# Properties

 Label 1225.4.a.j Level $1225$ Weight $4$ Character orbit 1225.a Self dual yes Analytic conductor $72.277$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.2773397570$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - 2q^{3} - 7q^{4} - 2q^{6} - 15q^{8} - 23q^{9} + O(q^{10})$$ $$q + q^{2} - 2q^{3} - 7q^{4} - 2q^{6} - 15q^{8} - 23q^{9} - 8q^{11} + 14q^{12} + 28q^{13} + 41q^{16} + 54q^{17} - 23q^{18} + 110q^{19} - 8q^{22} - 48q^{23} + 30q^{24} + 28q^{26} + 100q^{27} - 110q^{29} - 12q^{31} + 161q^{32} + 16q^{33} + 54q^{34} + 161q^{36} + 246q^{37} + 110q^{38} - 56q^{39} - 182q^{41} - 128q^{43} + 56q^{44} - 48q^{46} + 324q^{47} - 82q^{48} - 108q^{51} - 196q^{52} + 162q^{53} + 100q^{54} - 220q^{57} - 110q^{58} - 810q^{59} + 488q^{61} - 12q^{62} - 167q^{64} + 16q^{66} - 244q^{67} - 378q^{68} + 96q^{69} - 768q^{71} + 345q^{72} - 702q^{73} + 246q^{74} - 770q^{76} - 56q^{78} + 440q^{79} + 421q^{81} - 182q^{82} - 1302q^{83} - 128q^{86} + 220q^{87} + 120q^{88} - 730q^{89} + 336q^{92} + 24q^{93} + 324q^{94} - 322q^{96} + 294q^{97} + 184q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 −2.00000 −7.00000 0 −2.00000 0 −15.0000 −23.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.4.a.j 1
5.b even 2 1 49.4.a.b 1
7.b odd 2 1 175.4.a.b 1
15.d odd 2 1 441.4.a.i 1
20.d odd 2 1 784.4.a.g 1
21.c even 2 1 1575.4.a.e 1
35.c odd 2 1 7.4.a.a 1
35.f even 4 2 175.4.b.b 2
35.i odd 6 2 49.4.c.c 2
35.j even 6 2 49.4.c.b 2
105.g even 2 1 63.4.a.b 1
105.o odd 6 2 441.4.e.e 2
105.p even 6 2 441.4.e.h 2
140.c even 2 1 112.4.a.f 1
280.c odd 2 1 448.4.a.i 1
280.n even 2 1 448.4.a.e 1
385.h even 2 1 847.4.a.b 1
420.o odd 2 1 1008.4.a.c 1
455.h odd 2 1 1183.4.a.b 1
595.b odd 2 1 2023.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 35.c odd 2 1
49.4.a.b 1 5.b even 2 1
49.4.c.b 2 35.j even 6 2
49.4.c.c 2 35.i odd 6 2
63.4.a.b 1 105.g even 2 1
112.4.a.f 1 140.c even 2 1
175.4.a.b 1 7.b odd 2 1
175.4.b.b 2 35.f even 4 2
441.4.a.i 1 15.d odd 2 1
441.4.e.e 2 105.o odd 6 2
441.4.e.h 2 105.p even 6 2
448.4.a.e 1 280.n even 2 1
448.4.a.i 1 280.c odd 2 1
784.4.a.g 1 20.d odd 2 1
847.4.a.b 1 385.h even 2 1
1008.4.a.c 1 420.o odd 2 1
1183.4.a.b 1 455.h odd 2 1
1225.4.a.j 1 1.a even 1 1 trivial
1575.4.a.e 1 21.c even 2 1
2023.4.a.a 1 595.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2} - 1$$ $$T_{3} + 2$$ $$T_{19} - 110$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$2 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$8 + T$$
$13$ $$-28 + T$$
$17$ $$-54 + T$$
$19$ $$-110 + T$$
$23$ $$48 + T$$
$29$ $$110 + T$$
$31$ $$12 + T$$
$37$ $$-246 + T$$
$41$ $$182 + T$$
$43$ $$128 + T$$
$47$ $$-324 + T$$
$53$ $$-162 + T$$
$59$ $$810 + T$$
$61$ $$-488 + T$$
$67$ $$244 + T$$
$71$ $$768 + T$$
$73$ $$702 + T$$
$79$ $$-440 + T$$
$83$ $$1302 + T$$
$89$ $$730 + T$$
$97$ $$-294 + T$$