# Properties

 Label 1225.2.b.d Level $1225$ Weight $2$ Character orbit 1225.b Analytic conductor $9.782$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.78167424761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 1.00000i − 1.00000i
0 1.00000i 2.00000 0 0 0 0 2.00000 0
99.2 0 1.00000i 2.00000 0 0 0 0 2.00000 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 1225.2.b.d 2
5.b even 2 1 inner 1225.2.b.d 2
5.c odd 4 1 245.2.a.c 1
5.c odd 4 1 1225.2.a.e 1
7.b odd 2 1 175.2.b.a 2
15.e even 4 1 2205.2.a.e 1
20.e even 4 1 3920.2.a.ba 1
21.c even 2 1 1575.2.d.c 2
28.d even 2 1 2800.2.g.l 2
35.c odd 2 1 175.2.b.a 2
35.f even 4 1 35.2.a.a 1
35.f even 4 1 175.2.a.b 1
35.k even 12 2 245.2.e.a 2
35.l odd 12 2 245.2.e.b 2
105.g even 2 1 1575.2.d.c 2
105.k odd 4 1 315.2.a.b 1
105.k odd 4 1 1575.2.a.f 1
140.c even 2 1 2800.2.g.l 2
140.j odd 4 1 560.2.a.b 1
140.j odd 4 1 2800.2.a.z 1
280.s even 4 1 2240.2.a.k 1
280.y odd 4 1 2240.2.a.u 1
385.l odd 4 1 4235.2.a.c 1
420.w even 4 1 5040.2.a.v 1
455.s even 4 1 5915.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 35.f even 4 1
175.2.a.b 1 35.f even 4 1
175.2.b.a 2 7.b odd 2 1
175.2.b.a 2 35.c odd 2 1
245.2.a.c 1 5.c odd 4 1
245.2.e.a 2 35.k even 12 2
245.2.e.b 2 35.l odd 12 2
315.2.a.b 1 105.k odd 4 1
560.2.a.b 1 140.j odd 4 1
1225.2.a.e 1 5.c odd 4 1
1225.2.b.d 2 1.a even 1 1 trivial
1225.2.b.d 2 5.b even 2 1 inner
1575.2.a.f 1 105.k odd 4 1
1575.2.d.c 2 21.c even 2 1
1575.2.d.c 2 105.g even 2 1
2205.2.a.e 1 15.e even 4 1
2240.2.a.k 1 280.s even 4 1
2240.2.a.u 1 280.y odd 4 1
2800.2.a.z 1 140.j odd 4 1
2800.2.g.l 2 28.d even 2 1
2800.2.g.l 2 140.c even 2 1
3920.2.a.ba 1 20.e even 4 1
4235.2.a.c 1 385.l odd 4 1
5040.2.a.v 1 420.w even 4 1
5915.2.a.f 1 455.s even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1225, [\chi])$$:

 $$T_{2}$$ T2 $$T_{19} - 2$$ T19 - 2 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 25$$
$17$ $$T^{2} + 9$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T + 3)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 12)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2} + 81$$
$53$ $$T^{2} + 144$$
$59$ $$T^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 1)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 12)^{2}$$
$97$ $$T^{2} + 1$$