# Properties

 Label 483.2.h.b 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: $$x^{12} + 14 x^{10} + 67 x^{8} + 141 x^{6} + 129 x^{4} + 39 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{2} -\beta_{5} q^{3} + ( 1 - \beta_{7} ) q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} -\beta_{3} q^{6} + ( -\beta_{4} - \beta_{11} ) q^{7} - q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{8} q^{2} -\beta_{5} q^{3} + ( 1 - \beta_{7} ) q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} -\beta_{3} q^{6} + ( -\beta_{4} - \beta_{11} ) q^{7} - q^{8} - q^{9} + ( 2 \beta_{1} - \beta_{2} - \beta_{9} - \beta_{11} ) q^{10} -\beta_{4} q^{11} + ( -\beta_{5} + \beta_{6} ) q^{12} + ( -2 \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{13} + ( -\beta_{2} - \beta_{4} + \beta_{9} + \beta_{10} ) q^{14} + ( \beta_{4} - \beta_{10} ) q^{15} + ( -2 + 2 \beta_{7} - \beta_{8} ) q^{16} + ( -2 \beta_{1} - \beta_{2} - \beta_{9} - \beta_{11} ) q^{17} -\beta_{8} q^{18} + ( -2 \beta_{1} + \beta_{2} - \beta_{9} - \beta_{11} ) q^{19} + ( \beta_{1} - 2 \beta_{2} ) q^{20} + ( -\beta_{2} - \beta_{9} ) q^{21} + ( \beta_{9} + \beta_{10} - \beta_{11} ) q^{22} + ( -1 - \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{23} + \beta_{5} q^{24} + ( 3 - 3 \beta_{7} + 4 \beta_{8} ) q^{25} + ( \beta_{3} - 5 \beta_{5} + 2 \beta_{6} ) q^{26} + \beta_{5} q^{27} + ( \beta_{1} + \beta_{2} - \beta_{9} + 2 \beta_{10} ) q^{28} + ( 1 + \beta_{7} + \beta_{8} ) q^{29} + ( \beta_{4} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{30} + ( -\beta_{3} - 7 \beta_{5} + \beta_{6} ) q^{31} + ( 1 + \beta_{7} - 4 \beta_{8} ) q^{32} -\beta_{2} q^{33} -\beta_{1} q^{34} + ( -1 + 3 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{35} + ( -1 + \beta_{7} ) q^{36} + ( -\beta_{4} - 4 \beta_{10} ) q^{37} + ( -3 \beta_{1} + 2 \beta_{9} + 2 \beta_{11} ) q^{38} + ( 2 + \beta_{7} - 2 \beta_{8} ) q^{39} + ( -\beta_{1} + \beta_{2} ) q^{40} + ( -\beta_{3} + 7 \beta_{5} - \beta_{6} ) q^{41} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{11} ) q^{42} + ( -4 \beta_{4} - \beta_{10} ) q^{43} + ( -\beta_{4} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{44} + ( -\beta_{1} + \beta_{2} ) q^{45} + ( -7 + \beta_{4} + 2 \beta_{7} - \beta_{10} ) q^{46} + ( -2 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} ) q^{47} + ( \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{48} + ( -1 + \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{8} ) q^{49} + ( 9 - 4 \beta_{7} + 6 \beta_{8} ) q^{50} + ( \beta_{4} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{51} + ( -3 \beta_{3} + \beta_{5} - 3 \beta_{6} ) q^{52} + ( -\beta_{4} - 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{53} + \beta_{3} q^{54} + ( \beta_{3} + 5 \beta_{5} - 2 \beta_{6} ) q^{55} + ( \beta_{4} + \beta_{11} ) q^{56} + ( -\beta_{4} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{57} + ( 4 - \beta_{7} ) q^{58} + ( 3 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{59} + ( 2 \beta_{4} - \beta_{10} ) q^{60} + ( 3 \beta_{1} + 2 \beta_{2} + \beta_{9} + \beta_{11} ) q^{61} + ( -8 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{62} + ( \beta_{4} + \beta_{11} ) q^{63} + ( -7 + 2 \beta_{8} ) q^{64} + ( \beta_{4} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{65} + ( \beta_{1} - \beta_{9} - \beta_{11} ) q^{66} + ( -3 \beta_{4} - \beta_{10} ) q^{67} + ( 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{9} + 2 \beta_{11} ) q^{68} + ( -\beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{69} + ( 6 + 6 \beta_{3} + 7 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{70} + ( 6 + \beta_{7} - 2 \beta_{8} ) q^{71} + q^{72} + ( 3 \beta_{3} + \beta_{5} - 3 \beta_{6} ) q^{73} + ( 4 \beta_{4} + \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{74} + ( -4 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} ) q^{75} + ( \beta_{1} + 5 \beta_{2} ) q^{76} + ( -3 + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{8} ) q^{77} + ( -5 + 2 \beta_{7} + \beta_{8} ) q^{78} + ( -3 \beta_{4} + 3 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} ) q^{79} + ( -4 \beta_{1} + 5 \beta_{2} + \beta_{9} + \beta_{11} ) q^{80} + q^{81} + ( 8 \beta_{3} - 4 \beta_{5} + \beta_{6} ) q^{82} + ( 2 \beta_{1} + \beta_{2} ) q^{83} + ( 2 \beta_{1} - \beta_{4} - \beta_{10} + \beta_{11} ) q^{84} + ( -3 - \beta_{8} ) q^{85} + ( \beta_{4} + 4 \beta_{9} + 3 \beta_{10} - 4 \beta_{11} ) q^{86} + ( -\beta_{3} - \beta_{5} - \beta_{6} ) q^{87} + \beta_{4} q^{88} + ( -5 \beta_{1} - 2 \beta_{2} - 2 \beta_{9} - 2 \beta_{11} ) q^{89} + ( -2 \beta_{1} + \beta_{2} + \beta_{9} + \beta_{11} ) q^{90} + ( 4 \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{91} + ( 4 + \beta_{4} + 2 \beta_{7} - 5 \beta_{8} - \beta_{9} + \beta_{11} ) q^{92} + ( -7 + \beta_{7} - \beta_{8} ) q^{93} + ( -8 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{94} + ( -13 + 4 \beta_{7} - 3 \beta_{8} ) q^{95} + ( 4 \beta_{3} - \beta_{5} - \beta_{6} ) q^{96} + ( 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{9} + 2 \beta_{11} ) q^{97} + ( -10 + \beta_{3} + 4 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{98} + \beta_{4} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 8q^{4} - 12q^{8} - 12q^{9} + O(q^{10})$$ $$12q + 8q^{4} - 12q^{8} - 12q^{9} - 16q^{16} - 16q^{23} + 24q^{25} + 16q^{29} + 16q^{32} - 12q^{35} - 8q^{36} + 28q^{39} - 76q^{46} - 16q^{49} + 92q^{50} + 44q^{58} - 84q^{64} + 64q^{70} + 76q^{71} + 12q^{72} - 36q^{77} - 52q^{78} + 12q^{81} - 36q^{85} + 56q^{92} - 80q^{93} - 140q^{95} - 108q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 14 x^{10} + 67 x^{8} + 141 x^{6} + 129 x^{4} + 39 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{10} + 31 \nu^{8} + 78 \nu^{6} + 39 \nu^{4} - 8 \nu^{2} + 25$$$$)/14$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{10} + 23 \nu^{8} + 80 \nu^{6} + 124 \nu^{4} + 95 \nu^{2} + 12$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} - 15 \nu^{9} - 82 \nu^{7} - 209 \nu^{5} - 212 \nu^{3} - 27 \nu$$$$)/14$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{11} + 23 \nu^{9} + 73 \nu^{7} + 54 \nu^{5} - 66 \nu^{3} - 58 \nu$$$$)/7$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{11} - 45 \nu^{9} - 232 \nu^{7} - 501 \nu^{5} - 426 \nu^{3} - 95 \nu$$$$)/14$$ $$\beta_{6}$$ $$=$$ $$-\nu^{9} - 12 \nu^{7} - 43 \nu^{5} - 55 \nu^{3} - 19 \nu$$ $$\beta_{7}$$ $$=$$ $$($$$$-6 \nu^{10} - 69 \nu^{8} - 226 \nu^{6} - 239 \nu^{4} - 26 \nu^{2} + 20$$$$)/7$$ $$\beta_{8}$$ $$=$$ $$($$$$-5 \nu^{10} - 61 \nu^{8} - 228 \nu^{6} - 324 \nu^{4} - 143 \nu^{2} - 2$$$$)/7$$ $$\beta_{9}$$ $$=$$ $$($$$$21 \nu^{11} + 13 \nu^{10} + 259 \nu^{9} + 167 \nu^{8} + 994 \nu^{7} + 674 \nu^{6} + 1519 \nu^{5} + 1023 \nu^{4} + 896 \nu^{3} + 488 \nu^{2} + 203 \nu + 43$$$$)/28$$ $$\beta_{10}$$ $$=$$ $$($$$$-9 \nu^{11} - 121 \nu^{9} - 542 \nu^{7} - 1041 \nu^{5} - 830 \nu^{3} - 173 \nu$$$$)/14$$ $$\beta_{11}$$ $$=$$ $$($$$$-21 \nu^{11} + 13 \nu^{10} - 259 \nu^{9} + 167 \nu^{8} - 994 \nu^{7} + 674 \nu^{6} - 1519 \nu^{5} + 1023 \nu^{4} - 896 \nu^{3} + 488 \nu^{2} - 203 \nu + 43$$$$)/28$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{8} + \beta_{7} - \beta_{2} + 2 \beta_{1} - 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{11} + 3 \beta_{10} + \beta_{9} + 4 \beta_{6} - 7 \beta_{5} - 5 \beta_{4} - 5 \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{11} + 2 \beta_{9} + 11 \beta_{8} - 7 \beta_{7} + 9 \beta_{2} - 12 \beta_{1} + 23$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{11} - 13 \beta_{10} - 4 \beta_{9} - 11 \beta_{6} + 24 \beta_{5} + 14 \beta_{4} + 17 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$($$$$-19 \beta_{11} - 19 \beta_{9} - 86 \beta_{8} + 49 \beta_{7} - 64 \beta_{2} + 77 \beta_{1} - 134$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-57 \beta_{11} + 189 \beta_{10} + 57 \beta_{9} + 137 \beta_{6} - 324 \beta_{5} - 176 \beta_{4} - 236 \beta_{3}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$144 \beta_{11} + 144 \beta_{9} + 613 \beta_{8} - 337 \beta_{7} + 439 \beta_{2} - 514 \beta_{1} + 861$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$395 \beta_{11} - 1315 \beta_{10} - 395 \beta_{9} - 901 \beta_{6} + 2190 \beta_{5} + 1164 \beta_{4} + 1626 \beta_{3}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$-510 \beta_{11} - 510 \beta_{9} - 2122 \beta_{8} + 1151 \beta_{7} - 1496 \beta_{2} + 1740 \beta_{1} - 2871$$ $$\nu^{11}$$ $$=$$ $$($$$$-2711 \beta_{11} + 9025 \beta_{10} + 2711 \beta_{9} + 6058 \beta_{6} - 14857 \beta_{5} - 7847 \beta_{4} - 11139 \beta_{3}$$$$)/2$$

## 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
 1.64421i − 1.28628i − 1.64421i 1.28628i − 1.51343i − 0.167898i 1.51343i 0.167898i 2.61062i 0.712783i − 2.61062i − 0.712783i
−2.11491 1.00000i 2.47283 −1.98969 2.11491i 2.62846 0.302029i −1.00000 −1.00000 4.20801
160.2 −2.11491 1.00000i 2.47283 1.98969 2.11491i −2.62846 + 0.302029i −1.00000 −1.00000 −4.20801
160.3 −2.11491 1.00000i 2.47283 −1.98969 2.11491i 2.62846 + 0.302029i −1.00000 −1.00000 4.20801
160.4 −2.11491 1.00000i 2.47283 1.98969 2.11491i −2.62846 0.302029i −1.00000 −1.00000 −4.20801
160.5 0.254102 1.00000i −1.93543 −0.458377 0.254102i 1.07291 + 2.41844i −1.00000 −1.00000 −0.116474
160.6 0.254102 1.00000i −1.93543 0.458377 0.254102i −1.07291 2.41844i −1.00000 −1.00000 0.116474
160.7 0.254102 1.00000i −1.93543 −0.458377 0.254102i 1.07291 2.41844i −1.00000 −1.00000 −0.116474
160.8 0.254102 1.00000i −1.93543 0.458377 0.254102i −1.07291 + 2.41844i −1.00000 −1.00000 0.116474
160.9 1.86081 1.00000i 1.46260 −4.10256 1.86081i −0.663393 2.56123i −1.00000 −1.00000 −7.63407
160.10 1.86081 1.00000i 1.46260 4.10256 1.86081i 0.663393 + 2.56123i −1.00000 −1.00000 7.63407
160.11 1.86081 1.00000i 1.46260 −4.10256 1.86081i −0.663393 + 2.56123i −1.00000 −1.00000 −7.63407
160.12 1.86081 1.00000i 1.46260 4.10256 1.86081i 0.663393 2.56123i −1.00000 −1.00000 7.63407
 $$n$$: 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.b 12
3.b odd 2 1 1449.2.h.h 12
7.b odd 2 1 inner 483.2.h.b 12
21.c even 2 1 1449.2.h.h 12
23.b odd 2 1 inner 483.2.h.b 12
69.c even 2 1 1449.2.h.h 12
161.c even 2 1 inner 483.2.h.b 12
483.c odd 2 1 1449.2.h.h 12

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 4 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(483, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 4 T + T^{3} )^{4}$$
$3$ $$( 1 + T^{2} )^{6}$$
$5$ $$( -14 + 71 T^{2} - 21 T^{4} + T^{6} )^{2}$$
$7$ $$117649 + 19208 T^{2} - 1617 T^{4} - 784 T^{6} - 33 T^{8} + 8 T^{10} + T^{12}$$
$11$ $$( 56 + 53 T^{2} + 14 T^{4} + T^{6} )^{2}$$
$13$ $$( 2116 + 645 T^{2} + 47 T^{4} + T^{6} )^{2}$$
$17$ $$( -14 + 81 T^{2} - 52 T^{4} + T^{6} )^{2}$$
$19$ $$( -19166 + 2241 T^{2} - 84 T^{4} + T^{6} )^{2}$$
$23$ $$( 12167 + 4232 T - 437 T^{2} - 344 T^{3} - 19 T^{4} + 8 T^{5} + T^{6} )^{2}$$
$29$ $$( 26 - 7 T - 4 T^{2} + T^{3} )^{4}$$
$31$ $$( 65536 + 5889 T^{2} + 146 T^{4} + T^{6} )^{2}$$
$37$ $$( 157304 + 8941 T^{2} + 166 T^{4} + T^{6} )^{2}$$
$41$ $$( 51076 + 5601 T^{2} + 158 T^{4} + T^{6} )^{2}$$
$43$ $$( 240254 + 14431 T^{2} + 241 T^{4} + T^{6} )^{2}$$
$47$ $$( 153664 + 9408 T^{2} + 176 T^{4} + T^{6} )^{2}$$
$53$ $$( 270494 + 14151 T^{2} + 217 T^{4} + T^{6} )^{2}$$
$59$ $$( 33124 + 4005 T^{2} + 119 T^{4} + T^{6} )^{2}$$
$61$ $$( -2744 + 1493 T^{2} - 119 T^{4} + T^{6} )^{2}$$
$67$ $$( 39326 + 4799 T^{2} + 141 T^{4} + T^{6} )^{2}$$
$71$ $$( -134 + 105 T - 19 T^{2} + T^{3} )^{4}$$
$73$ $$( 2704 + 3249 T^{2} + 114 T^{4} + T^{6} )^{2}$$
$79$ $$( 734174 + 28625 T^{2} + 324 T^{4} + T^{6} )^{2}$$
$83$ $$( -3584 + 869 T^{2} - 54 T^{4} + T^{6} )^{2}$$
$89$ $$( -14 + 5359 T^{2} - 277 T^{4} + T^{6} )^{2}$$
$97$ $$( -251384 + 22205 T^{2} - 374 T^{4} + T^{6} )^{2}$$