# Properties

 Label 483.2.h.c Level $483$ Weight $2$ Character orbit 483.h Analytic conductor $3.857$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$483 = 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 483.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.85677441763$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 17 x^{10} + 92 x^{8} + 180 x^{6} + 92 x^{4} + 17 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + \beta_{9} q^{3} + ( 3 + \beta_{7} ) q^{4} -\beta_{5} q^{5} + \beta_{4} q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -3 - 3 \beta_{2} + \beta_{7} ) q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + \beta_{9} q^{3} + ( 3 + \beta_{7} ) q^{4} -\beta_{5} q^{5} + \beta_{4} q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -3 - 3 \beta_{2} + \beta_{7} ) q^{8} - q^{9} + ( -\beta_{1} + \beta_{5} - \beta_{6} ) q^{10} + ( \beta_{3} - \beta_{10} ) q^{11} + ( \beta_{8} + 3 \beta_{9} ) q^{12} + ( -\beta_{4} + 2 \beta_{9} ) q^{13} + ( 2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{11} ) q^{14} + \beta_{3} q^{15} + ( 6 + \beta_{2} + 2 \beta_{7} ) q^{16} + ( -\beta_{1} + \beta_{5} - \beta_{6} ) q^{17} + \beta_{2} q^{18} + ( \beta_{1} - \beta_{5} ) q^{19} + ( -\beta_{1} - 4 \beta_{5} + \beta_{6} ) q^{20} + ( \beta_{5} - \beta_{10} ) q^{21} + ( -2 \beta_{3} - 2 \beta_{10} + \beta_{11} ) q^{22} + ( 1 - 2 \beta_{3} - \beta_{7} - \beta_{10} ) q^{23} + ( 3 \beta_{4} + \beta_{8} - 3 \beta_{9} ) q^{24} + ( -1 + \beta_{2} ) q^{25} + ( 2 \beta_{4} - \beta_{8} - 5 \beta_{9} ) q^{26} -\beta_{9} q^{27} + ( 3 \beta_{1} + 4 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{10} - \beta_{11} ) q^{28} + ( -1 + \beta_{2} + \beta_{7} ) q^{29} + ( -\beta_{3} + \beta_{11} ) q^{30} + ( -\beta_{4} - \beta_{8} - 3 \beta_{9} ) q^{31} + ( -5 - 4 \beta_{2} - \beta_{7} ) q^{32} + ( -\beta_{1} + \beta_{5} ) q^{33} + ( -\beta_{1} - 6 \beta_{5} + \beta_{6} ) q^{34} + ( 1 + \beta_{2} - \beta_{4} - \beta_{7} + 4 \beta_{9} ) q^{35} + ( -3 - \beta_{7} ) q^{36} + ( \beta_{3} + 2 \beta_{10} + \beta_{11} ) q^{37} + ( \beta_{1} + 2 \beta_{5} - \beta_{6} ) q^{38} + ( -2 - \beta_{2} ) q^{39} + ( -4 \beta_{1} + 5 \beta_{5} - 2 \beta_{6} ) q^{40} + 5 \beta_{9} q^{41} + ( \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{10} ) q^{42} + ( -2 \beta_{3} - \beta_{11} ) q^{43} + ( 3 \beta_{3} - 2 \beta_{11} ) q^{44} + \beta_{5} q^{45} + ( 3 + \beta_{2} + \beta_{3} - \beta_{7} - 2 \beta_{10} - 2 \beta_{11} ) q^{46} + ( -3 \beta_{4} + \beta_{8} + 2 \beta_{9} ) q^{47} + ( -\beta_{4} + 2 \beta_{8} + 6 \beta_{9} ) q^{48} + ( -1 - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{8} + 2 \beta_{9} ) q^{49} + ( -5 + \beta_{2} - \beta_{7} ) q^{50} + ( -\beta_{3} + \beta_{11} ) q^{51} + ( -5 \beta_{4} + \beta_{8} + 9 \beta_{9} ) q^{52} + ( -\beta_{3} + \beta_{10} + \beta_{11} ) q^{53} -\beta_{4} q^{54} + ( -2 \beta_{4} - \beta_{8} + 5 \beta_{9} ) q^{55} + ( 3 \beta_{1} - 5 \beta_{3} + 4 \beta_{5} + \beta_{6} + 2 \beta_{10} + 2 \beta_{11} ) q^{56} + ( \beta_{3} - \beta_{10} ) q^{57} + ( -8 - \beta_{2} ) q^{58} + ( \beta_{4} + 2 \beta_{8} - 5 \beta_{9} ) q^{59} + ( 4 \beta_{3} + 2 \beta_{10} - \beta_{11} ) q^{60} + ( -\beta_{1} - 2 \beta_{6} ) q^{61} + ( -5 \beta_{4} - 2 \beta_{8} - 2 \beta_{9} ) q^{62} + ( -\beta_{1} - \beta_{3} ) q^{63} + ( 11 + 5 \beta_{2} - \beta_{7} ) q^{64} + ( 3 \beta_{3} - \beta_{11} ) q^{65} + ( -\beta_{1} - 2 \beta_{5} + \beta_{6} ) q^{66} -3 \beta_{3} q^{67} + ( -6 \beta_{1} + 7 \beta_{5} - 4 \beta_{6} ) q^{68} + ( -\beta_{1} - 2 \beta_{5} - \beta_{8} + \beta_{9} ) q^{69} + ( -2 + \beta_{2} + 4 \beta_{4} - 2 \beta_{7} - \beta_{8} - 5 \beta_{9} ) q^{70} + ( 2 + \beta_{2} + 2 \beta_{7} ) q^{71} + ( 3 + 3 \beta_{2} - \beta_{7} ) q^{72} + ( \beta_{4} + 5 \beta_{8} + \beta_{9} ) q^{73} + ( 6 \beta_{3} + 6 \beta_{10} + \beta_{11} ) q^{74} + ( -\beta_{4} - \beta_{9} ) q^{75} + ( 2 \beta_{1} - 3 \beta_{5} + 2 \beta_{6} ) q^{76} + ( -5 - 2 \beta_{2} + \beta_{7} - \beta_{8} + 4 \beta_{9} ) q^{77} + ( 5 + 2 \beta_{2} + \beta_{7} ) q^{78} + ( -\beta_{3} + 6 \beta_{10} - \beta_{11} ) q^{79} + ( -\beta_{1} - 9 \beta_{5} + 3 \beta_{6} ) q^{80} + q^{81} + 5 \beta_{4} q^{82} + ( 3 \beta_{1} - \beta_{5} + 3 \beta_{6} ) q^{83} + ( \beta_{1} - \beta_{3} + 4 \beta_{5} - \beta_{6} - 2 \beta_{10} - \beta_{11} ) q^{84} + ( -5 - 4 \beta_{2} - \beta_{7} ) q^{85} + ( -3 \beta_{3} - 2 \beta_{10} - 2 \beta_{11} ) q^{86} + ( -\beta_{4} + \beta_{8} - \beta_{9} ) q^{87} + ( -9 \beta_{3} + \beta_{11} ) q^{88} + ( -3 \beta_{1} + \beta_{6} ) q^{89} + ( \beta_{1} - \beta_{5} + \beta_{6} ) q^{90} + ( -\beta_{1} + \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{91} + ( -4 - \beta_{2} - 9 \beta_{3} - 6 \beta_{10} + \beta_{11} ) q^{92} + ( 3 - \beta_{2} + \beta_{7} ) q^{93} + ( 4 \beta_{4} - 2 \beta_{8} - 18 \beta_{9} ) q^{94} + ( 5 + 2 \beta_{2} - \beta_{7} ) q^{95} + ( 4 \beta_{4} - \beta_{8} - 5 \beta_{9} ) q^{96} + ( 7 \beta_{1} + 3 \beta_{5} - \beta_{6} ) q^{97} + ( 10 + \beta_{2} - 2 \beta_{4} + 2 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} ) q^{98} + ( -\beta_{3} + \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 4q^{2} + 36q^{4} - 24q^{8} - 12q^{9} + O(q^{10})$$ $$12q + 4q^{2} + 36q^{4} - 24q^{8} - 12q^{9} + 68q^{16} - 4q^{18} + 12q^{23} - 16q^{25} - 16q^{29} - 44q^{32} + 8q^{35} - 36q^{36} - 20q^{39} + 32q^{46} - 4q^{49} - 64q^{50} - 92q^{58} + 112q^{64} - 28q^{70} + 20q^{71} + 24q^{72} - 52q^{77} + 52q^{78} + 12q^{81} - 44q^{85} - 44q^{92} + 40q^{93} + 52q^{95} + 116q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 17 x^{10} + 92 x^{8} + 180 x^{6} + 92 x^{4} + 17 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-19 \nu^{10} - 322 \nu^{8} - 1730 \nu^{6} - 3310 \nu^{4} - 1478 \nu^{2} - 141$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$\nu^{10} + 17 \nu^{8} + 92 \nu^{6} + 180 \nu^{4} + 91 \nu^{2} + 11$$ $$\beta_{3}$$ $$=$$ $$-\nu^{11} - 17 \nu^{9} - 92 \nu^{7} - 180 \nu^{5} - 92 \nu^{3} - 16 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$6 \nu^{11} + 98 \nu^{9} + 485 \nu^{7} + 730 \nu^{5} - 63 \nu^{3} - 56 \nu$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$33 \nu^{10} + 554 \nu^{8} + 2920 \nu^{6} + 5340 \nu^{4} + 1966 \nu^{2} + 187$$$$)/10$$ $$\beta_{6}$$ $$=$$ $$($$$$93 \nu^{10} + 1554 \nu^{8} + 8110 \nu^{6} + 14450 \nu^{4} + 4526 \nu^{2} + 267$$$$)/20$$ $$\beta_{7}$$ $$=$$ $$5 \nu^{10} + 84 \nu^{8} + 443 \nu^{6} + 809 \nu^{4} + 292 \nu^{2} + 26$$ $$\beta_{8}$$ $$=$$ $$($$$$-63 \nu^{11} - 1054 \nu^{9} - 5510 \nu^{7} - 9830 \nu^{5} - 3046 \nu^{3} - 97 \nu$$$$)/20$$ $$\beta_{9}$$ $$=$$ $$($$$$83 \nu^{11} + 1394 \nu^{9} + 7350 \nu^{7} + 13430 \nu^{5} + 4886 \nu^{3} + 457 \nu$$$$)/20$$ $$\beta_{10}$$ $$=$$ $$($$$$-11 \nu^{11} - 184 \nu^{9} - 962 \nu^{7} - 1720 \nu^{5} - 550 \nu^{3} - 39 \nu$$$$)/2$$ $$\beta_{11}$$ $$=$$ $$9 \nu^{11} + 152 \nu^{9} + 811 \nu^{7} + 1528 \nu^{5} + 649 \nu^{3} + 69 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} + \beta_{8} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} + 2 \beta_{5} - \beta_{2} + \beta_{1} - 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{11} - 8 \beta_{9} - 4 \beta_{8} + 3 \beta_{4} - 8 \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{7} + 8 \beta_{6} - 17 \beta_{5} + 12 \beta_{2} - 2 \beta_{1} + 39$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{11} + \beta_{10} + 37 \beta_{9} + 11 \beta_{8} - 17 \beta_{4} + 30 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$($$$$-17 \beta_{7} - 64 \beta_{6} + 147 \beta_{5} - 112 \beta_{2} - 10 \beta_{1} - 291$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$143 \beta_{11} - 36 \beta_{10} - 672 \beta_{9} - 140 \beta_{8} + 319 \beta_{4} - 480 \beta_{3}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$196 \beta_{7} + 523 \beta_{6} - 1278 \beta_{5} + 985 \beta_{2} + 189 \beta_{1} + 2318$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-1324 \beta_{11} + 428 \beta_{10} + 5973 \beta_{9} + 997 \beta_{8} - 2832 \beta_{4} + 3979 \beta_{3}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$-974 \beta_{7} - 2176 \beta_{6} + 5540 \beta_{5} - 4254 \beta_{2} - 1012 \beta_{1} - 9565$$ $$\nu^{11}$$ $$=$$ $$($$$$11780 \beta_{11} - 4324 \beta_{10} - 52317 \beta_{9} - 7677 \beta_{8} + 24640 \beta_{4} - 33565 \beta_{3}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/483\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$346$$ $$442$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
160.1
 − 0.341398i − 2.92914i 0.341398i 2.92914i 2.14584i 0.466018i − 2.14584i − 0.466018i − 0.562016i − 1.77931i 0.562016i 1.77931i
−2.69639 1.00000i 5.27053 −2.58774 2.69639i −0.551006 + 2.58774i −8.81864 −1.00000 6.97756
160.2 −2.69639 1.00000i 5.27053 2.58774 2.69639i 0.551006 2.58774i −8.81864 −1.00000 −6.97756
160.3 −2.69639 1.00000i 5.27053 −2.58774 2.69639i −0.551006 2.58774i −8.81864 −1.00000 6.97756
160.4 −2.69639 1.00000i 5.27053 2.58774 2.69639i 0.551006 + 2.58774i −8.81864 −1.00000 −6.97756
160.5 1.17819 1.00000i −0.611859 −1.67982 1.17819i −2.04406 + 1.67982i −3.07728 −1.00000 −1.97916
160.6 1.17819 1.00000i −0.611859 1.67982 1.17819i 2.04406 1.67982i −3.07728 −1.00000 1.97916
160.7 1.17819 1.00000i −0.611859 −1.67982 1.17819i −2.04406 1.67982i −3.07728 −1.00000 −1.97916
160.8 1.17819 1.00000i −0.611859 1.67982 1.17819i 2.04406 + 1.67982i −3.07728 −1.00000 1.97916
160.9 2.51820 1.00000i 4.34132 −1.21729 2.51820i 2.34908 + 1.21729i 5.89592 −1.00000 −3.06538
160.10 2.51820 1.00000i 4.34132 1.21729 2.51820i −2.34908 1.21729i 5.89592 −1.00000 3.06538
160.11 2.51820 1.00000i 4.34132 −1.21729 2.51820i 2.34908 1.21729i 5.89592 −1.00000 −3.06538
160.12 2.51820 1.00000i 4.34132 1.21729 2.51820i −2.34908 + 1.21729i 5.89592 −1.00000 3.06538
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 160.12 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 inner
23.b odd 2 1 inner
161.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.h.c 12
3.b odd 2 1 1449.2.h.e 12
7.b odd 2 1 inner 483.2.h.c 12
21.c even 2 1 1449.2.h.e 12
23.b odd 2 1 inner 483.2.h.c 12
69.c even 2 1 1449.2.h.e 12
161.c even 2 1 inner 483.2.h.c 12
483.c odd 2 1 1449.2.h.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.h.c 12 1.a even 1 1 trivial
483.2.h.c 12 7.b odd 2 1 inner
483.2.h.c 12 23.b odd 2 1 inner
483.2.h.c 12 161.c even 2 1 inner
1449.2.h.e 12 3.b odd 2 1
1449.2.h.e 12 21.c even 2 1
1449.2.h.e 12 69.c even 2 1
1449.2.h.e 12 483.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - T_{2}^{2} - 7 T_{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(483, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 8 - 7 T - T^{2} + T^{3} )^{4}$$
$3$ $$( 1 + T^{2} )^{6}$$
$5$ $$( -28 + 33 T^{2} - 11 T^{4} + T^{6} )^{2}$$
$7$ $$117649 + 4802 T^{2} + 1519 T^{4} + 476 T^{6} + 31 T^{8} + 2 T^{10} + T^{12}$$
$11$ $$( 175 + 167 T^{2} + 25 T^{4} + T^{6} )^{2}$$
$13$ $$( 4 + 21 T^{2} + 23 T^{4} + T^{6} )^{2}$$
$17$ $$( -1792 + 685 T^{2} - 62 T^{4} + T^{6} )^{2}$$
$19$ $$( -175 + 167 T^{2} - 25 T^{4} + T^{6} )^{2}$$
$23$ $$( 12167 - 3174 T + 943 T^{2} - 292 T^{3} + 41 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$29$ $$( -50 - 19 T + 4 T^{2} + T^{3} )^{4}$$
$31$ $$( 100 + 329 T^{2} + 54 T^{4} + T^{6} )^{2}$$
$37$ $$( 1372 + 1053 T^{2} + 114 T^{4} + T^{6} )^{2}$$
$41$ $$( 25 + T^{2} )^{6}$$
$43$ $$( 1372 + 2281 T^{2} + 95 T^{4} + T^{6} )^{2}$$
$47$ $$( 71824 + 7228 T^{2} + 197 T^{4} + T^{6} )^{2}$$
$53$ $$( 7 + 1662 T^{2} + 86 T^{4} + T^{6} )^{2}$$
$59$ $$( 49 + 1220 T^{2} + 132 T^{4} + T^{6} )^{2}$$
$61$ $$( -229327 + 11938 T^{2} - 194 T^{4} + T^{6} )^{2}$$
$67$ $$( 20412 + 2673 T^{2} + 99 T^{4} + T^{6} )^{2}$$
$71$ $$( 128 - 53 T - 5 T^{2} + T^{3} )^{4}$$
$73$ $$( 2458624 + 59633 T^{2} + 450 T^{4} + T^{6} )^{2}$$
$79$ $$( 51772 + 23757 T^{2} + 386 T^{4} + T^{6} )^{2}$$
$83$ $$( -2903152 + 66741 T^{2} - 470 T^{4} + T^{6} )^{2}$$
$89$ $$( -17500 + 4457 T^{2} - 171 T^{4} + T^{6} )^{2}$$
$97$ $$( -4435312 + 107173 T^{2} - 614 T^{4} + T^{6} )^{2}$$