# Properties

 Label 1045.2.b.a Level $1045$ Weight $2$ Character orbit 1045.b Analytic conductor $8.344$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.34436701122$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + (\beta_{2} + \beta_1 - 1) q^{5} + (\beta_{3} + 2) q^{6} + 2 \beta_1 q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 - b1) * q^3 + (-b3 - 1) * q^4 + (b2 + b1 - 1) * q^5 + (b3 + 2) * q^6 + 2*b1 * q^7 + (-b2 - 2*b1) * q^8 - q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + (\beta_{2} + \beta_1 - 1) q^{5} + (\beta_{3} + 2) q^{6} + 2 \beta_1 q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - q^{9} + ( - \beta_{3} - \beta_1 - 2) q^{10} - q^{11} + ( - \beta_{2} + 3 \beta_1) q^{12} + ( - 3 \beta_{2} - \beta_1) q^{13} + ( - 2 \beta_{3} - 6) q^{14} + (\beta_{2} + \beta_1 + 4) q^{15} + 3 q^{16} + (3 \beta_{2} + \beta_1) q^{17} - \beta_1 q^{18} + q^{19} + (\beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{20} + (2 \beta_{3} + 4) q^{21} - \beta_1 q^{22} + ( - 3 \beta_{2} + \beta_1) q^{23} + ( - \beta_{3} - 6) q^{24} + ( - 2 \beta_{2} - 2 \beta_1 - 3) q^{25} + \beta_{3} q^{26} + ( - 2 \beta_{2} - 2 \beta_1) q^{27} + ( - 2 \beta_{2} - 8 \beta_1) q^{28} + ( - \beta_{3} + 2) q^{29} + ( - \beta_{3} + 4 \beta_1 - 2) q^{30} + (\beta_{3} - 4) q^{31} + ( - 2 \beta_{2} - \beta_1) q^{32} + (\beta_{2} + \beta_1) q^{33} - \beta_{3} q^{34} + ( - 2 \beta_{3} - 2 \beta_1 - 4) q^{35} + (\beta_{3} + 1) q^{36} + ( - 3 \beta_{2} + 3 \beta_1) q^{37} + \beta_1 q^{38} + (2 \beta_{3} - 8) q^{39} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{40} + ( - 3 \beta_{3} - 2) q^{41} + (2 \beta_{2} + 10 \beta_1) q^{42} - 2 \beta_1 q^{43} + (\beta_{3} + 1) q^{44} + ( - \beta_{2} - \beta_1 + 1) q^{45} + ( - \beta_{3} - 6) q^{46} + (5 \beta_{2} + \beta_1) q^{47} + ( - 3 \beta_{2} - 3 \beta_1) q^{48} + ( - 4 \beta_{3} - 5) q^{49} + (2 \beta_{3} - 3 \beta_1 + 4) q^{50} + ( - 2 \beta_{3} + 8) q^{51} + ( - 5 \beta_{2} + \beta_1) q^{52} + ( - 5 \beta_{2} - 3 \beta_1) q^{53} + (2 \beta_{3} + 4) q^{54} + ( - \beta_{2} - \beta_1 + 1) q^{55} + (4 \beta_{3} + 10) q^{56} + ( - \beta_{2} - \beta_1) q^{57} + ( - \beta_{2} - \beta_1) q^{58} - \beta_{3} q^{59} + ( - 4 \beta_{3} + \beta_{2} - 3 \beta_1 - 4) q^{60} - 2 q^{61} + (\beta_{2} - \beta_1) q^{62} - 2 \beta_1 q^{63} + (\beta_{3} + 7) q^{64} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 8) q^{65} + ( - \beta_{3} - 2) q^{66} + (3 \beta_{2} + 3 \beta_1) q^{67} + (5 \beta_{2} - \beta_1) q^{68} + (4 \beta_{3} - 4) q^{69} + (2 \beta_{3} - 2 \beta_{2} - 10 \beta_1 + 6) q^{70} + (\beta_{3} - 12) q^{71} + (\beta_{2} + 2 \beta_1) q^{72} + ( - \beta_{2} - 3 \beta_1) q^{73} + ( - 3 \beta_{3} - 12) q^{74} + (3 \beta_{2} + 3 \beta_1 - 8) q^{75} + ( - \beta_{3} - 1) q^{76} - 2 \beta_1 q^{77} + (2 \beta_{2} - 2 \beta_1) q^{78} + 2 \beta_{3} q^{79} + (3 \beta_{2} + 3 \beta_1 - 3) q^{80} - 11 q^{81} + ( - 3 \beta_{2} - 11 \beta_1) q^{82} + (2 \beta_{2} - 4 \beta_1) q^{83} + ( - 6 \beta_{3} - 20) q^{84} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 - 8) q^{85} + (2 \beta_{3} + 6) q^{86} - 4 \beta_{2} q^{87} + (\beta_{2} + 2 \beta_1) q^{88} + (2 \beta_{3} + 10) q^{89} + (\beta_{3} + \beta_1 + 2) q^{90} + 2 \beta_{3} q^{91} + ( - 7 \beta_{2} - 7 \beta_1) q^{92} + (6 \beta_{2} + 2 \beta_1) q^{93} + ( - \beta_{3} + 2) q^{94} + (\beta_{2} + \beta_1 - 1) q^{95} + (\beta_{3} - 6) q^{96} + (\beta_{2} + 3 \beta_1) q^{97} + ( - 4 \beta_{2} - 17 \beta_1) q^{98} + q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 - b1) * q^3 + (-b3 - 1) * q^4 + (b2 + b1 - 1) * q^5 + (b3 + 2) * q^6 + 2*b1 * q^7 + (-b2 - 2*b1) * q^8 - q^9 + (-b3 - b1 - 2) * q^10 - q^11 + (-b2 + 3*b1) * q^12 + (-3*b2 - b1) * q^13 + (-2*b3 - 6) * q^14 + (b2 + b1 + 4) * q^15 + 3 * q^16 + (3*b2 + b1) * q^17 - b1 * q^18 + q^19 + (b3 + b2 - 3*b1 + 1) * q^20 + (2*b3 + 4) * q^21 - b1 * q^22 + (-3*b2 + b1) * q^23 + (-b3 - 6) * q^24 + (-2*b2 - 2*b1 - 3) * q^25 + b3 * q^26 + (-2*b2 - 2*b1) * q^27 + (-2*b2 - 8*b1) * q^28 + (-b3 + 2) * q^29 + (-b3 + 4*b1 - 2) * q^30 + (b3 - 4) * q^31 + (-2*b2 - b1) * q^32 + (b2 + b1) * q^33 - b3 * q^34 + (-2*b3 - 2*b1 - 4) * q^35 + (b3 + 1) * q^36 + (-3*b2 + 3*b1) * q^37 + b1 * q^38 + (2*b3 - 8) * q^39 + (b3 + b2 + 2*b1 + 6) * q^40 + (-3*b3 - 2) * q^41 + (2*b2 + 10*b1) * q^42 - 2*b1 * q^43 + (b3 + 1) * q^44 + (-b2 - b1 + 1) * q^45 + (-b3 - 6) * q^46 + (5*b2 + b1) * q^47 + (-3*b2 - 3*b1) * q^48 + (-4*b3 - 5) * q^49 + (2*b3 - 3*b1 + 4) * q^50 + (-2*b3 + 8) * q^51 + (-5*b2 + b1) * q^52 + (-5*b2 - 3*b1) * q^53 + (2*b3 + 4) * q^54 + (-b2 - b1 + 1) * q^55 + (4*b3 + 10) * q^56 + (-b2 - b1) * q^57 + (-b2 - b1) * q^58 - b3 * q^59 + (-4*b3 + b2 - 3*b1 - 4) * q^60 - 2 * q^61 + (b2 - b1) * q^62 - 2*b1 * q^63 + (b3 + 7) * q^64 + (-2*b3 + 3*b2 + b1 + 8) * q^65 + (-b3 - 2) * q^66 + (3*b2 + 3*b1) * q^67 + (5*b2 - b1) * q^68 + (4*b3 - 4) * q^69 + (2*b3 - 2*b2 - 10*b1 + 6) * q^70 + (b3 - 12) * q^71 + (b2 + 2*b1) * q^72 + (-b2 - 3*b1) * q^73 + (-3*b3 - 12) * q^74 + (3*b2 + 3*b1 - 8) * q^75 + (-b3 - 1) * q^76 - 2*b1 * q^77 + (2*b2 - 2*b1) * q^78 + 2*b3 * q^79 + (3*b2 + 3*b1 - 3) * q^80 - 11 * q^81 + (-3*b2 - 11*b1) * q^82 + (2*b2 - 4*b1) * q^83 + (-6*b3 - 20) * q^84 + (2*b3 - 3*b2 - b1 - 8) * q^85 + (2*b3 + 6) * q^86 - 4*b2 * q^87 + (b2 + 2*b1) * q^88 + (2*b3 + 10) * q^89 + (b3 + b1 + 2) * q^90 + 2*b3 * q^91 + (-7*b2 - 7*b1) * q^92 + (6*b2 + 2*b1) * q^93 + (-b3 + 2) * q^94 + (b2 + b1 - 1) * q^95 + (b3 - 6) * q^96 + (b2 + 3*b1) * q^97 + (-4*b2 - 17*b1) * q^98 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{5} + 8 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^5 + 8 * q^6 - 4 * q^9 $$4 q - 4 q^{4} - 4 q^{5} + 8 q^{6} - 4 q^{9} - 8 q^{10} - 4 q^{11} - 24 q^{14} + 16 q^{15} + 12 q^{16} + 4 q^{19} + 4 q^{20} + 16 q^{21} - 24 q^{24} - 12 q^{25} + 8 q^{29} - 8 q^{30} - 16 q^{31} - 16 q^{35} + 4 q^{36} - 32 q^{39} + 24 q^{40} - 8 q^{41} + 4 q^{44} + 4 q^{45} - 24 q^{46} - 20 q^{49} + 16 q^{50} + 32 q^{51} + 16 q^{54} + 4 q^{55} + 40 q^{56} - 16 q^{60} - 8 q^{61} + 28 q^{64} + 32 q^{65} - 8 q^{66} - 16 q^{69} + 24 q^{70} - 48 q^{71} - 48 q^{74} - 32 q^{75} - 4 q^{76} - 12 q^{80} - 44 q^{81} - 80 q^{84} - 32 q^{85} + 24 q^{86} + 40 q^{89} + 8 q^{90} + 8 q^{94} - 4 q^{95} - 24 q^{96} + 4 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^5 + 8 * q^6 - 4 * q^9 - 8 * q^10 - 4 * q^11 - 24 * q^14 + 16 * q^15 + 12 * q^16 + 4 * q^19 + 4 * q^20 + 16 * q^21 - 24 * q^24 - 12 * q^25 + 8 * q^29 - 8 * q^30 - 16 * q^31 - 16 * q^35 + 4 * q^36 - 32 * q^39 + 24 * q^40 - 8 * q^41 + 4 * q^44 + 4 * q^45 - 24 * q^46 - 20 * q^49 + 16 * q^50 + 32 * q^51 + 16 * q^54 + 4 * q^55 + 40 * q^56 - 16 * q^60 - 8 * q^61 + 28 * q^64 + 32 * q^65 - 8 * q^66 - 16 * q^69 + 24 * q^70 - 48 * q^71 - 48 * q^74 - 32 * q^75 - 4 * q^76 - 12 * q^80 - 44 * q^81 - 80 * q^84 - 32 * q^85 + 24 * q^86 + 40 * q^89 + 8 * q^90 + 8 * q^94 - 4 * q^95 - 24 * q^96 + 4 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ v^3 + v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}$$ -v^3 + v^2 - v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (b3 - b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (-b3 - b2 + b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$$.

 $$n$$ $$496$$ $$761$$ $$837$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
419.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 2.00000i −3.82843 −1.00000 2.00000i 4.82843 4.82843i 4.41421i −1.00000 −4.82843 + 2.41421i
419.2 0.414214i 2.00000i 1.82843 −1.00000 + 2.00000i −0.828427 0.828427i 1.58579i −1.00000 0.828427 + 0.414214i
419.3 0.414214i 2.00000i 1.82843 −1.00000 2.00000i −0.828427 0.828427i 1.58579i −1.00000 0.828427 0.414214i
419.4 2.41421i 2.00000i −3.82843 −1.00000 + 2.00000i 4.82843 4.82843i 4.41421i −1.00000 −4.82843 2.41421i
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.b.a 4
5.b even 2 1 inner 1045.2.b.a 4
5.c odd 4 1 5225.2.a.d 2
5.c odd 4 1 5225.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.a 4 1.a even 1 1 trivial
1045.2.b.a 4 5.b even 2 1 inner
5225.2.a.d 2 5.c odd 4 1
5225.2.a.g 2 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 6T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1045, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$(T^{2} + 4)^{2}$$
$5$ $$(T^{2} + 2 T + 5)^{2}$$
$7$ $$T^{4} + 24T^{2} + 16$$
$11$ $$(T + 1)^{4}$$
$13$ $$T^{4} + 48T^{2} + 64$$
$17$ $$T^{4} + 48T^{2} + 64$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} + 72T^{2} + 784$$
$29$ $$(T^{2} - 4 T - 4)^{2}$$
$31$ $$(T^{2} + 8 T + 8)^{2}$$
$37$ $$(T^{2} + 72)^{2}$$
$41$ $$(T^{2} + 4 T - 68)^{2}$$
$43$ $$T^{4} + 24T^{2} + 16$$
$47$ $$T^{4} + 136T^{2} + 16$$
$53$ $$T^{4} + 144T^{2} + 3136$$
$59$ $$(T^{2} - 8)^{2}$$
$61$ $$(T + 2)^{4}$$
$67$ $$(T^{2} + 36)^{2}$$
$71$ $$(T^{2} + 24 T + 136)^{2}$$
$73$ $$T^{4} + 48T^{2} + 64$$
$79$ $$(T^{2} - 32)^{2}$$
$83$ $$T^{4} + 152T^{2} + 4624$$
$89$ $$(T^{2} - 20 T + 68)^{2}$$
$97$ $$T^{4} + 48T^{2} + 64$$
show more
show less