Properties

 Label 40.2.f.a Level $40$ Weight $2$ Character orbit 40.f Analytic conductor $0.319$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$40 = 2^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 40.f (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.319401608085$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 1) q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{3} - 1) q^{6} + ( - \beta_{2} - 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} - q^{9}+O(q^{10})$$ q + b1 * q^2 + b2 * q^3 + (b3 - 1) * q^4 + (-b3 - b2) * q^5 + (-b3 - 1) * q^6 + (-b2 - 2*b1) * q^7 + 2*b2 * q^8 - q^9 $$q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 1) q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{3} - 1) q^{6} + ( - \beta_{2} - 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} - q^{9} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{10} + 2 \beta_{3} q^{11} + ( - 2 \beta_{2} - 2 \beta_1) q^{12} + ( - \beta_{3} + 3) q^{14} + (\beta_{2} + 2 \beta_1 - 2) q^{15} + ( - 2 \beta_{3} - 2) q^{16} + (2 \beta_{2} + 4 \beta_1) q^{17} - \beta_1 q^{18} - 2 \beta_{3} q^{19} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 3) q^{20} + 2 \beta_{3} q^{21} + (4 \beta_{2} + 2 \beta_1) q^{22} + (\beta_{2} + 2 \beta_1) q^{23} + 4 q^{24} + ( - 2 \beta_{2} - 4 \beta_1 - 1) q^{25} - 4 \beta_{2} q^{27} + ( - 2 \beta_{2} + 2 \beta_1) q^{28} + (\beta_{3} - 2 \beta_1 - 3) q^{30} + 4 q^{31} + ( - 4 \beta_{2} - 4 \beta_1) q^{32} + ( - 2 \beta_{2} - 4 \beta_1) q^{33} + (2 \beta_{3} - 6) q^{34} + ( - 2 \beta_{3} + 3 \beta_{2}) q^{35} + ( - \beta_{3} + 1) q^{36} + 6 \beta_{2} q^{37} + ( - 4 \beta_{2} - 2 \beta_1) q^{38} + (2 \beta_{2} + 4 \beta_1 - 4) q^{40} + (4 \beta_{2} + 2 \beta_1) q^{42} - 3 \beta_{2} q^{43} + ( - 2 \beta_{3} - 6) q^{44} + (\beta_{3} + \beta_{2}) q^{45} + (\beta_{3} - 3) q^{46} + ( - 3 \beta_{2} - 6 \beta_1) q^{47} + 4 \beta_1 q^{48} + q^{49} + ( - 2 \beta_{3} - \beta_1 + 6) q^{50} - 4 \beta_{3} q^{51} - 4 \beta_{2} q^{53} + (4 \beta_{3} + 4) q^{54} + (2 \beta_{2} + 4 \beta_1 + 6) q^{55} + 4 \beta_{3} q^{56} + (2 \beta_{2} + 4 \beta_1) q^{57} + 6 \beta_{3} q^{59} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{60} - 2 \beta_{3} q^{61} + 4 \beta_1 q^{62} + (\beta_{2} + 2 \beta_1) q^{63} + 8 q^{64} + ( - 2 \beta_{3} + 6) q^{66} + 3 \beta_{2} q^{67} + (4 \beta_{2} - 4 \beta_1) q^{68} - 2 \beta_{3} q^{69} + ( - 3 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 3) q^{70} - 12 q^{71} - 2 \beta_{2} q^{72} + (2 \beta_{2} + 4 \beta_1) q^{73} + ( - 6 \beta_{3} - 6) q^{74} + (4 \beta_{3} - \beta_{2}) q^{75} + (2 \beta_{3} + 6) q^{76} - 6 \beta_{2} q^{77} - 4 q^{79} + (2 \beta_{3} - 4 \beta_1 - 6) q^{80} - 5 q^{81} + 7 \beta_{2} q^{83} + ( - 2 \beta_{3} - 6) q^{84} + (4 \beta_{3} - 6 \beta_{2}) q^{85} + (3 \beta_{3} + 3) q^{86} + ( - 4 \beta_{2} - 8 \beta_1) q^{88} + 6 q^{89} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{90} + (2 \beta_{2} - 2 \beta_1) q^{92} + 4 \beta_{2} q^{93} + ( - 3 \beta_{3} + 9) q^{94} + ( - 2 \beta_{2} - 4 \beta_1 - 6) q^{95} + (4 \beta_{3} - 4) q^{96} + ( - 2 \beta_{2} - 4 \beta_1) q^{97} + \beta_1 q^{98} - 2 \beta_{3} q^{99}+O(q^{100})$$ q + b1 * q^2 + b2 * q^3 + (b3 - 1) * q^4 + (-b3 - b2) * q^5 + (-b3 - 1) * q^6 + (-b2 - 2*b1) * q^7 + 2*b2 * q^8 - q^9 + (b3 - 2*b2 - b1 + 1) * q^10 + 2*b3 * q^11 + (-2*b2 - 2*b1) * q^12 + (-b3 + 3) * q^14 + (b2 + 2*b1 - 2) * q^15 + (-2*b3 - 2) * q^16 + (2*b2 + 4*b1) * q^17 - b1 * q^18 - 2*b3 * q^19 + (b3 + 2*b2 + 2*b1 + 3) * q^20 + 2*b3 * q^21 + (4*b2 + 2*b1) * q^22 + (b2 + 2*b1) * q^23 + 4 * q^24 + (-2*b2 - 4*b1 - 1) * q^25 - 4*b2 * q^27 + (-2*b2 + 2*b1) * q^28 + (b3 - 2*b1 - 3) * q^30 + 4 * q^31 + (-4*b2 - 4*b1) * q^32 + (-2*b2 - 4*b1) * q^33 + (2*b3 - 6) * q^34 + (-2*b3 + 3*b2) * q^35 + (-b3 + 1) * q^36 + 6*b2 * q^37 + (-4*b2 - 2*b1) * q^38 + (2*b2 + 4*b1 - 4) * q^40 + (4*b2 + 2*b1) * q^42 - 3*b2 * q^43 + (-2*b3 - 6) * q^44 + (b3 + b2) * q^45 + (b3 - 3) * q^46 + (-3*b2 - 6*b1) * q^47 + 4*b1 * q^48 + q^49 + (-2*b3 - b1 + 6) * q^50 - 4*b3 * q^51 - 4*b2 * q^53 + (4*b3 + 4) * q^54 + (2*b2 + 4*b1 + 6) * q^55 + 4*b3 * q^56 + (2*b2 + 4*b1) * q^57 + 6*b3 * q^59 + (-2*b3 + 2*b2 - 2*b1 + 2) * q^60 - 2*b3 * q^61 + 4*b1 * q^62 + (b2 + 2*b1) * q^63 + 8 * q^64 + (-2*b3 + 6) * q^66 + 3*b2 * q^67 + (4*b2 - 4*b1) * q^68 - 2*b3 * q^69 + (-3*b3 - 4*b2 - 2*b1 - 3) * q^70 - 12 * q^71 - 2*b2 * q^72 + (2*b2 + 4*b1) * q^73 + (-6*b3 - 6) * q^74 + (4*b3 - b2) * q^75 + (2*b3 + 6) * q^76 - 6*b2 * q^77 - 4 * q^79 + (2*b3 - 4*b1 - 6) * q^80 - 5 * q^81 + 7*b2 * q^83 + (-2*b3 - 6) * q^84 + (4*b3 - 6*b2) * q^85 + (3*b3 + 3) * q^86 + (-4*b2 - 8*b1) * q^88 + 6 * q^89 + (-b3 + 2*b2 + b1 - 1) * q^90 + (2*b2 - 2*b1) * q^92 + 4*b2 * q^93 + (-3*b3 + 9) * q^94 + (-2*b2 - 4*b1 - 6) * q^95 + (4*b3 - 4) * q^96 + (-2*b2 - 4*b1) * q^97 + b1 * q^98 - 2*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 $$4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 4 q^{10} + 12 q^{14} - 8 q^{15} - 8 q^{16} + 12 q^{20} + 16 q^{24} - 4 q^{25} - 12 q^{30} + 16 q^{31} - 24 q^{34} + 4 q^{36} - 16 q^{40} - 24 q^{44} - 12 q^{46} + 4 q^{49} + 24 q^{50} + 16 q^{54} + 24 q^{55} + 8 q^{60} + 32 q^{64} + 24 q^{66} - 12 q^{70} - 48 q^{71} - 24 q^{74} + 24 q^{76} - 16 q^{79} - 24 q^{80} - 20 q^{81} - 24 q^{84} + 12 q^{86} + 24 q^{89} - 4 q^{90} + 36 q^{94} - 24 q^{95} - 16 q^{96}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 4 * q^10 + 12 * q^14 - 8 * q^15 - 8 * q^16 + 12 * q^20 + 16 * q^24 - 4 * q^25 - 12 * q^30 + 16 * q^31 - 24 * q^34 + 4 * q^36 - 16 * q^40 - 24 * q^44 - 12 * q^46 + 4 * q^49 + 24 * q^50 + 16 * q^54 + 24 * q^55 + 8 * q^60 + 32 * q^64 + 24 * q^66 - 12 * q^70 - 48 * q^71 - 24 * q^74 + 24 * q^76 - 16 * q^79 - 24 * q^80 - 20 * q^81 - 24 * q^84 + 12 * q^86 + 24 * q^89 - 4 * q^90 + 36 * q^94 - 24 * q^95 - 16 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 1$$ b3 - 1 $$\nu^{3}$$ $$=$$ $$2\beta_{2}$$ 2*b2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/40\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$21$$ $$31$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 −0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i 0.707107 + 1.22474i
−0.707107 1.22474i 1.41421 −1.00000 + 1.73205i −1.41421 1.73205i −1.00000 1.73205i 2.44949i 2.82843 −1.00000 −1.12132 + 2.95680i
29.2 −0.707107 + 1.22474i 1.41421 −1.00000 1.73205i −1.41421 + 1.73205i −1.00000 + 1.73205i 2.44949i 2.82843 −1.00000 −1.12132 2.95680i
29.3 0.707107 1.22474i −1.41421 −1.00000 1.73205i 1.41421 + 1.73205i −1.00000 + 1.73205i 2.44949i −2.82843 −1.00000 3.12132 0.507306i
29.4 0.707107 + 1.22474i −1.41421 −1.00000 + 1.73205i 1.41421 1.73205i −1.00000 1.73205i 2.44949i −2.82843 −1.00000 3.12132 + 0.507306i
 $$n$$: 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
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.2.f.a 4
3.b odd 2 1 360.2.d.b 4
4.b odd 2 1 160.2.f.a 4
5.b even 2 1 inner 40.2.f.a 4
5.c odd 4 2 200.2.d.e 4
8.b even 2 1 inner 40.2.f.a 4
8.d odd 2 1 160.2.f.a 4
12.b even 2 1 1440.2.d.c 4
15.d odd 2 1 360.2.d.b 4
15.e even 4 2 1800.2.k.m 4
16.e even 4 2 1280.2.c.i 4
16.f odd 4 2 1280.2.c.k 4
20.d odd 2 1 160.2.f.a 4
20.e even 4 2 800.2.d.f 4
24.f even 2 1 1440.2.d.c 4
24.h odd 2 1 360.2.d.b 4
40.e odd 2 1 160.2.f.a 4
40.f even 2 1 inner 40.2.f.a 4
40.i odd 4 2 200.2.d.e 4
40.k even 4 2 800.2.d.f 4
60.h even 2 1 1440.2.d.c 4
60.l odd 4 2 7200.2.k.l 4
80.i odd 4 2 6400.2.a.co 4
80.j even 4 2 6400.2.a.cm 4
80.k odd 4 2 1280.2.c.k 4
80.q even 4 2 1280.2.c.i 4
80.s even 4 2 6400.2.a.cm 4
80.t odd 4 2 6400.2.a.co 4
120.i odd 2 1 360.2.d.b 4
120.m even 2 1 1440.2.d.c 4
120.q odd 4 2 7200.2.k.l 4
120.w even 4 2 1800.2.k.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.f.a 4 1.a even 1 1 trivial
40.2.f.a 4 5.b even 2 1 inner
40.2.f.a 4 8.b even 2 1 inner
40.2.f.a 4 40.f even 2 1 inner
160.2.f.a 4 4.b odd 2 1
160.2.f.a 4 8.d odd 2 1
160.2.f.a 4 20.d odd 2 1
160.2.f.a 4 40.e odd 2 1
200.2.d.e 4 5.c odd 4 2
200.2.d.e 4 40.i odd 4 2
360.2.d.b 4 3.b odd 2 1
360.2.d.b 4 15.d odd 2 1
360.2.d.b 4 24.h odd 2 1
360.2.d.b 4 120.i odd 2 1
800.2.d.f 4 20.e even 4 2
800.2.d.f 4 40.k even 4 2
1280.2.c.i 4 16.e even 4 2
1280.2.c.i 4 80.q even 4 2
1280.2.c.k 4 16.f odd 4 2
1280.2.c.k 4 80.k odd 4 2
1440.2.d.c 4 12.b even 2 1
1440.2.d.c 4 24.f even 2 1
1440.2.d.c 4 60.h even 2 1
1440.2.d.c 4 120.m even 2 1
1800.2.k.m 4 15.e even 4 2
1800.2.k.m 4 120.w even 4 2
6400.2.a.cm 4 80.j even 4 2
6400.2.a.cm 4 80.s even 4 2
6400.2.a.co 4 80.i odd 4 2
6400.2.a.co 4 80.t odd 4 2
7200.2.k.l 4 60.l odd 4 2
7200.2.k.l 4 120.q odd 4 2

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(40, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2T^{2} + 4$$
$3$ $$(T^{2} - 2)^{2}$$
$5$ $$T^{4} + 2T^{2} + 25$$
$7$ $$(T^{2} + 6)^{2}$$
$11$ $$(T^{2} + 12)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 24)^{2}$$
$19$ $$(T^{2} + 12)^{2}$$
$23$ $$(T^{2} + 6)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T - 4)^{4}$$
$37$ $$(T^{2} - 72)^{2}$$
$41$ $$T^{4}$$
$43$ $$(T^{2} - 18)^{2}$$
$47$ $$(T^{2} + 54)^{2}$$
$53$ $$(T^{2} - 32)^{2}$$
$59$ $$(T^{2} + 108)^{2}$$
$61$ $$(T^{2} + 12)^{2}$$
$67$ $$(T^{2} - 18)^{2}$$
$71$ $$(T + 12)^{4}$$
$73$ $$(T^{2} + 24)^{2}$$
$79$ $$(T + 4)^{4}$$
$83$ $$(T^{2} - 98)^{2}$$
$89$ $$(T - 6)^{4}$$
$97$ $$(T^{2} + 24)^{2}$$