# Properties

 Label 125.2.b.b Level $125$ Weight $2$ Character orbit 125.b Analytic conductor $0.998$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$125 = 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 125.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.998130025266$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/125\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
 − 1.61803i − 0.618034i 0.618034i 1.61803i
1.61803i 0.381966i −0.618034 0 0.618034 3.00000i 2.23607i 2.85410 0
124.2 0.618034i 2.61803i 1.61803 0 −1.61803 3.00000i 2.23607i −3.85410 0
124.3 0.618034i 2.61803i 1.61803 0 −1.61803 3.00000i 2.23607i −3.85410 0
124.4 1.61803i 0.381966i −0.618034 0 0.618034 3.00000i 2.23607i 2.85410 0
 $$n$$: 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 125.2.b.b 4
3.b odd 2 1 1125.2.b.f 4
4.b odd 2 1 2000.2.c.e 4
5.b even 2 1 inner 125.2.b.b 4
5.c odd 4 1 125.2.a.a 2
5.c odd 4 1 125.2.a.b yes 2
15.d odd 2 1 1125.2.b.f 4
15.e even 4 1 1125.2.a.c 2
15.e even 4 1 1125.2.a.d 2
20.d odd 2 1 2000.2.c.e 4
20.e even 4 1 2000.2.a.a 2
20.e even 4 1 2000.2.a.l 2
25.d even 5 2 625.2.e.d 8
25.d even 5 2 625.2.e.g 8
25.e even 10 2 625.2.e.d 8
25.e even 10 2 625.2.e.g 8
25.f odd 20 2 625.2.d.a 4
25.f odd 20 2 625.2.d.d 4
25.f odd 20 2 625.2.d.g 4
25.f odd 20 2 625.2.d.j 4
35.f even 4 1 6125.2.a.d 2
35.f even 4 1 6125.2.a.g 2
40.i odd 4 1 8000.2.a.d 2
40.i odd 4 1 8000.2.a.v 2
40.k even 4 1 8000.2.a.c 2
40.k even 4 1 8000.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.a.a 2 5.c odd 4 1
125.2.a.b yes 2 5.c odd 4 1
125.2.b.b 4 1.a even 1 1 trivial
125.2.b.b 4 5.b even 2 1 inner
625.2.d.a 4 25.f odd 20 2
625.2.d.d 4 25.f odd 20 2
625.2.d.g 4 25.f odd 20 2
625.2.d.j 4 25.f odd 20 2
625.2.e.d 8 25.d even 5 2
625.2.e.d 8 25.e even 10 2
625.2.e.g 8 25.d even 5 2
625.2.e.g 8 25.e even 10 2
1125.2.a.c 2 15.e even 4 1
1125.2.a.d 2 15.e even 4 1
1125.2.b.f 4 3.b odd 2 1
1125.2.b.f 4 15.d odd 2 1
2000.2.a.a 2 20.e even 4 1
2000.2.a.l 2 20.e even 4 1
2000.2.c.e 4 4.b odd 2 1
2000.2.c.e 4 20.d odd 2 1
6125.2.a.d 2 35.f even 4 1
6125.2.a.g 2 35.f even 4 1
8000.2.a.c 2 40.k even 4 1
8000.2.a.d 2 40.i odd 4 1
8000.2.a.u 2 40.k even 4 1
8000.2.a.v 2 40.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(125, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 1$$
$3$ $$T^{4} + 7T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 9)^{2}$$
$11$ $$(T + 3)^{4}$$
$13$ $$T^{4} + 27T^{2} + 81$$
$17$ $$T^{4} + 18T^{2} + 1$$
$19$ $$(T^{2} - 5 T + 5)^{2}$$
$23$ $$T^{4} + 12T^{2} + 16$$
$29$ $$(T^{2} - 45)^{2}$$
$31$ $$(T^{2} + T - 31)^{2}$$
$37$ $$T^{4} + 108T^{2} + 1296$$
$41$ $$(T + 3)^{4}$$
$43$ $$(T^{2} + 81)^{2}$$
$47$ $$T^{4} + 123T^{2} + 3721$$
$53$ $$T^{4} + 27T^{2} + 121$$
$59$ $$(T^{2} + 15 T + 45)^{2}$$
$61$ $$(T^{2} + T - 31)^{2}$$
$67$ $$T^{4} + 243T^{2} + 9801$$
$71$ $$(T + 3)^{4}$$
$73$ $$T^{4} + 27T^{2} + 81$$
$79$ $$(T^{2} + 10 T + 5)^{2}$$
$83$ $$T^{4} + 72T^{2} + 16$$
$89$ $$(T^{2} - 180)^{2}$$
$97$ $$T^{4} + 63T^{2} + 81$$