# Properties

 Label 725.4.a.b Level $725$ Weight $4$ Character orbit 725.a Self dual yes Analytic conductor $42.776$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 725.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$42.7763847542$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 29) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + ( 5 - 3 \beta ) q^{3} + ( -5 + 2 \beta ) q^{4} + ( -1 + 2 \beta ) q^{6} + ( 8 + 10 \beta ) q^{7} + ( -9 - 11 \beta ) q^{8} + ( 16 - 30 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + ( 5 - 3 \beta ) q^{3} + ( -5 + 2 \beta ) q^{4} + ( -1 + 2 \beta ) q^{6} + ( 8 + 10 \beta ) q^{7} + ( -9 - 11 \beta ) q^{8} + ( 16 - 30 \beta ) q^{9} + ( -13 + 37 \beta ) q^{11} + ( -37 + 25 \beta ) q^{12} + ( 13 - 26 \beta ) q^{13} + ( 28 + 18 \beta ) q^{14} + ( 9 - 36 \beta ) q^{16} + ( -30 + 18 \beta ) q^{17} + ( -44 - 14 \beta ) q^{18} + ( -110 - 32 \beta ) q^{19} + ( -20 + 26 \beta ) q^{21} + ( 61 + 24 \beta ) q^{22} + ( -26 + 48 \beta ) q^{23} + ( 21 - 28 \beta ) q^{24} + ( -39 - 13 \beta ) q^{26} + ( 125 - 117 \beta ) q^{27} -34 \beta q^{28} + 29 q^{29} + ( -147 + 63 \beta ) q^{31} + ( 9 + 61 \beta ) q^{32} + ( -287 + 224 \beta ) q^{33} + ( 6 - 12 \beta ) q^{34} + ( -200 + 182 \beta ) q^{36} + ( -156 - 56 \beta ) q^{37} + ( -174 - 142 \beta ) q^{38} + ( 221 - 169 \beta ) q^{39} + ( 20 - 138 \beta ) q^{41} + ( 32 + 6 \beta ) q^{42} + ( 161 + 171 \beta ) q^{43} + ( 213 - 211 \beta ) q^{44} + ( 70 + 22 \beta ) q^{46} + ( 65 - 207 \beta ) q^{47} + ( 261 - 207 \beta ) q^{48} + ( -79 + 160 \beta ) q^{49} + ( -258 + 180 \beta ) q^{51} + ( -169 + 156 \beta ) q^{52} + ( -501 - 122 \beta ) q^{53} + ( -109 + 8 \beta ) q^{54} + ( -292 - 178 \beta ) q^{56} + ( -358 + 170 \beta ) q^{57} + ( 29 + 29 \beta ) q^{58} + ( -450 - 248 \beta ) q^{59} + ( -474 + 178 \beta ) q^{61} + ( -21 - 84 \beta ) q^{62} + ( -472 - 80 \beta ) q^{63} + ( 59 + 358 \beta ) q^{64} + ( 161 - 63 \beta ) q^{66} + ( -160 + 484 \beta ) q^{67} + ( 222 - 150 \beta ) q^{68} + ( -418 + 318 \beta ) q^{69} + ( -330 + 34 \beta ) q^{71} + ( 516 + 94 \beta ) q^{72} + ( -324 - 640 \beta ) q^{73} + ( -268 - 212 \beta ) q^{74} + ( 422 - 60 \beta ) q^{76} + ( 636 + 166 \beta ) q^{77} + ( -117 + 52 \beta ) q^{78} + ( 129 + 341 \beta ) q^{79} + ( 895 - 150 \beta ) q^{81} + ( -256 - 118 \beta ) q^{82} + ( -606 + 64 \beta ) q^{83} + ( 204 - 170 \beta ) q^{84} + ( 503 + 332 \beta ) q^{86} + ( 145 - 87 \beta ) q^{87} + ( -697 - 190 \beta ) q^{88} + ( 380 - 522 \beta ) q^{89} + ( -416 - 78 \beta ) q^{91} + ( 322 - 292 \beta ) q^{92} + ( -1113 + 756 \beta ) q^{93} + ( -349 - 142 \beta ) q^{94} + ( -321 + 278 \beta ) q^{96} + ( -12 - 578 \beta ) q^{97} + ( 241 + 81 \beta ) q^{98} + ( -2428 + 982 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 10 q^{3} - 10 q^{4} - 2 q^{6} + 16 q^{7} - 18 q^{8} + 32 q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} + 10 q^{3} - 10 q^{4} - 2 q^{6} + 16 q^{7} - 18 q^{8} + 32 q^{9} - 26 q^{11} - 74 q^{12} + 26 q^{13} + 56 q^{14} + 18 q^{16} - 60 q^{17} - 88 q^{18} - 220 q^{19} - 40 q^{21} + 122 q^{22} - 52 q^{23} + 42 q^{24} - 78 q^{26} + 250 q^{27} + 58 q^{29} - 294 q^{31} + 18 q^{32} - 574 q^{33} + 12 q^{34} - 400 q^{36} - 312 q^{37} - 348 q^{38} + 442 q^{39} + 40 q^{41} + 64 q^{42} + 322 q^{43} + 426 q^{44} + 140 q^{46} + 130 q^{47} + 522 q^{48} - 158 q^{49} - 516 q^{51} - 338 q^{52} - 1002 q^{53} - 218 q^{54} - 584 q^{56} - 716 q^{57} + 58 q^{58} - 900 q^{59} - 948 q^{61} - 42 q^{62} - 944 q^{63} + 118 q^{64} + 322 q^{66} - 320 q^{67} + 444 q^{68} - 836 q^{69} - 660 q^{71} + 1032 q^{72} - 648 q^{73} - 536 q^{74} + 844 q^{76} + 1272 q^{77} - 234 q^{78} + 258 q^{79} + 1790 q^{81} - 512 q^{82} - 1212 q^{83} + 408 q^{84} + 1006 q^{86} + 290 q^{87} - 1394 q^{88} + 760 q^{89} - 832 q^{91} + 644 q^{92} - 2226 q^{93} - 698 q^{94} - 642 q^{96} - 24 q^{97} + 482 q^{98} - 4856 q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
−0.414214 9.24264 −7.82843 0 −3.82843 −6.14214 6.55635 58.4264 0
1.2 2.41421 0.757359 −2.17157 0 1.82843 22.1421 −24.5563 −26.4264 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
$$5$$ $$1$$
$$29$$ $$-1$$

## 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 725.4.a.b 2
5.b even 2 1 29.4.a.a 2
15.d odd 2 1 261.4.a.b 2
20.d odd 2 1 464.4.a.f 2
35.c odd 2 1 1421.4.a.c 2
40.e odd 2 1 1856.4.a.h 2
40.f even 2 1 1856.4.a.n 2
145.d even 2 1 841.4.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.a.a 2 5.b even 2 1
261.4.a.b 2 15.d odd 2 1
464.4.a.f 2 20.d odd 2 1
725.4.a.b 2 1.a even 1 1 trivial
841.4.a.a 2 145.d even 2 1
1421.4.a.c 2 35.c odd 2 1
1856.4.a.h 2 40.e odd 2 1
1856.4.a.n 2 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2 T_{2} - 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(725))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 2 T + T^{2}$$
$3$ $$7 - 10 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-136 - 16 T + T^{2}$$
$11$ $$-2569 + 26 T + T^{2}$$
$13$ $$-1183 - 26 T + T^{2}$$
$17$ $$252 + 60 T + T^{2}$$
$19$ $$10052 + 220 T + T^{2}$$
$23$ $$-3932 + 52 T + T^{2}$$
$29$ $$( -29 + T )^{2}$$
$31$ $$13671 + 294 T + T^{2}$$
$37$ $$18064 + 312 T + T^{2}$$
$41$ $$-37688 - 40 T + T^{2}$$
$43$ $$-32561 - 322 T + T^{2}$$
$47$ $$-81473 - 130 T + T^{2}$$
$53$ $$221233 + 1002 T + T^{2}$$
$59$ $$79492 + 900 T + T^{2}$$
$61$ $$161308 + 948 T + T^{2}$$
$67$ $$-442912 + 320 T + T^{2}$$
$71$ $$106588 + 660 T + T^{2}$$
$73$ $$-714224 + 648 T + T^{2}$$
$79$ $$-215921 - 258 T + T^{2}$$
$83$ $$359044 + 1212 T + T^{2}$$
$89$ $$-400568 - 760 T + T^{2}$$
$97$ $$-668024 + 24 T + T^{2}$$