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

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
 0.5 + 1.32288i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$S_{2}^{\mathrm{new}}(28, [\chi])$$.