# Properties

 Label 403.2.k.c Level 403 Weight 2 Character orbit 403.k Analytic conductor 3.218 Analytic rank 0 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

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

## 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
−0.809017 2.48990i 0.190983 0.587785i −3.92705 + 2.85317i 0.381966 −1.61803 −3.92705 + 2.85317i 6.04508 + 4.39201i 2.11803 + 1.53884i −0.309017 0.951057i
157.1 0.309017 + 0.224514i 1.30902 0.951057i −0.572949 1.76336i 2.61803 0.618034 −0.572949 1.76336i 0.454915 1.40008i −0.118034 + 0.363271i 0.809017 + 0.587785i
287.1 −0.809017 + 2.48990i 0.190983 + 0.587785i −3.92705 2.85317i 0.381966 −1.61803 −3.92705 2.85317i 6.04508 4.39201i 2.11803 1.53884i −0.309017 + 0.951057i
326.1 0.309017 0.224514i 1.30902 + 0.951057i −0.572949 + 1.76336i 2.61803 0.618034 −0.572949 + 1.76336i 0.454915 + 1.40008i −0.118034 0.363271i 0.809017 0.587785i
 $$n$$: 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.c 4
31.d even 5 1 inner 403.2.k.c 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + 6 T_{2}^{2} - 4 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 4 T^{2} + 2 T^{3} + 9 T^{4} + 4 T^{5} + 16 T^{6} + 8 T^{7} + 16 T^{8}$$
$3$ $$1 - 3 T + T^{2} + T^{3} + 4 T^{4} + 3 T^{5} + 9 T^{6} - 81 T^{7} + 81 T^{8}$$
$5$ $$( 1 - 3 T + 11 T^{2} - 15 T^{3} + 25 T^{4} )^{2}$$
$7$ $$1 + 9 T + 29 T^{2} + 33 T^{3} + 4 T^{4} + 231 T^{5} + 1421 T^{6} + 3087 T^{7} + 2401 T^{8}$$
$11$ $$1 - 3 T + 8 T^{2} - 51 T^{3} + 265 T^{4} - 561 T^{5} + 968 T^{6} - 3993 T^{7} + 14641 T^{8}$$
$13$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$17$ $$1 + T - 16 T^{2} - 33 T^{3} + 239 T^{4} - 561 T^{5} - 4624 T^{6} + 4913 T^{7} + 83521 T^{8}$$
$19$ $$1 + 10 T + 21 T^{2} - 190 T^{3} - 1519 T^{4} - 3610 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8}$$
$23$ $$1 - 12 T + 41 T^{2} - 36 T^{3} + 49 T^{4} - 828 T^{5} + 21689 T^{6} - 146004 T^{7} + 279841 T^{8}$$
$29$ $$1 - 10 T + 11 T^{2} + 120 T^{3} - 439 T^{4} + 3480 T^{5} + 9251 T^{6} - 243890 T^{7} + 707281 T^{8}$$
$31$ $$1 - 4 T + 46 T^{2} - 124 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 13 T + 115 T^{2} + 481 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$1 - 5 T - 31 T^{2} + 205 T^{3} + 476 T^{4} + 8405 T^{5} - 52111 T^{6} - 344605 T^{7} + 2825761 T^{8}$$
$43$ $$1 + 2 T - 19 T^{2} - 254 T^{3} + 619 T^{4} - 10922 T^{5} - 35131 T^{6} + 159014 T^{7} + 3418801 T^{8}$$
$47$ $$1 - T + 59 T^{2} - 177 T^{3} + 2264 T^{4} - 8319 T^{5} + 130331 T^{6} - 103823 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 28 T + 401 T^{2} + 4024 T^{3} + 32129 T^{4} + 213272 T^{5} + 1126409 T^{6} + 4168556 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 49 T^{2} + 270 T^{3} + 3211 T^{4} + 15930 T^{5} - 170569 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 - 11 T + 61 T^{2} )^{4}$$
$67$ $$( 1 + 9 T + 153 T^{2} + 603 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 - 2 T - 7 T^{2} + 466 T^{3} + 1865 T^{4} + 33086 T^{5} - 35287 T^{6} - 715822 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 11 T - 22 T^{2} + 845 T^{3} - 5609 T^{4} + 61685 T^{5} - 117238 T^{6} - 4279187 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 9 T - 48 T^{2} + 643 T^{3} - 195 T^{4} + 50797 T^{5} - 299568 T^{6} - 4437351 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 14 T + 53 T^{2} + 870 T^{3} + 14801 T^{4} + 72210 T^{5} + 365117 T^{6} + 8005018 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 11 T - 38 T^{2} - 1037 T^{3} - 5625 T^{4} - 92293 T^{5} - 300998 T^{6} + 7754659 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 12 T + 47 T^{2} - 600 T^{3} - 11759 T^{4} - 58200 T^{5} + 442223 T^{6} + 10952076 T^{7} + 88529281 T^{8}$$