# Properties

 Label 625.2.b.b Level $625$ Weight $2$ Character orbit 625.b Analytic conductor $4.991$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.99065012633$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + ( - 2 \beta_{2} + 1) q^{6} + (\beta_{3} + 3 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + (3 \beta_{2} + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b3 - b1) * q^3 + (b2 + 1) * q^4 + (-2*b2 + 1) * q^6 + (b3 + 3*b1) * q^7 + (b3 + 2*b1) * q^8 + (3*b2 + 1) * q^9 $$q + \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + ( - 2 \beta_{2} + 1) q^{6} + (\beta_{3} + 3 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + (3 \beta_{2} + 1) q^{9} + ( - \beta_{2} - 1) q^{11} + \beta_1 q^{12} + (\beta_{3} + 4 \beta_1) q^{13} + (2 \beta_{2} - 3) q^{14} + 3 \beta_{2} q^{16} + ( - 5 \beta_{3} - 4 \beta_1) q^{17} + (3 \beta_{3} - 2 \beta_1) q^{18} + ( - 3 \beta_{2} - 4) q^{19} + ( - 5 \beta_{2} + 2) q^{21} - \beta_{3} q^{22} + (3 \beta_{3} + 3 \beta_1) q^{23} + ( - 3 \beta_{2} + 1) q^{24} + (3 \beta_{2} - 4) q^{26} + (\beta_{3} + 2 \beta_1) q^{27} + (4 \beta_{3} + \beta_1) q^{28} + (2 \beta_{2} + 6) q^{29} + 2 q^{31} + (5 \beta_{3} + \beta_1) q^{32} - \beta_1 q^{33} + (\beta_{2} + 4) q^{34} + (\beta_{2} + 4) q^{36} - 3 \beta_{3} q^{37} + ( - 3 \beta_{3} - \beta_1) q^{38} + ( - 7 \beta_{2} + 3) q^{39} - 4 \beta_{2} q^{41} + ( - 5 \beta_{3} + 7 \beta_1) q^{42} + (6 \beta_{3} + 4 \beta_1) q^{43} + ( - \beta_{2} - 2) q^{44} - 3 q^{46} + ( - 10 \beta_{3} + \beta_1) q^{47} + ( - 3 \beta_{3} + 6 \beta_1) q^{48} + (3 \beta_{2} - 3) q^{49} + (3 \beta_{2} + 1) q^{51} + (5 \beta_{3} + \beta_1) q^{52} + (2 \beta_{3} - 4 \beta_1) q^{53} + (\beta_{2} - 2) q^{54} + (\beta_{2} - 7) q^{56} + ( - \beta_{3} - 2 \beta_1) q^{57} + (2 \beta_{3} + 4 \beta_1) q^{58} + ( - 8 \beta_{2} + 1) q^{59} + (3 \beta_{2} - 4) q^{61} + 2 \beta_1 q^{62} + (10 \beta_{3} - 3 \beta_1) q^{63} + (2 \beta_{2} - 1) q^{64} + ( - \beta_{2} + 1) q^{66} + (5 \beta_{3} - 4 \beta_1) q^{67} + ( - 9 \beta_{3} - 5 \beta_1) q^{68} - 3 \beta_{2} q^{69} + (7 \beta_{2} + 8) q^{71} + (7 \beta_{3} - \beta_1) q^{72} + ( - 5 \beta_{3} - 3 \beta_1) q^{73} + 3 \beta_{2} q^{74} + ( - 4 \beta_{2} - 7) q^{76} + ( - 4 \beta_{3} - \beta_1) q^{77} + ( - 7 \beta_{3} + 10 \beta_1) q^{78} + ( - 6 \beta_{2} - 3) q^{79} + (6 \beta_{2} + 4) q^{81} + ( - 4 \beta_{3} + 4 \beta_1) q^{82} - 8 \beta_1 q^{83} + (2 \beta_{2} - 3) q^{84} + ( - 2 \beta_{2} - 4) q^{86} + (4 \beta_{3} - 2 \beta_1) q^{87} + ( - 3 \beta_{3} - \beta_1) q^{88} + (\beta_{2} - 2) q^{89} + (5 \beta_{2} - 13) q^{91} + (6 \beta_{3} + 3 \beta_1) q^{92} + (2 \beta_{3} - 2 \beta_1) q^{93} + (11 \beta_{2} - 1) q^{94} + (3 \beta_{2} - 4) q^{96} + (9 \beta_{3} - \beta_1) q^{97} + (3 \beta_{3} - 6 \beta_1) q^{98} + ( - \beta_{2} - 4) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b3 - b1) * q^3 + (b2 + 1) * q^4 + (-2*b2 + 1) * q^6 + (b3 + 3*b1) * q^7 + (b3 + 2*b1) * q^8 + (3*b2 + 1) * q^9 + (-b2 - 1) * q^11 + b1 * q^12 + (b3 + 4*b1) * q^13 + (2*b2 - 3) * q^14 + 3*b2 * q^16 + (-5*b3 - 4*b1) * q^17 + (3*b3 - 2*b1) * q^18 + (-3*b2 - 4) * q^19 + (-5*b2 + 2) * q^21 - b3 * q^22 + (3*b3 + 3*b1) * q^23 + (-3*b2 + 1) * q^24 + (3*b2 - 4) * q^26 + (b3 + 2*b1) * q^27 + (4*b3 + b1) * q^28 + (2*b2 + 6) * q^29 + 2 * q^31 + (5*b3 + b1) * q^32 - b1 * q^33 + (b2 + 4) * q^34 + (b2 + 4) * q^36 - 3*b3 * q^37 + (-3*b3 - b1) * q^38 + (-7*b2 + 3) * q^39 - 4*b2 * q^41 + (-5*b3 + 7*b1) * q^42 + (6*b3 + 4*b1) * q^43 + (-b2 - 2) * q^44 - 3 * q^46 + (-10*b3 + b1) * q^47 + (-3*b3 + 6*b1) * q^48 + (3*b2 - 3) * q^49 + (3*b2 + 1) * q^51 + (5*b3 + b1) * q^52 + (2*b3 - 4*b1) * q^53 + (b2 - 2) * q^54 + (b2 - 7) * q^56 + (-b3 - 2*b1) * q^57 + (2*b3 + 4*b1) * q^58 + (-8*b2 + 1) * q^59 + (3*b2 - 4) * q^61 + 2*b1 * q^62 + (10*b3 - 3*b1) * q^63 + (2*b2 - 1) * q^64 + (-b2 + 1) * q^66 + (5*b3 - 4*b1) * q^67 + (-9*b3 - 5*b1) * q^68 - 3*b2 * q^69 + (7*b2 + 8) * q^71 + (7*b3 - b1) * q^72 + (-5*b3 - 3*b1) * q^73 + 3*b2 * q^74 + (-4*b2 - 7) * q^76 + (-4*b3 - b1) * q^77 + (-7*b3 + 10*b1) * q^78 + (-6*b2 - 3) * q^79 + (6*b2 + 4) * q^81 + (-4*b3 + 4*b1) * q^82 - 8*b1 * q^83 + (2*b2 - 3) * q^84 + (-2*b2 - 4) * q^86 + (4*b3 - 2*b1) * q^87 + (-3*b3 - b1) * q^88 + (b2 - 2) * q^89 + (5*b2 - 13) * q^91 + (6*b3 + 3*b1) * q^92 + (2*b3 - 2*b1) * q^93 + (11*b2 - 1) * q^94 + (3*b2 - 4) * q^96 + (9*b3 - b1) * q^97 + (3*b3 - 6*b1) * q^98 + (-b2 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 8 q^{6} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 + 8 * q^6 - 2 * q^9 $$4 q + 2 q^{4} + 8 q^{6} - 2 q^{9} - 2 q^{11} - 16 q^{14} - 6 q^{16} - 10 q^{19} + 18 q^{21} + 10 q^{24} - 22 q^{26} + 20 q^{29} + 8 q^{31} + 14 q^{34} + 14 q^{36} + 26 q^{39} + 8 q^{41} - 6 q^{44} - 12 q^{46} - 18 q^{49} - 2 q^{51} - 10 q^{54} - 30 q^{56} + 20 q^{59} - 22 q^{61} - 8 q^{64} + 6 q^{66} + 6 q^{69} + 18 q^{71} - 6 q^{74} - 20 q^{76} + 4 q^{81} - 16 q^{84} - 12 q^{86} - 10 q^{89} - 62 q^{91} - 26 q^{94} - 22 q^{96} - 14 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 + 8 * q^6 - 2 * q^9 - 2 * q^11 - 16 * q^14 - 6 * q^16 - 10 * q^19 + 18 * q^21 + 10 * q^24 - 22 * q^26 + 20 * q^29 + 8 * q^31 + 14 * q^34 + 14 * q^36 + 26 * q^39 + 8 * q^41 - 6 * q^44 - 12 * q^46 - 18 * q^49 - 2 * q^51 - 10 * q^54 - 30 * q^56 + 20 * q^59 - 22 * q^61 - 8 * q^64 + 6 * q^66 + 6 * q^69 + 18 * q^71 - 6 * q^74 - 20 * q^76 + 4 * q^81 - 16 * q^84 - 12 * q^86 - 10 * q^89 - 62 * q^91 - 26 * q^94 - 22 * q^96 - 14 * 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 $$\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
624.1
 − 1.61803i − 0.618034i 0.618034i 1.61803i
1.61803i 2.61803i −0.618034 0 4.23607 3.85410i 2.23607i −3.85410 0
624.2 0.618034i 0.381966i 1.61803 0 −0.236068 2.85410i 2.23607i 2.85410 0
624.3 0.618034i 0.381966i 1.61803 0 −0.236068 2.85410i 2.23607i 2.85410 0
624.4 1.61803i 2.61803i −0.618034 0 4.23607 3.85410i 2.23607i −3.85410 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 625.2.b.b 4
5.b even 2 1 inner 625.2.b.b 4
5.c odd 4 1 625.2.a.a 2
5.c odd 4 1 625.2.a.d yes 2
15.e even 4 1 5625.2.a.c 2
15.e even 4 1 5625.2.a.e 2
20.e even 4 1 10000.2.a.b 2
20.e even 4 1 10000.2.a.m 2
25.d even 5 2 625.2.e.e 8
25.d even 5 2 625.2.e.f 8
25.e even 10 2 625.2.e.e 8
25.e even 10 2 625.2.e.f 8
25.f odd 20 2 625.2.d.c 4
25.f odd 20 2 625.2.d.e 4
25.f odd 20 2 625.2.d.f 4
25.f odd 20 2 625.2.d.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.a 2 5.c odd 4 1
625.2.a.d yes 2 5.c odd 4 1
625.2.b.b 4 1.a even 1 1 trivial
625.2.b.b 4 5.b even 2 1 inner
625.2.d.c 4 25.f odd 20 2
625.2.d.e 4 25.f odd 20 2
625.2.d.f 4 25.f odd 20 2
625.2.d.i 4 25.f odd 20 2
625.2.e.e 8 25.d even 5 2
625.2.e.e 8 25.e even 10 2
625.2.e.f 8 25.d even 5 2
625.2.e.f 8 25.e even 10 2
5625.2.a.c 2 15.e even 4 1
5625.2.a.e 2 15.e even 4 1
10000.2.a.b 2 20.e even 4 1
10000.2.a.m 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(625, [\chi])$$:

 $$T_{2}^{4} + 3T_{2}^{2} + 1$$ T2^4 + 3*T2^2 + 1 $$T_{3}^{4} + 7T_{3}^{2} + 1$$ T3^4 + 7*T3^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 1$$
$3$ $$T^{4} + 7T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 23T^{2} + 121$$
$11$ $$(T^{2} + T - 1)^{2}$$
$13$ $$T^{4} + 42T^{2} + 361$$
$17$ $$T^{4} + 58T^{2} + 121$$
$19$ $$(T^{2} + 5 T - 5)^{2}$$
$23$ $$T^{4} + 27T^{2} + 81$$
$29$ $$(T^{2} - 10 T + 20)^{2}$$
$31$ $$(T - 2)^{4}$$
$37$ $$(T^{2} + 9)^{2}$$
$41$ $$(T^{2} - 4 T - 16)^{2}$$
$43$ $$T^{4} + 72T^{2} + 16$$
$47$ $$T^{4} + 223 T^{2} + 11881$$
$53$ $$T^{4} + 72T^{2} + 16$$
$59$ $$(T^{2} - 10 T - 55)^{2}$$
$61$ $$(T^{2} + 11 T + 19)^{2}$$
$67$ $$T^{4} + 138T^{2} + 841$$
$71$ $$(T^{2} - 9 T - 41)^{2}$$
$73$ $$T^{4} + 47T^{2} + 1$$
$79$ $$(T^{2} - 45)^{2}$$
$83$ $$T^{4} + 192T^{2} + 4096$$
$89$ $$(T^{2} + 5 T + 5)^{2}$$
$97$ $$T^{4} + 183T^{2} + 7921$$