# Properties

 Label 1183.2.c.g Level $1183$ Weight $2$ Character orbit 1183.c Analytic conductor $9.446$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.11667456256.1 Defining polynomial: $$x^{8} + 13 x^{6} + 44 x^{4} + 21 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{4} q^{3} + ( -1 + \beta_{2} + \beta_{3} ) q^{4} + ( -\beta_{1} + 2 \beta_{5} ) q^{5} + ( \beta_{1} + \beta_{5} ) q^{6} -\beta_{5} q^{7} + ( -3 \beta_{1} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{8} + ( 2 + \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{4} q^{3} + ( -1 + \beta_{2} + \beta_{3} ) q^{4} + ( -\beta_{1} + 2 \beta_{5} ) q^{5} + ( \beta_{1} + \beta_{5} ) q^{6} -\beta_{5} q^{7} + ( -3 \beta_{1} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{8} + ( 2 + \beta_{3} - \beta_{4} ) q^{9} + ( 3 + \beta_{2} - \beta_{3} ) q^{10} + \beta_{6} q^{11} + ( -3 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{12} -\beta_{2} q^{14} + ( -\beta_{1} - \beta_{5} - 2 \beta_{6} ) q^{15} + ( 4 - 4 \beta_{2} - \beta_{3} - \beta_{4} ) q^{16} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{17} + ( -2 \beta_{1} + \beta_{5} - \beta_{6} ) q^{18} + ( -3 \beta_{1} - 2 \beta_{6} + \beta_{7} ) q^{19} + ( 3 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{20} + \beta_{6} q^{21} + ( 1 - \beta_{2} ) q^{22} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{23} + ( -4 \beta_{1} - \beta_{6} + 2 \beta_{7} ) q^{24} + ( -2 - 3 \beta_{2} + \beta_{3} ) q^{25} + ( -7 + \beta_{2} + 3 \beta_{4} ) q^{27} + ( \beta_{1} + \beta_{5} - \beta_{7} ) q^{28} + ( \beta_{3} + \beta_{4} ) q^{29} + ( 1 - \beta_{3} ) q^{30} + ( \beta_{5} - \beta_{7} ) q^{31} + ( 4 \beta_{1} + 7 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{32} + ( -5 \beta_{5} - \beta_{6} - \beta_{7} ) q^{33} + ( 4 \beta_{1} + \beta_{6} - \beta_{7} ) q^{34} + ( 2 + \beta_{2} ) q^{35} + ( 9 - 2 \beta_{4} ) q^{36} + ( -\beta_{1} - 2 \beta_{5} + \beta_{6} ) q^{37} + ( 5 - 4 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{38} + ( -4 + \beta_{3} - \beta_{4} ) q^{40} + ( -2 \beta_{1} + 6 \beta_{5} + \beta_{7} ) q^{41} + ( 1 - \beta_{2} ) q^{42} + ( -1 - \beta_{2} ) q^{43} + ( 2 \beta_{1} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{44} + ( 2 \beta_{1} + 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{45} + ( -\beta_{1} + 6 \beta_{5} - \beta_{6} - \beta_{7} ) q^{46} + ( -3 \beta_{1} + \beta_{5} - \beta_{6} ) q^{47} + ( 1 - 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{48} - q^{49} + ( -2 \beta_{1} + 11 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{50} + ( -2 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{51} + ( 1 - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{53} + ( -5 \beta_{1} + \beta_{7} ) q^{54} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{55} + ( -1 + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{56} + ( -4 \beta_{1} + 5 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{57} + ( -2 \beta_{1} + 3 \beta_{5} - \beta_{6} ) q^{58} + ( -2 \beta_{1} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{59} + ( 2 \beta_{1} - 4 \beta_{5} - 3 \beta_{6} ) q^{60} + ( 2 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{61} + ( 2 + 4 \beta_{2} + \beta_{4} ) q^{62} + ( -2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{63} + ( -1 + 10 \beta_{2} + 2 \beta_{3} ) q^{64} + ( 1 - \beta_{2} + \beta_{4} ) q^{66} + ( 6 \beta_{1} + \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{67} + ( -7 + 4 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{68} + ( 4 + 2 \beta_{3} + 2 \beta_{4} ) q^{69} + ( \beta_{1} - 3 \beta_{5} + \beta_{7} ) q^{70} + ( 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( 3 \beta_{1} - 2 \beta_{6} ) q^{72} + ( 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{73} + ( 4 - 4 \beta_{2} - \beta_{3} ) q^{74} + ( 1 - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{75} + ( 11 \beta_{1} + 5 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{76} -\beta_{4} q^{77} + ( -6 - \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{79} + ( -2 \beta_{1} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{80} + ( 8 + \beta_{2} - 7 \beta_{4} ) q^{81} + ( 4 + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{82} + ( -4 \beta_{1} + \beta_{5} + \beta_{7} ) q^{83} + ( 2 \beta_{1} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{84} + ( -2 \beta_{1} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{85} + ( 3 \beta_{5} - \beta_{7} ) q^{86} + ( 3 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{87} + ( 4 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{88} + ( \beta_{1} + 2 \beta_{6} + \beta_{7} ) q^{89} + ( -7 - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{90} + ( 4 + 7 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{92} + ( \beta_{1} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{93} + ( 8 - \beta_{2} - 3 \beta_{3} ) q^{94} + ( -5 - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{95} + ( 6 \beta_{1} + 13 \beta_{5} - \beta_{7} ) q^{96} + ( -2 \beta_{1} + \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{97} -\beta_{1} q^{98} + ( \beta_{1} + 7 \beta_{5} + 6 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{3} - 10q^{4} + 14q^{9} + O(q^{10})$$ $$8q + 2q^{3} - 10q^{4} + 14q^{9} + 22q^{10} - 24q^{12} + 2q^{14} + 38q^{16} + 8q^{17} + 10q^{22} + 4q^{23} - 10q^{25} - 52q^{27} + 2q^{29} + 8q^{30} + 14q^{35} + 68q^{36} + 46q^{38} - 34q^{40} + 10q^{42} - 6q^{43} + 22q^{48} - 8q^{49} - 14q^{51} + 4q^{53} - 6q^{55} - 12q^{56} + 16q^{61} + 10q^{62} - 28q^{64} + 12q^{66} - 66q^{68} + 36q^{69} + 40q^{74} + 14q^{75} - 2q^{77} - 52q^{79} + 48q^{81} + 28q^{82} + 26q^{87} - 6q^{88} - 52q^{90} + 24q^{92} + 66q^{94} - 42q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 13 x^{6} + 44 x^{4} + 21 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 10 \nu^{4} + 22 \nu^{2} + 3$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 10 \nu^{4} - 14 \nu^{2} + 21$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{6} - 38 \nu^{4} - 122 \nu^{2} - 33$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} - 38 \nu^{5} - 122 \nu^{3} - 41 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{7} - 66 \nu^{5} - 230 \nu^{3} - 127 \nu$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$-\nu^{7} - 13 \nu^{5} - 43 \nu^{3} - 14 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} - \beta_{5} - 7 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{4} - 7 \beta_{3} - 10 \beta_{2} + 18$$ $$\nu^{5}$$ $$=$$ $$-10 \beta_{7} + 7 \beta_{6} + 15 \beta_{5} + 48 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$10 \beta_{4} + 48 \beta_{3} + 86 \beta_{2} - 117$$ $$\nu^{7}$$ $$=$$ $$86 \beta_{7} - 48 \beta_{6} - 152 \beta_{5} - 337 \beta_{1}$$

## Character values

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

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 2.74108i − 2.22001i − 0.710287i − 0.231361i 0.231361i 0.710287i 2.22001i 2.74108i
2.74108i 1.36482 −5.51353 0.741082i 3.74108i 1.00000i 9.63087i −1.13727 2.03137
337.2 2.22001i 0.549551 −2.92843 4.22001i 1.22001i 1.00000i 2.06113i −2.69799 9.36845
337.3 0.710287i 2.40788 1.49549 1.28971i 1.71029i 1.00000i 2.48280i 2.79790 −0.916066
337.4 0.231361i −3.32225 1.94647 2.23136i 0.768639i 1.00000i 0.913059i 8.03736 0.516249
337.5 0.231361i −3.32225 1.94647 2.23136i 0.768639i 1.00000i 0.913059i 8.03736 0.516249
337.6 0.710287i 2.40788 1.49549 1.28971i 1.71029i 1.00000i 2.48280i 2.79790 −0.916066
337.7 2.22001i 0.549551 −2.92843 4.22001i 1.22001i 1.00000i 2.06113i −2.69799 9.36845
337.8 2.74108i 1.36482 −5.51353 0.741082i 3.74108i 1.00000i 9.63087i −1.13727 2.03137
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.c.g 8
13.b even 2 1 inner 1183.2.c.g 8
13.d odd 4 1 1183.2.a.k 4
13.d odd 4 1 1183.2.a.l 4
13.f odd 12 2 91.2.f.c 8
39.k even 12 2 819.2.o.h 8
52.l even 12 2 1456.2.s.q 8
91.i even 4 1 8281.2.a.bp 4
91.i even 4 1 8281.2.a.bt 4
91.w even 12 2 637.2.g.j 8
91.x odd 12 2 637.2.h.h 8
91.ba even 12 2 637.2.h.i 8
91.bc even 12 2 637.2.f.i 8
91.bd odd 12 2 637.2.g.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 13.f odd 12 2
637.2.f.i 8 91.bc even 12 2
637.2.g.j 8 91.w even 12 2
637.2.g.k 8 91.bd odd 12 2
637.2.h.h 8 91.x odd 12 2
637.2.h.i 8 91.ba even 12 2
819.2.o.h 8 39.k even 12 2
1183.2.a.k 4 13.d odd 4 1
1183.2.a.l 4 13.d odd 4 1
1183.2.c.g 8 1.a even 1 1 trivial
1183.2.c.g 8 13.b even 2 1 inner
1456.2.s.q 8 52.l even 12 2
8281.2.a.bp 4 91.i even 4 1
8281.2.a.bt 4 91.i even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 13 T_{2}^{6} + 44 T_{2}^{4} + 21 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 21 T^{2} + 44 T^{4} + 13 T^{6} + T^{8}$$
$3$ $$( -6 + 16 T - 9 T^{2} - T^{3} + T^{4} )^{2}$$
$5$ $$81 + 217 T^{2} + 140 T^{4} + 25 T^{6} + T^{8}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$36 + 148 T^{2} + 101 T^{4} + 19 T^{6} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$( -53 + 60 T - 12 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$19$ $$250000 + 55000 T^{2} + 4025 T^{4} + 111 T^{6} + T^{8}$$
$23$ $$( 72 + 56 T - 38 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$29$ $$( -5 - 21 T - 22 T^{2} - T^{3} + T^{4} )^{2}$$
$31$ $$2916 + 2080 T^{2} + 485 T^{4} + 42 T^{6} + T^{8}$$
$37$ $$256 + 2480 T^{2} + 1137 T^{4} + 66 T^{6} + T^{8}$$
$41$ $$318096 + 219148 T^{2} + 11129 T^{4} + 186 T^{6} + T^{8}$$
$43$ $$( -2 - 8 T - 3 T^{2} + 3 T^{3} + T^{4} )^{2}$$
$47$ $$10000 + 10344 T^{2} + 2753 T^{4} + 106 T^{6} + T^{8}$$
$53$ $$( -1389 + 890 T - 140 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$59$ $$498436 + 193776 T^{2} + 13317 T^{4} + 218 T^{6} + T^{8}$$
$61$ $$( -100 + 176 T - 15 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$67$ $$121220100 + 5048176 T^{2} + 74149 T^{4} + 458 T^{6} + T^{8}$$
$71$ $$4129024 + 441920 T^{2} + 16128 T^{4} + 228 T^{6} + T^{8}$$
$73$ $$3139984 + 429480 T^{2} + 19113 T^{4} + 302 T^{6} + T^{8}$$
$79$ $$( -7680 - 1024 T + 134 T^{2} + 26 T^{3} + T^{4} )^{2}$$
$83$ $$181476 + 112720 T^{2} + 8557 T^{4} + 194 T^{6} + T^{8}$$
$89$ $$11664 + 18520 T^{2} + 5193 T^{4} + 143 T^{6} + T^{8}$$
$97$ $$246238864 + 10018344 T^{2} + 128417 T^{4} + 619 T^{6} + T^{8}$$