Properties

 Label 403.2.c.a Level 403 Weight 2 Character orbit 403.c 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.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + 2 q^{4} + 4 i q^{5} -2 i q^{7} + q^{9} +O(q^{10})$$ $$q + 2 q^{3} + 2 q^{4} + 4 i q^{5} -2 i q^{7} + q^{9} + i q^{11} + 4 q^{12} + ( -2 - 3 i ) q^{13} + 8 i q^{15} + 4 q^{16} -2 q^{17} -2 i q^{19} + 8 i q^{20} -4 i q^{21} -4 q^{23} -11 q^{25} -4 q^{27} -4 i q^{28} + 8 q^{29} -i q^{31} + 2 i q^{33} + 8 q^{35} + 2 q^{36} -3 i q^{37} + ( -4 - 6 i ) q^{39} -4 i q^{41} + 4 q^{43} + 2 i q^{44} + 4 i q^{45} -2 i q^{47} + 8 q^{48} + 3 q^{49} -4 q^{51} + ( -4 - 6 i ) q^{52} -4 q^{53} -4 q^{55} -4 i q^{57} + 6 i q^{59} + 16 i q^{60} + 10 q^{61} -2 i q^{63} + 8 q^{64} + ( 12 - 8 i ) q^{65} + 16 i q^{67} -4 q^{68} -8 q^{69} -6 i q^{71} -7 i q^{73} -22 q^{75} -4 i q^{76} + 2 q^{77} -16 q^{79} + 16 i q^{80} -11 q^{81} + 12 i q^{83} -8 i q^{84} -8 i q^{85} + 16 q^{87} -15 i q^{89} + ( -6 + 4 i ) q^{91} -8 q^{92} -2 i q^{93} + 8 q^{95} + 2 i q^{97} + i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} + 4q^{4} + 2q^{9} + O(q^{10})$$ $$2q + 4q^{3} + 4q^{4} + 2q^{9} + 8q^{12} - 4q^{13} + 8q^{16} - 4q^{17} - 8q^{23} - 22q^{25} - 8q^{27} + 16q^{29} + 16q^{35} + 4q^{36} - 8q^{39} + 8q^{43} + 16q^{48} + 6q^{49} - 8q^{51} - 8q^{52} - 8q^{53} - 8q^{55} + 20q^{61} + 16q^{64} + 24q^{65} - 8q^{68} - 16q^{69} - 44q^{75} + 4q^{77} - 32q^{79} - 22q^{81} + 32q^{87} - 12q^{91} - 16q^{92} + 16q^{95} + O(q^{100})$$

Character values

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

 $$n$$ $$249$$ $$313$$ $$\chi(n)$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
311.1
 − 1.00000i 1.00000i
0 2.00000 2.00000 4.00000i 0 2.00000i 0 1.00000 0
311.2 0 2.00000 2.00000 4.00000i 0 2.00000i 0 1.00000 0
 $$n$$: 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
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 403.2.c.a 2
13.b even 2 1 inner 403.2.c.a 2
13.d odd 4 1 5239.2.a.b 1
13.d odd 4 1 5239.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.c.a 2 1.a even 1 1 trivial
403.2.c.a 2 13.b even 2 1 inner
5239.2.a.b 1 13.d odd 4 1
5239.2.a.c 1 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{2}$$
$3$ $$( 1 - 2 T + 3 T^{2} )^{2}$$
$5$ $$( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$
$7$ $$1 - 10 T^{2} + 49 T^{4}$$
$11$ $$1 - 21 T^{2} + 121 T^{4}$$
$13$ $$1 + 4 T + 13 T^{2}$$
$17$ $$( 1 + 2 T + 17 T^{2} )^{2}$$
$19$ $$1 - 34 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$( 1 - 8 T + 29 T^{2} )^{2}$$
$31$ $$1 + T^{2}$$
$37$ $$1 - 65 T^{2} + 1369 T^{4}$$
$41$ $$1 - 66 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 4 T + 43 T^{2} )^{2}$$
$47$ $$1 - 90 T^{2} + 2209 T^{4}$$
$53$ $$( 1 + 4 T + 53 T^{2} )^{2}$$
$59$ $$1 - 82 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 10 T + 61 T^{2} )^{2}$$
$67$ $$1 + 122 T^{2} + 4489 T^{4}$$
$71$ $$1 - 106 T^{2} + 5041 T^{4}$$
$73$ $$1 - 97 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 16 T + 79 T^{2} )^{2}$$
$83$ $$1 - 22 T^{2} + 6889 T^{4}$$
$89$ $$1 + 47 T^{2} + 7921 T^{4}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$