# Properties

 Label 1183.1.t.a Level $1183$ Weight $1$ Character orbit 1183.t Analytic conductor $0.590$ Analytic rank $0$ Dimension $12$ Projective image $D_{7}$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1183.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.590393909945$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.17213603549184.1 Defining polynomial: $$x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.1655595487.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-25 \nu^{11} + 95 \nu^{9} - 361 \nu^{7} + 155 \nu^{5} - 30 \nu^{3} - 1563 \nu$$$$)/559$$ $$\beta_{3}$$ $$=$$ $$($$$$25 \nu^{10} - 95 \nu^{8} + 361 \nu^{6} - 155 \nu^{4} + 30 \nu^{2} + 1004$$$$)/559$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{10} - 20 \nu^{8} + 76 \nu^{6} - 139 \nu^{4} + 124 \nu^{2} - 24$$$$)/43$$ $$\beta_{5}$$ $$=$$ $$($$$$45 \nu^{10} - 171 \nu^{8} + 538 \nu^{6} - 279 \nu^{4} + 54 \nu^{2} + 242$$$$)/559$$ $$\beta_{6}$$ $$=$$ $$($$$$70 \nu^{11} - 266 \nu^{9} + 899 \nu^{7} - 434 \nu^{5} + 84 \nu^{3} + 1246 \nu$$$$)/559$$ $$\beta_{7}$$ $$=$$ $$($$$$114 \nu^{10} - 545 \nu^{8} + 2071 \nu^{6} - 2831 \nu^{4} + 3379 \nu^{2} - 95$$$$)/559$$ $$\beta_{8}$$ $$=$$ $$($$$$114 \nu^{11} - 545 \nu^{9} + 2071 \nu^{7} - 2831 \nu^{5} + 3379 \nu^{3} - 95 \nu$$$$)/559$$ $$\beta_{9}$$ $$=$$ $$($$$$-128 \nu^{10} + 710 \nu^{8} - 2698 \nu^{6} + 4483 \nu^{4} - 4402 \nu^{2} + 852$$$$)/559$$ $$\beta_{10}$$ $$=$$ $$($$$$-242 \nu^{11} + 1255 \nu^{9} - 4769 \nu^{7} + 7314 \nu^{5} - 7781 \nu^{3} + 1506 \nu$$$$)/559$$ $$\beta_{11}$$ $$=$$ $$($$$$-317 \nu^{11} + 1540 \nu^{9} - 5852 \nu^{7} + 8338 \nu^{5} - 9548 \nu^{3} + 1848 \nu$$$$)/559$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + 3 \beta_{8} + \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{9} + 2 \beta_{7} + 4 \beta_{4} - 2$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{11} - \beta_{10} + 9 \beta_{8} - 9 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-5 \beta_{5} + 9 \beta_{3} - 14$$ $$\nu^{7}$$ $$=$$ $$-5 \beta_{6} - 14 \beta_{2} - 28 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-28 \beta_{9} - 14 \beta_{7} - 19 \beta_{5} - 47 \beta_{4} + 28 \beta_{3} - 28$$ $$\nu^{9}$$ $$=$$ $$-47 \beta_{11} + 19 \beta_{10} - 89 \beta_{8} - 19 \beta_{6} - 47 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$-89 \beta_{9} - 42 \beta_{7} - 155 \beta_{4} + 42$$ $$\nu^{11}$$ $$=$$ $$-155 \beta_{11} + 66 \beta_{10} - 286 \beta_{8} + 286 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$-1$$ $$\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
699.1
 0.385418 − 0.222521i −1.56052 + 0.900969i −1.07992 + 0.623490i 1.07992 − 0.623490i 1.56052 − 0.900969i −0.385418 + 0.222521i 0.385418 + 0.222521i −1.56052 − 0.900969i −1.07992 − 0.623490i 1.07992 + 0.623490i 1.56052 + 0.900969i −0.385418 − 0.222521i
−1.56052 + 0.900969i 0 1.12349 1.94594i 0 0 0.866025 + 0.500000i 2.24698i −0.500000 + 0.866025i 0
699.2 −1.07992 + 0.623490i 0 0.277479 0.480608i 0 0 −0.866025 0.500000i 0.554958i −0.500000 + 0.866025i 0
699.3 −0.385418 + 0.222521i 0 −0.400969 + 0.694498i 0 0 0.866025 + 0.500000i 0.801938i −0.500000 + 0.866025i 0
699.4 0.385418 0.222521i 0 −0.400969 + 0.694498i 0 0 −0.866025 0.500000i 0.801938i −0.500000 + 0.866025i 0
699.5 1.07992 0.623490i 0 0.277479 0.480608i 0 0 0.866025 + 0.500000i 0.554958i −0.500000 + 0.866025i 0
699.6 1.56052 0.900969i 0 1.12349 1.94594i 0 0 −0.866025 0.500000i 2.24698i −0.500000 + 0.866025i 0
1161.1 −1.56052 0.900969i 0 1.12349 + 1.94594i 0 0 0.866025 0.500000i 2.24698i −0.500000 0.866025i 0
1161.2 −1.07992 0.623490i 0 0.277479 + 0.480608i 0 0 −0.866025 + 0.500000i 0.554958i −0.500000 0.866025i 0
1161.3 −0.385418 0.222521i 0 −0.400969 0.694498i 0 0 0.866025 0.500000i 0.801938i −0.500000 0.866025i 0
1161.4 0.385418 + 0.222521i 0 −0.400969 0.694498i 0 0 −0.866025 + 0.500000i 0.801938i −0.500000 0.866025i 0
1161.5 1.07992 + 0.623490i 0 0.277479 + 0.480608i 0 0 0.866025 0.500000i 0.554958i −0.500000 0.866025i 0
1161.6 1.56052 + 0.900969i 0 1.12349 + 1.94594i 0 0 −0.866025 + 0.500000i 2.24698i −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1161.6 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 $$\Q(\sqrt{-7})$$
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner
91.b odd 2 1 inner
91.n odd 6 1 inner
91.t odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.1.t.a 12
7.b odd 2 1 CM 1183.1.t.a 12
13.b even 2 1 inner 1183.1.t.a 12
13.c even 3 1 1183.1.b.a 6
13.c even 3 1 inner 1183.1.t.a 12
13.d odd 4 1 1183.1.n.a 6
13.d odd 4 1 1183.1.n.b 6
13.e even 6 1 1183.1.b.a 6
13.e even 6 1 inner 1183.1.t.a 12
13.f odd 12 1 1183.1.d.a 3
13.f odd 12 1 1183.1.d.b yes 3
13.f odd 12 1 1183.1.n.a 6
13.f odd 12 1 1183.1.n.b 6
91.b odd 2 1 inner 1183.1.t.a 12
91.i even 4 1 1183.1.n.a 6
91.i even 4 1 1183.1.n.b 6
91.n odd 6 1 1183.1.b.a 6
91.n odd 6 1 inner 1183.1.t.a 12
91.t odd 6 1 1183.1.b.a 6
91.t odd 6 1 inner 1183.1.t.a 12
91.bc even 12 1 1183.1.d.a 3
91.bc even 12 1 1183.1.d.b yes 3
91.bc even 12 1 1183.1.n.a 6
91.bc even 12 1 1183.1.n.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.1.b.a 6 13.c even 3 1
1183.1.b.a 6 13.e even 6 1
1183.1.b.a 6 91.n odd 6 1
1183.1.b.a 6 91.t odd 6 1
1183.1.d.a 3 13.f odd 12 1
1183.1.d.a 3 91.bc even 12 1
1183.1.d.b yes 3 13.f odd 12 1
1183.1.d.b yes 3 91.bc even 12 1
1183.1.n.a 6 13.d odd 4 1
1183.1.n.a 6 13.f odd 12 1
1183.1.n.a 6 91.i even 4 1
1183.1.n.a 6 91.bc even 12 1
1183.1.n.b 6 13.d odd 4 1
1183.1.n.b 6 13.f odd 12 1
1183.1.n.b 6 91.i even 4 1
1183.1.n.b 6 91.bc even 12 1
1183.1.t.a 12 1.a even 1 1 trivial
1183.1.t.a 12 7.b odd 2 1 CM
1183.1.t.a 12 13.b even 2 1 inner
1183.1.t.a 12 13.c even 3 1 inner
1183.1.t.a 12 13.e even 6 1 inner
1183.1.t.a 12 91.b odd 2 1 inner
1183.1.t.a 12 91.n odd 6 1 inner
1183.1.t.a 12 91.t odd 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$( 1 - T^{2} + T^{4} )^{3}$$
$11$ $$1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12}$$
$13$ $$T^{12}$$
$17$ $$T^{12}$$
$19$ $$T^{12}$$
$23$ $$( 1 + 2 T + 5 T^{2} + 3 T^{4} + T^{5} + T^{6} )^{2}$$
$29$ $$( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2}$$
$31$ $$T^{12}$$
$37$ $$1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12}$$
$41$ $$T^{12}$$
$43$ $$( 1 + 2 T + 5 T^{2} + 3 T^{4} + T^{5} + T^{6} )^{2}$$
$47$ $$T^{12}$$
$53$ $$( -1 - 2 T + T^{2} + T^{3} )^{4}$$
$59$ $$T^{12}$$
$61$ $$T^{12}$$
$67$ $$1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12}$$
$71$ $$1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12}$$
$73$ $$T^{12}$$
$79$ $$( -1 - 2 T + T^{2} + T^{3} )^{4}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$T^{12}$$