# Properties

 Label 605.2.b.d Level $605$ Weight $2$ Character orbit 605.b Analytic conductor $4.831$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 4\beta_1$$ b3 - 4*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
364.1
 − 1.93185i − 0.517638i 0.517638i 1.93185i
1.93185i 0.517638i −1.73205 −1.73205 1.41421i −1.00000 0.896575i 0.517638i 2.73205 −2.73205 + 3.34607i
364.2 0.517638i 1.93185i 1.73205 1.73205 + 1.41421i −1.00000 3.34607i 1.93185i −0.732051 0.732051 0.896575i
364.3 0.517638i 1.93185i 1.73205 1.73205 1.41421i −1.00000 3.34607i 1.93185i −0.732051 0.732051 + 0.896575i
364.4 1.93185i 0.517638i −1.73205 −1.73205 + 1.41421i −1.00000 0.896575i 0.517638i 2.73205 −2.73205 3.34607i
 $$n$$: 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 605.2.b.d 4
5.b even 2 1 inner 605.2.b.d 4
5.c odd 4 2 3025.2.a.y 4
11.b odd 2 1 605.2.b.e yes 4
11.c even 5 4 605.2.j.f 16
11.d odd 10 4 605.2.j.e 16
55.d odd 2 1 605.2.b.e yes 4
55.e even 4 2 3025.2.a.z 4
55.h odd 10 4 605.2.j.e 16
55.j even 10 4 605.2.j.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.d 4 1.a even 1 1 trivial
605.2.b.d 4 5.b even 2 1 inner
605.2.b.e yes 4 11.b odd 2 1
605.2.b.e yes 4 55.d odd 2 1
605.2.j.e 16 11.d odd 10 4
605.2.j.e 16 55.h odd 10 4
605.2.j.f 16 11.c even 5 4
605.2.j.f 16 55.j even 10 4
3025.2.a.y 4 5.c odd 4 2
3025.2.a.z 4 55.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{4} + 4T_{2}^{2} + 1$$ T2^4 + 4*T2^2 + 1 $$T_{19}^{2} + 2T_{19} - 26$$ T19^2 + 2*T19 - 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4T^{2} + 1$$
$3$ $$T^{4} + 4T^{2} + 1$$
$5$ $$T^{4} - 2T^{2} + 25$$
$7$ $$T^{4} + 12T^{2} + 9$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 18)^{2}$$
$17$ $$T^{4} + 16T^{2} + 16$$
$19$ $$(T^{2} + 2 T - 26)^{2}$$
$23$ $$T^{4} + 52T^{2} + 484$$
$29$ $$(T^{2} - 48)^{2}$$
$31$ $$(T^{2} + 14 T + 46)^{2}$$
$37$ $$T^{4} + 48T^{2} + 144$$
$41$ $$(T^{2} - 3)^{2}$$
$43$ $$T^{4} + 156T^{2} + 4761$$
$47$ $$T^{4} + 148T^{2} + 2209$$
$53$ $$T^{4} + 148T^{2} + 676$$
$59$ $$(T^{2} - 6 T + 6)^{2}$$
$61$ $$(T^{2} + 20 T + 97)^{2}$$
$67$ $$T^{4} + 228T^{2} + 1089$$
$71$ $$(T^{2} + 6 T - 18)^{2}$$
$73$ $$(T^{2} + 24)^{2}$$
$79$ $$(T^{2} - 4 T - 8)^{2}$$
$83$ $$(T^{2} + 98)^{2}$$
$89$ $$(T^{2} - 6 T - 3)^{2}$$
$97$ $$T^{4} + 84T^{2} + 36$$