# Properties

 Label 403.2.a.a Level 403 Weight 2 Character orbit 403.a Self dual yes Analytic conductor 3.218 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.618034 1.61803
0.381966 −2.00000 −1.85410 −2.23607 −0.763932 1.00000 −1.47214 1.00000 −0.854102
1.2 2.61803 −2.00000 4.85410 2.23607 −5.23607 1.00000 7.47214 1.00000 5.85410
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.a.a 2
3.b odd 2 1 3627.2.a.c 2
4.b odd 2 1 6448.2.a.r 2
13.b even 2 1 5239.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.a.a 2 1.a even 1 1 trivial
3627.2.a.c 2 3.b odd 2 1
5239.2.a.e 2 13.b even 2 1
6448.2.a.r 2 4.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$-1$$
$$31$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(403))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4}$$
$3$ $$( 1 + 2 T + 3 T^{2} )^{2}$$
$5$ $$1 + 5 T^{2} + 25 T^{4}$$
$7$ $$( 1 - T + 7 T^{2} )^{2}$$
$11$ $$1 + 2 T^{2} + 121 T^{4}$$
$13$ $$( 1 - T )^{2}$$
$17$ $$1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$( 1 - T + 19 T^{2} )^{2}$$
$23$ $$1 + 6 T + 50 T^{2} + 138 T^{3} + 529 T^{4}$$
$29$ $$1 - 6 T + 62 T^{2} - 174 T^{3} + 841 T^{4}$$
$31$ $$( 1 + T )^{2}$$
$37$ $$1 + 6 T + 38 T^{2} + 222 T^{3} + 1369 T^{4}$$
$41$ $$1 + 77 T^{2} + 1681 T^{4}$$
$43$ $$1 + 10 T + 66 T^{2} + 430 T^{3} + 1849 T^{4}$$
$47$ $$1 + 14 T^{2} + 2209 T^{4}$$
$53$ $$1 - 18 T + 182 T^{2} - 954 T^{3} + 2809 T^{4}$$
$59$ $$1 + 6 T + 107 T^{2} + 354 T^{3} + 3481 T^{4}$$
$61$ $$1 - 14 T + 126 T^{2} - 854 T^{3} + 3721 T^{4}$$
$67$ $$( 1 + 8 T + 67 T^{2} )^{2}$$
$71$ $$( 1 - 3 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 14 T + 73 T^{2} )^{2}$$
$79$ $$( 1 - 4 T + 79 T^{2} )^{2}$$
$83$ $$1 - 12 T + 122 T^{2} - 996 T^{3} + 6889 T^{4}$$
$89$ $$1 + 6 T + 182 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$1 - 4 T + 153 T^{2} - 388 T^{3} + 9409 T^{4}$$