# Properties

 Label 1225.4.a.h 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 25) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + 7 q^{3} - 7 q^{4} - 7 q^{6} + 15 q^{8} + 22 q^{9} + O(q^{10})$$ $$q - q^{2} + 7 q^{3} - 7 q^{4} - 7 q^{6} + 15 q^{8} + 22 q^{9} - 43 q^{11} - 49 q^{12} - 28 q^{13} + 41 q^{16} + 91 q^{17} - 22 q^{18} + 35 q^{19} + 43 q^{22} - 162 q^{23} + 105 q^{24} + 28 q^{26} - 35 q^{27} + 160 q^{29} - 42 q^{31} - 161 q^{32} - 301 q^{33} - 91 q^{34} - 154 q^{36} + 314 q^{37} - 35 q^{38} - 196 q^{39} + 203 q^{41} - 92 q^{43} + 301 q^{44} + 162 q^{46} + 196 q^{47} + 287 q^{48} + 637 q^{51} + 196 q^{52} - 82 q^{53} + 35 q^{54} + 245 q^{57} - 160 q^{58} + 280 q^{59} + 518 q^{61} + 42 q^{62} - 167 q^{64} + 301 q^{66} - 141 q^{67} - 637 q^{68} - 1134 q^{69} + 412 q^{71} + 330 q^{72} - 763 q^{73} - 314 q^{74} - 245 q^{76} + 196 q^{78} + 510 q^{79} - 839 q^{81} - 203 q^{82} + 777 q^{83} + 92 q^{86} + 1120 q^{87} - 645 q^{88} + 945 q^{89} + 1134 q^{92} - 294 q^{93} - 196 q^{94} - 1127 q^{96} + 1246 q^{97} - 946 q^{99} + O(q^{100})$$

## 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 7.00000 −7.00000 0 −7.00000 0 15.0000 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.h 1
5.b even 2 1 1225.4.a.i 1
7.b odd 2 1 25.4.a.a 1
21.c even 2 1 225.4.a.e 1
28.d even 2 1 400.4.a.s 1
35.c odd 2 1 25.4.a.b yes 1
35.f even 4 2 25.4.b.b 2
56.e even 2 1 1600.4.a.h 1
56.h odd 2 1 1600.4.a.bt 1
105.g even 2 1 225.4.a.c 1
105.k odd 4 2 225.4.b.f 2
140.c even 2 1 400.4.a.c 1
140.j odd 4 2 400.4.c.e 2
280.c odd 2 1 1600.4.a.i 1
280.n even 2 1 1600.4.a.bs 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 7.b odd 2 1
25.4.a.b yes 1 35.c odd 2 1
25.4.b.b 2 35.f even 4 2
225.4.a.c 1 105.g even 2 1
225.4.a.e 1 21.c even 2 1
225.4.b.f 2 105.k odd 4 2
400.4.a.c 1 140.c even 2 1
400.4.a.s 1 28.d even 2 1
400.4.c.e 2 140.j odd 4 2
1225.4.a.h 1 1.a even 1 1 trivial
1225.4.a.i 1 5.b even 2 1
1600.4.a.h 1 56.e even 2 1
1600.4.a.i 1 280.c odd 2 1
1600.4.a.bs 1 280.n even 2 1
1600.4.a.bt 1 56.h odd 2 1

## Hecke kernels

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

 $$T_{2} + 1$$ $$T_{3} - 7$$ $$T_{19} - 35$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-7 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$43 + T$$
$13$ $$28 + T$$
$17$ $$-91 + T$$
$19$ $$-35 + T$$
$23$ $$162 + T$$
$29$ $$-160 + T$$
$31$ $$42 + T$$
$37$ $$-314 + T$$
$41$ $$-203 + T$$
$43$ $$92 + T$$
$47$ $$-196 + T$$
$53$ $$82 + T$$
$59$ $$-280 + T$$
$61$ $$-518 + T$$
$67$ $$141 + T$$
$71$ $$-412 + T$$
$73$ $$763 + T$$
$79$ $$-510 + T$$
$83$ $$-777 + T$$
$89$ $$-945 + T$$
$97$ $$-1246 + T$$