# Properties

 Label 1225.2.b.g 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ 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 a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
2.41421i 0.414214i −3.82843 0 −1.00000 0 4.41421i 2.82843 0
99.2 0.414214i 2.41421i 1.82843 0 −1.00000 0 1.58579i −2.82843 0
99.3 0.414214i 2.41421i 1.82843 0 −1.00000 0 1.58579i −2.82843 0
99.4 2.41421i 0.414214i −3.82843 0 −1.00000 0 4.41421i 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.g 4
5.b even 2 1 inner 1225.2.b.g 4
5.c odd 4 1 245.2.a.h 2
5.c odd 4 1 1225.2.a.k 2
7.b odd 2 1 1225.2.b.h 4
7.c even 3 2 175.2.k.a 8
15.e even 4 1 2205.2.a.n 2
20.e even 4 1 3920.2.a.bq 2
35.c odd 2 1 1225.2.b.h 4
35.f even 4 1 245.2.a.g 2
35.f even 4 1 1225.2.a.m 2
35.j even 6 2 175.2.k.a 8
35.k even 12 2 245.2.e.e 4
35.l odd 12 2 35.2.e.a 4
35.l odd 12 2 175.2.e.c 4
105.k odd 4 1 2205.2.a.q 2
105.x even 12 2 315.2.j.e 4
140.j odd 4 1 3920.2.a.bv 2
140.w even 12 2 560.2.q.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 35.l odd 12 2
175.2.e.c 4 35.l odd 12 2
175.2.k.a 8 7.c even 3 2
175.2.k.a 8 35.j even 6 2
245.2.a.g 2 35.f even 4 1
245.2.a.h 2 5.c odd 4 1
245.2.e.e 4 35.k even 12 2
315.2.j.e 4 105.x even 12 2
560.2.q.k 4 140.w even 12 2
1225.2.a.k 2 5.c odd 4 1
1225.2.a.m 2 35.f even 4 1
1225.2.b.g 4 1.a even 1 1 trivial
1225.2.b.g 4 5.b even 2 1 inner
1225.2.b.h 4 7.b odd 2 1
1225.2.b.h 4 35.c odd 2 1
2205.2.a.n 2 15.e even 4 1
2205.2.a.q 2 105.k odd 4 1
3920.2.a.bq 2 20.e even 4 1
3920.2.a.bv 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 $$S_{2}^{\mathrm{new}}(1225, [\chi])$$:

 $$T_{2}^{4} + 6 T_{2}^{2} + 1$$ $$T_{19}^{2} - 8$$ $$T_{31} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T^{2} + T^{4}$$
$3$ $$1 + 6 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -4 - 4 T + T^{2} )^{2}$$
$13$ $$16 + 24 T^{2} + T^{4}$$
$17$ $$16 + 24 T^{2} + T^{4}$$
$19$ $$( -8 + T^{2} )^{2}$$
$23$ $$1 + 6 T^{2} + T^{4}$$
$29$ $$( -1 + T )^{4}$$
$31$ $$( 6 + T )^{4}$$
$37$ $$T^{4}$$
$41$ $$( 17 + 10 T + T^{2} )^{2}$$
$43$ $$529 + 54 T^{2} + T^{4}$$
$47$ $$( 4 + T^{2} )^{2}$$
$53$ $$64 + 48 T^{2} + T^{4}$$
$59$ $$( -56 - 8 T + T^{2} )^{2}$$
$61$ $$( -63 + 6 T + T^{2} )^{2}$$
$67$ $$14161 + 246 T^{2} + T^{4}$$
$71$ $$( -56 + 8 T + T^{2} )^{2}$$
$73$ $$16 + 24 T^{2} + T^{4}$$
$79$ $$( 136 + 24 T + T^{2} )^{2}$$
$83$ $$25921 + 326 T^{2} + T^{4}$$
$89$ $$( -23 - 6 T + T^{2} )^{2}$$
$97$ $$16 + 136 T^{2} + T^{4}$$