# Properties

 Label 1225.2.b.c Level $1225$ Weight $2$ Character orbit 1225.b Analytic conductor $9.782$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

# 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: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} + 3 i q^{8} + 3 q^{9} +O(q^{10})$$ $$q + i q^{2} + q^{4} + 3 i q^{8} + 3 q^{9} + 4 q^{11} - q^{16} + 3 i q^{18} + 4 i q^{22} -8 i q^{23} -2 q^{29} + 5 i q^{32} + 3 q^{36} -6 i q^{37} + 12 i q^{43} + 4 q^{44} + 8 q^{46} + 10 i q^{53} -2 i q^{58} -7 q^{64} + 4 i q^{67} + 16 q^{71} + 9 i q^{72} + 6 q^{74} -8 q^{79} + 9 q^{81} -12 q^{86} + 12 i q^{88} -8 i q^{92} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 6 q^{9} + O(q^{10})$$ $$2 q + 2 q^{4} + 6 q^{9} + 8 q^{11} - 2 q^{16} - 4 q^{29} + 6 q^{36} + 8 q^{44} + 16 q^{46} - 14 q^{64} + 32 q^{71} + 12 q^{74} - 16 q^{79} + 18 q^{81} - 24 q^{86} + 24 q^{99} + O(q^{100})$$

## 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
1.00000i 0 1.00000 0 0 0 3.00000i 3.00000 0
99.2 1.00000i 0 1.00000 0 0 0 3.00000i 3.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
5.b even 2 1 inner
35.c odd 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.c 2
5.b even 2 1 inner 1225.2.b.c 2
5.c odd 4 1 49.2.a.a 1
5.c odd 4 1 1225.2.a.c 1
7.b odd 2 1 CM 1225.2.b.c 2
15.e even 4 1 441.2.a.c 1
20.e even 4 1 784.2.a.f 1
35.c odd 2 1 inner 1225.2.b.c 2
35.f even 4 1 49.2.a.a 1
35.f even 4 1 1225.2.a.c 1
35.k even 12 2 49.2.c.a 2
35.l odd 12 2 49.2.c.a 2
40.i odd 4 1 3136.2.a.n 1
40.k even 4 1 3136.2.a.o 1
55.e even 4 1 5929.2.a.c 1
60.l odd 4 1 7056.2.a.bg 1
65.h odd 4 1 8281.2.a.d 1
105.k odd 4 1 441.2.a.c 1
105.w odd 12 2 441.2.e.d 2
105.x even 12 2 441.2.e.d 2
140.j odd 4 1 784.2.a.f 1
140.w even 12 2 784.2.i.f 2
140.x odd 12 2 784.2.i.f 2
280.s even 4 1 3136.2.a.n 1
280.y odd 4 1 3136.2.a.o 1
385.l odd 4 1 5929.2.a.c 1
420.w even 4 1 7056.2.a.bg 1
455.s even 4 1 8281.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 5.c odd 4 1
49.2.a.a 1 35.f even 4 1
49.2.c.a 2 35.k even 12 2
49.2.c.a 2 35.l odd 12 2
441.2.a.c 1 15.e even 4 1
441.2.a.c 1 105.k odd 4 1
441.2.e.d 2 105.w odd 12 2
441.2.e.d 2 105.x even 12 2
784.2.a.f 1 20.e even 4 1
784.2.a.f 1 140.j odd 4 1
784.2.i.f 2 140.w even 12 2
784.2.i.f 2 140.x odd 12 2
1225.2.a.c 1 5.c odd 4 1
1225.2.a.c 1 35.f even 4 1
1225.2.b.c 2 1.a even 1 1 trivial
1225.2.b.c 2 5.b even 2 1 inner
1225.2.b.c 2 7.b odd 2 1 CM
1225.2.b.c 2 35.c odd 2 1 inner
3136.2.a.n 1 40.i odd 4 1
3136.2.a.n 1 280.s even 4 1
3136.2.a.o 1 40.k even 4 1
3136.2.a.o 1 280.y odd 4 1
5929.2.a.c 1 55.e even 4 1
5929.2.a.c 1 385.l odd 4 1
7056.2.a.bg 1 60.l odd 4 1
7056.2.a.bg 1 420.w even 4 1
8281.2.a.d 1 65.h odd 4 1
8281.2.a.d 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 $$S_{2}^{\mathrm{new}}(1225, [\chi])$$:

 $$T_{2}^{2} + 1$$ $$T_{19}$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$144 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -16 + T )^{2}$$
$73$ $$T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$