# Properties

 Label 403.2.k.a Level 403 Weight 2 Character orbit 403.k Analytic conductor 3.218 Analytic rank 1 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.k (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$249$$ $$313$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
66.1
 −0.309017 − 0.951057i 0.809017 + 0.587785i −0.309017 + 0.951057i 0.809017 − 0.587785i
−0.500000 1.53884i −0.809017 + 2.48990i −0.500000 + 0.363271i −0.381966 4.23607 −2.30902 + 1.67760i −1.80902 1.31433i −3.11803 2.26538i 0.190983 + 0.587785i
157.1 −0.500000 0.363271i 0.309017 0.224514i −0.500000 1.53884i −2.61803 −0.236068 −1.19098 3.66547i −0.690983 + 2.12663i −0.881966 + 2.71441i 1.30902 + 0.951057i
287.1 −0.500000 + 1.53884i −0.809017 2.48990i −0.500000 0.363271i −0.381966 4.23607 −2.30902 1.67760i −1.80902 + 1.31433i −3.11803 + 2.26538i 0.190983 0.587785i
326.1 −0.500000 + 0.363271i 0.309017 + 0.224514i −0.500000 + 1.53884i −2.61803 −0.236068 −1.19098 + 3.66547i −0.690983 2.12663i −0.881966 2.71441i 1.30902 0.951057i
 $$n$$: 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
31.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.k.a 4
31.d even 5 1 inner 403.2.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.k.a 4 1.a even 1 1 trivial
403.2.k.a 4 31.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2 T_{2}^{3} + 4 T_{2}^{2} + 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} + 5 T^{3} + 11 T^{4} + 10 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8}$$
$3$ $$1 + T + 3 T^{2} + 5 T^{3} + 16 T^{4} + 15 T^{5} + 27 T^{6} + 27 T^{7} + 81 T^{8}$$
$5$ $$( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} )^{2}$$
$7$ $$1 + 7 T + 27 T^{2} + 95 T^{3} + 296 T^{4} + 665 T^{5} + 1323 T^{6} + 2401 T^{7} + 2401 T^{8}$$
$11$ $$1 - 5 T + 4 T^{2} - 25 T^{3} + 201 T^{4} - 275 T^{5} + 484 T^{6} - 6655 T^{7} + 14641 T^{8}$$
$13$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$17$ $$1 + 15 T + 118 T^{2} + 675 T^{3} + 3079 T^{4} + 11475 T^{5} + 34102 T^{6} + 73695 T^{7} + 83521 T^{8}$$
$19$ $$1 - 2 T - 15 T^{2} + 68 T^{3} + 149 T^{4} + 1292 T^{5} - 5415 T^{6} - 13718 T^{7} + 130321 T^{8}$$
$23$ $$1 + 8 T + T^{2} - 146 T^{3} - 651 T^{4} - 3358 T^{5} + 529 T^{6} + 97336 T^{7} + 279841 T^{8}$$
$29$ $$1 + 8 T + 5 T^{2} - 232 T^{3} - 1711 T^{4} - 6728 T^{5} + 4205 T^{6} + 195112 T^{7} + 707281 T^{8}$$
$31$ $$1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 13 T + 105 T^{2} + 481 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$1 + 19 T + 95 T^{2} - 599 T^{3} - 8776 T^{4} - 24559 T^{5} + 159695 T^{6} + 1309499 T^{7} + 2825761 T^{8}$$
$43$ $$1 - 6 T + 93 T^{2} - 350 T^{3} + 4851 T^{4} - 15050 T^{5} + 171957 T^{6} - 477042 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 11 T + 49 T^{2} - 547 T^{3} + 5964 T^{4} - 25709 T^{5} + 108241 T^{6} - 1142053 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 6 T + 83 T^{2} + 480 T^{3} + 6481 T^{4} + 25440 T^{5} + 233147 T^{6} + 893262 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 14 T + 37 T^{2} + 142 T^{3} + 3555 T^{4} + 8378 T^{5} + 128797 T^{6} + 2875306 T^{7} + 12117361 T^{8}$$
$61$ $$( 1 + 8 T + 133 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 3 T + 35 T^{2} - 201 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 - 34 T + 665 T^{2} - 8926 T^{3} + 86729 T^{4} - 633746 T^{5} + 3352265 T^{6} - 12168974 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 5 T + 12 T^{2} - 655 T^{3} + 8639 T^{4} - 47815 T^{5} + 63948 T^{6} - 1945085 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 21 T + 162 T^{2} + 1343 T^{3} + 15405 T^{4} + 106097 T^{5} + 1011042 T^{6} + 10353819 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 12 T + 61 T^{2} + 1146 T^{3} + 17149 T^{4} + 95118 T^{5} + 420229 T^{6} + 6861444 T^{7} + 47458321 T^{8}$$
$89$ $$1 + T - 38 T^{2} - 727 T^{3} + 5055 T^{4} - 64703 T^{5} - 300998 T^{6} + 704969 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 8 T + 87 T^{2} + 1270 T^{3} + 19901 T^{4} + 123190 T^{5} + 818583 T^{6} + 7301384 T^{7} + 88529281 T^{8}$$