# Properties

 Label 2025.4.a.q Level $2025$ Weight $4$ Character orbit 2025.a Self dual yes Analytic conductor $119.479$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$119.478867762$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.2292.1 Defining polynomial: $$x^{3} - x^{2} - 13 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 45) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 3 - \beta_{1} - \beta_{2} ) q^{4} + ( 14 - 2 \beta_{1} + \beta_{2} ) q^{7} + ( -8 + \beta_{1} + 2 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 3 - \beta_{1} - \beta_{2} ) q^{4} + ( 14 - 2 \beta_{1} + \beta_{2} ) q^{7} + ( -8 + \beta_{1} + 2 \beta_{2} ) q^{8} + ( 3 \beta_{1} + 11 \beta_{2} ) q^{11} + ( -18 - \beta_{1} - 13 \beta_{2} ) q^{13} + ( 17 - \beta_{1} + 25 \beta_{2} ) q^{14} + ( -5 + 6 \beta_{1} - 8 \beta_{2} ) q^{16} + ( -56 + \beta_{1} - 3 \beta_{2} ) q^{17} + ( -58 - 3 \beta_{1} - 7 \beta_{2} ) q^{19} + ( 112 - 11 \beta_{1} - 29 \beta_{2} ) q^{22} + ( 60 + 12 \beta_{1} - 3 \beta_{2} ) q^{23} + ( -140 + 13 \beta_{1} + \beta_{2} ) q^{26} + ( 166 - 9 \beta_{1} - 10 \beta_{2} ) q^{28} + ( 107 - 10 \beta_{1} - 4 \beta_{2} ) q^{29} + ( -138 - 11 \beta_{1} - 51 \beta_{2} ) q^{31} + ( -42 - 49 \beta_{2} ) q^{32} + ( -36 + 3 \beta_{1} - 59 \beta_{2} ) q^{34} + ( -126 + 17 \beta_{1} + 7 \beta_{2} ) q^{37} + ( -68 + 7 \beta_{1} - 33 \beta_{2} ) q^{38} + ( -49 + 23 \beta_{1} + 17 \beta_{2} ) q^{41} + ( 198 + 29 \beta_{1} - 37 \beta_{2} ) q^{43} + ( -286 + 5 \beta_{1} + 119 \beta_{2} ) q^{44} + ( -69 + 3 \beta_{1} - 9 \beta_{2} ) q^{46} + ( 202 - 5 \beta_{1} - 54 \beta_{2} ) q^{47} + ( 152 - 37 \beta_{1} + 63 \beta_{2} ) q^{49} + ( 116 + 7 \beta_{1} - 115 \beta_{2} ) q^{52} + ( -220 + 8 \beta_{1} + 62 \beta_{2} ) q^{53} + ( -219 + 18 \beta_{1} + 30 \beta_{2} ) q^{56} + ( -14 + 4 \beta_{1} + 171 \beta_{2} ) q^{58} + ( 72 + 36 \beta_{1} - 118 \beta_{2} ) q^{59} + ( -473 - 25 \beta_{1} + 45 \beta_{2} ) q^{61} + ( -528 + 51 \beta_{1} - 21 \beta_{2} ) q^{62} + ( -499 + \beta_{1} + 71 \beta_{2} ) q^{64} + ( 630 - 7 \beta_{1} + 48 \beta_{2} ) q^{67} + ( -210 + 51 \beta_{1} + 29 \beta_{2} ) q^{68} + ( 2 + 83 \beta_{1} - 7 \beta_{2} ) q^{71} + ( 108 - 64 \beta_{1} + 20 \beta_{2} ) q^{73} + ( 26 - 7 \beta_{1} - 235 \beta_{2} ) q^{74} + ( 80 + 57 \beta_{1} - 21 \beta_{2} ) q^{76} + ( -236 + \beta_{1} + 239 \beta_{2} ) q^{77} + ( -90 + 64 \beta_{1} + 48 \beta_{2} ) q^{79} + ( 118 - 17 \beta_{1} - 204 \beta_{2} ) q^{82} + ( -242 + 118 \beta_{1} - 13 \beta_{2} ) q^{83} + ( -494 + 37 \beta_{1} + 61 \beta_{2} ) q^{86} + ( 398 - 31 \beta_{1} - 203 \beta_{2} ) q^{88} + ( -629 - 80 \beta_{1} - 88 \beta_{2} ) q^{89} + ( -332 + 45 \beta_{1} - 331 \beta_{2} ) q^{91} + ( -588 - 87 \beta_{1} - 54 \beta_{2} ) q^{92} + ( -579 + 54 \beta_{1} + 286 \beta_{2} ) q^{94} + ( 120 + 20 \beta_{1} + 58 \beta_{2} ) q^{97} + ( 804 - 63 \beta_{1} + 311 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 11 q^{4} + 43 q^{7} - 27 q^{8} + O(q^{10})$$ $$3 q - q^{2} + 11 q^{4} + 43 q^{7} - 27 q^{8} - 14 q^{11} - 40 q^{13} + 27 q^{14} - 13 q^{16} - 166 q^{17} - 164 q^{19} + 376 q^{22} + 171 q^{23} - 434 q^{26} + 517 q^{28} + 335 q^{29} - 352 q^{31} - 77 q^{32} - 52 q^{34} - 402 q^{37} - 178 q^{38} - 187 q^{41} + 602 q^{43} - 982 q^{44} - 201 q^{46} + 665 q^{47} + 430 q^{49} + 456 q^{52} - 730 q^{53} - 705 q^{56} - 217 q^{58} + 298 q^{59} - 1439 q^{61} - 1614 q^{62} - 1569 q^{64} + 1849 q^{67} - 710 q^{68} - 70 q^{71} + 368 q^{73} + 320 q^{74} + 204 q^{76} - 948 q^{77} - 382 q^{79} + 575 q^{82} - 831 q^{83} - 1580 q^{86} + 1428 q^{88} - 1719 q^{89} - 710 q^{91} - 1623 q^{92} - 2077 q^{94} + 282 q^{97} + 2164 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 13 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{2} + 4 \nu - 11$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - 2 \nu - 9$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{2} + 2 \beta_{1} + 29$$$$)/3$$

## 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
 0.0765073 4.10645 −3.18296
−4.57358 0 12.9176 0 0 20.1145 −22.4912 0 0
1.2 −0.174985 0 −7.96938 0 0 −8.46371 2.79440 0 0
1.3 3.74857 0 6.05174 0 0 31.3492 −7.30318 0 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
$$3$$ $$-1$$
$$5$$ $$1$$

## 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 2025.4.a.q 3
3.b odd 2 1 2025.4.a.s 3
5.b even 2 1 405.4.a.j 3
9.d odd 6 2 225.4.e.c 6
15.d odd 2 1 405.4.a.h 3
45.h odd 6 2 45.4.e.b 6
45.j even 6 2 135.4.e.b 6
45.l even 12 4 225.4.k.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.b 6 45.h odd 6 2
135.4.e.b 6 45.j even 6 2
225.4.e.c 6 9.d odd 6 2
225.4.k.c 12 45.l even 12 4
405.4.a.h 3 15.d odd 2 1
405.4.a.j 3 5.b even 2 1
2025.4.a.q 3 1.a even 1 1 trivial
2025.4.a.s 3 3.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2025))$$:

 $$T_{2}^{3} + T_{2}^{2} - 17 T_{2} - 3$$ $$T_{7}^{3} - 43 T_{7}^{2} + 195 T_{7} + 5337$$ $$T_{11}^{3} + 14 T_{11}^{2} - 2816 T_{11} + 43548$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 - 17 T + T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$5337 + 195 T - 43 T^{2} + T^{3}$$
$11$ $$43548 - 2816 T + 14 T^{2} + T^{3}$$
$13$ $$-75364 - 2452 T + 40 T^{2} + T^{3}$$
$17$ $$156324 + 8920 T + 166 T^{2} + T^{3}$$
$19$ $$57316 + 7292 T + 164 T^{2} + T^{3}$$
$23$ $$61209 - 4833 T - 171 T^{2} + T^{3}$$
$29$ $$107067 + 27331 T - 335 T^{2} + T^{3}$$
$31$ $$-9860940 - 13932 T + 352 T^{2} + T^{3}$$
$37$ $$-3335284 + 24708 T + 402 T^{2} + T^{3}$$
$41$ $$-7228275 - 44597 T + 187 T^{2} + T^{3}$$
$43$ $$15444524 + 9956 T - 602 T^{2} + T^{3}$$
$47$ $$-2487483 + 95281 T - 665 T^{2} + T^{3}$$
$53$ $$3250536 + 106300 T + 730 T^{2} + T^{3}$$
$59$ $$127375896 - 354644 T - 298 T^{2} + T^{3}$$
$61$ $$55613497 + 589307 T + 1439 T^{2} + T^{3}$$
$67$ $$-208776159 + 1093677 T - 1849 T^{2} + T^{3}$$
$71$ $$-223775052 - 685460 T + 70 T^{2} + T^{3}$$
$73$ $$134927744 - 372928 T - 368 T^{2} + T^{3}$$
$79$ $$-138322584 - 387924 T + 382 T^{2} + T^{3}$$
$83$ $$-955843821 - 1160973 T + 831 T^{2} + T^{3}$$
$89$ $$-125506395 + 238491 T + 1719 T^{2} + T^{3}$$
$97$ $$16898264 - 67668 T - 282 T^{2} + T^{3}$$