Properties

 Label 791.1.x.a Level $791$ Weight $1$ Character orbit 791.x Analytic conductor $0.395$ Analytic rank $0$ Dimension $12$ Projective image $D_{28}$ CM discriminant -7 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$791 = 7 \cdot 113$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 791.x (of order $$28$$, degree $$12$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.394760424993$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{28})$$ Defining polynomial: $$x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{28}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{28} - \cdots)$$

$q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{28}^{3} + \zeta_{28}^{7} ) q^{2} + ( -1 + \zeta_{28}^{6} - \zeta_{28}^{10} ) q^{4} + \zeta_{28}^{8} q^{7} + ( \zeta_{28}^{3} - \zeta_{28}^{7} - \zeta_{28}^{9} + \zeta_{28}^{13} ) q^{8} -\zeta_{28}^{9} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{28}^{3} + \zeta_{28}^{7} ) q^{2} + ( -1 + \zeta_{28}^{6} - \zeta_{28}^{10} ) q^{4} + \zeta_{28}^{8} q^{7} + ( \zeta_{28}^{3} - \zeta_{28}^{7} - \zeta_{28}^{9} + \zeta_{28}^{13} ) q^{8} -\zeta_{28}^{9} q^{9} + ( -\zeta_{28}^{10} - \zeta_{28}^{12} ) q^{11} + ( -\zeta_{28} - \zeta_{28}^{11} ) q^{14} + ( 1 + \zeta_{28}^{2} - \zeta_{28}^{6} + \zeta_{28}^{10} + \zeta_{28}^{12} ) q^{16} + ( \zeta_{28}^{2} + \zeta_{28}^{12} ) q^{18} + ( -\zeta_{28} + \zeta_{28}^{3} + \zeta_{28}^{5} + \zeta_{28}^{13} ) q^{22} + ( \zeta_{28}^{8} + \zeta_{28}^{11} ) q^{23} + \zeta_{28}^{5} q^{25} + ( -1 + \zeta_{28}^{4} - \zeta_{28}^{8} ) q^{28} + ( \zeta_{28}^{4} - \zeta_{28}^{13} ) q^{29} + ( \zeta_{28} - \zeta_{28}^{3} - \zeta_{28}^{5} + \zeta_{28}^{7} + \zeta_{28}^{9} - \zeta_{28}^{13} ) q^{32} + ( \zeta_{28} - \zeta_{28}^{5} + \zeta_{28}^{9} ) q^{36} + ( -\zeta_{28}^{6} + \zeta_{28}^{9} ) q^{37} + ( -\zeta_{28}^{4} - \zeta_{28}^{13} ) q^{43} + ( \zeta_{28}^{2} + \zeta_{28}^{4} - \zeta_{28}^{6} - \zeta_{28}^{8} + \zeta_{28}^{10} + \zeta_{28}^{12} ) q^{44} + ( 1 - \zeta_{28} - \zeta_{28}^{4} - \zeta_{28}^{11} ) q^{46} -\zeta_{28}^{2} q^{49} + ( -\zeta_{28}^{8} + \zeta_{28}^{12} ) q^{50} + ( -\zeta_{28}^{8} - \zeta_{28}^{12} ) q^{53} + ( \zeta_{28} + \zeta_{28}^{3} - \zeta_{28}^{7} + \zeta_{28}^{11} ) q^{56} + ( -\zeta_{28}^{2} + \zeta_{28}^{6} - \zeta_{28}^{7} + \zeta_{28}^{11} ) q^{58} + \zeta_{28}^{3} q^{63} + ( -1 - \zeta_{28}^{2} - \zeta_{28}^{4} + \zeta_{28}^{6} + \zeta_{28}^{8} - \zeta_{28}^{10} - \zeta_{28}^{12} ) q^{64} + ( \zeta_{28}^{2} + \zeta_{28}^{13} ) q^{67} + ( \zeta_{28}^{10} - \zeta_{28}^{11} ) q^{71} + ( -\zeta_{28}^{2} - \zeta_{28}^{4} + \zeta_{28}^{8} - \zeta_{28}^{12} ) q^{72} + ( -\zeta_{28}^{2} + \zeta_{28}^{9} - \zeta_{28}^{12} - \zeta_{28}^{13} ) q^{74} + ( \zeta_{28}^{4} + \zeta_{28}^{6} ) q^{77} + ( -\zeta_{28}^{3} + \zeta_{28}^{10} ) q^{79} -\zeta_{28}^{4} q^{81} + ( -\zeta_{28}^{2} + \zeta_{28}^{6} + \zeta_{28}^{7} - \zeta_{28}^{11} ) q^{86} + ( \zeta_{28} - \zeta_{28}^{3} - 2 \zeta_{28}^{5} - \zeta_{28}^{7} + \zeta_{28}^{9} + \zeta_{28}^{11} - \zeta_{28}^{13} ) q^{88} + ( -1 - \zeta_{28}^{3} + \zeta_{28}^{4} + \zeta_{28}^{7} - \zeta_{28}^{8} - \zeta_{28}^{11} ) q^{92} + ( \zeta_{28}^{5} - \zeta_{28}^{9} ) q^{98} + ( -\zeta_{28}^{5} - \zeta_{28}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 12q^{4} - 2q^{7} + O(q^{10})$$ $$12q - 12q^{4} - 2q^{7} + 12q^{16} - 2q^{23} - 12q^{28} - 2q^{29} - 2q^{37} + 2q^{43} + 14q^{46} - 2q^{49} + 4q^{53} - 12q^{64} + 2q^{67} + 2q^{71} + 2q^{79} + 2q^{81} - 12q^{92} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/791\mathbb{Z}\right)^\times$$.

 $$n$$ $$227$$ $$568$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{28}^{9}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
111.1
 −0.433884 − 0.900969i 0.781831 − 0.623490i 0.781831 + 0.623490i −0.781831 − 0.623490i −0.781831 + 0.623490i 0.433884 + 0.900969i −0.974928 − 0.222521i 0.433884 − 0.900969i −0.974928 + 0.222521i 0.974928 − 0.222521i −0.433884 + 0.900969i 0.974928 + 0.222521i
−0.974928 1.22252i 0 −0.321552 + 1.40881i 0 0 −0.900969 + 0.433884i 0.626980 0.301938i −0.781831 0.623490i 0
258.1 0.433884 + 1.90097i 0 −2.52446 + 1.21572i 0 0 0.623490 + 0.781831i −2.19064 2.74698i −0.974928 0.222521i 0
279.1 0.433884 1.90097i 0 −2.52446 1.21572i 0 0 0.623490 0.781831i −2.19064 + 2.74698i −0.974928 + 0.222521i 0
286.1 −0.433884 + 1.90097i 0 −2.52446 1.21572i 0 0 0.623490 0.781831i 2.19064 2.74698i 0.974928 0.222521i 0
307.1 −0.433884 1.90097i 0 −2.52446 + 1.21572i 0 0 0.623490 + 0.781831i 2.19064 + 2.74698i 0.974928 + 0.222521i 0
454.1 0.974928 + 1.22252i 0 −0.321552 + 1.40881i 0 0 −0.900969 + 0.433884i −0.626980 + 0.301938i 0.781831 + 0.623490i 0
573.1 0.781831 0.376510i 0 −0.153989 + 0.193096i 0 0 −0.222521 + 0.974928i −0.240787 + 1.05496i −0.433884 + 0.900969i 0
622.1 0.974928 1.22252i 0 −0.321552 1.40881i 0 0 −0.900969 0.433884i −0.626980 0.301938i 0.781831 0.623490i 0
664.1 0.781831 + 0.376510i 0 −0.153989 0.193096i 0 0 −0.222521 0.974928i −0.240787 1.05496i −0.433884 0.900969i 0
692.1 −0.781831 0.376510i 0 −0.153989 0.193096i 0 0 −0.222521 0.974928i 0.240787 + 1.05496i 0.433884 + 0.900969i 0
734.1 −0.974928 + 1.22252i 0 −0.321552 1.40881i 0 0 −0.900969 0.433884i 0.626980 + 0.301938i −0.781831 + 0.623490i 0
783.1 −0.781831 + 0.376510i 0 −0.153989 + 0.193096i 0 0 −0.222521 + 0.974928i 0.240787 1.05496i 0.433884 0.900969i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 783.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
113.h even 28 1 inner
791.x odd 28 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 791.1.x.a 12
7.b odd 2 1 CM 791.1.x.a 12
113.h even 28 1 inner 791.1.x.a 12
791.x odd 28 1 inner 791.1.x.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
791.1.x.a 12 1.a even 1 1 trivial
791.1.x.a 12 7.b odd 2 1 CM
791.1.x.a 12 113.h even 28 1 inner
791.1.x.a 12 791.x odd 28 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(791, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$49 - 49 T^{2} + 49 T^{4} + 35 T^{6} + 21 T^{8} + 7 T^{10} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
$11$ $$( 7 + 14 T + 7 T^{2} + T^{6} )^{2}$$
$13$ $$T^{12}$$
$17$ $$T^{12}$$
$19$ $$T^{12}$$
$23$ $$1 - 6 T + 32 T^{2} - 70 T^{3} + 61 T^{4} - 2 T^{5} - 8 T^{6} - 8 T^{7} - 4 T^{8} + 2 T^{10} + 2 T^{11} + T^{12}$$
$29$ $$1 + 8 T + 39 T^{2} + 84 T^{3} + 82 T^{4} + 40 T^{5} - T^{6} - 22 T^{7} - 4 T^{8} + 2 T^{10} + 2 T^{11} + T^{12}$$
$31$ $$T^{12}$$
$37$ $$1 + 8 T + 39 T^{2} + 84 T^{3} + 82 T^{4} + 40 T^{5} - T^{6} - 22 T^{7} - 4 T^{8} + 2 T^{10} + 2 T^{11} + T^{12}$$
$41$ $$T^{12}$$
$43$ $$1 - 8 T + 39 T^{2} - 84 T^{3} + 82 T^{4} - 40 T^{5} - T^{6} + 22 T^{7} - 4 T^{8} + 2 T^{10} - 2 T^{11} + T^{12}$$
$47$ $$T^{12}$$
$53$ $$( 1 + 3 T + 2 T^{2} - T^{3} + 4 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$59$ $$T^{12}$$
$61$ $$T^{12}$$
$67$ $$1 + 6 T + 11 T^{2} - 14 T^{3} + 26 T^{4} - 68 T^{5} + 55 T^{6} - 20 T^{7} + 24 T^{8} - 14 T^{9} + 2 T^{10} - 2 T^{11} + T^{12}$$
$71$ $$1 - 8 T + 32 T^{2} - 42 T^{3} + 26 T^{4} + 2 T^{5} + 34 T^{6} - 34 T^{7} + 17 T^{8} + 2 T^{10} - 2 T^{11} + T^{12}$$
$73$ $$T^{12}$$
$79$ $$64 - 64 T + 32 T^{2} - 16 T^{4} + 16 T^{5} - 8 T^{6} + 8 T^{7} - 4 T^{8} + 2 T^{10} - 2 T^{11} + T^{12}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$T^{12}$$