# Properties

 Label 1925.2.b.n Level $1925$ Weight $2$ Character orbit 1925.b Analytic conductor $15.371$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1925 = 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1925.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.3712023891$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 385) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5 \beta_{1} - 7$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} - 9$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$$.

 $$n$$ $$276$$ $$1002$$ $$1751$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 0.403032 − 0.403032i −0.854638 + 0.854638i 1.45161 + 1.45161i 1.45161 − 1.45161i −0.854638 − 0.854638i 0.403032 + 0.403032i
2.67513i 2.48119i −5.15633 0 6.63752 1.00000i 8.44358i −3.15633 0
1849.2 1.53919i 1.17009i −0.369102 0 −1.80098 1.00000i 2.51026i 1.63090 0
1849.3 1.21432i 0.688892i 0.525428 0 −0.836535 1.00000i 3.06668i 2.52543 0
1849.4 1.21432i 0.688892i 0.525428 0 −0.836535 1.00000i 3.06668i 2.52543 0
1849.5 1.53919i 1.17009i −0.369102 0 −1.80098 1.00000i 2.51026i 1.63090 0
1849.6 2.67513i 2.48119i −5.15633 0 6.63752 1.00000i 8.44358i −3.15633 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1849.6 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 1925.2.b.n 6
5.b even 2 1 inner 1925.2.b.n 6
5.c odd 4 1 385.2.a.f 3
5.c odd 4 1 1925.2.a.v 3
15.e even 4 1 3465.2.a.bh 3
20.e even 4 1 6160.2.a.bn 3
35.f even 4 1 2695.2.a.g 3
55.e even 4 1 4235.2.a.q 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.f 3 5.c odd 4 1
1925.2.a.v 3 5.c odd 4 1
1925.2.b.n 6 1.a even 1 1 trivial
1925.2.b.n 6 5.b even 2 1 inner
2695.2.a.g 3 35.f even 4 1
3465.2.a.bh 3 15.e even 4 1
4235.2.a.q 3 55.e even 4 1
6160.2.a.bn 3 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 $$S_{2}^{\mathrm{new}}(1925, [\chi])$$:

 $$T_{2}^{6} + 11 T_{2}^{4} + 31 T_{2}^{2} + 25$$ $$T_{3}^{6} + 8 T_{3}^{4} + 12 T_{3}^{2} + 4$$ $$T_{19}^{3} + 6 T_{19}^{2} - 4 T_{19} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$25 + 31 T^{2} + 11 T^{4} + T^{6}$$
$3$ $$4 + 12 T^{2} + 8 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$( 1 + T^{2} )^{3}$$
$11$ $$( 1 + T )^{6}$$
$13$ $$4 + 476 T^{2} + 48 T^{4} + T^{6}$$
$17$ $$4 + 900 T^{2} + 60 T^{4} + T^{6}$$
$19$ $$( -8 - 4 T + 6 T^{2} + T^{3} )^{2}$$
$23$ $$400 + 384 T^{2} + 44 T^{4} + T^{6}$$
$29$ $$( 40 + 12 T - 10 T^{2} + T^{3} )^{2}$$
$31$ $$( -26 + 20 T + 10 T^{2} + T^{3} )^{2}$$
$37$ $$10000 + 5904 T^{2} + 152 T^{4} + T^{6}$$
$41$ $$( -54 - 36 T + T^{3} )^{2}$$
$43$ $$71824 + 7392 T^{2} + 188 T^{4} + T^{6}$$
$47$ $$24964 + 5780 T^{2} + 180 T^{4} + T^{6}$$
$53$ $$16 + 304 T^{2} + 104 T^{4} + T^{6}$$
$59$ $$( -74 + 14 T^{2} + T^{3} )^{2}$$
$61$ $$( -62 - 60 T - 10 T^{2} + T^{3} )^{2}$$
$67$ $$29584 + 13232 T^{2} + 228 T^{4} + T^{6}$$
$71$ $$( -800 + 80 T + 24 T^{2} + T^{3} )^{2}$$
$73$ $$36100 + 26484 T^{2} + 332 T^{4} + T^{6}$$
$79$ $$( -244 - 112 T + 8 T^{2} + T^{3} )^{2}$$
$83$ $$1201216 + 35376 T^{2} + 332 T^{4} + T^{6}$$
$89$ $$( -320 + 48 T + 20 T^{2} + T^{3} )^{2}$$
$97$ $$25600 + 36864 T^{2} + 384 T^{4} + T^{6}$$