# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.l (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/403\mathbb{Z}\right)^\times$$.

 $$n$$ $$249$$ $$313$$ $$\chi(n)$$ $$-1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 0.500000 + 0.866025i −1.00000 −1.50000 0.866025i 1.50000 0.866025i −1.50000 + 0.866025i 1.73205i 1.00000 1.73205i −1.50000 + 2.59808i
129.1 1.73205i 0.500000 0.866025i −1.00000 −1.50000 + 0.866025i 1.50000 + 0.866025i −1.50000 0.866025i 1.73205i 1.00000 + 1.73205i −1.50000 2.59808i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.l.a 2
13.b even 2 1 403.2.l.b yes 2
31.c even 3 1 403.2.l.b yes 2
403.l even 6 1 inner 403.2.l.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.l.a 2 1.a even 1 1 trivial
403.2.l.a 2 403.l even 6 1 inner
403.2.l.b yes 2 13.b even 2 1
403.2.l.b yes 2 31.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$:

 $$T_{2}^{2} + 3$$ $$T_{5}^{2} + 3 T_{5} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + 4 T^{4}$$
$3$ $$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$
$5$ $$1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$11$ $$1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4}$$
$13$ $$1 - 7 T + 13 T^{2}$$
$17$ $$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} )$$
$23$ $$( 1 + 23 T^{2} )^{2}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$1 - 4 T + 31 T^{2}$$
$37$ $$( 1 - 10 T + 37 T^{2} )( 1 + T + 37 T^{2} )$$
$41$ $$1 - 9 T + 68 T^{2} - 369 T^{3} + 1681 T^{4}$$
$43$ $$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$
$47$ $$1 - 82 T^{2} + 2209 T^{4}$$
$53$ $$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$
$59$ $$1 + 3 T + 62 T^{2} + 177 T^{3} + 3481 T^{4}$$
$61$ $$( 1 - 10 T + 61 T^{2} )^{2}$$
$67$ $$1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4}$$
$71$ $$1 - 15 T + 146 T^{2} - 1065 T^{3} + 5041 T^{4}$$
$73$ $$1 + 15 T + 148 T^{2} + 1095 T^{3} + 5329 T^{4}$$
$79$ $$1 - 5 T - 54 T^{2} - 395 T^{3} + 6241 T^{4}$$
$83$ $$1 + 21 T + 230 T^{2} + 1743 T^{3} + 6889 T^{4}$$
$89$ $$1 - 130 T^{2} + 7921 T^{4}$$
$97$ $$1 - 146 T^{2} + 9409 T^{4}$$