# Properties

 Label 1225.2.b.j Level $1225$ Weight $2$ Character orbit 1225.b Analytic conductor $9.782$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 245) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{6} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{6} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} + ( -3 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{11} + ( \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{13} -4 q^{16} + ( -3 \zeta_{8} - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{17} -4 \zeta_{8}^{2} q^{18} -6 q^{19} + ( -3 \zeta_{8} - 4 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{22} + ( \zeta_{8} - 6 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{23} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{24} + ( -2 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{26} + ( -\zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( 3 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29} + ( 6 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{31} + ( 5 \zeta_{8} + 7 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{33} + ( 6 + \zeta_{8} - \zeta_{8}^{3} ) q^{34} + ( -3 \zeta_{8} - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{37} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{38} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{39} + ( -2 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{41} -2 \zeta_{8}^{2} q^{43} + ( -2 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{46} + ( 3 \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{47} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{48} + ( -7 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{51} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{53} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{54} + ( 6 \zeta_{8} + 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{57} + ( 3 \zeta_{8} - 8 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{58} + ( -2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{59} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{61} + ( 6 \zeta_{8} - 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{62} -8 q^{64} + ( -10 - 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{66} + ( 3 \zeta_{8} - 4 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{67} + ( -4 - 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{69} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{71} -8 \zeta_{8}^{2} q^{72} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{73} + ( 6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{74} + ( -\zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{78} + ( 7 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{79} + ( -1 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{81} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{82} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{86} + ( \zeta_{8} + 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{87} + ( -6 \zeta_{8} - 8 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{88} + 8 q^{89} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{93} + ( -6 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{94} + ( -3 \zeta_{8} + 9 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{97} + ( 8 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{6} + O(q^{10})$$ $$4q + 8q^{6} - 12q^{11} - 16q^{16} - 24q^{19} + 16q^{24} - 8q^{26} + 12q^{29} + 24q^{31} + 24q^{34} - 4q^{39} - 8q^{41} - 8q^{46} - 28q^{51} + 8q^{54} - 8q^{59} - 32q^{64} - 40q^{66} - 16q^{69} - 24q^{71} + 24q^{74} + 28q^{79} - 4q^{81} + 32q^{89} - 24q^{94} + 32q^{99} + O(q^{100})$$

## 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
 −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i
1.41421i 0.414214i 0 0 0.585786 0 2.82843i 2.82843 0
99.2 1.41421i 2.41421i 0 0 3.41421 0 2.82843i −2.82843 0
99.3 1.41421i 2.41421i 0 0 3.41421 0 2.82843i −2.82843 0
99.4 1.41421i 0.414214i 0 0 0.585786 0 2.82843i 2.82843 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.j 4
5.b even 2 1 inner 1225.2.b.j 4
5.c odd 4 1 245.2.a.f yes 2
5.c odd 4 1 1225.2.a.p 2
7.b odd 2 1 1225.2.b.i 4
15.e even 4 1 2205.2.a.t 2
20.e even 4 1 3920.2.a.br 2
35.c odd 2 1 1225.2.b.i 4
35.f even 4 1 245.2.a.e 2
35.f even 4 1 1225.2.a.r 2
35.k even 12 2 245.2.e.g 4
35.l odd 12 2 245.2.e.f 4
105.k odd 4 1 2205.2.a.v 2
140.j odd 4 1 3920.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.e 2 35.f even 4 1
245.2.a.f yes 2 5.c odd 4 1
245.2.e.f 4 35.l odd 12 2
245.2.e.g 4 35.k even 12 2
1225.2.a.p 2 5.c odd 4 1
1225.2.a.r 2 35.f even 4 1
1225.2.b.i 4 7.b odd 2 1
1225.2.b.i 4 35.c odd 2 1
1225.2.b.j 4 1.a even 1 1 trivial
1225.2.b.j 4 5.b even 2 1 inner
2205.2.a.t 2 15.e even 4 1
2205.2.a.v 2 105.k odd 4 1
3920.2.a.br 2 20.e even 4 1
3920.2.a.bw 2 140.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 $$S_{2}^{\mathrm{new}}(1225, [\chi])$$:

 $$T_{2}^{2} + 2$$ $$T_{19} + 6$$ $$T_{31}^{2} - 12 T_{31} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$1 + 6 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 1 + 6 T + T^{2} )^{2}$$
$13$ $$49 + 22 T^{2} + T^{4}$$
$17$ $$289 + 38 T^{2} + T^{4}$$
$19$ $$( 6 + T )^{4}$$
$23$ $$1156 + 76 T^{2} + T^{4}$$
$29$ $$( -23 - 6 T + T^{2} )^{2}$$
$31$ $$( 18 - 12 T + T^{2} )^{2}$$
$37$ $$196 + 44 T^{2} + T^{4}$$
$41$ $$( -14 + 4 T + T^{2} )^{2}$$
$43$ $$( 4 + T^{2} )^{2}$$
$47$ $$81 + 54 T^{2} + T^{4}$$
$53$ $$( 18 + T^{2} )^{2}$$
$59$ $$( -14 + 4 T + T^{2} )^{2}$$
$61$ $$( -8 + T^{2} )^{2}$$
$67$ $$4 + 68 T^{2} + T^{4}$$
$71$ $$( 28 + 12 T + T^{2} )^{2}$$
$73$ $$( 72 + T^{2} )^{2}$$
$79$ $$( -23 - 14 T + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$( -8 + T )^{4}$$
$97$ $$3969 + 198 T^{2} + T^{4}$$