# Properties

 Label 1225.4.a.c Level $1225$ Weight $4$ Character orbit 1225.a Self dual yes Analytic conductor $72.277$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q - 2q^{2} - 7q^{3} - 4q^{4} + 14q^{6} + 24q^{8} + 22q^{9} + O(q^{10})$$ $$q - 2q^{2} - 7q^{3} - 4q^{4} + 14q^{6} + 24q^{8} + 22q^{9} - 5q^{11} + 28q^{12} + 14q^{13} - 16q^{16} + 21q^{17} - 44q^{18} + 49q^{19} + 10q^{22} + 159q^{23} - 168q^{24} - 28q^{26} + 35q^{27} + 58q^{29} + 147q^{31} - 160q^{32} + 35q^{33} - 42q^{34} - 88q^{36} - 219q^{37} - 98q^{38} - 98q^{39} + 350q^{41} + 124q^{43} + 20q^{44} - 318q^{46} - 525q^{47} + 112q^{48} - 147q^{51} - 56q^{52} - 303q^{53} - 70q^{54} - 343q^{57} - 116q^{58} - 105q^{59} - 413q^{61} - 294q^{62} + 448q^{64} - 70q^{66} - 415q^{67} - 84q^{68} - 1113q^{69} - 432q^{71} + 528q^{72} + 1113q^{73} + 438q^{74} - 196q^{76} + 196q^{78} - 103q^{79} - 839q^{81} - 700q^{82} - 1092q^{83} - 248q^{86} - 406q^{87} - 120q^{88} - 329q^{89} - 636q^{92} - 1029q^{93} + 1050q^{94} + 1120q^{96} + 882q^{97} - 110q^{99} + O(q^{100})$$

## 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
−2.00000 −7.00000 −4.00000 0 14.0000 0 24.0000 22.0000 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.c 1
5.b even 2 1 49.4.a.d 1
7.b odd 2 1 1225.4.a.d 1
7.c even 3 2 175.4.e.a 2
15.d odd 2 1 441.4.a.d 1
20.d odd 2 1 784.4.a.b 1
35.c odd 2 1 49.4.a.c 1
35.i odd 6 2 49.4.c.a 2
35.j even 6 2 7.4.c.a 2
35.l odd 12 4 175.4.k.a 4
105.g even 2 1 441.4.a.e 1
105.o odd 6 2 63.4.e.b 2
105.p even 6 2 441.4.e.k 2
140.c even 2 1 784.4.a.r 1
140.p odd 6 2 112.4.i.c 2
280.bf even 6 2 448.4.i.f 2
280.bi odd 6 2 448.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 35.j even 6 2
49.4.a.c 1 35.c odd 2 1
49.4.a.d 1 5.b even 2 1
49.4.c.a 2 35.i odd 6 2
63.4.e.b 2 105.o odd 6 2
112.4.i.c 2 140.p odd 6 2
175.4.e.a 2 7.c even 3 2
175.4.k.a 4 35.l odd 12 4
441.4.a.d 1 15.d odd 2 1
441.4.a.e 1 105.g even 2 1
441.4.e.k 2 105.p even 6 2
448.4.i.a 2 280.bi odd 6 2
448.4.i.f 2 280.bf even 6 2
784.4.a.b 1 20.d odd 2 1
784.4.a.r 1 140.c even 2 1
1225.4.a.c 1 1.a even 1 1 trivial
1225.4.a.d 1 7.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2} + 2$$ $$T_{3} + 7$$ $$T_{19} - 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$7 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$5 + T$$
$13$ $$-14 + T$$
$17$ $$-21 + T$$
$19$ $$-49 + T$$
$23$ $$-159 + T$$
$29$ $$-58 + T$$
$31$ $$-147 + T$$
$37$ $$219 + T$$
$41$ $$-350 + T$$
$43$ $$-124 + T$$
$47$ $$525 + T$$
$53$ $$303 + T$$
$59$ $$105 + T$$
$61$ $$413 + T$$
$67$ $$415 + T$$
$71$ $$432 + T$$
$73$ $$-1113 + T$$
$79$ $$103 + T$$
$83$ $$1092 + T$$
$89$ $$329 + T$$
$97$ $$-882 + T$$