# Properties

 Label 403.2.be.b Level 403 Weight 2 Character orbit 403.be 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.be (of order $$12$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character Values

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

 $$n$$ $$249$$ $$313$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
0.366025 0.366025i 2.36603 + 1.36603i 1.73205i −1.36603 0.366025i 1.36603 0.366025i 2.86603 0.767949i 1.36603 + 1.36603i 2.23205 + 3.86603i −0.633975 + 0.366025i
99.1 0.366025 + 0.366025i 2.36603 1.36603i 1.73205i −1.36603 + 0.366025i 1.36603 + 0.366025i 2.86603 + 0.767949i 1.36603 1.36603i 2.23205 3.86603i −0.633975 0.366025i
161.1 −1.36603 + 1.36603i 0.633975 0.366025i 1.73205i 0.366025 + 1.36603i −0.366025 + 1.36603i 1.13397 4.23205i −0.366025 0.366025i −1.23205 + 2.13397i −2.36603 1.36603i
398.1 −1.36603 1.36603i 0.633975 + 0.366025i 1.73205i 0.366025 1.36603i −0.366025 1.36603i 1.13397 + 4.23205i −0.366025 + 0.366025i −1.23205 2.13397i −2.36603 + 1.36603i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
403.be Even 1 yes

## Hecke kernels

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

 $$T_{2}^{4} + 2 T_{2}^{3} + 2 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{7}^{4} - 8 T_{7}^{3} + 41 T_{7}^{2} - 130 T_{7} + 169$$