# Properties

 Label 1225.2.b.f 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: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{2} ) q^{3} + ( -3 + \beta_{3} ) q^{4} + 4 q^{6} + ( -\beta_{1} + 4 \beta_{2} ) q^{8} + ( -1 - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{2} ) q^{3} + ( -3 + \beta_{3} ) q^{4} + 4 q^{6} + ( -\beta_{1} + 4 \beta_{2} ) q^{8} + ( -1 - \beta_{3} ) q^{9} + \beta_{3} q^{11} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{12} + ( \beta_{1} + 3 \beta_{2} ) q^{13} + ( 3 - 3 \beta_{3} ) q^{16} + ( \beta_{1} + 3 \beta_{2} ) q^{17} + ( -\beta_{1} - 4 \beta_{2} ) q^{18} + ( -4 + 2 \beta_{3} ) q^{19} + 4 \beta_{2} q^{22} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{23} + ( -4 + 4 \beta_{3} ) q^{24} + ( -2 - 2 \beta_{3} ) q^{26} + ( -\beta_{1} + 3 \beta_{2} ) q^{27} + ( -2 + 3 \beta_{3} ) q^{29} + ( \beta_{1} - 4 \beta_{2} ) q^{32} + ( -\beta_{1} - 5 \beta_{2} ) q^{33} + ( -2 - 2 \beta_{3} ) q^{34} + ( -1 + \beta_{3} ) q^{36} + 6 \beta_{2} q^{37} + ( -4 \beta_{1} + 8 \beta_{2} ) q^{38} + ( 4 + 3 \beta_{3} ) q^{39} + ( -2 + 2 \beta_{3} ) q^{41} + ( -2 \beta_{1} - 6 \beta_{2} ) q^{43} + ( 4 - 2 \beta_{3} ) q^{44} -8 q^{46} + ( -3 \beta_{1} + \beta_{2} ) q^{47} + 12 \beta_{2} q^{48} + ( 4 + 3 \beta_{3} ) q^{51} -2 \beta_{2} q^{52} -2 \beta_{1} q^{53} + ( 8 - 4 \beta_{3} ) q^{54} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{57} + ( -2 \beta_{1} + 12 \beta_{2} ) q^{58} -4 q^{59} + ( -6 + 6 \beta_{3} ) q^{61} + ( -3 - \beta_{3} ) q^{64} + 4 \beta_{3} q^{66} -4 \beta_{1} q^{67} -2 \beta_{2} q^{68} + ( 8 + 2 \beta_{3} ) q^{69} + 8 q^{71} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{72} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{73} + ( 6 - 6 \beta_{3} ) q^{74} + ( 20 - 8 \beta_{3} ) q^{76} + ( 4 \beta_{1} + 12 \beta_{2} ) q^{78} + ( 4 + \beta_{3} ) q^{79} -7 q^{81} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{82} + 4 \beta_{2} q^{83} + ( 4 + 4 \beta_{3} ) q^{86} + ( -\beta_{1} - 13 \beta_{2} ) q^{87} + 4 \beta_{1} q^{88} + ( 2 + 2 \beta_{3} ) q^{89} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{92} + ( 16 - 4 \beta_{3} ) q^{94} + ( 4 - 4 \beta_{3} ) q^{96} + ( 5 \beta_{1} + 7 \beta_{2} ) q^{97} + ( -4 - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 10q^{4} + 16q^{6} - 6q^{9} + O(q^{10})$$ $$4q - 10q^{4} + 16q^{6} - 6q^{9} + 2q^{11} + 6q^{16} - 12q^{19} - 8q^{24} - 12q^{26} - 2q^{29} - 12q^{34} - 2q^{36} + 22q^{39} - 4q^{41} + 12q^{44} - 32q^{46} + 22q^{51} + 24q^{54} - 16q^{59} - 12q^{61} - 14q^{64} + 8q^{66} + 36q^{69} + 32q^{71} + 12q^{74} + 64q^{76} + 18q^{79} - 28q^{81} + 24q^{86} + 12q^{89} + 56q^{94} + 8q^{96} - 20q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 5 \beta_{1}$$

## 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
 − 2.56155i − 1.56155i 1.56155i 2.56155i
2.56155i 1.56155i −4.56155 0 4.00000 0 6.56155i 0.561553 0
99.2 1.56155i 2.56155i −0.438447 0 4.00000 0 2.43845i −3.56155 0
99.3 1.56155i 2.56155i −0.438447 0 4.00000 0 2.43845i −3.56155 0
99.4 2.56155i 1.56155i −4.56155 0 4.00000 0 6.56155i 0.561553 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.f 4
5.b even 2 1 inner 1225.2.b.f 4
5.c odd 4 1 245.2.a.d 2
5.c odd 4 1 1225.2.a.s 2
7.b odd 2 1 175.2.b.b 4
15.e even 4 1 2205.2.a.x 2
20.e even 4 1 3920.2.a.bs 2
21.c even 2 1 1575.2.d.e 4
28.d even 2 1 2800.2.g.t 4
35.c odd 2 1 175.2.b.b 4
35.f even 4 1 35.2.a.b 2
35.f even 4 1 175.2.a.f 2
35.k even 12 2 245.2.e.i 4
35.l odd 12 2 245.2.e.h 4
105.g even 2 1 1575.2.d.e 4
105.k odd 4 1 315.2.a.e 2
105.k odd 4 1 1575.2.a.p 2
140.c even 2 1 2800.2.g.t 4
140.j odd 4 1 560.2.a.i 2
140.j odd 4 1 2800.2.a.bi 2
280.s even 4 1 2240.2.a.bh 2
280.y odd 4 1 2240.2.a.bd 2
385.l odd 4 1 4235.2.a.m 2
420.w even 4 1 5040.2.a.bt 2
455.s even 4 1 5915.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 35.f even 4 1
175.2.a.f 2 35.f even 4 1
175.2.b.b 4 7.b odd 2 1
175.2.b.b 4 35.c odd 2 1
245.2.a.d 2 5.c odd 4 1
245.2.e.h 4 35.l odd 12 2
245.2.e.i 4 35.k even 12 2
315.2.a.e 2 105.k odd 4 1
560.2.a.i 2 140.j odd 4 1
1225.2.a.s 2 5.c odd 4 1
1225.2.b.f 4 1.a even 1 1 trivial
1225.2.b.f 4 5.b even 2 1 inner
1575.2.a.p 2 105.k odd 4 1
1575.2.d.e 4 21.c even 2 1
1575.2.d.e 4 105.g even 2 1
2205.2.a.x 2 15.e even 4 1
2240.2.a.bd 2 280.y odd 4 1
2240.2.a.bh 2 280.s even 4 1
2800.2.a.bi 2 140.j odd 4 1
2800.2.g.t 4 28.d even 2 1
2800.2.g.t 4 140.c even 2 1
3920.2.a.bs 2 20.e even 4 1
4235.2.a.m 2 385.l odd 4 1
5040.2.a.bt 2 420.w even 4 1
5915.2.a.l 2 455.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 $$S_{2}^{\mathrm{new}}(1225, [\chi])$$:

 $$T_{2}^{4} + 9 T_{2}^{2} + 16$$ $$T_{19}^{2} + 6 T_{19} - 8$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 9 T^{2} + T^{4}$$
$3$ $$16 + 9 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -4 - T + T^{2} )^{2}$$
$13$ $$4 + 21 T^{2} + T^{4}$$
$17$ $$4 + 21 T^{2} + T^{4}$$
$19$ $$( -8 + 6 T + T^{2} )^{2}$$
$23$ $$256 + 36 T^{2} + T^{4}$$
$29$ $$( -38 + T + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 36 + T^{2} )^{2}$$
$41$ $$( -16 + 2 T + T^{2} )^{2}$$
$43$ $$64 + 84 T^{2} + T^{4}$$
$47$ $$1024 + 89 T^{2} + T^{4}$$
$53$ $$256 + 36 T^{2} + T^{4}$$
$59$ $$( 4 + T )^{4}$$
$61$ $$( -144 + 6 T + T^{2} )^{2}$$
$67$ $$4096 + 144 T^{2} + T^{4}$$
$71$ $$( -8 + T )^{4}$$
$73$ $$2704 + 168 T^{2} + T^{4}$$
$79$ $$( 16 - 9 T + T^{2} )^{2}$$
$83$ $$( 16 + T^{2} )^{2}$$
$89$ $$( -8 - 6 T + T^{2} )^{2}$$
$97$ $$7396 + 253 T^{2} + T^{4}$$