) \( -7 q^{49} \) \( -5 \beta q^{50} \) \( -10 q^{53} \) \( + ( -6 + 5 \beta ) q^{56} \) \( + 2 \beta q^{58} \) \( + ( 3 - 6 \beta ) q^{63} \) \( + ( 2 + 5 \beta ) q^{64} \) \( + ( -6 + 12 \beta ) q^{67} \) \( + ( -2 + 4 \beta ) q^{71} \) \( + ( -6 - 3 \beta ) q^{72} \) \( -6 \beta q^{74} \) \( + 14 q^{77} \) \( + ( 6 - 12 \beta ) q^{79} \) \( + 9 q^{81} \) \( + ( -8 + 2 \beta ) q^{86} \) \( + ( 12 - 10 \beta ) q^{88} \) \( + ( -4 - 6 \beta ) q^{92} \) \( + 7 \beta q^{98} \) \( + ( -6 + 12 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut 3q^{4} \) \(\mathstrut +\mathstrut 5q^{8} \) \(\mathstrut -\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut 3q^{4} \) \(\mathstrut +\mathstrut 5q^{8} \) \(\mathstrut -\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut 7q^{14} \) \(\mathstrut +\mathstrut q^{16} \) \(\mathstrut +\mathstrut 3q^{18} \) \(\mathstrut -\mathstrut 14q^{22} \) \(\mathstrut +\mathstrut 10q^{25} \) \(\mathstrut -\mathstrut 7q^{28} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 11q^{32} \) \(\mathstrut +\mathstrut 9q^{36} \) \(\mathstrut +\mathstrut 12q^{37} \) \(\mathstrut +\mathstrut 14q^{44} \) \(\mathstrut +\mathstrut 14q^{46} \) \(\mathstrut -\mathstrut 14q^{49} \) \(\mathstrut -\mathstrut 5q^{50} \) \(\mathstrut -\mathstrut 20q^{53} \) \(\mathstrut -\mathstrut 7q^{56} \) \(\mathstrut +\mathstrut 2q^{58} \) \(\mathstrut +\mathstrut 9q^{64} \) \(\mathstrut -\mathstrut 15q^{72} \) \(\mathstrut -\mathstrut 6q^{74} \) \(\mathstrut +\mathstrut 28q^{77} \) \(\mathstrut +\mathstrut 18q^{81} \) \(\mathstrut -\mathstrut 14q^{86} \) \(\mathstrut +\mathstrut 14q^{88} \) \(\mathstrut -\mathstrut 14q^{92} \) \(\mathstrut +\mathstrut 7q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
0.500000 + 1.32288i
0.500000 1.32288i
−0.500000 1.32288i 0 −1.50000 + 1.32288i 0 0 2.64575i 2.50000 + 1.32288i −3.00000 0
27.2 −0.500000 + 1.32288i 0 −1.50000 1.32288i 0 0 2.64575i 2.50000 1.32288i −3.00000 0
\(n\): 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
7.b Odd 1 CM by \(\Q(\sqrt{-7}) \) yes
4.b Odd 1 yes
28.d Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(28, [\chi])\).