# Properties

 Label 125.2.b.a 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: 4.0.4400.1 Defining polynomial: $$x^{4} + 7x^{2} + 11$$ x^4 + 7*x^2 + 11 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 11$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ b2 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 4\beta_1$$ b3 - 4*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
 1.54336i − 2.14896i 2.14896i − 1.54336i
2.49721i 1.54336i −4.23607 0 −3.85410 0.953850i 5.58394i 0.618034 0
124.2 1.32813i 2.14896i 0.236068 0 2.85410 3.47709i 2.96979i −1.61803 0
124.3 1.32813i 2.14896i 0.236068 0 2.85410 3.47709i 2.96979i −1.61803 0
124.4 2.49721i 1.54336i −4.23607 0 −3.85410 0.953850i 5.58394i 0.618034 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.a 4
3.b odd 2 1 1125.2.b.a 4
4.b odd 2 1 2000.2.c.c 4
5.b even 2 1 inner 125.2.b.a 4
5.c odd 4 2 125.2.a.c 4
15.d odd 2 1 1125.2.b.a 4
15.e even 4 2 1125.2.a.k 4
20.d odd 2 1 2000.2.c.c 4
20.e even 4 2 2000.2.a.o 4
25.d even 5 2 625.2.e.b 8
25.d even 5 2 625.2.e.h 8
25.e even 10 2 625.2.e.b 8
25.e even 10 2 625.2.e.h 8
25.f odd 20 4 625.2.d.k 8
25.f odd 20 4 625.2.d.l 8
35.f even 4 2 6125.2.a.o 4
40.i odd 4 2 8000.2.a.bj 4
40.k even 4 2 8000.2.a.bk 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 8T_{2}^{2} + 11$$ acting on $$S_{2}^{\mathrm{new}}(125, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 8T^{2} + 11$$
$3$ $$T^{4} + 7T^{2} + 11$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 13T^{2} + 11$$
$11$ $$(T - 2)^{4}$$
$13$ $$T^{4} + 32T^{2} + 176$$
$17$ $$T^{4} + 28T^{2} + 176$$
$19$ $$(T^{2} + 10 T + 20)^{2}$$
$23$ $$T^{4} + 17T^{2} + 11$$
$29$ $$(T^{2} - 5 T - 5)^{2}$$
$31$ $$(T - 2)^{4}$$
$37$ $$T^{4} + 68T^{2} + 176$$
$41$ $$(T^{2} + T - 31)^{2}$$
$43$ $$T^{4} + 107T^{2} + 1331$$
$47$ $$T^{4} + 43T^{2} + 11$$
$53$ $$T^{4} + 112T^{2} + 2816$$
$59$ $$(T^{2} - 20)^{2}$$
$61$ $$(T^{2} + T - 31)^{2}$$
$67$ $$T^{4} + 28T^{2} + 176$$
$71$ $$(T^{2} + 6 T - 116)^{2}$$
$73$ $$T^{4} + 352 T^{2} + 21296$$
$79$ $$(T^{2} + 10 T + 20)^{2}$$
$83$ $$T^{4} + 77T^{2} + 1331$$
$89$ $$(T^{2} - 15 T + 55)^{2}$$
$97$ $$T^{4} + 128T^{2} + 176$$