Properties

 Label 403.2.be.a 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 $$12$$, degree $$4$$, minimal)

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: $$1$$ Twist minimal: yes 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.

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

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
−1.36603 + 1.36603i 0.633975 + 0.366025i 1.73205i −1.36603 0.366025i −1.36603 + 0.366025i −4.23205 + 1.13397i −0.366025 0.366025i −1.23205 2.13397i 2.36603 1.36603i
99.1 −1.36603 1.36603i 0.633975 0.366025i 1.73205i −1.36603 + 0.366025i −1.36603 0.366025i −4.23205 1.13397i −0.366025 + 0.366025i −1.23205 + 2.13397i 2.36603 + 1.36603i
161.1 0.366025 0.366025i 2.36603 1.36603i 1.73205i 0.366025 + 1.36603i 0.366025 1.36603i −0.767949 + 2.86603i 1.36603 + 1.36603i 2.23205 3.86603i 0.633975 + 0.366025i
398.1 0.366025 + 0.366025i 2.36603 + 1.36603i 1.73205i 0.366025 1.36603i 0.366025 + 1.36603i −0.767949 2.86603i 1.36603 1.36603i 2.23205 + 3.86603i 0.633975 0.366025i
 $$n$$: 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.be even 12 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.be.a 4
13.d odd 4 1 403.2.be.b yes 4
31.e odd 6 1 403.2.be.b yes 4
403.be even 12 1 inner 403.2.be.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.be.a 4 1.a even 1 1 trivial
403.2.be.a 4 403.be even 12 1 inner
403.2.be.b yes 4 13.d odd 4 1
403.2.be.b yes 4 31.e odd 6 1

Hecke kernels

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

 $$T_{2}^{4} + 2 T_{2}^{3} + 2 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{7}^{4} + 10 T_{7}^{3} + 41 T_{7}^{2} + 104 T_{7} + 169$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} + 2 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8}$$
$3$ $$1 - 6 T + 20 T^{2} - 48 T^{3} + 91 T^{4} - 144 T^{5} + 180 T^{6} - 162 T^{7} + 81 T^{8}$$
$5$ $$( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} )( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} )$$
$7$ $$( 1 + 5 T + 7 T^{2} )^{2}( 1 + 2 T^{2} + 49 T^{4} )$$
$11$ $$1 - 6 T + 9 T^{2} + 66 T^{3} - 376 T^{4} + 726 T^{5} + 1089 T^{6} - 7986 T^{7} + 14641 T^{8}$$
$13$ $$1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4}$$
$17$ $$( 1 - 17 T^{2} + 289 T^{4} )^{2}$$
$19$ $$1 - 6 T + 45 T^{2} - 126 T^{3} + 704 T^{4} - 2394 T^{5} + 16245 T^{6} - 41154 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + 2 T + 20 T^{2} + 46 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4} )( 1 + 4 T - 13 T^{2} + 116 T^{3} + 841 T^{4} )$$
$31$ $$1 - 13 T^{2} + 961 T^{4}$$
$37$ $$1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 20 T + 164 T^{2} + 668 T^{3} + 2335 T^{4} + 27388 T^{5} + 275684 T^{6} + 1378420 T^{7} + 2825761 T^{8}$$
$43$ $$1 - 2 T - 56 T^{2} + 52 T^{3} + 1579 T^{4} + 2236 T^{5} - 103544 T^{6} - 159014 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 20 T + 200 T^{2} - 1460 T^{3} + 9982 T^{4} - 68620 T^{5} + 441800 T^{6} - 2076460 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 30 T + 477 T^{2} + 5310 T^{3} + 44420 T^{4} + 281430 T^{5} + 1339893 T^{6} + 4466310 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 22 T + 122 T^{2} - 1232 T^{3} - 20033 T^{4} - 72688 T^{5} + 424682 T^{6} + 4518338 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 182 T^{2} + 15291 T^{4} - 677222 T^{6} + 13845841 T^{8}$$
$67$ $$1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 82008 T^{5} + 614993 T^{6} - 2406104 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 16 T + 65 T^{2} + 1016 T^{3} - 15428 T^{4} + 72136 T^{5} + 327665 T^{6} - 5726576 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 6 T + 90 T^{2} - 564 T^{3} - 97 T^{4} - 41172 T^{5} + 479610 T^{6} + 2334102 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 36 T + 694 T^{2} + 9432 T^{3} + 96531 T^{4} + 745128 T^{5} + 4331254 T^{6} + 17749404 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 38 T + 650 T^{2} - 6944 T^{3} + 62479 T^{4} - 576352 T^{5} + 4477850 T^{6} - 21727906 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 16 T + 128 T^{2} + 1840 T^{3} + 25774 T^{4} + 163760 T^{5} + 1013888 T^{6} + 11279504 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 20 T + 200 T^{2} + 2460 T^{3} + 29582 T^{4} + 238620 T^{5} + 1881800 T^{6} + 18253460 T^{7} + 88529281 T^{8}$$