# Properties

 Label 2025.4.a.g Level $2025$ Weight $4$ Character orbit 2025.a Self dual yes Analytic conductor $119.479$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{2} + ( 1 + 3 \beta ) q^{4} + ( -2 - 3 \beta ) q^{7} + ( -17 + \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{2} + ( 1 + 3 \beta ) q^{4} + ( -2 - 3 \beta ) q^{7} + ( -17 + \beta ) q^{8} + ( 37 - 8 \beta ) q^{11} + ( -2 + 15 \beta ) q^{13} + ( 26 + 8 \beta ) q^{14} + ( 1 - 9 \beta ) q^{16} + ( -45 - 9 \beta ) q^{17} + ( -25 - 27 \beta ) q^{19} + ( 27 - 21 \beta ) q^{22} + ( -26 + 19 \beta ) q^{23} + ( -118 - 28 \beta ) q^{26} + ( -74 - 18 \beta ) q^{28} + ( -26 + \beta ) q^{29} + ( 20 + 3 \beta ) q^{31} + ( 207 + 9 \beta ) q^{32} + ( 117 + 63 \beta ) q^{34} + ( 52 - 54 \beta ) q^{37} + ( 241 + 79 \beta ) q^{38} + ( 17 + 98 \beta ) q^{41} + ( -47 + 6 \beta ) q^{43} + ( -155 + 79 \beta ) q^{44} + ( -126 - 12 \beta ) q^{46} + ( -154 - 91 \beta ) q^{47} + ( -267 + 21 \beta ) q^{49} + ( 358 + 54 \beta ) q^{52} + ( -108 + 162 \beta ) q^{53} + ( 10 + 46 \beta ) q^{56} + ( 18 + 24 \beta ) q^{58} + ( 467 - 136 \beta ) q^{59} + ( 272 - 105 \beta ) q^{61} + ( -44 - 26 \beta ) q^{62} + ( -287 - 153 \beta ) q^{64} + ( -461 - 66 \beta ) q^{67} + ( -261 - 171 \beta ) q^{68} + ( 612 + 144 \beta ) q^{71} + ( 349 - 243 \beta ) q^{73} + ( 380 + 56 \beta ) q^{74} + ( -673 - 183 \beta ) q^{76} + ( 118 - 71 \beta ) q^{77} + ( -556 + 309 \beta ) q^{79} + ( -801 - 213 \beta ) q^{82} + ( -460 + 107 \beta ) q^{83} + ( -1 + 35 \beta ) q^{86} + ( -693 + 165 \beta ) q^{88} + ( -234 + 72 \beta ) q^{89} + ( -356 - 69 \beta ) q^{91} + ( 430 - 2 \beta ) q^{92} + ( 882 + 336 \beta ) q^{94} + ( -317 - 102 \beta ) q^{97} + ( 99 + 225 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 5 q^{4} - 7 q^{7} - 33 q^{8} + O(q^{10})$$ $$2 q - 3 q^{2} + 5 q^{4} - 7 q^{7} - 33 q^{8} + 66 q^{11} + 11 q^{13} + 60 q^{14} - 7 q^{16} - 99 q^{17} - 77 q^{19} + 33 q^{22} - 33 q^{23} - 264 q^{26} - 166 q^{28} - 51 q^{29} + 43 q^{31} + 423 q^{32} + 297 q^{34} + 50 q^{37} + 561 q^{38} + 132 q^{41} - 88 q^{43} - 231 q^{44} - 264 q^{46} - 399 q^{47} - 513 q^{49} + 770 q^{52} - 54 q^{53} + 66 q^{56} + 60 q^{58} + 798 q^{59} + 439 q^{61} - 114 q^{62} - 727 q^{64} - 988 q^{67} - 693 q^{68} + 1368 q^{71} + 455 q^{73} + 816 q^{74} - 1529 q^{76} + 165 q^{77} - 803 q^{79} - 1815 q^{82} - 813 q^{83} + 33 q^{86} - 1221 q^{88} - 396 q^{89} - 781 q^{91} + 858 q^{92} + 2100 q^{94} - 736 q^{97} + 423 q^{98} + O(q^{100})$$

## 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
 3.37228 −2.37228
−4.37228 0 11.1168 0 0 −12.1168 −13.6277 0 0
1.2 1.37228 0 −6.11684 0 0 5.11684 −19.3723 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.g 2
3.b odd 2 1 2025.4.a.n 2
5.b even 2 1 81.4.a.d 2
9.c even 3 2 225.4.e.b 4
15.d odd 2 1 81.4.a.a 2
20.d odd 2 1 1296.4.a.u 2
45.h odd 6 2 27.4.c.a 4
45.j even 6 2 9.4.c.a 4
45.k odd 12 4 225.4.k.b 8
60.h even 2 1 1296.4.a.i 2
180.n even 6 2 432.4.i.c 4
180.p odd 6 2 144.4.i.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 45.j even 6 2
27.4.c.a 4 45.h odd 6 2
81.4.a.a 2 15.d odd 2 1
81.4.a.d 2 5.b even 2 1
144.4.i.c 4 180.p odd 6 2
225.4.e.b 4 9.c even 3 2
225.4.k.b 8 45.k odd 12 4
432.4.i.c 4 180.n even 6 2
1296.4.a.i 2 60.h even 2 1
1296.4.a.u 2 20.d odd 2 1
2025.4.a.g 2 1.a even 1 1 trivial
2025.4.a.n 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2025))$$:

 $$T_{2}^{2} + 3 T_{2} - 6$$ $$T_{7}^{2} + 7 T_{7} - 62$$ $$T_{11}^{2} - 66 T_{11} + 561$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-6 + 3 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$-62 + 7 T + T^{2}$$
$11$ $$561 - 66 T + T^{2}$$
$13$ $$-1826 - 11 T + T^{2}$$
$17$ $$1782 + 99 T + T^{2}$$
$19$ $$-4532 + 77 T + T^{2}$$
$23$ $$-2706 + 33 T + T^{2}$$
$29$ $$642 + 51 T + T^{2}$$
$31$ $$388 - 43 T + T^{2}$$
$37$ $$-23432 - 50 T + T^{2}$$
$41$ $$-74877 - 132 T + T^{2}$$
$43$ $$1639 + 88 T + T^{2}$$
$47$ $$-28518 + 399 T + T^{2}$$
$53$ $$-215784 + 54 T + T^{2}$$
$59$ $$6609 - 798 T + T^{2}$$
$61$ $$-42776 - 439 T + T^{2}$$
$67$ $$208099 + 988 T + T^{2}$$
$71$ $$296784 - 1368 T + T^{2}$$
$73$ $$-435398 - 455 T + T^{2}$$
$79$ $$-626516 + 803 T + T^{2}$$
$83$ $$70788 + 813 T + T^{2}$$
$89$ $$-3564 + 396 T + T^{2}$$
$97$ $$49591 + 736 T + T^{2}$$