# Properties

 Label 1925.2.b.e Level $1925$ Weight $2$ Character orbit 1925.b Analytic conductor $15.371$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 2 q^{4} - i q^{7} - 6 q^{9} +O(q^{10})$$ q + 3*i * q^3 + 2 * q^4 - i * q^7 - 6 * q^9 $$q + 3 i q^{3} + 2 q^{4} - i q^{7} - 6 q^{9} - q^{11} + 6 i q^{12} + 4 i q^{13} + 4 q^{16} + 2 i q^{17} + 6 q^{19} + 3 q^{21} + 5 i q^{23} - 9 i q^{27} - 2 i q^{28} - 10 q^{29} + q^{31} - 3 i q^{33} - 12 q^{36} - 5 i q^{37} - 12 q^{39} - 2 q^{41} + 8 i q^{43} - 2 q^{44} + 8 i q^{47} + 12 i q^{48} - q^{49} - 6 q^{51} + 8 i q^{52} + 6 i q^{53} + 18 i q^{57} - 3 q^{59} - 2 q^{61} + 6 i q^{63} + 8 q^{64} - 3 i q^{67} + 4 i q^{68} - 15 q^{69} + q^{71} - 10 i q^{73} + 12 q^{76} + i q^{77} - 6 q^{79} + 9 q^{81} - 12 i q^{83} + 6 q^{84} - 30 i q^{87} + 15 q^{89} + 4 q^{91} + 10 i q^{92} + 3 i q^{93} - 5 i q^{97} + 6 q^{99} +O(q^{100})$$ q + 3*i * q^3 + 2 * q^4 - i * q^7 - 6 * q^9 - q^11 + 6*i * q^12 + 4*i * q^13 + 4 * q^16 + 2*i * q^17 + 6 * q^19 + 3 * q^21 + 5*i * q^23 - 9*i * q^27 - 2*i * q^28 - 10 * q^29 + q^31 - 3*i * q^33 - 12 * q^36 - 5*i * q^37 - 12 * q^39 - 2 * q^41 + 8*i * q^43 - 2 * q^44 + 8*i * q^47 + 12*i * q^48 - q^49 - 6 * q^51 + 8*i * q^52 + 6*i * q^53 + 18*i * q^57 - 3 * q^59 - 2 * q^61 + 6*i * q^63 + 8 * q^64 - 3*i * q^67 + 4*i * q^68 - 15 * q^69 + q^71 - 10*i * q^73 + 12 * q^76 + i * q^77 - 6 * q^79 + 9 * q^81 - 12*i * q^83 + 6 * q^84 - 30*i * q^87 + 15 * q^89 + 4 * q^91 + 10*i * q^92 + 3*i * q^93 - 5*i * q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - 12 q^{9}+O(q^{10})$$ 2 * q + 4 * q^4 - 12 * q^9 $$2 q + 4 q^{4} - 12 q^{9} - 2 q^{11} + 8 q^{16} + 12 q^{19} + 6 q^{21} - 20 q^{29} + 2 q^{31} - 24 q^{36} - 24 q^{39} - 4 q^{41} - 4 q^{44} - 2 q^{49} - 12 q^{51} - 6 q^{59} - 4 q^{61} + 16 q^{64} - 30 q^{69} + 2 q^{71} + 24 q^{76} - 12 q^{79} + 18 q^{81} + 12 q^{84} + 30 q^{89} + 8 q^{91} + 12 q^{99}+O(q^{100})$$ 2 * q + 4 * q^4 - 12 * q^9 - 2 * q^11 + 8 * q^16 + 12 * q^19 + 6 * q^21 - 20 * q^29 + 2 * q^31 - 24 * q^36 - 24 * q^39 - 4 * q^41 - 4 * q^44 - 2 * q^49 - 12 * q^51 - 6 * q^59 - 4 * q^61 + 16 * q^64 - 30 * q^69 + 2 * q^71 + 24 * q^76 - 12 * q^79 + 18 * q^81 + 12 * q^84 + 30 * q^89 + 8 * q^91 + 12 * q^99

## 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
 − 1.00000i 1.00000i
0 3.00000i 2.00000 0 0 1.00000i 0 −6.00000 0
1849.2 0 3.00000i 2.00000 0 0 1.00000i 0 −6.00000 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 1925.2.b.e 2
5.b even 2 1 inner 1925.2.b.e 2
5.c odd 4 1 77.2.a.a 1
5.c odd 4 1 1925.2.a.h 1
15.e even 4 1 693.2.a.c 1
20.e even 4 1 1232.2.a.l 1
35.f even 4 1 539.2.a.c 1
35.k even 12 2 539.2.e.c 2
35.l odd 12 2 539.2.e.f 2
40.i odd 4 1 4928.2.a.bj 1
40.k even 4 1 4928.2.a.a 1
55.e even 4 1 847.2.a.b 1
55.k odd 20 4 847.2.f.i 4
55.l even 20 4 847.2.f.h 4
105.k odd 4 1 4851.2.a.j 1
140.j odd 4 1 8624.2.a.a 1
165.l odd 4 1 7623.2.a.j 1
385.l odd 4 1 5929.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a 1 5.c odd 4 1
539.2.a.c 1 35.f even 4 1
539.2.e.c 2 35.k even 12 2
539.2.e.f 2 35.l odd 12 2
693.2.a.c 1 15.e even 4 1
847.2.a.b 1 55.e even 4 1
847.2.f.h 4 55.l even 20 4
847.2.f.i 4 55.k odd 20 4
1232.2.a.l 1 20.e even 4 1
1925.2.a.h 1 5.c odd 4 1
1925.2.b.e 2 1.a even 1 1 trivial
1925.2.b.e 2 5.b even 2 1 inner
4851.2.a.j 1 105.k odd 4 1
4928.2.a.a 1 40.k even 4 1
4928.2.a.bj 1 40.i odd 4 1
5929.2.a.f 1 385.l odd 4 1
7623.2.a.j 1 165.l odd 4 1
8624.2.a.a 1 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1925, [\chi])$$:

 $$T_{2}$$ T2 $$T_{3}^{2} + 9$$ T3^2 + 9 $$T_{19} - 6$$ T19 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2} + 25$$
$29$ $$(T + 10)^{2}$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2} + 25$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 3)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 9$$
$71$ $$(T - 1)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T + 6)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 25$$